Coalition or decentralization: a game-theoretic analysis of a three-echelon supply chain network.
Mahdiraji, Hannan Amoozad ; Govindan, Kannan ; Zavadskas, Edmundas Kazimieras 等
Introduction
More than 50 years ago, Forrester (1958) introduced the elements of
a theory that today is called supply chain management (SCM). The concept
of the supply chain means that many experts believe that competition is
transferred from companies to chains. SCM is extensive enough, and
therefore large international corporations such as Cisco, Dell Computer,
Gillette, Kodak, LEGO, Motorola, Sony, 3M, Xerox and Wal-Mart
implemented it in the past decade. Moreover, international consultancy
firms like IDM business Consulting Services, A.T. Kearney, Cap Gemini,
etc. have adopted SCM as an important business area. Also, a large
number of universities and business schools have included SCM courses in
their curricula. Many scholars and experts gave different definitions
for SCM that depend on their viewpoints and attitudes (Walker 2005). The
role and importance of supply chain management have faced a number of
challenges and problems. Although a comprehensive model dealing with the
issues of the supply chain has not been explained, we have to indicate
that questions such as reviewing the theoretical foundations of
information systems, marketing, financial management, logistical and
organizational relations have been considered by many researchers (Wang
et al. 2007). There are many challenges that latent in the concept of
the supply chain. The decisions made in SCM are mainly about the flows
between chain stages. Therefore, many scholars express challenges and
problems, and SCM have tried to answer them (Chandra, Kamrani 2004;
Chopra, Miendel 2007; Simchi-Levi et al. 2004; Wisner et al. 2008). The
objective of supply chain management is to improve various activities
and components to increase the overall benefits of the supply chain
system. Many decisions are made at a different level of the supply
chain, which includes detailed and strategic decisions. Planning
important decisions in a multi-echelon supply chain will affect all
levels and the SC as a whole (Stadtler, Kilger 2007). If each level of
the supply chain makes their inventory, the decision on pricing and
advertising without considering other levels as well as the bullwhip
effect will occur and the advantage of supply chain competiveness
decrease (Lee et al. 1997). Between the components and different levels
of the supply chain, in order to achieve the overall objectives, many
contradictions may occur, including that these disorders, over time, may
result in the decreased strength and competitiveness of the supply
chain. Such conflicts, like marketing costs (advertising), pricing and
inventory decisions can occur during the life cycle of the supply chain.
For avoiding such loss in the SC, many coordination mechanisms have been
introduced in recent researches. There are many possible interactive
coordination mechanisms that can occur between different levels
(Esmaeili et al. 2008). A large part of these mechanisms are based on a
game theory approach. The game theory is concerned with the actions of
decision makers who are conscious that their actions affect each other.
The game theory approach is an appropriate tool for collaboration in the
supply chain. Beside contradiction with decision variables, different
levels of the supply chain may decide on acting independently or, in
some cases, trying to integrate with some other levels for gaining more
advantages. In this case, the main challenge is how to act in
multi-echelon supply chains taking into account the levels involved.
Making a choice independently or integrating with some or all levels
will be a critical decision, and therefore affects the overall profit of
the chain. This decision will have an influence on prices, inventory,
lot sizes and costs which will finally change the overall profit of the
supply chain. For solving this problem and finding a suitable answer, a
three- echelon unlimited supply chain with S suppliers, M manufacturers
and K retailers has been considered. In addition, decisions on pricing,
inventory and advertising are included as three main decision variables
in the proposed models. By using the definition of the Nash equilibrium
for continuous problems, the best responses for each level of the supply
chain in decentralization and integration situations are identified and
used in a simulated supply chain. By comparing the obtained results, the
best decisions are illustrated. The remainder of this paper is organized
as follows. First, supply chain management, the game theory and the Nash
equilibrium are introduced and the classification of researches on
similar topics is illustrated. Next, the assumptions and notations of
our proposed models are presented and the payoff functions of each
player in all situations (integration or decentralization) are designed.
After, the best response of each player is calculated. Finally, by using
the proposed models in the simulated SC, the results are compared and
the final conclusion is proposed.
