A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management

Accepted Manuscript

A Polymorphic Uncertain Equilibrium Model and Its Deterministic Equivalent Formulation for Decentralized Supply Chain Management Zhong Wan, Hao Wu, Lin Dai PII: DOI: Reference:

S0307-904X(17)30420-1 10.1016/j.apm.2017.06.028 APM 11832

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

25 November 2016 27 May 2017 17 June 2017

Please cite this article as: Zhong Wan, Hao Wu, Lin Dai, A Polymorphic Uncertain Equilibrium Model and Its Deterministic Equivalent Formulation for Decentralized Supply Chain Management, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.06.028

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Highlights • A polymorphic uncertain equilibrium model is constructed to capture the joint maximization of profits in a supply chain for its applicability.

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• By compromise programming, a deterministic equivalent formulation (DEF) of the uncertain model is obtained to find an equilibrium point. • A modified partially Jacobian smoothing algorithm is developed to solve the DEF.

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• Sensitivity analysis offers a number of useful managerial implications.

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A Polymorphic Uncertain Equilibrium Model and Its Deterministic Equivalent Formulation for Decentralized Supply Chain Management∗

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Zhong Wan,† Hao Wu,‡ Lin Dai§

Abstract

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Supply chain management is a multidisciplinary engineering problem. In this paper, a polymorphic uncertain equilibrium model (PUEM) is constructed to capture the joint maximization of the profits for the manufacturers and retailers in a supply chain network. To ensure applicability of the model in practice, the demand of the consumers is regarded as a continuous random variable, the holding cost of the retailer and the transaction cost between the manufacturer and retailer are described by fuzzy sets. For the PUEM, a deterministic equivalent formulation (DEF) is first derived by compromise programming approach such that the existing powerful algorithms in the standard smooth optimization are employed to find an approximate equilibrium point for the uncertain problem. Actually, the DEF turns out to be a nonlinear complementarity problem (NCP), a special variational inequality. Thus, a modified partially Jacobian smoothing algorithm is developed to solve the corresponding NCP, where the gradient information of the model is used to efficiently generate search direction. Sensitivity analysis offers a number of useful managerial implications based on practical applications of the model.

Keywords: complementarity problems; fuzzy model; compromising program; algorithm; supply chain management ∗

This work is supported by the National Science Foundation of China (Grant No. 71671190) and the Natural Science Foundation of Guangdong, China (Grant No. 2016A030310105) † Corresponding author, School of Mathematics and Statistics, Central South University, Changsha, China, [email protected] ‡ School of Finance and Statistics, Hunan University, Changsha, China § School of Mathematics and Statistics, Central South University, Changsha, China

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1

Introduction

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Supply chain management (SCM) is an interdisciplinary research field since it is involved in manufacturing, logistics and marketing, and has attracted extensive attentions from applied mathematicians, operation researcher and engineers. A large number of research results can be obtained from the academic search service. For the most recent research advance, one can see [1–3] and the references therein. In the context of decision makers, research on the SCM has been conducted from two different viewpoints. If only a single decision maker is in charge of the whole SCM, the supply chain is called a centralized one. On the contrary, in a decentralized supply chain, each firm in the supply chain network gives its own decision. Materials/products flows or cash flows among the firms are regulated as all the decision makers are to maximize their individual profits, or by a contract. Due to conflicting interests in the decentralized supply chain, an equilibrium model was first constructed in [4] to formulate a competitive supply chain network in a deterministic environment. Subsequently, in [5], a stochastic equilibrium model was presented for the competitive supply chain management problem with random demand in the same framework of [4]. Then, in [6], the model was extended to treat the risk caused by the random demand as well as maximizing the expectation of profit. Yang in [7] further extended the deterministic model in [4] to formulate the equilibrium problem of a general closed-loop supply chain network, which is associated with the raw material suppliers, manufacturers, retailers, consumers and recovery centers. In [8], with the help of variational inequality, a bi-criteria indicator was presented to assess the supply chain network performance for the critical needs under capacity and demand disruptions. It is noted that the model in [4] is also applicable for the management problem of transportation network [9]. To put in a nutshell, one of the main shinning points in the existing optimization models of SCM focuses on taking all the relevant factors into account such that some exogenous model parameters are treated as endogenous variables of the models in order to improve the applicability of the final integrated models. Another research trend on SCM is to consider the uncertainty existing in the practical industry management. For the decision-makers in SCM, they often want to know whether the consumer demand and the transportation cost are time-varying or not before their decision-making. If the demand or the cost is uncertain, then it is impossible to make an optimal decision in the meaning of standard optimization theory. Actually, in [5, 6, 10–14], the demands are assumed to be random. In [10], the demand is assumed to be a random variable subject to the Gamma distribution. In [3, 13], the demands are normally distributed random variables, and it has been shown that different representations of continuous random demands can generate serious affects on the practical 3

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managerial policies, especially in comparison with a deterministic demand known a prior. As done in [5], we assume in this paper that the random demand is uniformly distributed for computational simplicity. In [15–17], uncertain cost has been incorporated into the construction of models because it may be not realistic to assume that all the costs in a real-world problem are precisely known. For instance, if the cost is the traveling time between two nodes in a supply chain network, then the exact cost is rarely known in advance. In this case, it is more reasonable to provide a range of possible values or a possibility distribution for the cost. Although stochastic mathematic is a useful tool to describe the uncertainty of costs, they could not flexibly and effectively deal with the uncertainties caused by fuzziness, as pointed out in [15]. Actually, owing to lack of evidence or insufficient information, cost accounting in practice is often associated with subjective judgment in decision-making. Therefore, it is more helpful to apply the theory of fuzzy mathematics into treatment of the uncertain costs. Compared with an assumption of random costs, fuzzy cost does not require to know all the possible values of the costs in advance. Additionally, for a random variable, the distribution function must be given or estimated based on the statistical data. For a fuzzy subset, since all the possible results can be unknown prior to observation, its membership function may be determined by some subjective methods or individual experiences. Motivated by construction of a more practicable SCM model, we attempt to extend the framework of the approach described in [4, 5] into a more general case, where fuzzy numbers are used to describe the imprecise holding cost of the retailer and the transaction cost between the manufacturer and retailer, and the demand of the consumers is assumed to be a continuous random variable. Consequently, the management problem of the decentralized supply chain will be formulated as a polymorphic uncertain equilibrium model (PUEM), where the profits of the manufacturers and retailers are jointly maximized in a competitive approach. For the PUEM, we will present a compromising optimization method to find an equilibrium solution of the decentralized SCM. Note that a concept of polymorphic uncertain nonlinear program (PUNP) was first proposed in [18] as a new model to formulate the optimization problems with polymorphic uncertainty, originating from the structural design of mechanical elements. Then, more results have been obtained in [19–23]. However, up to the best of our knowledge, there does not exist a polymorphic uncertain equilibrium model in the literature for the management problem of a decentralized supply chain network. Owing to complexity in PUEM, we will first derive deterministic equivalent formulations (DEF) in virtue of compromising programming and expectation methods from

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Figure 1: Network structure of the supply chain

Polymorphic uncertain model and its DEF

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the polymorphic uncertain optimization problems of the manufacturers and the retailers, respectively. Then, a deterministic equilibrium model is obtained under some given satisfaction levels, which turns out to be a nonlinear complementarity problem (NCP). In this paper, different from the projection algorithms for solving a variational inequality in [4, 5], we attempt to analyze the gradient information in PUEM to develop a more efficient algorithm which can find an equilibrium solution of the NCP. The paper is organized as follows. In next section, we construct the supply chain network model with fuzzy transaction costs and random demand, and a DEF is derived. In Section 3, an efficient algorithm is developed to find a robust solution of the model. Section 4 is devoted to scenario analysis and sensitivity analysis. Some conclusions are drawn in the last section.

