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Simultaneous competitive supply chain network design with continuous attractiveness variables
DOI: Reference:
http://dx.doi.org/10.1016/j.cie.2017.03.020 CAIE 4672
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
4 September 2016 11 March 2017 15 March 2017
Please cite this article as: Fahimi, K., Seyedhosseini, S.M., Makui, A., Simultaneous competitive supply chain network design with continuous attractiveness variables, Computers & Industrial Engineering (2017), doi: http:// dx.doi.org/10.1016/j.cie.2017.03.020
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Simultaneous competitive supply chain network design with continuous attractiveness variables Kaveh Fahimi Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected] Seyed Mohammad Seyedhosseini* Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected] Ahmad Makui Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected]
Abstract A two-stage algorithm is developed for the competitive supply chain network design problem in which two competitors simultaneously enter the market with no existing rival chain, shape their networks, and set attractiveness of their opened distribution centres to maximize their profits. The customer behaviour is based on the Huff gravity-based rule. The proposed algorithm is constructed based on the Lemke and Howson algorithm and variational inequality formulation with the help of bi-level programming, the modified projection method, and the possibility theory. We derive the equilibrium condition and establish the finite dimensional variational inequality formulation, and provide properties of the equilibrium patterns in terms of the results of existence and uniqueness. Finally, we generate some instances and use a real world study to discuss the effect of the different structures of the competitors, namely centralized, decentralized, cooperative, or unknown modes, on the equilibrium solution. Keywords: simultaneous competitive supply chain network design; Lemke and Howson algorithm; variational inequality; bi-level programming
1
1- Introduction At present, competition has progressed from “firms against firms” to “supply chains versus supply chains”; globalization is a common phenomenon; developing countries are trying to omit monopoly to enrol in the World Trade Organization (WTO); and countries are ratifying different foreign investment strategies and policies to attract international investors. Thus, investors are presented with good opportunities to design their networks locally and obtain virgin markets encountered by simultaneous competitions, but they are faced with certain questions, such as: what is the best network design in this competitive mode? What is the best structure of the network (centralized, decentralized, or cooperative mode)? How much market shares can they obtain? What is their equilibrium condition? The aim of this paper is to find the solutions to these questions. A supply chain (SC) is an organized network of suppliers, manufacturers, warehouses, and retailers involved in the production and distribution of merchandise in the right quantities and delivered to the right locations at the right time in order to minimize total costs while satisfying service level requirements (Simchi-Levi et al. 2003). According to Deloitte Consulting (1999), “no longer will companies compete against other companies but total supply chains will compete against other supply chains.” Several examples of supply chain competition can be found in maritime shipping, the automotive and retail industries, and online bookstores (Farahani et al. 2014). Supply chain network design (SCND) specifies the physical structure of a chain and has a huge impact on the SC’s overall performance. Simchi-Levi et al. (2003) consider SCND as the primary and most important step for decreasing (increasing) the cost (the profit) of chains. According to the SCND literature (e.g., Shen 2007; Meixell and Gargeya 2005; Beamon 1998), several studies (Altiparmark et al. 2006; Torabi and Hassini 2008; Pishvaee and Rabbani 2011; Shankar et al. 2013; Badri et al. 2013, Özceylan et al. 2014, Vahdani and Mohamadi 2015, Yang et al 2015, Ardalan et al. 2016, Keyvanshokooh et al. 2016, Özceylan et al. 2016, Jeihoonian et al. 2017, Varsei and Polyakovskiy 2017) have been conducted on monopoly assumptions. Competitive SCND (CSCND) considers the impact of competitive markets in designing the network structure of a chain to improve its future competitiveness (see Farahani et al. 2014 for a review on CSCND). Players and customers are the most important factors in CSCND. If no rival exists, monopolistic competition occurs; when one existing rival reacts to a newcomer, duopolistic competition occurs; and when more than one active competitor exists, oligopolistic competition is established. Depending on the reaction of the rivals, three kinds of competition have been examined in the literature: 1) Static competition: In the case no rival reacts to the entry of a newcomer, static competition occurs (see Plastra 2001 for a review of static competition). This competition is usually modelled by mathematical programming (Berman and Krass 1998; Aboolian et al. 2007; Revelle et al. 2007; Aboolian et al. 2007). 2) Dynamic competition: If rivals show simultaneous reaction to their decisions, dynamic competition takes place, and it mostly occurs in the context of operational characteristics and is modelled by unconstrained programming, which is solved using differential systems (Xiao and Yang 2008; Zhang 2006; Godinho and Dias 2013; Godinho and Dias 2010, Sinha and Sarmah 2010, Friesz et al. 2011, Jain et al. 2014, Chen et al 2015, Nagurney et al. 2015, SantibanezGonzalez and Diabat 2016, Hjaila et al. 2017, Lipan et al 2017). 3
3) Competition with foresight: In this type of competition almost has occurred by time sequence between the reactions of the competitors and is modelled using bi-level or multi-level programing. It was named by Stackelberg or leader-follower game (Drezner and Drezner 1998; Plastra and Vanhaverbeke 2008; Kucukaydın et al. 2011; Kucukaydın et al. 2012, Zhang and Liu L.P 2013, Yue and You 2014, Zhu 2015, Drezner et al. 2015, Yang et al. 2015, Hjaila et al. 2016, Aydin et al. 2016, Genc and Giovanni 2017; for reviews, see Eiselt and Laporte 1997; Krass and Pesch 2012). he other important factors in CSC D are customer behaviour and customer demand function . According to the literature, customer demand can be inelastic or elastic, and elastic demand can depend on price, service, price and service, or price and distance (Farahani et al. 2014). Customer utility functions are mostly categorized into deterministic and random utility models. Hotelling (1929) first introduced deterministic utility functions, in which customers visit facilities that give them the highest utility. However, in the random utility function, which was first introduced by Huff (1964, 1966), customers visit each facility with a certain probability that is directly related to the attractiveness of the facility and inversely related to the attractiveness of other facilities. Further, three kinds of competition exist in the SC competition literature: horizontal competition, i.e., competition between firms of one tier of an SC (Nagurney et al. 2002, Dong et al. 2004, Cruz 2008; Zhang and Zhou 2012, Qiang et al 2013; Huseh 2015; Qiang 2015, Li and Nagurney 2016, Nagurney et al. 2016 ); vertical competition, i.e., competition between the firms of different tiers of an SC (Bernstein and Federgruen 2005; Anderson and Bao 2010, Chen et al. 2013, Wu 2013, Zhao and Wang 2015, Zhang et al. 2015, Bai et al. 2016, Bo and Li 2016, Li et al. 2016, Huang et al. 2016, Wang et al. 2017 Genc and Giovanni 2017, Chaeb and Rasti-Barzoki 2017); and SC versus SC, i.e., competition between SCs (Boyaci and Gallego 2004; Xiao and Yang 2008; Zhang 2006; Li et al 2013, Chung and Kwon 2016). There are very few studies focusing on CSCND in the literature. Rezapour and Farahani (2010) developed a centralized SCND in a price-dependent market where a rival exists, derived equilibrium conditions, and established a finite dimensional variational inequality formulation. Rezapour et al. (2011) developed a model for sequential duopolistic CSCND under deterministic price-dependent demand in the presence of a rival chain. Rezapour and Farahani (2014) presented a new bi-level model for CSCND while anticipating the price and service level competition in the presence of an existing rival. In their model, the inner level deals with operational variables and the outer level specifies the strategic variables. Rezapour et al. (2014) presented a bi-level model for CSCND in the presence of an existing rival where demand is elastic with respect to price and distance. Rezapour et al. (2015) presented a bi-level model for closedloop SCND in price-dependent market demand with an existing SC, which only has a forward direction, but the new chain was a closed-loop SC. Fallah et al. (2015) presented a competitive closed-loop SCND problem in a price-dependent market under uncertainty and investigated the impact of simultaneous and Stackelberg competition between the chains. Rezapour et al. (2016) proposed a supply chain network design problem under competition, which could suffer from possible disruption, and designed a piecewise linear method to solve the problem. In our proposed problem, two SCs are planning to enter the market simultaneously, in which no rival already exists, so they have difficulties to predict the needed parameters because of a high degree of uncertainty and an absence of historical data. We therefore use fuzzy mathematical programming to 4
handle the described environment as it matches the described circumstance and requires fewer computations than the scenario-based stochastic programming approach (Pishvaee and Torabi 2010). Fuzzy mathematical programming (Pishvaee et al. 2012) has two main classes, a) flexible programming: this relates to a flexible target value of goals and constraints that are mostly related by a subjective fuzzy set, and b) possibility programming: this is applied to cope with the lack of knowledge about parameters and is modelled using possibilistic distributions. 1-1 Contribution of this paper In this paper we have developed a CSCND problem in dynamic competition: two SCs are simultaneously planning to enter the market in which no rival exists. The chains determine the optimal locations and attractiveness of their facilities to maximize their profits. To the best of our knowledge, such a CSCND problem with two tiers of facilities, including plant and distribution centre (DC) levels and continuous attractiveness variables of the DCs, has not been reported yet. The developed model for the chains is similar, based on nonconvex mixed integer nonlinear programming, and we also propose a two-step algorithm to solve the model and find the equilibrium solution. As the chains enter the dynamic competition, we use variational inequality (VI) formulation to find the equilibrium attractiveness of the DCs, but VI is only applicable on the models with continuous variables and convex functions. However, our problem is nonconvex and has 0-1 variables, and hence, we use the Lemke and Howson algorithm in stage two of our proposed solution method. We define the needed strategies based on location variables of the DCs in the first stage of our algorithm and use a bi-level programming, in which the inner level gives the equilibrium attractiveness of the DCs with the help of VI formulation and the modified projection method and the outer level optimizes the model and specifies the optimum locations of plants and other variables. In stage two, the equilibrium locations of the DCs will be chosen. Since both chains are newcomers, they encounter a high degree of uncertainty and lack historical data, so we use fuzzy mathematical programming to cope with the uncertain parameters. To the best of our knowledge, the introduced problem with continuous attractiveness variables has not yet been reported. Our proposed approach toward the problem is similar to that reported by Rezapour and Farahani (2010) and Nagurney (1999) and the subsequent paper by the same author. The remainder of this paper is structured as follows: Section 2 describes our problem and its properties, Section 3 presents our solution approach, Section 4 gives the numerical results and discussion, and Section 5 is the conclusion. 2- Problem definition In this study, we develop a problem faced by two centralized SCs, including plant and DC levels, entering simultaneously into a market with no existing rival (Fig 1). The chains wish to determine the optimal location of the plants, DCs, their assignments, and equilibrium attractiveness of the opened DCs, so as to maximize their incomes while minimizing their cost. The joining factor of the competitors is the quality of the opened DCs, and the competitors design their network in dynamic competition. Both chains produce an identical product, the customer demand is inelastic, and the customer utility function is based on the Huff gravity rule model. Figure 1 inserted here
The described situation is full of uncertainty owing to the absence and lack of knowledge of the SCs because they are newcomers to the market, and this is a major obstacle to obtaining the random 5
distribution function for uncertain parameters. Liu and Iwamura (1998) maintained that uncertainty is categorized into two main classes: probability and possibility. If the distribution function is available or can be found experimentally, the uncertainty is probabilistic, and it can be modelled by stochastic programming approaches. However, in the case of insufficient information, there exist some unfamiliar parameters, for which possibilistic theory is used to measure the subjective beliefs to a fuzzy set. We therefore used possibilistic programming and fuzzy mathematical programming to cope with uncertainties that are set to our described circumstance. Imagine there are l demand points indexed by k , and s, s operating plants indexed by i, i and m, m candidate sites for opening DCs indexed by j , j (also alternatively indexed by g , g ) for SC 1 and SC2. Assume that SC1 opens a DC at location j , with a quality level of a j and d (1)2 jk , which is the Euclidian distance between the opened DC at
j and customer k . By utilizing the gravity-based rule, the
attractiveness of this facility for customer k is given by
aj d (1)2 jk
, and then, the total attractiveness of the m
aj
j 1
d (1)2 jk
opened DCs in SC 1 for customer k by the newly-opened DCs is given by
. Similarly, by
considering w j as the quality level of the facility j in SC 2 and d (2)2 j k , which is the Euclidian distance between the opened retailer at j and customer k , the attractiveness of this facility for customer k is m
w j
obtained by
d
, and the total attractiveness of the opened DCs in SC2 for customer k is
(2)2 j k
j 1
w j d (2)2 j k
.
(1) Thus, by considering probabilistic customer utility function proposed by Huff, the probability Atrjk that
customer k visits facility j of SC 1 (based on all opened DCs in SC1 and SC2) is expressed as its attractiveness divided by the summation of the attractiveness of all the existing facilities, and is expressed aj
as Atrjk(1)
2 d (1) jk
ag
d g
(1) 2 gk
g
. Similarly, this probability for facility
wg
j in market
k is equal to
d g( 2)k 2
w j Atrj(k2 )
d (j 2k) 2 ag
d g
(1) 2 gk
g
. It is worth noting that
wg
Atr
(1) jk
j
Atrj(2) k 1 in market k . Now, if we consider j
d g(k2 ) 2
d k as the demand of customer k , we can define the market share of facility j in market k by multiplying the demand of customer d k by the probability that the customer patronizes its demand to the facility dk Atrjk(1) , and then the total market share of SC 1 from the opened DCs in all the markets is given as
(d j
) Atrjk(1) . Similarly, the market share of facility j in market k is d k Atr j k , and the total market (2)
k
k
share of SC2 is equal to
(d j
k
) Atrj(2) k (similar to Kucukaydın et al. 2011; Kucukaydın et al. 2012).
k
6
It is worth revealing some properties of the market share of the chains. aj
Proposition 1)
d (1)2 jk ag
d g
(1)2 gk
g
wg
is strictly concave for any j, k
d g(2)2 k aj
Proof. Since the only variable in the term is a j , it suffices to show that L jk
d (1)2 jk ag
d g
(1)2 gk
g
wg
is concave
d g(2)2 k
for a j . The concavity of L jk is achieved if its second order derivate is always negative, so we compute the first and second order derivate of L jk , as follows: ag wg 1 1 aj (2)2 (1)2 (1)2 (1)2 (1)2 d jk d jk L jk d jk g d gk g d g k 2 a j ag wg (1)2 (2)2 g d g k g d gk
(1)
and ag wg 1 1 aj (2)2 (1)2 (1)2 (1)2 (1)2 d jk g d gk d jk d jk L jk g d g k ; 2 2 2 a j ag wg (1)2 (2)2 g d g k g d gk
(2)
2
then,
2 L jk a 2j
is always negative, so the proof is completed. aj
Proposition 2)
(d j
k
k
)
d (1)2 jk ag
d g
(1)2 gk
g
is concave in A (a1 , a2 ,...., am ) for A 0 T
wg d g(2)2 k
Proof) See the proof in Kucukaydın et al. 2011. The following assumptions, parameters, and variables are used to model the introduced problems: Assumptions: There are no common potential facilities between the chains. 7
The demand of each customer market is concentred at discrete points. Demand is inelastic. Customer’s utility function is in accordance to the Huff gravity rule. Products are identical.
