Strategic design of competing centralized supply chain networks for markets with deterministic demands

Advances in Engineering Software 41 (2010) 810–822

Contents lists available at ScienceDirect

Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

Strategic design of competing centralized supply chain networks for markets with deterministic demands Shabnam Rezapour *, Reza Zanjirani Farahani Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 19 September 2009 Received in revised form 22 December 2009 Accepted 30 December 2009 Available online 22 January 2010 Keywords: Supply chains Network design Price equilibrium Nash equilibrium

a b s t r a c t This paper develops an equilibrium model to design a centralized supply chain network operating in markets under deterministic price-depended demands and with a rival chain present. The two chains provide competitive products, either identical or highly substitutable, for some participating retailer markets. We model the optimizing behavior of these two chains, derive the equilibrium conditions, and establish the finite-dimensional variational inequality formulation, and solve it using a modified projection method. We provide properties of the equilibrium pattern in terms of the existence and uniqueness results. Our model also considers the impacts of the strategic facility location decisions on the tactical inventory and shipment decisions. Finally, we illustrate the model through a numerical example and discuss how the prices, costs, incomes, and profits behave with respect to key marketing activities, such as advertising, brand positioning, and brand loyalty. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Today competition is as fierce as ever as markets become saturated, barriers to enter new markets become lower, access to venture capital becomes easier, and the number of entrepreneurs increases. Studying other rivals in the markets can help entrants assess what competitive strategies they can utilize to help strengthen their competitive advantage. A supply chain (SC) consisting of independent parties with disparate agendas forms a chain of processes that transforms raw materials into products and delivers them to customers through specific activities, which includes the planning and coordination of tasks such as procuring raw materials, producing, storing, negotiating, receiving customer orders, and delivering products. Recent SC researches [9,39,37] emphasize that the nature of competition in the future will not be between companies but rather between SCs. According to the Deloitte Consulting’s (1999) report which is based on a survey of more than 200 large manufacturers and distributors in the United States and Canada, (including aerospace, automotive, consumer products, high-tech, etc. industries) ‘‘no longer will companies compete against other companies, but total SCs will compete against other SCs.” According to Taylor [35] ‘‘The classic model of company vs. company is starting to give way to a new model: SC vs. SC. In the 21st century, being the best at produc-

* Corresponding author. Address: Department of Industrial Engineering, Amirkabir University of Technology (AUT), 424 Hafez Ave., Tehran 15914, Iran. Tel.: +98 21 66937427; fax: +98 21 66929634. E-mail address: [email protected] (S. Rezapour). 0965-9978/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2010.01.001

ing or selling a superior product is no longer enough. Success now depends on assembling a team of companies that can rise above the win/lose negotiations of conventional trading relationships and work together to deliver the best products at the best price. Excellence in manufacturing is just the admission fee to be a player in the larger game of SC competition.” For example, Microsoft (software supplier) and HTC (device manufacturer) constitute a SC that competes with the SC consisting of Symbian (software supplier) and Nokia (device manufacturer) [37], and Dell and Wal-Mart Stores devastated their competition by reinventing their SCs [35]. How will these chains actually compete against each other? And what can practitioners do now in anticipation of this future? The answers to these questions remain still widely unknown. A great deal of supply chain management (SCM) literature assumes that the physical network structure of the SC is given, and the objective is to minimize (maximize) the system-wide cost (profit) by the best-planned movement of goods through the chain. However, the physical network structure of a SC completely influences its performance, and is one of the important factors impacting chain’s competitiveness. Supply chain network design (SCND) goes beyond SC planning by also considering location and capacity decisions of tiers of the SC. We refer the reader to Shen [31] for a review of SCND literature. SCND is viewed as the primary source of reducing costs and consequently increasing profit by many chains [33]. So fierce competition in today’s markets forces companies to better design and manage their SC networks. However, all of the previous works about SCND are based on models that ignore the competitive environment in which the chains will operate and how the future customer behavior will influences their profits.

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

These reasons (the importance of competitive network design in today’s B2B settings and lack of these kinds of problems in the literature) motivate us to develop a model for competitive SCND. This paper treats the design of a centralized SC network under an environment of competition among rival SCs. This paper is organized as follows. Section 2 presents the literature on competition, competition in SC, equilibrium models, and variational inequality (VI). Section 3 describes our problem, its assumptions, objectives, variables, and parameters. In Section 4, we present the SCs’ models, and derive their optimality conditions. Section 5 presents the governing equilibrium conditions of the models’ optimality conditions and derives the finite-dimensional VI formulation of the problem. Section 6 provides some properties of the equilibrium condition. Section 7 describes the algorithm used to solve the formulation. In Section 8, we apply the algorithm to a numerical example, and provide a discussion of the model and results. We conclude the paper with Section 9.

2. Literature review Many different factors must be taken into account when designing a new SC. One of the most important factors is the existence of other rivals in the markets providing the same goods. When no other rival exists, the new SC will have the monopoly of the markets in that area. However, if there already are other rivals in the area offering the same goods, then the new chain will have to compete for the market. Three types of competition have been investigated in the literature: (a) static competition: in this kind of competition, new entrant rival should make decision about its facilities, taking into account that other rivals already exist and the new entrant knows their characteristics (see for instance [20,15,1,40]); (b) competition with foresight: in this kind of competition, the rivals are not in the market yet but they will enter soon afterwards [32,11,30]; and (c) dynamic competition: this kind of competition is on the existence of Nash equilibriums in a scenario described as a game, in which rivals simultaneously compete in prices, locations, qualities, etc. In this case the rivals can change their decisions [36,12,5– 7,9,10,8,3]. In this paper we consider the duopolistic dynamic competition of the two rival chains. In SC context there are three different kinds of competitions: (a) competition among the firms of one tier of a SC [12,25,13,14,26,10,27,28,21]; (b) competition among the firms of different tiers of a SC [22,29,38,19]; and (c) competition between rival SCs [9,39,37]. As you can see there is little analytic work in the literature that studies the interaction of SCs and they all consider a predetermined network for the chain. Boyaci and Gallego [9] consider a market with two competing SCs, each consisting of one wholesaler and one retailer. They assume that the business environment forces SCs to charge similar prices and to compete strictly on the basis of customer service. They model customer service competition using game-theoretical concepts. They consider three competition scenarios between the SCs. Zhang [39] studies a supply chain economy (SCE) that comprises heterogeneous SC involving multiple products and competing for multiple markets. The proposed network model is built upon operation links and interface links, representing, respectively, substantial SC operations and coordination functions between the operations. Xiao and Yang [37] consider two competing SCs facing uncertain demands, where each SC consists of one risk-neutral supplier and one risk-averse retailer. Two retailers compete in retail price as well as service investment. They assume that each retailer has a long-term relationship with his supplier, which is assured by his

811

individual rationality constraint. The products of two suppliers are partially differentiated and each supplier sells products to customers through her retailer. In this paper, we simultaneously consider the SCND problem involving competition between two chains. Our paper is closely related to Nagurney et al. [25] and Nagurney and Toyasaki [27]. Nagurney et al. [25] study the equilibrium conditions of a three tier SC to simultaneously study the behaviors of its decision-makers (competition between the firms of one tier of the chain). They formulated this model as a VI. But our decision structure is different because we consider a SCND problem also we consider the future competition between this new chain and an already existing chain. The equilibrium model is drawn from economics and, in particular, from network economics. Due to the non-cooperative competition among SCs’ networks, Nash game-theoretic approach is able to perfectly quantity such a competition. Moreover, the steady consumption behavior of consumers for the product can be characterized by the well-known spatial price equilibrium conditions [23]. Many reports point out that pricing strategies in SC systems contribute to higher profitability and stronger competitive power. Swaminathan and Tayur [34] state that with the expansion of the Internet, pricing has become a crucial concern in SCM. In this work, we identify the equilibrium prices in the SCs under SC vs. SC competition and discuss how the equilibrium prices, costs, incomes, and profits behave with respect to key marketing activities, such as advertising, brand positioning, and brand loyalty. In this paper, we develop an equilibrium model to design a centralized SC network in markets with deterministic pricedependent demands and in the presence of a rival chain. These chains (existing and new chains) provide competitive products, either identical or highly substitutable, for some participating retailers’ markets. Our model also considers the impacts of the strategic facility location decisions on the tactical inventory and shipment decisions of the networks. To the best of our knowledge, such a model for SCND has not appeared heretofore in the literature. The approach towards the design problem is based upon VI which are described in Nagurney [23] and subsequent papers of this same author. Our paper makes a unique contribution to SCND in two ways. First, for simplicity almost all previous SCND papers ignored the rival chains of the markets. To the best of our knowledge, no previous SC research has studied both SCND and SC vs. SC competition simultaneously. Second, this paper connects the obtained sensitivity analysis results with marketing activities from a consumer behavior perspective. This serves as a theoretical foundation to identify pragmatic business implications that might assist real marketing management.

