Document Type: Research Paper
Authors
Department of Industrial Engineering, Iran University of Science & Technology, Terhan, Iran
Abstract
Keywords
Main Subjects
Article Title [Persian]
Authors [Persian]
این مقاله به معرفی مسئلة طراحی شبکة زنجیرة تأمین رقابتی نامتمرکز میپردازد که طی آن زنجیرههای تأمین نامتمرکز بهطور همزمان وارد بازار بدون رقیب میشود و طی بازی همزمان و غیرهمکارانه زنجیرة خود را شکل میدهد و قیمتهای عمده و خردهفروشی مشخص میکند. الگوریتمی سه مرحلهای برای حل این مسئله پیشنهاد شده است. مرحلة نخست، ساختارهای ممکن زنجیرهها را ایجاد میکند. مرحلة دوم، قیمتها و جریان مواد تعادلی را طی روش انعکاس اصلاحشده و فرمولبندی مشتقات نامساوی بهدست میآورد. مرحلة آخر، ساختار تعادلی شبکهها را با الگوریتم ویلسون محاسبه میکند. در آخر، مطالعهای موردی بههمراه آنالیز نتایج و بحث ارائه شده است.
Keywords [Persian]
Introduction
Today’s international business and open markets promote developing countries to omit monopoly and enroll in the World Trade Organization (WTO) to achieve benefits from open world trading, so they ratify different foreign investment strategies and policies to observe international investors. In these situations, the investors come across good opportunities to design their networks domestically and obtain intact markets encountered by simultaneous competitions. On the other hand, competition is promoted by firms against firms and supply chains versus supply chains. So, investors have questions such as the following: What is the best network design in this competitive mode? How many market shares can be obtained? What is their equilibrium condition? This paper aims to provide solutions to these questions.
According to the Supply Chain Network Design (SCND) literature many studies on monopoly assumptions (Altiparmak et al., 2006; Badri et al., 2013; Özceylan et al., 2014; Vahdani & Mohamadi, 2015; Yang et al., 2015 b; Ardalan et al., 2016; Keyvanshokooh et al., 2016; Özceylan et al., 2016; Jeihoonian et al., 2017; Varsei & Polyakovskiy, 2017) have been conducted. Several examples of supply chain (SC) competition can be found in maritime shipping, the automotive and retail industries, and online bookstores (see Farahani et al., 2014 for a review on competitive supply chain network design CSCND).
Players and customers are two integral elements in CSCND. Based on players’ reactions, a newcomer encounters monopoly competition (i.e., no rival exists, or existing rivals show no reactions to the newcomer), duopoly competition (i.e., just one rival shows reactions to the newcomer), and oligopoly competition (i.e., more than one rival show reactions to the newcomer). Based on players’ reactions, three types of competition have been discussed in the literature: Static competition (Berman & Krass, 1998; Aboolian et al., 2007a, 2007b; Revelle et al., 2007), dynamic competition (Sinha & Sarmah, 2010; Friesz et al., 2011; Jain et al., 2014; Chen et al., 2015; Nagurney et al., 2015; SantibanezGonzalez & Diabat, 2016; Hjaila et al., 2016 b; Lipan et al., 2017), and competition with foresight (Zhang & Liu, 2013; Yue & You, 2014; Zhu, 2015; Drezner et al., 2015; Yang et al., 2015 a; Hjaila et al., 2016 a; Aydin et al., 2016; Genc & Giovanni, 2017).