Parameters and variables
Parameters a lj
Minimum attractiveness level of DC in SC 1
a uj
Maximum attractiveness level of DC in SC 1
wlj
Minimum attractiveness level of DC in SC 2
wuj
Maximum attractiveness level of DC in SC 2
A1l alj
Matrix of minimum attractiveness level of DC in SC 1
A1u auj
Matrix of maximum attractiveness level of DC in SC 1
W2l wlj
Matrix of minimum attractiveness level of DC in SC 2
W2l wuj
Matrix of maximum attractiveness level of DC in SC 2
P (1)
Number of opened facilities in DC level of SC 1
q (1)
Number of opened facilities in plant level of SC 1
P(2)
Number of opened facilities in DC level of SC 2
q (2)
Number of opened facilities in plant level of SC 2
Fi (11)
Fixed cost of opening a plant at location i for SC1
Fj(12) ( Aj )
Cost of locating j th facility by DC in SC1 by quality level a j ; we assume that this function is continuous and convex, Dong et al (2004). i for
Fi(21)
Fixed cost of opening a plant at location
SC1
Fj(22) (W j )
Cost of locating j th facility by DC in SC2 by quality level w j ; we assume that this function is continuous and convex, Dong et al (2004). 8
and DC j for SC1
cij(11)
Unit transportation cost between plant
c (12) jk
Unit transportation cost between DC j and customer k for SC1
h (1) j
Unit holding cost at DC j for SC1
oi(1)
Unit production cost at plant
ci(21) j
Unit transportation cost between plant
c (22) j k
Unit transportation cost between DC j and customer k for SC1
h (2) j
Unit holding cost at DC j for SC1
oi(2)
Unit production cost at plant
Pijk(1)
Price of path ij to fulfil customer demand at market k in SC1
Pi (2) j k
Price of path i j to fulfil customer demand at market k in SC2
dk
Demand of customer k
m(1)
Amount of marginal profit for SC 1
m(2)
Amount of marginal profit for SC 2
i
i
i
for SC1 i
and DC j for SC1
for SC1
Variables xij(11)
Quantity of product shipped from plant i to DC j in SC1
X11 xij(11)
Vector of quantity of product shipped from plants to DCs in SC1
xi(21) j
Quantity of product shipped from plant i to DC j in SC2
X 21 xi(21) j
Vector of quantity of product shipped from plants to DCs in SC2
x (12) jk
Quantity of product shipped from DC j to customer k in SC1
X12 x(12) jk
Vector of quantity of product shipped from DCs to customers in SC1
x (22) j k
Quantity of product shipped from DC j to customer k in SC2
9
X 22 x(22) j k
Vector of quantity of product shipped from DCs to customers in SC2
aj
Quality level of DC at location j in SC 1
A1 a j
Vector of quality level of DCs in SC 1
w j
Quality level of DC at location j in SC 2
W2 w j
Vector of quality levels of DCs in SC 2
yi(11)
1 if SC1 opens a plant in location i 0 otherwise
y (12) j
1 if SC1 opens a DC in location j 0 otherwise
yi(21)
1 if SC 2 opens a plant in location i 0 otherwise
y (22) j
1 if SC 2 opens a DC in location j 0 otherwise
Y11
Vector of plants location variable in SC 1
Y12
Vector of DCs location variable in SC 1
Y21
Vector of plants location variable in SC 2
Y22
Vector of DCs location variable in SC 2
2-1 Modelling framework SC1 aj max Z1SC1 Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
(1)2 jk
(3)
y (12) j
yg(12) g
wg d g(2)2 k
yg(22)
(12) ( Fi (11) yi(11) cij(11) xij(11) c (12) jk x jk ( i
i
j
j
k
i
j
h (1) j 2
Fj(12) ( Aj ) y (12) j j ) xij(11) oi(1) xij(11) ) i
s .t
10
j
aj x (12) dk jk
d ag
d g
x
(11) ij
i
(1)2 gk
(1)2 jk
y (12) j
yg(12) g
wg
j,k
(4)
j
(5)
yg(22)
d g(2)2 k
x (12) jk k
P (1)
(6)
q (1)
(7)
alj a j auj
(8)
(11) xij(11) , x(12) , y (12) 0,1 jk , a j 0, yi j
(9)
y
(12) j
j
y
(11) i
i
SC 2 w j max Z1SC 2 Pi (2) j k ( d k ) j
k
i
d ag
d g
(1)2 gk
(2)2 j k
(10)
y (22) j
yg(12) g
wg d
(2)2 g k
yg(22)
(21) (22) (22) ( Fi (21) yi(21) ci(21) j xi j c j k x j k ( i
i
j
j
k
i
j
h (2) j 2
Fj(22) (W j ) y (22) j j (2) (21) ) xi(21) j oi xi j ) i
j
s .t w j x (22) j k d k
d ag
d g
x i
(21) i j
y (22) j
yg(12) g
wg d g(2)2 k
j,k
(11)
j
(12)
yg(22)
x (22) j k k
(22) j
P (2)
(13)
(21) i
q (2)
(14)
y j
y i
(1)2 gk
(2)2 j k
wlj w j wuj
(15)
11
(22) (21) (22) xi(21) j , x j k , w j 0,yi , y j 0,1
(16)
Terms 3, 10 represent the objective function of SC1 and SC2, which are composed of two essential terms. he first term represents total incomes by the opened DCs, and the rest are related to the SC’s costs, including location costs associated with opened DCs, fixed cost of opening plants and DCs, production cost at plants, transportation cost between plants and DCs, holding cost at DCs, and transportation cost between DCs and customers. Constraints 4,11 ensure that each DC should satisfy the amount of demands associated to it; constraints 5,12 ensure flow balance; constraints 6,13 and 7,14 ensure that only P(1) , P(2) (1) (2) DCs and q , q plants are opened; constraints 8,15 are related to the quality limits; and constraints 9,15
are related to the binary and non-negativity restriction on the corresponding decision variables. 3- Solution approach This section describes our solution method to the proposed non-convex mixed-integer nonlinear programing (MINLP) model for the dynamic CSCND problem by introducing a two-stage algorithm. Step one is constructed with a bi-level programming model, in which the inner level is used to obtain the equilibrium condition of the quality levels of the potential DCs with the help of variational inequality formulation and the modified projection method, see Nagurney et al. (2002) and the references therein, while the outer level finds the optimum plant’s locations and assignments with the help of a commercial optimization software. Step two finds Nash point(s) by the Lemke and Howson (1964) algorithm. Bi-level programming is a hierarchical optimization problem that is composed of two mathematical problems, here one of them is part of the constraints of the other and is mostly used for Stackelberg competition, in which the upper level is related to the mathematical model of the leader and the inner level expresses the mathematical level of the follower [refer to Colson et al., 2007, for more details]. We have adapted this modelling framework for our dynamic competition. Our proposed problem is a non-convex MINLP problem, for which there is no clear method for determining the optimum solution. In fact, the solution has two integral goals, i.e., maximizing the total income while minimizing the total costs. Owing to this fact and the structure of bi-level programming, we convert the goal of maximizing the total income minus the total location costs of the DCs to the constraint level of the original problem. We have thus convexified the problem and also created a bi-level model in which the inner level is related to maximizing the total income minus the total location costs of DCs and the outer level is to find the optimum locations of plants and their assignments to the potentially opened DCs through optimization techniques. Then, in the second stage, we obtain Nash point(s) by the Lemke and Howson (1964) algorithm, in which the equilibrium opened DCs and the network of the chains shaped simultaneously. In terms of the variables, it is clear that we encounter mainly two categories of variables including operational X11 , X12 , X 21 , X 22 and strategic Y11 , Y12 , Y21 , Y22 , A1 ,W2 variables. This differentiation in the variables helps us introduce our solution method. As the only joining factor is the quality of the DCs in the SCs and since the number of opened DCs in each chain, P1 , P 2 , are known beforehand, we can define possible strategies by fixing the location variables of the DCs, e.g., if SC 1 has 4 potential locations for opening 2 DCs and SC 2 has 3 potential locations for opening 1 DC, then a strategy is defined when SC1 opens DCs in location 1 and 2 and SC 2 opens a DC in location 3. Further, a total of
4 3 * 18 2 1
available
strategies are to be considered. We can now introduce our bi-level programming model that should be run 12
for each strategy in order to complete stage 1. We can also obtain the equivalent crisp model based on the method introduced by Inuiguchi and Ramik (2000), Liu and Iwamura (2006), Dubois and Prade (1987), Heilpern (1992), and Pishvaee et al. (2012) (see Appendix 1 for details). 3-1 Stage 1: Transforming of the proposed problem into a bi-level model We should actually define the following bi-level programming model for each combination of potential m m (1) * (2) P P
strategies in the first stage.
3-1-1 Inner level SC 1 m j (1) P
(17)
alj a j
m j (1) P
(18)
a j auj
m j (1) P
(19)
aj 0
m j (1) P
(20)
aj SC1 Z inner max Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
(1)2 jk
g
wg
Fj(12) ( Aj ) j
d g(2)2 k
s .t
SC 2 m j (2) P
(21)
wlj w j
m j (2) P
(22)
w j wuj
m j (2) P
(23)
w j SC 2 Z inner Pi (2) j k ( d k ) k
j
i
d ag
d g
(1)2 gk
(2)2 j k
g
wg
Fj(22) (W j ) j
d g(2)2 k
s .t
13
m j (2) P
wj 0
(24)
Terms 17, 21 represent the objective function of the inner level for SC 1 and SC 2, which includes the profits from selling the product to the customers minus the fixed cost of opening and setting the quality level of the facilities; constraints 19,20,22,23 are related to the quality limits; and constraints 20, 24 are related to non-negativity restrictions on the corresponding decision variables. 3-1-2 Outer level The outer level gives the outcome for the given quality levels from the inner level by solving the following model: SC1 aj SC1 Z outer max Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
Fj(12) ( Aj ) j
(1)2 jk
g
wg
i
j
j
k
(25)
d g(2)2 k
(12) ( Fi (11) cij(11) xij(11) c (12) jk x jk ( i
m j (1) P
i
h (1) j
j
2
) xij(11) oi(1) xij(11) ) i
j
m j (1) P
s .t (4-7) xij(11) , x(12) jk 0
m j (1) P
(26)
m j (2) P
(27)
SC2 w j SC 2 Z outer max Pi (2) j k ( d k ) j
k
i
d ag
d g
( F i
(21) i
c i
j
(21) (21) i j i j
x
c j
k
(1)2 gk
(22) j k
x
Fj(22) (W j ) j
(2)2 j k
g
(22) j k
wg d g(2)2 k
( i
j
h (2) j 2
(2) (21) ) xi(21) j oi xi j ) i
j
m j (2) P
s .t
14
m j (2) P
(11-14)
(22) xi(21) j , x j k 0
(28)
Terms 25, 26 represent the objective function of SC 1 and 2 in SC 1, in which quality levels are given from the inner part, and constraints 27, 28 are related to the non-negativity restrictions on the corresponding decision variables. 3-1-3 Gap analysis between the single level and its equivalent bi-level model When transforming the introduced single-level model into the proposed bi-level model, one might create a gap between the optimal solutions of the two models. In this section, we want to analyse this aspect more thoroughly. According to the proposed single-level model, for producing one unit of product in SC1 in plant i and delivering it to customer k through DC j , impose the variable costs as Cijk(1) cij(11) c (12) jk (
h (1) j 2
) oi(1) to the chain. This cost should be compromised with Pijk(1) as Pijk(1) Cijk(1) . This
consideration is made in the single-level model as all the terms simultaneously consider in a single objective, but in transforming the single-level model into the bi-level one, Cijk(1) is omitted from the inner level and leads to equilibrium quality levels A1 ,W2 , which might not be the equilibrium quality level of the single-level one. To solve this problem, we should modify Pijk(1) to Pijk(1) Cijk(1) in the inner level. This modification omits the possible gap between the proposed bi-level model and its equivalent single-level model. Henceforth, without loss of generality, we assume: Pijk(1) Cijk(1) m(1) . We can now write the objective function of the inner level as follows: aj
( Pijk(1) Cijk(1) )(dk ) k
j
i
d ag
d g
By considering m(1)
(1)2 gk
aj
(1)2 jk
g
wg
Fj(12) ( Aj ) m(1) (d k ) j k j i
d g(2)2 k
j
k
ag
g
(1)2 gk
g
d
1 , the objective function of the inner level is given as q (1)
d (1)2 jk
d
ag
g
aj
(dk )
d (1)2 jk
wg
Fj(12) ( Aj ) j
d g(2)2 k
Similarly, the inner level objective function in SC2 is as follows:
15
(1)2 gk
g
wg d g(2)2 k
Fj(12) ( Aj ) j
wj
(d j
k
)
k
Fj(22) (W j ) j
d (2)2 j k ag
d g
(1)2 gk
wg
d g(2)2 k
g
As an example, consider each chain has one plant and DC locations to open one plant and one DC. There exists one market by the demand equal to d1 900000 , and the required parameters of SC1 are (1) (1) (1)2 (1) Fi (11) 50, cij(11) 2.5, c(12) 1 . Similarly, we assume SC2 has the same jk 2.25, h j 1.2, oi 1.2, d jk 5, m
parameters as SC1. By solving the previous example with the introduced method, the equilibrium quality level and the total profits for the bi-level model are 148.5923 and 336556.5999, respectively, while the single-level model had equilibrium quality level and total profits of 149.501 and 336564.9798, respectively for both SCs. The results show that there exist 0.6% and 0.002% gaps between the equilibrium quality and total profits for the bi-level model in comparison with the single-level one. It is also worth noting that there is no solution method for obtaining the equilibrium results of the single-level model, and we therefore first solve the bi-level model. We then fix the quality and DC’s locations of each player to obtain the equilibrium results for the other player. This procedure functions correctly because if the bi-level model obtains the Nash equilibrium solution, no player in the single-level model has any motivation to change their decision without any changes in the decision of the other one because they gain no more profit. 3-1-4 Variational inequality formulations for the inner level In this part, we fist write Lagrange functions of the inner part, as follows: aj max L1 ( A, 1 , 1 ) (d k ) j
k
d ag
d g
(1)2 gk
(1)2 jk
g
wg
u (1) l Fj(12) ( Aj ) (1) j (a j a j ) j (a j a j ) j j j
j
k
d ag
d g
(1)2 gk
(29)
n m i, j (2) * (2) q P
(30)
d g(2)2 k
w j max L2 (W , 2 , 2 ) (d k )
n m i, j (1) * (1) q P
(2)2 j k
g
wg
u (2) l Fj(22) (W j ) (2) j (w j w j ) j (w j w j ) j j j
d g(2)2 k
(2) (1) (2) where the terms (1) j , j ( 1 j , 2 j ) are the Lagrange multipliers associated with conditions (2) (1) (2) (19),(23); and (1) j , j ( 1 j , 2 j ) are the Lagrange multipliers associated with conditions
(18),(22). We need to find the optimal quality levels of the DCs for the chains. The equilibrium is characterized by the fixed point describing the optimal quality levels of each SC. 3-1-5 Equilibrium condition Since SCs compete in a non-cooperative manner to maximize their profits, given the action of the other chain and with respect to the optimization problem (29), (30) as the optimization problems for the chains
16
and non-negativity assumption on the variables, the equilibrium condition of the inner level of the chains m m for any strategy of total (1) * (2) strategies can be expressed as follows, Nagurney (1999): P P
(d j
k
k
)
Atrjk(1) (a (1)* j ) a j
a
u j
a .
a
* j
a lj . j(1) j(1)*
* j
j
j
(1) j
(1)* j
Fj(12) (a (1)* j ) a j
jk(1)* jk(1)* . a j a*j
(31)
* Atrj(2) Fj(22) ( w*j ) k ( w j ) * ( d ) j(2)* (2)* k j . w j w j w w j k j j
w
(2)* w*j . j(2) j
w
(2)* wlj . j(2) j 0
j
j
u j
* j
j . k j . k j . k j . k j . k j . k
( A1 , 1 , 1 , W2 , 2 , 2 ) R
The equilibrium condition for the inner part of the chains coincides with the solution of VI (31). Definition 1 (Competitive attractiveness equilibrium of DCs in SCs). The equilibrium state of each strategy by the opened DCs for these two rival SCs is one where the quality levels of the related DCs satisfy the optimality condition (31). Theorem 1 (VI formulation). The equilibrium condition governing the attractiveness of DCs in SCs is equivalent to the solution of the VI formulation (31). Proof. The VI formulation method yields the inequality (31). The VI problem (31) can be rewritten in standard VI as follows, according to Nagurney (1999): determine x* k satisfying F ( X * ), X X * 0
x k
(32)
where x A1 , 1 , 1 ,W2 , 2 , 2 ,
(12) F ( X ) Atrjk(1) , Atrj(2) , Fj(22) k , F j
Further, the values of F are given by the functional terms preceding the multiplication signs in (32), the term .,. denotes the inner product in an n-dimensional Euclidean space. The quality levels of DCs and the related Lagrange multipliers according to the variables: A1 , 1 , 1 ,W2 , 2 , 2 can be obtained by solving the VI problem. 3-1-6 Qualitative properties 17
This section presents the existence and uniqueness of the above VI problem, with respect to the fact that the feasible set underlying the VI problem (31) is not compact, so it is not easy to derive the existence of a solution from the assumption of continuity of the functions. Therefore, by imposing a rather weak condition, we can generate the existence of a solution pattern, according to Nagurney et al. (2002). Let ( A1c , 1c , 1c ,W2c , 2c , 2c ) kc c c c c c c 0 A1 c1 , 0 1 c2 , 0 1 c3 , 0 W2 c4 , 0 2 c5 , 0 2 c6
(33)
j.k j.k j.k j .k j .k j .k Then, kc is a bounded, closed convex subset of R , and the following variational
inequality admits at least one solution x c k c F ( X c ), X X c 0
x K c
(34)
From the standard theory of VI, since kc is compact and F is continuous. Under the conditions in Theorem 1, we can chose c1 , c2 , c3 , c4 , c5 , c6 to be large enough that the restricted VI (34) will satisfy the boundless condition in (33) and the existence of a solution for the original VI problem will hold, according to Kinderlehrer and Stampacchia (1980). Theorem 2 (Existence). Suppose that there exist positive constants k with alj , wlj 0 such that:
Atrjk(1) (a (1)* j ) a j * Atrj(2) k ( w j )
w j
Fj(12) (a (1)* j ) a j
Fj(22) ( w*j ) w j
K
K
A1
W2
m with alj a j auj (j (1) ) P
(35)
m with wlj w j wuj (j (2) ) P
(36)
Then, VI (31) admits at least one solution. Proof. See the existence proof in Nagurney et al. (2001). (12) (2) * (22) * (a(1)* Lemma 1 (Monotonicity). Suppose that functions Atrjk(1) (a(1)* j ), Fj j ), Atrj k ( w j ), Fj ( w j ) for all
j, j , k are convex, then the vector function that enters the VI (31) is monotone, i.e.,
F ( X ) F ( X ), X X 0 x,x k
(37)
Proof. See the proof in Nagurney et al. (2002).