1. Literature review
This section includes the basics and concepts of the game theory,
different types of coordination contracts and reviews similar
researches. As a definition, the supply chain consists of all parties
involved, directly, or indirectly, in fulfilling a customer request
(Chopra, Meindel 2007) and mentioning all activities performed until a
raw material is delivered as the final good to a customer (Gumus, Guneri
2007). These activities and functions include new product development,
marketing, operations, distributions, financing and customer services. A
typical supply chain may involve a variety of stages such as customers,
retailers, wholesalers, distributors, manufacturers and raw material
suppliers (Chopra, Meindel 2007). Between the components and different
levels of the supply chain, in order to achieve the overall objectives,
many contradictions may occur, the contradictions that these disorders
take place over time result in the decreased strength and
competitiveness of the supply chain. One of the main tools for solving
the problem in the before mentioned situation are the game theory
approach. The essential elements of the game are players, actions,
payoffs and information (Chen 2009). These are collectively known as the
rules of the game, and the objective of the modeller is to describe the
situation in terms of the rules of the game so as to explain what will
happen in that situation. Trying to maximize their payoffs, the players
will devise plans known as strategies that pick actions depending on the
information that has arrived at each moment. The combination of
strategies chosen by each player is known as equilibrium while given
which modeller can see what actions come out of the conjunction of all
players' plans, which tells him the outcome of the game.
While information transaction is not possible between different
players (different layers of the SC), in such situations and by
considering the Nash definition, each player will stimulate competitor
believes or best responses, and, when these believes are correct, the
Nash equilibrium will occur (Osborne 2004). In the given two-player
game, the best responses are defined as (1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where [B.sub.i] stands for the best response for player i,
([S.sub.i], [S.sub.-i])--for the strategy chosen by player i & -i,
(i-i)--for two players of the game, ([S.sup.*.sub.i],[S.sup.*.sub.-1]
for the best strategies for players, [U.sub.i] ([S.sub.i],
[S.sub.-i])--for utility or payoff when players choose ([S.sub.i],
[S.sub.-i]) as their final decisions. By considering the definition of
the best response and continuous and discrete payoff functions, the Nash
equilibrium is computable by the derivation of utility or the payoff
function of each player regarding a specific decision variable. The Nash
game, definition and equilibrium are used in famous situations such as
the Bertrand duopoly model, Cornet model of duopoly, the final offer
arbitration and the problem of commons (Gibbons 2002).
Game theoretic analysis for supply chain networks includes a wide
range of research. Some scientists have focused on the use of the Nash
equilibrium in supply chain coordination by applying a profit sharing
contract (Feng et al. 2007; Jiazhen, Qin 2008; Feng 2008; Ying et al.
2007; Jaber et al. 2006; Bai, Wang 2008; Xu, Zhong 2011; Liu, Zhang
2006; Wang et al. 2009). The other contracts of supply chain
coordination are presented in Table 1 (Govindan, Nicoleta 2011).
Besides using coordination contracts, Nash and Stackelberg games
also can be used for coordinating the SC in complete situations with a
determined demand function (Leng, Parlar 2010; Arda, Hennet 2005).
Besides, Zandi et al. (2012) proposed a strategic theoretic approach of
a cooperative game to market segmentation. For a symmetric supply chain,
some researchers gave closed-form expressions of unique equilibrium
(Adida, Demiguel 2011). In addition, equilibrium for the Shapley value
and Eliasberg model for coordination and cooperation problems in the SC
are performable (Zhao et al. 2010; Leng, Zhu 2009). In some cases, while
incomplete information on decisions and payoffs dominate game
conditions, Bayesian--Nash Games are used (Wang, Zhao 2009; Cachon,
Lariviere 1999). Other optimization tools such as the queuing theory, a
Markov chain, backward induction, stochastic programming and a genetic
algorithm for solving coordination and cooperation problems in the
supply chain, mostly in the situations of incomplete information games,
are also performable (Cachon, Kok 2010; Hennet, Arda 2008; Stein,
Ginevicius 2010; Kaviani et al. 2011; Gupta, Weerawat 2006).
Considering uncertainty in the game theory approach to solving
different dilemmas is really noticeable, which eventuates to
multi-criteria decision making (MCDM) or multi-objective decision making
(MODM) usage in the game theory. Therefore, in 2005, Peldschus and
Zavadskas proposed a Fuzzy matrix game by multi-criteria modelling of
decision making in engineering projects (Peldschus, Zavadskas 2005).