Customer behavior and customer demand functions are important factors in CSCND. Customer demand can be elastic or inelastic, and in the case of elastic demand, it can depend on price, service, price and service, or price and distance (Farahani et al., 2014). Customer utility functions are mostly categorized into deterministic, introduced by Hotelling (1929), and random utility, introduced by Huff (1964, 1966) models. Moreover, three types of competition exist in the SC competition literature: Horizontal competition (Nagurney et al., 2002; Cruz, 2008; Zhang & Zhou, 2012; Qiang et al., 2013; Huseh, 2015; Qiang, 2015; Li & Nagurney, 2016; Nagurney et al., 2016), vertical competition (Chen et al., 2013; Wu, 2013; Zhao & Wang, 2015; Zhang et al., 2015; Bai et al., 2016; Bo & Li, 2016; Li & Nagurney, 2016; Huang et al., 2016; Wang et al., 2017; Genc & Giovanni, 2017; Chaeb & RastiBarzoki, 2017), and SC versus SC (Li et al., 2013; Chung & Kwon, 2016).
Contribution of This Paper
This paper addresses a simultaneous decentralized supply chain network design problem (SDSCND) in which decentralized SCs simultaneously enter a virgin market, shape their networks, set the wholesale and retail prices, and specify flows of products among the tiers in dynamic competition without any cooperation. This problem and this paper’s proposed approach to solve it, is most like that described in Rezapour and Farahani (2010), Nagurney (1999), and a subsequent paper of Nagurney. However, this paper’s main contribution is the addition of the location decision as 01 variables, which puts the problem in the class of mixedinteger, nonlinear programming models. Therefore, the proposed problem cannot be solved with the explicit solution algorithm. Thus, we propose a threestep algorithm to solve the problem and find an equilibrium solution. As the chains enter the dynamic competition, we use variational inequality formulation to find equilibrium results. However, VI is only applicable to models with continuous variables and convex functions; but, because our problem has 01 variables, we use the Wilson algorithm and specify some strategies related to the 01 variables to handle this matter. We define all the possible strategies based on location variables of the chains in the first stage of our algorithm and then use the VI formulation and modified projection method to obtain equilibrium results of continuous variables in the second stage. The third stage is constructed with the help of the Wilson algorithm, and we select the equilibrium locations in this step. With this threestep algorithm we are able to solve the problem of SDSCND in dynamic competition.
Problem Definition
This paper considers the problem of simultaneous decentralized competitive supply chain network design (SDCSCND) in which n SCs plan to enter into virgin market. Each chain has two independent tiers called plant and distribution center (DC) levels, which try to maximize the chain’s profits by selecting the best locations for their facilities and setting wholesale and retail prices. All the entities make decisions simultaneously in a noncooperative manner. The chains produce either identical or highly substitutable products, and customer demand is elastic and pricedependent.
Fig. 1. Simultanous decentralized competitive supply chain network design problem.
Before modelling the formulation, imagine there are incoming SCs indexed by ; then, th SC has potential locations for opening plants, indexed by and , and potential locations for opening DCs, indexed by . There exist demand points indexed by . Similar to the description provided by Tsay and Agrawal (2000), demand functions for th SC in market k can be defined as follows):
(1) 
The term is the potential market size and refers to selfprice sensitivity. The term represents the crossprice sensitivity. Since demand cannot be negative, we assume the following:
(2) 
Now we can introduce parameters and variables as follows:
Parameters and variables
parameters 


