A Dynamic Stackelberg Game of Supply Chain for a Corporate...

Research ArticleA Dynamic Stackelberg Game of Supply Chain fora Corporate Social Responsibility

Massimiliano Ferrara,1,2 Mehrnoosh Khademi,2 Mehdi Salimi,3 and Somayeh Sharifi2

1 Invernizzi Center for Research on Innovation, Organization, Strategy and Entrepreneurship (ICRIOS),Bocconi University, Milan, Italy2MEDAlics-Research Center, Universita per Stranieri Dante Alighieri, Reggio Calabria, Italy3Department of Mathematics, Tuyserkan Branch, Islamic Azad University, Tuyserkan, Iran

Correspondence should be addressed to Massimiliano Ferrara; [email protected]

Received 21 September 2016; Revised 21 December 2016; Accepted 27 December 2016; Published 2 February 2017

Academic Editor: Filippo Cacace

Copyright © 2017 Massimiliano Ferrara et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In this paper, we establish a dynamic game to allocate CSR (Corporate Social Responsibility) to the members of a supply chain.We propose a model of a supply chain in a decentralized state which includes a supplier and a manufacturer. For analyzing supplychain performance in decentralized state and the relationships between themembers of the supply chain, we formulate amodel thatcrosses through multiperiods with the help of a dynamic discrete Stackelberg game which is made under two different informationstructures. We obtain an equilibrium point at which both the profits of members and the level of CSR taken up by supply chainsare maximized.

1. Introduction

In recent years, companies and firms have been showingan ongoing interest in favor of CSR. This is mainly becauseof increasing consumer awareness of several CSR issues,for example, the environment, human rights, and safety.In addition, the firms are also forced to accept CSR dueto government policies and regulations. Recently CSR hasgained recognition and importance as field of research field[1, 2]. However, the research field still lacks a consistentdefinition of CSR and this has been the center of discussionfor several decades. Dahlsrud [3] presented an overview ofdifferent definitions of CSR and summarized the numberof dimensions included in each definition. There is a pos-itive correlation between CSR and profit [4, 5]. Moreover,CSR is an effective tool for supply chain management, forcoordination, purchasing, manufacturing, distribution, andmarketing functions [6]. According to previous studies, thelong-term investment on CSR is beneficial for a supply chain.Furthermore, a sustainable supply chain requires consider-ation of the social aspects of the business [7]. Carter et al.[8] established an effective approach and demonstrated that

environmental purchasing is significantly related to both netincome and cost of goods sold. Carter and Jennings [9] alsopointed towards the importance of CSR in the supply chain,in particular the role played by the purchasing managers insocially responsible activities and the effect of these activitieson the supply chain. Sethi [10] introduced a taxonomy inwhich a firm’s social activities include social obligations aswell as more voluntary social responsibility. And Carroll [11,12] developed a framework for CSR that consists of economic,legal, and ethical responsibilities.

A supply chain usually includes suppliers, manufacturers,distributors, wholesalers, and retailers. The members of asupply chain take their decisions based on maximizing theirindividual net benefits. When they need to accept CSR due togovernment’s regulations and policies or consumer’s concern,additional costs are forced on them. However, many firmsadopt CSR for enhancing their reputation due to publicconcerns over social issues or other benefits, such as increasedsales and tax returns and long-term profits in many respects,including stable suppliers, low cost delivery, and extendedmarket demand. These benefits are proved by some well-known case studies of large international companies, such as

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 8656174, 8 pageshttps://doi.org/10.1155/2017/8656174

https://doi.org/10.1155/2017/8656174

2 Discrete Dynamics in Nature and Society

Nike, Gap, H&M,Wal-Mart, andMattel [13].The situation inwhich the members of supply chain tend to gain individualbenefits (meanwhile they have to bear a level of CSR) isa conflict situation and this leads to an equilibrium status.Game theory is one of the most effective tools to deal withthis kind of management problems. A growing number ofresearch papers use game theoretical applications in supplychain management [14–17]. Cachon and Zipkin [18] discussNash equilibrium in noncooperative cases in a supply chainwith one supplier and multiple retailers. Hennet and Arda[19] presented a paper to evaluate the efficiency of differenttypes of contracts between the industrial partners of a supplychain. They applied game theory methods for decisionalpurposes. Tian et al. [20] presented a dynamics systemmodelbased on evolutionary game theory for green supply chainmanagement. Gadimi et al. [21] used cooperative games insupply chain management in order to find fair allocationschemes for dividing the total profit of grand coalition amongthe members.