Three years later, a new logarithmic normalization method in the game
theory was suggested (Zavadskas, Turskis 2008). Same time after, an
overview of MCDM methods and its application to economics based on the
game theory was figured by the same authors (Zavadskas, Turskis 2011).
Recent findings have illustrated the application of the MODM and game
theory approach where Peldschus and Zavadskas proposed an equilibrium
approach to construction processes by multi-objective decision making
for construction projects (Peldschus, Zavadskas 2012). As some of the
aforementioned researches indicate, in practice, the application of the
game theory in the fields of engineering and construction has further
developed (Zavadskas et al. 2004; Peldschus 2008; Kaplinski,
Tamosaitiene 2010; Peldschus et al. 2010).
This research mainly considers the use and application of the game
theory, especially non-cooperative Nash games, in supply chain
management. In Nash games, all levels of the SC or each player in the
game act simultaneously with complete information about the game,
players and equal power. Most of the games used in the SC are
non-repetitive, with two or finally three levels and limited to one or
finally two members at each level. Inventory, pricing and marketing
policies are included in researches. Our paper reflects on inventory
with shortage (backlog), incremental production, a nonlinear cost
production function for the manufacturer, a nonlinear demand function,
semi integrated (coalition) games and an unlimited supply chain, which
differs this research from others.
2. Basic model
A three-echelon supply network made of K retailers, M manufacturers
and S upstream suppliers where K [greater than or equal to] 1, M
[greater than or equal to] 1, S [greater than or equal to] 1, (Fig. 1)
has been considered. This network produces, distributes and sells
multiple products to the end customers (i.e. consumers):
[FIGURE 1 OMITTED]
In this supply network, we assume that all agents are able to
decide independently or, in some cases, integrate with the agents of
other levels of the network. The network follows a make-to-order pull
system in which orders first pass from retailers to manufacturers, and
then from manufacturers to suppliers. We assume that all agents have
complete information. In addition, shortages (and hence stock out) are
allowed for manufacturers. The cost of shortages will be considered for
the manufacturer during a shortage period, but no shortage is assumed
for retailers and suppliers. Consumer demand for the product depends on
both the retail price and marketing cost used for product advertisement.
Following literature (Lee 1993, Esmaeili et al. 2008; Jia et al. 2013),
we acknowledge a deterministic non-linear form of a consumer demand
function as given by (2):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where [D.sub.n] presents demand for product [sub.n], [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the selling price of
product n by retailer [sub.[gamma]], [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] points to marketing cost for product [sub.n, k a,
b,], are all strictly positive constants to represent the corresponding
coefficient in the demand function.
The total inventory cost of each manufacturer includes its shortage
cost and inventory holding cost and it is computable by (3), in which
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] stands for holding
cost estimated by the manufacturer, ([[gamma].sub.n] = 1 -
[D.sub.n]/[PC.sub.n]) denotes production rate, [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII] is production quantity, [B.sub.n] represents
manufacturer's shortage, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII] is the cost of manufacturer's shortage, [PC.sub.n] is
production capacity provided by the manufacturer (Jia et al. 2013):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
Notice that the model presented in (3) follows literature (Oganezov
2006; Wang, Tang 2009; Chakrabortty et al. 2010; Chang 2008; Pentico et
al. 2009; Jia et al. 2013).
As a remark, the final retail selling price and retail marketing
cost, production quantity and shortage quantity, besides the mass
production price for the manufacturer, and finally the raw material
price for suppliers are the main decision variables of this research.
The cost of unit production is nonlinear function [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] decreasing in demand and having
coefficients u > 0, [gamma] > 0 (Bazaraa et al. 1993). Regarding
this supply network, the scenario when each manufacturer supplies only a
specific product to a specific retailer n = r = m is considered.
However, upstream suppliers sell their raw materials to any manufacturer
when needed. This matches the scenario in many industries such as
fashion apparel in which many retailers (fashion retail brands) have
designed garment factories (manufacturers) to produce solely for them
while since there are relatively few upstream suppliers (for fabrics or
components such as zippers) in the market, most manufacturers will
source from the well-established ones (e.g. YKK is the zipper supplier
to all kinds of manufacturers and all kinds of retail brands). Following
the standard norm discussed in literature, we assume, throughout this
paper, that each player at any level of the supply network is fully
rational. To enhance the presentation, the notation employed in Table 2
of this paper is summarized.
