Research ArticleDemand Management Based on Model PredictiveControl Techniques

Yasser A Davizoacuten Rogelio Soto Joseacute de J Rodriacuteguez Ernesto Rodriacuteguez-LealCeacutesar Martiacutenez-Olvera and Carlos Hinojosa

School of Engineering Tecnologico de Monterrey 64849 Monterrey NL Mexico

Correspondence should be addressed to Yasser A Davizon a00534333itesmmx

Received 31 March 2014 Revised 30 May 2014 Accepted 30 May 2014 Published 26 June 2014

Academic Editor Hamid Reza Karimi

Copyright copy 2014 Yasser A Davizon et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Demand management (DM) is the process that helps companies to sell the right product to the right customer at the right timeand for the right price Therefore the challenge for any company is to determine how much to sell at what price and to whichmarket segment while maximizing its profits DM also helps managers efficiently allocate undifferentiated units of capacity to theavailable demand with the goal of maximizing revenue This paper introduces control system approach to demand managementwith dynamic pricing (DP) using themodel predictive control (MPC) technique In addition we present a proper dynamical systemanalogy based on active suspension and a stability analysis is provided via the Lyapunov direct method

1 Introduction

A supply chain is a complex system that involves suppliersproducers distribution centers retailers and finally the cus-tomers The main purpose of the supply chain management(SCM) is to optimize the entire chain In other words SCMmanages efficiently the flow of products and informationbetween elements of the chain in order to attain goals thatcannot be reached in an isolated manner The application ofDM to SCM allows us to efficiently balance the customerrsquosrequirements and the capabilities of the supply chain (eg see[1])

Nowadays in such dynamic business world the rapidlearning about the behaviour of demand and its yield isrequired The adequate management of these two variablesis an important source of competitive advantage for anycompany in the supply chain Moreover traditional yieldmanagement is switching into the more complex activity ofDM Therefore our research focuses on the integration ofyield or revenuemanagement into the framework of DM It iswell known that hoteliers airlines and car rentals companieshave implemented yield management activities successfullyAccording to [2] to include yield management activitiesinto DM enhances profitability and long run sustainability

Furthermore DM allows managers to sell the right productor service to the right customer at the right time and for theright price with the main objective of maximizing the profitsmargins This action can be made through the assignationof units of capacity to the available demand For examplewhen dealing with perishable products DM assumes a fixedcapacity However in the case of wafer fabrication facilities ifproduct mix is highly predictable or if all products use eachpiece of production equipment equally all overall productionforecast is needed to determine equipment requirementsHowever demand hasmultiple dimensions such as the differ-ent products sold the types of customers served (preferencesand purchase behavior) and time In this context decisionsmade about the price or quantity of a product may affect thedemand for related products andormay also affect the futuredemand for the same product This scenario represents anopportunity to develop a model that determines how muchto sell at what price and to which market segment Thuswe propose the application of DM to SCM as a helpful toolwhich integrates capacity inventory level and demand inDP framework Dynamic pricing (DP) is the core elementof DM and refers to fluid pricing between the buyer andthe seller rather than the more traditional fixed pricing [3]Based on this the companyrsquos main goal is to maximize its

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 702642 11 pageshttpdxdoiorg1011552014702642

2 Mathematical Problems in Engineering

total expected revenues bymaking the correct decision Fromhere our premise is to show that DP can be modeled applyingcontrol systems and specifically addressed by the techniquesof optimal control in real time as the well-known modelpredictive control (MPC) This research work introduces asecond order dynamical system which integrates capacityinventory level and biding price in the framework of DM viaDP We utilize a dynamical 14 active suspension system tomodel and prove the applicability of DP to SCM Moreoverby using the Lyapunov direct method we concluded that asufficient and necessary condition for stability is the presenceof inventory level along the supply chain

The remainder of this paperrsquos structure is as followsSection 2 describes DM in semiconductor manufacturingSection 3 presents sensitivity and stability analysis for DMSection 4 considers model predictive control formulationfor DM with respective simulations while conclusions areprovided in Section 5

2 Demand Management Sector Applications

21 Demand Management Sector Applications A Review ofthe Current Literature The techniques of DM are relativelynew and the first research that dealt directly with theseissues appeared less than 20 years ago The major sectorof application of DM has been in the airline industry anddifferent approaches have been presented in the literaturefor airlines and hotels [4ndash12] (see Table 1) manufacturingservices and transportation [13ndash22] (see Table 2) So muchresearch has been done in the DM field Please refer toTables 1 and 2 for an extended list of authors and a briefdescription of their work There are many different areas inwhichDMhas been applied successfully as shown in Figure 1To this day most of the research has been performed in theairline sector (which is justified because DM was developedto solve problems in this area) and in the area of hotelmanagement where by applying DM techniques it has beenpossible tomitigate some of the booking limit problems basedon reservations We present a novel approach for dynamicpricing-inventory level based on DM assuming that we havea pair of products subject to the same demand profile

22 From Supply to Demand Management The DampingEffect Synchronization between supply and demand is theoptimal scenario in any complex supply network In dynam-ical systems dissipation is considered the loss of energyover time this is because of damping It is common totake into account the damping effect in different situationslike economics finance and electromechanical systems andmuch more in biological systems In DM the notion ofdamping is related to inventory level basically to achievestability conditions over the supply chain

221 Finite Dimensional Formulation for Demand Man-agement The ordinary differential equation (ODE) for thedynamic system (DS) with damping is

1198721198892119909

1198891199052+ 120573

119889119909

119889119905+ 119870119909 = 119891 (119905) (1)

36

28

14

319

AirlinesServicesManufacturing

TransportationHotels

Figure 1 Demand management sector applications

The present model in (1) describes the dynamics of mass-spring-damper second order dynamical system Also pub-lished by us [23] this model proposes a dynamic pricing forDM applying fast model predictive control approach Thepresent work is an extension of our previously publishedwork

For purposes of demand management we present thefollowing model in the damping part of (1)

1198621198892119901

1198891199052+ [int

infin

0

120573 (119905) 119889119905]119889119901

119889119905+ 119870119901 = 119889 (119905) (2)

Equation (2) presents 119901mdashprice level 119862mdashcapacity 119889(119905)mdashdemand 119870mdashbiding price restitution coefficient and 120573(119905)mdashdamping coefficient-inventory level

The damping factor in (2) has the characteristic to be afunction over time and then Lipschitz property can be appliedand requires convergence in a scalar value (see Appendix A)

The demand management problem for optimal controlhas been addressed previously by different research groups[24] dynamic pricing for optimum inventory level has beenpreviously integrated as a first order ODE as shown in[25] Moreover [26] has extended this work by integratingdynamic pricing and demand level into a second orderODE The interaction of dynamic pricing and demand level[27] presents a dynamic pricing model on web service inwhich case the demand function is nonlinear Optimizationtechniques have been shown [28] and have been used toquantify demand response in energy industry applying bid-pricing approaches also integer linear programming hasbeen applied to model inventory level production level andcapacity for dynamic pricing in [29]

222 Infinite Dimensional Formulation for Demand Manage-ment An extended version of ODE in (2) is expressed as ageneralized wave equation first proposed in [30] for flexiblesystems such as

119862 (119901)1205972120593

1205971199052+ 120573

120597120593

120597119905+ 119870120593 = 119863 (119901 119905) (3)

Mathematical Problems in Engineering 3

Table 1 Demand management for airlines and hotels literature review

Author Sector application Methodology Contribution

Abdel Aziz et al [4] Hotels Research article (casestudies)

Application to address the problem of room pricing inhotels Optimization techniques are present Multiclassscheme similar to the one implemented in airlines isstudied

de Boer [5] Airlines Research article(theoretical)

Determination of a revenue management policy inresponse to the realization of the demand This is doneby matching supply and demand for seats

Hung and Chen [6] Airlines Research article (casestudies)

This study develops and tests two heuristics approachesthe dynamic seat rationing and the expected revenuegap to help the airline make a fulfillment or rejectiondecision when a customer arrives

Gosavi et al [7] AirlinesResearch article(simulation basedoptimization)

Application of simulation based optimization for seatallocation in airlines The model considers real lifeassumption such as cancelations and overbooking

Graf and Kimms [8] Airlines Research article (casestudies)

The optimal transfer prices are determined by anegotiation process Option-based capacity controlprocess results combined with transfer priceoptimization are compared with a first come first servedapproach

Kimms andMuller-Bungart [9] Airlines

Research article(simulation based

case study)

Application of heuristics and optimal networks inrevenue management Assuming stochastic demand

Netessine and Shumsky [10] Airlines SurveyAnalysis of fundamental concepts and trade-offs ofyield management to describe the parallels betweenyield management and inventory management

Rannou and Melli [11] Hotels Research article (casestudy)

Analysis and application of revenue management inhospitality industry The idea is to achieve aperformance measure for the return of investment

Walczak and Brumelle [12] Airlines Research article(theoretical)

Problem formulation of semi-Markov model fordynamic pricing and revenue management in airlineindustry assuming continuous demand

Equation (3) is called master damping equation for demandmanagement which can be written as partial differentialequation (PDE) as follows