18
(1) (12)* * (22) * ), Atrj(2) Lemma 2 (Strict monotonicity). Suppose that functions Atrjk(1) (a(1)* k ( w j ), Fj ( w j ) for j ), Fj (a j
all j, j , k are strictly convex, then the vector function that enters the VI (31) is strictly monotone with
respect to A1 ,W2 , i.e., for any x, x k with A1 ,W2 A1 ,W2 : F ( X ) F ( X ), X X 0 x,x k
(38)
Proof. See the proof in Nagurney et al. (2002). Lemma 2 implies the uniqueness of the quality levels A1 ,W2 at equilibrium; it is worth noting that no guarantee of unique 1 , 1 , 2 , 2 can be generally expected at equilibrium. Theorem 3 (Uniqueness). Under the condition of Lemma 2, there are unique quality levels
A1 ,W2
satisfying the equilibrium condition of VI in (32). This means if VI (32) admits a solution for quality levels, it should be the only existing solution. Proof. Under the strict monotonicity result of Lemma 2, uniqueness follows from the VI theory, according to Kinderlehrer and Stampacchia (1980). (12) (2) * (22) * (a(1)* Lemma 3 (Lipchitz continuity) Suppose that Atrjk(1) (a(1)* j ), Fj j ), Atrj k ( w j ), Fj ( w j ) functions have
bounded second-order derivatives for all j, j , k a, so function F that enters the VI problem (32) is Lipchitz continuous, i.e., F ( X ) F ( X ) L X X
x,x k, where L>0
(39)
Proof. The result can be achieved by directly applying a mid-value theorem calculus to the vector function F that enters the VI problem (82). 3-1-7 The algorithm for stage 1 This algorithm comprises two parts: the first part is based on the modified projection method, (Nagurney and Toyasaki, 2005; Rezapour and Farahani, 2010), which solves the inner part and leads to equilibrium quality levels of the DCs; the second part is related to the outer level that specifies the optimum locations of plants and optimum assignments by any commercial optimization software such as Barrons solver in GAMS. The modified projection method for the first part of the franchisees model is as follows: Modified projection method for solution of variational inequality of DCs (inner level) Step 0: initialization 0 0 0 0 0 0 Set ( A1 , 1 , 1 ,W2 , 2 , 2 ) k . Let t 1 ( t is the iteration counter), and set 0
Lipchitz constant for the problem (39). 19
1 , where L is the L
Step 1: computation Compute ( A1t , 1t , 1t ,W2t , 2t , 2t ) k by solving Atrjk(1) (atj1 ) Fj(12) (atj1 ) atj MAX 0, a tj1 (d k ) j(1),t 1 j(1),t 1 k a a j j
j
(40)
j
(41)
t t 1 (1), MAX 0, (1), atj1 alj j j
j
(42)
t 1 Atrj(2) Fj(22) ( wtj 1 ) k ( w j ) t 1 t 1 wtj MAX 0, wtj 1 (d k ) j(2), (2), j k w j w j
j
(43)
j(1),t MAX 0, j(1),t 1 auj atj1
j
(44)
j
(45)
j
(46)
t t 1 j(2), MAX 0, (2), wuj wtj 1 j
t t 1 (2), MAX 0, (2), wtj 1 wlj j j
Step 2: adaptation t t t t t t Compute ( A1 , 1 , 1 ,W1 , 2 , 2 ) k by solving
Atrjk(1) (atj11 ) Fj(12) (atj1 ) atj MAX 0, a tj1 (d k ) j(1),t 1 j(1),t 1 k a a j j
j
(47)
t t 1 (1), MAX 0, (1), atj1 alj j j
j
(48)
t 1 Atrj(2) Fj(22) ( wtj 1 ) k ( w j ) t 1 t 1 wtj MAX 0, wtj 1 (d k ) j(2), (2), j k w j w j
j
(49)
t t 1 (1), MAX 0, (1), auj atj1 j j
j
(50)
j
(51)
t t 1 (2), MAX 0, (2), wuj wtj 1 j j
t t 1 (2), MAX 0, (2), wtj 1 wlj j j
Step 3: convergence verification If
20
atj atj1 ,
t t 1 (1), (1), , j j
t t 1 (2), (2), , j j
t t 1 (1), (1), , j j
wtj wtj 1 ,
t t 1 (2), (2), j j
For all j , j with (a pre-specified tolerance), then stop; otherwise, set t t 1 and back to step 1. Theorem 4 (Convergence). If the function that enters VI (31) or (32) satisfies the conditions in Theorem 2, Lemma 1, and Lemma 3, then the described modified projection method converges to the solution of VI (31) or (32). Proof. See the proof in Nagurney et al. (2002). 3-2 Stage 2: shaping SC’s networks Up to this stage, we have found all the entities for constructing a bi-matrix in which we can select the equilibrium network for the chain. We now define the selection of the equilibrium networks as follows: 3-2-1 Bi-matrix for Stage 2 This part defines a bi-matrix for the simultaneous game of the SCs on locations and assignments. SC 1 m has (1) P
(m)! P !(m P (1) )! (1)
m any of (1) P
m pure strategies and SC 2 correspondingly has (2)
(m)! P !(m P (1) )! (1)
P
m strategies and SC 2 plays any of (2) P
(m)! . P (2) !(m P (2) )!
(m)! P !(m P (2) )! (2)
If SC 1 plays
strategies, the payoff to
the SC 1 and 2 in their related strategies in stage 1 can be calculated. The game is completely specified m m ). * (1) (2) P P
when the payoff for all related strategies is calculated. (A bi-matrix with the dimensions of
The calculated payoff matrix will lead the problem into pure or mixed strategies (in the case of the existence of one Nash equilibrium or more than one), we use the algorithm proposed by Lemke–Howson to find the equilibrium point(s). (Interested readers can refer to Lemke and Howson (1964)). 4- Numerical study and discussion In this section, the proposed problem and the solution approach are investigated through some numerical instances, and then, we introduce a real word problem and provide modelling frameworks, discussion, and results if the SCs wish to shape their networks in decentralized or cooperative modes. Section 4-1 explores the instances, Section 4-2 describes the case study and its solution, and Section 4-3 fully discusses the study. 4-1 Numerical study There is no benchmark problem on simultaneous competitive SCND in the literature. We therefore generate 15 random instances to test the proposed algorithm through run time. We consider demand points from k 5,10,15 , the number of candidate facility for plants as i, i 5,7 , and DCs as j, j 4,5,8 . The lower and upper quality levels are set to 10 and 500, the required parameters are
21
generated according to Table 1, and the cost functions for DCs in SCs are as Fj(12) ( Aj ) c1a 2j c2 a j and Fj(22) (W j ) c1w2j c2 w j . According to the table, the parameters are assumed to be trapezoidal fuzzy
numbers; uniform distribution is used to extract the four prominent points for each fuzzy trapezoidal number. The proposed algorithm was implemented in Matlab 2014a and carried out on a Pentium dualcore 2.6 GHz with 2 GB RAM. We solved the random instances in the worst case scenario of the Lemke– Howson algorithm, when every possible combination of pure strategies of the players needs to be enumerated and the required time must be reported. In the written computer code, the inner level of the proposed algorithm runs until one of the following criteria is fulfilled: The qualities reach the upper or lower level. The absolute value of the qualities between two successive iterations differ by no more than
106 .
Then, these given qualities go to the upper level to compute total costs of any possible combination of pure strategies by the outer level to shape the bi-matrix, and the next equilibrium Nash point will be chosen by the Lemke–Howson algorithm. Table 1 inserted here
n m n m , , , , k , which (1) (1) (2) (2) q P q P
Table 2 shows the achieved results. We show every instance by
represents the number of opened facilities from potential facilities of the chains and the number of m m and (2) show the potential pure strategies of each chain; and (1) P P
available customers;
m m (1) . (2) P P
is the whole available strategies of the simultaneous CSCND problem. We run each instance 10 times and report the average computational time in the worst case scenario in which the algorithm needs to consider all the possible strategies for the game. It is observable from the run time that by increasing the number of strategies, the CPU time will increase exponentially. Therefore, a meta heuristic is proposed for large-scale problem, but the proposed algorithm can reach the Nash equilibrium in a reasonable time for small and medium scale problems. Further, setting good quality levels is an important factor in managing the time since this is one of the stopping criteria in the lower level of the proposed bi-level model. Table 2 inserted here
In the next part, we introduce a real word problem and discuss different modelling frameworks for the introduced case study.
4-2 A real word problem In this part, we explore our problem and the solution approach with a case study using data inspired from a real world problem. Our case study is related to two international investors who want to simultaneously enter to the bearing industry in Iran; there is no company in Iran who can produce high quality bearings so no rival exists in the market, and based on the marketing research team, they can sell the entire 22
produced product easily to the virgin market. They are entering to the market simultaneously b y centralized structures; each chain has 5 and 4 potential locations for opening plants and DCs, and wants to open one plant and two DCs to provide products to two markets by inelastic demands. The parameters are assumed to be trapezoidal fuzzy numbers, and owing to confidentiality consideration, uniform distribution in a reasonable range is used to extract the four prominent points for each fuzzy trapezoidal number. All the monetary values are in the Iranian currency as well, i.e., Rial. The mentioned distributions are used to extract the required parameters (Table 1). The cost functions of the introduced problem are listed in Table 3. Table 3 inserted here
The achieved equilibrium results are shown in Table 4; all the variables that do not appear in the table are equal to zero. Table 4 inserted here
4-3 Discussion The proceeding case study investigates the CSCND problem in the centralized one player mode in which the whole SC has one owner. As this structure requires extremely high costs for shaping of the network, the CSCND problem may be determined in the decentralized or cooperative modes or without knowing the structure of the rival chain. In this section, these different modes and their effect on equilibrium results will be discussed. It is worth noting that as the quality levels are specified by VI, the only factor that the dominant player can improve is the location of the opened DCs that are specified by the Lemke– Howson algorithm. In other words, the dominant player will chose the locations of DCs in which the objective function are improved rather than the total SCs objective function. We can now introduce different modes, modelling, possible games, and the results. Decentralized mode: If each chain has individual owners for each tier that will be decided noncooperatively to optimize their objective functions, this structure will be shaped. In this mode, the dominant player will chose the locations of the DCs, and this mode is modelled by multi-level MINLP, and based on the dominant player of the chains, four possible games can occur:
Decentralized based on plant VS. decentralized based on plant Decentralized based on plant VS. decentralized based on DC Decentralized based on DC VS. decentralized based on plant Decentralized based on DC VS. decentralized based on DC
We can now express the modelling frame work of this structure as follows: Plants of SC 1 SC1 Z Plant min( Fi (11) yi(11) cij(11) xij(11) oi(1) xij(11) ) i
i
j
i
m j (1) P
j
23
(52)
s .t m j (1) P
(5,7)
xij(11) 0, yi(11) 0,1
(53)
DC of SC 1 aj 1 SC 2 Z DC max (d k ) k
j
(1)2 jk
d ag
d g
(1)2 gk
g
h (12) Fj(12) ( Aj ) c (12) ) xij(11) ) jk x jk ( 2 i j j j k (1) j
wg d g(2)2 k
m j (1) P
(54 )
s .t (4,6)
m j (1) P
x(12) jk 0
m j (1) P
(55 )
Plant of SC2 SC 2 (21) (2) (21) Z Plant min( Fi(21) yi(21) ci(21) j xi j oi xi j ) i
i
j
i
j
m j (2) P
(56)
s .t m j (2) P
(12-14)
(21) xi(21) 0,1 j 0 yi
(57)
DC of SC 2
24
w j SC 2 Z DC max (d k ) j
k
d ag
d g
(1)2 gk
(2)2 j k
g
wg d g(2)2 k
(22) Fj(22) (W j ) c (22) j k x j k j j k
m j (2) h P ( ) xi(21) j 2 i j (2) j
(58 )
m j (2) P
s .t
(11,13) m j (2) P
x(22) j k 0
(59 )
erms 52, 56 minimize the plant’s objective function of SC 1 and SC 2, including the fixed cost of opening plants and variable production cost and transportation cost, and the rest of the terms are related to its constraints. Constraints 53, 57 are related to the binary and non-negativity restrictions on the corresponding decision variables. erms 54, 58 maximize DC’s objective function of the chains including its market share minus fixed cost of opening DCs, holding and transportation cost, and the rest of the terms are related to its constraints. Constraints 55, 59 are related to the binary and non-negativity restrictions on the corresponding decision variables. Cooperative mode: Since each SC has two independent owners with different degrees of power and the tendency to cooperate with each other, this mode can be shaped such that the objective function of the chains is the weighted sum of each owners objective function, as follows: SC1 ZCO : min (
P
(
SC1 SC1* SC1* Z Plant Z Plant Z SC1 Z DC P p ( DC ) ) (1 ) )p SC1* SC1* Z Plant Z DC
m j (1) P
(60)
s .t m j (1) P
(4-7,26)
SC 2 ZCO : min (
P
(
SC 2 SC 2* SC 2* Z Plant Z Plant Z SC 2 Z DC P p ( DC ) ) (1 ) )p SC 2* SC 2* Z Plant Z DC
m j (2) P
s .t m j (2) P
(11-14,28)
25
(61)
erms 60, 61 minimize the distance between the DC’s and plant’s objective functions of the chains from their best one based on their power to reach to the best solution. It is worth noting that if was equal to 0.5, then this mode is equal to the centralized two players. Mixed structure: If both chains do not enter the market with the same mode, this structure can occur, e.g., SC1 enters the market in the decentralized mode based on the DC structure and SC2 enters the market in the centralized mode, which leads to the following games:
Centralized one player VS. decentralized based on plant Centralized one player VS. decentralized based on DC Centralized one player VS. centralized two players Centralized one player VS. cooperative Centralized two players VS. centralized one player Centralized two players VS. decentralized based on plant Centralized two players VS. decentralized based on DC Centralized two players VS. cooperative Decentralized based on plant VS. centralized one player Decentralized based on plant VS. centralized two players Decentralized based on plant VS. decentralized based on DC Decentralized based on plant VS. cooperative Decentralized based on DC VS. centralized one player Decentralized based on DC VS. centralized two players Decentralized based on DC VS. decentralized based on plant Decentralized based on DC VS. cooperative Cooperative VS. centralized one player Cooperative VS. centralized two players Cooperative VS. decentralized based on plant Cooperative VS. decentralized based on DC
Unknown mode: In this case, each chain does not know the exact structure of its rival, but the probability of each possible structure of the competitor is known, so that the expected cost can be calculated, e.g., when SC1 chooses the decentralized mode based on the plant, SC1 knows that SC2 can choose every possible structure by the determined probability. The expected value of each mode for SC 1 is calculated, and then, the best structure is chosen by SC1, and SC2 choose its best structure similarly. From all the introduced possible games investigated through the previous study, the results show that four different outcomes are possible for SCs, e.g., the results of the centralized one player VS. decentralized based on the plant are different from the results of the decentralized mode based on plant VS. centralized one player and are different from the results of decentralized based on plant VS. decentralized based on plant, and all these three structures have different results from the base mode (centralized one player VS. centralized one player). Further, in the cooperative mode, the best response of SCs depend on the amount of , and their results for 0.7 are different from 0.7 . It worth noting that 0 1 , and we assumed
26
that belongs to 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 . Therefore, we categorized the games based on their outcomes in four different scenarios, as follows: Centralized one player VS . centralized one player ; Centralized one player VS . DC ; Centralized one player VS . centralized two player ; Centralized one player VS . cooperative ( 0.7 ); Centralized two player VS . centralized one player ; Centralized two player VS . DC ; Centralized two player VS . centralized two player ; Centralized two player VS . cooperative ( 0.7); S1 : Decentralized based on DC VS . centralized one player ; Cooperative ( 0.7 ) VS . centralized one player ; Decentralized based on DC VS . cooperative ( 0.7 ); Cooperative ( 0.7 )VS . DC ; Cooperative ( 0.7) VS . centralized two player ; Cooperative ( 0.7) VS . cooperative ( 0.7); Decentralized based on DC VS . centralized two player ; Decentralized based on DC VS . decentralized based on DC Centralized one player VS . decentralized based on plant ; Centralized one player VS . cooperative ( 0.7); Centralized two player VS . decentralized based on plant ; Centralized two player VS . cooperative ( 0.7); S2 : Decentralized based on DC VS . decentralized based on plant ; Cooperative ( 0.7) VS . decentralized based on plant ; Cooperative ( 0.7) VS . cooperative( 0.7); Decentralized based on DC VS . cooperative ( 0.7) Decentralized based on plant VS . centralized one player ; Decentralized based on plant VS . decentralized based on DC ; Decentralized based on Plant VS . centralized two player ; Decentralized based on plant VS . cooperative ( 0.7 ); S3 : Cooperative ( 0.7) VS . centralized one player ; Cooperative ( 0.7) VS . centralized two player ; Cooperative ( 0.7) VS . decentralized based on DC ; Cooperative ( 0.7) VS . cooperative( £0.7)
Decentralized based on plant VS . decentralized based on plant ; Decentralized based on plant VS . cooperative ( 0.7); S4 : Cooperative ( 0.7) VS . decentralized based on plant ; Cooperative ( 0.7) VS . cooperative( 0.7)
The achieved results of S1 , S2 , S3 , S4 are shown in Tables 4, 5, 6, and 7. Table 5, 6, 7 inserted here
We can now discuss the scenarios to find the best incoming structure for the chains. Table 8 represents the scenarios, and we compare them with S1 , because this scenario includes centralized one player VS. centralized one player as the based structure mode. Table 8 inserted here
27
According to the table, the highest SC’s profit for SC2 will occur in S 2 where the chain enters the market by decentralized based on plant or cooperative by 0.7 . In this scenario the outcome of DC, plant, and SC for SC 2 are increased by 58.72%, 64.59%, and 33.32%, whereas these outcomes for SC1 will decrease by 33.04%, 31.34%, and 37.52%, respectively, and therefore, SC1 has no motivation to enter to the market by any structure that placed them into S 2 , but they will choose S3 as an incoming structure in which the outcomes can be improved by 43.17%, 46.08%, and 35.53% for DC, plant, and SC and the outcomes can be decreased by 48.44%, 40.88%, and 81.16% for SC2. However, by considering all the scenarios and outcomes, the best structure for SC 1 are decentralized based on plant or cooperative by 0.7 , and when SC1 chose one of these structures, SC2 will chose the decentralized based on plant or cooperative by 0.7 , which lead the game to S 4 . The outcomes of SC 1 will improve by 29.14%, 32.95%, and 19.12%, and those of SC2 decrease by 22.90%, 21.50%, and 29%, respectively. Table 9 ,10 inserted here
Tables 9,10 show the results of the unknown structure for SCs, when all structures have an equal chance to occur. These results also show that if the chain do not known the incoming structure of their rival entering to the market by decentralized based on plant or cooperative by 0.7 create the most expected value for them.