In 1934, Stackelberg introduced a concept of a hierarchicalsolution which is a simple dynamic game [22]. Stackelberggame involves players with asymmetric roles which are calledleader and follower. The leader chooses a strategy first andthe follower, with the knowledge of leader’s strategy, choosesa policy. The leader anticipates follower’s optimal responseand chooses the best possible point. There are differenttypes of dynamic Stackelberg solutions depending on theinformation’s structure, the open-loop, the feedback, and theglobal Stackelberg solution (that is a closed loop structureof information) structures. In an open-loop strategy, playerschoose their decisions at time 𝑡, with information of thestate at time zero. In contrast, in a closed loop informationstructure, leader has perfect knowledge of all the past andcurrent values and, in a feedback information structure,players use their knowledge of the current state at time tin order to formulate their decisions at time 𝑡 [23]. In thispaper, we consider feedback and closed loop structures. Adiscrete time version of the dynamic differential game hasbeen studied. The optimal control theory is the standard toolfor analyzing the differential game theory [24]. We formulateamodel and study the behavior for decentralized supply chainnetworks under CSR conditions with one leader and twofollowers. The Stackelberg game model is recommended andapplied here to find an equilibrium point at which the profitof the members of the supply chain is maximized and thelevel of CSR is adopted in the supply chain. We develop aStackelberg game by selecting the supplier as the leader andthe manufacturer as the follower. Using this approach, thesupplier, as a leader, can know the optimal reaction of hisfollower and utilizes such processes to maximize his ownprofit. The manufacturer, as a follower, tries to maximizehis profit by considering all the conditions. We propose aHamiltonian matrix to solve the optimal control problem inobtaining the equilibrium in this game.

The paper is organized as follows: Section 2 is devoted tothe mathematical model. Management policies and numer-ical examples are illustrated in Section 3 and Section 4contains a short conclusion.

2. Mathematical Model

Game theory has often been applied in diverse areas suchas business, economics, and management to solve problemsinvolving conflict and cooperation and it analyzes problemswhich are multicriteria and multidecision-makers. Supplychain management can be considered as a set of manage-ment processes. Consequently, game theory is an effectivemethod for supply chain management. Competition amongmembers in a supply chain network is one of the significanttopics which are emphasized in supply chain management.Furthermore, members in supply chain networks either arepressured to accept CRS by governments, organizations, andconsumer or have to bear at least some CSR under policiesand regulations. However, naturally, members in a supplychain network want to maximize their individual net profits;meanwhile they have to take the level of social responsibilityin entire supply chain network. These conditions providea challenge. In order to deal with this situation, we useStackelberg game model which is often applied to studydynamic problems. As it was mentioned, in a Stackelbergmodel, leader chooses a strategy first and then followerobserves this decision and makes his own strategy choice.Intuitively, the first player chooses the best possible pointbased on the second player’s best response function.

2.1. Problem Description and Assumptions. We establish aStackelberg game between the supplier as a leader and themanufacturer as the follower regarding the allocation of CSRto each member of a supply chain by dynamic Stackelberggame theory. The goal of each player is to maximize ownprofit with considering CSR condition.This model is a three-tier, decentralized vertical control supply chain network (seeFigure 1). All retailers and suppliers at the same levelmake thesame decision.Therefore, the general model can be simplifiedin a model that has only one supplier, one manufacturer, andone retailer. Moreover, we assume the manufacturer’s retailprice includes two parts: a fixed retailers’ profit of per-unitsale in addition to a per-unit lot sale charge so that we couldeliminate the retailer, who is not a decision-maker, from thegame. A Stackelberg differential game has two players playingthe game over a fixed finite horizon model.