In this section, we will construct an equilibrium model in an uncertain environment. Then, its deterministic equilibrium formulation is derived to find an approximate equilibrium for the uncertain problem. A general supply chain network can be described by Figure 1. In Figure 1, there are m manufacturers and n retailers. All the manufacturers produce homogeneous products, sell and transport them to the retailers who have ordered the products from the manufacturers. The objective of each manufacturer is to maximize the profit by determining optimal wholesale prices of the products, being associated with fuzzy transaction costs between the manufacturers and retailers. The goal of each retailer 5

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2.1

Notations

For readability, we first introduce the following notations. Indices

j(= 1, 2, . . . , n) : index of retailers. Parameters and symbols

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i(= 1, 2, . . . , m) : index of manufacturers.

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is to maximize the profit by choosing an optimal sale price and optimal order quantities from different manufacturers, where the holding cost of products is fuzzy, and the demand of market is random.

cej : the fuzzy holding cost function of Retailer j.

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cg ij (·) : the fuzzy function of transaction cost between Manufacturer i and Retailer j.

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cf ij : the fuzzy parameter in the transaction cost function between Manufacturer i and Retailer j.

dˆj (·) : the random demand function of Retailer j.

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fi (·) : the production cost function of Manufacturer i. E(·) : the expectation of random variable.

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Fj (·) : the demand distribution function of Retailer j. P os(·) : the possibility that a stochastic inequality holds.

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P of (·) : the possibility that a fuzzy inequality holds. αi : the satisfaction level of Manufacturer i.

βj : the satisfaction level of Retailer j. λ+ j : unit penalty of Retailer j for supply excess. λ− j : unit penalty of Retailer j for demand surplus. ∆+ j : the inventory quantity of Retailer j. ∆− j : the shortage quantity of Retailer j. 6

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Decision variables qij : the order quantity of Retailer j from Manufacturer i. ρ2j : the retail price of Retailer j.

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ρ1ij : the wholesale price given by Manufacturer i to Retailer j.

Optimization model of each manufacturer

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For Manufacturer i, if qij is the order quantity of Retailer j from Manufacturer i, then Manufacturer i can determine an optimal wholesale price, being referred to as ρ1ij , to maximize the profit. Denote fi (Q) the production cost of Manufacturer i, where Q is mn-dimensional row vector consisting of qij , i = 1, 2, . . . , m and j = 1, 2, . . . , n. Generally, fi is convex and continuous with respect to qij . Different from the existing models available in the literature, we take into consideration the transaction cost between Manufacturer i and Retailer j, and assume that it is a fuzzy function given by

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2 c^ f ij (qij ) = c ij qij , i = 1, 2, . . . , m, j = 1, 2, . . . , n,

(2.1)

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where cf ij is assumed to be a triangle fuzzy subset in this paper. By theory of fuzzy mathematics, cf f ij can be denoted by c ij = (lij , mij , uij ). If all the manufacturers compete in a noncooperative way. Then, the profit maximization for Manufacturer i is mathematically written as:

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max

n P

j=1

ρ1ij qij − fi (Q) −

n P c^ ij (qij )

(2.2)

j=1

s.t. qij ≥ 0, j = 1, 2, . . . , n.

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By compromising programming approach in [18, 20], for a given satisfaction level αi , the DEF of the fuzzy programming model (2.2) reads: max F mi s.t. P of

n P

j=1

ρ1ij qij − fi (Q) −

qij ≥ 0, j = 1, 2, . . . , n,

n P

j=1

2 cf ij qij ≥ F mi

!

≥ αi ,

(2.3)

where F mi , i = 1, 2, . . . , m, are auxiliary variables, and P of (·) represents the possibility that the following fuzzy inequality holds with a given satisfaction level αi : n X j=1

ρ1ij qij − fi (Q) − 7

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2 cf ij qij ≥ F mi .

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˜ we define (also see [24, 25]) For arbitrary two fuzzy subsets A˜ and B, ˜ = sup{min(µ ˜ (x), µ ˜ (y))|x, y ∈ R, x ≤ y}. P of (A˜ ≤ B) A B

(2.4)

Then, the first constraint in Model (2.3) reads

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n
X (1 − αi )lij qij2 + αi mij qij2 ≤ ρ1ij qij − fi (Q) − F mi . j=1

(2.5)

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n X

In virtue of (2.5), we obtain the following Lagrangian function corresponding to the constrained optimization model (2.3):

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L(F mi , qij ; γi , ηij ) ! n n n P P P = −F mi + γi ((1 − αi )lij + αi mij ) qij2 − ρ1ij qij + fi (Q) + F mi − ηij qij , j=1

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(2.6) where γi and ηij are the Lagrange multipliers being associated with all the constraints in Model (2.3). Thus, the Karush-Kuhn-Tucker (KKT) conditions for the optimal solution of Model (2.3) can be written as  ∂L  = −1 + γi = 0,    ∂F mi      ∂L ∂f i (Q)   = γi − ρ1ij + 2(1 − αi )lij qij + 2αi mij qij − ηij = 0, j = 1, 2, . . . , n,   ∂qij ∂qij      ηij ≥ 0, qij ≥ 0, ηij qij = 0,   n n 
P P  2 2  ≥ 0, (1 − α )l q + α m q γ ≥ 0, ρ q − f (Q) − F m −  i ij i ij i 1ij ij i i ij ij   j=1 j=1  !    n n
P P    γi ρ1ij qij − fi (Q) − F mi − (1 − αi )lij qij2 + αi mij qij2 = 0.  j=1

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(2.7) In other words, (2.7) is the necessary conditions of equilibrium point for the manufacturers in the decentralized supply chain management problem. Consequently, for Manufacturer i, the optimal production plan, qij∗ , satisfies the following complementary conditions:  qij ≥ 0,     ∂fi (Qi )  ηij = − ρ1ij + 2(1 − αi )lij qij + 2αi mij qij ≥ 0, (2.8) ∂q ij    i   ∂f (Q ) i   ηij qij = − ρ1ij + 2(1 − αi )lij qij + 2αi mij qij qij = 0, j = 1, 2, . . . , n. ∂qij

