competition and disruption in a dynamic urban supply chain

Transportation Research Part B 45 (2011) 1212–1231

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Transportation Research Part B

journal homepage: www.elsevier .com/ locate / t rb

Competition and disruption in a dynamic urban supply chain

Terry L. Friesz a,⇑, Ilsoo Lee a, Cheng-Chang Lin b

a Department of Industrial and Manufacturing Engineering, Pennsylvania State University, PA 16802, USAb Department of Transportation and Communication Management Science, National Cheng Kung University, Taiwan

a r t i c l e i n f o

Keywords:Dynamic supply chains
Supply chain networksUrban supply chainsCity logisticsDisruptionsDifferential variational inequalities
0191-2615/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.trb.2011.05.005

⇑ Corresponding author. Tel.: +1 814 863 2445.E-mail addresses: [email protected], [email protected]

a b s t r a c t

Rapid changes and complexities in business environments have stressed the importance ofinteractions between partners and competitors, leading supply chains to become the mostimportant element of contemporary business environments. There is a concomitant needfor foresight in describing supply chain performance in all operating environments, includ-ing those involving punctuated disruptions. Furthermore, the urban metropolis is nowwidely recognized to be an environment which is especially vulnerable to supply chain dis-ruptions and for which integrated supply chain decisions can produce very substantial netbenefits. Accordingly, this paper presents a dynamic supply chain network model formu-lated as a differential variational inequality; the model is fashioned to allow considerationof supply chain disruption threats to producers, freight carriers, and retail enterprises. TheDVI is solved using a fixed-point algorithm, and a simple numerical example, introduced toillustrate how the impacts of supply chain disruptions may be quantified, is presented.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Due to rapid changes in business environments and increasingly complex relationships among diverse business partnersand competitors, supply chains have become essential factors in the quest to improve profits and efficiency (Kogan and Tap-iero, 2007). A supply chain can be defined as a network of autonomous or semi-autonomous agents collectively responsiblefor the procurement, manufacturing and distribution activities associated with one or more sets of related outputs. In otherwords, a supply chain is the actual system charged with transforming natural resources, raw materials, and components intofinished products, and eventually moving a product or service from supplier to customer (Swaminathan et al., 1998). For themodern urban metropolis supply chains are quite literally the life-support system allowing them to function, evovle, andachieve the efficient use of resources. The various entities comprising a supply chain – typically consisting of intertwinedorganizations, people, technology, activities, information and resources – operate according to a multitude of constraintsand objectives that may be expressed mathematically as a dynamic noncooperative Nash game. In an urban setting the rel-evant infrastructures and agent behaviors interact in especially complex and important ways that must be captured whenmodeling. That is, the aformentioned objectives and constraints are generally interdependent from the points of view ofimproving on-time delivery, quality assurance and controlling costs. As a consequence, performance of any aspect of the ur-ban supply chain depends on the performance of the rest of the supply chain. Because of this organic relationship amongagents in the supply chain, a part of the supply chain or one of the agents in the supply chain may damage the overall per-formance of the supply chain. Like supply chain disruption, this breed of supply chain risk, known as supply chain uncer-tainty, yields a degree of vulnerability. As mentioned above, supply chains may be organically related to production

. All rights reserved.

(T.L. Friesz), [email protected] (I. Lee), [email protected] (C.-C. Lin).

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T.L. Friesz et al. / Transportation Research Part B 45 (2011) 1212–1231 1213

processes, transportation of goods, and market demands. Thus, supply chain disruptions and the related risks are major top-ics of interest in theoretical and applied research, as well as in industry.

Supply chain disruptions such as weather conditions, terrorist attacks and diseases, are unintended, unwanted situations,that result in diminished supply chain performance. Disruptions can be classified into five sources: (1) demand-side, (2) sup-ply-side, (3) regulatory, legal and bureaucratic, (4) infrastructure, and (5) catastrophic (Wagner and Bode, 2008). Supplychain vulnerability is an exposure to serious disturbance (Christopher and Peck, 2004). It is the capacity of supply chainsto anticipate, cope with, resist and recover from disruptions. Supply chain vulnerability can be represented as a functionof supply chain characteristics, which include density, complexity (base reduction and global sourcing) and network (nodeand link) connectivity criticality (Craighead et al., 2007). Thus, the goal of supply chain risk management is to design andimplement a supply chain capable of anticipating, coping with and rapidly recovering from disruptions.

Research has studied demand-side uncertainty in dynamic supply chain design (Adida and Perakis, 2010). To date, severalmajor disruptions have demonstrated the need to address supply-side disruption. For instance, in March of 2000, a fire at thePhillips Semiconductor plant in Albuquerque, New Mexico, caused a major customer, Ericsson, to lose $ 400 million in po-tential revenue. Another major customer, Nokia, managed to allocate alternative supplies and, therefore, mitigated the im-pact of the disruption (Latour, 2001). As a consequence, the goal of supply chain risk management is to alleviate theconsequences of disruptions and related risks or, simply put, to increase the robustness of the supply chain. In particular,a robust supply chain network configuration is critical (Snyder, 2003).

To study supply chain disruption, supply chain network flow equilibrium and supply chain network design and redesignmust be considered. Supply chain network flow equilibrium refers to the study of existing multi-level distribution networksin the hope of reaching an equilibrium of product, information and financial flows. In practice, supply chain network flowgaming involves tactical and operational decisions which can be fundamentally referred to as transportation decisions.On the other hand, supply chain network design and redesign aims to select optimal facility locations and sizes within a gi-ven network for a given objective, which is designated by the network authorities. Beyond location and size, the design andredesign of large-scale supply chain networks may include strategic decisions which involve the capacities required to fulfilthese activities, their allocation to specific product groups, and the control system incorporated to manage all activities.

This paper integrates strategic, tactical and operational considerations in proposing a dynamic supply chain network de-sign mathematical model, which manages disruption by formulating the maximization of each agent’s revenue and minimi-zation of supply chain disruption over finite time horizons based on the notion of a differential variational inequality (DVI).As supply chain uncertainty must be considered during all phases of supply chain planning, this paper also examines supplychain network design for lessening supply disruption based on strategic capacity, tactical inventory and operational trans-portation decisions. The model discussed is based on continuous-time differential game theory. Simply put, the dynamicsupply chain is a supply chain that is not in a steady state; rather, it is either in disequilibrium and moving toward an equi-librium or it has attained a moving equilibrium, a sequence of equilibrium states visited in succession without transitthrough any disequilibrium state. In dynamic supply chains, inventories of essential inputs as well as inventories of productsmade from those inputs are fundamental and critical state variables that devolve from the construction of competitive busi-ness strategies. As such, the technical literature on production planning and multiechelon inventory problems is directly rel-evant to modeling, planning and operating dynamic supply chains.

2. Supply chain network gaming with supply uncertainty

This paper considers supply disruption to be the motivating process for a type of supply chain uncertainty. To this end,supply uncertainty can be represented in a supply chain model by reflecting probability parameters of the primary cost func-tions of each agent involved in the supply chain. That is, suppliers’ transportation costs and manufacturers’ production effi-ciencies are directly influenced by supply chain uncertainty.

The oligopolistic firms of interest, embedded in a network economy, are in oligopolistic competition according to dynam-ics that describe the trajectories of inventories/backorders and flow conservation for each firm at each node of the network ofinterest. The oligopolistic firms, acting as shippers, compete as price-takers in the market for physical distribution services.That market can be considered perfectly competitive due to its involvement in other markets of the network economy. Thetime scale employed is neither short nor long; rather it is of sufficient length to allow output and shipping pattern adjust-ments, but not long enough for firms to re-locate or enter or leave the network economy. Thus this paper does not considermultiple production stages.

