在供应链管理中企业社会责任影响的多维决策方法

The impact of corporate social responsibility in supply chain management:Multicriteria decision-making approach
Jose M.Cruz ⁎
Department of Operations and Information Management,School of Business,University of Connecticut,Storrs,CT 06269-2041,United States
a b s t r a c t
a r t i c l e i n f o Article history:
Received 8August 2007
Received in revised form 8July 2009Accepted 30July 2009
Available online 8August 2009Keywords:
Supply chain networks Environment
Corporate social responsibility Risk management
Multicriteria optimization
This paper develops a decision support framework for modeling and analysis of supply chain networks with corporate social responsibility (CSR).We consider the multicriteria decision-making behavior of the various decision makers (manufacturers,retailers,and consumers),which includes the maximization of net return,the minimization of emission,and the minimization of risk.The emission and the risk are penalized by variable weights.The model allows one to investigate the interplay of the heterogeneous decision makers in the supply chain and to compute the resultant equilibrium pattern of product outputs,transactions,product prices,and levels of social responsibility activities.The results show that social responsibility activities can potentially reduce transaction costs,risk and environmental impact.
©2009Elsevier B.V.All rights reserved.
1.Introduction
In recent years,there has been a considerable shift in thinking with regard to improving the social and environmental performance of companies [81].On one hand,there are those that argue that the government should regulate the social and environmental perfor-mance of companies [69].On the other hand,there are those that believe that the private sector generally prefers the flexibility of self-designed voluntary standards [80].Many researchers have tried to understand business motivation to voluntarily adopt CSR programs [23,54].Swindley [79]argues that many firms regard CSR as cost of doing business though other firms may find CSR bene ficial.Firms engage in CSR activities as a way to enhance their reputation [30,31],preempt legal sanction [68],respond to NGO action [78],manage their risk [32,38],and to generate customer loyalty [4,5].Bowman [7]asserts that firms with proactive CSR that engage in managerial practices like environmental assessment and stakeholder manage-ment [84]tend to anticipate and reduce potential sources of business risk,such as potential governmental regulation,labor unrest,or environmental damage [67].
CSR has been a theme of many researchers.Carroll [9]traced the evolution of the CSR concept and found that the CSR construct originated in the 1950s.Carroll [10,11]integrated various streams of CSR research to de fine a model that extended corporate performance beyond traditional economic and legal considerations to include ethical and discretionary responsibilities.Wartick and Coghran [82]
traced the evolution of the corporate social performance model by focusing on three challenges to the concept of CSR:economic responsibility,public responsibility,and social responsiveness.They examined the management of social issues as a dimension of corporate social performance and concluded that the corporate social performance model is valuable for business and society.Carter and Jennings [15]indicated that CSR not only is synonymous with business ethics but also encompasses dimensions including philan-thropy,community,workplace diversity,safety,human rights,and environment.
CSR issues surrounding supply chains have only recently come to the fore,notably,in the context of conceptual and survey studies [13,15].Murphy and Poist [55]stated that although supply chain practitioners have been slow to adopt CSR considerations,social responsibility concepts in the supply chain are increasing in importance.Carter and Jennings [13,15]empirically established primary supply chain CSR categories of environment,diversity,human rights,philanthropy,and safety.Some researchers have examined individual elements of CSR in the supply chain.In response to growing CSR concerns,researchers have begun to deal with environmental risks [2,8,12,70,71],labor practices [25,73,74],pro-curement [13,14,37,72],and af firmative action purchasing [16].Moreover,organizations are expanding their responsibility for their products beyond their sales and delivery locations (cf.[6])and start managing the CSR of their partners within the supply chain [25,49].