Furthermore, if all the manufacturers make their production plans in a noncooperative way [27], then the equilibrium point of production quantities, being referred to as Q∗ ∈ Rmn , satisfies: F 1 (Q) ≥ 0, Q ≥ 0, F 1 (Q)QT = 0, (2.9) 8

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2.3

Optimization model of each retailer

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where the row vector function F 1 : Rmn 7→ Rmn is given by   F 1 (Q) = (Fij1 (Q))m×n , ∂f (Q)  Fij1 (Q) = i − ρ1ij + 2(1 − αi )lij qij + 2αi mij qij , i = 1, . . . , m, j = 1, . . . , n. ∂qij (2.10)

For Retailer j, we suppose that the demand dˆj is a random function, which depends on the retail price ρ2j and is perturbed by a random factor ξ. Specifically, dˆj is given by (2.11)

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−a dˆj (ρ2j ) = bj ρ2j j ξ, j = 1, 2, . . . , n,

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where ξ is called the stochastic perturbation of demand, aj is the sensitivity coefficient of price and bj is the fixed potential demand of the product. Clearly, as the price ρ2j increases, it follows from (2.11) that the demand dˆj is decreasing. However, even for the same price ρ2j , the demand is not a fixed value but a random variable. Let Fj (x, ρ2j ) be the distribution density of the random demand dˆj for a given ρ2j . Then, the probability distribution function of dˆj (ρ2j ) can be written as Z x ˆ P osj (x, ρ2j ) = P os(dj (ρ2j ) ≤ x) = Fj (t, ρ2j )dt. Denote

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sj =

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qij .

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Then, sj is the total order quantity of Retailer j from all the manufacturers. Since the consumers can not buy more than the retailer’s supply, the inventory and shortage quantities of Retailer j are given by ˆ 4+ j = max{0, sj − dj (ρ2j )}

and

ˆ 4− j = max{0, dj (ρ2j ) − sj },

respectively. In virtue of (2.11), we obtain the expected inventory and shortage quantities: Z sj + E(4j ) = (sj − x)Fj (x, ρ2j )dx + 0 Z0 sj = (sj − x)Fj (x, ρ2j )dx, 0 Z ∞ (2.12) E(4− (x − sj )Fj (x, ρ2j )dx j ) = 0+ Z ∞ sj = (x − sj )Fj (x, ρ2j )dx. sj

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− Denote λ+ j the unit penalty of Retailer j in the case of supply excess. Denote λj the loss of Retailer j brought about by unit demand surplus. Then, for Retailer j, the total expected loss, caused by either supply excess or demand surplus, is

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+ − − + + − − E(λ+ j 4j + λj 4j ) = λj E(4j ) + λj E(4j ).

Next, taking into account the above mentioned reasons in Section 1, we suppose that the holding cost of Retailer j depends on all the order quantities and is a fuzzy set. Specifically, the holding cost is given by !2 m X c] qij , (2.13) j (Q) = cej

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where cej = (lj , mj , uj ) is a triangle fuzzy subset. Therefore, the objective of Retailer j is to maximize the expected profit by choosing optimal order quantities, and the optimization model of Retailer j is constructed as follows:  
+ − − ˆ (q ; ρ ) = E ρ min{ d , s } − E λ+ max ΠR ij 2j 2j j j j j 4j + λ j 4j m m P P (2.14) − ρ1ij qij − c˜j (Q) i=1

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s.t. qij ≥ 0, i = 1, 2, . . . , m,

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where qij , i = 1, 2, . . . , m, are the order quantities, E(ρ2j min{dˆj , sj }) denotes the expected revenue of Retailer j, and dˆj is the abbreviated version of dˆj (ρ2j ). + From the definitions of 4− j and 4j , it follows that

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min{dˆj , sj } = dˆj − 4− j ,

E(ρ2j min{dˆj , sj }) = E(ρ2j (dˆj − 4− j )).

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Thus, Model (2.14) is rewritten as − − + + ˆ max ΠR j (qij ; ρ2j ) = ρ2j dj − (ρ2j + λj )E(4j ) − λj E(4j ) m m P P − ρ1ij qij − c] j (Q) i=1

(2.15)

i=1

s.t. qij ≥ 0, i = 1, 2, . . . , m.

By compromising programming approach, the fuzzy model (2.15) is transformed into max F rj     m m
P P + + − − ] ˆ s.t. Pof E ρ2j min{dj , sj } − E λj 4j + λj 4j − ρ1ij qij − cj (Q) ≥ F rj ≥ βj , i=1

qij ≥ 0, i = 1, 2, . . . , m,

i=1

(2.16)

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where βj is a given satisfaction level for the fuzzy objective function. By (2.4), Model (2.16) is equivalent to

s.t. ((1 − βj )lj + βj mj )

m P

i=1

qij

2

−

qij ≥ 0, i = 1, 2, . . . , m.

− + + ≤ ρ2j dˆj − (ρ2j + λ− j )E(4j ) − λj E(4j ) m P

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max F rj

ρ1ij qij − F rj ,

For (2.17), the corresponding Lagrangian function is:

µj

(ρ2j +

!2

− µj ρ2j dˆj +

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L(F rj , qij ; µj , νij ) = −F rj + µj ((1 − βj )lj + βj mj ) − λ− j )E(4j )

m X

+

+ λ+ j E(4j )

qij

i=1 m X

+

(2.17)

ρ1ij qij + F rj

i=1

!

−

m X

νij qij ,

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(2.18) where µj and νij are the Lagrangian multipliers. Similarly, the optimization model (2.17) of Retailer j can be reformulated into the following complementarity problem:  m − +  ν = 2 ((1 − β )l + β m ) P q + ρ + λ+ ∂E(4j ) + (ρ + λ− ) ∂E(4j ) ≥ 0, 2j ij j j j j ij 1ij j j ∂qij ∂qij i=1  qij ≥ 0, νij qij = 0, ∀i. (2.19) By direct calculation, we have

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∂E(4+ j ) = P osj (sj , ρ2j ), ∂qij

∂E(4− j ) = P osj (sj , ρ2j ) − 1. ∂qij

(2.20)

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Consequently, if all the retailers compete in a noncooperative way, then the optimal order quantities Q∗ ∈ Rmn satisfy the following complementarity problem: F 2 (Q; ρ2 ) ≥ 0, Q ≥ 0, F 2 (Q; ρ2 )QT = 0,

where the row vector function F 2 : Rmn 7→ Rmn is defined by  2 F (Q; ρ2 ) = (Fij2 (Q; ρ2 )),    m P Fij2 (Q; ρ2 ) = 2 ((1 − βj )lj + βj mj ) qij + ρ1ij + λ+ j P osj (sj , ρ2j )+  i=1   (ρ2j + λ− j )(P osj (sj , ρ2j ) − 1), i = 1, · · · , m, j = 1, . . . , n.