Continuous time is denoted by the scalar t 2 R1þ, initial time by t0 2 R1

þ, and final time by tf 2 R1þþ, with t0 < tf such that

t 2 ½t0; tf � � R1þ. The following sets are important in articulating models of oligopolistic network competition: F P for the set

of producing firms,A for directed arcs,N for nodes andW for origin-destination (OD) pairs. Subsets of these sets are denotedby the following subscripts: f for a specific firm, i for a specific node, and w for a specific OD pair.

Each producing firm f 2 F P has a facility at every node of the network, an assumption that is easily relaxed if needed;moreover, each firm controls production output rates

qf ¼ qfi : i 2 N

� �
as well as allocations of output to meet demand

1214 T.L. Friesz et al. / Transportation Research Part B 45 (2011) 1212–1231

cf ¼ cfi : i 2 N

� �
and shipping patterns
sf ¼ sfw : w 2 W

� �
Inventories
If ¼ Ifi : i 2 N

� �
are state variables determined by the controls qf, cf, and sf. The firm specific-controls may be concatenated to give
c 2 ðL2½t0; tf �ÞjN j�jFP j

q 2 ðL2½t0; tf �ÞjN j�jFP j

s 2 ðL2½t0; tf �ÞjWj�jFP j

Iðc; q; sÞ : ðL2½t0; tf �ÞjN j�jFP j � ðL2½t0; tf �ÞjN j�jFP j � ðL2½t0; tf �ÞjWj�jFP j ! ðH1½t0; tf �ÞjN j�jFP j

where L2[t0, tf] represents the space of square-integrable functions and H1½t0; tf � is a Sobolev space for the real interval½t0; tf � 2 R1

þ.

2.1. Reflecting supply chain uncertainty

As explained above, suppliers’ transportation costs and manufacturers’ production functions are impacted by supplyuncertainty. Please note it is outside the scope of this paper to consider retailers’ demand uncertainty. For the productionactivity of firm f 2 F P at node i, probability parameters ni and gi appear parametrically within the production and shippingcost functions with the corresponding cumulative distribution function denoted by Gi(ni) and Gi(gi), respectively. Thus, theexpected production, input flow cost and variable transportation cost functions for a producer of firm f 2 F P at node i are asfollows:

E eF fi hf

i

� �h i¼Z

ni

eF fi hf

i ; ni

� �dGiðniÞ

E eCfi hf

i

� �h i¼Z

ni

eCfi hf

i ;gi

� �dGiðgiÞ

E erij� �

¼Z

gi

erijðqij;giÞdGiðgiÞ

where eF ; eC ; er denote the production function depending parametrically on the random input factor flow, input flow costdepending parametrically on the random input factor flow, and transportation rate depending parametrically on the randomwholesale amount, respectively. For supplier at stage k, we apply probability parameter dk, reflecting the disruption effect onthe variable transportation cost with the corresponding cumulative distribution function given by Gk(dk). Thus, the expectedtransportation cost can be represented as follows:

E½eV kðukÞ� ¼Z

di

eV ðuk; dkÞdGkðdkÞ

Below, supply disruption risk minimization is considered, producing a multi-objective optimization problem. In particularwe will append variance to the objective function, following Tomlin (2006) and Silberberg and Suen (2000), by using a non-negative weight, bi, where a small (large) weight corresponds to risk-taking (risk aversion).

2.2. The General Firm’s objective and constraints

Each firm has the objective of maximizing net profit expressed as revenue less cost and can be thought of as allocatingoutput to meet demand, while also setting productions rates and shipment patterns. For each firm f 2 F P , net profit can berepresented as follows:

Uf ðcf ; qf ; sf ; c�f ; q�f Þ ¼Z tf

t0

e�qtXi2N

pi

Xg2FP

cgi ; t

!cf

i �Xi2N f

V fi ðq

f ; tÞ �X

w2Wf

rwðtÞsfw �

Xi2N

wfi If

i ; t� �8<:

9=;dt ð1Þ

where q 2 R1þþ is a constant nominal rate of discount, rw 2 R1

þþ is the freight rate (tariff) charged per unit of flow sw for ODpair w 2 W f ; wf

i is firm f’s inventory cost at node i, and Ifi is the inventory/backorder of firm f at node i. In (1), cf

i is the allo-cation of the output of firm f 2 F P at node i 2 N to consumption at that node. Our formulation is in terms of flows so weemploy the inverse demand functions pi(ci, t) where


ci ¼Xg2FP

cgi

is the total allocation of output to consumption for node i. Furthermore qfi is the output of firm f 2 F P at node i 2 N . Also

Vfi ðq; tÞ is the variable cost of production for firm f 2 F P at node i 2 N . Note that hf(cf, qf, sf; c�f, q�f) is a functional that is com-

pletely determined by the controls cf, qf and sf when non-own allocations to consumption and non-own production rates

c�f � ðcf 0 : f 0–f Þq�f � ðqf 0 : f 0–f Þ

are taken as exogenous data by firm f. The first term of the functional hf(cf, qf, sf; c�f, q�f) in expression (1) is the firm’s rev-enue; the second term is the firm’s cost of production; the third term is the firm’s shipping costs; and the last term is thefirms’s inventory or holding cost.

We also impose the terminal time inventory constraints

Ifi ðtf ÞP eK f

i 8f 2 F P; i 2 N f ð2Þ

where the eK fi 2 R1

þþ are exogenous. All consumption, production and shipping variables are non-negative and bounded fromabove; that is

Cf P cf P 0 ð3ÞQ f P qf P 0 ð4ÞSf P sf P 0 ð5Þ

where

Cf 2 RjFP jþþ

Q f 2 RjFP jþþ

Sf 2 RjWf jþþ

Constraints (3)–(5) are recognized as pure control constraints, while (2) are terminal conditions for the state space variables.Naturally,

Xf ¼ cf ; qf ; sf� �

: ð2Þ; ð3Þ; ð4Þ; ð5Þ�

is the set of feasible controls.Firm f solves an optimal control problem to determine its production qf, allocation of production to meet demand cf, and

shipping pattern sf – thereby also determining inventory If via dynamics we articulate momentarily – by maximizing its prof-it functional Uf(cf, qf, sf; c�f, q�f) subject to inventory dynamics expressed as flow balance equations and other pertinent pro-duction and inventory constraints. The inventory dynamics for firm f 2 F P , expressing simple flow conservation, obey

dIfi

dt¼ qf

i þX

w2Wdi

sfw �

Xw2Wo

i

sfw � cf

i 8i 2 N f ð6Þ

Ifi ðt0Þ ¼ Kf

i 8i 2 N f ð7ÞIfi ðtf ÞP eK f

i 8i 2 N f ð8Þ

where Kfi 2 R1

þþ and eK fi 2 R1

þ are exogenous, whileWdi is the set of OD pairs with destination node i andWo

i is the set of ODpairs with origin node i. Note that the transportation time for the flow of finished goods is not captured explicitly in theinventory dynamics; however, it is accounted for implicitly in the freight rate (tariff) charged per unit of flow. Furthermore,in addition to the terminal time inventory (state) constraints (8), the model is general enough to handle inventory con-straints over the entire planning horizon [t0, tf]. For instance, non-negativity of the inventory (state) variables could be im-posed to restrict firms from taking backorders. Consequently,

Iðc; q; sÞ ¼ argdIf

i

dt¼ qf

i þX

w2Wdi

sfw �

Xw2Wo

i

sfw � cf

i ; Ifi ðt0Þ ¼ Kf

i ; Ifi ðtf ÞP eK f

i 8f 2 F P; i 2 N f

8<:9=;

where we implicitly assume that the dynamics have solutions for all feasible controls.With the preceding development, we note that firm f’s problem is: with the c�f and q�f as exogenous inputs, compute cf, qf

and sf (thereby finding If) in order to solve the following extremal problems:

max Uf cf ; qf ; sf ; c�f ; q�f� �

s:t: cf ; qf ; sf� �

2 Xf

)8f 2 F P ð9Þ


where

Xf ¼ cf ; qf ; sf� �

: ð2Þ; ð3Þ; ð4Þ; ð5Þ hold�

also for all f 2 F P . That is, each firm is a Nash agent that knows and employs the current instantaneous values of the decisionvariables of other firms to make its own non-cooperative decisions. As such, (9) is a differential Nash game.