Nevertheless,decision support models that integrate CSR into supply chain management and design are surely needed.Within this new business environment,trade-offs between various objectives while providing resources to CSR activities,are becoming increasingly complex.The questions that arise when applying CSR to supply chain
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management and design are:(1)given that there is a vast array of decisions to be made on all levels(strategic,tactical,and operational), how does CSR govern and apply to those decisions,and(2)what are the potential conflicts that arise from CSR decision-making in supply chain management and design?To that end,this paper presents a decision support model that incorporates the challenges,opportuni-ties and constraints that managers face when deciding on the level of investment in CSR activities and the choice of trading partners (manufacturer or retailer)given their transaction cost,environmental consciousness and perceived riskiness.
In particular,we develop a multicriteria decision-making sup-ply chain network framework that captures the economic and CSR activities of manufacturing,retailer,and demand market.The models that yield the system optima associated the maximization of net profit,emission(waste)minimization,and the minimization of risk, with the weights associated with the environmental and risk criteria being distinct and variable for each such decision maker.This framework makes it possible to simulate different scenarios depend-ing on how concerned(or not)the decision makers are about en-vironmental issues,risk and CSR over all.Moreover,it allows for the explicit determination of the equilibrium levels of social responsibility activities between the decision makers,as well as,product transac-tions and prices.Hence,the resulting network model allows the decision makers to assess the impact of CSR activities on their key objectives,profit,environment and risk.
The network model presented is multilevel in structure and the flows are product transactions and levels of social responsibility activities.We consider both business-to-business(B2B)and business-to-consumer(B2C)transactions.Prices are associated with the nodes in the network which correspond to the different tiers of decision makers.Manufacturers are assumed to produce homogeneous product and to sell them either over physical or electronic links via the Internet to retailers and through electronic links directly to consumers.Retailers,in turn,can sell the products over physical or virtual links to consumers.Increasing levels of social responsibility activities are assumed to reduce transaction costs,risk,and environ-mental emissions.
This paper is organized as follows.In Section2,we develop the model and describe the decision makers'multicriteria decision-making behavior.We establish the governing equilibrium conditions along with the corresponding variational inequality formulation. The variables are the equilibrium prices,the equilibrium product flows,and the equilibrium levels of social responsibility activities.In Section3,we propose an algorithm,which is then applied to several illustrative numerical examples in Section4.In Section5,we provide managerial insights.In Section6,we discuss the role of decision support systems in CSR.We conclude the paper with Section7in which we summarize our results and suggest directions for future research.
2.The supply chain network sustainability equilibrium model
In this section,we develop the network model with manufac-turers,retailers,and demand markets in which we explicitly integrate levels of social responsibility activities between buyers and sellers. The model assumes that the manufacturingfirms are involved in the production of a homogeneous product and considers I manufacturers, and J retailers,which can be either physical or virtual,as in the case of electronic commerce.There are K demand markets for the homoge-neous product in the economy.We assume,for the sake of generality, that each manufacturer can transact directly electronically with the consumers at the demand market through the Internet and can also conduct transactions with the retailers either physically or electron-ically.We let l refer to a mode of transaction with l=1denoting a physical transaction and l=2denoting an electronic transaction via the Internet.
The top-tiered nodes in the supply chain network in Fig.1, enumerated by1,…,i…,I,represent the I manufacturers.We assume that each manufacturer seeks to determine his optimal production and his sales allocations of the product to the retailers and demand market in order to maximize his own profit.We also assume that each manufacturer seeks to minimize the total emission and risk associated with production and transportation to the retailers and demand markets.
Retailers,which are represented by the second-tiered nodes in Fig.1,function as intermediaries.The nodes corresponding to the retailers are enumerated as:1,…j,…,J with node j corresponding to retailer j.They purchase the product from the manufacturers and sell the product to the consumers at the different demand markets.We assume that the retailers compete with one another in a noncoop-erative manner.Also,we assume that the retailers are multicriteria decision makers with environmental and risk concerns and they also seek to minimize the emissions and risk associated with transacting (which can include transportation)with manufacturers and con-sumers as well as in operating their retail outlets.