3. Modelling/payoff functions
This section presents the analytical payoff function for each agent
of the supply network. Two different approaches are considered. First,
trio players, including suppliers, manufacturers and retailers, decide
independently without any coordination approach. Second, two parts of
the SC merge each other and act as one player to confront the
demonstration of the third party. In both approaches, mathematical
models are illustrated and the best responses of each player are
calculated based on the Nash equilibrium.
3.1. The payoff function and model of the retailer
For retailer r, it confronts holding and setup costs as well as
purchasing cost from getting supply from the manufacturer. In addition,
any retailer should have a positive margin to participate in the supply
network. The income of each retailer involves revenue achieved by
selling the final goods to the final customer. Considering the details
above, the payoff function of the retailer and a related optimization
model are shown in (4) (Jaafarnejad et al. 2012; Jia et al. 2013):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where the first constraint implies that the final retail selling
price is greater than the wholesale price paid to the manufacturer, and
the second and third constraints guarantee that demand should not be
negative or greater than production capacity. Decision variables
associated with (4) in the supply network include [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII].
3.2. The payoff function and model of the manufacturer
Manufacturer n confronts holding, setup, ordering and shortage
costs, purchasing cost and production cost. The manufacturer receives
revenue with the wholesale price. The payoff function of manufacturers
and model constraints are shown in (5) where the first constraint
implies that the wholesale price offered by the manufacturer to the
retailer is greater than that paid for raw materials and where the
second and third constraints ensure that demand should not be negative
or greater than the available production capacity (Jia et al. 2013).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
3.3. The payoff function and model of the supplier
A supplier has to bear holding, setup and purchasing costs. In
return, every supplier will gain revenue by selling raw materials to
manufacturers, and the total revenue depends on the amount of respective
production used by the manufacturer. By considering the above indicated
points, the payoff function of the supplier and its constraints are
shown in (6), where the first constraint implies that the selling price
of the raw material to the manufacturer should be greater than the
procurement cost of acquiring the raw material by the supplier, and the
second constraint guarantees that demand should not be negative
(Jaafarnejad et al. 2012; Jia et al. 2013):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
3.4. Channel integration
This section proposes that two levels of the SC make a joint
venture to confront the leadership of the third party. We have
considered two integration scenarios: manufacturer-retailer and
supplier-manufacturer. In the first case, inventory/holding cost issues
diminish to a single warehouse and the retailer does not have to enforce
marketing effort any more. The second one brings similar benefits. Under
each scenario, all parties involved simultaneously set their strategies.
3.4.1. Integration function (M-R Nash game) of the
manufacturer--retailer
Our research is aimed at finding the best way of two-agent
integration in the supply network to achieve the Nash equilibrium with
the largest profit. As a remark, it is known that strategic partnership
in a supply chain is most easily achievable and commonly seen in the
case between two agents. Multi-agent integration (more than two) is
relatively rare and usually done by involving a third part supply chain
coordinator. In this paper, we confine ourselves to two-agent
integration as commonly observed in the real world and assumed in the
mainstream literature.
To achieve this goal, under our analysis, three options are
possible for the three-echelon supply network: manufacturer- retailer
integration (MR), supplier--manufacturer integration (SM) and
supplier--retailer integration (SR). The purpose of this paper is to
find the best integration mode among these three.
By vertically integrating manufacturers with retailers, the number
of manufacturers is equal to the number of retailers, and the revenue of
the MR pair will be the revenue generated by the retail selling price
for the product sold. The incurred costs will include production costs,
shortage costs, setup costs and holding costs. We have to notice that
holding cost at this level includes one warehouse between manufacturers
and retailers. In addition, marketing cost will not occur while
manufacturers and retailers are bonded, because these two members are
vertically integrated and simply controlled by one level in a
centralized manner. A remark that marketing in our research is based on
the costs the retailer spends to sell product n of the related
manufacturer can be made. These costs, for example, include allocating a
suitable stage in a retailer's shop for representing the product.
However, when M and R integrate, this item will be eliminated because
they act as one. Mathematically, if we make the integrated function of M
and R, marketing cost write-off from the equation. By considering the
above rules, the final payoff function of the MR Player is described as
(7):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
Decision variables for this function are [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII] and constraints insist that demand should not
be negative. The objective function maximizes the manufacturer and
retailer when both act as one player. In this case, the overall income
from selling the final product to costumers should be maximized while
production, holding, stock out and purchasing costs should be minimized.
