a di erential stackelberg game for pricing on a freight...

Scientia Iranica E (2016) 23(5), 2391{2406

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

A di�erential Stackelberg game for pricing on a freighttransportation network with one dominant shipper andmultiple oligopolistic carriers

M. Mozafari, B. Karimi� and M. Mahootchi

Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, P.O. Box15916-34311, Iran.

Received 5 May 2015; received in revised form 12 August 2015; accepted 2 September 2015

KEYWORDSDynamic pricing;Di�erential game;Stackelberg-Nashequilibrium;Bi-level programming;Freight networkequilibrium.

Abstract. This paper studies dynamic pricing and freight network equilibrium problemon a system consisting of one dominant producer called the shipper and multipleoligopolistic carriers who serve the shipper's origin-destination orders. The shipper sellsa homogeneous commodity to spatially separated demand markets. The demand receivedby the shipper is price-sensitive, while the prices set for each market are in uenced by thepricing strategies of the oligopolistic carriers. We formulate the problem as a di�erentialStackelberg-Nash game to �nd the equilibrium production, price, and routing decisionsover a planning horizon. A �nite dimensional discretization method and a penalty functionalgorithm are proposed to solve the model. The existence and uniqueness of propertiesare also explored. Finally, some numerical examples are presented and a comprehensivesensitivity analysis on the critical parameters is conducted to show the e�ciency of theproposed model and solution method.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

With the deregulation of transport industry and theemergence of private carriers, including transportationand Third Party Logistics companies (3PLs), relation-ships among players of freight transportation servicechain have become extremely complex. In today's com-petitive market place, the key stockholders involvedin freight transportation networks, including shippers,carriers, and infrastructure companies, typically ex-press a pro�t maximization behavior. Hence, thenegotiation and pricing mechanisms applied by themnoticeably impact on freight ows over the network.

Urban freight transportation markets can be

*. Corresponding author. Tel.: +98 21 64545374;Fax: +98 21 66954569E-mail addresses: [email protected] (M. Mozafari);[email protected] (B. Karimi)

characterized by oligopolistic behavior of a few largecarriers or third party logistics providers who competeto win transportation contracts [1,2]. Chanut andPache [3] noted that the French freight market isoligopolistic as the top �ve 3PL hold a third of themarket. Arvidsson et al. [4] carried out a similarempirical study on Sweden cargo market and showedthat the two dominant players in the market typicallyhave controlled over 80% of the freight ows.

In contrast to passenger individual route choice,freight route choices re ect the collective e�ect of thedecisions made by multiple agents. The freight owprediction models are able to understand and analyzethe impact of freight route choices on the transporta-tion network planning. An early classi�cation of themodels predicting freight ows presented by Harker andFriesz [5] is distinguished in three categories:

1. The econometric model which uses time series

2392 M. Mozafari et al./Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 2391{2406

to predict the relationship between transportationsupply and demand;

2. The spatial price equilibrium model in which the actof distributing goods by shippers between spatiallyseparated markets to �nd minimum costs leads toan equilibrium of prices and transportation ows ona simpli�ed network;

3. The freight network equilibrium model which con-siders the interaction of shippers and carriers on areal network to predict the equilibrium freight ows.

Recently, Tavasszy et al. [6] have reviewed freight trans-portation demand models by integrating the classical4-step modelling approach into spatial computable gen-eral equilibrium models, supply chain choice models,and hyper-network models. In this paper, we focuson trip distribution on the network to estimate freight ows related to a certain commodity traded in a marketwhere a dominant producer tries to deliver goods to hiscustomers through oligopolistic carriers. Our modelcan be categorized as a freight network equilibriummodel integrated with supply chain decisions, includingproduction, inventory, price, and rout choices.

The �rst signi�cant predictive freight networkmodel was introduced by Kresge and Roberts [7] study-ing a multi-modal multi-commodity network, in whichshippers decide on mode choice and general routingis calculated by the shortest path. Friesz et al. [8]determined freight network equilibrium ows takinginto account the interaction between shippers andcarriers. This model was then extended by Friesz andHarker [9] and Harker and Freisz [10]. Friesz et al. [11]proposed a sequential shipper-carrier model in whichthe shippers are cost-minimization agents who decideon the purchase location, the products, and the carrierto transport the products to their destinations, afterwhich the carriers make routing decisions in responseto the shippers. Hurley and Petersen [12] presented anequilibrium solution approach for the freight networkproblem in a system consisting of multiple pro�t maxi-mizing shippers and carriers, where the carriers decideon tari�s and the shippers determine their productionlevels to minimize their costs. Altman and Wyn-ter [13] gave an overview of network equilibrium modelsand discussed pricing issues in transportation andtelecommunication networks. King and Topaloglu [14]modeled the freight transportation pricing problemwhich presents a linear sensitive demand function. Thisstudy was then extended by Topaloghlu and Powell [15]for the situation that the transportation demand isuncertain. Topal and Bing�ol [16] studied the inventoryreplenishment problem for a retailer who needs thetransportation services in the availability of a truckload(TL) carrier and a less-than-ruckload (LTL) carrier.Ulku and Bookbinder [17] investigated a logistic marketwith price- and time-sensitive demand, considering

that the delivery time is guaranteed by a Third PartyLogistics (3PL) provider. Lin and Lee [18] proposeda model to simultaneously determine the zone pricesand operational plan for a freight carrier with limitedcapacity. The carrier aims to maximize his pro�t whilemeeting the expected service level and operationalrequirements.