Number of opened facilities in plant level of SC


Number of opened facilities in DC level of SC


Cost structure of the problem 

Cost of locating in order to produce units in the th plant of SC , we assume these functions are continuous and convex (Dong et al., 2004). 

Cost of procurement, producing and handling units in the th plant of SC , we assume these functions are continuous and convex (Nagurney et al., 2002). 

Cost of transaction (ordering, transportation and other expenses) units between plant and DC of SC , we assume these functions are continuous and convex (Nagurney et al., 2002). 

Cost of locating in order to distribute units in the th DC of SC , we assume these functions are continuous and convex, (Dong et al., 2004). 

Unit holding cost at DC for SC 

Cost of transaction (ordering, transportation and other expenses) units between DC and customer for SC , we assume these functions are continuous and convex (Nagurney et al., 2002). 

Variables 


Wholesale price of plant to DC in the th SC 

Vector of wholesale price of plants to DCs for all SCs 

Quantity of product shipped from plant to DC in the th SC 

Vector of quantity of product shipped from plants to DCs 

Price of th SC in market 

Vector of price of SCs in market 

Amount of product that DC considers to satisfy for market in the th SC 

Vector of amount of product that DC considers to satisfy for market in the th SC 

Vector of location variables of plants 

Vector of location variables of DCs 

Modelling Framework
Plants model
(3) 




(4) 

(5) 
Term (3) represents the objective function of the plants for each chain that includes total revenue from selling the product to the DCs of the chain minus total location and transaction costs. Constraint (4) specifies the number of opened plants in the chain, and constraint (5) is related to the binary and nonnegativity restriction on the corresponding decision variables.
DC’s model
(6) 




(7) 

(8) 

(9) 
Term (6) represents the objective function of DCs of the chain which includes profits captured by selling the product to the customers minus total location and transaction costs. Constraint (7) is related to flow balance; constraint (8) specifies the number of opened DCs in each chain and constraint (9) is related to the binary and nonnegativity restriction on the corresponding decision variables.
Solution Approach
This section presents our proposed algorithm for solving the SDCSCND problem. The algorithm is essentially based on the chains’ decision variables. Each SC has two different types of decision variables, including continuous and discrete (01) variables. Continuous variables are related to wholesale and retail prices and the amount of shipments among the tiers while discrete variables are related to the locations of the facilities. Also, they are intrinsically different decisions, as location is strategic while the other variables are related to operational decisions. On the other hand, they are related to each other, as each opened location has its own costs that affect the chains’ prices, market shares, demands, and profits.
With the help of the Wilson algorithm, Wilson (1971), the variational inequality formulation, the modified projection method, and Nagurney et al. (2002), and references therein, we propose a threestep algorithm in which the first step defines basic strategies based on location variables. Each strategy contains a potential network design structure for the chains, as their location is fixed. In the second step, we use variational inequalities and the modified projection method to calculate the payoff for each potential network structure. After steps one and two, the payoff for all the possible structures of the chains is calculated. In step three and with the help of the Wilson algorithm, the Nash equilibrium locations can be found, and the chains’ equilibrium network design is obtained. The algorithm procedure is as follows:
1.1. Construct an empty polymatrix by considering all pure strategies of the players (any combination of the facilities to be opened from all the potential facilities of each player).
2.1. Develop VI formulation of the players’ problems in each strategy and solve it with the help of the modified projection method.
3.1. Fill the empty polymatrix with the obtained payoffs from the previous stage and find the best network structure using the Wilson algorithm.
To clarify the proposed algorithm, consider an example in which one SC, composed of plant and DC levels, is planning to enter one virgin market in a decentralized manner, shape its network, and set wholesale and retail prices and flows. Imagine both plant and DC have two potential locations, and they intend to open one facility to capture the market demand. Table 1 shows the cost functions of the entities related to the places.
Table 1. Cost functions
Plant1(fixed cost) 
Plant 2 to DC2(transportation cost) 

Plant2(fixed cost) 
DC1(fixed cost) 

Plant1(production cost) 
DC2(fixed cost) 

Plant2(production cost) 
DC1(holding cost) 

Plant 1 to DC1(transportation cost) 
DC2(holding cost) 

Plant 2 to DC1(transportation cost) 
DC1 to market 1(transportation cost) 

Plant 1 to DC2(transportation cost) 
DC2 to market 1(transportation cost) 
According to step one, each player has pure strategies; consequently, potential strategies are available. The opened plant and DC can be as follows: . Now, the algorithm can be applied to step two in which it can calculate the equilibrium prices and flows with the help of the VI formulation and the modified projection method. The results of this step are presented in Table 2.
Table 2. Nash equilibrium solution of each potential strategy

Amount of shipments between plant and DC 
Amount of shipments between DC and market 

Price of plant for DC 
Price of DC in market 
Market share 

Income of plant 
Cost of plant 
Profit of plant 
Income of DC 
Cost of DC 
Profit of DC 
78.83 
78.83 

698.93 
1280.8 
78.83 

55088 
27755 
24227 
100971 
23744 
77227 

66.61 
66.61 

591.40 
1288.9 
66.61 

39386 
22089 
17297 
85864 
23323 
62541 

70.13 
70.13 

768.48 
1286.6 
70.13 

53877 
27089 
26788 
90225 
18825 
71400 

60.29 
60.29 

661.33 
1293.1 
60.29 

39864 
20062 
19802 
77973 
19130 
58844 
Step one trough step two result in an equilibrium solution for the continuous variables in each strategy. The algorithm can then be applied to step three in which it can select the equilibrium locations of the facilities in order to finalize the network design with the help of the Wilson algorithm. It is worth noting that in the case of two existing players, this step can also be conducted using the Lemke and Howson’s (1964) algorithm. Table 3 presents this step. According to the constructed matrix, strategy is selected as the Nash equilibrium solution of the game.
Table 3. Final Nash equilibrium solution
DC’s pure strategies Plant’s pure strategies 
1 
2 
1 