(2.21)

(2.22)

Remark 1 Compared with the stochastic model in [5], construction of Models (2.2) and (2.14) takes into account the fuzziness of holding costs and transaction costs in practice, as 11

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2.4

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well as the random demand. Thus, compromising programming approach and expectation method have been used to transform the polymorphic uncertain optimization problem into a deterministic equivalent formulation. As the uncertain models are converted into standard optimization problems, we can use some analytic tools and powerful computational techniques in the classical optimization theory to find a robust solution for the uncertain models.

Equilibrium model of decentralized SCM

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We now discuss the equilibrium conditions of market for the management problem of a decentralized supply chain network. As stated in [26, 27], the order quantity of products between the manufacturer and retailer is exactly equal to the quantity of demand if and only if the equilibrium price is positive. In other words, if the supply quantity is greater than that needed by market, then the equilibrium price is zero. Mathematically, the following conditions hold.  m P  qij , if ρ2j > 0,  = i=1 ˆ dj (ρ2j ) j = 1, 2, . . . , n. (2.23) m P   < qij , if ρ2j = 0, ρ2 = (ρ21 , . . . , ρ2n ).

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Then, for the market of the whole supply chain network, the equilibrium selling prices of all the retailers satisfy: F 3 (Q, ρ2 ) ≥ 0, ρ2 ≥ 0, F 3 (Q, ρ2 )ρT2 = 0,

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where the row vector function F 3 : Rn 7→ Rn is specified by   F 3 (Q, ρ2 ) = (Fj3 (Q, ρ2j )), m P  Fj3 (Q, ρ2j ) = qij − dj , j = 1, 2, . . . , n,

(2.24)

(2.25)

i=1

and dj is the mean value of dˆj (ρ2j ). Combining (2.9), (2.21) and (2.24), we obtain an equilibrium model for the management problem of a decentralized supply chain network as follows.
(Q, ρ2 ) ≥ 0, F (Q, ρ2 ) = F 12 (Q, ρ2 ), F 3 (Q, ρ2 ) ≥ 0, F (Q, ρ2 )(Q, ρ2 )T = 0,

where

12 12 F 12 (Q, ρ2 ) = (F11 (Q, ρ2 ), . . . , Fij12 (Q, ρ2 ), . . . , Fmn (Q, ρ2 ))

12

(2.26)

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is given by m m P P ∂fi (Q) + 2(1 − αi )lij qij + 2αi mij qij + 2(1 − βj )lj qij + 2βj mj qij ∂qij i=1 i=1 − + λ+ j P osj (sj , ρ2j ) + (ρ2j + λj )(P osj (sj , ρ2j ) − 1). (2.27) It is noted that with a solution of Model (2.26), the wholesale prices of manufacturers, ρ1ij , can be directly calculated by

ρ1ij = −2 ((1 − βj )lj + βj mj )

m X i=1

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Fij12 (Q, ρ2 ) =

− qij − λ+ j P osj (sj , ρ2j ) − (ρ2j + λj )(P osj (sj , ρ2j ) − 1)

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in the case that qij∗ > 0 (also see [4]). By Model (2.26), we present the following concept.

Definition 1 For a polymorphic uncertain equilibrium model (PUEM) of supply chain network, a solution (Q∗ , ρ∗2 ) ∈ Rmn+n of Model (2.26) is called a compromising (robust) Nash equilibrium point of PUEM.

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Remark 2 Since the PUEM of the supply chain network in this paper is involved with random demand and fuzzy costs, there does not exist any optimal equilibrium solution for all the manufacturers and retailers from the viewpoint of standard optimization theory. In light of the presented compromising approach in Section 2, we will make an optimal decision by solving Model (2.26) for the management problem of uncertain supply chain network.

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Remark 3 Owing to the complexity of Problem (2.26), it is valuable to develop an efficient algorithm to solve Model (2.26), especially in the case that Model (2.26) is a largescale complementarity problem. In the existing results, different projection algorithms are often developed to solve Model (2.26) (see [4, 5]). However, for all the projection algorithms, it is required to assume that Model (2.26) can be transformed as a monotone variational inequality. Another drawback of these algorithms lies in that the gradient information of the model is not employed to generate a better approximate solution. Remark 4 Main differences between our model and any one available in the literature (see [4–7]) can be summarized as follows. (1) In our model, the holding costs of retailers and the transaction costs between the manufacturers and retailers are treated by fuzzy subsets, as well as being dependent on the decision variables of the model. It is more helpful to make the model capture the subjective judgment of decision-maker in practice, owing to lack of evidence or insufficient information. 13

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(2) Owing to polymorphic uncertainty in our model (not merely randomness), compromising programming approach is used to treat the complexity of the constructed model such that standard optimization techniques can be extended to find its solution. (3) The deterministic equivalent formulation of our model is a complementary problem, rather than a variational inequality as done in the existing results. It is more helpful to mine the gradient information of the model to develop an efficient algorithm for solving the model in the next section.

Gradient based algorithm and its efficiency

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In this section, an efficient algorithm is developed based on the gradient information in Model (2.26).

Development of gradient based algorithm

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Very recently, for a nonlinear complementarity problem with P0 function (not necessarily monotone), a new Jacobian Newton algorithm has been developed based on partially smoothing technique in [28]. Owing to the reported efficiency of the algorithm, as well as the established global and super-linear convergence, we now extend the algorithm in [28] into solving Model (2.26). By any NCP-function (function of nonlinear complementarity problem), we can write Model (2.26) as a nonsmooth system of equations. For example, define φmin : R2 → R, φmin (a, b) = min{a, b}. Then, φmin is the most popular NCP-function. Actually, with φmin , Model (2.26) is converted into   12 φmin (q11 , F11 (Q, ρ21 ))   ..   .     12  φmin (qij , Fij (Q, ρ2j ))    ..   .     12  φmin (qmn , Fmn (Q, ρ2n ))    = 0. Φ(Q, ρ2 ) ,  (3.1)  3 φ (ρ , F (Q, ρ )) 21  min 21 1    ..   .     3 φ (ρ , F (Q, ρ ))  min 2j j  2j   ..   .   3 φmin (ρ2n , Fn (Q, ρ2n ))

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Furthermore, with a partially smoothing function in [28]: ϕ : R+ × R2 → R, specified by (3.2)

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 a, if a < b, µ > 0,    a−b µ arctan + b, if a ≥ b, µ > 0, ϕ(µ, a, b) =  µ   min{a, b}, if µ = 0

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for any (µ, a, b) ∈ R+ × R2 , we obtained a smooth approximation to Problem (3.1):     12 ϕ(µ, q11 , F11 (Q, ρ21 )) ϕ12 11   .   .   ..   ..       12   12  ϕij   ϕ(µ, qij , Fij (Q, ρ2j ))    .   .   .   .   .   .   12   12  ϕmn   ϕ(µ, qmn , Fmn (Q, ρ2n ))   = 0.    (3.3) Φµ (Q, ρ2 ) ,  3  =   3 (Q, ρ )) ϕ(µ, ρ , F ϕ 21 21   1   1   .   .   ..   ..       3    ϕj   ϕ(µ, ρ2j , Fj3 (Q, ρ2j ))    .   .   ..   ..     ϕ(µ, ρ2n , Fn3 (Q, ρ2n )) ϕ3n

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Clearly, the Jacobian matrix of Φµ is

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12 12 3 3 3 ∇Φµ (Q, ρ2 ) = (∇ϕ12 11 , . . . , ∇ϕij , . . . , ∇ϕmn , ∇ϕ1 , . . . , ∇ϕj , . . . , ∇ϕm ).

