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Modelling inter-supply chain competition with resourcelimitation and demand disruptionZhaobo Chena, Chunxian Tengb, Ding Zhangc & Jiayi Sunb

a College of Economy and Management, Taiyuan University of Science and Technology,Taiyuan, Chinab Institute of System Engineering, Harbin University of Science and Technology, Harbin,Chinac School of Business, State University of New York, Oswego, NY, USAPublished online: 30 Aug 2014.

To cite this article: Zhaobo Chen, Chunxian Teng, Ding Zhang & Jiayi Sun (2014): Modelling inter-supply chaincompetition with resource limitation and demand disruption, International Journal of Systems Science, DOI:10.1080/00207721.2014.942499

To link to this article: http://dx.doi.org/10.1080/00207721.2014.942499

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International Journal of Systems Science, 2014http://dx.doi.org/10.1080/00207721.2014.942499

Modelling inter-supply chain competition with resource limitation and demand disruption

Zhaobo Chena,∗, Chunxian Tengb, Ding Zhangc and Jiayi Sunb

aCollege of Economy and Management, Taiyuan University of Science and Technology, Taiyuan, China; bInstitute of System Engineering,Harbin University of Science and Technology, Harbin, China; cSchool of Business, State University of New York, Oswego, NY, USA

(Received 15 December 2012; accepted 24 April 2013)

This paper proposes a comprehensive model for studying supply chain versus supply chain competition with resourcelimitation and demand disruption. We assume that there are supply chains with heterogeneous supply network structures thatcompete at multiple demand markets. Each supply chain is comprised of internal and external firms. The internal firms arecoordinated in production and distribution and share some common but limited resources within the supply chain, whereasthe external firms are independent and do not share the internal resources. The supply chain managers strive to developoptimal strategies in terms of production level and resource allocation in maximising their profit while facing competitionat the end market. The Cournot–Nash equilibrium of this inter-supply chain competition is formulated as a variationalinequality problem. We further study the case when there is demand disruption in the plan-execution phase. In such a case,the managers need to revise their planned strategy in order to maximise their profit with the new demand under disruptionand minimise the cost of change. We present a bi-criteria decision-making model for supply chain managers and developthe optimal conditions in equilibrium, which again can be formulated by another variational inequality problem. Numericalexamples are presented for illustrative purpose.

Keywords: supply chain versus supply chain competition; internal and external firms; resource allocation; demand disruption;variational inequality

1. Introduction

As the levels of complexity increase and supply chain in-terdependence becomes more prevalent, increased levelsof risk occur. For example, it was reported that the pro-duction of cars dropped in 2009 significantly because ofthe global economic downturn. As a consequence of thecollapse in demand in the automotive industry and the as-sociated overall reduction of demand for acrylonitrile, therehas been a significant reduction in the supply of acetoni-trile solvent, which is widely used in the pharmaceuticalindustry for laboratory use and in active pharmaceutical in-gredient (API) processes. This makes the price of acetoni-trile increase sharply; many companies switch to methanol,resulting in a higher price of it. Moreover, the available re-sources, such as human resources, machineries, that can beused to deal with such risks are limited.

This article aims at studying the strategic behaviour ofcompeting supply chains facing internal resource limitationand external demand disruption. We place our study on aplatform of supply chain versus supply chain competition.The competing supply chains may have heterogeneous sup-ply network structures and be comprised of internal firmsand external firms. The internal firms work together un-der a centralised supply chain management and share somecommon but limited resources. The external firms are third-

∗Corresponding author. Email: chenzb1983@hrbust.edu.cn

party independent firms, who may simultaneously partici-pate in more than one supply chain. A two-phase model isproposed to study this decision-making problem. In phase1, the model develops the optimal strategy for each com-peting supply chain in terms of production output and re-source allocation. The Cournot–Nash equilibrium of thisinter-supply chain competition is formulated as a varia-tional inequality problem. In phase 2, the model assumesthat a demand disruption is observed during the executionof the original plan. The supply chains then need to modifytheir plans in order to both maximise their revenues un-der the new demand and minimise their plan adjustmentcosts. This bi-criteria decision making is incorporated intothe second variational inequality formulation, which canbe solved for optimal production changes in the disruptionand for new market price at the end market. A numericalexample is provided to illustrate the model and computa-tional results for market chain flows and equilibrium marketprices are presented.

This paper contributes to the literature by integratingtwo subjects of current interest, supply chain competitionand management of demand disruption, into one mathe-matical model.

To date there have been much works studying supplychain competition problem. McGuire and Staelin (1983)

C© 2014 Taylor & Francis

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2 Z. Chen et al.

analysed various retail distribution structures in the contextof two competing manufacturers, and found that the produc-tion substitutability influences the equilibrium distributionstructure. Xiao and Yang (2008) developed a price and ser-vice competition model of two supply chains to investigatethe optimal decisions of players under demand uncertainty,and analysed the effects of the retailers’ risk sensitivity onthe players’ optimal strategies. Research works that studysupply chain competition under uncertainty are includedbut are not limited to Xiao and Yang (2008), Wu and Chen(2003), Wu, Baron, and Berman (2009), Shou, Huang, andLi (2009), Ha and Tong (2008) and Ha, Tong, and Zhang(2011). Particularly, Wu and Chen (2003) considered a sys-tem with two competing supply chains, each chain has onemanufacture and two exclusive retailers. They derived ex-plicit equilibrium of three different industry organisationstructures, and examined the impact of demand uncertainty.Wu et al. (2009) investigated the equilibrium behaviour oftwo competing supply chains in the presence of demand un-certainty. Shou et al. (2009) investigated the competition oftwo supply chains which are subject to supply uncertainty,and examined the decisions of the suppliers and retailersat operational level, design level, and strategic level, re-spectively. Ha and Tong (2008) investigate contracting andinformation sharing in two competing supply chains andcharacterise the equilibrium information sharing decisionsunder different investment costs. Furthermore, Ha et al.(2011) study the incentive for vertical demand informationsharing in competing supply chains with production disec-onomies. However, most of these works have been estab-lished only for the competition between two supply chainsthat have identical and much simplified structure such asone-manufacture one-retailer. In the context of supply chaineconomy, Zhang, Dong, and Nagurney (2003) and Zhang(2006) have presented a model that can be used to study thecompetition among any number of supply chains with com-plex and heterogeneous structures. But their model does notconsider the effect of unanticipated events such as demanddisruption in the plan-execution phase.

Another body of literature that is relevant to this paperis the literature on supply chain disruption management.Hendricks and Singhal (2003, 2005) investigated the short-and long-term effects of supply chain disruption, respec-tively. Kleindorfer and Saad (2005) provided a conceptualframework that reflects the joint activities of risk assess-ment and risk mitigation; both activities are fundamentalto disruption risk management in supply chains. Cerulloand Cerullo (2004), Tomlin (2006) and Skipper and Hanna(2009) examined the use of continuity planning to dealwith the disruption risk. In addition, an increasing numberof publications investigated the coordination mechanism incoping with the disruptive events in the supply chain. Li,Wang, and Cheng (2010) investigated the sourcing strat-egy of a retailer and the pricing strategies of two suppli-ers in a supply chain under supply disruption and devised

a coordination mechanism to maximise the profits of bothsuppliers. Deviation penalty has been incorporated into dis-ruption management models. Particularly, Qi, Bard, and Yu(2004) investigated a one-supplier-one-retailer supply chainthat experiences a demand disruption during the planninghorizon. Xiao, Yu, Sheng, and Xia (2005) studied the co-ordination of a one-manufacturer-two-retailer supply chainwith demand disruption. Xiao and Qi (2008) investigatedthe coordination of a supply chain with one manufacturerand two competing retailers with production disruption,and extended the model to the case with cost and demanddisruptions. Chen and Xiao (2009) considered a supplychain consisting of one manufacturer, one dominant retailerand multiple fringe retailers, and investigated the effective-ness of a linear quantity discount schedule and the Groveswholesale price schedule to coordinate the supply chain af-ter demand disruption. Most recently, Qiang and Nagurney(2012) proposed a humanitarian logistic model for sup-ply/distribution of critical needs in a disruption caused by anature disaster. They consider a general network structureand disruptions that may have an impact on both the networklink capacities and product demand. The problem is studiedin a bi-criteria system optimisation framework for networkperformance. The above-mentioned works all lie under theumbrella of intra supply chain disruption management, andthere is little work studying the disruption management inan inter-supply chain competition environment.