This model has a state variable and control variableslike any dynamic game. We define the state variable as thelevel of social responsibility taken up by companies, andthe control variables are the capital amounts invested whilefulfilling the social responsibility. Specifically, all of the socialresponsibilities taken up by the firm 𝑗 at period 𝑡 can beexpressed as the investment 𝐼𝑗𝑡 . We suppose that 𝑥𝑡 evolvesaccording to the following rule:

𝑥𝑡+1 = 𝑓 (𝐼𝑡, 𝑥𝑡) . (1)

More specifically we have the following assumptions.The function 𝐵𝑡(𝑥𝑡) represents social benefit which is

proportional to social responsibility taken up by the supplychain system [25]. The function 𝑇𝑡 = 𝜏𝐼𝑡[1 + 𝜃(𝐼𝑡)]measuresthe value of the tax return to the members of the supplychain [26]. Both 𝜏 and 𝜃 are tax return policy parameters.

Discrete Dynamics in Nature and Society 3

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SiS1 S2

Si: supplier

Ri: retailer

Period t + 1Period tPeriod 1

R1 R2

M: manufacturer

Ri

M

SiS1 S2

R1 R2 Ri

M

SiS1 S2

R1 R2 Ri

M

Figure 1: Three-tier supply chain network.

Specifically, 𝜏 is the rate of individual posttax return oninvestment (ROI), and 𝜃 is the rate of supply chain’s posttaxreturn on investment (ROI). The market inverse demand is𝑃𝑀(𝑞𝑡) = 𝑎 − 𝑏𝑞𝑡 [27]. The accumulation of the level ofsocial responsibility taken up by the firms is given by 𝑥𝑡+1 =𝛼𝑥𝑡 + 𝛽1𝐼𝑆𝑡 + 𝛽2𝐼𝑀𝑡 . Here, 𝛽1 is the rate of converting thesupplier’s capital investment in CSR to the amount of CSRtaken up by the supply chain and 𝛽2 is the rate of convertingthe manufacturer’s capital investment in CSR to the amountof CSR taken up by the supply chain [28].

2.2. Notations and Definitions. By facilitating the model,certain parameters and decision variables are used. Notationsand definitions we use in our model are shown as follows:

𝑡: Period 𝑡𝑇: Planning horizon𝑞𝑡: Demand quantity at period 𝑡𝑎: Market potential𝑏: Price sensitivity𝑥𝑡: State variable, degree of taking SR𝐻𝑆: Hamiltonian-Lagrangian function of the supplier

𝐽𝑆𝑡 : Objective function of the supplier

𝐽𝑀𝑡 : Objective function of the manufacturer

𝐵𝑀(𝑥𝑡): Social benefit of the manufacturer

𝐵𝑆(𝑥𝑡): Social benefit of the supplier𝑇𝑆(𝑥𝑡): Tax return of the supplier

𝑇𝑀(𝑥𝑡): Tax return of the manufacturer

𝐼𝑀𝑡 : The amount of investment done by the manufac-turer

𝐼𝑆𝑡 : The amount of investment done by the supplier

𝑑:The percentage of investment of the supplier payoff

𝑤: The price of supplier’s product which is deliveredto manufacturer

𝑐: The price of the supplier’s raw materials

𝛿: Parameter of the supplier’s social benefit

��: Parameter of the manufacturer’s social benefit

𝛼: Deteriorating rate of the level of current socialresponsibility

𝜏: The rate of individual posttax return on investment(ROI)

𝜃: The rate of supply chain’s posttax return on invest-ment (ROI)

𝛽1:The rate of converting the supplier’s capital invest-ment in CSR to the amount of CSR taken up by thesupply chain

𝛽2: The rate of converting the manufacturer’s capitalinvestment in CSR to the amount of CSR taken up bythe supply chain

2.3. Objective Functions. The objective functions are made todepend on the control vectors and the static variable. Themembers of the supply chain attempt to optimize their netprofits, which includes minimizing the cost of raw materialsand investment in social responsibility and maximizing sale


revenues and benefits from taking social responsibility as wellas tax returns. Thus, the objective function of the supplier is