Recently, many researchers have utilized gametheory to analyze the interactions between freightnetwork stockholders and studied the equilibrium ow.A dynamic freight network assignment model wasdeveloped by Agrawal and Ziliaskopoulos [19] in whichthe shippers choose the carrier with the lowest cost andthe market reaches the equilibrium when no shippercan reduce his costs by changing the carriers fora shipment. They applied a variational inequalitymethod to obtain the Nash equilibrium point. Xiaoand Yang [20] addressed the equilibrium ow in asystem consisting of shippers, carriers, and infrastruc-ture companies as pro�t maximizing agents. Theyconsidered the shipper as a non-cooperative agent,while the carriers and infrastructure companies maketheir decisions cooperatively. Friesz and Holguin-Veras [21], and Friesz et al. [22] studied a dynamicpricing and freight network equilibrium problem onan urban transportation market consisting of multipleshippers and carriers. They formulated the problemas a di�erential game and applied a nonlinear comple-mentarity problem to �nd the equilibrium of the non-cooperative game between the players. Mozafari andKarimi [23] investigated pricing decisions for truckloadcarriers in a duopoly market, considering two scenariosof non-cooperative and cooperative games. Price andfrequency competitions between freight carriers wereformulated by Shah and Bruckner [24] using gametheoretic models. Naimi Sadigh et al. [25] exam-ined channel pricing problem comparing two scenar-ios of centralized and decentralized structures usinggame theory. Mutlu and Cetinkaya [26] analyzedtwo di�erent structures of a carrier-retailer supplychain with common-carriage option. In the former,the carrier and the retailer simultaneously competeand choose strategies to optimize their own bene�t;in the latter, both the carrier and retailer try tomaximize the total channel payo�. Lee et al. [27]modeled the oligopolistic behavior of the carriers inmaritime fright transportation networks using a multi-level hierarchical game theoretical approach. Moza-fari et al. [28] proposed a generalized Nash equilib-rium problem to analyze dynamic pricing and eetplanning decisions of oligopolistic freight carriers whocompete to win transportation contracts on a net-work. Nagurney et al. [29] developed both static anddynamic supply chain network models with multiplemanufacturers and freight service providers compet-ing on price and quality. They analyzed the Nash

M. Mozafari et al./Scientia Iranica, Transactions E: Industrial Engineering 23 (2016) 2391{2406 2393

equilibrium of the market using variational inequalityproblem.

However, all the researches above studied thesimultaneous decisions of the freight network stock-holders, where all the agents have almost the samedecision power, and there are comparatively fewerstudies which concern the sequential behavior of theshipper-carrier relationship when one of the agentspossesses the dominant power in the transportationservice chain. Brotcorne et al. [30] focused on freighttari�-setting problem where the leader is one amonga group of competing carriers and the follower is theshipper while ignoring the allocation of the freight owsamong the rival carriers. Zhang et al. [31] establishedan equilibrium problem with equilibrium constraintfor a container transportation super-network in whichthe upper level ports are involved in non-cooperativecompetition, while the lower level shippers competefor the path of minimum cost. Holgu��n-Veras etal. [32] discussed the theoretical and empirical evidenceon the freight mode choice problem, considering theinteractions between shippers and carriers. Theycompared the experiment of perfect cooperation withthe setting in that the shipper selects the shipmentsize as the leader through game theory framework.Friesz et al. [33] studied a dynamic shipper carrierproblem in which the carrier acts as the leader andseveral shippers compete on the sale of products asthe followers under the Cournot-Nash behavior. Abi-level modeling approach that captures hierarchicalrelationships between shippers and carriers in maritimefreight transportation networks was proposed by Lee etal. [34] where the carriers are the leaders and shippersare the followers.

This paper addresses the dynamic pricing behav-ior of the main stockholders in a freight transportationservice chain, where a dominant shipper acts as a pro�tmaximizing leader and multiple oligopolistic carriersact as Cournot-Nash followers who try to capturetransportation service demand from the transactionsbetween the shipper and spatially separated demandmarkets. The shipper decides on the productionlevel and the sale's price of a homogeneous product,while the carriers compete on prices and make routingdecisions simultaneously to optimize their own pro�ts.Also, all the players' strategies can be changed contin-uously regarding the time. In other words, there existscontinuous-time dynamic equilibrium points for thefreight network game while the time value of money isconsidered. We propose a di�erential Stackelberg-Nashgame using a bi-level programming approach to modelthe problem in such an environment. Then, the bi-levelmodel is transformed into a single level optimizationmodel by including the equilibrium conditions of thecarriers' Nash game at the lower level as a set ofconstraints for the shipper's model at the upper level.

A �nite dimensional time discretization method isproposed to approximate the model as a mathematicalprogram with linear constraints. The existence anduniqueness of properties are investigated. Finally, apenalty function algorithm is presented to solve thesingle level mathematical model.

The rest of the paper is organized as follows: Sec-tion 2 describes the problem in detail while declaringthe assumptions, notations, variables, and parameters.In Section 3, the di�erential Stackelberg-Nash pricinggame between the shipper and carriers is formulatedas a bi-level optimal control problem. In Section 4,we explore the equilibrium conditions at the lowerlevel using a �nite dimensional discretization methodand the equivalent single level model is formulated.Existence and uniqueness of properties are examinedin this section. Section 5 is devoted to numerical studyand sensitivity analysis. Finally, concluding remarksare discussed in Section 6.

2. Problem de�nition

In this section, a detailed description of the problem in-cluding the structure of the network, the assumptions,and notations is presented.

2.1. Network structureWe study a freight transportation market structure inwhich N + 1 decision makers interact with each other.The set of players includes a dominant shipper who isresponsible for producing and shipping a homogeneouscommodity to the demand markets within the networkand N competing carriers who are providing freighttransportation services to the shipper. The transporta-tion network consists of, L, nodes representing thelocations of the demand markets, or where the shipper'sand carriers' facilities have been placed, and, A, arcswhich connect the origin and destination nodes. Allthe players are pro�t maximizing agents and try tooptimize their own objectives by choosing their decisionvariables dynamically over a planning horizon.

In the network of Figure 1, it is assumed thatthe shipper has two production sites at nodes 1 and 6.Each node has a market with a predetermined potentialdemand for the commodity produced by the shipper.

Figure 1. A schematic view of an instance of the network.


Furthermore, there are several carriers which competeto get transportation service demand of the shipperat the shipper's locations by pricing decisions. Eachselected carrier makes routing decisions to optimize hisbene�t. For example, to serve a transportation orderfrom node 1 to node 3, one carrier may choose the route1-2-3 or 1-4-3.

2.2. AssumptionsThe model presented in this paper is based upon thefollowing assumptions:1. Customers are price sensitive and the quantity

demanded at each market will decrease signi�cantlyas price increases;

2. The shipper possesses As di�erent locations, eachproducing the same commodity, but with di�erentproduction capacity in the network;

3. Regardless of the site of shipping the commodity,each market has its own price which could bedi�erent from other markets;

4. The shipper as the leader, taking into accountthe followers' reactions, declares his strategies tooptimize his own bene�t function over a time-continuous planning horizon;

5. The e�ect of time value of money is investigated;6. There is a competition among the followers (i.e.

carriers) to achieve more demand from the trans-actions between the shipper at di�erent productionsites and customers at di�erent demand markets;

7. The demand captured by each carrier betweenorigins and destinations is in uenced by his ownprice as well as his competitors' prices;

8. The routing decisions made by the carriers collec-tively determine the freight ows in the market overthe planning horizon;

9. Each carrier is able to hire extra eets wheneverthe demand is more than his own eet, thusthe transportation eet capacity of each carrier isassumed to be unlimited.