The rest of the paper is organised as follows. In Section2, we present the basic model for supply chain versus sup-ply chain competition in which each supply chain has itsown internal firms and that there are external firms whichcan participate in different supply chains. There are certainand limited resources for each supply chain to be allocatedamong its internal firms. In Section 3, we propose a varia-tional inequality formulation for the Cournot–Nash equilib-rium of the competitive supply chains. Section 4 addressesthe concerns of demand disruption in the execution phase ofthe production plan and provides a model for optimal pro-duction changes. In Section 5, an algorithm is presented tosolve variational inequality problem, and some numericalexamples are presented for illustrative purpose in Section 6.Finally, we give concluding remarks in Section 7.

2. Model elements

Consider that the multiple heterogeneous supply chainscompete at multiple product markets. The supply chainsmay have different supply network structures and differ-ent component firms, which are a combination of internaland external firms. The component firms of a supply chainare coordinated in a cooperative way to produce and de-liver products to end consumer market. We assume that thehomogenous products made by different supply chains areregarded identical at the market.

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International Journal of Systems Science 3

Figure 1 Example of inter-supply chain competition.

We will use m to index the supply chain, where m =1, 2, . . . ,M . The set of firms related to supply chain m isdenoted by Am, which can be partitioned into the set AI

m

for its internal firms and the set AOm for its external firms,

obviously we have Am = AIm ∪ AO

m . We denote a demandmarket by j and a final product by l, where j = 1, 2, . . . , J

and l = 1, 2, . . . , L.For illustrative purpose, we consider a competition

model constituted by two supply chains, as shown inFigure 1. A1 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15} is theset of component firms of supply chain 1, where AI

1 ={6, 7, 8, 9} and AO

1 = {1, 2, 3, 4, 5, 10, 14, 15} are the setsof its internal and external firms, respectively. {1, 2, 3, 4, 5}is the set of its suppliers, {10} is the third-party enterprisewhich transforms the input product p = 5 into the outputproduct p = 7, {14, 15} are the consignees of product 9and product 7 at demand markets 1 and 2. Supply chain 1produces two kinds of final products, product p = 7 andproduct p = 9. While supply chain 2 only produces prod-uct p = 9; furthermore, we assume that the product p = 9from both supply chains are homogeneous.

In order to analyse this problem clearly, we first presentthe definition of market chain, which is analogous to theconcept of a product-market chain in Lakhal, Martel, Ket-tani, and Oral (1999) and M-chain in Zhang (2006).

Definition 1: Market chain: A market chain is a networkof different firms involved in procurement, distribution,and vendition that are associated with the production anddelivery of a final product to the end market.

As shown in Figure 1, the networks which areconsisted of the firms {1, 3, 4, 5, 6, 7, 8, 9, 14} and{2, 3, 4, 5, 6, 7, 8, 9, 14} are two different market chains ofsupply chain 1. Apparently, a market chain only delivers aparticular kind of final product to an end market.

We use S to denote the set of all market chains in thecompetition model and Sm to denote the set of market chainsbelonging to supply chain m, furthermore, sk

m is the kthmarket chain of supply chain m, sk

m ∈ Sm. Use Xskm

to denotethe chain flow of market chain sk

m, and group Xskm

of allmarket chains into the column vector X. In a supply chain,a firm uses the products of immediately preceding firmsas inputs, and then provides products for the immediatelysucceeding firms. We denote the output quantity (in termsof semi-product or final product) of firm a by xa . However,the chain flow in this paper refers to the volume of finalproduct that a market chain delivers to a market; therefore,the chain flow on a market chain may be different from thequantity of output of a firm. One can track back to determinethe quantity of output of a firm from the end product output

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4 Z. Chen et al.

of a market chain. We recall the definition of process rate ofZhang (2006). λask

mis the amount of output of firm a, that is

needed to make one unit of the final product on market chainskm. Take the industry of automobile as an example, if firm

a is responsible for the production of wheels for a marketchain sk

m of family car, then λaskm

= 4. Mathematically, onehas the following equations for xa and Xsk

m:

xa =∑

skm∈Sm

δaskmλask

mXsk

m,

λaskm

={

1, if a participates in skm,

0, otherwise∀a ∈ AI

m

xa =∑skm∈S

δaskmλask

mXsk

m,

λaskm

={

1, if a participates in skm,

0, otherwise∀a ∈ AO

m.

(1)

The firms in a supply chain need some durable re-sources, such as the human resources, machineries, andfactory buildings, etc., to transform input into output. Themeaning of durable resources in this paper is similar tothe definition in Lakhal et al. (1999) and Lakhal, Martel,Kettani, and Oral (2001). Therefore, the amount of finalproduct is not only directly affected by the price of rawmaterials but also by the cost of using durable resources.

In this paper, we assume that it is probable for the in-ternal firms of a supply chain to share each other’s durableresources because of the partnerships of these firms. LetRm denote the set of all distinct durable resources of supplychain m. The amount of durable resource r possessed bysupply chain m during the decision horizon is denoted byh0

arm. We assume that the enterprise in a supply chain cannot

alter its durable resources base during the decision horizon.This assumption is different from Lakhal et al. (2001) andLakhal (2006). Let garm

denote the amount of durable re-source r used by firm a, a ∈ AI

m, to produce a unit of outputproduct. Hence, the amount of durable resource r neededfor firm a to accomplish its work is

harm= garm

xa, ∀a ∈ AIm, r ∈ Rm. (2)

The required amount of durable resource r for supplychain m is

hrm=

∑a∈AI

m

harm=∑a∈AI

m

garmxa =

∑a∈AI

m

∑skm∈Sm

garmδask

mλask

mXsk

m,

r ∈ Rm. (3)

Therefore, for a given resource r , r ∈ Rm, the amount ofresource available for supply chain m must satisfy the fol-lowing constrain

hrm≤ h0

rm, r ∈ Rm. (4)

Particularly, for an external firm a of supply chain m,a ∈ AO

m , we assume that the firm is obliged to meet the

supply chain’s requirement and has the capacity to do so.Let ca (xa) denote the total revenue of the external firm a,a ∈ AO

m , which depends on the amount of output of the firm.ca (xa) is the payout of the supply chains to the external firma, it can also be seen as the cost of these supply chains thatemploy this common external firm. In our model, an exter-nal firm is allowed to participate in more than one marketchain and even in more than one supply chain. Therefore,some cost sharing mechanism must be introduced to im-pute the cost ca (xa) to the market chains it participates in.Here, we adopt the proportional allocation mechanism asin Lakhal et al. (2001) and Zhang (2006) to impute ca (xa)to a particular market chain. In the proportional imputationmechanism, for firm a, a ∈ AO

m , the cost imputed to marketchain sk

m can be expressed as

Caskm

= ca (xa)

xa

δaskmλask

mXsk

m,

δaskm

={

1, if a participates in skm

0, otherwise. (5)

From the above description, the competition between supplychains in this model can be understood in the followingthree perspectives. First, according to Equations (1) and (5),the cost Cask

mdepends upon the production pattern of the

interactive supply chains, consequently, these supply chainsnot only compete with each other at the end product marketbut also interact in the procurement of raw material andin the use of third-party enterprises. Second, the differentproducts made in a same supply chain also compete witheach other for limited durable resource within the supplychain, such as product 7 and product 9 from supply chain1 in Figure 1. Third, the different products from differentsupply chains may compete with each other in acquiring rawmaterials. In our example, product 7 from supply chain 1 iscompeting with product 9 from supply chain 2 in acquiringthe same raw material.