3. Problem definition We consider a centralized SC network comprising one or more producers, distribution centers (DCs), and retailers in markets with deterministic demands and in the presence of a rival chain. These chains (existing and new chains) provide competitive products, either identical or highly substitutable, for some participating markets (Fig. 1). The problem environment and assumptions of this study are as follows: 3.1. Specifications of facilities in these networks  We assume that the candidate locations of the DCs of the new network are known.

812

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Producers of Existing network

Our model’s decision variables include:

DCs of Existing network

1

1

2

2

. . .

qij(1)

. . .

I

s (jk1)

J

Participated Retailers Markets 1

1

2

2

. . .

. . .

K

K

Existing supply chain

New supply chain 1 2 . . .

1 2

qij( 2 )

. . .

I′ Available producers of New network

s (jk2 )

J′ Candidate DCs locations of New network

Fig. 1. Structure of these two supply chains.

 There are some available producers which can be selected to use in the new network.  All the producers have infinite capacity (their capacities are large enough).  Several DCs will be opened in new network and each DC is served directly by the producers of that network and distributes products to retailers.  DCs of the networks combine the orders from different retailers and order from the producers.  There are no common facilities in these two chains, except their participating retailers.  Facilities (numbers, locations, and capacities) of existing chain’s layers are known. 3.2. Demand of markets  The two chains have a participating retailer for each market and each market has a deterministic price-dependent demand.  The demand of the two chains in each market is subject to the products’ prices at the related retailer. 3.3. Cost functions of networks  Location costs are incurred when DCs with specific capacities are established in new network.  Transportation costs are incurred in each network for shipments from the producers to the DCs of that network.  Local transportation costs of each network are incurred in moving the goods from the DCs of that network to the retailers.  Inventory costs are incurred at each DC and consist of the carrying cost for the average inventory used over a period of time.  The retailers of these networks maintain only a minimal amount of inventory, which is ignored in the models of the networks.  Each producer has a specific production cost. 3.4. Competition of chains  Each network maximizes its profit, given the action of the other network so they compete in a non-cooperative manner.

 Which producers should new network use?  How many DCs should new network have and where should it locate them?  How does new network set the capacity of DCs at each location?  What quantities should each network produce and store at these locations?  What quantities should be moved from location to location in each network?  What is the optimal price of products at each selected market of both networks that simultaneously maximizes the total profit of these rival networks? Our model objective for each network is to maximize its total profit in these competing markets and in presence of complete information. Our model parameters include:  Cost components of both chains.  Candidate locations of DCs in the new network.  Available producers which can be selected to use in the new network.  Capacities of DCs in the existing network.  Deterministic price-dependent demand of the chains in each market. 4. Model To begin modeling the problem, let us use the following notations throughout the paper: 4.1. Sets and indices I: Set of existing network’s producers I = {1, 2, . . ., jIj}, i 2 I. I0 : Set of new network’s available producers I0 = {1, 2, . . ., jI0 j}, i0 2 I0 . J: Set of existing network’s DC locations J = {1, 2, . . ., jJj}, j 2 J. J0 : Set of new network’s candidate DC locations J0 = {1, 2, . . ., jJ0 j}, j 0 2 J0 . K: Set of networks’ available participated retailers K = {1, 2, . . ., jKj}, k 2 K. 4.2. Decision variables ð1Þ

qij : Amounts of goods shipped from ith producer to jth DC in the existing network. h i ð1Þ Q 11 ¼ qij : Vector of goods shipped from producers to DCs in the existing network. ð1Þ sjk : Amounts of goods shipped from jth DC to kth retailer in the existing network. ð1Þ

Q 12 ¼ ½sjk ]: Vector of goods shipped from DCs to retailers in the existing network. ð1Þ pk : Price of existing network’s goods in kth market. h i ð1Þ P 1 ¼ pk : Vector of prices of existing network in the markets. ð2Þ

qi0 j0 : Amounts of goods shipped from i0 th producer to j0 th DC in the new network. h i ð2Þ Q 21 ¼ qi0 j0 : Vector of goods shipped from producers to DCs in the new network. ð2Þ sj0 k : Amounts of goods shipped from j0 th DC to kth retailer in the new network. h i ð2Þ Q 22 ¼ sj0 k : Vector of goods shipped from DCs to retailers in the new network.

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822 ð2Þ

yj0 : Capacity of DC located in j0 th candidate location in the new network. h i ð2Þ Y 2 ¼ yj0 : Vector of capacities of DCs located in candidate locations in the new network. ð2Þ pk : Price of new network goods in kth market. h i ð2Þ P2 ¼ pk : Vector of prices of the new network in the markets.   ð1Þ ð2Þ Pk ¼ pk ; pk : Vector of both networks’ prices in market k.

4.3. Parameters ð1Þ

aj : Capacity of jth DC in the existing network; h i ð1Þ A1 ¼ aj : Vector of capacities of DCs in the existing network.

4.4. Cost components of networks   ð1Þ ð1Þ F i qi : Cost of procurement, producing and handling qi units in ith producer of the existing network; we assume that these P ð1Þ ð1Þ functions are continuous and convex [13], qi ¼ J qij .   ð2Þ ð2Þ F i0 qi0 : Cost of procurement, producing and handling qi0 units in i0 th producer of the new network; we assume that these P ð2Þ ð2Þ functions are continuous and convex, qi0 ¼ J0 qi0 j0 .   ð1Þ C ij qij : Cost of transaction (ordering, transportation and other ð1Þ

expenses) qij

units between ith producer and jth DC of the

existing network; we assume that these functions are continuous and convex [25].   ð2Þ C i0 j0 qi0 j0 : Cost of transaction (ordering, transportation, and ð2Þ

other expenses) qi0 j0 units between i0 th producer and j0 th DC of the new network; we assume that these functions are continuous and convex.   ð2Þ ð2Þ Lj0 yj0 : Cost of locating a DC with capacity yj0 at j0 th candidate place of the new network; we assume that these functions are continuous and convex.   ð1Þ Hj qj : Inventory holding cost at jth DC of the existing net  P ð1Þ ð1Þ ð1Þ ð1Þ ¼ ðhj =2Þ  qj ; qj ¼ I qij ; where work; we assume Hj qj hj is the product’s unit inventory holding cost at jth DC.   ð2Þ Hj0 qj0 : Inventory holding cost at j0 th DC of the new network;     P ð2Þ ð2Þ ð2Þ ð2Þ we assume Hj0 qj0 ¼ hj0 =2  qj0 ; qj0 ¼ I0 qi0 j0 ; where hj0 is the product’s unit inventory holding cost at j0 th DC.   ð1Þ T jk sjk : Cost of transaction (ordering, transportation, and other expenses)

ð1Þ sjk

units between jth DC and kth retailer of

the existing network; we assume that these functions are continuous and convex [13].   ð2Þ T j0 k sj0 k : Cost of transaction (ordering, transportation, and ð2Þ

813

ð1Þ

dk ðP k Þ: Demand at kth market of the existing network; we     ð1Þ ð1Þ ð2Þ assume dk ðPk Þ ¼ v k  b  pk þ d  pk . ð2Þ

dk ðP k Þ: Demand at kth market of the new network; we assume     ð2Þ ð2Þ ð1Þ dk ðP k Þ ¼ v k  b  pk þ d  pk .

vk: The market base for market k. d: The substitutability coefficient of the two networks’ products in the markets, 0 < d < 1. In our model, d is also understood as the cross-brand price sensitivity between the product of existing and new chains. According to Bawa and Shoemaker [4] and Ailawadi [2], more marketing activities lead to more frequent brand switching, causing d to increase. b: The parameter b represents the self-price sensitivity. Note that marketing literature explains that self-price sensitivity is negatively associated with the level of brand loyalty: as loyalty for a particular brand increases, the brand becomes less pricesensitive. Note that magnitude of self-price sensitivity is greater than that of cross-price sensitivity: 0 6 d 6 b. Hanssens et al. [16] find that customers’ self-price sensitivity is stronger than their cross-price sensitivity. It is worth noting the reasons that our model uses a linear demand function. First, a linear function is tractable. Next, a linear demand function often achieves a satisfactory fit to the given data set. Third, we follow a tradition of microeconomics analysis and marketing research about brand management and pricing [18]. 4.6. Mathematical model The total cost incurred in existing network, is equal to the sum of that network’s producers’ production costs, the total transaction costs, DCs’ inventory holding costs in that network. The total revenue of this network is equal to the price that the retailers charge for the product of that network (and the customers are willing to pay) times the quantity of the product is purchased from the retailers by all customers. We can express the criterion of profit maximization for this network as:

MAX Z 1 ðQ 11 ; Q 12  P 1 Þ X ð1Þ X ð1Þ X  ð1Þ  ¼ pk  sjk  F i qi K



J

XX I

X

S:T:

J

ð1Þ

qij P

I

X

ð1Þ

I

  Xh X X X  ð1Þ  j ð1Þ ð1Þ C ij qij  qij  T jk sjk  2 I J J K X

ð1Þ

sjk

ð8jÞ

ð1Þ ð2Þ

K ð1Þ

qij 6 aj

ð8jÞ

ð3Þ

I ð1Þ

ð1Þ

qij ; sjk P 0 ð8i; j; kÞ

ð4Þ

other expenses) sj0 k units between j0 th DC and kth retailer of the new network; we assume that these functions are continuous and convex.