By direction calculation, we have

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∇Φµ (Q, ρ2 ) = Da (Q, ρ2 ) + ∇F (Q, ρ2 )Db (Q, ρ2 ), Da (Q, ρ2 ) = diag{a11 (Q, ρ2 ), . . . , aij (Q, ρ2 ), . . . , amn (Q, ρ2 ), a1 (Q, ρ2 ), . . . , aj (Q, ρ2 ), . . . , an (Q, ρ2 )}, Db (Q, ρ2 ) = diag{b11 (Q, ρ2 ), . . . , bij (Q, ρ2 ), . . . , bmn (Q, ρ2 ), b1 (Q, ρ2 ), . . . , bj (Q, ρ2 ), . . . , bn (Q, ρ2 )},

where

  1,

 if i, j ∈ i, j : qij < Fij12 (Q, ρ2 ) , α(Q, ρ2 ),

µ2  2 , if i, j ∈ / α(Q, ρ2 ), µ + (qij − Fij12 (Q, ρ2 ))2  if i, j ∈ α(Q, ρ2 ),  0, 2 µ bij (Q, ρ2 ) =  1− 2 , if i, j ∈ / α(Q, ρ2 ), µ + (qij − Fij12 (Q, ρ2 ))2 aij (Q, ρ2 ) =

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  1,

 if j ∈ j : ρ2j < Fj3 (Q, ρ2j ) , α(Q, ρ2 ),

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µ2  2 , if j ∈ / α(Q, ρ2 ), µ + (ρ2j − Fj3 (Q, ρ2 ))2  if j ∈ α(Q, ρ2 ),  0, 2 µ bj (Q, ρ2 ) =  1− 2 , if j ∈ / α(Q, ρ2 ). µ + (ρ2j − Fj3 (Q, ρ2 ))2

aj (Q, ρ2 ) =

In addition, the Clark generalized gradients of φmin (qij , Fij12 (Q, ρ2j )) and φmin ( q2j , Fj3 (Q, ρ2j ) ) are if qij < Fij12 (Q, ρ2 ), if qij = Fij12 (Q, ρ2 ), if qij ≥ Fij12 (Q, ρ2 ), if ρ2j < Fj3 (Q, ρ2 ), if ρ2j = Fj3 (Q, ρ2 ), if ρij ≥ Fj3 (Q, ρ2 ),

(3.4)

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   ei , 12 ∂C φmin (qij , Fij (Q, ρ2 )) = conv(ei , ∇Fij12 (Q, ρ2 )),   ∇Fij12 (x),    ei , 3 ∂C φmin (ρ2j , Fj (Q, ρ2 )) = conv(ei , ∇Fj3 (Q, ρ2 )),   ∇Fj3 (x),

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where ei is a vector with the i-th component being 1, the others being 0, and conv(·, ·) represents the convex combination of two vectors. Thus, ∂C Φ(Q, ρ2 ) is obtained. Before statement of algorithm, we first define merit functions ψ : Rn → R and ψµ : Rn → R, given by 1 1 ψ(Q, ρ2 ) = kΦ(Q, ρ2 )k2 , ψµ (Q, ρ2 ) = kΦµ (Q, ρ2 )k2 , 2 2

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(3.5)

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respectively. With the above preparation, we are in a position to develop an efficient algorithm to solve Model (2.26).

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Algorithm 1 (Extended Jacobian Smoothing Newton Algorithm)
, κ > 0 and γ ∈ (0, +∞). Take an initial Step 0. Choose ρ, α, η ∈ (0, 1), σ ∈ 0, 1−α 2 0 0 mn+m 0 0 0 point (Q , ρ2 ) ∈ R . Compute β0 = kΦ(Q , ρ )k, and µ0 = αβ . Set k := 0. 2κ Step 1. Solve the following linear system of equations: (k)

(k)

∇Φµk (Q(k) , ρ2 )T d = −Φ(Q(k) , ρ2 ).

(3.6)

The solution is referred to as d(k) . (k+1) (k) Step 2. Set λ(k) := ρmk , (Q(k+1) , ρ2 ) := (Q(k) , ρ2 ) + λ(k) d(k) , where mk is the smallest nonnegative integer m such that (k)

(k)

(k)

ψµk ((Q(k) , ρ2 ) + ρm d(k) ) − ψµk (Q(k) , ρ2 ) ≤ −2σρm ψ(Q(k) , ρ2 ). 16

(3.7)

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(k+1)

Step 3. If kΦ(Q(k+1) , ρ2 Step 4. If (k+1)

kΦ(Q

(k+1) , ρ2 )k

n

)k = 0, the algorithm stops. Otherwise, go to Step 4.

≤ max ηβk , α

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o

(k+1) (k+1) (k+1) (k+1) , ρ2 ) − Φµk (Q , ρ2 ) ,

Φ(Q (k+1)

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then update βk by βk+1 = kΦ(Q(k+1) , ρ2 )k, and choose µk+1 such that the following inequalities are satisfied:   αβk+1 µk 0 < µk+1 ≤ min , , (3.9) 2κ 4        (k+1) (k+1) (k+1) dist ∇Φµk+1 Q(k+1) , ρ2 , ∂C Φ Q(k+1) , ρ2 , F Q(k+1) , ρ2 ≤ γβk+1 . (3.10) Otherwise, set βk+1 := βk , µk+1 := µk , go to Step 5. Step 5. Set k := k + 1. Return to Step 1.

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Remark 5 In [3], on the basis of the gradient information of the constructed model, a modified Topkis-Veinott algorithm has been developed to solve the management problem of a global supply chain. By a large number of numerical simulation, it has been shown that the gradient based algorithm outperforms the heuristic ones. For the supply chain network equilibrium model, the solution methods in the existing results often first transform the original model into a variational inequality (VI) (see, for example, [5, 6]). Then, by assuming that the corresponding VI is monotone, a fixed point principle based projection algorithm is developed to find the equilibrium solution. However, this type of methods don’t employ the gradient information of the original model. Compared with the existing results, development of Algorithm 1 sufficiently mines the gradient information in the constructed model.