We assume that xa �= 0 for a given external firm a. Thisassumption can be understood as follows: every externalfirm will take some effective measures to ensure its sur-vival in the competition. Although every supply chain mayproduce more than one kind of final products, the same kindof products made by different supply chains is consideredhomogeneous. Let ρl

j denote the retail price for the final

product l at market j . βj

skm

and ηlskm

are zero-one variables,where

βj

skm

={

1, if end market of market chain sjmis j

0, otherwise

ηlskm

={

1, if end market of market chain sjmis j

0, otherwise.

(6)

For convenience, we let ρskm

denote the retail price forthe final product on market chain sk

m, mathematically, the

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International Journal of Systems Science 5

relationship between ρskm

and ρlj can be expressed as

ρskm

=J∑

j=1

L∑l=1

βj

skmηl

skmρl

j . (7)

Therefore, the total revenue for supply chain m can besimplified as

∑skm∈Sm

Xskmρsk

m. In the next section, we will

establish the equilibrium conditions for the multiple supplychain competition model and develop the optimal produc-tion and sale plan without demand disruption.

3. Equilibrium of supply chain competition

Before the selling season, every supply chain will developthe optimal production and sale plan to maximise its profit,and the optimisation problem for supply chain m can beexpressed as

max um =∑

skm∈Sm

Xskmρsk

m−∑a∈AO

m

∑skm∈Sm

ca (xa)

xa

δaskmλask

mXsk

m

−∑r∈Rm

γrm

⎛⎝∑

a∈AIm

∑skm∈Sm

garmδask

mλask

mXsk

m

⎞⎠ ,

s.t. Xskm

≥ 0, skm ∈ Sm

hrm≤ h0

rm, r ∈ Rm,

(8)

where xa = ∑skm∈S δask

mλask

mXsk

m, ∀a ∈ AO

m , γrmdenotes the

cost of using a unit of durable resource r in supply chainm. The objective function of this optimisation problem ex-presses that the profit of supply chain m is the differencebetween its total revenue minus the payout to its exter-nal enterprises and the cost of using durable resources.Xsk

m≥ 0, sk

m ∈ Sm are the non-negativity constraints for allmarket chains. The constraints hrm

≤ h0rm

, r ∈ Rm expressthat the resource available for supply chain cannot surpassthe amount it owns.

We assume that these supply chains compete in a non-cooperative fashion, and each one is facing the same opti-misation problem like (8) and trying to maximise its ownprofit, given the actions of the other supply chains. We alsoassume that all imputed cost functions Cask

mare continu-

ous and convex, and then the optimality conditions for allsupply chains can be expressed as the following variationalinequality: determine X∗

skm

∈ � satisfying

∑skm∈S

⎡⎣∑

a∈AOm

(∂ca

(x∗

a

)∂Xsk

m

δaskmλask

mX∗

skm

x∗a

+ ca

(x∗

a

) δaskm

λaskm

(x∗

a −δaskm

λaskm

X∗skm

)(x∗

a )2

⎞⎠

+∑r∈Rm

γrm

⎛⎝∑

a∈AIm

garmδask

mλask

m

⎞⎠− ρsk

m

⎤⎦

×[Xsk

m− X∗

skm

]≥ 0,∀Xsk

m∈ �,

(9)

where � = {Xskm|xsk

m≥ 0,

∑a∈AI

m

∑skm∈Sm

garmδask

mλask

mXsk

m

≤ h0rm

,∀skm ∈ S,∀r ∈ Rm,m = 1, . . . ,M} is the feasible

set for this problem.The optimality conditions as expressed by (9) have a

nice economic interpretation. With the limitation durableresources, there will be a positive flow on market chain sk

m,if the price that the consumer can accept is equal to themarginal production cost associated with it. However, if themarginal production cost exceeds the willing payment forthe product, the flow on the market chain will be zero.

Next, we will discuss how to determine the retail pricefor the products at the end markets. Since the same finalproduct from different supply chains is homogeneous, nosingle supply chain can increase the price above the pricecharged by the other supply chains at the market. However,it is also possible for a supply chain to affect the marketprice by changing the quantity of its supplies. Hence, letρl

j and dlj (ρl

j ) denote the price and demand for the productl at the end market j , the demand function is continuousand monotonically decrease with ρl

j . According to the spa-tial price equilibrium conditions in Nagurney, Dong, andZhang (2002), the market equilibrium conditions can takethe form: for all product l and market j , l = 1, 2, . . . , L,j = 1, 2, . . . , J satisfy

dlj

(ρl∗

j

)⎧⎪⎪⎨⎪⎪⎩

≤∑skm∈S

X∗skmβ

j

skmηl

skm, ifρl∗

j = 0

=∑skm∈S

X∗skmβ

j

skmηl

skm, ifρl∗

j > 0. (10)

The equilibrium conditions in turn can be expressed asa variational inequality problem: determine ρl∗

j ∈ RJL+ such

that

J∑j=1

L∑l=1

⎡⎣∑

skm∈S

X∗skmβ

j

skmηl

skm

− dlj

(ρl∗

j

)⎤⎦× [

ρlj − ρl∗

j

] ≥ 0, ∀ρlj ∈ RJL

+ , (11)

where ρ is the JL-dimensional column vector of the retailerprice of these products. In the equilibrium conditions of themultiple supply chain competition model without demanddisruption, the supply chains and the demand markets mustsatisfy the variational inequalities (9) and (11), respectively.Same as the theory given in Nagurney et al. (2002), we canobtain the following theory for the model.

Theorem 1: The multiple supply chains are in equilibriumconditions, if and only if the following variational inequality

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6 Z. Chen et al.

is satisfied

∑skm∈S

⎡⎣∑

a∈AOm

(∂ca

(x∗

a

)∂Xsk

m

δaskmλask

mX∗

skm

x∗a

+ ca

(x∗

a

) δaskmλask

m

(x∗

a − δaskmλask

mX∗

skm

)(x∗

a

)2

⎞⎠

+∑r∈Rm

γrm

⎛⎝∑

a∈AIm

garmδask

mλask

m

⎞⎠− ρ∗

skm

⎤⎦

×[Xsk

m− X∗

skm

]+

J∑j=1

L∑l=1

⎡⎣∑

skm∈S

X∗skmβ

j

skmηl

skm

− dlj

(ρl∗

j

)⎤⎦×[ρl

j − ρl∗j

]≥ 0, ∀ρl

j ∈ RJL+ , Xsk

m∈ �.

(12)

Proof: The summation of (9) and (11), yields inequality(12). �

The variational inequality (12) can be written as: deter-mine y∗ ∈ K such that

F (y∗) , (y − y∗) ≥ 0,∀y ∈ K,

where

F (y) =(

F1 (y)F2 (y)

)y =

(X

ρ

)

F1 (y) =(

− ∂um

∂Xskm

: skm ∈ S

)

F2 (y) =(∑

skm∈S

X∗skmβ

j

skmηl

skm

− dlj

(ρl

j

): l = 1, 2, . . . , L;

j = 1, 2, . . . , J

)K = {

(x, ρ)T |X ∈ �, ρ ∈ RJL+}.

The following two theorems guarantee the existence anduniqueness of the solution to variational inequality (12).

Theorem 2: In (1) there exists a sufficiently large W , suchthat for any sk

m ∈ S, − ∂um

∂Xskm

> 0 for all Xskm

with Xskm

≥W . (2) for any sk

m ∈ S, a ∈ AOm and m = 1, 2, . . . ,M ,

ca (xa )xa

δaskmλask

mXsk

mhave continuous first derivatives respect

to Xskm

on K , then the variational inequality (12) admits atleast one solution.