5- Conclusion In this paper, we consider the simultaneous CSCND problem with different structures: centralized, decentralized, cooperative, or unknown modes, and the continuous attractiveness levels of the DCs, so as to try to maximize their market shares while minimizing their costs. Our proposed model is nonconvex MINLP, and we therefore propose a two-step algorithm to reach a global optimum solution, in which step one is constructed by a bi-level programming where the inner level gives the equilibrium attractiveness of any possible combinations of opened DCs by VI formulation and the modified projection method, and the outer level calculates the optimum solution for each combination. Stage two selects the best response solution by the Lemke and Howson algorithm. We also derive the qualitative properties of the equilibrium pattern, such as the existence and the uniqueness of the solution in the inner level. Finally, we generate random instances, use a real word study and fully discuss the model to clarify the effect of different structures of the players on their outcomes and the equilibrium solution. For further research, this model can be developed by stochastic programming approach or by considering robust, green, or closed loop CSCND problems.
28
Appendix 1 Imagine a is a trapezoidal fuzzy number that has the following membership function: f a ( x) 1 a ( x) ha ( x) 0
if
a(1) x a(2) ,
if
a(2) x a(3) ,
if
a(3) x a(4) ,
if
a(4)
x or x
(62)
a(1)
The expected value [EV] and expected interval [EI] of a can be defined as follows: EI [a(2)
a( 2)
a(1)
f a ( x)dx, a(3)
a( 4)
a( e )
(63)
ha ( x)dx],
(64)
a( 2) a( 4) 1 EV [a(2) f a ( x)dx a(3) ha ( x)dx] a(1) a( e ) 2
If a has the following possibility distribution function,
then EI (a) [
a(1) a(2) a(3) a(4) , ], 2 2
(65)
a(1) a(2) a(3) a(4)
(66)
EV (a) [
4
]
By considering k a real number, the possibility (Pos) and necessity (Nec) of a k can be defined as follows (Dubois and Prade 1987; Dubois 1987; Pishvaee et al. 2012): 29
1 k a(1) Pos (a k ) a(2) a(1) 0
if
if
a(1) k
1 k a(3) Nec(a k ) a(4) a(3) 0
if
a(4) k ,
if
if if
a(2) k ,
(67)
a(1) k a(2) ,
(68)
a(3) k a(4) , a(3) k
Assuming is the degree of feasibility, now if
0.5 , we have:
Pos(a k ) k (1 )a(1) ( )a(2) ,
(69)
Nec(a k ) k (1 )a(3) ( )a(4)
(70)
As Inuiguchi and Ramik (2000) and Pishvaee et al. (2012) reported, necessity is more meaningful for application in the fuzzy model to obtain its equivalent crisp model. Now consider the following impact form: min Z fy cx s.t Ax d Sx Ny
(71)
x 0, y 0,1
where the vectors f , c, d are related to the imprecise parameters (like fixed costs, variable costs, and demands), matrixes A, S , N are the coefficient matrixes of the constraints, and x, y are the vector of continuous and binary variables. Now imagine that f , c, d , N are imprecise parameters with trapezoidal possibility distributions, and the rest of the parameters are crisp. According to possibility chance constraint programming [Inuiguchi and Ramik 2000, Dubois and Prade 1987 and Pishvaee et al. 2012] and the described method, we can replace the expected value of the imprecise parameters in the objective function and use the necessary measure to write the equivalent crisp constraints, as follows:
30
f (1) f (2) f (3) f (4) min Z 4 s.t Ax (1 )d (3) d (4)
c(1) c(2) c(3) c(4) y 4
x
(72)
Sx (1 ) N (2) N (1) y x 0, y 0,1
As this procedure is easy to handle, requires less computations, matches our described circumstance, and is a powerful tool to cope with uncertainty, we use it in our model to write the equivalent crisp model of our proposed model.
31
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38
CSCND problem SC2
SC1
Plant
DC
customers
Figure 1 CSCND problem
39
DC
Plant
Table 1 Distribution of parameters (11)
Fi ,
(21) i
F
oi(1) , oi(2)
( (1500,2000), (2000,2500), (2500,3000), (3000,4000)) u u u u
(u (2,2.5), u (2.5,2.75), u (2.75,3), u (3,3.5))
cij(11) , ci(21) j
(u(0.9,1.5), u(1.5, 2.1), u(2.1, 2.5), u(2.5,3.12))
(22) c (12) jk , c j k
(u(1.5, 2), u(2, 2.5), u(2.5,3), u(3,3.5))
(2) h (1) j , hj
(u(1.25,1.5), u(1.5,1.75), u(1.75, 2), u(2, 2.25))
dk
(u(12000,13000), u(13000,15000), u(15000,17000), u(17000,19000))
c1
(u(0.5,1.5), u(1.5,3), u(3, 4.5), u(4.5,6))
c2
(u(1.5, 2.5), u(2.5,3.5), u(3.5, 4.5), u(4.5,5.5))
40
Table 2 Numerical instances Examples n m n m , , , ,k q (1) P(1) q (2) P(2)
1
2
3
4
5
6
7
Pure strategies of the chains
Opened plants
Opened DCs
Equilibrium qualities
Total profit
CPU (Seconds) Average time(10 runs)
5 4 5 4 , , , ,5 1 3 1 1
SC1
4
(3)
(1,3,4)
(66.90, 52.15, 43.61)
495028.89
SC2
4
(5)
(2)
(71.55)
172528.52
5 4 5 4 , , , ,5 1 2 1 2
SC1
6
(2)
(3,4)
(57.9, 49.56)
380068.43
SC2
6
(2)
(1,2)
(53.32, 35.01)
314475.11
7 5 7 5 , , , ,10 1 5 1 5
SC1
1
(4)
(1,2,3,4,5)
(46.13, 38.58, 53.13, 37.39, 41.68)
877225.93
SC2
1
(2)
(1,2,3,4,5)
(169.47, 104.74, 10.09, 28.70, 10.05)
366944.84
7 5 7 5 , , , ,15 1 5 1 5
SC1
1
(7)
(1,2,3,4,5)
(88.55, 71.56, 96.44, 75.98, 78.15)
1058413.16
SC2
1
(2)
(1,2,3,4,5)
(311.29, 232.66, 10, 10)
632230.76
7 5 7 5 , , , ,15 1 1 1 5
SC1
5
(6)
(3)
(269.23)
503070.88
SC2
1
(1)
(1,2,3,4,5)
(101.22, 94.68, 122.78, 91.64, 96.62)
615993.51
7 5 7 5 , , , ,10 1 1 1 1
SC1
5
(3)
(5)
(92.35)
721262.06
SC2
5
(5)
(4)
(175.04)
428446.18
7 5 7 5 , , , ,10 1 1 1 3
SC1
5
(2)
(5)
(196.96)
540377.86
SC2
10
(7)
(3,4,5)
(85.90, 80.13, 100.14)
576897.37
41
1.54
3.63
0.37
0.43
0.71
2.93
15.44
Examples n m n m (1) , (1) , (2) , (2) , k q P q P
8
9
10
11
12
13
14
15
Pure strategies of the chains
Opened plants
Opened DCs
Equilibrium qualities
Total profit
CPU (Seconds) Average time(10 runs)
7 5 7 5 , , , ,10 1 2 1 2
SC1
10
(1)
(3,5)
(85.83, 70.75)
556489.33
SC2
10
(1)
(4,5)
(197.22, 159.18)
296986.17
7 5 7 5 , , , ,15 1 2 1 2
SC1
10
(5)
(1,3)
(108.13, 85.93)
948532.37
SC2
10
(1)
(1,2)
(246.90, 215.52)
546076.36
7 8 7 8 , , , ,5 2 1 2 1
SC1
8
(1,5)
(3)
(66.28)
216495.7799
SC2
8
(1,2)
(6)
(119.71)
233038.6878
7 8 7 8 , , , ,10 1 1 1 1
SC1
8
(1,5)
(3)
(102.36)
527341.80
SC2
8
(1,2)
(6)
(175.04)
403733.49
7 8 7 8 , , , ,15 2 1 2 1
SC1
8
(4,7)
(3)
(127.60)
859697.31
SC2
8
(1,6)
(1)
(166.83)
665573.29
7 8 7 8 , , , ,5 1 1 1 2
SC1
8
(1,4)
(4)
(127.51)
186352.14
SC2
28
(2,7)
(4,8)
(63.94, 71)
325820.56
7 8 7 8 , , , ,10 1 1 1 2
SC1
8
(3,5)
(3)
(186.1)
484500.8
SC2
28
(5,6)
(6,7)
(89.26, 103.67)
575841.94
7 8 7 8 , , , ,15 1 1 1 2
SC1
8
(2,4)
(2)
(235.59)
652347.9
SC2
28
(4,5)
(4,7)
(111.47, 132.21)
958556.30
42
55.69
91.87
15.92
48.84
56.99
178.94
393.3
707.75
Table 3 Cost functions of the DCs
SC 1
SC 2
Fixed cost of DC 1
F1 (12)( A1 ) 2a12 2.5a1
F1(22) (W1 ) 1.5w12 3w1
Fixed cost of DC 2
F2(12) ( A2 ) 2.5a22 3a2
F2(22) (W2 ) 2.8w22 4w2
Fixed cost of DC 3
F3(12) ( A3 ) 1.5a32 4a3
F3(22) (W3 ) 3w32 1w3
Fixed cost of DC 4
F4(12) ( A4 ) 2.2a42 5a4
F4(22) (W4 ) 4w42 5w4
Table 4 Computational results for SCs
Amount of shipments between plant 2 and DC2 Amount of shipments between plant 2 and DC3 Amount of shipments between plant 1 and DC3 Amount of shipments between plant 1 and DC4 Amount of shipments between DC2 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC2 and market 2 Amount of shipments between DC3 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC2 Equilibrium quality of DC3 Equilibrium quality of DC4 Total incomes Total costs
SC1 5599.41 6647.462 0 0 2819.269 3000.776 0 2780.141 3646.686 0 38.86 35.97 0 167491.7634 143617.2034
SC2 0 0 11438.979 14.095 4245.964 5.940 0 7193.015 8.155 0 50.98 0.08 125812 111788.06
Table 5 Compeutational results of S 2
S2 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC2 Amount of shipments between plant 2 and DC2 Amount of shipments between plant 2 and DC3 Amount of shipments between DC1 and market 1 Amount of shipments between DC2 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC2 and market 2 Amount of shipments between DC3 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC2 Equilibrium quality of DC3 Total incomes Total costs
SC1
SC2
0 0 4042.546 4257.25 0 1702.514 1580.370 0 2340.032 2676.880 0 35.48 28.65 114071.31 99154.86
11254.145 7214.885 0 0 4745.280 4076.476 0 6508.865 3138.409 0 56.95 29.14 0 188148.11 169441.31
43
Table 6 Compeutational results of S3
S3 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC3 Amount of shipments between plant 1 and DC4 Amount of shipments between DC1 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC3 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC3 Equilibrium quality of DC4 Total incomes Total costs
SC1
SC2
7493.11 0 10239.535 2174.047 0 7040.694 5319.063 0 3198.841 38.54 0 34.47 226268.33 193910.42
0 5874.221 0 0 1412.515 0 0 4461.706 0 0 35.56 0 64584.35 61941.47
Table 7 Compeutational results of S 4
S4 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC4 Amount of shipments between plant 2 and DC1 Amount of shipments between plant 2 and DC2 Amount of shipments between DC1 and market 1 Amount of shipments between DC2 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC2 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC2 Equilibrium quality of DC4 Total incomes Total costs
SC1
SC2
6839.835 9254.821
0 0 6567.275 2062.37 2316.839 1016.083 0 4250.436 1046.287 0 40.076 10.47 0 89603.6123 79640.5823
0 0 1824.979 0 6127.868 5014.856 0 3126.953 38.37 0 35.58 204782.0127 176342.2527
Table 8 Comparasion analysis
S1 S2 S3
SC1
DC objective function 86686.51
Plant objective function 62811.95
SC objective function 23874.56
SC2 SC1 SC2 SC1 SC2 SC1 SC2
74748.82 58041.45 118639.81 124111.19 38542.05 111948.71 57628.78
60716.88 43124.99353 99933.01 91753.280 35899.17 83508.95 47665.76
14031.94 14916.45 18706.80 32357.91 2642.88 28439.76 9963.03
44
Gain/ loss of DC (%)
Gain/ loss of plant (%)
Gain/ loss of SC (%)
-33.0444 58.71797 43.17244 -48.4379 29.14202 -22.9034
-31.3427 64.58851 46.07615 -40.8745 32.95074 -21.495
-37.5216 33.31585 35.53301 -81.1653 19.12161 -28.9975
Table 9 SC1 VS unknown structures
SC1
Decentral ized based on DC
Decentral ized based on plant
Two player centralize d
One player centralized
Cooperative
0.1
0.2
0.3
.04
0.5
0.6
0.7
0.8
0.9
Expected value
Decentralized based on DC
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
Decentralized based on plant
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
One player centralized
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
Two player centralized
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.1
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.2
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.3
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.4
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.5
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.6
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.7
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.8
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
0.9
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
Cooperative
45
Table 10 SC2 VS unknown structures
SC2
Decentral ized based on DC
Decentral ized based on plant
Two player centralize d
One player centralized
Cooperative
0.1
0.2
0.3
.04
0.5
0.6
0.7
0.8
0.9
Expected value
Decentralized based on DC
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
Decentralized based on plant
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
One player centralized
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
Two player centralized
14031.94
28439.76
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
28439.76
28439.76
17356.82
0.1
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.2
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.3
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.4
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.5
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.6
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.7
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.8
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
0.9
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
Cooperative
46
Highlights
We analyse the simultaneous competitive supply chain network design problem. We discuss the effect of different modes on the problem. A two-stage algorithm is proposed to solve the problem. Bi-level programming is used in the proposed algorithm.