The bottom-tiered nodes in Fig.1represent the demand markets, which can be distinguished from one another by their geographic locations or the type of associated consumers such as whether they correspond,for example,to businesses or to households.There are K bottom-tiered nodes with node k corresponding to demand market k.
The structure of the network in Fig.1guarantees that the conservation offlow equations associated with the production and distribution is satisfied.Theflows on the links joining the manufacturers with the retailers and demand market nodes are denoted respectively by the components of the vectors Q1and Q2.Theflows on the links joining the retailer nodes with the demand markets are given by the respective components of the vector:Q3.The variables for this model are given in Table1.All vectors are assumed to be column vectors.
We now turn to the description of the functions.Wefirst discuss the production cost,transaction cost,handling,and unit transaction cost functions given in Table2.Each manufacturer is faced with a certain production cost function that may depend,in general,on the entire vector of production outputs.Furthermore,each manufacturer and each retailer are faced with transaction costs.The transaction costs are affected/influenced by the amount of the product transacted and the levels of social responsibility activities.
Each retailer is also faced with what we term a handling/con-version cost(cf.Table2),which may include,for example,the cost of handling and storing the product.The handling/conversion cost of a retailer is a function of how much he has obtained of the product from the various manufacturers in what transaction mode.
The consumers at each demand market are faced with a unit transaction cost.As in the case of the manufacturers and the retailers, higher level of social responsibility activities may potentially
reduce Fig.1.The structure of the supply chain network with electronic commerce.
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J.M.Cruz/Decision Support Systems48(2009)224–236
transaction costs,which means that they can lead to quanti fiable cost reductions.The unit transaction costs depend on the amounts of the product that the retailers and the manufacturers transact with the demand markets as well as on the vectors of social responsibility activities established with the demand markets.The generality of the unit transaction cost function structure enables the modeling of competition on the demand side.Moreover,it allows for information exchange between the consumers at the demand markets who may inform one another as to their social responsibility activities which,in turn,can be re flected in the transaction costs.We assume that the production cost,the transaction cost,and the handling cost functions are convex and continuously differentiable and that the unit cost functions are continuous.
We now turn to the description of the social responsibility acti-vities production cost,emission functions and,finally,the risk functions and the demand functions.We assume that the social re-sponsibility activities production cost functions as well as the emission and the risk functions are convex and continuously differentiable.The demand functions are assumed to be continuous.
We start by describing the social responsibility activities produc-tion cost functions that are given in Table 3.We assume that each manufacturer may spend money,for example,in the form of time/service,investment in new technology,training employees,and information sharing in order to promote a sound environmental policy.Here social responsibility activities are activities that promote quality assurance,environmental preservation,and compliance.Ac-cording to Simpson [77],positive relationships have been established between environmental performance and improvements to the man-ufacturing quality management [45,46],lean manufacturing practice [44,46,76]and worker involvement [34,45,75].Furthermore,each retailer may actively try to achieve a certain relationship level with a manufacturer and/or demand market.
These social responsibility activities production cost functions may be distinct for each such combination.Their speci fic functional forms may be in fluenced by such factors as the willingness of retailers or demand markets to establish/maintain a level of social responsibility activities as well as the level of previous activities that exist.Hence,we assume that these production cost functions are also affected and in fluenced by the levels of social responsibility activities.We assume that these levels of social responsibility activities (cf.Table 1)take on a value that lies in the range [0,1].No social responsibility activity is indicated by a level of zero and the strongest possible level of social responsibility activity is indicated by a level of one.The levels of social responsibility activities,along with the product ows,are endoge-nously determined in the model.
We now describe the emission functions as presented in Table 4.We also assume that the emission functions depend on the volume of transactions between the particular pair via the particular mode,and on the levels of social responsibility activities between decision makers (see,e.g.,[20,29,34,36,51]).We assume that each manufac-turer seeks to minimize the total emission (waste)generated in the production process as well as in the process of product delivery to the next tier of decision makers.We also assume that the retailers follow similar behavior.