2.3. Inverse demands

Agreements are in place that prevent producers from selling directly to consumers, so retailers and consumers will facesubtly distinct inverse demand functions for the finished good of interest. To understand this, again let F P be the set of firmswith producers at node i 2 N and F R the set of retailers, potentially occupying every network node. Also let wj refer to thewholesale price paid by retailers r 2 F R for the producers’ output in market j 2 N . We note that the following identity holds:

Djðwj þ ajwjÞ ¼Xr2FR

crj ð10Þ

where Dj(�) is the market demand function at node j 2 N for the finished good of interest, and crj is the consumption at node

j 2 N of the goods flow from retailer r 2 F R; furthermore, aj 2 R1þþ is the retailers’ margin at node j 2 N . It can be assumed

that each such demand function has an inverse denoted by Hj(�, �) such that

wj ¼ Hj

Xr2FR

crj ; aj

!8j 2 N ð11Þ

which is called the wholesalers’ inverse demand. Next, the consumers’ inverse demand for the finished good at node j 2 N isdenoted by Wj(�). While the inverse is obtained from (10), it is not identical to the wholesalers’ inverse demand (11). In par-ticular, the consumers’ inverse demand takes the form

wj þ ajwj ¼ Wj

Xr2FR

crj

!8j 2 N ð12Þ

and wj + ajwj is the retail price paid by consumers for the finished good at node j 2 N . We assume that the inverse demandWj(�) exists for every retail market j 2 N . Clearly an alternative form of (12) is

wj ¼1

1þ ajWj

Xr2FR

crj

!8r 2 F R; j 2 N ð13Þ

Expressions (11) and (13) make very clear that

Hj

Xr2FR

crj ; aj

!¼ 1

1þ ajWj

Xr2FR

crj

!8j 2 N ð14Þ

2.4. Firms’ extremal problem

To facilitate the situation described above, the following state dynamics for producers are employed:

dIfi

dt¼ E eF f

i hfi

� �h iþXj2N

sfji �

Xj2N

sfij �

Xr2FR

Xj2N

qfrij 8f 2 F P ; 8r 2 F R; i 2 N ð15Þ

where E eF fi ð�Þ

h iis the expected single factor production function for production by firm f 2 F P at node i 2 N and F P is the set

of firms producing the homogeneous product of interest. In addition, hfi is the flow of the single factor to node i 2 N for use

by firm f 2 F P . Furthermore, N represents the final echelon (stage) of supplying the single factor to producers at nodes i 2 N .The set of retailers we consider is F R. Additionally, qfr

ij denotes the sales by firm f 2 F P from inventory or new production atnode i 2 N to retailer r 2 F R at node j 2 N . Moreover, the aggregate input factor flow uN is disaggregated into individualflows hf

i used by each firm f 2 F P at each node i 2 N where

uN ¼Xf2FP

Xi2N

hfi

Naturally we employ the notation

hf ¼ hfi : i 2 N

� �
to describe the vector of factor allocations controlled by firm f 2 F P with producers at node i 2 N . Contracts between eachproducing firm f 2 F P and the supply chain agent are written for every node at which firm f 2 F P has a presence. Recalling


that we have, for simplicity of notation, assumed every firm has a presence at every node, we now assume such contracts fixthe price of factor inputs subject to random, spot-market delivery costs obeying an unknown probability distribution. Such

circumstances act to establish an expected cost function E eCfi hf

i ; t� �h i

for the instantaneous cost to acquire the input flow hfi

at each node i 2 N . Additionally, upper and lower bounds are established by those contracts for factor flow to each producerat node i of firm f 2 F P:

Af 6Xi2N

hfi 6 Bf 8f 2 F P ð16Þ

where

½Af ;Bf � � R1þþ 8f 2 F P ð17Þ

In light of the preceding development, the extremal problem for each firm f 2 F P can be expressed as follows:

max JPf ðqf ; sf ;hf

; cÞ ¼Z tf

t0

e�qtXr2FR

Xi2N

Xj2N

11þ ar

j

Wj

Xg2FP

cgj

!qfr

ij �Xi2N

E eCfi hf

i

� �h i�Xi2N

Xj2N

E½erij� sfij þ

Xr2FR

qfrij

!(

�Xi2N

wfi If

i ; t� �

�Xi2N

bfi Var eF f

i hfi

� �� þ Var eCf

i hfi

� �þXj2N

Var erij� �

sfij þ

Xr2FR

qfrij

!" #)dt ð18Þ

subject to

dIfi

dt¼ E eF f

i hfi

� �h iþXj2N

sfji �

Xj2N

sfij �

Xr2FR

Xj2N

qfrij 8f 2 F P ; 8r 2 F R; i 2 N ð19Þ

Ifi ðt0Þ ¼ Kf

i 8i 2 N ð20ÞIfi ðtf ÞP eK f

i 8i 2 N ð21Þ0 6 qfr

ij 6eqfr

ij i; j 2 N ; r 2 F r ð22Þ

0 6 sfij 6

esfij i; j 2 N ð23Þ

Af 6Xi2N

hfi 6 Bf 8f 2 F P ð24Þ

where qf, sf, and hf are vectors of output, shipping and input factor flows under the control of firm f 2 F P with producers. Asdiscussed above, shipments to retailers are free onboard and thus paid by producers of firm f 2 F P . Furthermore, rij is thefreight rate (tariff) for the origin–destination (OD) pair (i, j), while inventory cost at node i 2 N experienced by firmf 2 F P is wf

i ð�; �Þ, and Kfi is the initial inventory at node i 2 N held by firm f 2 F P . Please note that (21) describes the terminal

time inventory constraint while (22) and (23) are constraints expressing upper bounds on outputs and shipments, whereeK fi ; eqfr

ij and esfij are exogenous fixed parameters for all firms f 2 F P and i; j 2 N . Constraint (24) forms upper and lower bounds

on aggregate factor flows to producers at node i 2 N . Note that in the above

qfij ¼ qfr

ij : r 2 F R

� �ð25Þ

qf ¼ qfij : i; j 2 N

� �ð26Þ

q�f ¼ qg : g 2 F P n fð Þ ð27Þ

sf ¼ sfij : i; j 2 N

� �ð28Þ

hf ¼ hfi : i 2 N

� �ð29Þ

h ¼ hf: f 2 F P

� �ð30Þ

h�f ¼ hg: g 2 F P n f

� �ð31Þ

cr ¼ crj : j 2 N

� �ð32Þ

c ¼ ðcr : r 2 F RÞ ð33Þ

2.5. Retailers’ extremal problem

It is assumed that producers will always be able to deliver the goods that retailers request albeit at fluctuating prices dueto particular circumstances. Such a perspective is expressed through probability parameters in the model. By stipulating that


only retailers may sell finished goods, the single homogeneous finished good of interest must be obtained from producers,leading to the following dynamics for retailers:

dRrj

dt¼Xf2FP

Xi2N

qfrij � cr

j 8r 2 F R; j 2 N ð34Þ

where Rrj denotes the inventory of retailer r 2 F R at node j 2 N ;F R denotes the set of retailers, N denotes the set of nodes at

which retailer r is located and crj denotes the consumption activity served by retailer r 2 F R at node j 2 N . Therefore, the