4.5. Demand functions in the markets We assume that retail price is the only important factor affecting the market demand. Thus, similar to Tsay and Agarwal [36], we assume that the demand function of market k is:

The total cost incurred in new network, is equal to the sum of that network’s selected producers’ production costs, the total transaction costs, located DCs’ inventory holding costs, and costs of locating facilities (DCs) in that network. The total revenue of this network is equal to the price that the retailers charge for the product of that network (and the customers are willing to pay) times the quantity of the product is purchased from the retailers by all customers. We can express the criterion of profit maximization for this network as:

814

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

MAX Z 2 ðQ 21 ; Q 22 ; Y 2 ; P2 Þ X ð2Þ X ð2Þ X  ð2Þ  ¼ pk  s j0 k  F i0 qi0 J0

K

I0

  X h0 X j ð2Þ ð2Þ  C i0 j0 qi0 j0  q0 0  2 I0 i j I0 J0 J0   X   XX ð2Þ ð2Þ  T j0 k s j0 k  Lj0 yj0 XX

J0

X

S:T: X I

0

I0

K

X

0

ð2Þ

ð 8j Þ

s j0 k

ð6Þ

K

ð2Þ q i0 j 0

ð2Þ

ð2Þ

q i 0 j0 P

ð5Þ

J0

6

ð2Þ

0

ð2Þ yj0

ð8j Þ

ð7Þ 0

ð2Þ

0

qi0 j0 ; sj0 k ; yj0 P 0 ð8i ; j ; kÞ

ð8Þ

The first terms of these objective functions include the revenue of networks. The second terms are related to production costs. The third terms include costs of shipments from producers to DCs. The fourth terms are related to inventory holding costs in DCs, the fifth terms of these objective functions include products shipment costs between DCs and retailers and the last term of second objective function is related to facility locating costs. Objective functions (1), (5) express that the difference between the revenues minus the total cost of each network should be maximized. Constraints (2), (6) simply express that retailers cannot purchase more from a DC than is held in stock and constraints (3), (7) express that the amounts of products enters the DCs of each network cannot exceed their capacities. The Lagrange function of the existing network model is:

MAX L1 ðQ 11 ; Q 12 ; P1 ; c1 ; b1 Þ X ð1Þ X ð1Þ X  ð1Þ  ¼ pk  sjk  F i qi K

J

J ð1Þ

ð1Þ

I

ð1Þ

ð1Þ

K

J

I

ð9Þ

The Lagrange function of the new network model is:

MAX L2 ðQ 21 ; Q 22 ; Y 2 ; P2 ; c2 ; b2 Þ X ð2Þ X ð2Þ X  ð2Þ  pk  s j0 k  F i0 q i0 ¼ J0



X I0

ð2Þ ð2Þ ð2Þ qi0 j0 ; sj0 k ; yj0 ;

ð2Þ

qi0 j0 

ð2Þ

s j0 k

J0

þ

X

0

ð2Þ

ð2Þ

bj0  yj0 

0

P 0 ð8i ; j ; kÞ

ð1Þ

ð2Þ

where the terms cj ; cj0



h

X I0

ð1Þ aj 

J

h

ð2Þ ; c 2 ¼ c j0 c1 ¼ cð1Þ j

i

ð1Þ qij

#

h i ð1Þ ð1Þ  bj  bj

I

    2 3 ð2Þ ð2Þ h i @C i0 j0 qi0 j0 X X @F i0 qi0 ð2Þ ð2Þ 5 ð2Þ ð2Þ 4  qi0 j0  qi0 j0 þ þ þ hj0 =2  cj0 þ bj0 ð2Þ ð2Þ @qi0 j0 @qi0 j0 I0 J0   2 3 ð2Þ h i X X @T j0 k sj0 k ð2Þ ð2Þ ð2Þ ð2Þ 4 þ cj0  pk 5  sj0 k  sj0 k þ ð2Þ @sj0 k K J0 " # i X X ð2Þ X ð2Þ h ð2Þ ð2Þ þ qi0 j0  sj0 k  cj0  cj0

J0

!

I0

K



ð2Þ yj0



I0

0

0

0

0

0

0

jjJ jþjJ jjKjþjJ jþjJ jþjJ j 8ðQ 11 ;Q 12 ; c1 ;b1 ;Q 21 ;Q 22 ;Y 2 ; c2 ;b2 Þ 2 RjIjjJjþjJjjKjþjJjþjJjþjI þ

ð2Þ

qi0 j0

ð11Þ

ð10Þ i

K

X

3 h i X @Lj0 ð2Þ ð2Þ ð2Þ 4 þ  bj0 5  yj0  yj0 ð2Þ @yj0 J0 " # i X ð2Þ X ð2Þ h ð2Þ ð2Þ þ yj 0  qi0 j0  bj0  bj0 P0

J0

J0

K

ð2Þ ð2Þ ; bj0 j0

c

X

!

I

"

2

  X h0 X j ð2Þ ð2Þ  C i0 j0 qi0 j0  qi0 j0  2 0 0 0 0 I J J I   X   X XX ð2Þ ð2Þ  T j0 k sj0 k  L j0 y j0 þ cð2Þ j0 K

þ

X

J0

I0

XX

J0

Since the demand for processed materials in each market is elastic depending on price, the equilibrium conditions of the system can be divided into two parts: system–equilibrium conditions assuming fixed demands (fixed prices), and equilibrium conditions characterized by the demand functions. Here, we assume that these networks compete in a non-cooperative manner so that each maximizes its profits, given the action of the other network. We now consider the optimality conditions of the system assuming that these existing and new networks are faced with the optimization problems (9), (10) and the non-negativity assumption on the variables. So the system–equilibrium condition of both networks can be mathematically expressed as follows [23]:   2  ð1Þ  3 ð1Þ h i @C ij qij X X @F i qi ð1Þ ð1Þ 5 ð1Þ ð1Þ 4 þ þ h =2  c þ b  qij  qij j j j ð1Þ ð1Þ @qij @qij I J   2 3 ð1Þ h i X X @T jk sjk ð1Þ ð1Þ 5 ð1Þ ð1Þ 4 þ cj  pk þ  sjk  sjk ð1Þ @sjk J K " # i X X ð1Þ X ð1Þ h ð1Þ ð1Þ þ qij  sjk  cj  cj J

P 0 ð8i; j; kÞ

K

5. Equilibrium condition

I

  Xh X X X  ð1Þ  j ð1Þ ð1Þ C ij qij  qij  T jk sjk   2 I J J I J K ! ! X ð1Þ X ð1Þ X ð1Þ X ð1Þ X ð1Þ ð1Þ þ cj  qij  sjk þ bj  aj  qij XX

qij ; sjk ; cj ; bj

among the facilities, and retailers’ sale price) under the condition that the demand at each market is subject to the prices of existing and new networks’ goods in the retailers. The equilibrium is characterized by the fixed point describing the optimal material flow each facility processes under the optimal demand of markets at which the offered optimal equilibrium prices of both networks are accepted.

are the Lagrange

multipliers, respectively, associated with first constraints (2) of the existing network for DC j and first constrains (6) of the new net h i h i ð1Þ ð2Þ ð1Þ ð2Þ are the Lawork for DC j0 . The terms bj ; bj0 b1 ¼ bj ; b2 ¼ bj0 grange multipliers, respectively, associated with second constraints (3) of the existing network for DC j and second constrains (7) of the new network for DC j0 . The problem of interest is to find the new network’s optimal design (place and capacity of DCs, amounts of product shipments

The system–equilibrium condition for both networks coincides with the solution of the VI (11). In (11) we have not had the prices charged be variables. They become endogenous variables in the complete equilibrium model. ð1Þ ð2Þ If the equilibrium prices, pk ; pk , that the consumers are willing to pay in demand market k are positive, then the amount of P ð1Þ P ð2Þ product supplied by the retailers of both networks J sjk ; J0 sj0 k ð1Þ    ð2Þ    will be equal to the demands dk pk ; dk pk in that market, respectively. Otherwise, the demand is less than or equal to the total amount of commodity available in that market. These conditions correspond to the following spatial price equilibrium conditions [23]:

815

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

8 P ð1Þ > > < ¼ J sjk   ð1Þ dk Pk P ð1Þ > > : 6 sjk

> 0 8ðkÞ

ð1Þ

¼ 0 8ðkÞ

if pk

J

8 P ð2Þ > ¼ s j0 k > < J0 ð2Þ  dk ðPk Þ P ð2Þ > > : 6 sj0 k J

ð1Þ

if pk

ð2Þ

> 0 8ðkÞ

ð2Þ

¼ 0 8ðkÞ

if pk

if pk

0

ð12Þ

For easy reference in the subsequent sections, VI problem (14) can be rewritten in standard VI form as follows [23]: Determine x* 2 j satisfying