Proofs of all the theorems are given in Appendix 1.

Theorem 3: For every skm ∈ S, a ∈ AI

m, and m =1, 2, . . . , M , if the derivative of

δaskm

λaskm

ca (xa )

xarespect to Xsk

mis

continuous and strictly monotone in Xskm

on the set K , thereis a unique product vector X and a unique demand pricevector ρ satisfying the equilibrium conditions of these com-

peting supply chains, in other words, there exists a uniquesolution to the problem (12).

Once we have solved the variational inequality (12), wecan get the solution that maximises the decision-makingproblem (8) for every supply chain. Thus, every supplychain in this competition model can develop its optimal pro-duction and sale plan accordingly. The following questionsare also solved, such as the supply chain should producewhat kinds of products, and how to produce these products,as well as the optimal amount of these products.

But in the plan-execution phase, various unanticipatedevents may disrupt the equilibrium conditions and makethe sound plan deviate from its intended course. Hence, thesupply chain should adjust the optimal plan in real timeto cope with these disruptions. Our aim is to develop ascheme for revising a production plan after a disruptionhas occurred, rather than proposing new ways of dealingwith demand uncertainties in the planning stage. Of course,formulating a good plan based on certain probability as-sumptions is important, but realistically, it is not possiblefor the decision-maker to anticipate all contingencies. In thenext section, we will analyse how to adjust the productionplan to reach the profit goal while minimising the nega-tive impact caused by the disruptions in the supply chaincompetition environment.

4. Adjusted equilibrium under demand disruption

In this section, we turn to study how supply chains adjusttheir production plans in a competitive environment underdemand disruption. When any unanticipated event occursin the plan-execution phase causing demand disruption, theoriginal optimal strategy of a supply chain developed inSection 3 will be no longer optimal. The supply chain man-agers will then need to revise their planned productions inaccount for the disrupted demand. We note that such sit-uation is more complex than that of resolving the originalequilibrium condition with the updated new demand. Thisis because the original plans are already in the executionphase. The managers need not only to cope with the newdemand but also with what they have already done in ex-ecuting the old plan, such as contracts signed with theirsuppliers, materials already purchased, facility expanded,workers hired. It costs to alter those actions and some canbe very expensive or even impossible.

To manage supply chain disruption, Yu and Qi (2004)argues the importance to seek a balance between the max-imisation of profit and minimisation of the deviation to theoriginal plan. In the same spirit we propose a bi-criteriamodel for supply chain managers in coping with demanddisruption and inter-chain competition with two objectivesfor profit maximisation under the new demand and minimi-sation of the cost in adjustment of the original plan.

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International Journal of Systems Science 7

For convenience, we denote the variables in the case ofdisruption with a hat to distinguish them from the corre-sponding variables in the original plan defined in Section2 and Section 3. The first criterion is the profit maximi-sation of the new plan under the disrupted demand. Wepropose our second criterion as the minimisation of theaggregated adjustment costs on all the internal firms of asupply chain. The proposed criterion is designed to captureadjustment cost in complex situations. First, the cost of pro-duction adjustment is actually incurred by the internal firmsthat comprise the supply chain and, because of differencesamong these firms, the cost of production adjustment maybe different. Hence, evaluating the adjustment cost at thefirm level is more appropriate than evaluating the deviationon the chain level. Second, some internal firms may par-ticipate in two market chains of which one increases andthe other decreases output in the disruption, the deviationeffects can be cancelled out on this common firm and resultin no real adjustment of production, thus no cost at all. Yet,when evaluating the chain adjustment cost on chain level,it would have accounted for double effects.

For an internal firm a, let Za be the adjustment costfor firm a as a function of the production change on thefirm, which means Za is a function of xa − x∗

a . Let X∗

denote the equilibrium production pattern of all the com-peting supply chains before demand disruption, which isdetermined by variatonal inequality (12). X∗ determines aunique production pattern x∗

a of firm through the flow con-servation equation (1), simultaneously, the amount of re-sources allocated to each firm is also uniquely determinedby (3). Our second criterion is to minimise the aggregatedadjustment costs on all the internal firms that participatein the supply chain. The multi-criteria decision-makingproblem for supply chain m, m = 1, 2, . . . , M , can beexpressed by

max um =∑

skm∈Sm

Xskmρsk

m−∑a∈AO

m

∑skm∈Sm

ca (xa)

xa

δaskmλask

mXsk

m

−∑r∈Rm

γrm

⎛⎝∑

a∈AIm

∑skm∈Sm

garmδask

mλask

mXsk

m

⎞⎠

−∑a∈AI

m

Za

(xa − x∗

a

)s.t. Xsk

m≥ 0, sk

m ∈ Sm

hrm≤ h0

rm, r ∈ Rm.

(13)

Suppose that the supply chains compete in a non-cooperative manner and all Cask

mand Za are continuous and

convex, and then the equilibrium strategies of all the sup-ply chains in a Cournot–Nash sense should satisfy the fol-lowing variational inequality problem, determine X∗ ∈ �

satisfying

∑skm∈S

⎡⎣∑

a∈AOm

(∂ca

(x∗

a

)∂Xsk

m

δaskmλask

mX∗

skm

x∗a

+ ca

(x∗

a

) δaskmλask

m

(x∗

a − δaskmλask

mX∗

skm

)(x∗

a

)2

⎞⎠

+∑a∈AI

m

∂Za

(x∗

a − x∗a

)∂X∗

skm

+∑r∈Rm

γrm

⎛⎝∑

a∈AIm

garmδask

mλask

m

⎞⎠− ρsk

m

⎤⎦

×[Xsk

m− X∗

skm

]≥ 0, ∀Xsk

m∈ �,

(14)

where � = {Xskm|Xsk

m≥ 0,

∑a∈AI

m

∑skm∈Sm

garmδask

mλask

m

Xskm

≤ h0rm

,∀skm ∈ S,∀r ∈ Rm,m = 1, 2, . . . , M} is the

feasible set of the variational inequality.Variational inequality (14) has a nice economic inter-

pretation. If there is a positive flow Xskm

> 0 on market chainskm, then the price that the consumer at the market is will-

ing to pay, ρskm, should be equal to the marginal production

cost plus the production adjustment cost under the demanddisruption associated with this market chain.

We use dlj to denote the demand disruption of product

l at end market j , a positive dlj represents the increased

market demand and a negative dlj represents the decreased

market demand. Assume [dl−j , dl+

j ] is the range of demand

disruption dlj , and that dl−

j + dlj (0) > 0, the market equi-

librium conditions under demand disruption are, product l

at market j , l = 1, 2, . . . , L, j = 1; , 2, . . . , J

dlj + dl

j

(ρl∗

j

)⎧⎪⎪⎨⎪⎪⎩

≤∑skm∈S

X∗skmβ

j

skmηl

skm, if ρl∗

j = 0

=∑skm∈S

X∗skmβ

j

skmηl

skm, if ρl∗

j > 0. (15)

Taking X∗ ∈ � as exogenous variables, the equilibriumconditions (15) can be expressed as a variational inequalityproblem: determine ρ ∈ RJL

+ such that

J∑j=1

L∑l=1

⎡⎣∑

skm∈S

X∗skmβ

j

skmηl

skm

− (dl

j + dlj

(ρl∗

j

))⎤⎦[ρl

j − ρl∗j

] ≥ 0,∀ρlj ∈ RJL

+ , (16)

where ρ is the JL-dimensional column vector of ρlj .