2 


It is worth noting that since steps one and three are based on the players’ potential strategies, and step two is based on variational inequality and the modified projection method. With respect to the fact that pure strategies are finite and the modified projection method has a convergence criterion, the proposed procedure converges to an equilibrium solution (see the convergence proof of the Wilson algorithm; Wilson (1971); the variational inequality formulation and the modified projection method; and Nagurney et al. (2002) and references therein).
Stage one
This stage defines the number of strategies and shapes the matrix that should be filled in the next stage. By examining the 01 decision variables in each chain, it is understandable that each plant has pure strategies, and each DC has pure strategies. So the dimension of the matrix is .
Stage two
In this stage we should optimize the following models for the opened plants and DCs in each strategy to calculate payoff for the game.
Modified plants model
(10) 




(11) 
Term (10) represents the objective function of the opened plants for each chain that includes total revenue from selling the product to the DCs of the chain minus total location and transaction costs and; constraint (11) is related to the nonnegativity restriction on the corresponding decision variables.
Modified DC’s model
(12) 




(13) 

(14) 
Term (12) represents the objective function of opened DCs of the chain which includes profits captured by selling the product to the customers minus total transaction costs. Constraint 15 is related to flow balance between opened plants and DCs; constraint 16 is related to the nonnegativity restrictions on the corresponding decision variables.
The modified plant and DC model should be formulated as VI model (according to Rezapour & Farahani (2010), Nagurney (1999), and a subsequent paper of Nagurney). The VI is solvable by several algorithms as modified projection method, like Nagurney and Toyasaki (2005), Rezapour and Farahani (2010), the Eulertype model (Nagurney et al., 2003; SantibanezGonzalez & Diabat, 2016), some evolutionary algorithm (Majig et al., 2007), and extended mathematical programming (EMP) of GAMS (SantibanezGonzalez & Diabat, 2016). In this article, we use modified projection method to solve the model.
Stage three
Now, we can find the Nash equilibrium locations and shape the network structure of the chains by the specified prices and amount of shipments. This step is conducted with the help of the Wilson algorithm (Wilson, 1971).
Computational Results
This section presents a realworld problem occurring in the Iranian capacitor industry and inspired us to propose the problem and solution described in this paper. The first subsection describes the problem environment, and the second subsection provides discussions of the results.
Case study
As a consulter group, we study a realworld problem in which two SCs are planning to enter the capacitor industry in the Iranian market. These two SCs want to produce a special type of capacitor that is used in refrigerators. This type of capacitor is solely imported, and there is no domestic producer for it, so they decided to enter this virgin market by shaping their SC domestically. They want to open one plant and two DCs from two and four potential locations and shape their networks in a decentralized manner in a dynamic competition and set the wholesale and retail prices. Table 4 represents their cost structures, and Table 5 represents the achieved results. Both chains open a plant at location one and DCs in location one and two to obtain Nash equilibrium locations. The proposed algorithm was implemented in Matlab 2014a. The convergence criterion used was that the absolute value of the flows and prices between two successive iterations differed by no more than , and the computational time is negligible.
Table 4. Cost structure of the Chains
Cost functions 
SC1 
SC2 
Plant1(fixed cost) 

Plant2(fixed cost) 

Plant1(production cost) 

Plant2(production cost) 

Plant 1 to DC1(transportation cost) 

Plant 2 to DC1(transportation cost) 

Plant 1 to DC2(transportation cost) 

Plant 2 to DC2(transportation cost) 

Plant 1 to DC3(transportation cost) 

Plant 2 to DC3(transportation cost) 

Plant 1 to DC4(transportation cost) 

Plant 2 to DC4(transportation cost) 