Table 5describes the risk functions.We note that the risk functions in our model are functions of both the product transactions and the levels of social responsibility activities.Juttner et al.[40]suggest that supply chain-relevant risk sources fall into three categories:environ-mental risk sources (e.g.,fire,social –political actions,or acts of God),organizational risk sources (e.g.,production uncertainties),and
Table 1
Variables in the supply chain network sustainability model.Notation De finition q I -dimensional vector of the amounts of the product produced by the manufacturers with component i denoted by q i .
Q 12IJ -dimensional vector of the amounts of the product transacted between the manufacturers with the retailers via the two modes with component ijl denoted by q ijl Q 2I K -dimensional vector of the amounts of the product transacted between the manufacturers and the demand markets with component ik denoted by q ik
Q 32JK -dimensional vector of the amounts of the product transacted between the retailers and the demand markets via the two modes with component jkl denoted by q jkl η12IJ -dimensional vector of the levels social responsibility activities between the manufacturers and the retailers/mode combinations with component ijl denoted by ηijl η2IK -dimensional vector of the levels social responsibility activities between the manufacturers and the demand market combinations with component ik denoted by ηik η32JK -dimensional vector of the levels social responsibility activities between the retailers and the demand market/mode combinations with component jkl denoted by ηjkl ρ1ijl Price associated with the product transacted between manufacturer i and retailer j via mode l ρ1ik Price associated with the product transacted between manufacturer i and demand market k
ρ2jkl Price associated with the product transacted between retailer j and demand market k via mode l
ρ3
K -dimensional vector of the demand market prices of the product at the demand markets with component k denoted by ρ3k
Table 2
Production,handling,transaction,and unit transaction cost functions.Notation De finition f i (q i )
The
production cost function of manufacturer i
c ijl (q ijl ,ηijl )The transaction cost function of manufacturer i transacting with retailer j via mode l
c ik (q ik ,ηik )The transaction cost function of manufacturer i transacting with deman
d market k via th
e Internet
c j (q j )The handling/conversion cost function of retailer j .q j =P I i =1P 2
l =1q ijl
c ijhl (q ijl ,ηijl )The transaction cost function of retailer j transacting with manufacturer i via mode l c jkl (q jkl ,ηjkl )The transaction cost function of retailer j transacting with deman
d market k via mod
e l
c ik (Q 2,Q 3,η2,η3)The unit transaction cost function associate
d with consumers at demand market k in obtaining th
e product from manufacturer i
c jkl (Q 2,Q 3,η2,η3)
The
unit transaction cost function associated with consumers at demand market k in obtaining the product from retailer j via mode l
Table 3
Social responsibility activities production cost functions.Notation De finition b ijl (ηijl )The social responsibility activities production cost function associated with manufacturer i and retailer j transacting in mode l b ik (ηik )The social responsibility activities production cost function associated with manufacturer i and demand market k
b ijl
(ηijl )The social responsibility activities production cost function associated with retailer j transacting with manufacturer i via mode l
b jkl (ηjkl )
The social
responsibility
activities
production
cost
function
associated
with
retailer j and demand market k in transacting via mode l
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J.M.Cruz /Decision Support Systems 48(2009)224–236
network-related risk sources.Johnson[39]and Norrman and Jansson [66]argue that network-related risk arises from the interaction between organizations within the supply chain,e.g.,due to insuffi-cient interaction and cooperation.Here,we model supply chain organizational risk,environmental risk,and network-related risk by defining the risk as a function of productflows as well as the levels of social responsibility activities.We use levels of social responsibility activities(levels of cooperation)as a way of possibly mitigating these supply risk sources.Indeed,high levels of social responsibility activities are assumed to reduce risk and transactional uncertainty. The results in Feldman et al.[27]suggest that adopting a more environmentally proactive posture has,in addition to any direct environmental and cost reduction benefits,a significant and favorable impact on thefirm's perceived riskiness to investors and,accordingly, its cost of equity capital and value in the market place.