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(4)

where first derivative of price in time is V119901and second

derivative in time is 119886119901 See Appendix B for a detailed proof

of (3) and (4)

3 Robust and Stability Analysis forDemand Management

DM can be stated as follows how to select the productrsquos mixamount and price in order to satisfy a fluctuating demandwith the main goal of maximizing the profit For this reasonin this document we propose the use of MPC by comparingDM and DP using an active suspension with damping

31 Physical Modeling in DM-DP Problem The use of aphysical model represents an approach to obtain a suitablemodel to be controlled According to [31] DM is the creationacross the supply chain and its markets of a coordinated flowof demand Also the role of DM is to decrease demandThe application of the active suspension analogy to trackboth problems is suitable because the price level (whichrepresents outputs) and demand (which represents input)converge to a stability condition that requires inventory levelin the dissipation of the dynamical system DM assumes fixedcapacity (which in terms of active suspension is the mass)inventory level (damping of the system) and biding price(restitution coefficient) In this case the idea is to present amodel in which by using DM we can assess how demanddoes affect prices We have chosen an active suspensionsystem to approach the system model see Figure 2

The dynamic equations for the system are as followsFor the capacity fixed 119862

119891

119862119891

119901119891= 119896119891(119908 minus 119901

119891) minus 119896119906(119901119891minus 119901119906)

minus 120573 ( 119901119891minus 119901119906) minus 120572 sdot 119889

(5)

4 Mathematical Problems in Engineering

Table 2 Demand management for manufacturing and services literature review

Author Sector application Methodology Contribution

Bertsimas and Shioda [13] Services Research article (casestudies)

Application of revenue management in restaurantsUsing stochastic optimization via dynamicprogramming Computational experiments arepresented for reservation and without reservation casestudies

Boyd and Bilegan [14] Services Research article (casestudy in e-commerce)

Analysis of revenue management applied toe-commerce in the airline industry Optimizationtechniques are shown in inventory control mechanism

Defregger and Kuhn [15] Manufacturing Research article (casestudies)

Application of revenue management to a make-to-ordermanufacturing company with limited inventorycapacity Classifying the underlying Markov decisionprocess by introducing a heuristic procedure

Kimes [16] Services Survey Revenue management analysis via the application ofoptimization forecasting and overbooking

Kumar and Frederik [17] Services Survey

Application of revenue management in constructionindustry The benefits of revenue management can berealized at manufacturing companies Uncertaindemand and pricing of available capacity are analyzedvia this approach

Lee et al [18] Transportation Research article (casestudies)

Analysis and application of heuristic to solve a singlerevenue management problem with postponementarising from the sea cargo industry Optimization basedcase studies are addressed

Nair and Bapna [19] Services Research articleThis paper studies optimal policies for allocatingmodems capacity among segments of customers using acontinuous time Markov decision

Spengler et al [20] Manufacturing Research article (casestudies)

Revenue management approach for companies in theiron and steel industry is developed The aim is toimprove short-term order selection

Shin and Park [21] Manufacturing Research article(theoretical)

Yield improvement is analyzed in semiconductormanufacturing via machine learning

Steinhardt and Gonsch [22] Services Research article (casestudies)

This paper addresses the problem of integrating revenuemanagement capacity control with upgrade decisionmaking A new structural property for an integrateddynamic programming formulation is present

ku

Cu

Cf

d

d

pf

pu

wkf

120573

Figure 2 Active suspension analogy for DM-DP model

For the capacity unfixed subject to active suspension perfor-mance 119862

119906

119862119906119901119906= 119896119906(119901119891minus 119901119906) + 120573 ( 119901

119891minus 119901119906) + (1 minus 120572) 119889 (6)

Variables definition is as follows

119862119891 capacity fixed

119862119906 capacity fixed subject to active suspension

119901119891 price level for the 119862

119891capacity fixed

119901119906 price level for the 119862

119906subject to active suspension

119908 price level disturbance in the market119889 demand119896119891 biding price restitution coefficient based on price

for 119862119891

119896119906 biding price restitution coefficient based on price

for 119862119906

Mathematical Problems in Engineering 5

120573 damping-inventory level in the system

120572 embedding parameter 0 le 120572 le 1

As it is stated in DM 119862119891and 119862

119906are assumed fixed

the characteristic of this system is such that the dampingcoefficient is 120573 The nature of the input signal is the demandand the output is the price level for the capacity fixed andfor the price level subject to active suspension There exists arelation between biding prices and restitution coefficients 119896

119891

and 119896119906 this means the amount of product per unit of price

Themodel presented in the active suspension for demand andprice modeling from Figure 2 is based on the assumptionthat the analysis is for two products in the system We planto extend our model to a half-car active suspension withbalanced capacity and unbalanced capacity for four productsFinally in future work we intend to extend it to a full-carmodel for a multiproduct system and analyze if the systemis balanced (with the same amount of capacity in each activesuspension)

32 Robust Policies in DM A Sensitivity Analysis The follow-ing approach can be used for the design of robust controllersas suggested in [32] which analyzed the robustness for mass-spring systemwithout damping Consider a representation ofan uncertain linear dynamical system in state-space form

= 119860 (119902) 119911 + 119861 (119902) 119906 (7)

where 119911 isin R119899 119906 isin R119898 and 119902 is a vector of uncertainparameters The sensitivity of the states with respect toparameter 119902 is

119889

119889119902119894

=120597119860

120597119902119894

119911 + 119860120597119911

120597119902119894

+120597119861

120597119902119894

119906 (8)

Consider the quarter active suspension system fromFigure 2 the equations of motion assuming that the uncer-tain parameter is the damping coefficient (120573) and the partialderivative of (5) and (6) with respect to 120573 are

120597 119901119891

120597120573=

1

119862119891

[(119906minus 119891) + 120573(

120597119906

120597120573minus

120597119891

120597120573)] (9)

120597 119901119906

120597120573= minus

1

119862119906

[(119906minus 119891) minus 120573(

120597119891

120597120573minus120597119906

120597120573)] (10)

Both equations are not independent of one another thesensitivity states are expressed as

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (11)

The following boundary conditions increase the robust-ness

120597119901119891

120597120573=120597119901119906

120597120573=

120597 119901119891

120597120573=120597 119901119906

120597120573= 0 (12)

Integrating (11) in time and subject to boundary conditions(12) finally the relationship is such that

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (13)

Substituting (13) into (10) permits achieving the general statesensitivity equation of the form

120597 119901119906

120597120573= minus

1

119862119906

(119906minus 119891) + (

120573

119862119891

+120573

119862119906

)(120597119906

120597120573) (14)

The sensitivity analysis presented in Section 32 considersvariations in the inventory level which gives a capacityequivalent 119862eq = 119862

119906119862119891(119862119906+ 119862119891) to be used in the

dynamical system for modeling consideration and it refers torobust policies We conclude that a robust inventory policy isrepresented by the following equation 120573119862eq = 120573((1119862

119891) +

(1119862119906))

33 Stability Analysis for DM For the behavior of the damp-ing effect in DM this section proposes the stability criteriavia Lyapunov direct method

Definition 1 A linear system is stabilizable if all unstablemodes are controllable

DM-ODE equation (2) is necessary to analyze the stabilityperformance Assume the DM-ODE equation has the struc-ture = 119860119909 + 119861119906

Theorem 2 For the dynamical system (2) and consideringthat (A B) are controllable a necessary and sufficient conditionfor stability in DM is 120573 gt 0

Proof The following condition in the demand of the systemis assumed

119889 (119905) = 0 (15)

Equation (2) can be formulated as

119862eq + 120573 + 119896119901 = 0 (16)

From (16) we use energy as the Lyapunov function

119881 (119901 ) =1

2119862eq2+1

21198961199012 (17)

119889119881

119889119905=120597119881

120597119901

119889119901

119889119905+120597119881

120597

119889

119889119905 (18)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (19)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (

minus120573

119862eq minus

119896

119862eq119901) (20)

Eliminating terms from (20) the following condition isachieved minus120573 sdot 2 le 0 Finally the presence of inventory levelin the system 120573 gt 0 is a necessary and sufficient condition forstability purposes

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

total expected revenues bymaking the correct decision Fromhere our premise is to show that DP can be modeled applyingcontrol systems and specifically addressed by the techniquesof optimal control in real time as the well-known modelpredictive control (MPC) This research work introduces asecond order dynamical system which integrates capacityinventory level and biding price in the framework of DM viaDP We utilize a dynamical 14 active suspension system tomodel and prove the applicability of DP to SCM Moreoverby using the Lyapunov direct method we concluded that asufficient and necessary condition for stability is the presenceof inventory level along the supply chain

The remainder of this paperrsquos structure is as followsSection 2 describes DM in semiconductor manufacturingSection 3 presents sensitivity and stability analysis for DMSection 4 considers model predictive control formulationfor DM with respective simulations while conclusions areprovided in Section 5