𝐽𝑆 = 𝑇∑𝑡=1

𝑃𝑆𝑡 𝑞𝑡 − 𝑐𝑞𝑡 + 𝐵𝑆𝑡 (𝑥𝑡) + 𝑇𝑆𝑡 (𝐼𝑆𝑡 , 𝐼𝑡) − 𝐼𝑆𝑡 + 𝑑𝐼𝑀𝑡= 𝑇∑𝑡=1

𝑤𝑞𝑡 − 𝑐𝑞𝑡 + 𝛿𝑥2𝑡 + 𝜏𝐼𝑆𝑡 (1 + 𝜃 (𝐼𝑆𝑡 + 𝐼𝑀𝑡 )) − 𝐼𝑆𝑡+ 𝑑𝐼𝑀𝑡 ,

(2)

where 𝑃𝑆𝑡 is the price of the supplier’s raw material. Suppose𝑃𝑆𝑡 = 𝑤 and 𝐵𝑆𝑡 (𝑥𝑡) = 𝛿𝑥2𝑡 is the social benefit of the sup-plier, 𝛿 is the parameter of the supplier’s social benefit, and𝑇S𝑡 (𝐼𝑆𝑡 , 𝐼𝑡) is the tax return of the supplier. Similarly, the

objective function of the manufacturer is

𝐽𝑀 = 𝑇∑𝑡=1

𝑃𝑀𝑡 (𝑞𝑡) 𝑞𝑡 − 𝑃𝑆𝑡 𝑞𝑡 + 𝐵𝑀 (𝑥𝑡) + 𝑇𝑀 (𝐼𝑀𝑡 , 𝐼𝑡)− 𝐼𝑀𝑡

= 𝑇∑𝑡=1

(𝑎 − 𝑏𝑞𝑡) 𝑞𝑡 − 𝑤𝑞𝑡 + ��𝑥2𝑡+ 𝜏𝐼𝑀𝑡 (1 + 𝜃 (𝐼𝑆𝑡 + 𝐼𝑀𝑡 )) − 𝐼𝑀𝑡 ,

(3)

where 𝑃𝑀𝑡 (𝑞𝑡) is the retail price of the product of themanufacturer.𝐵𝑀𝑡 (𝑥𝑡) = ��𝑥2𝑡 is the social benefit of themanu-facturer, �� is the parameter of the manufacturer’s socialbenefit, and 𝑇𝑀𝑡 (𝐼𝑀𝑡 , 𝐼𝑡) is the tax return of the manufacturer.

2.4. The Feedback Solution. In Stackelberg game, underthe feedback structure of information assumption that theplayers use their knowledge of the current state at time 𝑡 inorder to formulate their decisions at time 𝑡. Players selecttheir strategies on current time and they do not depend onthe initial condition. Hence, feedback strategies are subgame-perfect. To solve Stackelberg game, under the feedbackstructure of information assumption use the dynamic pro-gramming method with appropriate value functions [29].In fact, feedback equilibrium strategies at any time t arefunctions of the values of the state variables at that time. Ina feedback Stackelberg game the advantage of the leader overthe follower is instantaneous not global, as the differentialgame could be viewed as the limit of the discrete time gameas the number of stages becomes unbounded. Therefore,corresponding to the leader’s instantaneous strategy, thefollower will make an instantaneous response which dependson the current state and the leader’s current action.

Let 𝑇 be the last period of the problem. We solve a Stack-elberg game by backward induction for last period of game,which is to substitute the follower’s response function derivedfrom solving the optimization problem of the follower, giventhe leader’s response to the leader’s objective function in

the last period. For fixed 𝐼𝑆𝑇, the reaction function of themanufacturer is directly given by argmax𝐼𝑀

𝑇

𝐽𝑀T , that is:

𝐼𝑀∗𝑇 = 1 − 𝜏𝜃𝐼𝑆𝑡 − 𝜏2𝜏𝜃 . (4)