ð13Þ

We now establish the following: Theorem 1 (VI formulation). The equilibrium condition governing these two rival SCs’ networks model are equivalent to the solution of the following VI problem:   8 Q 11 ; Q 12 ; c1 ; b1 ; P 1 ; Q 21 ; Q 22 ; Y 2 ; c2 ; b2 ; P2 2 j Determine satisfying

  2  ð1Þ  3 ð1Þ h i @C ij qij X X @F i qi ð1Þ ð1Þ ð1Þ ð1Þ 4 þ þ hj =2  cj þ bj 5  qij  qij ð1Þ ð1Þ @qij @qij I J   2 3 ð1Þ h i X X @T jk sjk ð1Þ ð1Þ 5 ð1Þ ð1Þ 4  sjk  sjk þ þ cj  pk ð1Þ @sjk J K " # i X X ð1Þ X ð1Þ h ð1Þ ð1Þ  cj  cj þ qij  sjk þ

X

"

I

K

ð1Þ

aj 

X

J

#

ð1Þ

qij

h i ð1Þ ð1Þ  bj  bj

I

    3 ð2Þ ð2Þ @C i0 j0 qi0 j0 X X @F i0 qi0 ð2Þ ð2Þ 4 þ þ hj0 =2  cj0 þ bj0 5 þ ð2Þ ð2Þ @qi0 j0 @qi0 j0 I0 J0 h i ð2Þ ð2Þ  qi0 j0  qi0 j0   2 3 ð2Þ h i X X @T j0 k sj0 k ð2Þ ð2Þ ð2Þ ð2Þ 4 þ þ cj0  pk 5  sj0 k  sj0 k ð2Þ @sj0 k K J0 " # i X X ð2Þ X ð2Þ h ð2Þ ð2Þ þ qi0 j0  sj0 k  cj0  cj0 2

J0

I0

K

J0

þ

J

K

J0

k

k

k

8ðQ 11 ; Q 12 ; c1 ; b1 ; P1 ; Q 21 ; Q 22 ; Y 2 ; c2 ; b2 ; P2 Þ 2 j

j  fðQ 11 ; Q 12 ; c1 ; b1 ; P1 ; Q 21 ; Q 22 ; Y 2 ; c2 ; b2 ; P2 Þ jðQ 11 ; Q 12 ; c1 ; b1 ; P1 ; Q 21 ; Q 22 ; Y 2 ; c2 ; b2 ; P2 Þ 0

0

0

0

0

0

jjJ jþjJ jjKjþjJ jþjJ jþjJ jþjKj 2 RjIjjJjþjJjjKjþjJjþjJjþjKjþjI þ

In this section, we highlight some properties (the existence and uniqueness results) of the above VI problem (14). Since the feasible set underlying the VI problem (14) is not compact so it is not easy to derive existence of a solution from the assumption of continuity of the functions. So in this section we impose a rather weak condition to guarantee the existence of a solution pattern [25]. Let

jb ¼

n  Q b11 ; Q b12 ; cb1 ; bb1 ; Pb1 ; Q b21 ; Q b22 ; Y b2 ; cb2 ; bb2 ; P b2

j0 6 :Q b11 6 b1 ; 0 6 Q b12 6 b2 ; 0 6 cb1 6 b3 ; 0 6 bb1 6 b4 ; 0 6 Pb1 6 b5 ; 0 6 Q b21 6 b6 ; 0 6 Q b22 6 b7 ; 0 6 Y b2 6 b8 ; 0 o 6 cb2 6 b9 ; 0 6 bb2 6 b10 ; 0 6 Pb2 6 b11

hFðxb Þ; x  xb i P 0 8x 2 jb

ð17Þ

admits at least one solution xb 2 jb, from the standard theory of VIs, since jb is compact and F is continuous. Under the conditions in Theorem 2 we can choose b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, and b11 large enough so the restricted VI (17) satisfy the condition of boundedness implied in (16) and the existence of a solution for the original VI problem will hold [17]. Theorem 2 (Existence). Suppose that there exist positive constants W, S, T with T > 0 such that:

ð2Þ

ð14Þ

ð16Þ

Then jb is 0 a0 0 bounded, closed convex subset of jjJ jþjJ jjKjþjJ 0 jþjJ 0 jþjJ 0 jþjKj . Since jb is compact and F is RjIjjJjþjJjjKjþjJjþjJjþjKjþjI þ continuous, according to the VIs theory, the following VI admits at least one solution xb 2 jb

  ð1Þ @C ij qij

ð1Þ

2 3 h i X X ð2Þ ð2Þ ð2Þ ð2Þ  4 P0 s 0  d ðP Þ5  p  p þ k

6. Qualitative properties

ð1Þ

ð1Þ

dk ðPk Þ 6 S 8Pk with pk > T ð8kÞ     ð2Þ ð2Þ @F i0 qi0 @C i0 j0 qi0 j0 0 0 ð2Þ þ P W 8Q 21 with qi0 j0 P S ð8i ; j Þ ð2Þ ð2Þ @qi0 j0 @qi0 j0   ð2Þ @T j0 k sj0 k 0 ð2Þ P W 8Q 22 with sj0 k P S ð8j ; kÞ ð2Þ @sj0 k   ð2Þ @Lj0 yj0 0 ð2Þ P W 8Y 2 with yj0 P S ð8j Þ ð2Þ @yj0

# h i ð1Þ    ð1Þ ð1Þ ð1Þ sjk  dk Pk  pk  pk

jk

and F are given by the functional terms preceding the multiplication signs in (14). The term h.,.i denotes the inner product in Ndimensional Euclidean space.

þ P W 8Q 11 with qij P S ð8i; jÞ ð1Þ ð1Þ @qij @qij   ð1Þ @T jk sjk ð1Þ P W 8Q 12 with sjk P S ð8j; kÞ ð1Þ @sjk

I0

K

ð15Þ

x ¼ ðQ 11 ; Q 12 ; c1 ; b1 ; P1 ; Q 21 ; Q 22 ; Y 2 ; c2 ; b2 ; P2 Þ;   ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ ð2Þ FðxÞ ¼ F ij ; F jk ; F 1j ; F 2j ; F k ; F i0 j0 ; F j0 k ; F 1j0 ; F 2j0 ; F 3j0 ; F k

  ð1Þ @F i qi

2  ð2Þ  3 h i X @Lj0 yj0 ð2Þ ð2Þ ð2Þ 4  bj0 5  yj0  yj0 ð2Þ @yj0 J0 " # i X ð2Þ X ð2Þ h ð2Þ ð2Þ þ y j0  qi0 j0  bj0  bj0 " X X

hFðx Þ; x  x i P 0 8x 2 j where

Definition 1 (Competitive SCs equilibrium). The equilibrium state of these two rival SCs’ networks is one where the product flows between the tiers of these networks coincide and the product flows and prices satisfy the sum of the optimality conditions (11)–(13).

J

Proof. The summation of (11)–(13), yields the above inequality (14). h

dk ðPk Þ 6 S 8Pk

ð2Þ

with pk > T ð8kÞ

Then VI (14) admits at least one solution.

ð18Þ

ð19Þ ð20Þ ð21Þ

ð22Þ

ð23Þ ð24Þ

816

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Proof. See the proof of existence in Nagurney et al. [24]. h

7.1. Step 0: Initialization

  ð1Þ ð1Þ Lemma 1 (Monotonicity). Suppose that F i ðqi Þ or C ij qij ;           ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ T jk sjk ; F i0 qi0 or C i0 j0 qi0 j0 ; T j0 k sj0 k , and Lj0 yj0 functions for

Set ðQ 011 ; Q 012 ; c01 ; b01 ; P01 ; Q 021 ; Q 022 ; Y 02 ; c02 ; b02 ; P 02 Þ 2 j. Let s = 1 (s denotes an iteration counter) and set d such that 0 < d < 1L , L is the Lipschitz constant for the problem (27).

ð1Þ

ð2Þ

all i, j, k, i0 , j0 are convex and dk ðPk Þ; dk ðP k Þ functions for all k are ð1Þ

ð2Þ

monotone decreasing functions of the generalized prices pk ; pk , respectively, then the vector function F that enters the VI (15) is monotone, that is,

hFðxÞ  Fðx0 Þ; x  x0 i P 0 8x; x0 2 j

7.2. Step 1: Computation Compute solving:

8 <

ð25Þ

  ð1Þ or Lemma 2 (Strict monotonicity). Suppose that F i qi             ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ C ij qij ; T jk sjk ; F i0 qi0 or C i0 j0 qi0 j0 ; T j0 k sj0 k ; Lj0 yj0 functions

ð1Þ;s1

þhj =2  cj

ð26Þ

Lemma 2 implies the uniqueness of product shipments, Q11, Q12, Q21, Q22, the DCs’ capacities, Y2, and the prices at the demand markets, P1, P2, at the equilibrium. We note also that no guarantee of a unique c1, b1, c2, b2, can be generally expected at the equilibrium. Theorem 3 (Uniqueness). Under the condition of Lemma 2, there is unique product shipments, Q 11 ; Q 12 ; Q 21 ; Q 22 , unique DCs’ capacities, Y 2 , and the prices at the demand markets, P 1 ; P 2 , satisfying the equilibrium condition of the system. In other words, if the VI (15) admits a solution, that should be the only solution in Q11, Q12, Q21, Q22, Y2, P1, P2.