Combining variational inequalities (14) and (16), wehave the following integrated variational inequality problemas a complete formulation for the equilibrium productionstrategy for every supply chain in the competition and the

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8 Z. Chen et al.

equilibrium market prices under demand disruption

∑skm∈S

⎡⎣∑

a∈AOm

(∂ca

(x∗

a

)∂Xsk

m

δaskmλask

mX∗

skm

x∗a

+ ca

(x∗

a

) δaskmλask

m

(x∗

a − δaskmλask

mX∗

skm

)(x∗

a

)2

⎞⎠

+∑a∈AI

m

∂Za

(x∗

a − x∗a

)∂Xsk

m

+∑r∈Rm

⎛⎝γrm

∑a∈AI

m

garmδask

mλask

m

⎞⎠− ρ∗

skm

⎤⎦×

[Xsk

m− X∗

skm

]

+J∑

j=1

L∑l=1

⎡⎣∑

skm∈S

X∗skmβ

j

skmηl

skm

− (dl

j + dlj

(ρl∗

j

))⎤⎦× [

ρlj − ρl∗

j

] ≥ 0,∀Xskm

∈ �,∀ρlj ∈ RJL

+ .

(17)

Denoting d = (d1

1 ,d21 , . . . ,dL

J

)Tas a col-

umn vector of demand disruption, the variational in-equality (17) can be written as a standard parametricvariational inequality V I (F (y, d) , K) form: for anygiven d ∈ B, where B = {d|dl−

j ≤ dlj ≤ dl+

j , j =1, 2, . . . , J, l = 1, 2, . . . , L}, determine y∗ ∈ K suchthat

F (y∗,d) , (y − y∗) ≥ 0, ∀y ∈ K, (18)

where

F (y, d) =(

F1 (y, d)F2 (y, d)

)y =

(X

ρ

)

F1 (y, d) =(

− ∂um

∂Xskm

: skm ∈ S

)

F2 (y, d) =(∑

skm∈S

Xskmβ

j

skmηl

skm

− (dl

j + dlj

(ρl

j

)):

l = 1, 2, . . . , L; j = 1, 2, . . . , J

),

K = {(X, ρ)T |X ∈ �, ρ ∈ RJL+ }.

By solving the parametric variational inequality (18),we will obtain a solution y∗ of optimal production strat-egy for each supply chain that maximises the bi-criteriadecision-making problem (13) and the market equilibriumprices under the demand disruption for any given d.

The feasible set K for parametric variational inequality(18) is independent of d. Then the parametric variationalinequality (18) is much easier for analysis. Furthermore, d

is used to depict the disruption of market demand, and the

variational inequality depicts the Nash games between thesupply chains. Hence, the parametric variational inequal-ity (18) integrates two subjects of current interest, supplychain versus supply chain competition and management ofdemand disruption, into one mathematical model. The fol-lowing theorem guarantees the uniqueness of the solutionto VI(F (y, d) , K).

Theorem 4: For the parametric variational inequality(18), if for any d ∈ B, F (y, d) is strictly monotone iny, and for all d ∈ B and F (y, d) is local Lipschitz con-tinuous, then there exists an implicit function y = ϕ (d)which is the unique solution to (18). Furthermore, the im-plicit function determined by the variational inequality iscontinuous on the compact and convex set B.

From Theorem 4, we can derive the following theoremfor the equilibrium model of supply chain competition withdemand disruption and resource limitation.

Theorem 5: For every skm ∈ S, a ∈ AI

m, and m =1, 2, . . . ,M , if (1) the derivative of

δaskm

λaskm

ca (xa )

xaand

Za

(xa − x∗

a

)respect to Xsk

mis continuous and strictly

monotone in Xskm

on the set K , dlj

(ρl

j

)is continuous and

strictly monotone in ρlj for every j = 1, 2, . . . , J and l =

1, 2, . . . , L. (2) for all d ∈ B and y ∈ K , the derivative

ofδaskm

λaskm

ca (xa )

xaand Za

(xa − x∗

a

)respect to Xsk

mis local

Lipschitz continuous, then there exists an implicit functionwhich is the unique solution to (17). Furthermore, the im-plicit function determined by the variational inequality iscontinuous on the compact and convex set B.

In conclusion to this section, when a demand disruptiond is observed or predicted, one can derive the new produc-tion plan for every supply chain by solving the parametricvariational inequality problem (17), which maximises thebi-criteria decision-making problem (13) for each supplychain accounting not only the revenue maximisation butalso the plan adjustment cost minimisation. The parametricvariational inequality problem (17) will also give the newequilibrium market price under demand disruption.

5. The algorithm

In this section, an algorithm is presented which can beapplied to solve variational inequality problem V I (F,K)(12).

Step 0. Initialisation – set X0 ∈ K , α and ε.Step 1. Computation – solve the following convex

quadratic programming problem at iteration τ :

Xτ+1 = arg minX∈K

1

2XT X − (Xτ + αF (Xτ ))T X.

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Table 1. Bills of process rates.

The firm

The process rate of every firm associate to market chain skm

Market chain λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10 λ11 λ12 λ13 λ14 λ15

Supply chain 1 s11 1 – 1 1 1 1 1 1 1 – – – – 1 –

s21 – 1 1 1 1 1 1 1 1 – – – – 1 –

s31 1 – 1 1 1 1 – 1 1 1 – – – 1 –

s41 – 1 1 1 1 1 – 1 1 1 – – – 1 –

s51 1 – 1 1 – 1 1 – – – – – – – 1

s61 – 1 1 1 – 1 1 – – – – – – – 1

Supply chain 2 s12 1 – 1 1 1 – – – – 1 1 1 1 1 –

s22 – 1 1 1 1 – – – – 1 1 1 1 1 –

Step 2. Convergence verification – if Xτ+1 − Xτ ≤ ε,then stop; otherwise, set τ := τ + 1, and go toStep 1.

We now state the convergence result for the algorithmfor this model.

Theorem 6: (Convergence). For every skm ∈ S, a ∈ AI

m,and m = 1, 2, . . . ,M , assume that the derivative ofδaskm

λaskm

ca (xa )

xarespect to Xsk

mis strictly monotone and Lips-

chitz continuous on the set K . Then the algorithm describedabove converges to the solution of the variational inequal-ity (12).

The detail of the algorithm’s convergence can be re-ferred to Nagurney and Zhang (1996). Furthermore, forany given d ∈ B, the parametric variational inequality(18) can also be solved in the same way.

6. Numerical examples

In this section, we will present a numerical example to illus-trate the model. In Section 6.1, we will focus on how supplychains develop their original production plan. In Section6.2, we present how to cope with the demand disruption.

As shown in Figure 1, there are two competing supplychains, supply chain 1 produces two products p = 7 andp = 9, supply chain 2 only produces product 9. Externalfirms 1, 2, 3, 4, and 5 are the raw material suppliers of thetwo supply chains, a third-party enterprise 10 is responsibleto transform the input product p = 5 to output productp = 7, and firms 14 and 15 are the consignees of the twosupply chains at markets 1 and 2, respectively. We assumethat the cost functions of these external firms are

c1 (x1) = 210 + 10.7x1, c2 (x2) = 280 + 10.1x2,

c3 (x3) = 400 + 9x3, c4 (x4) = 250 + 14x4,

c5 (x5) = 180 + 7.2x5, c10(x10) = 7x10,

c14(x14) = 4x14, c15(x15) = 3x15.

s11 , s

21 , s

31 , s4

1 , s51 , s

61 are the market chains belonging to sup-

ply chain 1, particularly, they are comprised of the followingsets of firms, respectively

s11 = {1, 3, 4, 5, 6, 7, 8, 9, 14} , s2

1 = {2, 3, 4, 5, 6, 7, 8, 9, 14} ,

s31 = {1, 3, 4, 5, 6, 8, 9, 10, 14} ,

s41 = {2, 3, 4, 5, 6, 8, 9, 10, 14} , s5

1 = {1, 3, 4, 6, 7, 15} ,

s61 = {2, 3, 4, 6, 7, 15} .

The two market chains of supply chain 2 are

s12 = {1, 3, 4, 5, 10, 11, 12, 13, 14} ,

s22 = {2, 3, 4, 5, 10, 11, 12, 13, 14} .