2 Demand Management Sector Applications

21 Demand Management Sector Applications A Review ofthe Current Literature The techniques of DM are relativelynew and the first research that dealt directly with theseissues appeared less than 20 years ago The major sectorof application of DM has been in the airline industry anddifferent approaches have been presented in the literaturefor airlines and hotels [4ndash12] (see Table 1) manufacturingservices and transportation [13ndash22] (see Table 2) So muchresearch has been done in the DM field Please refer toTables 1 and 2 for an extended list of authors and a briefdescription of their work There are many different areas inwhichDMhas been applied successfully as shown in Figure 1To this day most of the research has been performed in theairline sector (which is justified because DM was developedto solve problems in this area) and in the area of hotelmanagement where by applying DM techniques it has beenpossible tomitigate some of the booking limit problems basedon reservations We present a novel approach for dynamicpricing-inventory level based on DM assuming that we havea pair of products subject to the same demand profile

22 From Supply to Demand Management The DampingEffect Synchronization between supply and demand is theoptimal scenario in any complex supply network In dynam-ical systems dissipation is considered the loss of energyover time this is because of damping It is common totake into account the damping effect in different situationslike economics finance and electromechanical systems andmuch more in biological systems In DM the notion ofdamping is related to inventory level basically to achievestability conditions over the supply chain

221 Finite Dimensional Formulation for Demand Man-agement The ordinary differential equation (ODE) for thedynamic system (DS) with damping is

1198721198892119909

1198891199052+ 120573

119889119909

119889119905+ 119870119909 = 119891 (119905) (1)

36

28

14

319

AirlinesServicesManufacturing

TransportationHotels

Figure 1 Demand management sector applications

The present model in (1) describes the dynamics of mass-spring-damper second order dynamical system Also pub-lished by us [23] this model proposes a dynamic pricing forDM applying fast model predictive control approach Thepresent work is an extension of our previously publishedwork

For purposes of demand management we present thefollowing model in the damping part of (1)

1198621198892119901

1198891199052+ [int

infin

0

120573 (119905) 119889119905]119889119901

119889119905+ 119870119901 = 119889 (119905) (2)

Equation (2) presents 119901mdashprice level 119862mdashcapacity 119889(119905)mdashdemand 119870mdashbiding price restitution coefficient and 120573(119905)mdashdamping coefficient-inventory level

The damping factor in (2) has the characteristic to be afunction over time and then Lipschitz property can be appliedand requires convergence in a scalar value (see Appendix A)

The demand management problem for optimal controlhas been addressed previously by different research groups[24] dynamic pricing for optimum inventory level has beenpreviously integrated as a first order ODE as shown in[25] Moreover [26] has extended this work by integratingdynamic pricing and demand level into a second orderODE The interaction of dynamic pricing and demand level[27] presents a dynamic pricing model on web service inwhich case the demand function is nonlinear Optimizationtechniques have been shown [28] and have been used toquantify demand response in energy industry applying bid-pricing approaches also integer linear programming hasbeen applied to model inventory level production level andcapacity for dynamic pricing in [29]

222 Infinite Dimensional Formulation for Demand Manage-ment An extended version of ODE in (2) is expressed as ageneralized wave equation first proposed in [30] for flexiblesystems such as

119862 (119901)1205972120593

1205971199052+ 120573

120597120593

120597119905+ 119870120593 = 119863 (119901 119905) (3)

Mathematical Problems in Engineering 3

Table 1 Demand management for airlines and hotels literature review

Author Sector application Methodology Contribution

Abdel Aziz et al [4] Hotels Research article (casestudies)

Application to address the problem of room pricing inhotels Optimization techniques are present Multiclassscheme similar to the one implemented in airlines isstudied

de Boer [5] Airlines Research article(theoretical)

Determination of a revenue management policy inresponse to the realization of the demand This is doneby matching supply and demand for seats

Hung and Chen [6] Airlines Research article (casestudies)

This study develops and tests two heuristics approachesthe dynamic seat rationing and the expected revenuegap to help the airline make a fulfillment or rejectiondecision when a customer arrives

Gosavi et al [7] AirlinesResearch article(simulation basedoptimization)

Application of simulation based optimization for seatallocation in airlines The model considers real lifeassumption such as cancelations and overbooking

Graf and Kimms [8] Airlines Research article (casestudies)

The optimal transfer prices are determined by anegotiation process Option-based capacity controlprocess results combined with transfer priceoptimization are compared with a first come first servedapproach

Kimms andMuller-Bungart [9] Airlines

Research article(simulation based

case study)

Application of heuristics and optimal networks inrevenue management Assuming stochastic demand

Netessine and Shumsky [10] Airlines SurveyAnalysis of fundamental concepts and trade-offs ofyield management to describe the parallels betweenyield management and inventory management

Rannou and Melli [11] Hotels Research article (casestudy)

Analysis and application of revenue management inhospitality industry The idea is to achieve aperformance measure for the return of investment

Walczak and Brumelle [12] Airlines Research article(theoretical)

Problem formulation of semi-Markov model fordynamic pricing and revenue management in airlineindustry assuming continuous demand

Equation (3) is called master damping equation for demandmanagement which can be written as partial differentialequation (PDE) as follows

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(4)

where first derivative of price in time is V119901and second

derivative in time is 119886119901 See Appendix B for a detailed proof

of (3) and (4)

3 Robust and Stability Analysis forDemand Management

DM can be stated as follows how to select the productrsquos mixamount and price in order to satisfy a fluctuating demandwith the main goal of maximizing the profit For this reasonin this document we propose the use of MPC by comparingDM and DP using an active suspension with damping

31 Physical Modeling in DM-DP Problem The use of aphysical model represents an approach to obtain a suitablemodel to be controlled According to [31] DM is the creationacross the supply chain and its markets of a coordinated flowof demand Also the role of DM is to decrease demandThe application of the active suspension analogy to trackboth problems is suitable because the price level (whichrepresents outputs) and demand (which represents input)converge to a stability condition that requires inventory levelin the dissipation of the dynamical system DM assumes fixedcapacity (which in terms of active suspension is the mass)inventory level (damping of the system) and biding price(restitution coefficient) In this case the idea is to present amodel in which by using DM we can assess how demanddoes affect prices We have chosen an active suspensionsystem to approach the system model see Figure 2

The dynamic equations for the system are as followsFor the capacity fixed 119862

119891

119862119891

119901119891= 119896119891(119908 minus 119901

119891) minus 119896119906(119901119891minus 119901119906)

minus 120573 ( 119901119891minus 119901119906) minus 120572 sdot 119889

(5)

4 Mathematical Problems in Engineering

Table 2 Demand management for manufacturing and services literature review

Author Sector application Methodology Contribution

Bertsimas and Shioda [13] Services Research article (casestudies)

Application of revenue management in restaurantsUsing stochastic optimization via dynamicprogramming Computational experiments arepresented for reservation and without reservation casestudies

Boyd and Bilegan [14] Services Research article (casestudy in e-commerce)

Analysis of revenue management applied toe-commerce in the airline industry Optimizationtechniques are shown in inventory control mechanism

Defregger and Kuhn [15] Manufacturing Research article (casestudies)

Application of revenue management to a make-to-ordermanufacturing company with limited inventorycapacity Classifying the underlying Markov decisionprocess by introducing a heuristic procedure

Kimes [16] Services Survey Revenue management analysis via the application ofoptimization forecasting and overbooking

Kumar and Frederik [17] Services Survey

Application of revenue management in constructionindustry The benefits of revenue management can berealized at manufacturing companies Uncertaindemand and pricing of available capacity are analyzedvia this approach

Lee et al [18] Transportation Research article (casestudies)

Analysis and application of heuristic to solve a singlerevenue management problem with postponementarising from the sea cargo industry Optimization basedcase studies are addressed

Nair and Bapna [19] Services Research articleThis paper studies optimal policies for allocatingmodems capacity among segments of customers using acontinuous time Markov decision

Spengler et al [20] Manufacturing Research article (casestudies)

Revenue management approach for companies in theiron and steel industry is developed The aim is toimprove short-term order selection

Shin and Park [21] Manufacturing Research article(theoretical)

Yield improvement is analyzed in semiconductormanufacturing via machine learning

Steinhardt and Gonsch [22] Services Research article (casestudies)

This paper addresses the problem of integrating revenuemanagement capacity control with upgrade decisionmaking A new structural property for an integrateddynamic programming formulation is present

ku

Cu

Cf

d

d

pf

pu

wkf

120573

Figure 2 Active suspension analogy for DM-DP model

For the capacity unfixed subject to active suspension perfor-mance 119862

119906

119862119906119901119906= 119896119906(119901119891minus 119901119906) + 120573 ( 119901

119891minus 119901119906) + (1 minus 120572) 119889 (6)

Variables definition is as follows

119862119891 capacity fixed

119862119906 capacity fixed subject to active suspension

119901119891 price level for the 119862

119891capacity fixed

119901119906 price level for the 119862

119906subject to active suspension

119908 price level disturbance in the market119889 demand119896119891 biding price restitution coefficient based on price

for 119862119891

119896119906 biding price restitution coefficient based on price

for 119862119906

Mathematical Problems in Engineering 5

120573 damping-inventory level in the system

120572 embedding parameter 0 le 120572 le 1

As it is stated in DM 119862119891and 119862

119906are assumed fixed

the characteristic of this system is such that the dampingcoefficient is 120573 The nature of the input signal is the demandand the output is the price level for the capacity fixed andfor the price level subject to active suspension There exists arelation between biding prices and restitution coefficients 119896