DC1(fixed cost) 

DC2(fixed cost) 

DC3(fixed cost) 

DC4(fixed cost) 

DC1(holding cost) 

DC2(holding cost) 

DC3(holding cost) 

DC4(holding cost) 

DC1 to market 1(transportation cost) 

DC2 to market 1(transportation cost) 

DC3 to market 1(transportation cost) 

DC4 to market 1(transportation cost) 

DC1 to market 2(transportation cost) 

DC2 to market 2(transportation cost) 

DC3 to market 2(transportation cost) 

DC4 to market 2(transportation cost) 
Table 5. Computational results for SCs

SC1 
SC2 
Amount of shipments between plant 1 and DC1 
4694.3 
4472.9 
Amount of shipments between plant 1 and DC2 
4165.3 
4518.2 
Amount of shipments between DC1 and market 1 
1574.5 
1583 
Amount of shipments between DC1 and market 2 
3119.8 
2889.9 
Amount of shipments between DC2 and market 1 
1546.2 
1595.8 
Amount of shipments between DC2 and market 2 
2619.2 
2922.4 
Price of plant 1 for DC1 
41315.09012 
38919.48382 
Price of plant 1 for DC2 
36327.07227 
39223.30433 
Price of DCs in market 1 
52844.97657 
52823.45377 
Price of DCs in market 2 
64094.66474 
64067.50132 
Market share 1 
3120.678925 
3178.79216 
Market share 2 
5739.003739 
5812.346629 
Income of plant 
345259305.3 
351300806.8 
Cost of plant 
195122565.1 
175674633.4 
Profit of plant 
150136740.1 
175626173.3 
Income of DC 
532750281.5 
540295763.9 
Cost of DC 
94475846.25 
94517840.4 
Profit of DC 
438274435.2 
445777923.5 
Landa 1 
41315 
38919 
Landa 2 
36327 
39223 
Discussion
The case study presented in this paper reflects an SDCSCND problem that has nonlinear, fixed production and transaction costs related to the producers and DCs. Moreover, the demand function at each market is related to the retail prices of the chains and the prices relate to the costs of the players; therefore, the chains can use different locations for their facilities or marketing activities to influence the costs of the chains and parameter values of the demand function. Here, we discuss the sensitivity analysis for SCs with respect to the crossprice and selfprice parameters.
Tables 6 and 7 represent the sensitivity analysis for SCs with respect to cross price effect while the selfprice parameter is set to 1.2. Tables 8 and 9 represent the sensitivity analysis for SCs with respect to selfprice effect while the crossprice parameter is set to 1.5.
Table 6. Sensitivity analysis for SC 1 with respect to cross price effect
beta,SC1 
1 
0.9 
0.85 
0.8 
0.75 
0.7 
0.65 
0.6 
0.4 
0.2 
0.1 
0.05 
0.005 
Price of plant 1 for DC1 
26978 
22990 
21407 
20029 
18817 
17743 
16785 
15926 
13218 
11298 
10533 
10188 
9896 
Price of plant 1 for DC2 
23722 
20215 
18824 
17612 
16546 
15602 
14760 
14004 
11624 
9262 
9262 
8959 
8702 
Price of DCs in market 1 
34088 
28934 
26899 
25130 
23578 
22206 
20985 
19891 
16457 
14032 
13069 
12636 
12269 
Price of DCs in market 2 
42273 
36137 
33694 
31561 
29683 
28016 
26527 
25188 
20959 
17948 
16745 
16202 
15743 
Market share 1 
1941 
1627 
1505 
1400 
1308 
1227 
1156 
1092 
894 
756 
702 
678 
657 
Market share 2 
3844 
3302 
3085 
2895 
2727 
2577 
2443 
2323 
1940 
1666 
1556 
1506 
1464 
Income of plant 
147212824 
106897379 
92685914 
81130824 
71609105 
63670228 
56981864 
51294533 
35333885 
25810276 
22431937 
20986276 
19801434 
Cost of plant 
83201732 
60417960 