extremal problem faced by each retailer r 2 F R is the following:

maxJRr ðcr ; c�r; qrÞ ¼

Z tf

t0

e�qtXi2N

Xj2N

crj �

11þ ar

j

Xi2N

qfrij

!Wj

Xg2FP

cgj

!�Xj2N

/rj ðR

rj ; tÞ

" #dt ð35Þ

dRrj

dt¼Xf2F P

Xi2N

qfrij � cr

j 8r 2 F R; j 2 N ð36Þ

0 6 crj 6 ecr

j 8r 2 F R; j 2 N ð37ÞRr

j ðt0Þ ¼ Q rj 8r 2 F R; j 2 N ð38Þ

Rrj ðtf Þ ¼ eQ r

j 8r 2 F R; j 2 N ð39Þ

where /ri Rr

j ; t� �

represents the inventory costs, Q rj refers to the initial inventory, ~Q r

j refers to the terminal time inventory and~cr

j refers to the upper bound on consumption for retailer r 2 F R at node j 2 N . Please note that

cr ¼ crj : j 2 N

� �ð40Þ

c�r ¼ ðcg : g 2 F P � frgÞ ð41Þ

qrij ¼ qfr

ij : f 2 F P

� �ð42Þ

qr ¼ qrij : i; j 2 N

� �ð43Þ

Thus, the constraints of this extremal problem depend on the vector qr, which is exogenous, since output allocations aredetermined by the producers.

2.6. Suppliers’ extremal problem

Now let us consider a multi-echelon supply chain stretching from unrefined raw materials to factor flows ready for use byproducers. We use uk to denote the flow of the input factor exiting stage k (that is, the flow from stage k to stage k + 1). If weuse Sk to denote the inventory at stage k of the supply chain, we may write

dSk

dt¼ uk�1 � uk k ¼ 1; . . . ;N

To account for supply chain disruption, each supplier is assumed to not make any material; rather, each supplier has an ini-tial inventory and does not possess a production function. Please recall that there are contracts in place specifying a random

fee schedule Cfi hf

i ; t� �

and guarantied upper and lower bounds for the factor flow to each producer of firm f 2 F p at node

i 2 N at time t 2 [t0, tf], where the allocations hf are controlled by producer f 2 F P . The controls available to the supply chainagent are captured by the vector

u ¼ uk : k ¼ 1; . . . ;Nð Þ ð44Þ

As a consequence, the supplier seeks to minimize total cost by solving the following optimal control problem:

min JSðuÞ ¼Z tf

t0

e�qtXN

k¼1

E eV kðuk; tÞh i

þukðSk; tÞ þ bk � Var eV kðuk; tÞ� �h i

dt ð45Þ

subject to

dSk

dt¼ uk�1 � uk k ¼ 1; . . . ;N ð46Þ

Skð0Þ ¼ S0k k ¼ 1; . . . ;N ð47Þ

uN ¼Xf2FP

Xi2N

hfi ð48Þ

0 6 uk 6 Uk k ¼ 1; . . . ;N ð49Þ

where E eV kð�; �Þh i

is the expected variable cost of preparing the stage k flow, uk(�, �) is the inventory cost function, and Uk is thetechnological upper bound of input factor flow at node k of the supply chain. Please note that the constraints of this extremal


problem depend on the vector hf which is determined exogenously since the producers decide factor flows to their produc-tion facilities within the bounds set by the contracts they hold with the supply chain manager.

3. The differential variational inequality for supply chains under uncertainty

In this section we give an overview of how the relevant differential variational inequality for our combined producer–re-tailer–supply chain game may be formed.

3.1. Maximum principle for the firms

With

cq�f

�ð50Þ

as exogenous, each firm f 2 F P solves

max JPf qf ; sf ;hf

; c� �

s:t: qf ; sf ;hf� �

2 KfPðh

�f Þð51Þ

where

KfPðh

�f Þ �qf

sf

hf

0B@1CA : ð19Þ; ð20Þ; ð21Þ; ð22Þ; ð23Þ; ð24Þ hold

8><>:9>=>; ð52Þ

The corresponding Hamiltonian is

HfP qf ; sf ;uf ; If ; kf ; c� �

¼ e�qtXr2FR

Xi2N

Xj2N

11þ ar

j

Wj

Xg2FP

cgj

!qfr

ij �Xi2N

Xh2N

E eCfi hf

i ; t� �h i

�Xi2N

Xj2N

E erij� �

sfij þ

Xr2FR

qfrij

!(

�Xi2N

wfi If

i ; t� �

� bfi Var eF f

i ðhfi Þ

� �þ Var eCf

i

�uf

i

� �þ Var erij

� �sf

ij þXr2FR

qfrij

!" #)

þXi2N

kfi E eF f

i hfi

� �h iþXj2N

sfji �

Xj2N

sfij �

Xr2FR

Xj2N

qfrij

" #ð53Þ

where If ¼ Ifi : i 2 N

� �, and kf ¼ kf

i : i 2 N� �

is a vector of adjoint variables. We will use the notation

rzHf�P ¼ rzH

fP qf�; sf�;hf�

; If�; kf�; c��

ð54Þ

to denote the gradient of the Hamiltonian (53) with respect to the control vector

z ¼qf

sf

hf

0B@1CA

of firm f evaluated at a Nash equilibrium. The maximum principle for firm f 2 F P with producers at node i 2 N leads to:

rqf Hf�P

h iTðqf � qf�Þ þ rsf Hf�

P

h iTðsf � sf�Þ þ rhf Hf�

P

h iTðhf � hf�Þ 6 0 ð55Þ

qf

sf

hf

0B@1CA; qf�

sf�

hf�

0B@1CA 2 Kf

Pðh�f Þ ð56Þ

3.2. Maximum principle for the retailers

With

c�r

qr

�ð57Þ

as exogenous, each retailer r 2 F R solves


JRr ðcr; c�r ; qrÞ

s:t: cr 2 KrRðqrÞ

ð58Þ

where

KrRðqrÞ � cr : ð36Þ; ð37Þ; ð38Þ; ð39Þ holdf g ð59Þ


HrR cr ;Rr; cr; c�r; qrð Þ ¼ e�qt

Xi2N

Xj2N

crj �

11þ ar

j

Xi2N

qfrij

!Wj

Xg2FR

cgj

!�Xj2N

/rj ðR

rj ; tÞ

" #þXj2N

crj

Xf2FP

Xi2N

qfrij � cr

j

!ð60Þ

where Rr ¼ Rrj : j 2 N

� �and cr ¼ cr

j : j 2 N� �

is a vector of adjoint variables. We will use the notation

rcr Hr�R ¼ rcr Hr

R cr�;Rr�; cr�; c�r�; qr�ð Þ ð61Þ

to denote the gradient of the Hamiltonian (60) with respect to the controls of retailer r evaluated at a Nash equilibrium. Themaximum principle for retailer r 2 F R leads to:

rcr Hr�R

� �T cr � cr�ð Þ 6 0cr; cr� 2 KrRðqr�Þ ð62Þ

3.3. Minimum principle for the suppliers

With h as exogenous, the supply chain manager solves

min JSðuÞs:t: u 2 KSðhf Þ

ð63Þ

and

KSðhf Þ � h : ð46Þ; ð47Þ; ð48Þ; ð49Þ holdf g ð64Þ


HSðu; S; fÞ ¼ e�qtXN

k¼1

E eV kðuk; tÞh i

þukðSk; tÞ þ bk � Var eV kðuk; tÞ� �h i

þXN

k¼1

fk½uk�1 � uk� ð65Þ

where S = (Sk: k 2 [1, N]) and f = (fk: k 2 [1, N]) are a vector of adjoint variables. We will use the notation