The demand functions as given in Table6are associated with the bottom-tiered nodes of the supply chain network.The demand of consumers for the product at a demand market depends,in general, not only on the price of the product at that demand market but also on the prices of the product at the other demand markets.Consequently, consumers at a demand market,in a sense,also compete with consumers at other demand markets.
We now turn to describing the behavior of the various economic decision makers.The model is presented,for ease of exposition,for the case of a single homogeneous product.It can also handle multiple products through a replication of the links and added notation.We first focus on the manufacturers.We then turn to the retailers,and, subsequently,to the consumers at the demand markets.An equilibrium solution is denoted by⁎.
2.1.Decision-makers'multicriteria behavior
Multicriteria decision-making methodology is widely used in supply chain research.For background on decision-making in general and on multicriteria decision-making,in particular,see Karwan et al.
[41].Recently,various researchers have argued that multicriteria decision-making with equally weighted objective functions might not adequately reveal an agents preference.Choo and Wedley[18]sur-veyed procedures for estimating implied criterion weights.See also Ballestero and Romero[1],Weber and Borcherding[83],Yu[86],and Ma et al.[53].Subsequently,Choo et al.[19]provided interpretations of criteria and their appropriate roles in distinct multicriteria decision-making models.Dong and Nagurney[24]and Nagurney and Ke[60]introduced state-dependent weights for the modeling of a sector's bicriteria decision-making problem in the context of a financial network.Nagurney et al.[61]also considered variable weights in the context of a multicriteria network equilibrium model but the model was single-tiered and not supply chain.In this paper, we introduce a class of objective functions with variable weights for multicriteria decision-making in supply chain network equilibrium framework.
2.1.1.The behavior of the manufacturers and their optimality conditions
The manufacturers are involved in the production of a homogeneous product and in transacting with the retailers physically or electronically as well as directly with the demand markets electronically.Further-more,they are also involved in establishing social responsibility activities.The quantity of the product produced by manufacturer i must satisfy the following conservation offlow equation:
q i=
X J
j=1
X2
l=1
q ijl+
X K
k=1
q ik;ð1Þ
which states that the quantity of the product produced by manufac-turer i is equal to the sum of the quantities transacted between the manufacturer and all retailers(via the two modes)and the demand markets.Hence,in view of Eq.(1),and as noted in Table2,we have that for each manufacturer i the production cost function is denoted by f i(q i).Furthermore,each manufacturer may actively try to achieve a certain level of social responsibility activity with a retailer and/or a demand market.
Each manufacturer i tries to maximize his profits.He faces total costs that equal to the sum of his production cost plus the total transaction costs and the costs that he incurs in establishing social responsibility activities.His revenue,in turn,is equal to the sum of the price multiplied by quantities of the product transacted.
Noting the conservation offlow Eq.(1)one can express the criterion of profit maximization for manufacturer i as:
Maximize z1i=
X J
j=1
X2
l=1
ρ1ijl q ijl+
X K
k=1
ρ1ik q ik−f i q iðÞ
−
X I
j=1
X2
l=1
c ijl q ijl;n ijl
−
X K
k=1
c ik q ik;n ik
ðÞ
−
X J
j=1
X2
l=1
b ijl n ijl
−
X K
k=1
b ik n ik
ðÞ:
ð2Þ
Note that in Eq.(2),thefirst two terms represent the revenue whereas the subsequentfive terms represent the various costs.
In addition to the criterion of profit maximization,we assume that each manufacturer also seeks to minimize the total emissions(waste) generated in the production of the product as well as its delivery to the next tier of decision makers,whether retailers or consumers at the demand markets.Here,we also assume that the emission function depends on the volume of transactions between the particular pair via the particular mode,and on the levels of social responsibility activities between decision makers.