119891

and 119896119906 this means the amount of product per unit of price

Themodel presented in the active suspension for demand andprice modeling from Figure 2 is based on the assumptionthat the analysis is for two products in the system We planto extend our model to a half-car active suspension withbalanced capacity and unbalanced capacity for four productsFinally in future work we intend to extend it to a full-carmodel for a multiproduct system and analyze if the systemis balanced (with the same amount of capacity in each activesuspension)

32 Robust Policies in DM A Sensitivity Analysis The follow-ing approach can be used for the design of robust controllersas suggested in [32] which analyzed the robustness for mass-spring systemwithout damping Consider a representation ofan uncertain linear dynamical system in state-space form

= 119860 (119902) 119911 + 119861 (119902) 119906 (7)

where 119911 isin R119899 119906 isin R119898 and 119902 is a vector of uncertainparameters The sensitivity of the states with respect toparameter 119902 is

119889

119889119902119894

=120597119860

120597119902119894

119911 + 119860120597119911

120597119902119894

+120597119861

120597119902119894

119906 (8)

Consider the quarter active suspension system fromFigure 2 the equations of motion assuming that the uncer-tain parameter is the damping coefficient (120573) and the partialderivative of (5) and (6) with respect to 120573 are

120597 119901119891

120597120573=

1

119862119891

[(119906minus 119891) + 120573(

120597119906

120597120573minus

120597119891

120597120573)] (9)

120597 119901119906

120597120573= minus

1

119862119906

[(119906minus 119891) minus 120573(

120597119891

120597120573minus120597119906

120597120573)] (10)

Both equations are not independent of one another thesensitivity states are expressed as

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (11)

The following boundary conditions increase the robust-ness

120597119901119891

120597120573=120597119901119906

120597120573=

120597 119901119891

120597120573=120597 119901119906

120597120573= 0 (12)

Integrating (11) in time and subject to boundary conditions(12) finally the relationship is such that

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (13)

Substituting (13) into (10) permits achieving the general statesensitivity equation of the form

120597 119901119906

120597120573= minus

1

119862119906

(119906minus 119891) + (

120573

119862119891

+120573

119862119906

)(120597119906

120597120573) (14)

The sensitivity analysis presented in Section 32 considersvariations in the inventory level which gives a capacityequivalent 119862eq = 119862

119906119862119891(119862119906+ 119862119891) to be used in the

dynamical system for modeling consideration and it refers torobust policies We conclude that a robust inventory policy isrepresented by the following equation 120573119862eq = 120573((1119862

119891) +

(1119862119906))

33 Stability Analysis for DM For the behavior of the damp-ing effect in DM this section proposes the stability criteriavia Lyapunov direct method

Definition 1 A linear system is stabilizable if all unstablemodes are controllable

DM-ODE equation (2) is necessary to analyze the stabilityperformance Assume the DM-ODE equation has the struc-ture = 119860119909 + 119861119906

Theorem 2 For the dynamical system (2) and consideringthat (A B) are controllable a necessary and sufficient conditionfor stability in DM is 120573 gt 0

Proof The following condition in the demand of the systemis assumed

119889 (119905) = 0 (15)

Equation (2) can be formulated as

119862eq + 120573 + 119896119901 = 0 (16)

From (16) we use energy as the Lyapunov function

119881 (119901 ) =1

2119862eq2+1

21198961199012 (17)

119889119881

119889119905=120597119881

120597119901

119889119901

119889119905+120597119881

120597

119889

119889119905 (18)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (19)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (

minus120573

119862eq minus

119896

119862eq119901) (20)

Eliminating terms from (20) the following condition isachieved minus120573 sdot 2 le 0 Finally the presence of inventory levelin the system 120573 gt 0 is a necessary and sufficient condition forstability purposes

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Table 1 Demand management for airlines and hotels literature review

Author Sector application Methodology Contribution

Abdel Aziz et al [4] Hotels Research article (casestudies)

Application to address the problem of room pricing inhotels Optimization techniques are present Multiclassscheme similar to the one implemented in airlines isstudied

de Boer [5] Airlines Research article(theoretical)

Determination of a revenue management policy inresponse to the realization of the demand This is doneby matching supply and demand for seats

Hung and Chen [6] Airlines Research article (casestudies)

This study develops and tests two heuristics approachesthe dynamic seat rationing and the expected revenuegap to help the airline make a fulfillment or rejectiondecision when a customer arrives

Gosavi et al [7] AirlinesResearch article(simulation basedoptimization)

Application of simulation based optimization for seatallocation in airlines The model considers real lifeassumption such as cancelations and overbooking

Graf and Kimms [8] Airlines Research article (casestudies)

The optimal transfer prices are determined by anegotiation process Option-based capacity controlprocess results combined with transfer priceoptimization are compared with a first come first servedapproach

Kimms andMuller-Bungart [9] Airlines

Research article(simulation based

case study)

Application of heuristics and optimal networks inrevenue management Assuming stochastic demand

Netessine and Shumsky [10] Airlines SurveyAnalysis of fundamental concepts and trade-offs ofyield management to describe the parallels betweenyield management and inventory management

Rannou and Melli [11] Hotels Research article (casestudy)

Analysis and application of revenue management inhospitality industry The idea is to achieve aperformance measure for the return of investment

Walczak and Brumelle [12] Airlines Research article(theoretical)

Problem formulation of semi-Markov model fordynamic pricing and revenue management in airlineindustry assuming continuous demand

Equation (3) is called master damping equation for demandmanagement which can be written as partial differentialequation (PDE) as follows

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(4)

where first derivative of price in time is V119901and second

derivative in time is 119886119901 See Appendix B for a detailed proof

of (3) and (4)

3 Robust and Stability Analysis forDemand Management

DM can be stated as follows how to select the productrsquos mixamount and price in order to satisfy a fluctuating demandwith the main goal of maximizing the profit For this reasonin this document we propose the use of MPC by comparingDM and DP using an active suspension with damping

31 Physical Modeling in DM-DP Problem The use of aphysical model represents an approach to obtain a suitablemodel to be controlled According to [31] DM is the creationacross the supply chain and its markets of a coordinated flowof demand Also the role of DM is to decrease demandThe application of the active suspension analogy to trackboth problems is suitable because the price level (whichrepresents outputs) and demand (which represents input)converge to a stability condition that requires inventory levelin the dissipation of the dynamical system DM assumes fixedcapacity (which in terms of active suspension is the mass)inventory level (damping of the system) and biding price(restitution coefficient) In this case the idea is to present amodel in which by using DM we can assess how demanddoes affect prices We have chosen an active suspensionsystem to approach the system model see Figure 2

The dynamic equations for the system are as followsFor the capacity fixed 119862

119891

119862119891

119901119891= 119896119891(119908 minus 119901

119891) minus 119896119906(119901119891minus 119901119906)

minus 120573 ( 119901119891minus 119901119906) minus 120572 sdot 119889

(5)

4 Mathematical Problems in Engineering

Table 2 Demand management for manufacturing and services literature review

Author Sector application Methodology Contribution

Bertsimas and Shioda [13] Services Research article (casestudies)

Application of revenue management in restaurantsUsing stochastic optimization via dynamicprogramming Computational experiments arepresented for reservation and without reservation casestudies

Boyd and Bilegan [14] Services Research article (casestudy in e-commerce)

Analysis of revenue management applied toe-commerce in the airline industry Optimizationtechniques are shown in inventory control mechanism

Defregger and Kuhn [15] Manufacturing Research article (casestudies)

Application of revenue management to a make-to-ordermanufacturing company with limited inventorycapacity Classifying the underlying Markov decisionprocess by introducing a heuristic procedure

Kimes [16] Services Survey Revenue management analysis via the application ofoptimization forecasting and overbooking

Kumar and Frederik [17] Services Survey

Application of revenue management in constructionindustry The benefits of revenue management can berealized at manufacturing companies Uncertaindemand and pricing of available capacity are analyzedvia this approach

Lee et al [18] Transportation Research article (casestudies)

Analysis and application of heuristic to solve a singlerevenue management problem with postponementarising from the sea cargo industry Optimization basedcase studies are addressed

Nair and Bapna [19] Services Research articleThis paper studies optimal policies for allocatingmodems capacity among segments of customers using acontinuous time Markov decision

Spengler et al [20] Manufacturing Research article (casestudies)

Revenue management approach for companies in theiron and steel industry is developed The aim is toimprove short-term order selection

Shin and Park [21] Manufacturing Research article(theoretical)

Yield improvement is analyzed in semiconductormanufacturing via machine learning

Steinhardt and Gonsch [22] Services Research article (casestudies)

This paper addresses the problem of integrating revenuemanagement capacity control with upgrade decisionmaking A new structural property for an integrateddynamic programming formulation is present

ku

Cu

Cf

d

d

pf

pu

wkf

120573

Figure 2 Active suspension analogy for DM-DP model

For the capacity unfixed subject to active suspension perfor-mance 119862

119906

119862119906119901119906= 119896119906(119901119891minus 119901119906) + 120573 ( 119901