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Remark 6 Similar to [28], the convergence theory of Algorithm 1 can be established, even if (2.26) is not monotone.

3.2

Test of efficiency

As Algorithm 1 is used to solve the nonlinear complementarity problem (2.27), the gradient information inherent in Model (2.27) can be employed to improve the computational efficiency of searching for the equilibrium point, as well as the assumption of monotonicity can be removed theoretically. Actually, since the Jacobian matrix of the function Φµk is used to generate an approximate Newton direction in Step 1 of Algorithm 1, fast convergence has been theoretically and numerically proved. Compared with the Quasi-Newton algorithm presented in [29],

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a different merit function is constructed in this paper to choose a suitable step length in Step 2 of Algorithm 1, which can reduce the iteration number of algorithm. In order to compare the efficiency between Algorithm 1 and the modified projection method (called MPM) in [5], we solve Model (2.26), which corresponds to the established case in [5]. The computer codes of the algorithms are written in MATLAB2012b, and run in the operation system of Windows 7 with PC i3-390M 2.66GHz CPU and 2.00GB RAM. In Table 3.1, the elapsed CPU time of the algorithms is reported.

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Table 3.1: Comparison of numerical efficiency Algorithm Algorithm 1 MPM Tolerance of solution 10−6 10−7 10−8 10−6 10−7 10−8 time (s) 2.380 2.834 2.954 4.954 16.657 46.299

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From the numerical results in Table 3.1, it is clear that:

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• Algorithm 1 finds an equilibrium point of Model (2.26) more rapidly than MPM with the same tolerance. • For different requirements of tolerance, Algorithm 1 appears more robust than MPM.

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To further verify the efficiency of Algorithm 1, we solve all the six real equilibrium problems of supply chain networks presented in [5], especially in comparison with the modified projection method in [5]. Since polymorphic uncertainty is taken into consideration for a number of model parameters in this paper, we need to specify the features of the parameters as follows.

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• The transaction cost cij = c˜ij qij2 , where c˜ij is a triangular fuzzy set, given by c˜ij = (0, 0.5, 1). !2 !2 2 2 P P • The holding costs c1 (Q) = c˜j qi1 and c2 (Q) = c˜j qi2 , where c˜j is a j=1

j=1

triangular fuzzy set, given by c˜j = (0, 0.5, 1).

• The demand dˆj (ρ2j ) = bj ρ−1 2j ξ, where bj is set up as done in [5], and ξ is a random variable uniformly distributed in [0, 1]. In addition, we choose the satisfaction levels β1 = β2 = 0.9. Suppose that the fixed potential demands of products b1 = b2 = 10, and the penalty of unit supply excess and 18

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− the loss of unit demand surplus λ+ j = λj = 1, j = 1, 2. For a fair comparison between the two types of solution methods, we choose the same initial approximate solutions, and both of the algorithms stop with the same stopping criterion:

(N +1) (N +1) (N ) (N ) ρ2 − Q ρ2 ≤ 10−6 .

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As the algorithms found the same equilibrium point, their elapsed CPU times for the six problems are reported in Table 3.2, where Ex. i represents an example corresponding to the i-th example in [5].

of efficiency between Algorithm 1 and MPM Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 2.943 2.933 3.388 1.274 1.489 18.511 37.384 22.554 28.476 41.656

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Table 3.2: Comparison Ex. 1 Algorithm 1 2.380 4.954 MPM

Scenario analysis and sensitivity analysis

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The results in Table 3.2 indicate that numerical performance of our algorithm (Algorithm 1) is robust for all the test problems in finding the equilibrium point, as well as being faster than the modified projection method (see [5]).

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In this section, we apply Algorithm 1 to solve a management problem of supply chain network in the environment of polymorphic uncertainty. We focus on the impacts of uncertainty on the decision-making. For the sake of simplicity, we construct a series of scenarios by using the following setting: there are two manufacturers and two retailers in the supply chain network. For the manufacturers, the functions of production costs are specified by f1 (q) = 2.5q12 + q1 q2 + 2q1 ,

respectively, where qi =

2 P

f2 (q) = 2.5q22 + q1 q2 + 2q2 ,

qij , i = 1, 2. The transaction cost between Manufacturer i

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and Retailer j is given by

2 c^ f ij (qij ) = c ij qij ,

where cf ij , i = 1, 2 and j = 1, 2, are assumed to be triangular fuzzy subsets. Particularly, we choose cf f cf f 11 = c 12 = (4, 4.5, 5), 21 = c 22 = (4, 4.5, 5). 19

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The holding cost of Retailer j is specified by c] j (Q) = cej

m X i=1

qij

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,

−a dˆj (ρ2j ) = bj ρ2j j ξ,

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where the fuzzy parameters ce1 = (2, 2.3, 2.6) and ce2 = (2, 2.3, 2.6). As done in [5], the demand is assumed to be a uniformly distributed random variable. In this paper, we assume that the demand of Retailer j is

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where a1 = a2 = 1, b1 = b2 = 10, and ξ is uniformly distributed in [0, 1]. In addition, the unit penalty of supply excess and the loss of unit demand surplus are chosen as + − − λ+ 1 = λ2 = 1 and λ1 = λ2 = 1, respectively. The satisfaction levels α1 = α2 = 0.8 and β1 = β2 = 0.8. In virtue of Model (2.26) and Algorithm 1, we attempt to answer the following questions by sensitivity analysis:

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• How do fuzziness of costs affect the equilibrium?

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• What are the impacts of the demand randomness and sensitivity coefficient of price on the equilibrium point?

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• How does the satisfaction level affect the optimal profits?

Impacts of fuzziness inherent in transaction costs

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Since it is inappropriate to regard the transaction cost as a constant in practice, it is interesting to investigate what are the impacts of fuzziness inherent in transaction costs. Without loss of generality, we only need to proceed with the discussion by gradually enlarging the support set of the fuzzy parameter cf 11 to describe the different degrees of fuzziness. In Figures 2, 3 and 4, we have presented the impacts of fuzziness inherent in cf 11 on the maximal profits of all the players, the optimal order quantities of the retailers and the optimal wholesale and retail prices, respectively. From Figures 2, 3 and 4, it is easy to see that: • As the fuzziness of cf 11 is enhanced, i.e. the support set of the fuzzy subset is enlarged, the total order quantity and the maximal profit increase for Manufacturer 1. The optimal wholesale price of Manufacturer 1 decreases for Retailer 1, but it rises up for Retailer 2. 20

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Figure 2: Impact of fuzzy transaction cost cf 11 on maximal profits

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• For an increasing fuzziness of cf 11 , the optimal wholesale price of the product from Manufacturer 1 to Retailer 1 reduces. However, the optimal wholesale price becomes greater to Retailer 2, i.e., the case with less fuzziness. Owing to the competition from Manufacturer 1, the total order quantity and the maximal profit decrease for Manufacturer 2.