2.3. NotationsThe notations used to formulize the game theoreticalmodel are listed in the following:

Sets:L The set of nodes in the network;A The set of arcs connecting nodes in the

network;Ls The set of locations for the shipper's

production facilities;Rij The set of paths between origin i and

destination j;C The set of competing carriers;T The planning horizon T = [t0; tf ].

Shipper's variables:

pj(t) The price charged by the shipper todemand market placed on node j 2 Lat time t;

dj(t) The amount of demand received frommarket j 2 L at time t;

qi(t) The production quantity at theshipper's facility placed on node i 2 LSat time t;

Ii(t) The inventory held by the shipper atthe facility placed on node i 2 LS attime t.

Carriers' variables:'cij(t) The price charged by carrier c to the

shipper for providing transportationservices between origin i 2 LS anddestination j 2 L at time t;

vcij(t) The ow of shipments received bycarrier c for transportation servicesbetween origin i 2 LS and destinationj 2 L at time t;

hcr;ij(t) The ow of shipments from carrier con path r between origin i 2 LS anddestination j 2 L at time t;

fca(t) The ow of shipments from carrier con arc a 2 A in the network at time t.

Parameters:� The constant nominal discount rate of

future cash ows representing the timevalue of money;

i(t) The inventory cost of the shipper atlocation i 2 LS and time t;

�i(t) The production cost of the shipper atlocation i 2 LS and time t;

Capi The maximum production capacity ofthe shipper at his facility placed onnode i 2 LS ;

�Dj(t) The potential market demand at nodej 2 L and time t;

�j The price elasticity of the marketdemand at node j 2 L;

�Ii The initial inventory at node i 2 LS ;kca The travel cost on arc a at time t;�a;r This parameter is equal to 1 if arc a

exists in the path r, and 0 otherwise; c The elasticity of demand for carrier c

to his own price;�c;�c The elasticity of demand for carrier c

to his rivals' prices.


3. The di�erential Stackelberg-Nash gamemodel

In this section, we apply a bi-level modeling approachto formulize the Stackelberg-Nash game of the shipper-carrier problem de�ned in the previous section. Sincethe decisions made by the players are changing contin-uously by time, the corresponding game is a di�erentialStackelberg-Nash game.

3.1. The shipper's objective functional,dynamics, and constraints

The shipper in the upper level acts as the leader andtries to maximize his net pro�t over the �nite planninghorizon, T = [t0; tf ], considering the followers' reac-tions. The shipper dynamically determines the priceof the commodity charged to each demand market,the quantity produced at each production site, andthe quantity shipped from each production site toeach market. The quantity shipped to each marketis a�ected by the price of the commodity set for thatdemand market, as well as the price charged by thecarriers for the transportation. The shipper's model isde�ned as follows:

maxp;d;q

tfZt0

e��t:(Xj2L

pj(t):dj(t)� Xi2LS

i(Ii(t); t)

�Xi2LS

�i(qi(t); t)�Xi2LS

Xj2L

Xc2C

'cij(t):vcij(t)

�dt:

(1)

Subject to:

dIi(t)dt

= qi(t)�Xj2L

Xc2C

vcij(t) 8i 2 Ls; (2)

Ii(t0) = Ii 8i 2 Ls; (3)

Ii(t) � 0 8i 2 Ls; (4)

dj(t) = �Dj(t)� �j :pj(t) 8j 2 L; (5)

0 � qi(t) � Capi 8i 2 Ls; (6)

pj(t) � 0 8j 2 L: (7)

Eq. (1) represents the objective functional of theshipper as its revenue minus the inventory, production,and transportation costs, respectively, where the trans-portation term is determined by the followers' reactionsat the lower level. Constraint (2) shows the statedynamics as the changes in the inventory level of eachproduction site i 2 LS . Constraint (3) determines theinitial level of the inventory at the beginning of theplanning horizon t = t0. Constraint (4) ensures posi-tive inventory and prohibits shortage or backlog. The

demand is de�ned as a linear price-sensitive function byConstraint (5). Constraint (6) guarantees the capacityof each production site. Finally, boundaries on decisionvariables are de�ned by Constraint (7).

3.2. The carriers' objective functional,dynamics, and constraints

The carriers in the lower level are in oligopolisticCournot-Nash competition. They simultaneously setthe price of transportation services and make therouting decisions for di�erent origins and destinationson the network. This is with respect to the decisionsmade by the shipper with the goal of maximizingtheir own net pro�ts over a �nite planning horizon,t 2 [t0; tf ]. The routing decisions made by the carrierscollectively determine the overall freight ow on thenetwork. Each carrier tries to achieve more demandfrom the transactions between the shipper and hiscustomers at di�erent demand markets. We de�nethe transportation demand for each carrier as a linearfunction in which the potential demand is in uencedby the carrier's own price as well as the competitors'price given as:

vcij(t) = �Dj(t)� c'cij(t) +X

g2C�fcg�c;g:'gij(t): (8)

No shortage or backlog is allowed, thus the wholedemand of each market must be shipped by the setof carriers. The sub-model of each carrier is de�ned asfollows:

max Jc';v;h

=

tfZt0

e��t:(Xi2Ls

Xj2L

'cij(t):vcij(t)

�Xa2A

kca:fca(t)

)dt: (9)

Subject to:

vcij(t) = �Dj(t)� c'cij(t) +X

g2C�fcg�c;g:'gij(t)

8i 2 Ls; 8j 2 L; (10)

f ca(t) =Xi2Ls

Xj2L

Xr2Rij

�a;r:hcr;ij(t) 8a 2 A; (11)

Xr2Rij

hcr;ij(t) = vcij(t) 8i 2 Ls; 8j 2 L; (12)

Xc2C

Xi2Ls

vcij(t) = dj(t) 8i 2 Ns; 8j 2 L; (13)

'cij(t) � 0; vcij(t) � 0 8i 2 Ls; 8j 2 L; (14)

f ca(t) � 0 8a 2 A; (15)


hcr;ij(t) � 0 8i 2 Ls; 8j 2 L; 8r 2 Rij : (16)

Eq. (9) represents the objective functional of eachcarrier as its revenue minus the transportation costs.Constraint (10) shows the transportation demand func-tion. Constraint (11) de�nes the ow on di�erent arcsof the network. Constraint (12) states that the sum of ows on the paths connecting origin i to destinationj must be equal to the total transportation demandbetween i and j. Constraint (13) ensures that thedemand of each market is collectively shipped by thecarriers. Finally, Constraints (14) to (16) de�ne theboundaries on the decision variables.