ruH�S ¼ ruHS u�; S�; f�ð Þ ð66Þ

to denote the gradient of the Hamiltonian (65) with respect to supply chain controls evaluated at a Nash equilibrium. Theminimum principle for the single supply chain manager leads to:

ruH�S� �Tðu� u�ÞP 0 u; u� 2 KSðhf�Þ ð67Þ

3.4. The DVI

Note that the variational inequalities derived above hold for each instant of continuous time. So we may integrate theindividual variational inequalities over time and sum them over discrete agent indices to obtain a single necessary condition:the solution

q�

s�

hf�

c�

u�

0BBBBBB@

1CCCCCCA 2 X ð68Þ

satisfies
Xf2FP
Z tf

t0

�rqf Hf�P

h iTðqf � qf�Þ þ �rsf Hf�

P

h iTðsf � sf�Þ þ �ruf Hf�

P

h iTðhf � hf�Þ

� dt

þZ tf

t0

Xr2FR

�rcr Hr�R

� �Tðcr � cr�Þdt þZ tf

t0

ruH�S� �Tðu� u�Þdt P 0 ð69Þ


for all

q

s

hf

c

u

0BBBBBB@

1CCCCCCA 2 X ð70Þ

where

X ¼ K� C ð71ÞK ¼ KSðh�Þ �

Yf2FP

KfPðh

�f Þ �Y

r2FR

KrRðqr�Þ ð72Þ

and C is the set of adjoint variables determined by the adjoint equations and the transversality conditions.

4. Fixed point algorithm

A method to solve differential variational inequalities must first be devised before differential variational inequalities canbe applied to the previously developed supply chain model. Generally, there exists an equivalent functional fixed point prob-lem corresponding to a given differential variational inequality. The following formulation provides an immediate, simpleand often quite effective algorithm for solving DVI(F, f0, U, C, x0). In this section we present in summary form certain key re-sults on continuous-time fixed point computation from Friesz (2010).

4.1. Formulation

To solve DVI(F, f0, U, C, x0), we will rely on the following notion of regularity:

Definition 1. Regularity of DVI(F, f0, U, C, x0). We call DVI(F, f0, U, C, x0) regular if:

R1. u 2 U # (L2[t0, s])m

R2. xðu; tÞ : ðL2½t0; tf �Þm ! ðH1½t0; tf �Þn exists and is unique, strongly continuous and G-differentiable for all admissible u;R3. C(x, t): is continuously differentiable with respect to x and t;

R4. F(x, u, t) is continuous with respect to x and u;R5. f0(x, u, t) is continuously differentiable with respect to x and u;R6. U is convex and compact; andR7. there is a constant dual vector t 2 Rr for the terminal constraints C[x(tf), tf] = 0.

In particular, we are now ready to state and prove the following result:

Theorem 1. Fixed point formulation of DVI(F, f0, U, C, x0). When regularity in the sense of Definition 1 holds, DVI(F, f0, U, C, x0) isequivalent to the following fixed point problem:

u ¼ PU ½u� aFðxðu; tÞ;u; tÞ�

where PU[�] is the minimum norm projection onto U # (L2[t0, s])m and a 2 R1þþ is an arbitrary positive constant.

Proof. See Friesz, 2010. h

4.2. The algorithm

Naturally there is an associated fixed point algorithm based on the iterative scheme

ukþ1 ¼ PU uk � aFðxðuk; tÞ; uk; tÞ� �

The positive scalar a may be chosen empirically to assist convergence and may even be changed as the algorithm progresses.The detailed structure of the fixed point algorithm is given below:

Fixed point algorithm

Step 0. Initialization. Identify an initial feasible solution u0 2 U and set k = 0.Step 1. Solve optimal control problem. Solve the following optimal control problem:


minv

JkðvÞ ¼ cTC½xðtf Þ; tf � þZ tf

t0

12½uk � aFðxk; uk; tÞ � v�2 dt ð73Þ

s:t:dxdt¼ f0ðx; v; tÞ; xðt0Þ ¼ x0 ð74Þ

v 2 U ð75Þ

Call the solution uk+1.Step 2. Stopping test. If kuk+1 � ukk 6 e1 where e1 2 R1

þþ is a preset tolerance, stop and declare u⁄ uk+1. Otherwise setk = k + 1 and go to Step 1.

The convergence of this algorithm is guaranteed by the following result:

Theorem 2. When DVI(F, f0, U, C, x0) is regular in the sense of Definition 1, while additionally F(x, u, t) is strongly monotonic foru 2 U, the fixed point algorithm presented above converges.

Proof. See Friesz, 2010. h

4.3. Solving the sub-problems

It is important to realize that the fixed point algorithm of Section 4 can be carried out in continuous time provided weemploy a continuous time representation of the solution of each subproblem (73)–(75) from Step 1 of the fixed point algo-rithm. This may be done using a continuous time gradient projection method. For our present circumstances, that algorithmmay be stated as

Descent algorithm in Hilbert space for the projection sub-problems

Step 0. Initialization. Pick vk,0(t) 2 U and set j = 0.Step 1. Finding State Variables. Solve the state dynamics

dxdt¼ f0 x;vk;j; t

� �ð76Þ

xðt0Þ ¼ x0 ð77Þ

Call the solution xk,j(t). In the event a discrete time method is used to solve the state dynamics (76) and (77), curve fitting isused to obtain the continuous time state vector xk,j(t).Step 2. Finding adjoint variables. Solve the adjoint dynamics

ð�1Þdkdt¼ rxHk

��v¼vk;j

x¼xk;jð78Þ

kðtf Þ ¼@C½xk;jðtf Þ; tf �

@xðtf Þð79Þ

where

Hk ¼ 12½uk � aFðxk;uk; tÞ � v �2 þ kT f0ðx; vk;j; tÞ

Call the solution kk,j(t). In the event a discrete time method is used to solve the adjoint dynamics (78) and (79), curve fitting isused to obtain the continuous time adjoint vector kk,j(t).Step 3. Finding the gradient. Determine

rv Jk;jðtÞ ¼ rvHk

Step 4. Stopping test. For a fixed and suitably small fixed step size

hk 2 R1þþ

determine

vk;jþ1ðtÞ ¼ PU ½vk;jðtÞ � hkrv Jk;j� ð80Þ

In the event a discrete time method is used to solve the above projection subproblem, curve fitting is used to obtain the con-tinuous time control vector (80).Step 5. Stopping test. For e2 2 R1

þþ, a pre-set tolerance, stop if kvk,j+1 � vk,jk < e2 and declare vk⁄ vk,j+1. Otherwise set j = j + 1and go to Step 1.


5. Numerical example

This numerical example deals with both the deterministic case and the case considering uncertain production and trans-portation functions. While network topologies for suppliers, firms with producers at node i 2 N , and retailers are the samefor both cases , suppliers’ transportation costs and producers’ production and transportation costs differ reflecting the net-work disruption risk. In this example, the network consists of 2 arcs and 3 nodes for suppliers and 9 arcs and 7 nodes forproducers and retailers. In addition, there are 5 arcs between suppliers and producers (Fig. 1). Firm 1 with producers (at nodei = 1, 2, 3) and firm 2 with producers (at node i = 4, 5) have activities located at nodes i = 1, 2, 3, 4, 5. Retailer 1 is located atnode 6 and retailer 2 is located at node 7. The time interval of interest is [0, 10], meaning t0 = 0 and tf = 10.