The objective function of supplier in period 𝑇 is

𝐽𝑆𝑇 = 𝑃𝑆𝑇𝑞𝑇 − 𝑐𝑞𝑇 + 𝐵𝑆𝑡 (𝑥𝑇) + 𝑇𝑆𝑇 (𝐼𝑆𝑇, 𝐼𝑇) − 𝐼𝑆𝑇 + 𝑑𝐼𝑀𝑇= 𝑤𝑞𝑇 − 𝑐𝑞𝑇 + 𝛿𝑥2𝑇 + 𝜏𝐼𝑆𝑇 (1 + 𝜃 (𝐼𝑆𝑇 + 𝐼𝑀𝑇 )) − 𝐼𝑆𝑇+ 𝑑𝐼𝑀𝑇 ;

(5)

substituting the value of 𝐼𝑀𝑇 given by (4) to (5) the maximumof 𝐽𝑆𝑇 is obtained when

𝐼𝑆∗𝑇 = 1 − 𝜏 + 𝑑2𝜏𝜃 . (6)

After some algebra we get, for the last period, resolution tothe problem at the period 𝑇 − 1. Let 𝑉𝑀(𝑇 − 1, 𝑇) denote thevalue functions for the period𝑇−1 to 𝑇. For any given policyby the leader the follower’s value function equation is

𝑉𝑀 (𝑇 − 1, 𝑇) = argmax𝐼𝑀𝑇

[𝐽𝑀𝑇−1 + 𝐽𝑀𝑇 ] , (7)

where 𝐽𝑀𝑇 is known. Using the definition 𝑥𝑇 = 𝛽1𝐼𝑆𝑇−1+𝛽2𝐼𝑀𝑇−1into 𝐽𝑀𝑇 and maximizing the value function for any fixed 𝐼𝑆𝑇−1gives an optimal action for the manufacturer for the period𝑇 − 1. Subsequently, the leader’s value function equationdefined by

𝑉𝑆 (𝑇 − 1, 𝑇) = argmax𝐼𝑆𝑇

[𝐽𝑆𝑇−1 + 𝐽𝑆𝑇] ; (8)

using again the state equation definition 𝑥𝑇 and value of the𝐼𝑀∗𝑇−1 into above equation, we can obtain 𝐼𝑆∗𝑇−1.This system can be solved in obtaining value functions at

any time 𝑡 and the feedback Stackelberg strategies.With somebackward-forward equations we can get the values of 𝐼𝑀𝑡 , 𝐼𝑆𝑡 ,and 𝑥𝑡.2.5.TheGlobal Stackelberg Solution. In this section, the struc-ture of information is considered closed loop model of BasarandOlsder [29].This is the derivation of (global) Stackelberg’ssolutions when the leader has access to dynamic information.In this structure the leader has a perfect knowledge of allthe past and current values of the state and controls and theleader tries to find an incentive strategy such that he canreach his global optimum.The idea of declaring a reward (orpunishment) for a decision-maker according to his particularchoice of action in order include a certain desired behavioron the part of decision-maker is known as an incentive (or incase of the punishment, as a threat) [29].


The global optimum of the supplier is assumed to beunique. The objective function of the supplier is

𝐽𝑆 = 𝑇∑𝑡=1

𝑃𝑆𝑡 𝑞𝑡 − 𝑐𝑞𝑡 + 𝐵𝑆𝑡 (𝑥𝑡) + 𝑇𝑆𝑡 (𝐼𝑆𝑡 , 𝐼𝑡) − 𝐼𝑆𝑡 + 𝑑𝐼𝑀𝑡= 𝑇∑𝑡=1

𝑤𝑞𝑡 − 𝑐𝑞𝑡 + 𝛿𝑥2𝑡 + 𝜏𝐼𝑆𝑡 (1 + 𝜃 (𝐼𝑆𝑡 + 𝐼𝑀𝑡 )) − 𝐼𝑆𝑡+ 𝑑𝐼𝑀𝑡 .