Proof. Under the strict monotonicity result of Lemma 2, uniqueness follows from the VI theory [17]. h

ð1Þ

@qij

Þ

ð1Þ

@sjk

ð28Þ

þc

ð1Þ;s1 j



ð1Þ;s1 pk

!)

8j; k

ð29Þ (

X

s ð1Þ;s1 c~ð1Þ; ¼ MAX 0; cj d j

ð1Þ;s1

qij

X



!) ð1Þ;s1

8j

sjk

ð30Þ

K

( !) X ð1Þ;s1 ð1Þ;s ð1Þ;s1 ð1Þ ~ ¼ MAX 0; bj  d aj  qij 8j bj

ð31Þ

I

(

X

s ð1Þ;s1 ~ð1Þ; ¼ MAX 0; pk d p k

ð1Þ;s1 sjk



ð1Þ dk



s1

Pk

!) 

8k

ð32Þ

J

8 <

0

s ð2Þ;s1 ~ð2Þ; q ¼ MAX 0; qi0 j0  d@ i0 j 0 :

s ~sð2Þ; j0 k

  ð2Þ;s1 @F i0 qi0 ð2Þ

@qi0 j0

ð2Þ;s1

þ bj0

o

þ

  ð2Þ;s1 @C i0 j0 qi0 j0 ð2Þ

@qi0 j0

8i0 ; j0

ð33Þ

8 0 19 ð2Þ;s1 < = @T j0 k ðsj0 k Þ ð2Þ;s1 ð2Þ; s 1 ð2Þ; s 1 A ¼ MAX 0; sj0 k  d@ þ cj0  pk ð2Þ : ; @s 0 jk

Lemma 3 (Lipschitz continuity). Suppose that

              ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ ð2Þ F i qi ; C ij qij ; T jk sjk ; F i0 qi0 ; C i0 j0 qi0 j0 ; T j0 k sj0 k ; Lj0 yj0 functions have bounded second-order derivatives for all i, j, i0 , j0 , k, and ð1Þ ð2Þ dk ðPk Þ; dk ðP k Þ functions for all k have bounded first-order derivatives so the function F that enters the VI problem (15) is Lipschitz continues, that is:

kFðxÞ  Fðx Þk 6 Lkx  x k 8x; x 2 j; where L > 0

þ

by

  ð1Þ;s1 @C ij qij

8i; j

ð1Þ;s1

ð2Þ;s1

0

o

@T jk ðsjk

s ð1Þ;s1 ~sð1Þ; ¼ MAX 0; sjk d jk

þhj0 =2  cj0

0

ð1Þ;s1

Þ

I

Proof. See the proof in Nagurney et al. [25]. h

0

ð1Þ;s1

þ bj

(

ð2Þ

for all i, j, k, i0 , j0 are strictly convex and dk ðP k Þ; dk ðPk Þ functions for all k are strictly monotone decreasing then the vector function F that enters the VI (15) is strictly monotone with respect to (Q11, Q12, P1, Q21, Q22, Y2, P2), that is for any x, x0 2 j with   ðQ 11 ; Q 12 ; P1 ; Q 21 ; Q 22 ; Y 2 ; P2 Þ – Q 011 ; Q 012 ; P 01 ; Q 021 ; Q 022 ; Y 02 ; P 02 :

hFðxÞ  Fðx0 Þ; x  x0 i > 0 8x; x0 2 j

0

@F i ðqi s ð1Þ;s1 ~ð1Þ; q ¼ MAX 0; qij  d@ ij ð1Þ : @qij

Proof. See the proof in Nagurney et al. [25]. h

ð1Þ

es ;c es ; Q es ; Y es ; Q es ; Q es; c es Þ 2 j ~s ; P ~s ; P ~s1 ; b ~s2 ; b ðQ 11 12 1 1 21 22 2 2 2

ð27Þ

8j 0 ; k

ð34Þ

8 0 19 ð2Þ;s1 < = @Lj0 ðyj0 Þ ð2Þ;s ð2Þ;s1 ð2Þ; s 1 A ~ 0 ¼ MAX 0; y 0  d@  bj0 8j 0 y ð2Þ j j : ; @y 0

ð35Þ

j

( s ~ð2Þ; j0

c

¼ MAX 0; c

ð2Þ;s1 j0

d

X I0

ð2Þ;s1 qi0 j0



X

!) ð2Þ;s1 sj0 k

8j 0

ð36Þ

K

Proof. The result is direct by applying a mid-value theorem from calculus to the vector function F that enters the VI problem (15). h

( !) X ð2Þ;s1 ð2Þ;s ð2Þ;s1 ð2Þ;s1 ~  d yj0  qi0 j0 8j0 bj0 ¼ MAX 0; bj0

7. The algorithm

s ~ð2Þ; p k

ð37Þ

I0

8 0 19 <   = X ð2Þ;s1 ð2Þ ð2Þ;s1 s 1 A ¼ MAX 0; pk  d@ sj0 k  dk Pk 8k : ; 0

ð38Þ

J

In this section, we propose the algorithm to compute the VI (15). The algorithm we use is the modified projection method presented by Nagurney and Toyasaki [27]. This algorithm only requires monotonicity of F(x) and with Lipschitz continuity condition holding. The statement of this algorithm for our model is as follows:

7.3. Step 2: Adaptation Compute solving:

 s  Q 11 ; Q s12 ; cs1 ; bs1 ; P s1 ; Q s21 ; Q s22 ; Y s2 ; cs2 ; bs2 ; Ps2 2 j

by

817

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

  8 0 ð1Þ;s1 < ~ijð1Þ;s1 @C ij q ~ @F ð q Þ i ð1Þ;s ð1Þ;s1 i qij ¼ MAX 0; qij  d@ þ ð1Þ ð1Þ : @q @q ij

8.1. Numerical example

ij

o

~ð1Þ;s1 ~jð1Þ;s1 þ b þhj =2  c 8i; j ð39Þ j   8 9 0 1 ð1Þ;s1 < = @T jk ~sjk ð1Þ;s ð1Þ;s1 ð1Þ;s1 ð1Þ;s1 A ~ ~ sjk ¼ MAX 0; sjk  d@ þ c  p j k ð1Þ : ; @sjk

8j; k

(

s ð1Þ;s1 cð1Þ; ¼ MAX 0; cj d j

ð1Þ;s1 0; bj

¼ MAX

d

ð1Þ;s1 0; pk

¼ MAX

K

ð1Þ aj



X

ð1Þ;s1 ~sjk

d

X

s1 ~sð1Þ; jk

ð1Þ dk



8j

ð41Þ

8j



e s1 P k

ð42Þ

!) 

o

ij

8k

ð43Þ

ð1Þ

þ 8i ; j ð44Þ þhj0 =2  c   8 9 0 1 ð2Þ;s1 < = @T j0 k ~sj0 k ð2Þ;s ð2Þ;s1 s1 @ ~ð2Þ; ~kð2Þ;s1 A d þc p sj0 k ¼ MAX 0; sj0 k 0 ð2Þ j : ; @s 0 jk

8j ; k

8 0  ð2Þ;s1  19 < = ~0 @Lj0 y j ð2Þ;s ð2Þ;s1 ð2Þ; s 1 ~0 A  d@ b 8j 0 yj0 ¼ MAX 0; yj0 ð2Þ j : ; @y 0 j

( ð2Þ;s j0

c

ð2Þ;s1 j0

¼ MAX 0; c

I

( ð2Þ;s

bj0

d

X

ð2Þ;s1

¼ MAX 0; bj0

0

s1 ~ð2Þ; q  i0 j0

s1 ~ð2Þ; d y  j0

X 0

ð45Þ ð46Þ

!) s1 ~sð2Þ; j0 k

K

I

ð2Þ;s pk

X

s1 ~ð2Þ; q i0 j0

8j0

ð1Þ

ð1Þ d2 ðP2 Þ

0

0

The demand functions at the demand markets of the existing network were: ð2Þ

d1 ðP1 Þ ¼ 8000  p1 þ ð0:9Þ  p1

ij

0

ð1Þ

a1 ¼ 10; 000 ð1Þ

    8 0 s1 s1 ~ð2Þ; ~ð2Þ; < @F i0 q @C i0 j0 q i0 i0 j0 ð2Þ;s ð2Þ;s1  d@ þ qi0 j0 ¼ MAX 0; qi0 j0 ð2Þ ð2Þ : @q 0 0 @q 0 0 s1 ~ð2Þ; b j0

The cost functions of these two networks were listed in Table 1:

a2 ¼ 11; 000

J

s1 ~ð2Þ; j0

Example. This numerical example consisted of two rival networks; existing network has two producers and two DCs, and new network has two available producers and three candidate DC locations, as depicted in Fig. 2.

The capacities of DCs of the existing network were given by:

!)