119891minus 119901119906) + (1 minus 120572) 119889 (6)

Variables definition is as follows

119862119891 capacity fixed

119862119906 capacity fixed subject to active suspension

119901119891 price level for the 119862

119891capacity fixed

119901119906 price level for the 119862

119906subject to active suspension

119908 price level disturbance in the market119889 demand119896119891 biding price restitution coefficient based on price

for 119862119891

119896119906 biding price restitution coefficient based on price

for 119862119906

Mathematical Problems in Engineering 5

120573 damping-inventory level in the system

120572 embedding parameter 0 le 120572 le 1

As it is stated in DM 119862119891and 119862

119906are assumed fixed

the characteristic of this system is such that the dampingcoefficient is 120573 The nature of the input signal is the demandand the output is the price level for the capacity fixed andfor the price level subject to active suspension There exists arelation between biding prices and restitution coefficients 119896

119891

and 119896119906 this means the amount of product per unit of price

Themodel presented in the active suspension for demand andprice modeling from Figure 2 is based on the assumptionthat the analysis is for two products in the system We planto extend our model to a half-car active suspension withbalanced capacity and unbalanced capacity for four productsFinally in future work we intend to extend it to a full-carmodel for a multiproduct system and analyze if the systemis balanced (with the same amount of capacity in each activesuspension)

32 Robust Policies in DM A Sensitivity Analysis The follow-ing approach can be used for the design of robust controllersas suggested in [32] which analyzed the robustness for mass-spring systemwithout damping Consider a representation ofan uncertain linear dynamical system in state-space form

= 119860 (119902) 119911 + 119861 (119902) 119906 (7)

where 119911 isin R119899 119906 isin R119898 and 119902 is a vector of uncertainparameters The sensitivity of the states with respect toparameter 119902 is

119889

119889119902119894

=120597119860

120597119902119894

119911 + 119860120597119911

120597119902119894

+120597119861

120597119902119894

119906 (8)

Consider the quarter active suspension system fromFigure 2 the equations of motion assuming that the uncer-tain parameter is the damping coefficient (120573) and the partialderivative of (5) and (6) with respect to 120573 are

120597 119901119891

120597120573=

1

119862119891

[(119906minus 119891) + 120573(

120597119906

120597120573minus

120597119891

120597120573)] (9)

120597 119901119906

120597120573= minus

1

119862119906

[(119906minus 119891) minus 120573(

120597119891

120597120573minus120597119906

120597120573)] (10)

Both equations are not independent of one another thesensitivity states are expressed as

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (11)

The following boundary conditions increase the robust-ness

120597119901119891

120597120573=120597119901119906

120597120573=

120597 119901119891

120597120573=120597 119901119906

120597120573= 0 (12)

Integrating (11) in time and subject to boundary conditions(12) finally the relationship is such that

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (13)

Substituting (13) into (10) permits achieving the general statesensitivity equation of the form

120597 119901119906

120597120573= minus

1

119862119906

(119906minus 119891) + (

120573

119862119891

+120573

119862119906

)(120597119906

120597120573) (14)

The sensitivity analysis presented in Section 32 considersvariations in the inventory level which gives a capacityequivalent 119862eq = 119862

119906119862119891(119862119906+ 119862119891) to be used in the

dynamical system for modeling consideration and it refers torobust policies We conclude that a robust inventory policy isrepresented by the following equation 120573119862eq = 120573((1119862

119891) +

(1119862119906))

33 Stability Analysis for DM For the behavior of the damp-ing effect in DM this section proposes the stability criteriavia Lyapunov direct method

Definition 1 A linear system is stabilizable if all unstablemodes are controllable

DM-ODE equation (2) is necessary to analyze the stabilityperformance Assume the DM-ODE equation has the struc-ture = 119860119909 + 119861119906

Theorem 2 For the dynamical system (2) and consideringthat (A B) are controllable a necessary and sufficient conditionfor stability in DM is 120573 gt 0

Proof The following condition in the demand of the systemis assumed

119889 (119905) = 0 (15)

Equation (2) can be formulated as

119862eq + 120573 + 119896119901 = 0 (16)

From (16) we use energy as the Lyapunov function

119881 (119901 ) =1

2119862eq2+1

21198961199012 (17)

119889119881

119889119905=120597119881

120597119901

119889119901

119889119905+120597119881

120597

119889

119889119905 (18)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (19)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (

minus120573

119862eq minus

119896

119862eq119901) (20)

Eliminating terms from (20) the following condition isachieved minus120573 sdot 2 le 0 Finally the presence of inventory levelin the system 120573 gt 0 is a necessary and sufficient condition forstability purposes

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 2 Demand management for manufacturing and services literature review

Author Sector application Methodology Contribution

Bertsimas and Shioda [13] Services Research article (casestudies)

Application of revenue management in restaurantsUsing stochastic optimization via dynamicprogramming Computational experiments arepresented for reservation and without reservation casestudies

Boyd and Bilegan [14] Services Research article (casestudy in e-commerce)

Analysis of revenue management applied toe-commerce in the airline industry Optimizationtechniques are shown in inventory control mechanism

Defregger and Kuhn [15] Manufacturing Research article (casestudies)

Application of revenue management to a make-to-ordermanufacturing company with limited inventorycapacity Classifying the underlying Markov decisionprocess by introducing a heuristic procedure

Kimes [16] Services Survey Revenue management analysis via the application ofoptimization forecasting and overbooking

Kumar and Frederik [17] Services Survey

Application of revenue management in constructionindustry The benefits of revenue management can berealized at manufacturing companies Uncertaindemand and pricing of available capacity are analyzedvia this approach

Lee et al [18] Transportation Research article (casestudies)

Analysis and application of heuristic to solve a singlerevenue management problem with postponementarising from the sea cargo industry Optimization basedcase studies are addressed

Nair and Bapna [19] Services Research articleThis paper studies optimal policies for allocatingmodems capacity among segments of customers using acontinuous time Markov decision

Spengler et al [20] Manufacturing Research article (casestudies)

Revenue management approach for companies in theiron and steel industry is developed The aim is toimprove short-term order selection

Shin and Park [21] Manufacturing Research article(theoretical)

Yield improvement is analyzed in semiconductormanufacturing via machine learning

Steinhardt and Gonsch [22] Services Research article (casestudies)

This paper addresses the problem of integrating revenuemanagement capacity control with upgrade decisionmaking A new structural property for an integrateddynamic programming formulation is present

ku

Cu

Cf

d

d

pf

pu

wkf

120573

Figure 2 Active suspension analogy for DM-DP model

For the capacity unfixed subject to active suspension perfor-mance 119862

119906

119862119906119901119906= 119896119906(119901119891minus 119901119906) + 120573 ( 119901

119891minus 119901119906) + (1 minus 120572) 119889 (6)

Variables definition is as follows

119862119891 capacity fixed

119862119906 capacity fixed subject to active suspension

119901119891 price level for the 119862

119891capacity fixed

119901119906 price level for the 119862

119906subject to active suspension

119908 price level disturbance in the market119889 demand119896119891 biding price restitution coefficient based on price

for 119862119891

119896119906 biding price restitution coefficient based on price

for 119862119906

Mathematical Problems in Engineering 5

120573 damping-inventory level in the system

120572 embedding parameter 0 le 120572 le 1

As it is stated in DM 119862119891and 119862

119906are assumed fixed

the characteristic of this system is such that the dampingcoefficient is 120573 The nature of the input signal is the demandand the output is the price level for the capacity fixed andfor the price level subject to active suspension There exists arelation between biding prices and restitution coefficients 119896

119891

and 119896119906 this means the amount of product per unit of price

Themodel presented in the active suspension for demand andprice modeling from Figure 2 is based on the assumptionthat the analysis is for two products in the system We planto extend our model to a half-car active suspension withbalanced capacity and unbalanced capacity for four productsFinally in future work we intend to extend it to a full-carmodel for a multiproduct system and analyze if the systemis balanced (with the same amount of capacity in each activesuspension)

32 Robust Policies in DM A Sensitivity Analysis The follow-ing approach can be used for the design of robust controllersas suggested in [32] which analyzed the robustness for mass-spring systemwithout damping Consider a representation ofan uncertain linear dynamical system in state-space form

= 119860 (119902) 119911 + 119861 (119902) 119906 (7)

where 119911 isin R119899 119906 isin R119898 and 119902 is a vector of uncertainparameters The sensitivity of the states with respect toparameter 119902 is

119889

119889119902119894

=120597119860

120597119902119894

119911 + 119860120597119911

120597119902119894

+120597119861

120597119902119894

119906 (8)

Consider the quarter active suspension system fromFigure 2 the equations of motion assuming that the uncer-tain parameter is the damping coefficient (120573) and the partialderivative of (5) and (6) with respect to 120573 are

120597 119901119891

120597120573=

1

119862119891

[(119906minus 119891) + 120573(

120597119906

120597120573minus

120597119891

120597120573)] (9)

120597 119901119906

120597120573= minus

1

119862119906

[(119906minus 119891) minus 120573(

120597119891

120597120573minus120597119906

120597120573)] (10)

Both equations are not independent of one another thesensitivity states are expressed as