(9)

There exists a couple (𝐼∗𝑀𝑡 , 𝐼∗𝑆𝑡 ), ∀𝑡 ∈ [1, 𝑇], such thatmaximum of 𝐽𝑆𝑡 given by (𝐼∗𝑀𝑡 , 𝐼∗𝑆𝑡 ), ∀𝑡 ∈ [1, 𝑇]. Based onBasar and Olsder [29] max𝐼𝑆,𝐼𝑀𝐽𝑆(𝐼𝑀, 𝐼𝑆) is possible only if𝐽𝑆(𝐼𝑀, 𝐼S) is strictly concave in 𝐼𝑀 and 𝐼𝑆 and if there is nosingularity. In order to avoid this singularity, we need to adda constraint on 𝐼𝑆 or 𝐼𝑀 and this is a zero-point constraint.One should guess that this global optimum of supplier intime 𝑡 will be reached when the profit of the manufactureris zero (profit of manufacturer from playing game in time 𝑡).Therefore the optimum is obtained for the supplier under akind of zero-point constraint.

Recall that

𝐽𝑀 = (𝑎 − 𝑏𝑞𝑡) 𝑞𝑡 − 𝑤𝑞𝑡 + ��𝑥2𝑡+ 𝜏𝐼𝑀𝑡 (1 + 𝜃 (𝐼𝑆𝑡 + 𝐼𝑀𝑡 )) − 𝐼𝑀𝑡 . (10)

The profit of manufacturer with playing game in the time 𝑡 is𝑇𝑀𝑡 (𝐼𝑆𝑡 , 𝐼𝑡) − 𝐼𝑀𝑡 that should be zero and involves

𝐼𝑀𝑡 = 0or 𝐼𝑀𝑡 = 1 − 𝜏𝜃𝐼𝑆𝑡 − 𝜏𝜏𝜃 . (11)

As 𝐼𝑀𝑡 = 0 is not the good choice, so the other one will bechosen.

Since we consider this dynamic game as an optimalcontrol problem, the Hamiltonian function is a practical wayfor us in solving the game [10].

The Hamiltonian-Lagrangian for the supplier is definedby

𝐿𝑆𝑡 = 𝐽𝑆𝑡 + 𝑃𝑆𝑡+1 (𝑥𝑡+1) + 𝜆𝑡 (1 − 𝜏𝜃𝐼𝑆𝑡 − 𝜏𝜏𝜃 − 𝐼𝑀𝑡 ) , (12)

where 𝐽𝑆𝑡 is the objective function of the supplier and, toobtain the Stackelberg strategy of the supplier, we maximizethe objective function of the supplier by its Hamiltonianfunction. The maximization problem of the supplier, over 𝐼𝑆𝑡and 𝐼𝑀𝑡 , gives us the solution of the following set of first-orderconditions.

𝑥𝑡+1 = 𝜕𝐿𝑆𝑡𝜕𝑃𝑆𝑡+1 = 𝛼𝑥𝑡 + 𝛽1𝐼𝑆𝑡 + 𝛽2𝐼𝑀𝑡 . (13)

We also have

𝑃𝑆𝑡 = 𝜕𝐿𝑆𝑡𝜕𝑥𝑡 = 2𝛿𝑥𝑡 + 𝛼𝑃𝑆𝑡+1. (14)

Consequently, we can obtain the unique optimal response ofthe players as follows:

𝜕𝐿𝑆𝑡𝜕𝐼𝑀𝑡 = 𝜏𝜃𝐼𝑆𝑡 + 𝑑 − 𝜆𝑡 + 𝛽2𝑃𝑆𝑡+1 = 0,𝜕𝐿𝑆𝑡𝜕𝐼𝑆𝑡 = 𝜏𝜃 (2𝐼𝑆𝑡 + 𝐼𝑀𝑡 ) + 𝜏 − 𝜆𝑡 + 𝛽1𝑃𝑆𝑡+1 = 0.