47
http://dx.doi.org/10.1016/j.cie.2017.03.020 CAIE 4672
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
4 September 2016 11 March 2017 15 March 2017
Please cite this article as: Fahimi, K., Seyedhosseini, S.M., Makui, A., Simultaneous competitive supply chain network design with continuous attractiveness variables, Computers & Industrial Engineering (2017), doi: http:// dx.doi.org/10.1016/j.cie.2017.03.020
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Simultaneous competitive supply chain network design with continuous attractiveness variables Kaveh Fahimi Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected] Seyed Mohammad Seyedhosseini* Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected] Ahmad Makui Department of Industrial Engineering, Iran University of Science & Technology, Narmak, Terhan, Islamic Republic of Iran [email protected]
Abstract A two-stage algorithm is developed for the competitive supply chain network design problem in which two competitors simultaneously enter the market with no existing rival chain, shape their networks, and set attractiveness of their opened distribution centres to maximize their profits. The customer behaviour is based on the Huff gravity-based rule. The proposed algorithm is constructed based on the Lemke and Howson algorithm and variational inequality formulation with the help of bi-level programming, the modified projection method, and the possibility theory. We derive the equilibrium condition and establish the finite dimensional variational inequality formulation, and provide properties of the equilibrium patterns in terms of the results of existence and uniqueness. Finally, we generate some instances and use a real world study to discuss the effect of the different structures of the competitors, namely centralized, decentralized, cooperative, or unknown modes, on the equilibrium solution. Keywords: simultaneous competitive supply chain network design; Lemke and Howson algorithm; variational inequality; bi-level programming
1
1- Introduction At present, competition has progressed from “firms against firms” to “supply chains versus supply chains”; globalization is a common phenomenon; developing countries are trying to omit monopoly to enrol in the World Trade Organization (WTO); and countries are ratifying different foreign investment strategies and policies to attract international investors. Thus, investors are presented with good opportunities to design their networks locally and obtain virgin markets encountered by simultaneous competitions, but they are faced with certain questions, such as: what is the best network design in this competitive mode? What is the best structure of the network (centralized, decentralized, or cooperative mode)? How much market shares can they obtain? What is their equilibrium condition? The aim of this paper is to find the solutions to these questions. A supply chain (SC) is an organized network of suppliers, manufacturers, warehouses, and retailers involved in the production and distribution of merchandise in the right quantities and delivered to the right locations at the right time in order to minimize total costs while satisfying service level requirements (Simchi-Levi et al. 2003). According to Deloitte Consulting (1999), “no longer will companies compete against other companies but total supply chains will compete against other supply chains.” Several examples of supply chain competition can be found in maritime shipping, the automotive and retail industries, and online bookstores (Farahani et al. 2014). Supply chain network design (SCND) specifies the physical structure of a chain and has a huge impact on the SC’s overall performance. Simchi-Levi et al. (2003) consider SCND as the primary and most important step for decreasing (increasing) the cost (the profit) of chains. According to the SCND literature (e.g., Shen 2007; Meixell and Gargeya 2005; Beamon 1998), several studies (Altiparmark et al. 2006; Torabi and Hassini 2008; Pishvaee and Rabbani 2011; Shankar et al. 2013; Badri et al. 2013, Özceylan et al. 2014, Vahdani and Mohamadi 2015, Yang et al 2015, Ardalan et al. 2016, Keyvanshokooh et al. 2016, Özceylan et al. 2016, Jeihoonian et al. 2017, Varsei and Polyakovskiy 2017) have been conducted on monopoly assumptions. Competitive SCND (CSCND) considers the impact of competitive markets in designing the network structure of a chain to improve its future competitiveness (see Farahani et al. 2014 for a review on CSCND). Players and customers are the most important factors in CSCND. If no rival exists, monopolistic competition occurs; when one existing rival reacts to a newcomer, duopolistic competition occurs; and when more than one active competitor exists, oligopolistic competition is established. Depending on the reaction of the rivals, three kinds of competition have been examined in the literature: 1) Static competition: In the case no rival reacts to the entry of a newcomer, static competition occurs (see Plastra 2001 for a review of static competition). This competition is usually modelled by mathematical programming (Berman and Krass 1998; Aboolian et al. 2007; Revelle et al. 2007; Aboolian et al. 2007). 2) Dynamic competition: If rivals show simultaneous reaction to their decisions, dynamic competition takes place, and it mostly occurs in the context of operational characteristics and is modelled by unconstrained programming, which is solved using differential systems (Xiao and Yang 2008; Zhang 2006; Godinho and Dias 2013; Godinho and Dias 2010, Sinha and Sarmah 2010, Friesz et al. 2011, Jain et al. 2014, Chen et al 2015, Nagurney et al. 2015, SantibanezGonzalez and Diabat 2016, Hjaila et al. 2017, Lipan et al 2017). 3
3) Competition with foresight: In this type of competition almost has occurred by time sequence between the reactions of the competitors and is modelled using bi-level or multi-level programing. It was named by Stackelberg or leader-follower game (Drezner and Drezner 1998; Plastra and Vanhaverbeke 2008; Kucukaydın et al. 2011; Kucukaydın et al. 2012, Zhang and Liu L.P 2013, Yue and You 2014, Zhu 2015, Drezner et al. 2015, Yang et al. 2015, Hjaila et al. 2016, Aydin et al. 2016, Genc and Giovanni 2017; for reviews, see Eiselt and Laporte 1997; Krass and Pesch 2012). he other important factors in CSC D are customer behaviour and customer demand function . According to the literature, customer demand can be inelastic or elastic, and elastic demand can depend on price, service, price and service, or price and distance (Farahani et al. 2014). Customer utility functions are mostly categorized into deterministic and random utility models. Hotelling (1929) first introduced deterministic utility functions, in which customers visit facilities that give them the highest utility. However, in the random utility function, which was first introduced by Huff (1964, 1966), customers visit each facility with a certain probability that is directly related to the attractiveness of the facility and inversely related to the attractiveness of other facilities. Further, three kinds of competition exist in the SC competition literature: horizontal competition, i.e., competition between firms of one tier of an SC (Nagurney et al. 2002, Dong et al. 2004, Cruz 2008; Zhang and Zhou 2012, Qiang et al 2013; Huseh 2015; Qiang 2015, Li and Nagurney 2016, Nagurney et al. 2016 ); vertical competition, i.e., competition between the firms of different tiers of an SC (Bernstein and Federgruen 2005; Anderson and Bao 2010, Chen et al. 2013, Wu 2013, Zhao and Wang 2015, Zhang et al. 2015, Bai et al. 2016, Bo and Li 2016, Li et al. 2016, Huang et al. 2016, Wang et al. 2017 Genc and Giovanni 2017, Chaeb and Rasti-Barzoki 2017); and SC versus SC, i.e., competition between SCs (Boyaci and Gallego 2004; Xiao and Yang 2008; Zhang 2006; Li et al 2013, Chung and Kwon 2016). There are very few studies focusing on CSCND in the literature. Rezapour and Farahani (2010) developed a centralized SCND in a price-dependent market where a rival exists, derived equilibrium conditions, and established a finite dimensional variational inequality formulation. Rezapour et al. (2011) developed a model for sequential duopolistic CSCND under deterministic price-dependent demand in the presence of a rival chain. Rezapour and Farahani (2014) presented a new bi-level model for CSCND while anticipating the price and service level competition in the presence of an existing rival. In their model, the inner level deals with operational variables and the outer level specifies the strategic variables. Rezapour et al. (2014) presented a bi-level model for CSCND in the presence of an existing rival where demand is elastic with respect to price and distance. Rezapour et al. (2015) presented a bi-level model for closedloop SCND in price-dependent market demand with an existing SC, which only has a forward direction, but the new chain was a closed-loop SC. Fallah et al. (2015) presented a competitive closed-loop SCND problem in a price-dependent market under uncertainty and investigated the impact of simultaneous and Stackelberg competition between the chains. Rezapour et al. (2016) proposed a supply chain network design problem under competition, which could suffer from possible disruption, and designed a piecewise linear method to solve the problem. In our proposed problem, two SCs are planning to enter the market simultaneously, in which no rival already exists, so they have difficulties to predict the needed parameters because of a high degree of uncertainty and an absence of historical data. We therefore use fuzzy mathematical programming to 4
handle the described environment as it matches the described circumstance and requires fewer computations than the scenario-based stochastic programming approach (Pishvaee and Torabi 2010). Fuzzy mathematical programming (Pishvaee et al. 2012) has two main classes, a) flexible programming: this relates to a flexible target value of goals and constraints that are mostly related by a subjective fuzzy set, and b) possibility programming: this is applied to cope with the lack of knowledge about parameters and is modelled using possibilistic distributions. 1-1 Contribution of this paper In this paper we have developed a CSCND problem in dynamic competition: two SCs are simultaneously planning to enter the market in which no rival exists. The chains determine the optimal locations and attractiveness of their facilities to maximize their profits. To the best of our knowledge, such a CSCND problem with two tiers of facilities, including plant and distribution centre (DC) levels and continuous attractiveness variables of the DCs, has not been reported yet. The developed model for the chains is similar, based on nonconvex mixed integer nonlinear programming, and we also propose a two-step algorithm to solve the model and find the equilibrium solution. As the chains enter the dynamic competition, we use variational inequality (VI) formulation to find the equilibrium attractiveness of the DCs, but VI is only applicable on the models with continuous variables and convex functions. However, our problem is nonconvex and has 0-1 variables, and hence, we use the Lemke and Howson algorithm in stage two of our proposed solution method. We define the needed strategies based on location variables of the DCs in the first stage of our algorithm and use a bi-level programming, in which the inner level gives the equilibrium attractiveness of the DCs with the help of VI formulation and the modified projection method and the outer level optimizes the model and specifies the optimum locations of plants and other variables. In stage two, the equilibrium locations of the DCs will be chosen. Since both chains are newcomers, they encounter a high degree of uncertainty and lack historical data, so we use fuzzy mathematical programming to cope with the uncertain parameters. To the best of our knowledge, the introduced problem with continuous attractiveness variables has not yet been reported. Our proposed approach toward the problem is similar to that reported by Rezapour and Farahani (2010) and Nagurney (1999) and the subsequent paper by the same author. The remainder of this paper is structured as follows: Section 2 describes our problem and its properties, Section 3 presents our solution approach, Section 4 gives the numerical results and discussion, and Section 5 is the conclusion. 2- Problem definition In this study, we develop a problem faced by two centralized SCs, including plant and DC levels, entering simultaneously into a market with no existing rival (Fig 1). The chains wish to determine the optimal location of the plants, DCs, their assignments, and equilibrium attractiveness of the opened DCs, so as to maximize their incomes while minimizing their cost. The joining factor of the competitors is the quality of the opened DCs, and the competitors design their network in dynamic competition. Both chains produce an identical product, the customer demand is inelastic, and the customer utility function is based on the Huff gravity rule model. Figure 1 inserted here
The described situation is full of uncertainty owing to the absence and lack of knowledge of the SCs because they are newcomers to the market, and this is a major obstacle to obtaining the random 5
distribution function for uncertain parameters. Liu and Iwamura (1998) maintained that uncertainty is categorized into two main classes: probability and possibility. If the distribution function is available or can be found experimentally, the uncertainty is probabilistic, and it can be modelled by stochastic programming approaches. However, in the case of insufficient information, there exist some unfamiliar parameters, for which possibilistic theory is used to measure the subjective beliefs to a fuzzy set. We therefore used possibilistic programming and fuzzy mathematical programming to cope with uncertainties that are set to our described circumstance. Imagine there are l demand points indexed by k , and s, s operating plants indexed by i, i and m, m candidate sites for opening DCs indexed by j , j (also alternatively indexed by g , g ) for SC 1 and SC2. Assume that SC1 opens a DC at location j , with a quality level of a j and d (1)2 jk , which is the Euclidian distance between the opened DC at
j and customer k . By utilizing the gravity-based rule, the
attractiveness of this facility for customer k is given by
aj d (1)2 jk
, and then, the total attractiveness of the m
aj
j 1
d (1)2 jk
opened DCs in SC 1 for customer k by the newly-opened DCs is given by
. Similarly, by
considering w j as the quality level of the facility j in SC 2 and d (2)2 j k , which is the Euclidian distance between the opened retailer at j and customer k , the attractiveness of this facility for customer k is m
w j
obtained by
d
, and the total attractiveness of the opened DCs in SC2 for customer k is
(2)2 j k
j 1
w j d (2)2 j k
.
(1) Thus, by considering probabilistic customer utility function proposed by Huff, the probability Atrjk that
customer k visits facility j of SC 1 (based on all opened DCs in SC1 and SC2) is expressed as its attractiveness divided by the summation of the attractiveness of all the existing facilities, and is expressed aj
as Atrjk(1)
2 d (1) jk
ag
d g
(1) 2 gk
g
. Similarly, this probability for facility
wg
j in market
k is equal to
d g( 2)k 2
w j Atrj(k2 )
d (j 2k) 2 ag
d g
(1) 2 gk
g
. It is worth noting that
wg
Atr
(1) jk
j
Atrj(2) k 1 in market k . Now, if we consider j
d g(k2 ) 2
d k as the demand of customer k , we can define the market share of facility j in market k by multiplying the demand of customer d k by the probability that the customer patronizes its demand to the facility dk Atrjk(1) , and then the total market share of SC 1 from the opened DCs in all the markets is given as
(d j
) Atrjk(1) . Similarly, the market share of facility j in market k is d k Atr j k , and the total market (2)
k
k
share of SC2 is equal to
(d j
k
) Atrj(2) k (similar to Kucukaydın et al. 2011; Kucukaydın et al. 2012).
k
6
It is worth revealing some properties of the market share of the chains. aj
Proposition 1)
d (1)2 jk ag
d g
(1)2 gk
g
wg
is strictly concave for any j, k
d g(2)2 k aj
Proof. Since the only variable in the term is a j , it suffices to show that L jk
d (1)2 jk ag
d g
(1)2 gk
g
wg
is concave
d g(2)2 k
for a j . The concavity of L jk is achieved if its second order derivate is always negative, so we compute the first and second order derivate of L jk , as follows: ag wg 1 1 aj (2)2 (1)2 (1)2 (1)2 (1)2 d jk d jk L jk d jk g d gk g d g k 2 a j ag wg (1)2 (2)2 g d g k g d gk
(1)
and ag wg 1 1 aj (2)2 (1)2 (1)2 (1)2 (1)2 d jk g d gk d jk d jk L jk g d g k ; 2 2 2 a j ag wg (1)2 (2)2 g d g k g d gk
(2)
2
then,
2 L jk a 2j
is always negative, so the proof is completed. aj
Proposition 2)
(d j
k
k
)
d (1)2 jk ag
d g
(1)2 gk
g
is concave in A (a1 , a2 ,...., am ) for A 0 T
wg d g(2)2 k
Proof) See the proof in Kucukaydın et al. 2011. The following assumptions, parameters, and variables are used to model the introduced problems: Assumptions: There are no common potential facilities between the chains. 7
The demand of each customer market is concentred at discrete points. Demand is inelastic. Customer’s utility function is in accordance to the Huff gravity rule. Products are identical.