4. The Stackelberg-Nash equilibrium

In this section, we investigate the equilibrium condi-tions for the proposed di�erential Stackelberg gamebetween the shipper and the carriers; we explore theexistence and uniqueness of properties of the equilib-rium.

4.1. The Cournot-Nash price equilibriumcondition at the carriers' level

The game among competing carriers at the lower levelis a Generalized Nash Equilibrium Problem (GNEP)with joint constraint, in which both the objectivefunction and the constraint set for each player dependon the strategies taken by rival players [35,36]. To �ndthe best responses, we �rst take the time discretizationapproach for the optimal control problem of eachcarrier. The optimal control models (9)-(16) becometime discretized by assigning a discrete instant of timetz = t0 +z�t, where �t is the length of each time step.The number of time discretization can be calculatedby M = tf�t0

�t and tM = tf . In this way, we convertthe optimal control problem to a �nite dimensionalmathematical program for which the optimal solutioncan be achieved through Karush-Kuhn-Tucker (KKT)optimality conditions. When the optimal control modelis a convex model, the discretized solution will be agood approximation of the optimal controls.

Proposition 1. (Convexity) For every carrier (c 2C), the objective function is concave and the feasibleset is convex.

Proof. See Appendix A.A necessary and su�cient condition for a given

solution to be an optimal one for each carrier's subproblem is that a suitable constraint quali�cation holdsand there exists Lagrangian multiplier vectors (�tj , �

c;tij )

and (�c;tzij ; $c;tzr;ij ) � ~0 which satisfy the following system

of equations:

Lc =MXz=0

e��tz(Xi2Ls

Xj2L

'c;tzij

:

�Dtzj � c'c;tzij +

Xg2C�fcg

�c;g:'g;tzij

!�Xa2A

kca:Xi2Ls

Xj2L

Xr2Rij

�a;r:hc;tzr;ij +Xi2Ls

Xj2L

�c;tzij

:

0@ �Dtzj � c'c;tzij +

Xg2C�fcg

�c;g:'g;tzij �Xr2Rij

hc;tzr;ij

1A+Xj2L

�tzj :

�Dtzj ��j :ptzj �

Xi2Ls

Xc2C

�Dtzj � c'c;tzij

+X

g2C�fcg�c;g:'g;tzij

!!+Xi2Ls

Xj2L

�c;tzij :'c;tzij

+Xi2Ls

Xj2L

Xr2Rij

$c;tzr;ij :h

c;tzr;ij

):

(17)

@Lc@'c;tzij

= e��tz :(

�Dtzj � 2 c'c;tzij +

Xg2C�fcg

�c;g:'g;tzij

� c:�c;tzij + �tzj :

0@ c� Xg2C�fcg

�g;c

1A+ �c;tzij

)=0;

(18)

@Lc@hc;tzr;ij

= e��tz :(�Xa2A

�a;r:kca+$c;tzr;ij ��c;tzij

)= 0;

(19)


Xg2C�fcg


hc;tzr;ij = 0;(20)

�Dtzj � �j :ptzj �

Xi2Ls

Xc2C


+X


!= 0; (21)

�c;tzij :'c;tzij = 0; (22)

$c;tzr;ij :h

c;tzr;ij = 0; (23)

where Eq. (17) de�nes the Lagrange function, Eqs. (18)and (19) show stationary conditions, and Eqs. (20)to (23) are complementary slackness equations.


Proposition 2. The Slater constraint quali�cationholds for every carrier's sub problem.

Proof. See Appendix B.When the GNEP satis�es the convexity as-

sumption (proposition 1), the vector ( �'c;tzij ; �hc;tzr;ij ; ��tj ;��c;tij ; ��c;tzij ; �$c;tz

r;ij ) which solves the system of equationsobtained by concatenating the KKT optimality con-ditions of all carriers is an equilibrium point of theGNEP [35,37]. According to Proposition 1, X(�c) isde�ned as the feasible set of each carrier depending onprice strategies of the rival carriers, which is convexfor every carrier. Therefore, if X(�c) is closed, thenthere exists at least one Nash equilibrium point for theCournot GNEP among carriers.

Proposition 3. (Existence property) The GNEP ofthe carriers has an equilibrium point.

Proof. See Appendix C.

4.2. The equivalent single level optimizationproblem

A solution for the lower level Nash game can beexpressed implicitly as a function of the upper levelproblem's controls. If the uniqueness of solution for thelower level problem is guaranteed, we can transformthe bi-level di�erential problem into an equivalentsingle level dynamic model by adding the optimalityconditions of the lower level Nash game to the shipper'smodel constraints set.

Proposition 4. (Uniqueness of property) The equi-librium point of GNEP among rival carriers is unique.