(15)

And we have

𝜕𝐿𝑆𝑡𝜕𝜆𝑡 =1 − 𝜏𝜏𝜃 − 𝐼𝑆𝑡 − 𝐼𝑀𝑡 = 0. (16)

After some algebra, we obtain

𝜆𝑡 = 𝑃𝑆𝑡+1 (𝛽1 − 𝛽2) − 𝑑𝜏𝜃 . (17)

Since we use closed loop information, the structure variablesdepend on the current time variable and the initial statevariables. 𝑥1 is given. Furthermore, the boundary conditionis 𝑃𝑆𝑡+1 = 0.2.6. Augmented Discrete Hamiltonian Matrix. In the relatedliterature, one can find a lot methods to solve the optimalcontrol problem. We have chosen an algorithm given byMedanic and Radojevic which is based on an augmenteddiscrete Hamiltonian matrix [30]. Firstly, we assume

[𝑥𝑡+1𝑃𝑆𝑡 ] = [𝑎 𝑏𝑐 𝑑][

𝑥𝑡𝑃𝑆𝑡+1] + [

𝑑𝑒] , (18)

with the boundary conditions 𝑃𝑆𝑇+1 = 0 and 𝑥1 given.We can get the values of the matrices

[𝑥𝑡+1𝑃𝑆𝑡 ]

= [[[𝛼 2𝛽2 (−𝛽1 + 𝛽2)𝜏𝜃 + (𝛽1 − 𝛽2)2(𝜏𝜃)22𝛿 𝛼

]]][ 𝑥𝑡𝑃𝑆𝑡+1]

+ [[(−𝛽1𝑑 − 𝛽2𝜏)𝜏𝜃 + 𝑑 (𝛽2 − 𝛽1)(𝜏𝜃)20

]].

(19)

We assume a linear relation between 𝑃𝑆𝑡 and 𝑥𝑡; thus, theoptimal controls can be determined at each time step basedon the current estimated state

𝑃𝑆𝑡 = 𝐾𝑡𝑥𝑡 − 𝑔𝑡, (20)


where 𝐾𝑡 and 𝑔𝑡 are determined by the backward equationswith the following boundary conditions:

𝑃𝑆𝑇+1 = 0,𝑥1 = 1,𝐾𝑇 = 2𝛿,

𝐾𝑇+1 = 0,𝑔𝑇 = 0,

𝑔𝑇+1 = 0.

(21)

Once we get the different values of𝐾𝑡 and 𝑔𝑡 by the backwardloop, then we compute the optimal sequences {𝑥∗𝑡 }, {𝑝𝑆∗𝑡 },{𝜆∗𝑡 }, {𝐼𝑆∗𝑡 }, and {𝐼𝑀∗𝑡 }, 𝑡 = 1, . . . , 𝑇. With {𝐼𝑆∗𝑡 } and {𝐼𝑀∗𝑡 }, 𝑡 =1, . . . , 𝑇, the optimum value of the supplier can be obtainedunder the zero-profit constraint.

3. Management Policies and Reflexes:A Numerical Example

A supply chain structure usually includes a certain numbersof players. Our model falls in this field. In particular in ourwork the model is a three-tier, decentralized vertical controlsupply chain network. All retailers and suppliers at the samelevel make the same decision. Therefore, consequently themodel has only one supplier, one manufacturer, and oneretailer. The members of this supply chain-type take theirdecisions based on maximizing their individual net benefitswith a constraint: a given level of CSR thatmust be reached bythe network.This situation leads to an equilibrium status thathas relevantmanagement policies reflexes individual both forall players and for the supply chain network therein.

As we can observe by the following numerical example,the model which was elaborated gives us the opportunity toset up the mechanism design of the relationships among thedifferent levels and the players of our Management Game.This output is not simply theoretical but in our opinioncontains important issues in solving Management Decisions.As it is well-known, business logistics management refers tothe production and distribution process within the company,while supply chain management includes suppliers, manu-facturers, and retailers that distribute the product to the endcustomer. Supply chains include every business that comes incontact with a particular product, including companies thatassemble and deliver component parts to the manufacturer.This aspect is strictly correlated with the Corporate SocialResponsibility which we discussed by our model. In a visionwhich includes the social mission of the firms but looking forreaching profits which are fundamental for the economicalsustainability of the business, we find a model by which wecapture the existing correlations between these two issues.