Parameters and variables
Parameters a lj
Minimum attractiveness level of DC in SC 1
a uj
Maximum attractiveness level of DC in SC 1
wlj
Minimum attractiveness level of DC in SC 2
wuj
Maximum attractiveness level of DC in SC 2
A1l alj
Matrix of minimum attractiveness level of DC in SC 1
A1u auj
Matrix of maximum attractiveness level of DC in SC 1
W2l wlj
Matrix of minimum attractiveness level of DC in SC 2
W2l wuj
Matrix of maximum attractiveness level of DC in SC 2
P (1)
Number of opened facilities in DC level of SC 1
q (1)
Number of opened facilities in plant level of SC 1
P(2)
Number of opened facilities in DC level of SC 2
q (2)
Number of opened facilities in plant level of SC 2
Fi (11)
Fixed cost of opening a plant at location i for SC1
Fj(12) ( Aj )
Cost of locating j th facility by DC in SC1 by quality level a j ; we assume that this function is continuous and convex, Dong et al (2004). i for
Fi(21)
Fixed cost of opening a plant at location
SC1
Fj(22) (W j )
Cost of locating j th facility by DC in SC2 by quality level w j ; we assume that this function is continuous and convex, Dong et al (2004). 8
and DC j for SC1
cij(11)
Unit transportation cost between plant
c (12) jk
Unit transportation cost between DC j and customer k for SC1
h (1) j
Unit holding cost at DC j for SC1
oi(1)
Unit production cost at plant
ci(21) j
Unit transportation cost between plant
c (22) j k
Unit transportation cost between DC j and customer k for SC1
h (2) j
Unit holding cost at DC j for SC1
oi(2)
Unit production cost at plant
Pijk(1)
Price of path ij to fulfil customer demand at market k in SC1
Pi (2) j k
Price of path i j to fulfil customer demand at market k in SC2
dk
Demand of customer k
m(1)
Amount of marginal profit for SC 1
m(2)
Amount of marginal profit for SC 2
i
i
i
for SC1 i
and DC j for SC1
for SC1
Variables xij(11)
Quantity of product shipped from plant i to DC j in SC1
X11 xij(11)
Vector of quantity of product shipped from plants to DCs in SC1
xi(21) j
Quantity of product shipped from plant i to DC j in SC2
X 21 xi(21) j
Vector of quantity of product shipped from plants to DCs in SC2
x (12) jk
Quantity of product shipped from DC j to customer k in SC1
X12 x(12) jk
Vector of quantity of product shipped from DCs to customers in SC1
x (22) j k
Quantity of product shipped from DC j to customer k in SC2
9
X 22 x(22) j k
Vector of quantity of product shipped from DCs to customers in SC2
aj
Quality level of DC at location j in SC 1
A1 a j
Vector of quality level of DCs in SC 1
w j
Quality level of DC at location j in SC 2
W2 w j
Vector of quality levels of DCs in SC 2
yi(11)
1 if SC1 opens a plant in location i 0 otherwise
y (12) j
1 if SC1 opens a DC in location j 0 otherwise
yi(21)
1 if SC 2 opens a plant in location i 0 otherwise
y (22) j
1 if SC 2 opens a DC in location j 0 otherwise
Y11
Vector of plants location variable in SC 1
Y12
Vector of DCs location variable in SC 1
Y21
Vector of plants location variable in SC 2
Y22
Vector of DCs location variable in SC 2
2-1 Modelling framework SC1 aj max Z1SC1 Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
(1)2 jk
(3)
y (12) j
yg(12) g
wg d g(2)2 k
yg(22)
(12) ( Fi (11) yi(11) cij(11) xij(11) c (12) jk x jk ( i
i
j
j
k
i
j
h (1) j 2
Fj(12) ( Aj ) y (12) j j ) xij(11) oi(1) xij(11) ) i
s .t
10
j
aj x (12) dk jk
d ag
d g
x
(11) ij
i
(1)2 gk
(1)2 jk
y (12) j
yg(12) g
wg
j,k
(4)
j
(5)
yg(22)
d g(2)2 k
x (12) jk k
P (1)
(6)
q (1)
(7)
alj a j auj
(8)
(11) xij(11) , x(12) , y (12) 0,1 jk , a j 0, yi j
(9)
y
(12) j
j
y
(11) i
i
SC 2 w j max Z1SC 2 Pi (2) j k ( d k ) j
k
i
d ag
d g
(1)2 gk
(2)2 j k
(10)
y (22) j
yg(12) g
wg d
(2)2 g k
yg(22)
(21) (22) (22) ( Fi (21) yi(21) ci(21) j xi j c j k x j k ( i
i
j
j
k
i
j
h (2) j 2
Fj(22) (W j ) y (22) j j (2) (21) ) xi(21) j oi xi j ) i
j
s .t w j x (22) j k d k
d ag
d g
x i
(21) i j
y (22) j
yg(12) g
wg d g(2)2 k
j,k
(11)
j
(12)
yg(22)
x (22) j k k
(22) j
P (2)
(13)
(21) i
q (2)
(14)
y j
y i
(1)2 gk
(2)2 j k
wlj w j wuj
(15)
11
(22) (21) (22) xi(21) j , x j k , w j 0,yi , y j 0,1
(16)
Terms 3, 10 represent the objective function of SC1 and SC2, which are composed of two essential terms. he first term represents total incomes by the opened DCs, and the rest are related to the SC’s costs, including location costs associated with opened DCs, fixed cost of opening plants and DCs, production cost at plants, transportation cost between plants and DCs, holding cost at DCs, and transportation cost between DCs and customers. Constraints 4,11 ensure that each DC should satisfy the amount of demands associated to it; constraints 5,12 ensure flow balance; constraints 6,13 and 7,14 ensure that only P(1) , P(2) (1) (2) DCs and q , q plants are opened; constraints 8,15 are related to the quality limits; and constraints 9,15
are related to the binary and non-negativity restriction on the corresponding decision variables. 3- Solution approach This section describes our solution method to the proposed non-convex mixed-integer nonlinear programing (MINLP) model for the dynamic CSCND problem by introducing a two-stage algorithm. Step one is constructed with a bi-level programming model, in which the inner level is used to obtain the equilibrium condition of the quality levels of the potential DCs with the help of variational inequality formulation and the modified projection method, see Nagurney et al. (2002) and the references therein, while the outer level finds the optimum plant’s locations and assignments with the help of a commercial optimization software. Step two finds Nash point(s) by the Lemke and Howson (1964) algorithm. Bi-level programming is a hierarchical optimization problem that is composed of two mathematical problems, here one of them is part of the constraints of the other and is mostly used for Stackelberg competition, in which the upper level is related to the mathematical model of the leader and the inner level expresses the mathematical level of the follower [refer to Colson et al., 2007, for more details]. We have adapted this modelling framework for our dynamic competition. Our proposed problem is a non-convex MINLP problem, for which there is no clear method for determining the optimum solution. In fact, the solution has two integral goals, i.e., maximizing the total income while minimizing the total costs. Owing to this fact and the structure of bi-level programming, we convert the goal of maximizing the total income minus the total location costs of the DCs to the constraint level of the original problem. We have thus convexified the problem and also created a bi-level model in which the inner level is related to maximizing the total income minus the total location costs of DCs and the outer level is to find the optimum locations of plants and their assignments to the potentially opened DCs through optimization techniques. Then, in the second stage, we obtain Nash point(s) by the Lemke and Howson (1964) algorithm, in which the equilibrium opened DCs and the network of the chains shaped simultaneously. In terms of the variables, it is clear that we encounter mainly two categories of variables including operational X11 , X12 , X 21 , X 22 and strategic Y11 , Y12 , Y21 , Y22 , A1 ,W2 variables. This differentiation in the variables helps us introduce our solution method. As the only joining factor is the quality of the DCs in the SCs and since the number of opened DCs in each chain, P1 , P 2 , are known beforehand, we can define possible strategies by fixing the location variables of the DCs, e.g., if SC 1 has 4 potential locations for opening 2 DCs and SC 2 has 3 potential locations for opening 1 DC, then a strategy is defined when SC1 opens DCs in location 1 and 2 and SC 2 opens a DC in location 3. Further, a total of
4 3 * 18 2 1
available
strategies are to be considered. We can now introduce our bi-level programming model that should be run 12
for each strategy in order to complete stage 1. We can also obtain the equivalent crisp model based on the method introduced by Inuiguchi and Ramik (2000), Liu and Iwamura (2006), Dubois and Prade (1987), Heilpern (1992), and Pishvaee et al. (2012) (see Appendix 1 for details). 3-1 Stage 1: Transforming of the proposed problem into a bi-level model We should actually define the following bi-level programming model for each combination of potential m m (1) * (2) P P
strategies in the first stage.
3-1-1 Inner level SC 1 m j (1) P
(17)
alj a j
m j (1) P
(18)
a j auj
m j (1) P
(19)
aj 0
m j (1) P
(20)
aj SC1 Z inner max Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
(1)2 jk
g
wg
Fj(12) ( Aj ) j
d g(2)2 k
s .t
SC 2 m j (2) P
(21)
wlj w j
m j (2) P
(22)
w j wuj
m j (2) P
(23)
w j SC 2 Z inner Pi (2) j k ( d k ) k
j
i
d ag
d g
(1)2 gk
(2)2 j k
g
wg
Fj(22) (W j ) j
d g(2)2 k
s .t
13
m j (2) P
wj 0
(24)
Terms 17, 21 represent the objective function of the inner level for SC 1 and SC 2, which includes the profits from selling the product to the customers minus the fixed cost of opening and setting the quality level of the facilities; constraints 19,20,22,23 are related to the quality limits; and constraints 20, 24 are related to non-negativity restrictions on the corresponding decision variables. 3-1-2 Outer level The outer level gives the outcome for the given quality levels from the inner level by solving the following model: SC1 aj SC1 Z outer max Pijk(1) (d k ) k
j
i
d ag
d g
(1)2 gk
Fj(12) ( Aj ) j
(1)2 jk
g
wg
i
j
j
k
(25)
d g(2)2 k
(12) ( Fi (11) cij(11) xij(11) c (12) jk x jk ( i
m j (1) P
i
h (1) j
j
2
) xij(11) oi(1) xij(11) ) i
j
m j (1) P
s .t (4-7) xij(11) , x(12) jk 0
m j (1) P
(26)
m j (2) P
(27)
SC2 w j SC 2 Z outer max Pi (2) j k ( d k ) j
k
i
d ag
d g
( F i
(21) i
c i
j
(21) (21) i j i j
x
c j
k
(1)2 gk
(22) j k
x
Fj(22) (W j ) j
(2)2 j k
g
(22) j k
wg d g(2)2 k
( i
j
h (2) j 2
(2) (21) ) xi(21) j oi xi j ) i
j
m j (2) P
s .t
14
m j (2) P
(11-14)
(22) xi(21) j , x j k 0
(28)
Terms 25, 26 represent the objective function of SC 1 and 2 in SC 1, in which quality levels are given from the inner part, and constraints 27, 28 are related to the non-negativity restrictions on the corresponding decision variables. 3-1-3 Gap analysis between the single level and its equivalent bi-level model When transforming the introduced single-level model into the proposed bi-level model, one might create a gap between the optimal solutions of the two models. In this section, we want to analyse this aspect more thoroughly. According to the proposed single-level model, for producing one unit of product in SC1 in plant i and delivering it to customer k through DC j , impose the variable costs as Cijk(1) cij(11) c (12) jk (
h (1) j 2
) oi(1) to the chain. This cost should be compromised with Pijk(1) as Pijk(1) Cijk(1) . This
consideration is made in the single-level model as all the terms simultaneously consider in a single objective, but in transforming the single-level model into the bi-level one, Cijk(1) is omitted from the inner level and leads to equilibrium quality levels A1 ,W2 , which might not be the equilibrium quality level of the single-level one. To solve this problem, we should modify Pijk(1) to Pijk(1) Cijk(1) in the inner level. This modification omits the possible gap between the proposed bi-level model and its equivalent single-level model. Henceforth, without loss of generality, we assume: Pijk(1) Cijk(1) m(1) . We can now write the objective function of the inner level as follows: aj
( Pijk(1) Cijk(1) )(dk ) k
j
i
d ag
d g
By considering m(1)
(1)2 gk
aj
(1)2 jk
g
wg
Fj(12) ( Aj ) m(1) (d k ) j k j i
d g(2)2 k
j
k
ag
g
(1)2 gk
g
d
1 , the objective function of the inner level is given as q (1)
d (1)2 jk
d
ag
g
aj
(dk )
d (1)2 jk
wg
Fj(12) ( Aj ) j
d g(2)2 k
Similarly, the inner level objective function in SC2 is as follows:
15
(1)2 gk
g
wg d g(2)2 k
Fj(12) ( Aj ) j
wj
(d j
k
)
k
Fj(22) (W j ) j
d (2)2 j k ag
d g
(1)2 gk
wg
d g(2)2 k
g
As an example, consider each chain has one plant and DC locations to open one plant and one DC. There exists one market by the demand equal to d1 900000 , and the required parameters of SC1 are (1) (1) (1)2 (1) Fi (11) 50, cij(11) 2.5, c(12) 1 . Similarly, we assume SC2 has the same jk 2.25, h j 1.2, oi 1.2, d jk 5, m
parameters as SC1. By solving the previous example with the introduced method, the equilibrium quality level and the total profits for the bi-level model are 148.5923 and 336556.5999, respectively, while the single-level model had equilibrium quality level and total profits of 149.501 and 336564.9798, respectively for both SCs. The results show that there exist 0.6% and 0.002% gaps between the equilibrium quality and total profits for the bi-level model in comparison with the single-level one. It is also worth noting that there is no solution method for obtaining the equilibrium results of the single-level model, and we therefore first solve the bi-level model. We then fix the quality and DC’s locations of each player to obtain the equilibrium results for the other player. This procedure functions correctly because if the bi-level model obtains the Nash equilibrium solution, no player in the single-level model has any motivation to change their decision without any changes in the decision of the other one because they gain no more profit. 3-1-4 Variational inequality formulations for the inner level In this part, we fist write Lagrange functions of the inner part, as follows: aj max L1 ( A, 1 , 1 ) (d k ) j
k
d ag
d g
(1)2 gk
(1)2 jk
g
wg
u (1) l Fj(12) ( Aj ) (1) j (a j a j ) j (a j a j ) j j j
j
k
d ag
d g
(1)2 gk
(29)
n m i, j (2) * (2) q P
(30)
d g(2)2 k
w j max L2 (W , 2 , 2 ) (d k )
n m i, j (1) * (1) q P
(2)2 j k
g
wg
u (2) l Fj(22) (W j ) (2) j (w j w j ) j (w j w j ) j j j
d g(2)2 k
(2) (1) (2) where the terms (1) j , j ( 1 j , 2 j ) are the Lagrange multipliers associated with conditions (2) (1) (2) (19),(23); and (1) j , j ( 1 j , 2 j ) are the Lagrange multipliers associated with conditions
(18),(22). We need to find the optimal quality levels of the DCs for the chains. The equilibrium is characterized by the fixed point describing the optimal quality levels of each SC. 3-1-5 Equilibrium condition Since SCs compete in a non-cooperative manner to maximize their profits, given the action of the other chain and with respect to the optimization problem (29), (30) as the optimization problems for the chains
16
and non-negativity assumption on the variables, the equilibrium condition of the inner level of the chains m m for any strategy of total (1) * (2) strategies can be expressed as follows, Nagurney (1999): P P
(d j
k
k
)
Atrjk(1) (a (1)* j ) a j
a
u j
a .
a
* j
a lj . j(1) j(1)*
* j
j
j
(1) j
(1)* j
Fj(12) (a (1)* j ) a j
jk(1)* jk(1)* . a j a*j
(31)
* Atrj(2) Fj(22) ( w*j ) k ( w j ) * ( d ) j(2)* (2)* k j . w j w j w w j k j j
w
(2)* w*j . j(2) j
w
(2)* wlj . j(2) j 0
j
j
u j
* j
j . k j . k j . k j . k j . k j . k
( A1 , 1 , 1 , W2 , 2 , 2 ) R
The equilibrium condition for the inner part of the chains coincides with the solution of VI (31). Definition 1 (Competitive attractiveness equilibrium of DCs in SCs). The equilibrium state of each strategy by the opened DCs for these two rival SCs is one where the quality levels of the related DCs satisfy the optimality condition (31). Theorem 1 (VI formulation). The equilibrium condition governing the attractiveness of DCs in SCs is equivalent to the solution of the VI formulation (31). Proof. The VI formulation method yields the inequality (31). The VI problem (31) can be rewritten in standard VI as follows, according to Nagurney (1999): determine x* k satisfying F ( X * ), X X * 0
x k
(32)
where x A1 , 1 , 1 ,W2 , 2 , 2 ,
(12) F ( X ) Atrjk(1) , Atrj(2) , Fj(22) k , F j
Further, the values of F are given by the functional terms preceding the multiplication signs in (32), the term .,. denotes the inner product in an n-dimensional Euclidean space. The quality levels of DCs and the related Lagrange multipliers according to the variables: A1 , 1 , 1 ,W2 , 2 , 2 can be obtained by solving the VI problem. 3-1-6 Qualitative properties 17
This section presents the existence and uniqueness of the above VI problem, with respect to the fact that the feasible set underlying the VI problem (31) is not compact, so it is not easy to derive the existence of a solution from the assumption of continuity of the functions. Therefore, by imposing a rather weak condition, we can generate the existence of a solution pattern, according to Nagurney et al. (2002). Let ( A1c , 1c , 1c ,W2c , 2c , 2c ) kc c c c c c c 0 A1 c1 , 0 1 c2 , 0 1 c3 , 0 W2 c4 , 0 2 c5 , 0 2 c6
(33)
j.k j.k j.k j .k j .k j .k Then, kc is a bounded, closed convex subset of R , and the following variational
inequality admits at least one solution x c k c F ( X c ), X X c 0
x K c
(34)
From the standard theory of VI, since kc is compact and F is continuous. Under the conditions in Theorem 1, we can chose c1 , c2 , c3 , c4 , c5 , c6 to be large enough that the restricted VI (34) will satisfy the boundless condition in (33) and the existence of a solution for the original VI problem will hold, according to Kinderlehrer and Stampacchia (1980). Theorem 2 (Existence). Suppose that there exist positive constants k with alj , wlj 0 such that:
Atrjk(1) (a (1)* j ) a j * Atrj(2) k ( w j )
w j
Fj(12) (a (1)* j ) a j
Fj(22) ( w*j ) w j
K
K
A1
W2
m with alj a j auj (j (1) ) P
(35)
m with wlj w j wuj (j (2) ) P
(36)
Then, VI (31) admits at least one solution. Proof. See the existence proof in Nagurney et al. (2001). (12) (2) * (22) * (a(1)* Lemma 1 (Monotonicity). Suppose that functions Atrjk(1) (a(1)* j ), Fj j ), Atrj k ( w j ), Fj ( w j ) for all
j, j , k are convex, then the vector function that enters the VI (31) is monotone, i.e.,
F ( X ) F ( X ), X X 0 x,x k
(37)
Proof. See the proof in Nagurney et al. (2002).