Proof. See Appendix D.As it is shown in Propositions 3 and 4, there exists

a unique equilibrium point for the followers' GNEP atthe lower level. Therefore, we can convert the bi-leveldynamic Stackelberg-Nash game between shipper andcarriers to the following time discretized single leveldynamic optimization problem:

max Jsp;q;';h

MXz=0

e��tz :(Xj2L

ptzj :� �Dtz

j � �j :ptzj �� Xi2LS

tzi :Itzi �

Xi2LS

�tzi :qtzi �

Xi2LS

Xj2L

Xc2C

'c;tzij

:


Xg2C�fcg

�c;g:'g;tzij

!): (24)

Subject to:

Itzi =Itz�1i + qtzi �

Xj2L

Xc2C

( �Dtzj � c'c;tzij

+X

g2C�fcg�c;g:'g;tzij )

8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (25)

It0i = Ii 8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (26)

�Dtzj ��j :ptzj � 0 8j 2 L; 8z 2 f0; 1; � � � ;Mg; (27)

0 � qtzi � Capi 8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (28)

ptzj � 0 8j 2 L; 8z 2 f0; 1; � � � ;Mg; (29)

Itzi � 0 8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (30)


Xg2C�fcg

�c;g:'g;tzij � c:�c;tzij

+ �tzj :

c � X

g2C�fcg�g;c

!+ �c;tzij = 0

8i2Ls; 8j2L; 8c2C; 8z2f0; 1; � � � ;Mg;(31)

$c;tzr;ij �

Xa2A

�a;r:kca � �c;tzij = 0

8i 2 Ls; 8j 2 L; 8r 2 Rij ; 8c 2 C;8z 2 f0; 1; � � � ;Mg; (32)


Xg2C�fcg


hc;tzr;ij = 0

8i 2 Ls; 8j 2 L; 8c 2 C;8z 2 f0; 1; � � � ;Mg; (33)


Xi2Ls

Xc2C


+X


!= 0

8j 2 L; 8c 2 C; 8z 2 f0; 1; � � � ;Mg; (34)

�c;tzij :'c;tzij = 0 8i 2 Ls; 8j 2 L; 8c 2 C;8z 2 f0; 1; � � � ;Mg; (35)

$c;tzr;ij :h

c;tzr;ij = 0 8i 2 Ls; 8j 2 L; 8c 2 C;

8r2Rij ; 8z2f0; 1; � � � ;Mg; (36)


�c;tzij �0; 'c;tzij �0; $c;tzr;ij �0; hc;tzr;ij �0: (37)

Note that the state variables Itzi in the discretizedmodel are intermediate variables and can be easilyreplaced by the control variables qtzi and 'c;tzij asfollows:

Itzi =It0i +zX

n=1

(qtni �

Xj2L

Xc2C

�Dtnj � c'c;tnij

+X

g2C�fcg�c;g:'g;tnij

!): (38)

Since setting the transportation price equal to zeromakes the objective function of the carrier negative, wecan be sure that at the equilibrium point of the Nashgame, all the carriers choose non-zero transportationprices and 'c;tzij � 0. Hence, we can �x the Lagrangemultiplier, �c;tzij = 0, and remove Constraint (34).According to Propositions 3 and 4, the GNEP of thecarriers at the lower level has a unique equilibriumsolution for any given (�ptzj ; �qtzj ) from the shipper, andalso the shipper's objective function is concave and thefeasible set is convex regarding the shipper's decisionvariables (see Appendix E for the proof). Then, itis concluded that the Stackelberg-Nash problem hasa unique equilibrium point. This equilibrium can beobtained by solving the model (24)-(37).

4.3. Penalty function methodTo handle nonlinear constraints, we employ a penaltyfunction method which appends the nonlinear con-straints to the objective function. It assigns a penaltywhen a given nonlinear constraint is violated andapproximates the solution of the original problem [38].So in this case, a penalty vector:

� = (�c1;tzr;ij ; �c2;tzr;ij ; � � � ; �cNr;ij ; tz);

is de�ned, where �c;tzr;ij 2 R+ is a large number. When�c;tzr;ij approaches in�nity, the approximation becomesincreasingly accurate. Then, we can reformulate themodel (24) to (36) as a nonlinear programming modelwith linear constraints as follows:

max Jsp;q;';h

MXz=0

e��tz :(Xj2L

ptzj :� �Dtz

j � �j :ptzj ��Xi2LS

tzi :

It0i +

zXn=1

(qtni �

Xj2L

Xc2C


+X


!)!� Xi2LS

�tzi :qtzi

�Xi2LS

Xj2L

Xc2C

'c;tzij :


Xg2C�fcg

�c;g:'g;tzij

!

�Xc2C

Xi2LS

Xj2L

Xr2Rij

�c;tzr;ij :($c;tzr;ij :h

c;tzr;ij )

): (39)

Subject to:

�Dtzj ��j :ptzj � 0 8j 2 L; 8z 2 f0; 1; � � � ;Mg; (40)

0 � qtzi � Capi 8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (41)

ptzj � 0 8j 2 L; 8z 2 f0; 1; � � � ;Mg; (42)

It0i +zX

n=1

(qtni �

Xj2L

Xc2C


+X


!)� 0

8i 2 Ls; 8z 2 f0; 1; � � � ;Mg; (43)


Xg2C�fcg

�c;g:'g;tzij � c:�c;tzij

+ �tzj :

0@ c � Xg2C�fcg

�g;c

1A+ �c;tzij = 0

8i2Ls; 8j2L; 8c2C; 8z2f0; 1; � � � ;Mg; (44)

$c;tzr;ij �

Xa2A

�a;r:kca��c;tzij =0 8i2Ls; 8j2L;

8r 2 Rij ; 8c 2 C; 8z 2 f0; 1; � � � ;Mg; (45)


Xg2C�fcg


hc;tzr;ij = 0

8i2Ls; 8j2L; 8c2C; 8z2f0; 1; � � � ;Mg; (46)


Xi2Ls

Xc2C


+X


!= 0

8j 2 L; 8c 2 C; 8z 2 f0; 1; � � � ;Mg; (47)

'c;tzij � 0; $c;tzr;ij � 0; hc;tzr;ij � 0: (48)


We employ an initial penalty parameter vector anditeratively increase the penalty parameters until thealgorithm converges. In each iteration, we apply anoptimization technique to solve the model (39) to (48)by starting from the optimum solution of the previousiteration. The steps of the penalty function methodare listed below:

- Step 1: Set the iteration counter K = 0, choosean initial solution S0 = (pt

0zj ; q

t0zi ; '

c;t0zij ; hc;t

0z

r;ij ; $c;t0zr;ij )

which is the optimal solution of the model (24)to (37), ignore Constraint (36). Also, set the initialvalues for penalty parameters vector, �0;

- Step 2: Solve the model (39) to (48) and set SK+1 =(ptz�j ; qtz�i ; 'c;tz�ij ; hc;tz�r;ij ; $

c;tz�r;ij );

- Step 3. If the penalty function is equal to or lessthan a prede�ned " > 0, i.e.:Xc2C

Xi2LS

Xj2L

Xr2Rij

�c;tKz

r;ij :($c;tzK+1r;ij :hc;tzK+1

r;ij ) � ";

then stop, the optimal solution is found, otherwiseset:

�c;tzK+1r;ij =�:�c;t

Kz

r;ij ; 8i2Ls; j2L; r2Rij ;c 2 C; z 2 f0; 1; � � � ;Mg;

where � > 1. Also, set K = K + 1 and go to Step 2.