~ijð1Þ;s1 q

I

( ð1Þ;s pk

X

~ijð1Þ;s1  q

I

( ð1Þ;s bj

X

ð40Þ

!)

The algorithm was implemented in MATLAB R2006. The convergence criterion used was that the absolute value of the flows and prices between two successive iterations differed by no more than 104. Computation time of this example is negligible.

ð1Þ

ð2Þ

¼ 12; 000  p2 þ ð0:9Þ  p2

The demand functions at the demand markets of the new network were: ð2Þ

ð2Þ

ð1Þ

d1 ðP1 Þ ¼ 8000  p1 þ ð0:9Þ:p1 ð2Þ d2 ðP2 Þ

¼ 12; 000 

ð2Þ p2

ð1Þ

þ ð0:9Þ:p2

The parameter d in the modified projection method was set to 0.05. The modified projection method converged in 695 iterations and yielded the following equilibrium results (Table 2): So the networks of this example are as follows (Fig. 3):

ð47Þ 8.2. Discussion

!)

8j 0

8 0 19 <   = X ð2Þ;s1 ð2Þ ð2Þ;s1 e s1 A ~s 0 ¼ MAX 0; pk  d@  dk P 8k k jk : ; 0

ð48Þ

ð49Þ

J

7.4. Step 3: Convergence verification>

The preceding example presents the type of designing a SC network in markets with a rival chain. We note that this example had nonlinear production costs associated with the producers, nonlinear transaction costs between the producers and the DCs, and nonlinear transaction costs between the DCs and retailers. Moreover,

Producers of Existing network

DCs of Existing network

1

1

       ð1Þ;s  ð1Þ;s  ð1Þ;s ð1Þ;s1  ð1Þ;s1  ð1Þ;s1  If qij  qij  6 e; sjk  sjk  6 e; cj  cj       ð1Þ;s  ð1Þ;s ð1Þ;s1  ð1Þ;s1  6 e; bj  bj  6 e; pk  pk       ð2Þ;s    ð2Þ;s1 ð2Þ;s ð2Þ;s1 6 e; qi0 j0  qi0 j0  6 e; sj0 k  sj0 k       ð2Þ;s  ð2Þ;s ð2Þ;s1  ð2Þ;s1  6 e; yj0  yj0  6 e; ;cj0  cj0       ð2Þ;s  ð2Þ;s ð2Þ;s1  ð2Þ;s1  6 e; bj0  bj0  6 e; pk  pk 6e

qij(1) 2

s (jk1)

Existing supply chain

New supply chain

for all i, j, k, i0 , j0 with e > 0 (a pre-specified tolerance), then stop; otherwise set s = s + 1 and back to step 1.

1

8. Numerical example and discussion

2

In this section, we apply the modified projection method to a numerical example and also provide a discussion of the results. Section 8.1 describes the example and its solution and Section 8.2 fully discusses the model in the context of the example solved.

2

1

qij( 2 )

Available producers of New network

Participated Retailers Markets 1

1

2

2

s (jk2 )

2 3 Candidate DC locations of New network

Fig. 2. Structure of existing and new supply chain networks in the numerical example.

818

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Table 1 Cost functions of networks in numerical example. Cost functions

Existing network

New network

Production cost of

   2 ð1Þ ð1Þ ð1Þ F 1 q1 þ ð0:15Þ  q1 ¼ ð0:1Þ  q1     ð1Þ ð1Þ 2 ð1Þ ¼ ð0:2Þ  q2 F 2 q2 þ ð0:1Þ  q2

   2 ð2Þ ð2Þ ð2Þ F 1 q1 þ ð0:1Þ  q1 ¼ ð0:2Þ  q1     ð2Þ ð2Þ 2 ð2Þ ¼ ð0:1Þ  q2 F 2 q2 þ ð0:1Þ  q2

    ð1Þ ð1Þ 2 ð1Þ c11 q11 ¼ ð0:2Þ  q11 þ ð0:3Þ  q11    2 ð1Þ ð1Þ ð1Þ c12 q12 ¼ ð0:25Þ  q12 þ ð0:25Þ  q12

    ð2Þ ð2Þ 2 ð2Þ c11 q11 ¼ ð0:1Þ  q11 þ ð0:2Þ  q11    2 ð2Þ ð2Þ ð2Þ c12 q12 ¼ ð0:15Þ  q12 þ ð0:2Þ  q12    2 ð2Þ ð2Þ ð2Þ c13 q13 ¼ ð0:7Þ  q13 þ ð0:8Þ  q13    2 ð2Þ ð2Þ ð2Þ c21 q21 ¼ ð0:2Þ  q21 þ ð0:1Þ  q21    2 ð2Þ ð2Þ ð2Þ c22 q22 ¼ ð0:2Þ  q22 þ ð0:2Þ  q22     ð2Þ ð2Þ 2 ð2Þ c23 q23 ¼ ð0:8Þ  q23 þ ð0:85Þ  q23

Producer 1 Producer 2

Transaction cost between

Producer 1 and DC 1 Producer 1 and DC 2 Producer 1 and DC 3

–

Producer 2 and DC 1

    ð1Þ ð1Þ 2 ð1Þ c21 q21 ¼ ð0:3Þ  q21 þ ð0:2Þ  q21    2 ð1Þ ð1Þ ð1Þ c22 q22 ¼ ð0:1Þ  q22 þ ð0:3Þ  q22

Producer 2 and DC 2

Locating cost of

Holding cost of

Producer 2 and DC 3

–

DC 1

–

DC 2

–

DC 3

–

DC 1 DC 2

Local transaction cost between

    ð2Þ ð2Þ 2 L1 y1 ¼ ð0:3Þ  y1     ð2Þ ð2Þ 2 ¼ ð0:3Þ  y2 L2 y2     ð2Þ ð2Þ 2 L3 y3 ¼ ð1:3Þ  y3

  ð1Þ ð1Þ H 1 q1 ¼ ð0:3=2Þ  q1   ð1Þ ð1Þ ¼ ð0:35=2Þ  q2 H 2 q2

DC 3

–

DC 1 and retailer 1

   2 ð1Þ ð1Þ ð1Þ T 11 s11 ¼ ð0:3Þ  s11 þ ð0:35Þ  s11    2 ð1Þ ð1Þ ð1Þ T 12 s12 ¼ ð0:35Þ  s12 þ ð0:3Þ  s12    2 ð1Þ ð1Þ ð1Þ T 21 s21 ¼ ð0:4Þ  s21 þ ð0:2Þ  s21    2 ð1Þ ð1Þ ð1Þ T 22 s22 ¼ ð0:3Þ  s22 þ ð0:2Þ  s22

DC 1 and retailer 2 DC 2 and retailer 1 DC 2 and retailer 2 DC 3 and retailer 1

–

DC 3 and retailer 2

–

  ð2Þ ð2Þ H1 q1 ¼ ð0:2=2Þ  q1   ð2Þ ð2Þ ¼ ð0:3=2Þ  q2 H2 q2   ð2Þ ð2Þ ¼ ð0:7=2Þ  q3 H3 q3    2 ð2Þ ð2Þ T 11 s11 ¼ ð0:2Þ  s11    2 ð2Þ ð2Þ T 12 s12 ¼ ð0:3Þ  s12    2 ð2Þ ð2Þ T 21 s21 ¼ ð0:3Þ  s21    2 ð2Þ ð2Þ T 22 s22 ¼ ð0:1Þ  s22    2 ð2Þ ð2Þ T 31 s31 ¼ ð0:8Þ  s31    2 ð2Þ ð2Þ T 32 s31 ¼ ð0:9Þ  s32

ð2Þ

þ ð0:2Þ  s11

ð2Þ

þ ð0:1Þ  s12

ð2Þ

þ ð0:1Þ  s21

ð2Þ

þ ð0:1Þ  s22

ð2Þ

þ ð0:9Þ  s31

ð2Þ

þ ð0:9Þ  s32

Table 2 Computational results of numerical example. Variables Amounts of good shipped between

Producer Producer Producer Producer Producer Producer

Lagrange multipliers of the first constraints (c1, c2)

For DC 1 For DC 2 For DC 3

Lagrange multipliers of the second constraints (b1, b2)

For DC 1 For DC 2 For DC 3 DC 1 DC 2 DC 3

Located DCs capacities

1 1 2 2 3 3

1 1 1 2 2 2

Amounts of good shipped between

DC DC DC DC DC DC

and and and and and and

Price of goods in

Retailer 1 Retailer 2

the demand function at each of the demand market was depended to both networks’ prices.

and and and and and and

DC DC DC DC DC DC

1 2 3 1 2 3

Existing network

New network

6164.5 4842.1 – 3698.9 5649.2 –

3440.4 3438.6 1176.1 3440.6 4011.7 1176.1

3699.3 3390.1 –

6193.2 6877.6 5881.8

0 0 – – – – retailer retailer retailer retailer retailer retailer

1 2 1 2 1 2

4128.6 4470.2 3763.5 6881 7450.2 2352.2

4603.9 5259.5 3839.7 6651.6 – –

3698.7 3182.3 1325.3 6124.9 1118.9 1233.3

6462 7381.3

7672.9 8102.7

In the actual business, producers often vary their marketing actions, such as advertising and promotions. Such modifications