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• For Retailer 1, the whole order quantity is increasing as the fuzziness of cf 11 increases. The maximal profit becomes greater although the optimal retail price goes down.

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• However, for Retailer 2, the total order quantity has a slight reduction as the fuzziness of cf 11 increases, although the order quantity from Manufacturer 2 increases, while the order quantity from Manufacturer 1 reduces. The maximal profit of Retailer 2 has a slight decrease even if the optimal retail price is increasing.

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In summary, the fuzziness inherent in the transaction costs cf ij can generate positive effects on the direct players (Manufacturer i and Retailer j), while for the other indirect players (all the other participants except Manufacturer i and Retailer j), there exist some negative influences. Similarly, we can study the impact of fuzziness of the holding cost cej . In Figures 5, 6 and 7, we have presented the relation between the fuzziness of the holding cost ce1 and the maximal profits of all the players, the relation between the fuzziness and the optimal order quantities of the retailers, and the relation between the fuzziness and the optimal wholesale and retail prices, respectively. It is concluded from Figures 5, 6 and 7 that the fuzziness of the holding cost ce1 can seriously affect the decision-making. Specifically, as the holding cost ce1 becomes more fuzzy, it is true that: • The total order quantity, the optimal wholesale prices and the maximal profits are 21

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Figure 3: Impact of fuzzy transaction cost cf 11 on optimal order quantity

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increasing for the manufacturers. Furthermore, the wholesale price from Manufacturer 1 has a greater growth rate than that from Manufacturer 2.

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• For Retailer 1, its total order quantity goes up, and the optimal retail price and the maximal profit are deceasing. In contrast, for Retailer 2, Both of the total order quantity and the profit have a slight reduction, even if the retail price of Retailer 2 slightly increases.

Impacts of the fixed potential demand and sensitivity coefficient of price

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In a netshell, the fuzziness of the holding cost cej can generate positive effects on all the manufacturers. However, for the retailers, the effects seem to be complex.

Next, we analyze influences of the fixed potential demand and sensitivity coefficient of price on the decisions of all the players. Since it is often difficult to determine the values of these parameters as any model is applied to solve a real problem, we proceed with the sensitivity analysis on these parameters such that some practical managerial implications can be revealed. For this, we only change the values of the sensitivity coefficient of price a1 as an example (the same analysis can be made on a2 ). In Figures 8, 9 and 10, we have obtained the dependance relations between the sensitivity coefficient of price a1 and the optimal equilibrium solution. The change of the

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Figure 4: Impact of fuzzy transaction cost cf 11 on optimal prices

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Figure 5: Impact of fuzzy holding cost ce1 on maximal profits

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sensitivity coefficient a1 generates an important influence on the equilibrium solution. As a1 becomes greater, it is clear that: • For all the manufacturers, the profits and the total order quantities are increasing, even if the wholesale price of Manufacturer 1 has a significant decline. • For Retailer 1, all of the total order quantity, the optimal retail price and the profit go down. However, for Retailer 2, the order quantity and profit are going up, although the retail price has a slight reduction.

We are now in a position to investigate what happens if the potential demand of a retailer becomes greater for a given sensitivity coefficient of price. Similarly, we only change the value of the fixed potential demand b1 as an example. Our aim is to reveal the impacts of the time-varying b1 on the strategies of all the players in 23

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Figure 6: Impact of fuzzy holding cost ce1 on optimal order quantities

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the supply chain network. The corresponding equilibrium solutions are shown in Figures 11, 12 and 13. The results in Figures 11, 12 and 13 demonstrate that the time-varying fixed potential demand b1 causes a significant change of the equilibrium solution:

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• For Manufacturers, the wholesale prices and the corresponding profits are increasing as the potential demand becomes larger. Though the wholesale prices are different, their profits are same.

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• As expected, the total order quantity of Retailer 1 increases if the potential demand becomes larger. In addition, the retail price and the total profit of Retailer 1 also go up.

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• For Retailer 2, if the potential demand of Retailer 1 is large enough (b1 > 100.8956), the order quantities gradually reduce to zero. The corresponding profit also falls to zero, even if the retail price seems to rise up.

Therefore, we can conclude that for a retailer, development of market potential is helpful to bring profits to himself/herself and the manufacturers, but it is harmful to his/her competitors.

4.3

Impact of satisfaction levels

In general, owing to the uncertainty of parameters, there doesn’t exist any optimal solution for any uncertain optimization model. In Section 2, we have treated the uncertainty 24

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Figure 7: Impact of fuzzy holding cost ce1 on optimal prices

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Figure 8: Impact of price sensitivity coefficient a1 on the maximal profits

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by deriving the DEF of the uncertain optimization model if the satisfaction level of the equilibrium solution are given (see Definition 1). As shown in (2.5) and (2.17), the choices of the satisfaction levels (αi and βj ) may play a critical role in decision-making. In practice, different satisfaction levels often reflect the preferences of decision-makers towards the risk of uncertainty. In Figure 14-16, the changes of the optimal solutions have been presented as the satisfaction level α1 becomes larger and larger. Actually, higher α1 means that the constructed model pay more attention to whether the manufacturers maximize their profits or not, compared with the retailers in the supply chain network. Figure 14, 15 and 16 indicate that: • As the predetermined satisfaction level increases, the order quantities decrease, but the wholesale and retail prices increase. 25

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Figure 9: Impact of price sensitivity coefficient a1 on the optimal ordering

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Figure 10: Impact of price sensitivity coefficient a1 on the optimal prices

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• Though the profits of the manufacturers reduce for a higher satisfaction level, the accuracy of decision-making, hence the possibility of earning the profits, is improved. As a result, all the retailers can earn more profits owing to more accurate ordering and lower penalties of shortage or surplus as the satisfaction levels of retailers, βj , are fixed.

Similar conclusions can be drawn if the satisfaction levels of retailers become greater for a fixed satisfaction level of manufacturers.

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Conclusions

In this paper, we have constructed an equilibrium model for the management problem of supply chain networks in a polymorphic uncertain environment. Taking into consideration 26

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Figure 11: Impact of fixed potential demand b1 on the maximal profits

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Figure 12: Impact of fixed potential demand b1 on the optimal ordering

Figure 13: Impact of fixed potential demand b1 on the optimal prices 27

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Figure 14: Impact of satisfaction level α1 on the maximal profits

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Figure 15: Impact of satisfaction level α1 on the optimal ordering

Figure 16: Impact of satisfaction level α1 on the optimal prices 28

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the practical situation, the transaction costs and holding costs are regarded as fuzzy parameters, and the demands are described by continuous random variables. Especially, a DEF has been defined for the uncertain equilibrium problem if the satisfaction levels are given. Owing to complexity in the DEF, an efficient algorithm has been developed to find the equilibrium point, where the gradient information in the model is used to efficiently generate search direction, and new merit function is defined to obtain a suitable step length. Numerical results have shown the efficiency of the developed algorithm in this paper, particularly in comparison with the existing solution methods in the literature. Finally, many practical managerial implications of the model are obtained from the sensitivity analysis: • The fuzziness of his/her transaction cost has positive effect for the directly relevant players, but is harmful to the other players in the supply chain network.