To solve the mathematical model in Step 2, weemploy a multi-start global optimization algorithm inGAMS software using the MINOS solver. Bringingtogether all we discussed, the penalty function algo-rithm will converge to the equilibrium point of theStackelberg-Nash game among a dominant shipper andmultiple competing carriers.

5. Numerical results and sensitivity analysis

In this section, we present several examples to demon-strate the applicability of the proposed di�erentialStackelberg-Nash game model and examine the e�ec-tiveness of the proposed solution approach. Further-more, we conduct a comprehensive sensitivity analysison the main parameters, which illustrates some im-portant features of the model and highlights severalmanagerial aspects.

In order to solve each example, we transformthe bi-level di�erential Stackelberg-Nash game intoa single level optimization model by including theequilibrium conditions of the lower level Nash game inthe shipper's problem as a set of constraints. A �nitedimensional time discretization and a penalty functionmethod are then applied to approximate the modelas a mathematical program with linear constraints.Finally, the penalty function algorithm is coded inGAMS software solved using the MINOS solver as amulti start global optimization tool.

Example 1. Consider a network consisting of 8 nodesand 13 bidirectional arcs. There is one dominantshipper who possesses two production sites with limitedcapacities over the nodes 3 and 8. Also, there are twofreight carriers who serve transportation demands fromthe shipper's nodes to customers' nodes on the network.The carriers compete with each other by setting theirpricing decisions simultaneously in an oligopolisticmarket. Every node on the network is assumed as aparticular demand market for the commodity producedby the shipper. Figure 2 illustrates the schematicnetwork of the example.

The planning time interval is considered tobe [0,10]. As can be seen in Figure 2, the origin anddestination of a commodity ow may be connectedthrough di�erent routes. The relationships betweenarcs and routes for O-D pairs are summarized inTable 1. The transportation costs for carrier c ondi�erent network arcs are generated randomly from theuniform distribution kca � U(1; 10). Table 2 shows thevalues of the other parameters used in this example.

In the Stackelberg-Nash equilibrium point, theNet Present Value (NPV) of bene�t for the shipper isJ�s = 67655161:32, while the carriers c1 and c2 gain netpresent values of bene�t equal to J�c1 = 14065980:17and J�c2 = 7956465:29, respectively. Figure 3 shows theinventory and production trajectories for the shipperat di�erent production sites. As it can be seen, thetrajectories start from the initial inventory levels Iiat time t0, then the shipper gradually reduces his

Figure 2. Transportation network structure ofExample 1.

Figure 3. Inventory and production trajectories (I�i (t)and q�i (t)) of the shipper at production nodes.


Table 1. Route-arc relationship for O-D pairs.

O-D pairs Rijs' arc sequences O-D pairs Rijs' arc sequences

(3,1) f3.1g (8,1) f8.6,6.3,3.1g,f8.5,5.2,2.1g,f8.7,7.4,4.1g(3,2) f3.2g (8,2) f8.5,5.2g(3,3) f3.3g (8,3) f8.6,6.3g(3,4) f3.4g (8,4) f8.7,7.4g(3,5) f3.2,2.5g,f3.6,6.5g (8,5) f8.5g(3.6) f3.6g (8.6) f8.6g(3.7) f3.4,4.7g,f3.6,6.7g (8.7) f8.7g

inventory levels to keep the inventory holding cost lowand meet the terminal inventory condition Itfi = 0.In contrast, the production quantity levels at di�erentproduction sites start from zero at time t0 to allowthe shipper to sell o� his existing inventories, and thenincrease to meet the quantity of commodity demandedby di�erent markets up to prede�ned production capac-ities Capi. The capacity constraints force the shipperto produce in advance for future demand, when thequantity demanded in future will be more than theavailable production capacity.

Figures 4 illustrates the price trajectories for thecommodity sold to di�erent demand nodes and therealized demand received by the shipper from eachmarket. As expected, the demand market at node 8received higher prices from the shipper, because itsdemand is less price sensitive regarding the parameter�8 = 1, and there is no competitor for the shipper

Table 2. Values of the parameters.Parameter Value Parameter Value

�j � U(1; 5) � 0.01�Dtj � U(1000; 5000) c1 15�ti � U(8; 30) c2 17 ti 0:2 � �ti �c1;c2 0.4

Capi(i=3;8) f6000,5500g �c2;c1 0.6�Ii(i=3;8) f9000,9000g � 1

at the market; however, the quantity demanded fromnode 8 is about the demand from other nodes.

The transportation price trajectories for di�erentOrigin-Destinations (ODs) and the allocations of de-mand to freight carriers in the equilibrium point ofthe GNEP are illustrated in response to the shipper'sactions in Figures 5 and 6. To enhance clarity of these�gures, the transportation demands originated fromnode 8 are ignored.

As can be observed in Figures 5 and 6, carrierc1 o�ers higher prices and captures more demandfrom transactions between the shipper and the demandmarkets. The reason is the lower elasticity to ownprice as well as the rival's price that carrier c1 has.The lower price elasticity factor typically denotes thehigher reputation for the carrier in the market, whichenables him to set higher prices, get more transporta-tion demand, and gain more bene�t at the end ofthe planning horizon. In the following, we presentseveral examples to analyze the sensitivity to the mainparameters involved in the proposed model.

In order to investigate the sensitivity of theStackelberg-Nash equilibrium to the price elasticityparameter of the demand nodes, we change parameter�j at the interval [0:6�j ; 1:5�j ] for Example 1, whileall the other parameters are �xed.

As it can be seen in Figure 7, when �j increases,the quantity demanded by the shipper decreases; the

Figure 4. Price and demand trajectories (p�j (t) and d�j (t)) of the shipper's commodity for di�erent markets.


Figure 5. Transportation price trajectories, 'c�ij (t), between di�erent O-D pairs for the two carriers.

Figure 6. Transportation ow trajectories, vc�ij (t), on di�erent O-D pairs for the two carriers.