The demand functions for products 9 and 7 at the end mar-kets are

d91

(ρ9

1

) = 1000 − 0.2ρ91 , d7

2

(ρ7

2

) = 300 − 0.1ρ72 .

The bills of process rates and durable resources of each firmrespecting to market chain sk

m are shown in Tables 1 and 2.

6.1. Original schedule before disruption

The chain flows of each market chain are

Xs11

= 78.8793, Xs21

= 82.5840, Xs31

= 85.1243,

Xs41

= 85.3215

Xs51

= 72.6370, Xs12

= 163.9510, Xs22

= 164.1309.

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10 Z. Chen et al.

Table 2. Bill of durable resources.

The resource

Supply chain 1 Supply chain 2

The firm RES1 RES2 RES3 RES4 RES1 RES2 RES3 RES4

The needed resources for one-unit of output garm Firm 6 2 1 – – – – – –Firm 7 – – 2 1 – – – –Firm 8 – 1 – 2 – – – –Firm 9 1 – – 1 – – – –Firm 10 – – – – 2 1 – –Firm 11 – – – – – 1 – 2Firm 12 – – – – 1 – – 1

Cost of using RES r γ r 2.5 3.5 4.5 5.5 2.5 3.5 4.5 5.5Available RES h0

r 2000 1500 1000 2000 1500 800 – 1500

Note: RES is short for resource.

Table 3. Original production plans of the two supply chains.

The internal firms

Firm 6 Firm 7 Firm 8 Firm 9 Firm 11 Firm 12 Firm 13

Output of firms xa 474.0226 303.5768 326.9091 326.9091 328.0819 328.0819 328.0819

Allocation of durable resources RES1 948.0452 – – 326.9091 656.1638 – 328.0819RES2 474.0226 – 326.9091 – 328.0819 328.0819 –RES3 – 607.1536 – – – – –RES4 – 303.5768 653.8182 326.9091 – 656.1638 328.0819

The demand prices of products 9 and 7 on the demandmarket are

ρ91 = 1725.0440,

ρ72 = 1528.8640.

The equilibrium solution determines the output of allfirms and the amount of resources allocated to each firmthrough the flow conservation equations (1) and (3), re-spectively. The optimal original production plans of thetwo supply chains are shown in Table 3.

Furthermore, supply chain 1 should purchase 231.6406units material 1 from supplier 1, 242.3820 units material1 from supplier 2, 474.0226 units material 2 from supplier3, 474.0226 units material 3 from supplier 4, and 326.9091units material 6 from supplier 5. Supply chain 2 shouldpurchase 163.951units material 1 from supplier 1, 164.1309units material 1 from supplier 2, 328.0819 units material 2from supplier 3, 328.0819 units material 3 from supplier 4,and 328.0819 units material 6 from supplier 5. Particularly,the volume of product 9 and product 7 produced by supplychain 1 is 326.9091 and 147.1136. The volume of product9 produced by supply chain 2 is 328.0819.

6.2. Reschedule under demand disruption

When the demand disruption occurs in the plan-executionphase, the supply chains need to reschedule their optimalproduction and sale plan to reflect the change. In this sub-section, we will use the same numerical example to illus-trate the reschedule problem under demand disruption. Weassume that the disruption range for of d9

1 and d72 are

[−400, 400] and [−100, 100] , respectively. Consequently,the demand functions of products 9 and 7 at the end marketsbecome

d91

(ρ9

1

) = 1000 + d91 − 0.2ρ9

1 ,

d72

(ρ7

2

) = 300 + d72 − 0.1ρ7

2 .

We assume that the adjustment cost functionZa(xa − x∗

a ) = (xa − x∗a )2, for every internal firm of the

two supply chains. The Euler method is used again to com-pute the market chain flows and the profits of the two supplychains, by choosing d9

1 ∈ {−400,−240, 240, 400} andd7

2 ∈ {−80, 0, 80}, as shown in Table 4 and 5.Some economic implications of the solution can be ob-

served from comparing the equilibrium solutions in the twotables above. If the demand of product 9 is constant, thedemand of product 7 increases, while the output of product9 of supply chain 1 decreases and the flows of s5

1 , s61 , s1

2 ,

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International Journal of Systems Science 11

Table 4. The market chain flow and supply chain profit with demand disruptions.

The demand disruption of product 9 and product 7(d91 , d7

1 )

(−400, −80) (−400, 0) (−400, 80) (−240, −80) (−240, 0) (−240, 80)

The flow of market chain Xs11

73.2468 53.3434 20.4041 76.708 56.8138 23.8991Xs2

197.9425 95.9194 92.0393 98.0137 95.9671 92.0569

Xs31

32.259 39.8882 55.3598 49.8227 57.4646 72.9517Xs4

132.8157 40.8876 57.1366 50.3014 58.3753 74.6144

Xs51

57.2813 73.3789 88.6821 55.8817 71.9851 87.3011Xs6

162.4729 82.3658 103.8916 60.3654 80.2533 101.7664

Xs12

115.5642 116.4564 116.9461 134.5673 135.4562 135.9445Xs2

2116.0425 117.2968 118.4025 134.9938 136.2508 137.3573

The profit of supply chains with adjusting supply chain 1 216,930 324,620 462,420 372,750 477,220 611,910supply chain 2 106,690 113,960 119,960 258,970 267,410 274,380

The profit of supply chains without adjusting supply chain 1 −20,335 97,356 21,505 241,190 358,880 476,570supply chain 2 −116,060 −116,060 −116,060 146,410 146,410 146,410

Table 5. The market chain flow and supply chain profit with demand disruptions.

The demand disruption of product 9 and product 7 (d91 , d7

1 )

(240, −80) (240, 0) (240, 80) (400, −80) (400, 0) (400, 80)

The flow of market chain Xs11

92.7821 67.648 34.7531 98.0721 65.2952 32.3822Xs2

198.6249 95.6832 91.6778 106.1754 102.2549 98.3324

Xs31

99.3998 110.1944 125.7422 115.2741 130.9824 146.7429Xs4

199.5314 110.8419 127.0987 115.4584 131.8479 148.3297

Xs51

51.987 67.8918 83.2321 49.5218 64.8466 80.1458Xs6

153.2209 73.8266 95.321 51.2361 72.6872 94.1752

Xs12

191.6298 192.4768 192.9567 199.9129 199.5982 199.2767Xs2

2191.7537 193.0797 194.1929 200.0871 200.4018 200.7233

The profit of supply chain with adjusting supply chain 1 981,580 1,076,300 1,201,600 1,277,600 1,373,300 1,498,300supply chain 2 854,500 866,000 875,860 1,083,100 1,092,300 1,101,500

The profit of supply chain without adjusting supply chain 1 92,580 1,043,500 1,161,200 1,187,300 1,305,000 1,410,500supply chain 2 833,800 833,800 833,800 1,016,300 1,016,300 1,016,300

and s22 increase. This can be explained as follows. The sup-

ply chain 1 tends to produce more product 7 because of itsincreasing market demand, which leads to supply chain l’sless production and weaker competitiveness on product 9 atmarket 1 due to the resources limitation. As the demand ofproduct 7 increased, the profit of supply chain 1 and supplychain 2 also increased. Hence, in the supply chain com-petition environment, a local disruptive event will cause aglobal disruption of the planned production and make greateffect on the revenue of the supply chains. The above phe-nomenon is the so-called ripple effects in the supply chain.The preceding examples demonstrate the type of supplychain competition with demand disruption problems thatcan be solved using the two-phase model given in this pa-per. Furthermore, by a comparison to the case of keeping

the original production decisions, reschedule is proofed tobe more effective when demand disruption occurs. This nu-merical example also shows the rationality of the two-phasemodel.