Another aspect that was deeply studied is related to theinformation structure of the network. As we can see by thenumerical example which follows, if we choose feedbackand closed loop information structures among the players,the global environment of the supply chain promotes the

FC

00

100000

200000

300000

400000

500000

600000

2 4 6 8 10 12

CL

Figure 2: Evolution of 𝐽𝑆𝑡 , where FD and CL are feedback and closedloop solutions, respectively.

FD

00

50000100000150000200000250000300000350000400000450000

2 4 6 8 10 12

CL

Figure 3: Evolution of 𝐽𝑀𝑡 .

interaction between all the members of the network that arenaturally oriented in playing the game, that is, strengtheningthe immaterial but productive structure represented by thesupply chain.

3.1. Numerical Example. In this section we provide a numer-ical example. We run the following numerical simulationswith mathematical 8. The results presented here are obtainedfor the following values of the parameters: 𝑎 = 6, 𝑏 = 0.00001,𝑞𝑡 = 100000, 𝑑 = 0.5, 𝑐 = 2.4, 𝑤 = 3.6, 𝛼 = 2, 𝜏 = 0.2,and 𝜃 = 0.01. We set 𝛽1 = 0.3, 𝛽2 = 0.5, 𝛿 = 0.2, and�� = 0.2. We assume that the time horizon is 𝑇 = 10. Theinitial level of social responsibility is supposed to be 𝑥1 = 1.We draw the results of themodel, in feedback and closed loopStackelberg dynamic game. Figures 2 and 3 show the trendof supplier’s and manufacturer’s profits from periods one toten in a feedback and closed loop Stackelberg game. Forsupplier, which is the leader, the results of game in feedbackand closed loop solutions have same trend and during theperiod the profit of supplier has increased. However, for themanufacturer, which is the follower, closed loop solutioninvolves zero-profits. That is our closed loop solution whichis based on a nonprofit constraint. Moreover, in feedbacksolution the profit of manufacturer has risen over time.


Table 1: Cumulated payoffs.

Solutions 𝐽𝑀𝑡 𝐽𝑆𝑡Feedback (FD) 1778490 1717240Closed loop (CL) 0 1210050

FD JSOCL0

200000400000600000800000

100000012000001400000160000018000002000000

Figure 4: Comparison of the supplier’s profits, playing game andwithout playing any game in feedback and closed loop solutions.

Obviously, both manufacturer and supplier gain extra profitfrom playing the game in feedback solution. In addition, itis apparent that playing game in the closed loop solution isbeneficial to supplier which is the leader of the game.

Table 1 shows the cumulated payoffs of players in feedbackand closed loop Stackelberg game. Figures 4 and 5 comparethe cumulated profits of the supply chain’s members. JSOis supplier’s profit without playing the game and JMO ismanufacturer’s profit without playing the game. In sum, forthis case in which supplier is the leader of game, both supplierandmanufacturer are motivated to play the game in feedbacksolution because their benefits have increased; in closed loopsituation some incentive strategies are needed in order thatfollower plays game while profit of the leader has risen in thissolution as well.

4. Conclusion

We investigated a decentralized three-tire supply chainconsisting of supplier and manufacturer with the aim ofallocating CSR to members of the supply chain system overtime.We considered a Stackelberg game consisting of a leaderand a follower with two information structures.Themembersof a supply chain play games with each other to maximizetheir own profits; thus, the model used was a long-termcoinvestment game model. The equilibrium point in a timehorizon was determined at where the profit of supply chain’smembers was maximized and CSR was implemented amongmembers of the supply chain. We applied control theory andused an algorithm (augmented discrete Hamiltonian matrix)to obtain an optimal solution for the dynamic game model.

Competing Interests

The authors declare that they have no competing interests.

FD JMOCL0

200000400000600000800000

100000012000001400000160000018000002000000

Figure 5: Comparison of the manufacturer’s profits, playing gameandwithout playing any game in feedback and closed loop solutions.

Acknowledgments

The present research was supported by the MEDAlics,Research Center at Universita per Stranieri Dante Alighieri,Reggio Calabria, Italy.

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