The following are some reasons whyfirms may decide to minimize their emissions.First,because of public concern regarding environ-mental issues.Promoting environmental care can enhance a compa-ny's image.Second,by minimizing emissions,firms can reduce their transactions costs in dealing with regulators,local communities, environmental groups,and other external stakeholders[48].Finally, the main factor driving companies to improve their environmental performance is the risk of being held liable,or found negligent,for accidents or incident with significant real or perceived environmental damage.Klassen and McLaughlin[47],estimate that losses in share-holder value for large publicly tradedfirms from environmental incidents can be on the order of hundreds of millions of dollars per
Table4
Emission functions.
Notation Definition
e i(η1,η2,Q1,Q2)The emission function associated with manufacturer i e j(η1,η3,Q1,Q3)The emission function associated with retailer j
Table5
Risk functions.
Notation Definition
r i(Q1,Q2,η1,η2)The risk incurred by manufacturer i in his transactions r j(Q1,Q3,η1,η3)The risk incurred by retailer j in his transactions
Table6
Demand functions.
Notation Definition
d k(ρ3)Th
e demand for the product at demand market k as a function o
f the
demand market price vector 227
J.M.Cruz/Decision Support Systems48(2009)224–236
incident.To limit liability and negligence claims,a company may choose to implement strict emission reduction mechanisms.Environ-mental improvements,in turn,can lead to economic bene fits for companies.
We assume that the emission function for manufacturer i is convex and continuously differentiable and given by the function e i =e i (Q 1,Q 2,η1,η2)(cf.Table 4).
Hence,the second criterion of each manufacturer can be expressed mathematically as:
Minimize z 2i =e i
Q 1;Q 2
;η1;n 2 :
ð3Þ
Finally,we analyze the effects of CSR on risk.We assume that each
manufacturer is concerned with risk minimization.Risk is de fined as the possibility for companies to suffer harm or loss for their activities and also for the activities of their partners in the supply chain.In terms of CSR risk,companies may be found liable for pollution,non compliance with regulation,dangerous operations,use of hazardous raw materials,production of hazardous waste,and for health and safety issues.Firms with proactive CSR programs tend to anticipate and reduce potential sources of business risk,such as potential governmental regulation,labor unrest,or environmental damage [67].Moreover,in addition to any direct environmental and cost reduction bene fits,CSR activities have a signi ficant and favorable impact on the firm's perceived riskiness to investors and,accordingly,its cost of equity capital and value in the market place [27].
The third criterion faced by manufacturer i ,thus,corresponds to risk (cf.Table 5)minimization and can be expressed mathematically as:Minimize z 3i =r i Q 1;Q 2
;η1;η2 :
ð4Þ
We note that optimization problems of net revenue maximization and risk minimization are fairly standard [24].Here,use a general risk function and assume that it depends on the volume of transactions between the particular pair via the particular mode,and on the levels of social responsibility activities between decision makers.However,we can also use variance –covariance matrices for measuring risk and return relationship.
The problem that each decision maker will meet is the value trade-off.That is,the decision maker is faced with a problem of trading off the gain of one objective against another objective.The essence of the issue is,“How much achievement on objective z 1i is the decision maker willing to give up in order to improve achievement on objective z 2i or/and objective z 3i by some amount?”It is rational to assume that most decision makers will not weight all the objectives the same.For example,a risk-averse decision maker may be willing to accept a portfolio with a little lower mean return if the portfolio has lower risk.In other words,the risk-averse decision maker would be willing to take certain risks only if the risk return is much higher.As discussed in Dong and Nagurney [24],more attention will generally be given to reduce the risk when the risk is high and this kind of decision rationality argues that the objective function should penalize the states with high risk by imposing a greater weight to z 3i of high risks than to those z 3i with low risks.Thus,we apply the weights which are usually higher than 1for penalization of the emission and the risk objectives.