52386438 
45856103 
40474886 
35988183 
32208187 
28993914 
19973383 
14590741 
12681297 
11864199 
11194512 
Profit of plant 
64011092 
46479419 
40299476 
35274720 
31134219 
27682045 
24773676 
22300620 
15360502 
11219535 
9750640 
9122078 
8606922 
Income of DC 
228664090 
166418094 
144431952 
126535207 
111772510 
99452183 
89063436 
80222414 
55370805 
40507524 
35227392 
32966521 
31112881 
Cost of DC 
41033043 
29983401 
26066489 
22871675 
20231465 
18024326 
16160325 
14571745 
10093382 
7403927 
6446059 
6035468 
5698621 
Profit of DC 
187631047 
136434693 
118365463 
103663532 
91541045 
81427857 
72903111 
65650669 
45277422 
33103597 
28781333 
26931053 
25414260 
Table 7. Sensitivity analysis for SC 2 with respect to cross price effect
beta,SC2 
1 
0.9 
0.85 
0.8 
0.75 
0.7 
0.65 
0.6 
0.4 
0.2 
0.1 
0.05 
0.005 
Price of plant 1 for DC1 
25414 
21657 
20166 
18867 
17725 
16714 
15811 
15002 
12451 
10641 
9920 
9595 
9320 
Price of plant 1 for DC2 
25612 
21825 
20322 
19014 
17863 
16844 
15934 
15118 
12548 
10724 
9997 
9670 
9393 
Price of DCs in market 1 
34073 
28921 
26886 
25118 
23567 
22195 
20975 
19881 
16447 
14023 
13061 
12627 
12261 
Price of DCs in market 2 
42254 
36120 
33678 
31546 
29668 
28001 
26513 
25174 
20947 
17936 
16733 
16191 
15731 
Market share 1 
1978 
1659 
1535 
1427 
1334 
1251 
1179 
1114 
912 
771 
716 
691 
670 
Market share 2 
3892 
3343 
3123 
2931 
2760 
2609 
2473 
2351 
1964 
1686 
1575 
1524 
1482 
Income of plant 
149780530 
108758285 
94297726 
82540132 
72851566 
64773591 
57968049 
52181085 
35940810 
26250210 
22812537 
21341451 
20135749 
Cost of plant 
74906086 
54392624 
47161416 
41281810 
36436816 
32397199 
28993866 
26099880 
17978154 
13131727 
11412442 
10676696 
10073674 
Profit of plant 
74874444 
54365662 
47136310 
41258322 
36414750 
32376392 
28974183 
26081205 
17962656 
13118483 
11400095 
10664755 
10062075 
Income of DC 
231874114 
168744755 
146447143 
128297111 
113325673 
100831283 
90295871 
81330115 
56128280 
41055677 
35701108 
33408326 
31528496 
Cost of DC 
41060166 
30004662 
26085363 
22888467 
20246438 
18037704 
16172299 
14582478 
10100362 
7408408 
6449579 
6038559 
5701350 
Profit of DC 
190813948 
138740094 
120361780 
105408644 
93079235 
82793579 
74123572 
66747636 
46027918 
33647269 
29251529 
27369767 
25827146 
Table 8. Sensitivity analysis for SC 1 with respect to selfprice effect
alpha,SC1 
1.8 
2 
2.2 
2.5 
2.8 
3 
3.5 
4 
4.5 
5 
5.5 
6 
7 
Price of plant 1 for DC1 
22989 
17742 
14445 
11296 
9274 
8285 
6541 
5404 
4603 
4009 
3551 
3187 
2644 
Price of plant 1 for DC2 
20214 
15601 
12702 
9933 
8155 
7286 
5753 
4753 
4049 
3526 
3123 
2803 
2326 
Price of DCs in market1 
28933 
22205 
18010 
14030 
11489 
10251 
8075 
6660 
5668 
4932 
4366 
3916 
3247 
Price of DCs in market2 
36136 
28014 
22877 
17944 
14763 
13203 
10444 
8639 
7366 
6420 
5690 
5108 
4242 
Market share 1 
1627 
1227 
983 
756 
614 
545 
426 
349 
296 
257 
227 
203 
168 
Market share 2 
3302 
2577 
2114 
1665 
1374 
1231 
976 
809 
690 
602 
534 
480 
398 
Income of plant 
106887635 
63660351 
42195734 
25801375 
17389832 
13878621 
8650209 
5902527 
4282538 
3248072 
2547554 
2051324 
1411767 
Cost of plant 
60412453 
35982601 
23851547 
14585710 
9831432 
7846793 