18
(1) (12)* * (22) * ), Atrj(2) Lemma 2 (Strict monotonicity). Suppose that functions Atrjk(1) (a(1)* k ( w j ), Fj ( w j ) for j ), Fj (a j
all j, j , k are strictly convex, then the vector function that enters the VI (31) is strictly monotone with
respect to A1 ,W2 , i.e., for any x, x k with A1 ,W2 A1 ,W2 : F ( X ) F ( X ), X X 0 x,x k
(38)
Proof. See the proof in Nagurney et al. (2002). Lemma 2 implies the uniqueness of the quality levels A1 ,W2 at equilibrium; it is worth noting that no guarantee of unique 1 , 1 , 2 , 2 can be generally expected at equilibrium. Theorem 3 (Uniqueness). Under the condition of Lemma 2, there are unique quality levels
A1 ,W2
satisfying the equilibrium condition of VI in (32). This means if VI (32) admits a solution for quality levels, it should be the only existing solution. Proof. Under the strict monotonicity result of Lemma 2, uniqueness follows from the VI theory, according to Kinderlehrer and Stampacchia (1980). (12) (2) * (22) * (a(1)* Lemma 3 (Lipchitz continuity) Suppose that Atrjk(1) (a(1)* j ), Fj j ), Atrj k ( w j ), Fj ( w j ) functions have
bounded second-order derivatives for all j, j , k a, so function F that enters the VI problem (32) is Lipchitz continuous, i.e., F ( X ) F ( X ) L X X
x,x k, where L>0
(39)
Proof. The result can be achieved by directly applying a mid-value theorem calculus to the vector function F that enters the VI problem (82). 3-1-7 The algorithm for stage 1 This algorithm comprises two parts: the first part is based on the modified projection method, (Nagurney and Toyasaki, 2005; Rezapour and Farahani, 2010), which solves the inner part and leads to equilibrium quality levels of the DCs; the second part is related to the outer level that specifies the optimum locations of plants and optimum assignments by any commercial optimization software such as Barrons solver in GAMS. The modified projection method for the first part of the franchisees model is as follows: Modified projection method for solution of variational inequality of DCs (inner level) Step 0: initialization 0 0 0 0 0 0 Set ( A1 , 1 , 1 ,W2 , 2 , 2 ) k . Let t 1 ( t is the iteration counter), and set 0
Lipchitz constant for the problem (39). 19
1 , where L is the L
Step 1: computation Compute ( A1t , 1t , 1t ,W2t , 2t , 2t ) k by solving Atrjk(1) (atj1 ) Fj(12) (atj1 ) atj MAX 0, a tj1 (d k ) j(1),t 1 j(1),t 1 k a a j j
j
(40)
j
(41)
t t 1 (1), MAX 0, (1), atj1 alj j j
j
(42)
t 1 Atrj(2) Fj(22) ( wtj 1 ) k ( w j ) t 1 t 1 wtj MAX 0, wtj 1 (d k ) j(2), (2), j k w j w j
j
(43)
j(1),t MAX 0, j(1),t 1 auj atj1
j
(44)
j
(45)
j
(46)
t t 1 j(2), MAX 0, (2), wuj wtj 1 j
t t 1 (2), MAX 0, (2), wtj 1 wlj j j
Step 2: adaptation t t t t t t Compute ( A1 , 1 , 1 ,W1 , 2 , 2 ) k by solving
Atrjk(1) (atj11 ) Fj(12) (atj1 ) atj MAX 0, a tj1 (d k ) j(1),t 1 j(1),t 1 k a a j j
j
(47)
t t 1 (1), MAX 0, (1), atj1 alj j j
j
(48)
t 1 Atrj(2) Fj(22) ( wtj 1 ) k ( w j ) t 1 t 1 wtj MAX 0, wtj 1 (d k ) j(2), (2), j k w j w j
j
(49)
t t 1 (1), MAX 0, (1), auj atj1 j j
j
(50)
j
(51)
t t 1 (2), MAX 0, (2), wuj wtj 1 j j
t t 1 (2), MAX 0, (2), wtj 1 wlj j j
Step 3: convergence verification If
20
atj atj1 ,
t t 1 (1), (1), , j j
t t 1 (2), (2), , j j
t t 1 (1), (1), , j j
wtj wtj 1 ,
t t 1 (2), (2), j j
For all j , j with (a pre-specified tolerance), then stop; otherwise, set t t 1 and back to step 1. Theorem 4 (Convergence). If the function that enters VI (31) or (32) satisfies the conditions in Theorem 2, Lemma 1, and Lemma 3, then the described modified projection method converges to the solution of VI (31) or (32). Proof. See the proof in Nagurney et al. (2002). 3-2 Stage 2: shaping SC’s networks Up to this stage, we have found all the entities for constructing a bi-matrix in which we can select the equilibrium network for the chain. We now define the selection of the equilibrium networks as follows: 3-2-1 Bi-matrix for Stage 2 This part defines a bi-matrix for the simultaneous game of the SCs on locations and assignments. SC 1 m has (1) P
(m)! P !(m P (1) )! (1)
m any of (1) P
m pure strategies and SC 2 correspondingly has (2)
(m)! P !(m P (1) )! (1)
P
m strategies and SC 2 plays any of (2) P
(m)! . P (2) !(m P (2) )!
(m)! P !(m P (2) )! (2)
If SC 1 plays
strategies, the payoff to
the SC 1 and 2 in their related strategies in stage 1 can be calculated. The game is completely specified m m ). * (1) (2) P P
when the payoff for all related strategies is calculated. (A bi-matrix with the dimensions of
The calculated payoff matrix will lead the problem into pure or mixed strategies (in the case of the existence of one Nash equilibrium or more than one), we use the algorithm proposed by Lemke–Howson to find the equilibrium point(s). (Interested readers can refer to Lemke and Howson (1964)). 4- Numerical study and discussion In this section, the proposed problem and the solution approach are investigated through some numerical instances, and then, we introduce a real word problem and provide modelling frameworks, discussion, and results if the SCs wish to shape their networks in decentralized or cooperative modes. Section 4-1 explores the instances, Section 4-2 describes the case study and its solution, and Section 4-3 fully discusses the study. 4-1 Numerical study There is no benchmark problem on simultaneous competitive SCND in the literature. We therefore generate 15 random instances to test the proposed algorithm through run time. We consider demand points from k 5,10,15 , the number of candidate facility for plants as i, i 5,7 , and DCs as j, j 4,5,8 . The lower and upper quality levels are set to 10 and 500, the required parameters are
21
generated according to Table 1, and the cost functions for DCs in SCs are as Fj(12) ( Aj ) c1a 2j c2 a j and Fj(22) (W j ) c1w2j c2 w j . According to the table, the parameters are assumed to be trapezoidal fuzzy
numbers; uniform distribution is used to extract the four prominent points for each fuzzy trapezoidal number. The proposed algorithm was implemented in Matlab 2014a and carried out on a Pentium dualcore 2.6 GHz with 2 GB RAM. We solved the random instances in the worst case scenario of the Lemke– Howson algorithm, when every possible combination of pure strategies of the players needs to be enumerated and the required time must be reported. In the written computer code, the inner level of the proposed algorithm runs until one of the following criteria is fulfilled: The qualities reach the upper or lower level. The absolute value of the qualities between two successive iterations differ by no more than
106 .
Then, these given qualities go to the upper level to compute total costs of any possible combination of pure strategies by the outer level to shape the bi-matrix, and the next equilibrium Nash point will be chosen by the Lemke–Howson algorithm. Table 1 inserted here
n m n m , , , , k , which (1) (1) (2) (2) q P q P
Table 2 shows the achieved results. We show every instance by
represents the number of opened facilities from potential facilities of the chains and the number of m m and (2) show the potential pure strategies of each chain; and (1) P P
available customers;
m m (1) . (2) P P
is the whole available strategies of the simultaneous CSCND problem. We run each instance 10 times and report the average computational time in the worst case scenario in which the algorithm needs to consider all the possible strategies for the game. It is observable from the run time that by increasing the number of strategies, the CPU time will increase exponentially. Therefore, a meta heuristic is proposed for large-scale problem, but the proposed algorithm can reach the Nash equilibrium in a reasonable time for small and medium scale problems. Further, setting good quality levels is an important factor in managing the time since this is one of the stopping criteria in the lower level of the proposed bi-level model. Table 2 inserted here
In the next part, we introduce a real word problem and discuss different modelling frameworks for the introduced case study.
4-2 A real word problem In this part, we explore our problem and the solution approach with a case study using data inspired from a real world problem. Our case study is related to two international investors who want to simultaneously enter to the bearing industry in Iran; there is no company in Iran who can produce high quality bearings so no rival exists in the market, and based on the marketing research team, they can sell the entire 22
produced product easily to the virgin market. They are entering to the market simultaneously b y centralized structures; each chain has 5 and 4 potential locations for opening plants and DCs, and wants to open one plant and two DCs to provide products to two markets by inelastic demands. The parameters are assumed to be trapezoidal fuzzy numbers, and owing to confidentiality consideration, uniform distribution in a reasonable range is used to extract the four prominent points for each fuzzy trapezoidal number. All the monetary values are in the Iranian currency as well, i.e., Rial. The mentioned distributions are used to extract the required parameters (Table 1). The cost functions of the introduced problem are listed in Table 3. Table 3 inserted here
The achieved equilibrium results are shown in Table 4; all the variables that do not appear in the table are equal to zero. Table 4 inserted here
4-3 Discussion The proceeding case study investigates the CSCND problem in the centralized one player mode in which the whole SC has one owner. As this structure requires extremely high costs for shaping of the network, the CSCND problem may be determined in the decentralized or cooperative modes or without knowing the structure of the rival chain. In this section, these different modes and their effect on equilibrium results will be discussed. It is worth noting that as the quality levels are specified by VI, the only factor that the dominant player can improve is the location of the opened DCs that are specified by the Lemke– Howson algorithm. In other words, the dominant player will chose the locations of DCs in which the objective function are improved rather than the total SCs objective function. We can now introduce different modes, modelling, possible games, and the results. Decentralized mode: If each chain has individual owners for each tier that will be decided noncooperatively to optimize their objective functions, this structure will be shaped. In this mode, the dominant player will chose the locations of the DCs, and this mode is modelled by multi-level MINLP, and based on the dominant player of the chains, four possible games can occur:
Decentralized based on plant VS. decentralized based on plant Decentralized based on plant VS. decentralized based on DC Decentralized based on DC VS. decentralized based on plant Decentralized based on DC VS. decentralized based on DC
We can now express the modelling frame work of this structure as follows: Plants of SC 1 SC1 Z Plant min( Fi (11) yi(11) cij(11) xij(11) oi(1) xij(11) ) i
i
j
i
m j (1) P
j
23
(52)
s .t m j (1) P
(5,7)
xij(11) 0, yi(11) 0,1
(53)
DC of SC 1 aj 1 SC 2 Z DC max (d k ) k
j
(1)2 jk
d ag
d g
(1)2 gk
g
h (12) Fj(12) ( Aj ) c (12) ) xij(11) ) jk x jk ( 2 i j j j k (1) j
wg d g(2)2 k
m j (1) P
(54 )
s .t (4,6)
m j (1) P
x(12) jk 0
m j (1) P
(55 )
Plant of SC2 SC 2 (21) (2) (21) Z Plant min( Fi(21) yi(21) ci(21) j xi j oi xi j ) i
i
j
i
j
m j (2) P
(56)
s .t m j (2) P
(12-14)
(21) xi(21) 0,1 j 0 yi
(57)
DC of SC 2
24
w j SC 2 Z DC max (d k ) j
k
d ag
d g
(1)2 gk
(2)2 j k
g
wg d g(2)2 k
(22) Fj(22) (W j ) c (22) j k x j k j j k
m j (2) h P ( ) xi(21) j 2 i j (2) j
(58 )
m j (2) P
s .t
(11,13) m j (2) P
x(22) j k 0
(59 )
erms 52, 56 minimize the plant’s objective function of SC 1 and SC 2, including the fixed cost of opening plants and variable production cost and transportation cost, and the rest of the terms are related to its constraints. Constraints 53, 57 are related to the binary and non-negativity restrictions on the corresponding decision variables. erms 54, 58 maximize DC’s objective function of the chains including its market share minus fixed cost of opening DCs, holding and transportation cost, and the rest of the terms are related to its constraints. Constraints 55, 59 are related to the binary and non-negativity restrictions on the corresponding decision variables. Cooperative mode: Since each SC has two independent owners with different degrees of power and the tendency to cooperate with each other, this mode can be shaped such that the objective function of the chains is the weighted sum of each owners objective function, as follows: SC1 ZCO : min (
P
(
SC1 SC1* SC1* Z Plant Z Plant Z SC1 Z DC P p ( DC ) ) (1 ) )p SC1* SC1* Z Plant Z DC
m j (1) P
(60)
s .t m j (1) P
(4-7,26)
SC 2 ZCO : min (
P
(
SC 2 SC 2* SC 2* Z Plant Z Plant Z SC 2 Z DC P p ( DC ) ) (1 ) )p SC 2* SC 2* Z Plant Z DC
m j (2) P
s .t m j (2) P
(11-14,28)
25
(61)
erms 60, 61 minimize the distance between the DC’s and plant’s objective functions of the chains from their best one based on their power to reach to the best solution. It is worth noting that if was equal to 0.5, then this mode is equal to the centralized two players. Mixed structure: If both chains do not enter the market with the same mode, this structure can occur, e.g., SC1 enters the market in the decentralized mode based on the DC structure and SC2 enters the market in the centralized mode, which leads to the following games:
Centralized one player VS. decentralized based on plant Centralized one player VS. decentralized based on DC Centralized one player VS. centralized two players Centralized one player VS. cooperative Centralized two players VS. centralized one player Centralized two players VS. decentralized based on plant Centralized two players VS. decentralized based on DC Centralized two players VS. cooperative Decentralized based on plant VS. centralized one player Decentralized based on plant VS. centralized two players Decentralized based on plant VS. decentralized based on DC Decentralized based on plant VS. cooperative Decentralized based on DC VS. centralized one player Decentralized based on DC VS. centralized two players Decentralized based on DC VS. decentralized based on plant Decentralized based on DC VS. cooperative Cooperative VS. centralized one player Cooperative VS. centralized two players Cooperative VS. decentralized based on plant Cooperative VS. decentralized based on DC
Unknown mode: In this case, each chain does not know the exact structure of its rival, but the probability of each possible structure of the competitor is known, so that the expected cost can be calculated, e.g., when SC1 chooses the decentralized mode based on the plant, SC1 knows that SC2 can choose every possible structure by the determined probability. The expected value of each mode for SC 1 is calculated, and then, the best structure is chosen by SC1, and SC2 choose its best structure similarly. From all the introduced possible games investigated through the previous study, the results show that four different outcomes are possible for SCs, e.g., the results of the centralized one player VS. decentralized based on the plant are different from the results of the decentralized mode based on plant VS. centralized one player and are different from the results of decentralized based on plant VS. decentralized based on plant, and all these three structures have different results from the base mode (centralized one player VS. centralized one player). Further, in the cooperative mode, the best response of SCs depend on the amount of , and their results for 0.7 are different from 0.7 . It worth noting that 0 1 , and we assumed
26
that belongs to 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 . Therefore, we categorized the games based on their outcomes in four different scenarios, as follows: Centralized one player VS . centralized one player ; Centralized one player VS . DC ; Centralized one player VS . centralized two player ; Centralized one player VS . cooperative ( 0.7 ); Centralized two player VS . centralized one player ; Centralized two player VS . DC ; Centralized two player VS . centralized two player ; Centralized two player VS . cooperative ( 0.7); S1 : Decentralized based on DC VS . centralized one player ; Cooperative ( 0.7 ) VS . centralized one player ; Decentralized based on DC VS . cooperative ( 0.7 ); Cooperative ( 0.7 )VS . DC ; Cooperative ( 0.7) VS . centralized two player ; Cooperative ( 0.7) VS . cooperative ( 0.7); Decentralized based on DC VS . centralized two player ; Decentralized based on DC VS . decentralized based on DC Centralized one player VS . decentralized based on plant ; Centralized one player VS . cooperative ( 0.7); Centralized two player VS . decentralized based on plant ; Centralized two player VS . cooperative ( 0.7); S2 : Decentralized based on DC VS . decentralized based on plant ; Cooperative ( 0.7) VS . decentralized based on plant ; Cooperative ( 0.7) VS . cooperative( 0.7); Decentralized based on DC VS . cooperative ( 0.7) Decentralized based on plant VS . centralized one player ; Decentralized based on plant VS . decentralized based on DC ; Decentralized based on Plant VS . centralized two player ; Decentralized based on plant VS . cooperative ( 0.7 ); S3 : Cooperative ( 0.7) VS . centralized one player ; Cooperative ( 0.7) VS . centralized two player ; Cooperative ( 0.7) VS . decentralized based on DC ; Cooperative ( 0.7) VS . cooperative( £0.7)
Decentralized based on plant VS . decentralized based on plant ; Decentralized based on plant VS . cooperative ( 0.7); S4 : Cooperative ( 0.7) VS . decentralized based on plant ; Cooperative ( 0.7) VS . cooperative( 0.7)
The achieved results of S1 , S2 , S3 , S4 are shown in Tables 4, 5, 6, and 7. Table 5, 6, 7 inserted here
We can now discuss the scenarios to find the best incoming structure for the chains. Table 8 represents the scenarios, and we compare them with S1 , because this scenario includes centralized one player VS. centralized one player as the based structure mode. Table 8 inserted here
27
According to the table, the highest SC’s profit for SC2 will occur in S 2 where the chain enters the market by decentralized based on plant or cooperative by 0.7 . In this scenario the outcome of DC, plant, and SC for SC 2 are increased by 58.72%, 64.59%, and 33.32%, whereas these outcomes for SC1 will decrease by 33.04%, 31.34%, and 37.52%, respectively, and therefore, SC1 has no motivation to enter to the market by any structure that placed them into S 2 , but they will choose S3 as an incoming structure in which the outcomes can be improved by 43.17%, 46.08%, and 35.53% for DC, plant, and SC and the outcomes can be decreased by 48.44%, 40.88%, and 81.16% for SC2. However, by considering all the scenarios and outcomes, the best structure for SC 1 are decentralized based on plant or cooperative by 0.7 , and when SC1 chose one of these structures, SC2 will chose the decentralized based on plant or cooperative by 0.7 , which lead the game to S 4 . The outcomes of SC 1 will improve by 29.14%, 32.95%, and 19.12%, and those of SC2 decrease by 22.90%, 21.50%, and 29%, respectively. Table 9 ,10 inserted here
Tables 9,10 show the results of the unknown structure for SCs, when all structures have an equal chance to occur. These results also show that if the chain do not known the incoming structure of their rival entering to the market by decentralized based on plant or cooperative by 0.7 create the most expected value for them.