7. Conclusions

In this paper, we study supply chain versus supply chaincompetition with resource limitation and demand disrup-tion. The supply chains may have heterogeneous structuresand are comprised of internal firms and external firms.The internal firms work together under a centralised man-agement and share some common but limited resources.The external firms are the third-party independent firmsthat may simultaneously participate in more than one

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12 Z. Chen et al.

supply chains. A two-phase model is proposed to studythis decision-making problem. In phase 1, the model devel-ops the optimal strategy for each competing supply chainin terms of production output and resource allocation. TheCournot–Nash equilibrium of this inter-supply chain com-petition is formulated as a variational inequality problem.In phase 2, the model assumes that a demand disruption isobserved in the execution of the original plan. The supplychains then need to modify their plans in order to both max-imise their revenues under the new demand and minimisetheir plan adjustment costs. This bi-criteria decision mak-ing is incorporated into the second variational inequalityformulation, which can be solved for optimal productionchanges in the disruption and for new market price at theend market. A numerical example is provided to illustratethe model and computational results are presented for mar-ket chain flows and equilibrium market prices. Althoughthe example seems a little simple that only two supplychains with heterogeneous structures are considered, thetwo-phase model is also available to real life problems withlarger scale. One of the crucial pre-conditions to apply thismodel in practice is the acquisition of the cost informationof other supply chains which is quite difficult; however, thedeveloping information technologies are making such in-formation access to one supply chain in various ways, forexample, many competitors information can be acquiredfrom BusinessObjects (an SAP company).

AcknowledgementsWe would like to thank the anonymous referees and the editor fortheir many helpful suggestions and insightful comments, whichhave significantly improved the content and presentation of thispaper. The third author acknowledges the two Chinese institutionsfor hosting his sabbatical visit. He was appointed as a Zi Jin scholarchair professor at Nanjing University of Science and Technologyand as a Zi Qiang scholar chair professor at Shanghai Universityin China during the course when this work was completed.

FundingThis research was supported in part by the National Natural Sci-ence Foundation of China [grant number 71171069], [grant num-ber 70871031]; the Humanities and Social Sciences Foundationof Heilongjiang Province [grant number 12522052]; the Doc-toral Foundation of Taiyuan University of Science and Technology[grant number W20122001].

Notes on contributorsZhaobo Chen received the MSc and PhDdegrees from School of Management inHarbin University of Science and Technol-ogy, Harbin, China, in 2009 and 2011, re-spectively. He is currently a tutor of MS atthe College of Economy and Management,Taiyuan University of Science and Technol-ogy, Taiyuan, China. His research interestsinclude applied optimisation and applica-

tions in supply chain management and other managerial areas.

Chunxian Teng received the master’s de-gree in system engineering optimisationfrom the Harbin Institute of Technology,Harbin, China, in 1987. He has been a pro-fessor and a tutor of PhD at the Harbin Uni-versity of Science and Technology, Harbin,China, since 2003. His research interests in-clude supply chain management and optimi-sation theory.

Ding Zhang is a professor of managementscience and director of the MBA programmein the School of Business at the State Uni-versity of New York at Oswego. He receivedBS in mathematics from the University ofScience and Technology of China, an MSin operations research from Tsinghua Uni-versity, in Beijing, China, and a PhD in in-dustrial engineering from the University of

Massachusetts Amherst. He was an assistant professor at ShanghaiJiao Tong University (China), a research associate at the Univer-sity of Massachusetts Amherst, and a research fellow at the HongKong Polytechnic University. His research interests include: trans-portation science, supply chain management, spatial economics,and other competitive network problems.

Jiayi Sun received the master’s degree in ba-sic mathematics from the Harbin Universityof Science and Technology, Harbin, China,in 2010, where she is currently pursuingthe PhD degree in management science andengineering. Her research interests includeclosed-loop supply chain management andoptimisation theory.

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Appendix 1

Proof of Theorem 2: (1) It is easy to verify that the feasible set Kis convex. Furthermore, it is reasonable to expect that the profit ofany supply chain m, um, would decrease whenever its output hasbecome sufficiently large, that is, when um is differentiable, ∂um

∂Xskm

is negative for sufficiently large Xskm

. Hence, the optimal output ofthese supply chains will not exceed W . Therefore, the conditionsof the first part in Theorem 2 are enough to restrict variationalinequality (12) to satisfy the feasible set boundedness condition.

(2) Because for any skm ∈ S, a ∈ AO

m , and m = 1, 2, . . . , M ,ca (xa )

xaδask

mλask

mXsk

mhave continuous first derivatives respect to Xsk

m

on K , it is easy to see F1 (y) is continuous on K . By Theorem 3.1in Harker and Pang (1990), Theorem 2 is proved. �Proof of Theorem 3: For any y ′ and y ′′ in K ,

[F(y ′)− F

(y ′′)]T [y ′ − y ′′]

=∑skm∈S

∑a∈AO

m

[∂ca

(x ′

a

)∂Xsk

m

δaskmλask

mX′

skm

x ′a

− ∂ca

(x ′′

a

)∂Xsk

m

δaskmλask

mX′′

skm

x ′′a

]

×[X′

skm

− X′′skm

]+∑skm∈S

∑a∈AO

m

δaskmλask

m

×⎡⎣ca

(x ′

a

) (x ′

a − δaskmλask

mX′

skm

)(x ′

a

)2

−ca

(x ′′

a

) (x ′′

a − δaskmλask

mX′′

skm

)(x ′′

a

)2

⎤⎦[X′

skm

− X′′skm

]+

∑skm∈S

[ρ ′′

skm

− ρ ′skm

] [X′

skm

− X′′skm

]

+J∑

j=1

L∑l=1

∑skm∈S

[X′

skmβ

j

skmηl

skm

− X′′skmβ

j

skmηl

skm

] [ρl′

j − ρl′′j

]

−J∑

j=1

L∑l=1

[dl

j

(ρl′

j

)− dl

j

(ρl′′

j

)] [ρl′

j − ρl′′j

].

Since, ρskm

= ∑Jj=1

∑Ll=1 β

j

skmηl

skmρl

j , we obtain

∑skm∈S

[ρ ′′

skm

− ρ ′skm

] [X′

skm

− X′′skm

]

+J∑

j=1

L∑l=1

∑skm∈S

[X′

skmβ

j

skmηl

skm

− X′′skmβ

j

skmηl

skm

] [ρl′

j − ρl′′j

]= 0.

Because dlj

(ρl

j

)is continuous and strictly monotonically de-

crease in ρlj for every j = 1, 2, . . . , J and l = 1, 2, . . . , L, this

implies that if for every skm ∈ S,a ∈ AI

m, and m = 1, 2, . . . , M , the

derivative ofδaskm

λaskm

ca (xa )

xarespect to Xsk

mis continuous and strictly

monotone in Xskm

on the set K , F (y) is strictly monotone in y.

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14 Z. Chen et al.

From the Corollary 3.2 in Harker and Pang (1990), Theorem 3 isproved. �

Proof of Theorem 4: (1) For any given d ∈ B, since K iscompact and convex and F (y, d) is continuous on K , thenby the uniqueness theorem of variational inequality, parametric

variational inequality V I(F (y, d) , K

)has solutions. In terms

of strict monotonicity of the mapping F in y, it is well known that

V I(F (y, d) , K

)has a unique solution for any given d ∈

B. Therefore, for all d ∈ B, the implicit function y = ϕ (d)

determined by V I(F (y, d) , K

)is well defined.