4891405 
3338152 
2422314 
1837451 
1441368 
1160771 
799093 
Profit of plant 
46475182 
27677750 
18344187 
11215665 
7558400 
6031828 
3758805 
2564375 
1860224 
1410620 
1106187 
890553 
612673 
Income of DC 
166402867 
99436696 
66062896 
40493516 
27337986 
21836891 
13632051 
9312135 
6761835 
5131704 
4026974 
3243928 
2234019 
Cost of DC 
29980640 
18021502 
12022637 
7401355 
5011986 
4009756 
2510608 
1718664 
1250021 
949926 
746263 
601734 
415081 
Profit of DC 
136422227 
81415194 
54040259 
33092162 
22326000 
17827135 
11121443 
7593471 
5511814 
4181778 
3280711 
2642194 
1818938 
Table 9. Sensitivity analysis for SC 2 with respect to selfprice effect
alpha,SC2 
1.8 
2 
2.2 
2.5 
2.8 
3 
3.5 
4 
4.5 
5 
5.5 
6 
7 
Price of plant 1 for DC1 
21657 
16715 
13609 
10643 
8738 
7806 
6164 
5092 
4338 
3778 
3346 
3003 
2492 
Price of plant 1 for DC2 
21826 
16845 
13715 
10726 
8806 
7867 
6212 
5132 
4372 
3808 
3373 
3027 
2512 
Price of DCs in market1 
28922 
22197 
18004 
14026 
11486 
10248 
8073 
6659 
5666 
4931 
4365 
3915 
3246 
Price of DCs in market2 
36122 
28004 
22869 
17939 
14759 
13199 
10441 
8637 
7365 
6419 
5689 
5107 
4241 
Market share 1 
1659 
1251 
1003 
772 
626 
556 
435 
357 
302 
262 
231 
207 
171 
Market share 2 
3343 
2609 
2140 
1686 
1391 
1246 
988 
819 
699 
610 
541 
486 
403 
Income of plant 
108768184 
64783629 
42942146 
26259252 
17699271 
14125971 
8804891 
6008358 
4359492 
3306545 
2593488 
2088360 
1437311 
Cost of plant 
54397573 
32402219 
21479543 
13136250 
8855073 
7067843 
4406280 
3007347 
2182444 
1655622 
1298824 
1046047 
720204 
Profit of plant 
54370610 
32381410 
21462603 
13123003 
8844198 
7058128 
4398611 
3001012 
2177048 
1650923 
1294664 
1042313 
717107 
Income of DC 
168760176 
100846973 
67001154 
41069884 
27727921 
22148768 
13827315 
9445850 
6859130 
5205673 
4085106 
3290817 
2266377 
Cost of DC 
30007422 
18040529 
12036735 
7410985 
5018988 
4015567 
2514507 
1721469 
1252139 
951586 
747600 
602835 
415866 
Profit of DC 
138752754 
82806444 
54964419 
33658898 
22708933 
18133201 
11312808 
7724381 
5606991 
4254088 
3337507 
2687982 
1850511 
It is worth noting that in our case, the change of selfprice and crossprice parameters have no effects on location decision variables, but changes in location decision variables by change in these parameters are possible, and in these circumstances the shape of the networks will change.
Conclusion
This paper presents an important realworld problem in which decentralized SCs simultaneously enter the virgin market to shape their networks, set the wholesale and retail prices, and specify their market shares in dynamic competition. This problem is essential, as several developing countries are trying to omit monopoly and open their markets to international investors. These investors then encounter virgin markets and competition simultaneously.
We propose a threestep algorithm to reach a Nash equilibrium network design in which step one constructs all the potential network structures; step two computes the related decisions in dynamic competition for all the potential structures through VI formulation and the modified projection method, and step three determines the Nash structures for the SDCSCND problem with the help of the Wilson algorithm.