5.1. Deterministic case (without uncertainty)

The initial inventories at each node for producers i = 1, 2, 3, 4, 5 of firm f 2 F Pðf ¼ 1;2Þ and retailers r 2 F R ðr ¼ 1;2Þ at nodesj = 6, 7 are the following:

I11ð0Þ ¼ 2 I1

2ð0Þ ¼ 3 I13ð0Þ ¼ 2 I2

4ð0Þ ¼ 3 I25ð0Þ ¼ 2 R1

6ð0Þ ¼ 1 R27ð0Þ ¼ 1

The initial inventories for suppliers are the following:

S0ð0Þ ¼ 10 S1ð0Þ ¼ 3 S2ð0Þ ¼ 1

The discount rate qis assumed to be 0.05 and the retailers’ margins at nodes j = 6 and j = 7 are 0.1 and 0.08, respectively.Keeping in mind that the retailers occupy distinct nodes, the consumers’ inverse demand functions for nodes j = 6 andj = 7 are assumed to be the following:

W6ðc6; c7Þ ¼ 11� c6 W7ðc6; c7Þ ¼ 11� 1:5 � c7

where cj is the consumption at node j from the sole retailer located there. The production cost functions at nodesi = 1, 2, 3, 4, 5 are the following:

F11 h1

1

� �¼ 1 � h1

1ð Þ22 F1

2 h12

� �¼ 0:2 � h1

2ð Þ22 F1

3 h13

� �¼ 0:3 � h1

3ð Þ22

F24 h2

4

� �¼ 0:4 � h2

4ð Þ22 F2

5 h25

� �¼ 0:6 � h2

5ð Þ22

where hfi denotes the input factor flow from the final (3rd) stage of the supply chain to producer i of firm f 2 F Pðf ¼ 1;2Þ. The

inventory cost functions for producers i = 1, 2, 3, 4, 5 of firm f 2 F Pðf ¼ 1;2Þ are as follows:

w11 I1

1; t� �

¼ 3 � I11ð Þ22 w1

2 I12; t

� �¼ 3 � I1

2ð Þ22 w1

3 I13; t

� �¼ 3 � I1

3ð Þ22

w24 I2

4; t� �

¼ 4 � I24ð Þ22 w2

5 I25; t

� �¼ 3 � I2

5ð Þ22

The inventory cost functions for retailers r 2 F R (r = 1, 2) at nodes j = 6, 7, respectively, are

1 2 3

1

2

3

6

7

5

4

S12

S23

S13

q26

S45q27

q37

q56

q47

Suppliers

Producers/Retailer

Firm 1(Producers)

Firm 2(Producers)

Retailer 1and 2

u1 u2

h1

h2 h4

h5

h3

u3 = h1 + h2 + h3 + h4 + h5

Fig. 1. Supply chain network.


/16 R1

6; t� �

¼ 6 �R1

6

� �2

2/2

7 R27; t

� �¼ 5 �

R27

� �2

2

The inventory cost functions for stages k = 1, 2, 3 of the supply chain are as follows:

u1ðS1; tÞ ¼ 5 � ðS1Þ2 u2ðS2; tÞ ¼38� ðS2Þ2 u3ðS3; tÞ ¼ ðS3Þ2

The freight rates rij from node i to node j are constant and are shown below:

r12 ¼ 2 r13 ¼ 2 r23 ¼ 0:5 r45 ¼ 0:5 r26 ¼ 4 r27 ¼ 3 r37 ¼ 5 r56 ¼ 4 r47 ¼ 3

The costs to producers at nodes i = 1, 2, 3, 4, 5 of firm f 2 F Pðf ¼ 1;2Þ for acquiring input flows are the following:

C11 h1

1; t� �

¼ 2 � h11 C1

2 h12; t

� �¼ 2:5 � h1

2

C13 h1

3; t� �

¼ 2:1 � h13 C2

4 h24; t

� �¼ 3 � h2

4 C25 h2

5; t� �

¼ 2:1 � h25

Finally, the variable costs of preparing the stage k = 1, 2, 3 supply chain flows are presented below:

V1ðu1; tÞ ¼ 1:2 � u1 V2ðu2; tÞ ¼ 1:3 � u2 V3ðu3; tÞ ¼ 2 � u3

The following upper bounds on control variables are imposed:

eq ¼1010101010

26666664

37777775 ~s ¼

10101010

2666437775 eh ¼ Bf ¼

88888

26666664

37777775 ec ¼ 1010

� � eu ¼ 151515

264375

Keeping in mind that the subnetworks for producers and retailers are disjoint, the initial value problems that constituteinventory dynamics for producers i = 1, 2, 3, 4, 5 of firm f 2 F Pðf ¼ 1;2Þ are the following flow balance equations:

dI11

dt ¼ F11 h1

1

� �� s1

12 � s113 I1

1ðt0Þ ¼ I01 ¼ 2

dI12

dt ¼ F12 h1

2

� �þ s1

12 � s123 � q11

26 I12ðt0Þ ¼ I0

2 ¼ 3

dI13

dt ¼ F13 h1

3

� �þ s1

13 þ s123 � q12

37 I13ðt0Þ ¼ I0

3 ¼ 2

dI24

dt ¼ F24 h2

4

� �� s2

45 � q2247 I2

4ðt0Þ ¼ I04 ¼ 3

dI25

dt ¼ F25 h2

5

� �þ s2

45 � q2156 I2

5ðt0Þ ¼ I05 ¼ 2

where qfrij is the flow from a producer at node i of firm f 2 F Pðf ¼ 1;2Þ to a retailer r 2 F R (r = 1, 2) at node j and sf

ij is theshipment flow from a producer at node i to a producer at node j of firm f 2 F Pðf ¼ 1;2Þ. Inventory dynamics for retailerr 2 F R ðr ¼ 1;2Þ at nodes j = 6, 7 are the following flow balance equations:

dR16

dt ¼ q1126 þ q21

56 � c6 R06 ¼ 1

dR27

dt ¼ q1227 þ q12

37 þ q2247 � c7 R0

7 ¼ 1

Inventory dynamics for the supply chain stages k = 1, 2, 3 are the following flow balance equations:

dS1dt ¼ u0 � u1 S0

1 ¼ 10dS2dt ¼ u1 � u2 S0

2 ¼ 3dS3dt ¼ u2 � u3 S0

3 ¼ 1

u3 ¼ h11 þ h1

2 þ h13 þ h2

4 þ h25

The corresponding Hamiltonian for producer firm f 2 F P (f = 1, 2) is

H1P q1; s1;h1

; I1; k1; cr� �

¼ e�qt 11þ a1

6

W6ðc6Þq1126 þ

11þ a2

7

W7ðc7Þ q1227 þ q12

37

� �� 2h1

1 � 2:5h12 � 2:1h1

3 � 2s112 � 2s1

13

�� 0:5s1

23 � 4q1126 � 3q12

27 � 5q1237 �

X3

i¼1

w1i I1

i ; t� �

þ k1 F11 h1

1

� �� s1

12 � s113

h iþ k2 F1

2 h12

� �þ s1

12 � s123 � q11

26 � q1227

h iþ k3 F1

3 h13

� �þ s1

13 þ s123 � q12

37

h i


H2P q2; s2; h2

; I2; k2; cr� �

¼ e�qt 11þ a1

6

W6ðc6Þq2156 þ

11þ a2

7

W7ðc7Þ þ q2247 � 3h2

4 � 2:1h25 � 0:5s2

45 � 3q2247 � 4q21

56 �X5

i¼4

w2i ðI

2i ; tÞ

( )þ k4 F2

4 h24

� �� s2

45 � q2247

h iþ k5 F2

5 h25

� �þ s2

45 � q2156

h i
We will use the notation
Hf�P ¼ Hf

P qf�; sf�; hf�; If�; kf�; c�

� �
to denote the Hamiltonian evaluated at a Nash equilibrium where If ¼ If
i : i ¼ 1;2;3;4;5� �

and kf ¼ kfi : i ¼ 1;2;3;4;5

� �are

vectors of adjoint variables. The pertinent gradients of the Hamiltonians for producers are shown below:

rq1 H1�P ¼

rq1126

H1�P

rq1227

H1�P

rq1237

H1�P

2666437775 ¼

e�qt 11þa1

6W6 c�6� �

� 4� �

� k2

e�qt 11þa2

7W7 c�7� �

� 3� �

� k2

e�qt 11þa2

7W7 c�7� �

� 5� �

� k3

2666664

3777775

rs1 H1�P ¼

rs112

H1�P

rs113

H1�P

rs123

H1�P

2666437775 ¼

e�qtð�2Þ � k1 þ k2

e�qtð�2Þ � k1 þ k3

e�qtð�0:5Þ � k2 þ k3

264375

rh1 H1�P ¼

rh11H1�

P

rh12H2�

P

rh13H3�

P

2666437775 ¼

e�qtð�2Þ þ k1h1�1

e�qtð�2:5Þ þ 0:4k2h1�2

e�qtð�2:1Þ þ 0:6k3h1�3

26643775

rq2 H2�P ¼

rq2156

Hf�P

rq2247

Hf�P

24 35 ¼ e�qt 11þa1

6W6 c�6� �

� 4� �

� k5

e�qt 11þa2

7W7 c�7� �

� 3� �

� k4

264375

rs2 H2�P ¼ rs2

45H2�

P

h i¼ e�qtð�0:5Þ � k4 þ k5½ �

rh2 H2�P ¼

rh24H2�

P

rh25H2�

P

24 35 ¼ e�qtð�3Þ þ 0:4k4h2�4

e�qtð�2:1Þ þ 0:6k5h2�5

" #

The Hamiltonian for retailer r 2 F R ðr ¼ 1;2Þ is as follows:

H1R c1;R1; c1; c�1; qf� �

¼ e�qt c6 �1

1þ a16

q1126 þ q21

56

� � �W6ðc6Þ � /1

6 R16; t

� �� þ c6 q11

26 þ q2156 � c6

� �

H2R c2;R2; c2; c�2; qf� �

¼ e�qt c7 �1

1þ a27

q1227 þ q12

37 þ q2247

� � �W7ðc7Þ � /2

7 R27; t

� �� þ c7 q12

27 þ q1237 þ q22

47 � c7� �

We will use the notation

Hr�R ¼ cr�;Rr�; cr�; c�r�; qr�ð Þ

to denote the Hamiltonian evaluated at a Nash equilibrium where Rr ¼ Rrj : j ¼ 6;7

� �and cr ¼ cr

j : j ¼ 6;7� �

is are vectorsof adjoint variables. The pertinent gradients of the Hamiltonians for retailers are shown below:

rc1 H1�R ¼ rc6 Hr�

R

� �¼ e�qt 1

1þ a16

q1126 þ q21

56

� �� 2c6 þ 11

�� c6

� �rc2 H2�

R ¼ rc7 Hr�R

� �¼ e�qt 1:5

1þ a27

q1227 þ q12

37 þ q2247

� �� 3c7 þ 11

�� c7

� �
The Hamiltonian for the supply chain is as follows:
HSðu; S; f; hÞ ¼ e�qt 1:2u1 þ 1:3u2 þ 2u3 þ 5S21 þ

38

S22 þ S2

3

�þ f1ðu0 � u1Þ þ f2ðu1 � u2Þ þ f3ðu2 � u3Þ


where S = (Sk: k = 1, 2, 3) and f = (fk: k = 1, 2, 3) are vectors of adjoint variables. The pertinent gradients of the Hamiltoniansfor supplier are shown below:

ruH�S ¼

ru0 H�S

ru1 H�S

ru2 H�S

ru3 H�S

26666664

37777775 ¼f1

e�qtð1:2Þ � f1 þ f2

e�qtð1:3Þ � f1 þ f2

e�qtð1:2Þ � f3

26666664

37777775
As a result, we seek
q�

s�

h�

c�

u�

2666666664

37777777752 X

satisfying
Xf2F P
Z 10

0�rqf Hf�

P

h iTðqf � qf�Þ þ �rsf Hf�

P

h iTðsf � sf�Þ þ �rhf Hf�

P

h iTðhf � hf�Þ

� dt

þZ 10

0

Xr2FR

�rcr Hr�R

� �Tðcr � cr�Þdt þZ 10

0ruH�S� �Tðu� u�Þdt P 0

for all

q

s

h

c

u

2666666664

37777777752 X

where

X ¼ K� C

K ¼ KSðh�Þ �Y

f2FP

KfPðh

�f Þ �Y

r2FR

KrR qr�ð Þ

5.2. With-disruption case (with uncertainty)

In the case in which network disruptions do occur, initial inventories at each node for producers i = 1, 2, 3, 4, 5 of firmf 2 F P (f = 1, 2) and retailers r 2 F R (r = 1, 2) at node j = 6, 7 are the same as the deterministic case:

I11ð0Þ ¼ 2 I1

2ð0Þ ¼ 3 I13ð0Þ ¼ 2 I2

4ð0Þ ¼ 3 I25ð0Þ ¼ 2 R1

6ð0Þ ¼ 1 R27ð0Þ ¼ 1

The initial inventories for suppliers are the following:

S0ð0Þ ¼ 10 S1ð0Þ ¼ 3 S2ð0Þ ¼ 1

The discount rate q is assumed to be 0.05 and the retailers’ margins at nodes j = 6 and j = 7 are assumed to be 0.1 and 0.08,respectively. Keeping in mind that the retailers occupy distinct nodes, the consumers’ inverse demand functions for nodesi = 6 and i = 7 are the following:

W6ðc6; c7Þ ¼ 11� c6 W7ðc6; c7Þ ¼ 11� 1:5 � c7

where cj is the consumption at j from the sole retailer located there. The production functions at nodes i = 1, 2, 3, 4, 5 are thefollowing:

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

TIME

Who

le sa

le b

y Pr

oduc

ers

q26q56q27q37q47

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

TIME

q26q56q27q37q47

Who

le sa

le b

y Pr

oduc

ers

Fig. 2. Wholesale goods flow from producers to retailers.


F11ðh

11Þ ¼ 1 � ðh

11Þ

2

2� 0:1n1h1

1

F12ðh

12Þ ¼ 0:2 � ðh

12Þ

2

2� 0:05n2h1

2

F13ðh

13Þ ¼ 0:3 � ðh

13Þ

2

2� 0:05n3h1

3

F24ðh

24Þ ¼ 0:4 � ðh

24Þ

2

2� 0:05n4h2

4

F25ðh

25Þ ¼ 0:6 � ðh

25Þ

2

2� 0:1n5h2

5

where hfi denotes the input factor flow from the final (3rd) stage of the supply chain to producer i of firm f 2 F P (f = 1, 2) with

all ni following the uniform distribution ni ½0;2�.

E½F11ðh

11Þ� ¼ 1: � ðh

11Þ

2

2 � 0:1h11 VarðF1

1ðh11ÞÞ ¼ 0:01

3 ðh11Þ

2

E½F12ðh

12Þ� ¼ 0:2 � ðh

12Þ

2

2 � 0:05h12 VarðF1

2ðh12ÞÞ ¼ 0:0025

3 ðh12Þ

2

E½F13ðh

13Þ� ¼ 0:3 � ðh

13Þ

2

2 � 0:05h13 VarðF1

3ðh13ÞÞ ¼ 0:0025

3 ðh13Þ

2

E½F24ðh

24Þ� ¼ 0:4 � ðh

24Þ

2

2 � 0:05h24 VarðF2

4ðh24ÞÞ ¼ 0:0025

3 ðh24Þ

2

E½F25ðh

25Þ� ¼ 0:6 � ðh

25Þ

2

2 � 0:1h25 VarðF2

5ðh25ÞÞ ¼ 0:01

3 ðh25Þ

2

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

TIME

Ship

men

t flo

ws

s12s13s23s45

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

TIME

s12s13s23s45

Ship

men

t flo

ws

Fig. 3. Shipment trajectories of producers at nodes 1, 2, 3, 4, 5.