Figure 7. The impact of �j on the players' bene�t in the equilibrium point.

shipper chooses lower rates to keep his demand level,and his bene�t will decline. On the other side, bothcarriers gain degraded demand according to the smallermarket of the shipper. We have enhanced �j up to 50%and observed that carriers c1 and c2 lose up to 8% and16% of their bene�ts, respectively. The reason is thelower pricing power which carrier c2 possesses in thefreight transportation market regarding parameter c2 .

To investigate the sensitivity of the Stackelberg-Nash equilibrium to the potential demand factor, wechange parameter �Dt

j at the interval [0:6 �Dtj ; 1:5 �Dt

j ] inExample 1, while all the other parameters are �xed.

As it is observed in Figure 8, the shipper chooseshigher prices when he faces with a larger potentialmarket demand and he gains more bene�t at the endof the planning horizon. Since the carriers' marketis de�ned by the shipper's transactions, the realizeddemand for both carriers' increases for a larger �Dt

j .Here, we examine whether any changes in each

carrier's own price elasticity can in uence the equilib-rium point of the game or not. We �x c1 at its value inExample 1 and alter c2 at the interval [0:6 c2 ; 1:5 c2 ].A growth in parameter c2 implies a reduction in thepricing power of carrier c2. Hence, the lower pricing


Figure 8. The impact of �Dtj on the players' bene�t in the equilibrium point.

Figure 9. The impact of c on the players' bene�t in the equilibrium point.

power carrier c2 has, the lower prices he may chooseand the less bene�t he will gain. On the other side,by decreasing the pricing power of c2, c1 becomesmuch powerful in the market, captures more demandfrom the shipper's transactions, and gets more bene�t.When c2 approaches c1 , carriers achieve almost thesame demand shares (Figure 9).

We examine whether any changes in the rivals'price elasticity can in uence the equilibrium point ofthe game. We �x �c2;c1 at its value in Example 1and change �c1;c2 at the interval [1=5�c1;c2 ; 5�c1;c2 ].Regarding the demand function de�ned in Eq. (8), alarger �c1;c2 implies that a small increment in the pricesset by carrier c2 leads to a signi�cant enhancement inthe transportation service demand, and consequentlyto a higher bene�t for carrier c1. Since the carriers

compete for a constant demand de�ned by the shipper;the demand for carrier c1 will decrease as the demandfor carrier c2 increases. Therefore, carrier c2 is encour-aged to reduce his prices for keeping his demand share,therefore, he will gain lower bene�t (Figure 10).

Example 2. In order to investigate the impact ofthe competitive environment on the equilibrium pointof the game, here we add a new rival carrier servicingon the network of Example 1. Since the new carrierhas lower reputation in the market, we assume higherprice elasticity for him:

c3 = 19 and �c;g =

0@ 0 0:4 0:40:6 0 0:60:7 0:7 0

1A :

Figure 10. The impact of �c1;c2 on the players' bene�t in the equilibrium point.


In the Stackelberg-Nash equilibrium point of Exam-ple 2, the shipper achieves J�s = 66865521:04, whilethe carriers c1, c2, and c3 gain J�c1 = 13718644:67,J�c2 = 7486998:65, and J�c3 = 1523108:71, respectively.As it is expected, when a new carrier is added to themarket, a certain demand share will be allocated tohim regarding his prices. Thus, the previous carriersmay lose some demand and their bene�t will decreaseconsequently.

6. Concluding remarks and future work

This paper addressed a shipper-carrier problem inwhich a dominant shipper aims at selling a homoge-neous commodity to several spatially separated mar-kets with price sensitive demand and pursues the goalof maximizing his bene�t over a �nite planning horizon.The shipper requires transportation services to delivergoods to his customers, and multiple oligopolisticcarriers compete to capture these service demands ona geographic network. The freight carriers dynamicallyset their prices for origin-destination services and maketheir routing decisions to gain more transportationdemand from the transactions between the shipperand his customers. All the players' strategies canbe changed dynamically regarding the time; thereexists continuous-time dynamic equilibrium for thefreight network game while the time value of moneyis considered.

The problem has been formulated as a di�erentialStackelberg-Nash game to �nd the equilibrium pricetrajectories and routing decisions over the planninghorizon. A �nite dimensional discretization approachhas been applied to append the equilibrium conditionsof the carriers to the shipper's model and to trans-form the bi-level model to an equivalent single levelmathematical program; a penalty function algorithmhas been proposed to solve the resulting model. Somenumerical studies have been done to show how themathematical model and the proposed solution ap-proach can approximate the equilibrium trajectories ofthe di�erential Stackelberg-Nash game of the shipper-carrier problem. Finally, a comprehensive sensitivityanalysis has been conducted on the critical parametersand some managerial highlights have been discussed.For future research, one can consider nonlinear demandfunctions and the uncertainty of the demand functionparameters. Furthermore, investigating other decisionsof the carriers such as mode choice or eet assignmentare worthwhile.

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Appendix A

Time discretizing of each carrier's optimal control prob-lem and substituting Eqs. (10) and (11) into Eq. (9)give that:

Jc =MXz=0

e��tz :(Xi2Ls

Xj2L

'c;tzij :


+X


!�Xa2A

kca

:Xi2Ls

Xj2L

Xr2Rij

�a;r:hc;tzr;ij

): (A.1)


If a function f is convex, then �f is concave [39].The Gradient of �Jc('c;tzij ; hc;tzr;ij ) can be calculated asfollows:

r��Jc �'c;tzij ; hc;tzr;ij��

=

264�@Jc(';h)@'

�@Jc(';h)@h

375

=

266664�


Pg2C�fcg

�c;g:'g;tzij

!Pa2A

�a;r:kca

377775 :(A.2)

The Hessian of �Jc('c;tzij ; hc;tzr;ij ) can be given as:

H��Jc �'c;tzij ; hc;tzr;ij

��=

264�@2Jc(';h)@'2 �@2Jc(';h)

@'@h

�@2Jc(';h)@h@'

�@2Jc(';h)@h2

375=�2 c 00 0

�:

(A.3)

Then, we have:

('; h):hH��Jc('; h)

�i:('; h)T

= ('; h):�2 c 00 0

�:('; h)T = 2 c'2 � 0: (A.4)

Thus, the Hessian matrix, H(�Jc('c;tzij ; hc;tzr;ij )) , ispositive-semi-de�nite, �Jc is convex, and consequentlyJc is concave. Since all the constraints are linear, thefeasible set of each carrier is convex and the proof iscompleted.�

Appendix B

If the feasible set is convex and there exists a feasiblepoint such that every inequality constraint is satis-�ed strictly, then the Slater's constraint quali�cationholds, [39].