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (11)

The following boundary conditions increase the robust-ness

120597119901119891

120597120573=120597119901119906

120597120573=

120597 119901119891

120597120573=120597 119901119906

120597120573= 0 (12)

Integrating (11) in time and subject to boundary conditions(12) finally the relationship is such that

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (13)

Substituting (13) into (10) permits achieving the general statesensitivity equation of the form

120597 119901119906

120597120573= minus

1

119862119906

(119906minus 119891) + (

120573

119862119891

+120573

119862119906

)(120597119906

120597120573) (14)

The sensitivity analysis presented in Section 32 considersvariations in the inventory level which gives a capacityequivalent 119862eq = 119862

119906119862119891(119862119906+ 119862119891) to be used in the

dynamical system for modeling consideration and it refers torobust policies We conclude that a robust inventory policy isrepresented by the following equation 120573119862eq = 120573((1119862

119891) +

(1119862119906))

33 Stability Analysis for DM For the behavior of the damp-ing effect in DM this section proposes the stability criteriavia Lyapunov direct method

Definition 1 A linear system is stabilizable if all unstablemodes are controllable

DM-ODE equation (2) is necessary to analyze the stabilityperformance Assume the DM-ODE equation has the struc-ture = 119860119909 + 119861119906

Theorem 2 For the dynamical system (2) and consideringthat (A B) are controllable a necessary and sufficient conditionfor stability in DM is 120573 gt 0

Proof The following condition in the demand of the systemis assumed

119889 (119905) = 0 (15)

Equation (2) can be formulated as

119862eq + 120573 + 119896119901 = 0 (16)

From (16) we use energy as the Lyapunov function

119881 (119901 ) =1

2119862eq2+1

21198961199012 (17)

119889119881

119889119905=120597119881

120597119901

119889119901

119889119905+120597119881

120597

119889

119889119905 (18)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (19)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (

minus120573

119862eq minus

119896

119862eq119901) (20)

Eliminating terms from (20) the following condition isachieved minus120573 sdot 2 le 0 Finally the presence of inventory levelin the system 120573 gt 0 is a necessary and sufficient condition forstability purposes

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

120573 damping-inventory level in the system

120572 embedding parameter 0 le 120572 le 1

As it is stated in DM 119862119891and 119862

119906are assumed fixed

the characteristic of this system is such that the dampingcoefficient is 120573 The nature of the input signal is the demandand the output is the price level for the capacity fixed andfor the price level subject to active suspension There exists arelation between biding prices and restitution coefficients 119896

119891

and 119896119906 this means the amount of product per unit of price

Themodel presented in the active suspension for demand andprice modeling from Figure 2 is based on the assumptionthat the analysis is for two products in the system We planto extend our model to a half-car active suspension withbalanced capacity and unbalanced capacity for four productsFinally in future work we intend to extend it to a full-carmodel for a multiproduct system and analyze if the systemis balanced (with the same amount of capacity in each activesuspension)

32 Robust Policies in DM A Sensitivity Analysis The follow-ing approach can be used for the design of robust controllersas suggested in [32] which analyzed the robustness for mass-spring systemwithout damping Consider a representation ofan uncertain linear dynamical system in state-space form

= 119860 (119902) 119911 + 119861 (119902) 119906 (7)

where 119911 isin R119899 119906 isin R119898 and 119902 is a vector of uncertainparameters The sensitivity of the states with respect toparameter 119902 is

119889

119889119902119894

=120597119860

120597119902119894

119911 + 119860120597119911

120597119902119894

+120597119861

120597119902119894

119906 (8)

Consider the quarter active suspension system fromFigure 2 the equations of motion assuming that the uncer-tain parameter is the damping coefficient (120573) and the partialderivative of (5) and (6) with respect to 120573 are

120597 119901119891

120597120573=

1

119862119891

[(119906minus 119891) + 120573(

120597119906

120597120573minus

120597119891

120597120573)] (9)

120597 119901119906

120597120573= minus

1

119862119906

[(119906minus 119891) minus 120573(

120597119891

120597120573minus120597119906

120597120573)] (10)

Both equations are not independent of one another thesensitivity states are expressed as

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (11)

The following boundary conditions increase the robust-ness

120597119901119891

120597120573=120597119901119906

120597120573=

120597 119901119891

120597120573=120597 119901119906

120597120573= 0 (12)

Integrating (11) in time and subject to boundary conditions(12) finally the relationship is such that

120597 119901119891

120597120573= minus

119862119906

119862119891

(120597119906

120597120573) (13)

Substituting (13) into (10) permits achieving the general statesensitivity equation of the form

120597 119901119906

120597120573= minus

1

119862119906

(119906minus 119891) + (

120573

119862119891

+120573

119862119906

)(120597119906

120597120573) (14)

The sensitivity analysis presented in Section 32 considersvariations in the inventory level which gives a capacityequivalent 119862eq = 119862

119906119862119891(119862119906+ 119862119891) to be used in the

dynamical system for modeling consideration and it refers torobust policies We conclude that a robust inventory policy isrepresented by the following equation 120573119862eq = 120573((1119862

119891) +

(1119862119906))

33 Stability Analysis for DM For the behavior of the damp-ing effect in DM this section proposes the stability criteriavia Lyapunov direct method

Definition 1 A linear system is stabilizable if all unstablemodes are controllable

DM-ODE equation (2) is necessary to analyze the stabilityperformance Assume the DM-ODE equation has the struc-ture = 119860119909 + 119861119906

Theorem 2 For the dynamical system (2) and consideringthat (A B) are controllable a necessary and sufficient conditionfor stability in DM is 120573 gt 0

Proof The following condition in the demand of the systemis assumed

119889 (119905) = 0 (15)

Equation (2) can be formulated as

119862eq + 120573 + 119896119901 = 0 (16)

From (16) we use energy as the Lyapunov function

119881 (119901 ) =1

2119862eq2+1

21198961199012 (17)

119889119881

119889119905=120597119881

120597119901

119889119901

119889119905+120597119881

120597

119889

119889119905 (18)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (19)

119889119881

119889119905= 119896119901 sdot + 119862eq sdot (

minus120573

119862eq minus

119896

119862eq119901) (20)

Eliminating terms from (20) the following condition isachieved minus120573 sdot 2 le 0 Finally the presence of inventory levelin the system 120573 gt 0 is a necessary and sufficient condition forstability purposes

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

pr pk

k

dk

k + 1 k + N

Price level (predicted outputs)

Demand (manipulated output)

Figure 3 Schematic receding horizon control philosophy

4 Model Predictive ControlAnalysis for Demand Management

41Model Predictive Control TowardsOptimal Policies inDMModel predictive control (MPC) is a real time optimal controlstrategy that has been applied in process control aerospaceautomotive management science and robotic applicationsBesides these applications MPC has other advantages suchas good tracking performance physical constraints handlingand extension to nonlinear systems [33] MPC has been usedto address production-inventory systems For example inadaptive MPC the adapted model along with a smoothedestimation of the future customer demand is used to predictinventory levels over the optimization horizon [34] MPCpolicy shows improved performance greater flexibility andhigher functionality relative to an advanced order-up-topolicy based on control engineering principles found in theliterature [35] The philosophy of MPC also known as reced-ing horizon control (RHC) has the advantage of handlingcontrol and state constraints [36] The RHC strategy takesinto account an objective function and many constraints seeFigure 3 In RHC to define an optimal strategy the first steprequires calculating a control law and repeating this processindefinitely each sampling time

With MPC an optimization problem is solved at eachtime step to determine a plan of action over a fixed timehorizon [37] MPC is a nonlinear control policy that handlesinput and output constraints as well as various controlobjectives Using MPC a system can be controlled near itsphysical limits often outperforming linear control It is wellknown that the computation of predictive control laws is acrucial task in every MPC application to systems with fastdynamics due to the fact that an optimization problem hasto be solved online [38] Other approaches applying MPCtechniques have been developed such as simplified MPCalgorithm for Markov jump systems [39] and constrainedrobust MPC effective for uncertain Markov jump systems aspreviously shown [40]

For the active suspension analogy to DM-DP we will usea linear MPC for purposes of analysis and control Based onthis we are interested in optimal control problems of the form

min119889119896 119889119896+119873minus1

119869 =

119873minus1

sum

119896=0

1003817100381710038171003817119901119896 minus 1199011199031003817100381710038171003817

2

+1003817100381710038171003817119889119896 minus 119889119903

1003817100381710038171003817

2

(21a)

Table 3 MPC design parameters for optimal and robust policies

Policy case 119879119904

119873119901

119873119906

Optimal policies 005 10 2Robust policies 003 10 2

st 119909119896+1

= 119860 sdot 119909119896+ 119861 sdot 119889

119896

119901119896= 119862 sdot 119909

119896+ 119863 sdot 119889

119896

119889min le 119889119896 le 119889max

119901min le 119901119896 le 119901max

119909119896= 119909 (119896) 119896 = 0 119873 minus 1

(21b)

Here 119909 R rarr R119899 denotes the state 119889 R rarr R119899 thecontrol input and 119901 R rarr R119899 the output of the system