5- Conclusion In this paper, we consider the simultaneous CSCND problem with different structures: centralized, decentralized, cooperative, or unknown modes, and the continuous attractiveness levels of the DCs, so as to try to maximize their market shares while minimizing their costs. Our proposed model is nonconvex MINLP, and we therefore propose a two-step algorithm to reach a global optimum solution, in which step one is constructed by a bi-level programming where the inner level gives the equilibrium attractiveness of any possible combinations of opened DCs by VI formulation and the modified projection method, and the outer level calculates the optimum solution for each combination. Stage two selects the best response solution by the Lemke and Howson algorithm. We also derive the qualitative properties of the equilibrium pattern, such as the existence and the uniqueness of the solution in the inner level. Finally, we generate random instances, use a real word study and fully discuss the model to clarify the effect of different structures of the players on their outcomes and the equilibrium solution. For further research, this model can be developed by stochastic programming approach or by considering robust, green, or closed loop CSCND problems.
28
Appendix 1 Imagine a is a trapezoidal fuzzy number that has the following membership function: f a ( x) 1 a ( x) ha ( x) 0
if
a(1) x a(2) ,
if
a(2) x a(3) ,
if
a(3) x a(4) ,
if
a(4)
x or x
(62)
a(1)
The expected value [EV] and expected interval [EI] of a can be defined as follows: EI [a(2)
a( 2)
a(1)
f a ( x)dx, a(3)
a( 4)
a( e )
(63)
ha ( x)dx],
(64)
a( 2) a( 4) 1 EV [a(2) f a ( x)dx a(3) ha ( x)dx] a(1) a( e ) 2
If a has the following possibility distribution function,
then EI (a) [
a(1) a(2) a(3) a(4) , ], 2 2
(65)
a(1) a(2) a(3) a(4)
(66)
EV (a) [
4
]
By considering k a real number, the possibility (Pos) and necessity (Nec) of a k can be defined as follows (Dubois and Prade 1987; Dubois 1987; Pishvaee et al. 2012): 29
1 k a(1) Pos (a k ) a(2) a(1) 0
if
if
a(1) k
1 k a(3) Nec(a k ) a(4) a(3) 0
if
a(4) k ,
if
if if
a(2) k ,
(67)
a(1) k a(2) ,
(68)
a(3) k a(4) , a(3) k
Assuming is the degree of feasibility, now if
0.5 , we have:
Pos(a k ) k (1 )a(1) ( )a(2) ,
(69)
Nec(a k ) k (1 )a(3) ( )a(4)
(70)
As Inuiguchi and Ramik (2000) and Pishvaee et al. (2012) reported, necessity is more meaningful for application in the fuzzy model to obtain its equivalent crisp model. Now consider the following impact form: min Z fy cx s.t Ax d Sx Ny
(71)
x 0, y 0,1
where the vectors f , c, d are related to the imprecise parameters (like fixed costs, variable costs, and demands), matrixes A, S , N are the coefficient matrixes of the constraints, and x, y are the vector of continuous and binary variables. Now imagine that f , c, d , N are imprecise parameters with trapezoidal possibility distributions, and the rest of the parameters are crisp. According to possibility chance constraint programming [Inuiguchi and Ramik 2000, Dubois and Prade 1987 and Pishvaee et al. 2012] and the described method, we can replace the expected value of the imprecise parameters in the objective function and use the necessary measure to write the equivalent crisp constraints, as follows:
30
f (1) f (2) f (3) f (4) min Z 4 s.t Ax (1 )d (3) d (4)
c(1) c(2) c(3) c(4) y 4
x
(72)
Sx (1 ) N (2) N (1) y x 0, y 0,1
As this procedure is easy to handle, requires less computations, matches our described circumstance, and is a powerful tool to cope with uncertainty, we use it in our model to write the equivalent crisp model of our proposed model.
31
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Zhang C.T, Liu L.P (2013) Research on coordination mechanism in three-level green supply chain under non-cooperative game, Applied Mathematical Modeling 37:3369-3379 Zhang L, Zhou Y (2012), A new approach to supply chain network equilibrium models, Computers & Industrial Engineering 63:82-88 Zhu S 2015. Integration of capacity, pricing, and lead-time decisions in a decentralized supply chain. . Int J Prod Econ 164: 14–23 Zhao J, Wang L (2015) Pricing and retail service decisions in fuzzy uncertainty environments, Applied Mathematics and Computation 250:580-592 Wang L, Song H, Wang Y (2017). Pricing and service decisions of complementary products in a dualchannel supply chain. Computer and industrial engineering, In press Wu D (2013) Coordination of competing supply chains with news-vendor and buyback contract. Int J Prod Econ 144:1-13
38
CSCND problem SC2
SC1
Plant
DC
customers
Figure 1 CSCND problem
39
DC
Plant
Table 1 Distribution of parameters (11)
Fi ,
(21) i
F
oi(1) , oi(2)
( (1500,2000), (2000,2500), (2500,3000), (3000,4000)) u u u u
(u (2,2.5), u (2.5,2.75), u (2.75,3), u (3,3.5))
cij(11) , ci(21) j
(u(0.9,1.5), u(1.5, 2.1), u(2.1, 2.5), u(2.5,3.12))
(22) c (12) jk , c j k
(u(1.5, 2), u(2, 2.5), u(2.5,3), u(3,3.5))
(2) h (1) j , hj
(u(1.25,1.5), u(1.5,1.75), u(1.75, 2), u(2, 2.25))
dk
(u(12000,13000), u(13000,15000), u(15000,17000), u(17000,19000))
c1
(u(0.5,1.5), u(1.5,3), u(3, 4.5), u(4.5,6))
c2
(u(1.5, 2.5), u(2.5,3.5), u(3.5, 4.5), u(4.5,5.5))
40
Table 2 Numerical instances Examples n m n m , , , ,k q (1) P(1) q (2) P(2)
1
2
3
4
5
6
7
Pure strategies of the chains
Opened plants
Opened DCs
Equilibrium qualities
Total profit
CPU (Seconds) Average time(10 runs)
5 4 5 4 , , , ,5 1 3 1 1
SC1
4
(3)
(1,3,4)
(66.90, 52.15, 43.61)
495028.89
SC2
4
(5)
(2)
(71.55)
172528.52
5 4 5 4 , , , ,5 1 2 1 2
SC1
6
(2)
(3,4)
(57.9, 49.56)
380068.43
SC2
6
(2)
(1,2)
(53.32, 35.01)
314475.11
7 5 7 5 , , , ,10 1 5 1 5
SC1
1
(4)
(1,2,3,4,5)
(46.13, 38.58, 53.13, 37.39, 41.68)
877225.93
SC2
1
(2)
(1,2,3,4,5)
(169.47, 104.74, 10.09, 28.70, 10.05)
366944.84
7 5 7 5 , , , ,15 1 5 1 5
SC1
1
(7)
(1,2,3,4,5)
(88.55, 71.56, 96.44, 75.98, 78.15)
1058413.16
SC2
1
(2)
(1,2,3,4,5)
(311.29, 232.66, 10, 10)
632230.76
7 5 7 5 , , , ,15 1 1 1 5
SC1
5
(6)
(3)
(269.23)
503070.88
SC2
1
(1)
(1,2,3,4,5)
(101.22, 94.68, 122.78, 91.64, 96.62)
615993.51
7 5 7 5 , , , ,10 1 1 1 1
SC1
5
(3)
(5)
(92.35)
721262.06
SC2
5
(5)
(4)
(175.04)
428446.18
7 5 7 5 , , , ,10 1 1 1 3
SC1
5
(2)
(5)
(196.96)
540377.86
SC2
10
(7)
(3,4,5)
(85.90, 80.13, 100.14)
576897.37
41
1.54
3.63
0.37
0.43
0.71
2.93
15.44
Examples n m n m (1) , (1) , (2) , (2) , k q P q P
8
9
10
11
12
13
14
15
Pure strategies of the chains
Opened plants
Opened DCs
Equilibrium qualities
Total profit
CPU (Seconds) Average time(10 runs)
7 5 7 5 , , , ,10 1 2 1 2
SC1
10
(1)
(3,5)
(85.83, 70.75)
556489.33
SC2
10
(1)
(4,5)
(197.22, 159.18)
296986.17
7 5 7 5 , , , ,15 1 2 1 2
SC1
10
(5)
(1,3)
(108.13, 85.93)
948532.37
SC2
10
(1)
(1,2)
(246.90, 215.52)
546076.36
7 8 7 8 , , , ,5 2 1 2 1
SC1
8
(1,5)
(3)
(66.28)
216495.7799
SC2
8
(1,2)
(6)
(119.71)
233038.6878
7 8 7 8 , , , ,10 1 1 1 1
SC1
8
(1,5)
(3)
(102.36)
527341.80
SC2
8
(1,2)
(6)
(175.04)
403733.49
7 8 7 8 , , , ,15 2 1 2 1
SC1
8
(4,7)
(3)
(127.60)
859697.31
SC2
8
(1,6)
(1)
(166.83)
665573.29
7 8 7 8 , , , ,5 1 1 1 2
SC1
8
(1,4)
(4)
(127.51)
186352.14
SC2
28
(2,7)
(4,8)
(63.94, 71)
325820.56
7 8 7 8 , , , ,10 1 1 1 2
SC1
8
(3,5)
(3)
(186.1)
484500.8
SC2
28
(5,6)
(6,7)
(89.26, 103.67)
575841.94
7 8 7 8 , , , ,15 1 1 1 2
SC1
8
(2,4)
(2)
(235.59)
652347.9
SC2
28
(4,5)
(4,7)
(111.47, 132.21)
958556.30
42
55.69
91.87
15.92
48.84
56.99
178.94
393.3
707.75
Table 3 Cost functions of the DCs
SC 1
SC 2
Fixed cost of DC 1
F1 (12)( A1 ) 2a12 2.5a1
F1(22) (W1 ) 1.5w12 3w1
Fixed cost of DC 2
F2(12) ( A2 ) 2.5a22 3a2
F2(22) (W2 ) 2.8w22 4w2
Fixed cost of DC 3
F3(12) ( A3 ) 1.5a32 4a3
F3(22) (W3 ) 3w32 1w3
Fixed cost of DC 4
F4(12) ( A4 ) 2.2a42 5a4
F4(22) (W4 ) 4w42 5w4
Table 4 Computational results for SCs
Amount of shipments between plant 2 and DC2 Amount of shipments between plant 2 and DC3 Amount of shipments between plant 1 and DC3 Amount of shipments between plant 1 and DC4 Amount of shipments between DC2 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC2 and market 2 Amount of shipments between DC3 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC2 Equilibrium quality of DC3 Equilibrium quality of DC4 Total incomes Total costs
SC1 5599.41 6647.462 0 0 2819.269 3000.776 0 2780.141 3646.686 0 38.86 35.97 0 167491.7634 143617.2034
SC2 0 0 11438.979 14.095 4245.964 5.940 0 7193.015 8.155 0 50.98 0.08 125812 111788.06
Table 5 Compeutational results of S 2
S2 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC2 Amount of shipments between plant 2 and DC2 Amount of shipments between plant 2 and DC3 Amount of shipments between DC1 and market 1 Amount of shipments between DC2 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC2 and market 2 Amount of shipments between DC3 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC2 Equilibrium quality of DC3 Total incomes Total costs
SC1
SC2
0 0 4042.546 4257.25 0 1702.514 1580.370 0 2340.032 2676.880 0 35.48 28.65 114071.31 99154.86
11254.145 7214.885 0 0 4745.280 4076.476 0 6508.865 3138.409 0 56.95 29.14 0 188148.11 169441.31
43
Table 6 Compeutational results of S3
S3 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC3 Amount of shipments between plant 1 and DC4 Amount of shipments between DC1 and market 1 Amount of shipments between DC3 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC3 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC3 Equilibrium quality of DC4 Total incomes Total costs
SC1
SC2
7493.11 0 10239.535 2174.047 0 7040.694 5319.063 0 3198.841 38.54 0 34.47 226268.33 193910.42
0 5874.221 0 0 1412.515 0 0 4461.706 0 0 35.56 0 64584.35 61941.47
Table 7 Compeutational results of S 4
S4 Amount of shipments between plant 1 and DC1 Amount of shipments between plant 1 and DC4 Amount of shipments between plant 2 and DC1 Amount of shipments between plant 2 and DC2 Amount of shipments between DC1 and market 1 Amount of shipments between DC2 and market 1 Amount of shipments between DC4 and market 1 Amount of shipments between DC1 and market 2 Amount of shipments between DC2 and market 2 Amount of shipments between DC4 and market 2 Equilibrium quality of DC1 Equilibrium quality of DC2 Equilibrium quality of DC4 Total incomes Total costs
SC1
SC2
6839.835 9254.821
0 0 6567.275 2062.37 2316.839 1016.083 0 4250.436 1046.287 0 40.076 10.47 0 89603.6123 79640.5823
0 0 1824.979 0 6127.868 5014.856 0 3126.953 38.37 0 35.58 204782.0127 176342.2527
Table 8 Comparasion analysis
S1 S2 S3
SC1
DC objective function 86686.51
Plant objective function 62811.95
SC objective function 23874.56
SC2 SC1 SC2 SC1 SC2 SC1 SC2
74748.82 58041.45 118639.81 124111.19 38542.05 111948.71 57628.78
60716.88 43124.99353 99933.01 91753.280 35899.17 83508.95 47665.76
14031.94 14916.45 18706.80 32357.91 2642.88 28439.76 9963.03
44
Gain/ loss of DC (%)
Gain/ loss of plant (%)
Gain/ loss of SC (%)
-33.0444 58.71797 43.17244 -48.4379 29.14202 -22.9034
-31.3427 64.58851 46.07615 -40.8745 32.95074 -21.495
-37.5216 33.31585 35.53301 -81.1653 19.12161 -28.9975
Table 9 SC1 VS unknown structures
SC1
Decentral ized based on DC
Decentral ized based on plant
Two player centralize d
One player centralized
Cooperative
0.1
0.2
0.3
.04
0.5
0.6
0.7
0.8
0.9
Expected value
Decentralized based on DC
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
Decentralized based on plant
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
One player centralized
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
Two player centralized
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.1
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.2
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.3
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.4
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.5
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.6
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.7
23874.56
14916.45
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
23874.56
14916.45
14916.45
21807.30
0.8
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
0.9
32357.91
28439.76
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
32357.91
28439.76
28439.76
31453.72
Cooperative
45
Table 10 SC2 VS unknown structures
SC2
Decentral ized based on DC
Decentral ized based on plant
Two player centralize d
One player centralized
Cooperative
0.1
0.2
0.3
.04
0.5
0.6
0.7
0.8
0.9
Expected value
Decentralized based on DC
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
Decentralized based on plant
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
One player centralized
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
Two player centralized
14031.94
28439.76
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
28439.76
28439.76
17356.82
0.1
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.2
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.3
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.4
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.5
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.6
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.7
14031.94
2642.88
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
14031.94
2642.88
2642.88
11403.70
0.8
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
0.9
18706.80
9963.03
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
18706.80
9963.03
9963.03
16689.01
Cooperative
46
Highlights
We analyse the simultaneous competitive supply chain network design problem. We discuss the effect of different modes on the problem. A two-stage algorithm is proposed to solve the problem. Bi-level programming is used in the proposed algorithm.
47