De finition 1.(Criterion-dependent weight).A weight ωht =ωht (z ht )is called a criterion-dependent weight for criterion h and decision maker i,if it is strictly increasing,convex,smooth,and nonnegative.
In this paper,the weighted criteria of concern will be that of total emission and risk minimization so we will have that h =2,3in the
case of the manufacturers and the retailers.Also,since we have that all decision makers are faced with variable weights,in the case of the manufacturers:t =i ;i =1,…I .In the case of the retailers,we will have:t =j ;j =1,…,J .
Hence,let ω2i (z 2i )denote the emission-penalizing weight which depends on the amount of emissions generated per unit of product produced and transacted by manufacturer i .Moreover let ω3i (z 3i )denote the risk-penalizing weight which depends on the value of risk objective associated with manufacturer i .Furthermore,according to De finition 1ωhi ,where h =2,3,are strictly increasing,convex,smooth,and nonnegative functions.We now state the following de finition.De finition 2.(Emission and risk-penalizing value function of manufacturer i ).A value function U i for manufacturer i is called emission and risk-penalizing value function if Maximize U i =z 1i −ω2i z 2i ðÞz 2i −ω3i z 3i ðÞz 3i ;
ð5Þ
where ω2i (z 2i )z 2i and ω3i (z 3i )z 3i are as indicated in De finition 1.Thus,the multicriteria decision-making problem of manufacturer i can be expressed as:
Maximize X J j =1X 2l =1
ρ1ijl q ijl +
X K k =1
ρ1ik q ik −f i q i ðÞ
−X J j =1X 2l =1c ijl q ijl ;ηijl −X K k =1
c ik q ik ;ηik ÀÁ
−X J j =1X 2l =1
b ijl ηijl
−
X K k =1
b ik ηik ÀÁ−ω2i e i e i Q 1;Q 2;η1;n 2 −ω3i r i r i Q 1;Q 2
;η1;η2
ð6Þ
subject to:q ijl ≥0,q ik ≥0,0≤ηijl ≤1,0≤ηik ≤1;∀j ,k ,l .
Thus,the expression consisting of the first seven terms to the right-hand side of the equal sign in Eq.(6)represents the net revenue (which is to be maximized),whereas the last two terms in Eq.(6)represent the weighted dollar values of total emission and risk which are to be minimized by manufacturer i .
We note that value functions have been studied extensively and used for decision problems with multiple criteria (cf.[24,28,42,85,87]).Of course,a special example of a constant weight value function is the one with equal weights (see,e.g.,[22,63]).We now prove a theorem and then derive the optimality conditions of the manufacturers.The proof is similar to that found in Dong and Nagurney [24].
Theorem 1.(Concavity).The value function U i de fined in Eq.(6)is strictly concave with respect to (Q 1,Q 2,η1,η2)∈K 1,∀i where
K 1
≡Q 1;Q 2;η1;η2
j q ijl ≥0;q ik ≥0;0≤ηijl ≤1;0≤ηik ≤1;∀i ;j ;k ;l h i
:
Proof.Let g i (z hi )=ωhi (z hi )z hi .
Since ωhi (z hi )is assumed to be convex,strictly increasing,and nonnegative,and z hi >0,we have dg i z hi ðÞdz hi =d ωhi z hi ðÞ
dz hi z hi +ωhi z hi ðÞ>0ð7Þ
d 2g i z hi ðÞdz hi =d 2ωhi z hi ðÞdz hi
z hi +2d ωhi z hi ðÞ
dz hi >0:ð8Þ
Combining Eqs.(7)and (8),we know that g i is increasing and
strictly convex.z hi is convex with respect to (Q 1,Q 2,η1,η2)according to De finition 1.Hence the composition of G i ≡−g i ○z hi is strictly
228J.M.Cruz /Decision Support Systems 48(2009)224–236。