As they act as one player, only a single warehouse is considered and
shared by both for the final products.
3.4.2. Integration function (S-M Nash game) of the
supplier--manufacturer
While suppliers and manufacturers are integrating, their coalition
will affect their costs and benefits. In this situation, the only income
will supply from the mass price from manufacturers to retailers. On the
other hand, the costs of a new integrated level will include production
costs, shortage costs, setup costs and holding costs. By considering the
above rules, the final payoff function for the MR Player is described as
(8):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
Decision variables for this function are [MATHEMATICAL EXPRESSION
NOT REPRODUCIBLE IN ASCII] and constraints insist that demand should not
be negative. The objective function maximizes the manufacturer and
supplier when both act as one player. In this case, the overall income
from the selling product to retailers should be maximized while
production, holding, stock out and purchasing costs should be minimized.
3.5. Best responses
Based on the Nash definition of equilibrium, the best responses of
each player should be estimated by others. As the objective function of
each player is a nonlinear mathematical model, the best responses of
each player, due to its decision variable, are calculated by partial
differentiation and the first derivative. By considering the best
response definition, continuous and discrete payoff functions, Nash
equilibrium N(G) is computable by the derivation of utility or the
payoff function of each player [U.sub.i] ([S.sub.i], [S.sub.-i]) = f
([S.sub.i], [S.sub.-i]) regarding a specific decision variable. The best
response for the discrete function is calculated by (9) and that for
continuous payoff functions is illustrated in (10) (Rasmusen 2005):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)
3.5.1. Best responses for the decentralized game
Each player in three-level supply chains will make the best
decision when playing a game in the SC, i.e. they move simultaneously
and there is no Stackelberg leader. Make a remark that our research is
based on non-cooperative game situations; thus, in the game theory,
orders and demand follow the pull system, but pricing and other
agreements may not. For example, in Stackelberg games, leadership could
happen at any level of the SC, which contradicts the pull system. The
game theory is a pre-action approach and leads the players to order a
quantity which maximizes their and other levels. While SC information
and the demand function are completely accessible, any level enables to
model the SC profit function and make the best decision. By considering
the reasonable behaviour of each player and the Nash best response
principle, the best decisions on each player in the three-echelon SC
will be concluded by the derivation of the payoff function to decision
variables. The first order condition of each payoff function is used for
the best response, and the second condition is applied for concavity
analysis. By calculating the determinant of the Hessian matrix of each
player and with regard to its decision variables, it has been concluded
that all models are concave to their decision variables, and therefore
optimal solutions to the proposed models are definable. Table 3
represents the best response of each player if the Nash principle is
used by calculating the first order condition of the payoff functions of
each player regarding their decision variables. The details of the
results presented in the table are mentioned in Appendixes 1 to 3.
[TABLE 3 OMITTED]
3.5.2. Best MR responses
In this situation, two players or two levels of the supply chain
are considered, i.e. suppliers and the MR level. Suppliers are similar
to the decentralized model with the same best response. For a MR Player
(it is assumed that MR becomes one unit), by a change in the payoff
function, the best responses will vary as (17). We have to mention that
integration affects the relation between manufacturers and retailers;
hence, mass price from the manufacturer to the retailer as well as
marketing cost will be eliminated from the final results: [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
3.5.3. Best SM responses
In this case, two players or two levels of the supply chain are
taken into account, i.e. retailers and the SM level. Retailers are
similar to the decentralized model with the same best response. For a SM
Player, by a change in the payoff function, the best responses will vary
as (18). We have to mention that integration affects the relation
between manufacturers and suppliers, and therefore mass price from the
supplier to the manufacturer will be eliminated from the final results.