The conventional linear MPC law is based on the follow-ing algorithm

Algorithm (linear MPC)

(1) Achieve the new state 119901119896

(2) Solve the optimization problem (21a) and (21b)(3) Calculate the law 119889(119896) = 119889(119896 + 0 | 119896)(4) 119896 larr 119896 + 1 Go to (1)

Computational Complexity Analysis The complexity of thesolver for the optimization problem (21a) and (21b) dependson the choice of the performance index the model andconstraints For our algorithm the optimization problem isa quadratic program (QP)

Based on the nature of the proposed algorithm whichis a QP related to an active set method each iterationhas cost 119874(1198992) floating point operations (flops) where 119899 isthe number of decision variables and 119899 is proportional tohorizon119873

119901 It is important to notice that active set methods

are exponential in the worst case but show good practicalperformance Also active set methods work best for smalland medium size problems

42 Simulations Results After the presented analysis twoscenario simulations are shown for optimal policies in Figures4 and 5 The robust policies for DM are presented in Figures6 and 7 The main goal is to contribute in the performanceof MPC for optimal policies and robust policies in DM andto compare it with linear quadratic regulator (LQR) Theapplication of tracking MPC in DM approaches achievesa suitable response for optimal policies In Figure 6 pricelevel is shown taking into account capacity fixed and unfixedresponses It is important to notice that the demand responsein Figure 7 is constrained

MPC design parameters for optimal and robust policiesare present in Table 3 It is important to note that 119879

119904is the

sampling time 119873119901is the prediction horizon and 119873

119906is the

control horizon where119873119906lt 119873119901

In Figure 4 the MPC simulation reflects under optimalpolicies a performance with small overshoot in prices and

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 4 Design parameter for DM-active suspension dynamical system

Policy case 119862119906

119862119891

119896119906

119896119891

120573

Optimal policies 300 50 2000 2000 500Robust policies 500 300 10000 10000 1000

0 05 1 15 2 25 3 35 4 45 50

05

1

15

2

25

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 4 Simulation response for optimal policy price level-tracking MPC

0 05 1 15 2 25 3 35 4 45 5

0

5

10

15

20

25

Time scale

Dem

and

leve

l (un

its)

Optimal policies for demand management

Demand

minus5

Figure 5 Optimal policies for demand management

considerable fluctuations in demand It is important toremember that we have the price level as output for thedynamical system and demand as input

The MPC simulation shows in Figure 6 the performancein the application of the robust policies obtained from theanalysis showed in Section 32 It is important to note thatthe price level in both responses is reduced by half comparedwith scenario 1 from Figure 10This reduction is based on thenotion that there is a relation between inventory level and

0 05 1 15 2 25 30

02

04

06

08

1

12

14

Time scalePr

ice l

evel

(dlls

)

Robust policies for capacity

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 6 Simulation response for robust policy price level-trackingMPC

0 05 1 15 2 25 30

5

10

15

20

25

30

35

40

45

Time scale

Dem

and

leve

l (un

its)

Robust policies for demand management

Demand

Figure 7 Robust policies for demand management

capacity equivalent for the robust policy Under variationsin this relation the price level is reduced and the demandpresents small variations

Design parameters for DM-active suspension are shownin Table 4 for optimal and robust policies It is important tonote that these parameters are the same over both policies forpurposes of control via MPC and LQR

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

35

4

Time scale

Pric

e lev

el (d

lls)

Optimal policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 8 Optimal policies for capacity via LQR

In summary from demand curve in economics whichestablish that if price is lower consumers are ready to buymore based on this context with an optimal policy the pricesare higher and demand is low However for robust policiesthe prices are lower than in optimal policies and the demandis higher In this case the relation between the price and thedemand is satisfied from economic theory

Linear Quadratic Regulator Consider that MPC is an onlinesolution of a LQR optimization problem The approach hereis to introduce the LQR methodology and compare it withthe MPC results In the state-feedback version of the LQRproblem we assume that the whole state 119909 can be measuredand therefore it is available to control [41]The state-feedbackcontrol law 119906(119905) = minus119870119909(119905)minimizes the cost function

119869LQR = intinfin

0

(119909119879119876119909 + 119906

119879119877119906) 119889119905 (22)

where 119870 = 119877minus1119861119879119875 and 119875 is solved by an algebraic Riccati

equation The results achieved by the application of LQR inthe optimal and robust policies are presented in Figures 8 and9

Once LQR is applied for optimal and robust policiessimulations present more oscillations when compared toMPC using the same parameters from Table 4 In Figure 8for optimal policies the price level achieves more overshootand a larger setting time Also in Figure 9 the demand levelpresents higher variation than in MPC optimal policies Thedemand level is reduced at the end of time horizon as it hasbeen proposed in DM theory

Robust policies achieve an oscillatory performance in theprice level as it is noted in Figure 10 however the price levelis reduced considerably when compared to optimal policiesvia LQR In Figure 11 demand level achieves a low value inunits at the end of time horizon which is expected but theoscillation is persistent

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

Time scale

Dem

and

(uni

ts)

Optimal policies for demand management-LQR

Demand

Figure 9 Optimal policies for demand management via LQR

0 05 1 15 2 25 3 35 4 45 50

01

02

03

04

05

06

07

Time scale

Pric

e lev

el (d

lls)

Robust policies for capacity-LQR

Capacity fixed-Cf

Capacity unfixed-Cu

Figure 10 Robust policies for capacity via LQR

In summaryMPC simulations fromFigures 4 to 7 presentbetter performance than LQR which are shown in Figures 8to 11 for optimal and robust policies

Based on this context and these results we aim to exploreother control oriented approaches for DM-active suspensionanalogy via dynamic pricing such as robust control tech-niques as developed in [42 43] with the goal to compare itsperformance with MPC techniques Also we plan to exploreadaptive control techniques [44] output feedback control[45] and sample data control present [46] and to develop atrade-off with optimal control techniques

5 Conclusions

This research work presents a novel DM-DP approach basedon MPC for second order systems with damping which

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 05 1 15 2 25 3 35 4 45 50

5

10

15

20

25

30

Time scale

Dem

and

(uni

ts)

Robust policies for demand management-LQR

Demand

Figure 11 Robust policies for demand management via LQR

in terms of DM tell us that dissipation process is presentwith inventory storage level in the system Our results ofthe stability analysis from Lyapunov direct method showthat an inventory level is needed in the system The one-quarter active suspension model presents the analogy fora DM-DP process system behavior Besides the sensitivityanalysis shows us that an equivalent capacity level willprovide reduced price level Simulations results show thatunder robust policies the price level is reduced by half thisis because of the ratio between inventory level and capacityin the active suspension dynamical system For future workthe intention is to develop a full active suspension modelfor multiple product system as well as to achieve inventorylevel dynamics via first order ordinary differential equationsAnother future research area will focus on the exploration ofa full active suspensionmodel for one ormultiple products ina multistage supply chain

Appendices

A Inventory Level Derivation

The derivation of the value of 120573 requires the followinganalysis

Let 120573 (119905) = intinfin

0

119890minus119905120573

119889119905 Finally 120573 (119905) = 120573 (A1)

B Infinite Dimensional Analysis forDemand Management

We propose the following generalized wave equation whichdescribes the behavior of flexible systems presented first in[30]

119898(119909)1205972119906 (119909 119905)

1205971199052

+ 120573120597119906 (119909 119905)

120597119905+ 119870119906 (119909 119905) = 119865 (119909 119905) (B1)

The purpose is to obtain a PDE description for demandmanagement via dynamic pricing Based on this

119862 (119901)1205972120593 (119901 119905)

1205971199052

+ 120573[120597120593 (119901 119905)

120597119901+ 120572

120597120593 (119901 119905)

120597119905]

+ 119896120593 (119901 119905) = 119863 (119901 119905)

(B2)

where

120597120593

120597119905=120597120593

120597119901

120597119901

120597119905= V119901

120597120593

120597119901 (B3)

substituting (B3) in (B2) which gives

119862 (119901) [120597

120597119905(V119901

120597120593 (119901 119905)

120597119901)] + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B4)

Developing partial derivatives

119862 (119901)(V119901

1205972120593 (119901 119905)

120597119901120597119905+ 119886119901

120597120593 (119901 119905)

120597119901) + 120573

120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B5)

Define 119862(119901) equiv 1 and 0 le 120572 le 1The PDE description for demand management is

V119901

1205972120593 (119901 119905)

120597119901120597119905+ (120573 + 119886

119901)120597120593 (119901 119905)

120597119901

+ 120572120573120597120593 (119901 119905)

120597119905+ 119870120593 (119901 119905) = 119863 (119901 119905)

(B6)

where 120573 + 119886119901gt 0

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank Professor Leopoldo ECardenas-Barron for his exhaustive reviews and constructivecomments which have helped to improve the contents ofthis paper This work was supported in part by the ResearchChair e-Robots and the Laboratorio Nacional del Noreste yCentro de Mexico supported by Conacyt and Tecnologico deMonterrey

References

[1] K L Croxton D M Lambert S J Garcıa-Dastugue and DS Rogers ldquoThe demand management processrdquo InternationalJournal of Logistics Management vol 13 no 2 pp 51ndash66 2002