As it is shown in Proposition 1, the feasible setfor every carrier is convex. Moreover, we can easilyde�ne a price vector �'tzij = ( �'c1;tzij ; �'c2;tzij ; � � � ; �'cN ;tzij ) =("; "; � � � ; "), where " is a small positive number, thenthe transportation demand will be divided betweenrival carriers and vcij(t) > 0, 8c 2 C. In addition, eachcarrier is able to decompose his own origin-destinationtransportation demand into the possible paths, then wehave hcr;ij(t) > 0, 8r 2 Rij and consequently f ca(t) > 0,8a 2 A. Therefore, the interior feasible set of everycarrier is nonempty with regard to the rival carriers'action sets, and the Slater's constraint quali�cationholds. So, the proof is completed.�

Appendix C

We can easily de�ne conceptual upper bounds for thedecision variables. According to Eq. (8), if 'c;tij goesto in�nity, the transportation demand for carrier c willbe negative which is not allowed; thus, the variableis bounded from above and we have 0 � 'c;tij � �'.In addition, according to Eqs. (12) and (13), it isconcluded that hc;tr;ij and vc;tij are bounded from aboveand we have 0 � hc;tr;ij � vc;tij and 0 � vc;tij � dj . Since:

f c;ta =Xi2Ls

Xj2L

Xr2Rij

�a;r:hc;tr;ij ;

then f c;ta is also a bounded variable. Thereby, thefeasible set of every carrier is closed and the proof iscompleted.�

Appendix D

De�ning a functional vector F (x) on X, such that:

F (x) :=�rxc Jc(x)

�c=Nc=1

;

and X is the joint feasible set of all players, if F (x)is monotone, then the GNEP has a unique equilibriumpoint which can be obtained by concatenating KKTsystem of equations for all the players [35].

According to Facchinei and Kanzow [35], for amaximization problem, F (x) will be monotone if forany two x; y 2 X, it holds that:

h(F (x)� F (y)); (x� y)i � 0: (D.1)

For the GNEP of the carriers, F (x) can be formed as:

F (x) =

0BBBBBBB@�Dtzj � 2 c1'

c1ij ; tz +

�c1 ;gPg2C�fc1g

:'g;tzij

...

�Dtzj � 2 cN'

cN ;tzij +

�cN ;gPg2C�fcNg

:'g;tzij

� Pa2A

�a;r:kc1a...

� Pa2A

�a;r:kcNa

1CCCA : (D.2)

Let S0 = ('c;t0z

ij ; hc;t0z

r;ij ) and S00 = ('c;t00z

ij ; hc;t00z

r;ij ) be twosolutions in X, then for Inequality (D.1), we have:


h(F (S0)� F (S00)); (S0 � S00)i

=MXz=0

e��tz :Xc2C

Xi2LS

Xj2L

�Dtzj � 2 c'

c;t0zij

+X


!�

�Dtzj � 2 c'

c;t00zij

+X


!!:('c;t

0z

ij � 'c;t00z

ij )

� Xi2LS

Xj2L

Xr2Rij

Xa2A

�a;r:kca �Xa2A

�a;r:kca

!

:�hc;t

0z

r;ij � hc;t00z

r;ij

�=

MXz=0

e��tz :Xc2C

Xi2LS

Xj2L

(�2 c)

:�'c;t

0z

ij � 'c;t00z

ij

�2 � 0: (D.3)

Since c > 0, we conclude that Eq. (D.3) is negative.Therefore, the GNEP has a unique equilibrium point,and the proof is completed.�

Appendix E

Proposition E.1. The shipper's objective functionis concave and his feasible set is convex regarding hisown decision variables.

Considering the objective function of the shipper'smodel in Eq. (1), we can immediately remove the lastterm from the objective function, because the decisionvariables 'c;tzij and vc;tzij are obtained uniquely from thelower level's KKT system of Eqs. (43) to (47). Thesecond and the third terms are linear; consequently,they are concave regarding qtzi . For the �rst term,the second order optimality condition regarding ptzj isde�ned by:

�@2Js(p; q; '; h)@p2

j= �2�j < 0: (E.1)

Since �j > 0, Eq. (E.1) is negative and the �rst termis concave. Regarding the fact that the sum of concave

functions is a concave function, we can conclude thatthe shipper's objective is concave with respect to hisdecision variables. In addition, since all the constraintsof the model are linear, the feasible set is convexregarding the shipper's variables. Then, the proof iscompleted.�

Biographies

Marzieh Mozafari is a PhD candidate in IndustrialEngineering at Amirkabir University of Technology.She received her MSc degree in Industrial Engineeringfrom Amirkabir University of Technology and BScdegree in Industrial Engineering from K. N. ToosiUniversity of Technology. Her research interests aredynamic pricing and revenue management, game the-ory, freight transportation, and supply chain optimiza-tion.

Behrooz Karimi is a Professor of Faculty of Indus-trial Engineering and Management Systems at Amirk-abir University of Technology in Iran. He received hisPhD degree in Industrial Engineering from AmirkabirUniversity of Technology and his MSc degree in Indus-trial Engineering from Iran University of Science andTechnology. He has also received his BSc in IndustrialEngineering from Amirkabir University of Technology.His research interest areas include supply chain andlogistics management, inventory control & productionplanning, and meta-heuristics algorithms. He haspublished books, several papers, and book chapters inreputable journals. He serves as the editor/editorialboard member for a number of journals.

Masoud Mahootchi is an Assistant Professor offaculty of Industrial Engineering and Management Sys-tems at Amirkabir University of Technology in Iran. Hereceived his PhD degree in Systems Design Engineeringfrom University of Waterloo in Canada. He received hisMSc and BSc degrees in Industrial Engineering bothfrom Amirkabir University of Technology, in Iran. Hisresearch interest areas include stochastic programming,storage management, reinforcement learning, and port-folio optimization. He has published several papers andbook chapters in reputable journals.

a di erential stackelberg game for pricing on a freight...

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