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• For a retailer, the fuzziness of his/her holding cost has a positive effect on the manufacturers, and can enlarge the market share of the directly relevant players.

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• For any retailer, development of market potential can bring profits to himself/herself and manufacturers, but generates a negative impact to the other retailers.

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• Higher satisfaction level of manufacturers (retailers) can improve the his/her decisionmaking accuracy, though the his/her expected profit becomes smaller. However, higher satisfaction level of manufacturers (retailers) is helpful to the retailers (manufacturers).

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Although the constructed model in this paper has offered a deep understanding of equilibrium strategy in a polymorphic uncertain SCM, the structure of the supply chain network may be more complicate in practice. Actually, development of a multi-product equilibrium model and its solution methods remain to be explored for the supply chain network in a polymorphic uncertain environment. In addition, with the pressing requirement of green supply chain, it needs further investigation to extend the proposed modelling techniques in this paper into solving the management problems raised from closed-loop supply chain networks.

Acknowledgements The authors would like to express their thanks to the anonymous referees for their constructive comments on the paper, which have greatly improved its presentation.

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References

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[1] L.E. Hombach, C. Cambero, T. Sowlati, G. Walther, Optimal design of supply chains for second generation biofuels incorporating European biofuel regulations, Journal of Cleaner Production 2016 133(1) 565-575. [2] B.P. Michael, W.T. Ulrich, Reference points in revenue sharing contracts-How to design optimal supply chain contracts, European Journal of Operational Research, 2016 249(3) 1033-1049.

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[3] X.B. Zhang, S. Huang, Z. Wan, Optimal pricing and ordering in global supply chain management with constraints under random demand, Applied Mathematical Modelling, 2016 40 10105-10130. [4] A. Nagurney, J. Dong, D. Zhang, A supply chain network equilibrium model, Transportation Research Part E: Logistics and Transportation Review, 2002 38(5) 281-303.

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[5] J. Dong, D. Zhang, A. Nagurney, A supply chain network equilibrium model with random demands, European Journal of Operational Research, 2004 156(1) 194-212.

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[6] J. Dong, D. Zhang, H. Yan, Multitiered supply chain networks: multicriteria decisionmaking under uncertainty, Annals of Operations Research, 2005 135(1) 155-178.

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[7] G. Yang, Z. Wang, X. Li, The optimization of the closed-loop supply chain network, Transportation Research Part E: Logistics and Transportation Review, 2009 45(1) 16-28.

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[8] P. Qiang, A. Nagurney, A bi-criteria indicator to assess supply chain network performance for critical needs under capacity and demand disruptions, Transportation Research Part A: Policy and Practice, 2012 46(5) 801-812.

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[9] A. Nagurney, On the relationship between supply chain and transportation network equilibria: A supernetwork equivalence with computations, Transportation Research Part E: Logistics and Transportation Review, 2006 42(4) 293-316.

[10] H.C. Tijms, A First Course in Stochastic Models, John Wiley and Sons, 2003. [11] T. D. Dimitrakos, E. G. Kyriakidis, A single vehicle routing problem with pickups and deliveries, continuous random demands and predefined customer order, European Journal of Operational Research, 244(30) 2015 990-993.

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[12] D. Wang, Z. F. Qin, S. Kar, A novel single-period inventory problem with uncertain random demand and its application, Applied Mathematics and Computation, 269(15) 2015 133-145.

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[13] X.B. Zhang, S. Huang, Z. Wan, Stochastic programming approach to global supply chain management under random additive demand, Operational Research: An International Journal, DOI: 10.1007/s12351-016-0269-2. [14] S. Gamou, R. Yokoyama, K. Ito, Optimal unit sizing of cogeneration systems in consideration of uncertain energy demands as continuous random variables, Energy Conversion and Management, 43(9-12) 2002 1349-1361.

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[15] M. Sadeghi, H. M. Hosseini, Energy supply planning in Iran by using fuzzy linear programming approach (regarding uncertainties of investment costs), Energy Policy, 2006 34(9) 993-1003.

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[16] A. Ojha, B. Das, S. Mondal, M. Maiti, An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality, Mathematical and Computer Modelling, 2009 50(1-2) 166-178.

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[17] K. Adam, M. Kulej, Choosing robust solutions in discrete optimization problems with fuzzy costs, Fuzzy Sets and Systems, 160(5) 2009 667-682.

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[18] Z. Wan, S. J. Zhang, K. L. Teo, Two-step based sampling method for maximizing the capacity of V-belt driving in polymorphic uncertain environment, Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 2012 226(1) 177-191.

AC

CE

[19] S. J. Zhang, Z. Wan, Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive, Journal of Industrial and Management Optimization, 2012 8(2) 493-505. [20] Z. Wan, S. J. Zhang, K. L. Teo, Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving, Optimization and Engineering, 2014 15(1) 267-292. [21] X. Zhang, F. Zhang, X. Chen, Z. Wan, Polymorphic uncertain linear programming for generalized production planning problems, Journal of Optimization, 2014, doi:10.1155/2014/896756. [22] H. Jiang, Z. Wan, Polymorphic uncertain optimization to renewable energy planning in the city cluster, International Journal of Applied Mathematics and Statistics, 2013 50(20) 224-237. 31

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[23] Z. Wan, S. J. Zhang, Y. Zhou, Interval solution for nonlinear programming of maximizing the fatigue life of V-belt under polymorphic uncertain environment, Mathematical Problems in Engineering, 2013, http://dx.doi.org/10.1155/2013/712825.

CR IP T

[24] C. V. Negoita, L. A. Zadeh, H. J. Zimmermann, Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1978 1 3-28. [25] A. N. Gani, S. N. M. Assarudeen, A new operation on triangular fuzzy number for solving fuzzy linear programming problem, Applied Mathematical Sciences, 2012 6(11) 525-532.

AN US

[26] A. Nagurney, Network economics: A variational inequality approach, Springer Netherlands, 1993. [27] T. Fujiwara-Greve, Non-Cooperative Game Theory. Springer Japan, 2015, DIO: 10.1007/978-4-431-55645-9.

M

[28] Z. Wan, M. Yuan, C. Wang, A partially smoothing Jacobian method for nonlinear complementarity problems with P0 function, Journal of Computational and Applied Mathematics. 2015 286 158-171.

AC

CE

PT

ED

[29] Q. Meng, Y.K. Huang, R.L. Cheu, A note on supply chain network equilibrium models, Transportation Research Part E. 2007, 43 60-71.

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