819

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Producers of Existing network

DCs of Existing network

1

1

2

2

price of existing chain in first retailer price of existing chain in second retailer price of new chain in first retailer price of new chain in second retailer

Existing supply chain

1

1

2

2

8000 7000 6000

Price

New supply chain 1

1

9000

Participated Retailers Markets

5000 4000 3000

2

2000

2

3

Producers of New network

1000

DCs of New network

0

0.9

0.8

0.6

0.4

0.2

0.05

0.005

Amount of parameter d

Fig. 3. Supply chain networks of the numerical example. Fig. 4. The sensitivities of the optimal prices with respect to cross-price effect parameter in both chains.

influence the parameter values of the demand function. Here we discuss the sensitivity analysis of the equilibrium prices, market shares, total cost, total income, and total profit of each chain with respect to parameters: b and d, which represent various marketing decisions. In this part, we study the behavior of the equilibrium prices, market shares, total cost, total income, and total profit of each chain with respect to the cross-price effect parameters, d. Here we explain the change of the equilibrium by representing several numerical examples (we use the above solved example (b = 1) but vary the amount of parameter d in them). Table 3 represents the change of the optimal prices and market shares with respect to cross-price effect parameter. Fig. 4 shows the sensitivities of the optimal prices with respect to cross-price effect parameter in both chains. In this graph, we show that cross-price effect has a positive impact on equilibrium prices. As shown in Table 3, by decreasing the amount of parameter d, the total amount of market share in both chains decreases, so the incomes of both chains should decrease. By decreasing the market share of these chains, the total cost of these chains also decreases. In these examples the rate of reduction in incomes is more than the rate of the reduction in the total costs, so by decreasing the cross-price effect, the amount of profits in these chains decreases.

Table 4 represents the change of the optimal total cost, total income, and total profit with respect to cross-price effect parameter in both chains. Figs. 5 and 6 shows the sensitivities of the optimal total cost, total income, and total profits with respect to cross-price effect parameter in existing and new chains, respectively. The increase in customer demand as the overall level of service offered by the available facilities increases is called market expansion [1]. The increase in the level of service can come either as a result of new facilities being added or through design improvements to the existing facilities. The market expansion of existing chain with respect to cross-price effect is represented in Table 5 and graphically shown in Fig. 7. As shown in Fig. 7, by decreasing the amount of parameter d, the total amount of market expansion in existing chain decreases. In this part, we study the behavior of the equilibrium prices, market shares, total cost, total income, and total profit of each chain with respect to the self-price effect parameters, b. Here we explain the change of the equilibrium by representing several numerical examples (we use the above solved example (d = 0.8) but vary the amount of parameter b in them). Table 6 represents the change of the optimal prices and market shares with respect to self-price effect parameter.

Table 3 The change of the optimal prices and market shares with respect to cross-price effect parameter. Market share of new network in

Market share of existing chain in

Price of

Retailer 1

Retailer 2

Total

Retailer 1

Retailer 2

Total

Retailer 1

Retailer 2

6142.9

10540.47

16683.37

8443.61

11911.13

20,354.73

5635.32

9935.1

15,570.42

7742.94

11,170.8

18,913.74

4823.7

8930.9

13,754.6

6601.78

9946.42

16,548.2

4209.26

8130.1

12339.36

5711.18

8972.06

14,683.24

3733.68

7475.36

11,209.04

4995.12

8174.72

13,169.84

3445.43

7057.745

10503.175

4542.365

7519.3

12061.66

3368.42

6942.71

10,311.142

4417.948

7522.299

11,940.247

6462 7672.9 5968.9 7139.8 5162.5 6273.8 4529.9 5602.7 4018.9 5070.1 3694.6 4739.3 3605.3 4649.6

7381.3 8102.7 6892 7578.5 6086 6720.7 5447.5 6048.9 4927.3 5510.1 4594.9 5172 4503.1 5079.8

In

d

Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain

0.9 0.8 0.6 0.4 0.2 0.05 0.005

820

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Table 4 The change of the optimal total cost, total income, and total profits with respect to cross-price effect parameter. Total profit of

Total income of

Total cost of

d

New chain

Existing chain

New chain

Existing chain

New chain

Existing chain

51,016,123.67 44,480,213.08 34,780,828.69 28,044,382.89 23,183,112.1 20,378,583.53 19,646,442.05

56,905,231.689 49,223,988.166 37,831,601.37 29,912,371.132 24,175,185.624 20,178,653.299 19,966,692.551

132,540,123.67 115,528,213.08 90,284,828.69 72,761,382.892 60,120,112.104 52,831,583.539 50,929,442.051

142,482,231.68 123,205,988.16 94,615,601.37 74,746,371.132 60,354,185.624 51,332,653.299 49,801,692.551

81,524,000 71,048,000 55,504,000 44,717,000 36,937,000 32,453,000 31,283,000

85,577,000 73,982,000 56,784,000 44,834,000 36,179,000 31,154,000 29,835,000

market expansion of existing chain

Total income of existing chain

Total cost of existing chain

9000

Total profit of exsiting chain

160000000

8000

Market expansion

140000000 120000000 100000000 80000000 60000000

7000 6000 5000 4000 3000

40000000

2000

20000000

1000

0

0.9 0.8 0.6 0.4 0.2 0.05 0.005

0.9

0.8

0.6

0.4

0.2

0.05

0.005

0 0.9

Amount of parameter d

0.8

0.6

0.4

0.2

0.05

0.005

Amount of parameter d

Fig. 5. The sensitivities of the optimal total cost, total income, and total profits with respect to cross-price effect parameter in existing chain.

Fig. 7. The sensitivities of the market expansion with respect to cross-price effect parameter in existing chain.

Fig. 6. The sensitivities of the optimal total cost, total income, and total profits with respect to cross-price effect parameter in new chain.

In this graph, we show that self-price effect has a negative impact on equilibrium prices. As shown in Table 6, by increasing the amount of parameter b, the total amount of market share in both chains decreases, so the incomes of both chains should decrease. By decreasing the market share of these chains, the total cost of these chains also decreases. In these examples the rate of reduction in incomes is more than the rate of the reduction in the total cost, so by increasing the self-price effect, the amount of profits in these chains decreases. Table 7 represents the change of the optimal total cost, total income, and total profit with respect to self-price effect parameter in both chains. Figs. 9 and 10 shows the sensitivities of the optimal total cost, total income, and total profits with respect to self-price effect parameter in existing and new chains, respectively. The market share of existing and new chains with respect to different self-price effect is graphically shown in Fig. 11. As you can see, by decreasing the self-price effect of each chain, the market share of that chain increases. Now, we derive several managerial implications from these sensitivity analyses:

Fig. 8 shows the sensitivities of the optimal prices with respect to self-price effect parameter in both chains.

(1) For increasing the market share, the manager of both chains should try to reduce the self-price effect parameter by developing brand loyalty for their product.

Total cost of new chain Total income of new chain Total profit of new chain 140000000 120000000 100000000 80000000 60000000 40000000 20000000 0

0.9

0.8

0.6

0.4

0.2

0.05

0.005

amount of parameter d

Table 5 The sensitivities of the market expansion with respect to cross-price effect parameter in existing chain. 0.005 28.947

0.05 150.362

0.2 1258.54

0.4 2771.94

0.6 4636.9

0.8 7002.44

0.9 8443.43

d Market expansion of existing chain

821

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822 Table 6 The change of the optimal prices and market shares with respect to self-price effect parameter. Market share of new network in

Market share of existing chain in

Price of

Retailer 1

Retailer 2

Total

Retailer 1

Retailer 2

Total

Retailer 1

Retailer 2

7280.24

11719.76

19,000

8719.76

12,280.24

21,000

6552.304

10,944.98

17,497.28

8705.875

12,246.5

20,952.375

6198.88

10564.16

16763.04

8386.78

11886.25

20,273.03

5906.105

10238.78

16,144.885

8052.83

11,872.33

19,925.16

5636.12

9935.1

15,570.42

7742.94

11,170.8

18,913.74

7900.2 8799.9 6765.7 8070.9 6428.2 7715.2 6190.2 7416.9 5968.9 7139.8

8829.8 9180.1 7690.8 8479.6 7359.1 8136.8 7117.5 7847.6 6892 7578.5

Price of existing chain in first retailer price of existing chain in second chain

price of new chain in first retailer price of new chain in second retailer

200000000

9000

180000000

8000

160000000

7000

140000000

6000

120000000

5000

100000000

4000

80000000

3000

60000000

2000

40000000

1000

20000000 0.8

0.85

0.9

0.95

0

1

0.8

0.85

amount of parameter b Fig. 8. The sensitivities of the optimal prices with respect to self-price effect parameter in both chains.