The inventory cost functions for producers i = 1, 2, 3, 4, 5 of firm f 2 F P (f = 1, 2), retailers r 2 F R (r = 1, 2) at nodes j = 6, 7,respectively and stages k = 1, 2, 3 of the supply chain are all the same as those employed in the deterministic case. The freightrates rij from node i to node j are assumed to be as follows:

r12 ¼ 2þ g1 r13 ¼ 2þ 2g1 r23 ¼ 0:5þ 0:5g2

r45 ¼ 0:5þ 3g4 r26 ¼ 4þ g2 r27 ¼ 3þ 2g2

r37 ¼ 5þ g3 r56 ¼ 4þ 0:5g5 r47 ¼ 3þ 0:5g4

where gi follows normal distribution N (0, 4).

E½r12� ¼ 2 Varðr12Þ ¼ 4 E½r13� ¼ 2 Varðr13Þ ¼ 16 E½r23� ¼ 0:5 Varðr23Þ ¼ 1E½r45� ¼ 0:5 Varðr45Þ ¼ 36 E½r26� ¼ 4 Varðr26Þ ¼ 4 E½r27� ¼ 3 Varðr27Þ ¼ 16E½r37� ¼ 5 Varðr37Þ ¼ 4 E½r56� ¼ 4 Varðr56Þ ¼ 1 E½r47� ¼ 3 Varðr47Þ ¼ 1

The costs to producers at nodes i = 1, 2, 3, 4, 5 of firm f 2 F P (f = 1,2) for acquiring input flow are the following:

C11ðh

11; tÞ ¼ ð2þ g1Þ � h

11 E½C1

1ðh11; tÞ� ¼ 2 � h1

1 VarðC11Þ ¼ 4 � ðh1

1Þ2

C12ðh

12; tÞ ¼ ð2:5þ g2Þ � h

12 E½C1

2ðh12; tÞ� ¼ 2:5 � h1


2Þ2

C13ðh

13; tÞ ¼ ð2:1þ g3Þ � h

13 E½C1

3ðh13; tÞ� ¼ 2:1 � h1


3Þ2

C24ðh

24; tÞ ¼ ð3þ g4Þ � h

24 E½C2

4ðh24; tÞ� ¼ 3 � h2


4Þ2

C25ðh

25; tÞ ¼ ð2:1þ g5Þ � h

25 E½C2

5ðh25; tÞ� ¼ 2:1 � h2


5Þ2

Finally, the variable costs of preparing stages k = 1, 2, 3 supply chain flows are the following:


V1ðu1; tÞ ¼ 1:2 � u1 þ 3d1u1 E½V1� ¼ 4:2 � u1 VarðV1Þ ¼ 3ðu1Þ2

V2ðu2; tÞ ¼ 1:3 � u2 þ 2d2u2 E½V2� ¼ 3:3 � u2 VarðV2Þ ¼ 43 ðu2Þ2

V3ðu3; tÞ ¼ 2 � u3 þ d3u3 E½V3� ¼ 3 � u3 VarðV3Þ ¼ 13 ðu3Þ2

where the parameter dh is assumed to follow a uniform distribution U [0, 2]. Upper bounds imposed on control variablesare the same as those employed in the deterministic case. Finally, the risk aversion parameters are:b1

i ¼ 0:6 ði ¼ 1;2;3Þ; b2i ¼ 0:2 ði ¼ 4;5Þ, and bh = 1.2 (h = 1, 2, 3).

5.3. Numerical example result summary

In both numerical scenarios considered, the relevant production functions exhibit increasing returns to scalewhile the demand function is linear. The differential variational inequality describing the above problem is solved usinga fixed point algorithm. GAMS/PATHNLP and Matlab are used to solve the subproblems. Results are displayed below inFigs. 2–5.

The numerical results reveal the following:

1. Decreased production in the face of supply chain disruption. This is caused by a productivity arising directly diminished pro-duction resulted in fewer shipments among producers as well as fewer shipments from producers to retailers. See Fig. 2.

2. Wholesale good flows decline in the face of supply chain disruption see also Fig. 2. Diminished production ultimatelydecreases the demand for consumption.

3. Shipping flows decrease in the face of disruption. See Figs. 3 and 4. Input factor flow temporarily increases as does the var-iance in production. See Fig. 4.

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

TIME

Inpu

t flo

ws h1

h2h3h4h5

0 1 2 3 4 5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

TIME

h1h2h3h4h5

Inpu

t flo

ws

Fig. 4. Input factor flows from supplier to producers.

0 1 2 3 4 5 6 7 8 9 100

5

10

15

TIME

Row

mat

eria

l flo

ws

u1u2u3

0 1 2 3 4 5 6 7 8 9 100

5

10

15

TIME

u1u2u3

Row

mat

eria

l flo

ws

Fig. 5. Raw input flow among suppliers.


4. The raw materials flows among suppliers also temporarily increase but with their own peaks. See Fig. 5. This occurs becausesuppliers are sensitive to the variance of raw material flows. In order to minimize supply costs, a high value of risk aver-sion factor required the suppliers to narrow the variance of raw material flows. This resulted in a gradually change in theraw material flows.

6. Concluding remarks

This paper has explored a supply chain network model wherein supply chain uncertainty is represented by variance,which the producing firms wish to minimize. The model is formulated as a complex dynamic Nash game expressed as dif-ferential variational inequality (DVI). That formulation was crafted to integrate the complete supply–production–distribu-tion–retail activity chain. Risk has been considered in an expedient and narrow way, by introducing variance into eachproducer’s objective function. Moreover, the comparison of two numerical example problems indicates that the impact ofdisruptions determined by the DVI make sense in that they accord with intuition. This also demonstrates that supply chaindisruptions may potentially interrupt the normal supplies to consumption markets within an, urban metropolis. Thus, thestage is set for us to consider a much more complete representation of uncertainty and to explicitly compare stochasticgames to robust games.

References

Adida, E., Perakis, G., 2010. Dynamic pricing and inventory control: uncertainty and competition. Operations Research 58 (2), 289–302.Christopher, M., Peck, H., 2004. Building the resilient supply chain. International Journal of Logistics Management 15 (2), 1–13.Craighead, C.W., Blackhurst, J., Rungtusanatham, M.J., 2007. The severity of supply chain disruptions: design characteristics and mitigation capabilities.

Decision Sciences 38 (1), 131–156.Friesz, T.L., 2010. Dynamic Optimization and Differential Games. Springer-Verlag.Kogan, K., Tapiero, C.S., 2007. Supply Chain Games: Operations Management and Risk Valuation. Springer-Verlag, New York.


Latour, A., 2001. Trial by fire: A blaze in albuquerque sets off major crisis for cell-phone giants. Wall Street Journal, January 29.Silberberg, E., Suen, W., 2000. The Structure of Economics: A Mathemathical Analysis, third ed. McGraw-Hill, Irwin.Snyder, L.V., 2003. Supply Chain Robustness and Reliability: Models and Algorithm. Ph.D. thesis. Northwestern University, Evanston, Illinois.Swaminathan, J., Smith, S., Sadeh, N., 1998. Modeling supply chain dynamics: a multiagent approach. Decision Sciences 29 (3), 607–632.Tomlin, B.T., 2006. On the value of mitigation and contingency strategies for managing supply chain disruption risks. Management Science 52 (5), 639–657.Wagner, S.M., Bode, C., 2008. An empirical examination of supply chain performance along several dimensions of risk. Journal of Business Logistics 29 (1),

307–326.

competition and disruption in a dynamic urban supply chain

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