^{}Department of Industrial Engineering, Iran University of Science & Technology, Terhan, Iran

Abstract

This paper discusses a problem in which decentralized supply chains enter the market simultaneously with no existing rival chains, shape the supply chains’ networks, and set wholesale and retail prices in a noncooperative manner. All the chains produce either identical or highly substitutable products. Customer demand is elastic and price-dependent. A three-step algorithm is proposed to solve this problem. Step one considers the supply chains’ potential network structures. Step two is based on a finite-dimensional variational inequality formulation and is solved by a modified projection method to determine equilibrium prices. Step three selects the equilibrium locations to shape the chains’ equilibrium network structure with the help of the Wilson algorithm. Finally, this approach is applied to a real-world scenario, and the results are discussed. Moreover, sensitivity analyses are conducted.

طراحی همزمان شبکة زنجیرة تأمین رقابتی نامتمرکز تحت بازی چند قطبی

Abstract [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]

بازی همزمان، تعادل نش، طراحی شبکة زنجیرة تأمین رقابتی نامتمرکز، مشتقات نامساوی

Full Text

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; Santibanez-Gonzalez & 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 & Rasti-Barzoki, 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 (SD-SCND) 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 0-1 variables, which puts the problem in the class of mixed-integer, nonlinear programming models. Therefore, the proposed problem cannot be solved with the explicit solution algorithm. Thus, we propose a three-step 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 0-1 variables, we use the Wilson algorithm and specify some strategies related to the 0-1 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 three-step algorithm we are able to solve the problem of SD-SCND in dynamic competition.

Problem Definition

This paper considers the problem of simultaneous decentralized competitive supply chain network design (SD-CSCND) 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 price-dependent.

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 self-price sensitivity. The term represents the cross-price 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 non-negativity 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 non-negativity restriction on the corresponding decision variables.

Solution Approach

This section presents our proposed algorithm for solving the SD-CSCND problem. The algorithm is essentially based on the chains’ decision variables. Each SC has two different types of decision variables, including continuous and discrete (0-1) 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 three-step 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:

Initialize the whole strategies for the players:

1.1. Construct an empty poly-matrix by considering all pure strategies of the players (any combination of the facilities to be opened from all the potential facilities of each player).

Calculate Nash equilibrium prices and flows for all players in the chains in the defied strategies:

2.1. Develop VI formulation of the players’ problems in each strategy and solve it with the help of the modified projection method.

Find the best response of all the players:

3.1. Fill the empty poly-matrix 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 0-1 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 non-negativity 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 non-negativity 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 Euler-type model (Nagurney et al., 2003; Santibanez-Gonzalez & Diabat, 2016), some evolutionary algorithm (Majig et al., 2007), and extended mathematical programming (EMP) of GAMS (Santibanez-Gonzalez & 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 real-world 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 real-world 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 SD-CSCND 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 cross-price and self-price parameters.

Tables 6 and 7 represent the sensitivity analysis for SCs with respect to cross price effect while the self-price parameter is set to 1.2. Tables 8 and 9 represent the sensitivity analysis for SCs with respect to self-price effect while the cross-price 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 self-price 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 self-price 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 self-price and cross-price 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 real-world 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 three-step 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 SD-CSCND problem with the help of the Wilson algorithm.

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