(2) We claim that y = ϕ (d) is continuous on B. From theProposition 1.5.9 of Facchinei and Pang (2003), for any given d ,

a vector y is a solution of V I(F (y, d) , K

), if and only if there

exists a vector z such that g (z, d) = ∏K (z) − F

(∏K (z) ,d

)and y = ∏

K (z), where∏

K (z) is the Euclidean projection of z

onto K . For any given d0, d1 ∈ B, we obtain

||y1 − y0|| = ||∏K

(z1)−

∏K

(z0) || ≤ ||z1 − z0||

= ||g (z1, d1)− g

(z0, d0

) ||= ||g (z1, d1

)− g(z0, d1

)+ g(z0,d1

)−g

(z0, d0

) ||≤ ||g (z1, d1

)− g(z0, d1

) ||+||g (z0, d1

)− g(z0,d0

) ||.On the one hand,

||g (z1,d1)− g

(z0,d1

) ||2=∥∥∥∥∥∥∏K

(z1)−

∏K

(z0)− F

⎛⎝∏

K

(z1),d1

⎞⎠

+F

⎛⎝∏

K

(z0),d1

⎞⎠∥∥∥∥∥∥

2

=⟨∏

K

(z1)−

∏K

(z0)− F

⎛⎝∏

K

(z1),d1

⎞⎠

+F

⎛⎝∏

K

(z0),d1

⎞⎠ ,

∏K

(z1)−

∏K

(z0)− F

⎛⎝∏

K

(z1),d1

⎞⎠+ F

⎛⎝∏

K

(z0),d1

⎞⎠⟩

=∥∥∥∥∥∥∏K

(z1)−

∏K

(z0) ∥∥∥∥∥∥

2

− 2

⟨∏K

(z1)−

∏K

(z0),

F

⎛⎝∏

K

(z1),d1

⎞⎠− F

⎛⎝∏

K

(z0),d1

⎞⎠⟩

+∥∥∥∥∥∥F

⎛⎝∏

K

(z1),d1

⎞⎠− F

⎛⎝∏

K

(z0),d1

⎞⎠∥∥∥∥∥∥

2

.

Since F (y, d) is strictly monotone in y for any d ∈ B,there exists a constant α > 0 such that

⟨∏K

(z1)−

∏K

(z0), F

⎛⎝∏

K

(z1), d1

⎞⎠

−F

⎛⎝∏

K

(z0), d1

⎞⎠⟩

≥ α

∥∥∥∥∥∥∏K

(z1)−

∏K

(z0) ∥∥∥∥∥∥.

Moreover, if F (y, d) is local Lipschitz continuous for all d ∈B and y ∈ K , there exists a neighbourhood V and a constant β

such that, for any y, y ′ ∈ K and any d , d ′ ∈ B ∩V ,

||F (y, d) − F(y ′, d ′) || ≤ β

(||y − y ′|| + ||d − d ′||) .We obtain that

∥∥∥∥∥∥F⎛⎝∏

K

(z1), d1

⎞⎠− F

⎛⎝∏

K

(z0), d1

⎞⎠∥∥∥∥∥∥

2

≤ β2

∥∥∥∥∥∥∏K

(z1)−

∏K

(z0) ∥∥∥∥∥∥

2

≤ β2||z1 − z0||2.

Consequently, we obtain

||g (z1,d1)− g

(z0, d1

) || ≤√

1 − 2α + β2||z1 − z0||.

On the other hand,

||g (z0, d1)− g

(z0, d0

) ||= ||

∏K

(z0)−

∏K

(z0)− F

⎛⎝∏

K

(z0), d1

⎞⎠

+F

⎛⎝∏

K

(z0),d0

⎞⎠ ||

= ||F⎛⎝∏

K

(z0), d1

⎞⎠− F

⎛⎝∏

K

(z0), d0

⎞⎠ ||

≤ β||d1 − d0||.

This implies that

||y1 − y0|| ≤ ||z1 − z0|| ≤√

1 − 2α + β2||z1 − z0||+β||d1 − d0||,

which easily implies

||z1 − z0|| ≤ β

1 −√

1 − 2α + β2||d1 − d0||.

We obtain

||y1 − y0|| ≤ β

1 −√

1 − 2α + β2||d1 − d0||.

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International Journal of Systems Science 15

That implies that if d1 → d0, then y1 → y0. Hence, y =ϕ (d) is continuous on the compact and convex set B. �

Proof of Theorem 5: For any given d ∈ B, and any y ′ and y ′′

in K ,

[F (d, y ′) − F (d, y ′′)]T [y ′ − y ′′]

=∑skm∈S

∑a∈AO

m

[∂ca

(x ′

a

)∂Xsk

m

δaskmλask

mX′

skm

x ′a

− ∂ca

(x

′′a

)∂Xsk

m

δaskmλask

mX

′′skm

x′′a

]

×[X′

skm

− X′′skm

]+∑skm∈S

∑a∈AO

m

δaskmλask

m

×⎡⎣ca

(x ′

a

) (x ′

a − δaskmλask

mX′

skm

)(x ′

a

)2

−ca

(x

′′a

) (x

′′a − δask

mλask

mX

′′skm

)(x

′′a

)2

⎤⎦[X′

skm

− X′′skm

]∑skm∈S

∑a∈AI

m

[∂Za

(x ′

a − x∗a

)∂Xsk

m

− ∂Za

(x

′′a − x∗

a

)∂Xsk

m

] [X′

skm

− X′′skm

]

+∑skm∈S

[ρ

′′skm

− ρ ′skm

] [X′

skm

− X′′skm

]+

J∑j=1

L∑l=1

∑skm∈S

[X′

skmβ

j

skmηl

skm

−X′′skmβ

j

skmηl

skm

] [ρl′

j − ρl′′

j

]

−J∑

j=1

L∑l=1

[dl

j

(ρl′

j

)− dl

j

(ρl

′′

j

)] [ρl′

j − ρl′′

j

].

Since ρskm

= ∑Jj=1

∑Ll=1 β

j

skmηl

skmρl

j , we obtain

∑skm∈S

[ρ

′′skm

− ρ ′skm

] [X′

skm

− X′′skm

]

+J∑

j=1

L∑l=1

∑skm∈S

[X′

skmβ

j

skmηl

skm

− X′′skmβ

j

skmηl

skm

] [ρl′

j − ρl′′

j

]= 0.

This implies that if for every skm ∈ S, a ∈ AI

m, and m =1, 2, . . . , M , the derivative of

δaskm

λaskm

ca (xa )

xaand ∂Za

(xa − x∗

a

)re-

spect to Xskm

is continuous and strictly monotone in Xskm

on the

set K ,dlj

(ρl

j

)is continuous and strictly monotone in ρl

j for every

j = 1, 2, . . . , J and l = 1, 2, . . . , L, F (y, d) is strictly mono-tone in y for any d ∈ B.

On the other hand, if for all d ∈ B and y ′, y ′′ ∈ K , the

derivative ofδaskm

λaskm

ca (xa )

xaand ∂Za

(xa − x∗

a

)respect to Xsk

mis

local Lipschitz continuous, then it is easily valid that there existsa constant β1 and β2 such that

F1(y ′,d ′)− F1

(y ′′,d ′′)2 ≤ β2

1 y ′ − y ′′2

F2(y ′,d ′)− F2

(y ′′,d ′′)2

≤ β22

(y ′ − y ′′2 + d ′ − d ′′2) .

Since

F(y ′, d ′)− F

(y ′′, d ′′)2

= F1(y ′, d ′)− F1

(y ′′, d ′′)2

+F2(y ′, d ′)− F2

(y ′′, d ′′)2

≤ (β1 + β2)2 (y − y ′′ + d ′ − d ′′)2.

This implies F (y, d) is local Lipschitz continuous. FromTheorem 4, the statement is valid. �Proof of Theorem 6: The proof is patterned after arguments thatare now standard in the theory of stochastic algorithms. Accordingto Kusher and Clark (1978) and Nagurney and Zhang (1996),the above algorithms converge to the solution of the variationalinequality problem (12), provided that the function F that entersthe variational inequality is monotone and Lipschitz continuousand that a solution exists. Existence of a solution follows fromTheorem 2. Monotonicity follows Theorem3. Lipschitz continuity,in turn, follows from Theorem 5. The proof is completed. �

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