(2) For increasing the market share, the manager of both chains should employ marketing activities that lead to more frequent brand switching (causing d to increase). (3) Reducing the self-price effect parameter by developing brand loyalty for product can lead to more profit in these chains, if the rate of increasing in incomes is more than the rate of the increasing in the total cost. (4) Increasing the cross-price effect parameter by employing marketing activities can lead to more profit in these chains, if the rate of increasing in incomes is more than the rate of the increasing in the total cost. (5) Increasing the cross-price effect parameter always leads to more market expansion in existing chain.

9. Conclusion This paper has developed an equilibrium model for strategic designing of a centralized supply chain network in markets with

b

Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain Existing chain New chain

0.8 0.85 0.9 0.95 1.0

Total cost of existing chain Total income of existing chain Total profit of existing chain

10000

0

In

0.9

0.95

1

Amount of parameter b Fig. 9. The sensitivities of the optimal total cost, total income, and total profits with respect to self-price effect parameter in existing chain.

a rival chain. Prices associated with retailers of these networks are endogenous, as are the product shipments and consumption flows. We also derive qualitative properties of the equilibrium pattern such as the existence and the uniqueness of a solution. The modified projection method was proposed for the computation of the equilibrium prices and product shipments. Finally one illustrative example was considered in the computations and we discuss the sensitivity analysis of the equilibrium prices, market shares, total cost, total income, and total profit of each chain with respect to parameters: b and d, which represent various marketing decisions. We conclude that for increasing the market share, the manager of both chains should try to reduce the self-price effect parameter by developing brand loyalty for their product and should employ marketing activities that lead to more frequent brand switching. Reducing the self-price effect parameter and increasing the crossprice effect parameter can lead to more profit in these chains, if the rate of increasing in incomes is more than the rate of the

Table 7 The change of the optimal total cost, total income, and total profits with respect to self-price effect parameter. Total profit of

Total income of

Total cost of

b

New chain

Existing chain

New chain

Existing chain

New chain

Existing chain

66,033,952.8 56,063,042.8 51,491,056.1 47,793,840.1 44,485,924.9

86,237,911.1 62,424,720.7 56,470,001.6 55,189,937 49,223,988.2

171,653,952.8 145,692,042.8 133,784,056.1 124,154,840.1 115,533,924.9

177,319,911.1 153,086,720.7 141,384,001.6 134,349,937 123,205,988.2

105,620,000 89,629,000 82,293,000 76,361,000 71,048,000

91,082,000 90,662,000 84,914,000 79,160,000 73,982,000

0.8 0.85 0.9 0.95 1.0

822

S. Rezapour, R.Z. Farahani / Advances in Engineering Software 41 (2010) 810–822

Total cost of new chain Total income of new chain Total profit of new chain 200000000 180000000 160000000 140000000 120000000 100000000 80000000 60000000 40000000 20000000 0

0.8

0.85

0.9

0.95

1

Amount of parameter b Fig. 10. The sensitivities of the optimal total cost, total income, and total profits with respect to self-price effect parameter in new chain.

Market share of existing chain Market share of new chain 25000

Market share

20000 15000 10000 5000 0 0.8

0.85

0.9

0.95

1

Amount of parameter b Fig. 11. The sensitivities of the market share with respect to self-price effect parameter in both chains.

increasing in the total cost. For future research, this model can be adapted to include random demands at the retailer level. Also this model can be extended to be multi-criteria. This method can be used for designing the networks of reverse, close loop, and open loop SCs. One can extend this model to include more competitor SCs in the markets.

References [1] Aboolian R, Berman O, Krass D. Competitive facility location and design problem. Eur J Oper Res 2007;182:40–62. [2] Ailawadi KL. The retail power-performance conundrum: what have we learned? J Retailing 2001;77(3):299–318. [3] Anderson EJ, Bao Y. Price competition with integrated and decentralized supply chains. Eur J Oper Res 2009;200:227–34. [4] Bawa K, Shoemaker RW. The coupon-prone consumer: some findings based on purchase behavior across product classes. J Mark 1987;51(4):99–110. [5] Bernstein F, Federgruen A. Pricing and replenishment strategies in a distribution system with competing retailers. Oper Res 2003;51(3):409–26.

[6] Bernstein F, Federgruen A. A general equilibrium model for industries with price and service competition. Oper Res 2004;52(6):868–86. [7] Bernstein F, Federgruen A. Decentralized supply chain with competing retailers under demand uncertainty. Manage Sci 2005;51(1):18–29. [8] Bernstein F, Federgruen A. Coordination mechanisms for supply chains. Manuf Serv Oper Manage 2007;9(3):242–62. [9] Boyaci T, Gallego G. Supply chain coordination in a market with customer service competition. Prod Oper Manage 2004;13(1):3–22. [10] Carr C, Karmarkar US. Competition in multi-echelon assembly supply chain. Manage Sci 2005;51(1):45–59. [11] Chawla S, Rajan U, Ravi R, Sinha A. Min-max payoffs in a two-player location game. Oper Res Lett 2006;34:499–507. [12] Corbett CJ, Karmarkar US. Competition and structure serial supply chains with deterministic demand. Manage Sci 2001;47:966–78. [13] Dong J, Zhang D, Nagurney A. A supply chain network equilibrium model with random demands. Eur J Oper Res 2004;156:194–212. [14] Dong J, Zhang D, Yan H, Nagurney A. Multi-tiered supply chain networks: multi-criteria decision-making under uncertainty. Ann Oper Res 2005;135:155–78. [15] Fernandez J, Pelegrin B, Plastria F, Toth B. Solving a huff-like competitive location and design model for profit maximization in the plane. Eur J Oper Res 2007;179:1274–87. [16] Hanssens DM, Parsons LJ, Schultz RL. Market response models: econometric and time series analysis. 2nd ed. Boston, MA: Kluwer Academic Publishers; 2001. [17] Kinderlehrer D, Stampacchia G. An introduction to variational inequalities and their application. New York: Academic Press; 1980. [18] Kurata H, Yao DQ, Liu JJ. Pricing policies under direct vs. indirect channel competition and national vs. store brand competition. Eur J Oper Res 2007;180:262–81. [19] Majumder P, Srinivasan A. Leadership and competition in network supply chains. Manage Sci 2008;54:1189–204. [20] McGarvey RG, Cavalier TM. Constrained location of competitive facilities in the plane. Comput Oper Res 2005;32:359–78. [21] Meng Q, Huang YK, Cheu RL. A note on supply chain network equilibrium models. Transport Res Part E 2007;43:60–71. [22] Mishra BK, Raghunathan S. Retailer vs. vender managed inventory and brand competition. Manage Sci 2004;50:445–57. [23] Nagurney A. Network economics: a variational inequality approach. 2nd and revised ed. Dordrecht, 1999: Kluwer Academic Publishers; 1999. [24] Nagurney A, Dong J, Zhang D. Multicriteria spatial price networks: statics and dynamics. In: Daniele P, Maugeri A, Giannessi F, editors. Equilibrium problems and variational models. Dordrecht: Kluwer Academic Publishers; 2001. p. 1. [25] Nagurney A, Dong J, Zhang D. A supply chain network equilibrium model. Transport Res E 2002;38:281–303. [26] Nagurney A, Matsypura D. Global supply chain network dynamics with multicriteria decision-making under risk and uncertainty. Transport Res E 2005;41:585–612. [27] Nagurney A, Toyasaki T. Reserve supply chain management and electronic waste recycling: a multi-tiered network equilibrium framework for e-cycling. Transport Res E 2005;41:1–28. [28] Nagurney A. On the relationship between supply chain and transportation network equilibria: a supernetwork equivalence with computations. Transport Res E 2006;42:293–316. [29] Narayanan VG, Raman A. Agency costs in a supply chain demand uncertainty and price equilibrium. Manage Sci 2005;51(1):120–32. [30] Plastria F, Vanhaverbeke L. Discrete models for competitive location with foresight. Comput Oper Res 2008;35:683–700. [31] Shen ZJ. Integrated supply chain design models: a survey and future research directions. J Ind Manage Opt 2007;3(1):1–27. [32] Shiode S, Drezner Z. A competitive facility location problem on a tree network with stochastic weights. Eur J Oper Res 2003;149:47–52. [33] Simchi-Levi D, Kaminsky P. Managing the supply chain: the definitive guide for the business professional. Boston: Irwin McGraw-Hill; 2004. [34] Swaminathan JM, Tayur SR. Models for supply chain in e-business. Manage Sci 2003;49(10):1387–406. [35] Taylor DA. Supply chains: a management guides. Boston: Pearson Education; 2003. [36] Tsay AA, Agrawal N. Channel dynamics under price and service competition. Manuf Serv Oper Manage 2000;2(4):372–91. [37] Xiao T, Yang D. Price and service competition of supply chains with risk-averse retailers under demand uncertainty. Int J Prod Econ 2008;114(1):187–200. [38] Xiao T, Qi X. Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer and two competing retailers. Omega 2008;36:741–53. [39] Zhang D. A network economic model for supply chain versus supply chain competing. Omega 2006;34:283–95. [40] Zhang L, Rushton G. Optimizing the size and locations of facilities in competitive multi-site service systems. Comput Oper Res 2008;35:327–38.