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[2] F-S Chou and M Parlar ldquoOptimal control of a revenue man-agement system with dynamic pricing facing linear demandrdquoOptimal Control Applications ampMethods vol 27 no 6 pp 323ndash343 2006

[3] M Trufelli ldquoDynamic pricing new game new rules newmindsetrdquo Journal of Revenue and Pricing Management vol 5no 1 pp 81ndash82 2006

[4] H Abdel Aziz M Saleh M H Ramsy and H ElShishinyldquoDynamic room pricing model for hotel revenue managementsystemsrdquo Egyptian Informatics Journal vol 12 pp 177ndash183 2011

[5] S de Boer ldquoThe impact of dynamic capacity management onairline seat inventory controlrdquo Journal of Revenue and PricingManagement vol 2 no 4 pp 315ndash330 2004

[6] Y-F Hung and C-H Chen ldquoAn effective dynamic decisionpolicy for the revenuemanagement of an airline flightrdquo Interna-tional Journal of Production Economics vol 144 no 2 pp 440ndash450 2013

[7] A Gosavi E Ozkaya and A F Kahraman ldquoSimulation opti-mization for revenuemanagement of airlines with cancellationsand overbookingrdquo OR Spectrum vol 29 no 1 pp 21ndash38 2007

[8] MGraf andAKimms ldquoTransfer price optimization for option-based airline alliance revenue managementrdquo International Jour-nal of Production Economics vol 145 no 1 pp 281ndash293 2013

[9] A Kimms and M Muller-Bungart ldquoSimulation of stochasticdemand data streams for network revenue management prob-lemsrdquo OR Spectrum vol 29 no 1 pp 5ndash20 2007

[10] S Netessine and R Shumsky ldquoIntroduction to the theoryand practice of yield managementrdquo INFORMS Transactions onEducation no 3 1 pp 34ndash44 2002

[11] B Rannou and D Melli ldquoMeasuring the impact of revenuemanagementrdquo Journal of Revenue and PricingManagement vol2 no 3 pp 261ndash270 2003

[12] DWalczak and S Brumelle ldquoSemi-Markov information modelfor revenue management and dynamic pricingrdquo OR Spectrumvol 29 no 1 pp 61ndash83 2007

[13] D Bertsimas and R Shioda ldquoRestaurant revenuemanagementrdquoOperations Research vol 51 no 3 pp 472ndash486 2003

[14] E A Boyd and I C Bilegan ldquoRevenue management and e-commercerdquoManagement Science vol 49 no 10 pp 1363ndash13862003

[15] F Defregger and H Kuhn ldquoRevenue management for a make-to-order company with limited inventory capacityrdquo OR Spec-trum vol 29 no 1 pp 137ndash156 2007

[16] S E Kimes ldquoRevenue management a retrospectiverdquo CornellHotel and Restaurant Administration Quarterly vol 44 no 5-6 pp 131ndash138 2003

[17] S Kumar and J L Frederik ldquoRevenue management for a homeconstruction products manufacturerrdquo Journal of Revenue andPricing Management vol 5 no 4 pp 256ndash270 2007

[18] L H Lee E P Chew and M S Sim ldquoA heuristic to solve a seacargo revenue management problemrdquoOR Spectrum vol 29 no1 pp 123ndash136 2007

[19] S K Nair and R Bapna ldquoAn application of yield managementfor internet service providersrdquoNaval Research Logistics vol 48no 5 pp 348ndash362 2001

[20] T Spengler S Rehkopf and T Volling ldquoRevenue managementin make-to-order manufacturingmdashan application to the ironand steel industryrdquoOR Spectrum vol 29 no 1 pp 157ndash171 2007

[21] C K Shin and S C Park ldquoAmachine learning approach to yieldmanagement in semiconductor manufacturingrdquo International

Journal of Production Research vol 38 no 17 pp 4261ndash42712000

[22] C Steinhardt and J Gonsch ldquoIntegrated revenue managementapproaches for capacity control with planned upgradesrdquo Euro-pean Journal of Operational Research vol 223 no 2 pp 380ndash391 2012

[23] Y A Davizon J J Lozoya and R Soto ldquoDemand managementin semiconductor manufacturing a dynamic pricing approachbased on fast model predictive controlrdquo in Proceedings ofthe 7th IEEE Electronics Robotics and Automotive MechanicsConference (CERMA rsquo10) pp 534ndash539 October 2010

[24] P L Sacco ldquoOptimal control of input-output systems viademandmanagementrdquoEconomics Letters vol 39 no 3 pp 269ndash273 1992

[25] D N Burghes and A D Wood Mathematical Models in theSocial Management and Life Sciences Ellis Horwood 1980Mathematics and its Applications

[26] E A Ibijola and A A Obayomi ldquoNon-standard method forthe solution of ordinary differential equations arising frompricing policy for optimal inventory levelrdquoCanadian Journal onComputing in Mathematics Natural Sciences Engineering andMedicine vol 3 no 7 2012

[27] W Pan L Yu S Wang G Hua G Xie and J Zhang ldquoDynamicpricing strategy of provider with different QoS levels in webservicerdquo Journal of Networks vol 4 no 4 pp 228ndash235 2009

[28] A G Vlachos and P N Biskas ldquoDemand response in a real-time balancing market clearing with pay-as-bid pricingrdquo IEEETransactions on Smart Grid vol 4 no 4 pp 1966ndash1975 2013

[29] V Jayaraman and T Baker ldquoThe internet as an enabler fordynamic pricing of goodsrdquo IEEE Transactions on EngineeringManagement vol 50 no 4 pp 470ndash477 2003

[30] M J Balas ldquoFeedback control of flexible systemsrdquo IEEE Trans-actions on Automatic Control vol AC-23 no 4 pp 673ndash6791978

[31] J T Mentzer M A Moon D Estampe and G W MargolisDemand Management Handbook of Global Supply Chain Man-agement SAGE 2006

[32] M Vossler and T Singh ldquoMinimax stateresidual-energy con-strained shapers for flexible structures linear programmingapproachrdquo Journal of Dynamic Systems Measurement andControl vol 132 no 3 pp 1ndash8 2010

[33] X-M Tang H-C Qu H-F Xie and P Wang ldquoModel predic-tive control of linear systems over networks with state and inputquantizationsrdquoMathematical Problems in Engineering vol 2013Article ID 492804 8 pages 2013

[34] E Aggelogiannaki P Doganis and H Sarimveis ldquoAn adap-tive model predictive control configuration for production-inventory systemsrdquo International Journal of Production Eco-nomics vol 114 no 1 pp 165ndash178 2008

[35] J D Schwartz and D E Rivera ldquoA process control approachto tactical inventory management in production-inventorysystemsrdquo International Journal of Production Economics vol125 no 1 pp 111ndash124 2010

[36] Y-F Wang F-C Sun Y-A Zhang H-P Liu and H MinldquoGetting a suitable terminal cost and maximizing the terminalregion for MPCrdquo Mathematical Problems in Engineering vol2010 Article ID 853679 16 pages 2010

[37] J Mattingley Y Wang and S Boyd ldquoReceding horizon controlautomatic generation of high-speed solversrdquo IEEE ControlSystems Magazine vol 31 no 3 pp 52ndash65 2011

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[38] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[39] Y Yin Y Liu and H R Karimi ldquoA simplified predictive controlof constrained Markov jump system with mixed uncertaintiesrdquoAbstract and Applied Analysis vol 2014 Article ID 475808 7pages 2014

[40] T Yang andH R Karimi ldquoLMI-basedmodel predictive controlfor a class of constrained uncertain fuzzy Markov jump sys-temsrdquo Mathematical Problems in Engineering vol 2013 ArticleID 963089 13 pages 2013

[41] B Yang Y He J Han and G Liu ldquoRotor-flying manipulatormodeling analysis and controlrdquo Mathematical Problems inEngineering vol 2014 Article ID 492965 13 pages 2014

[42] H R Karimi and H Gao ldquoNew delay-dependent exponential119867infinsynchronization for uncertain neural networks with mixed

time delaysrdquo IEEE Transactions on Systems Man and Cybernet-ics B Cybernetics vol 40 no 1 pp 173ndash185 2010

[43] H R Karimi ldquoRobust delay-dependent 119867infin

control of uncer-tain time-delay systems with mixed neutral discrete anddistributed time-delays and Markovian switching parametersrdquoIEEE Transactions on Circuits and Systems I vol 58 no 8 pp1910ndash1923 2011

[44] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding-modecontrol for nonlinear active suspension vehicle systems using T-S fuzzy approachrdquo IEEE Transactions on Industrial Electronicsvol 60 no 8 pp 3328ndash3338 2013

[45] H Li X Jing and H R Karimi ldquoOutput-feedback-based 119867infin

control for vehicle suspension systems with control delayrdquo IEEETransactions on Industrial Electronics vol 61 no 1 pp 436ndash4462014

[46] H Li X Jing H-K Lam and P Shi ldquoFuzzy sampled-datacontrol for uncertain vehicle suspension systemsrdquo IEEE Trans-actions on Cybernetics vol 44 no 7 pp 1111ndash1126 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of