In all situations, the demand function and production rate equality
always affect the obtained results:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)
4. Numerical example
With reference to the methodology of our research and to the
evaluated results, the verification and validation of the investigated
outputs have been analysed applying to a numerical example of a
hypothetical supply chain. Due to a lack of numerical and historical
information, the design of the experimental approach has been performed
to produce suitable data. Each experiment has been assessed by the
proposed models, and the total profit of the SC has bred out based on
coalition or decentralization circumstances. Following sensitivity
analysis, the overall SC profits of each method have been compared, and
finally, an appropriate approach has been suggested. Through a numerical
experiment, 17 design cases (through the performance of experiments)
have been developed, and the total profit of the supply chain under each
scenario has been evaluated. Statistical tests on the conducted 17
experiments have disclosed that the decentralized system performs
significantly worse than the integration of the supplier with the
manufacturer, whereas no significant difference in other combinations
can be observed. Jia et al. (2013) used the same numerical example based
on the Stackelberg game. They considered and compared three types of
leadership and concluded that retailer leadership would beget the
highest profit for the supply chain. Regarding the novelty of our
research consisting of channel integration and the best responses of the
proposed coalition, the above mentioned numerical example has been
solved by our new non-cooperative game theory approach and coalition vs.
decentralization. Conclusively, the profit achieved from the coalition
was higher than employing decentralization and leadership methods.
4.1. Definition of the problem
Considering the above mentioned models for sensitivity analysis and
leadership selection, the three-echelon supply chain, including 2
suppliers, 2 manufacturers and 2 retailers has been designed. Table 4
indicates the numerical amounts of parameters proposed in the supply
chain.
For sensitivity analysis, five constants, including a, b, g, k, u,
have been chosen. The lower and upper bounds of these five elements are
shown in Table 5.
While using design of experiment (DOE) and 2k-p experiments, and
including one central point in each block, 17 different tests have been
designed employing MINITAB 16.5 software (see Table 6).
4.2. Results
All experiments designed in Table 5 are performed for the
decentralization game, MR game and SM game. For the decentralized game,
the models have been coded, debugged and solved by LINGO 11. On the
other hand, for MR and SM games, multi nonlinear equations have been
solved using the Levenberg--Marquardt algorithm through fsolve
application in MATLAB software. The calculated overall profit of the
supply chain taking into account different types of problems is
displayed in Table 7.
By taking a two-paired test, the results of the three types of
decision making used in the supply chain have been compared. The
findings obtained employing MINITAB 16.5 software is presented in Table
8. It is obvious that coalition, integration and semi centralization
bring better results and profit to the SC.
The main effects of each situation considering five critical
elements are calculated using MINITAB 16.5 software and shown in the
figures of Table 9. In all models, Gama has the least noticeable effect,
whereas K, Alpha and Beta have the strongest one, which means that
changes in unit production cost will impact SC profit less than demand
oscillation. K and Beta directly affect the overall profit of the SC in
all three situations, and Alpha makes an impact inversely on
decentralized and SM games. In the MR semi-integrated game, U parameter
has direct effects on the overall profit of the SC. Thus, it can be
concluded that higher marketing cost increases SC profit as a lower
retailer price do for decentralized and SM games. Also, Beta parameter
does not exist in the MR game, and therefore marketing cost is not
included. As Beta parameter does not exist in the MR game, marketing
cost is not included. We also conclude that changes in unit production
cost will impact SC profit less than demand oscillation. In addition,
higher marketing cost increases SC profit as a lower retailer price does
for decentralized and SM games.
Conclusions
The conducted research has demonstrated coordination in
multi-echelon supply chains in which the non-cooperative game theory
approach is used as a suitable tool for coordinating pricing, inventory
and marketing expenditure policies in the unlimited three-level supply
chain, since a different level acts independently (decentralization) or,
in some cases, integrates with other levels (coalition). For this
matter, first of all, the objective function of each player and
constraints has been modelled. Next, the best response of each player
has been obtained based on the definition of the Nash equilibrium.
Finally, two scenarios of decentralization and coalition have been
modelled and analysed conducting an experiment. Among this, the
concavities of the proposed models have been obtained and sensitivity
analyses for different situation have been illustrated. A remark that a
nonlinear demand and production cost function, besides unlimited levels
and stock out situation taking into account the inventory system of the
manufacturer, has been considered. Through the numerical experiment, 17
design cases (performed experiments) have been developed and the total
profit of the supply chain under each scenario has been evaluated.
According to statistical tests on the above introduced 17 experiments,
the authors have found that the decentralized system performs
significantly worse than the integration of the supplier with the
manufacturer, whereas no significant difference can be observed for
other combinations. To sum up, coalition and semi integration
circumstances are more effective and beget higher profit for the total
supply chain. In addition, based on the considered SC, coalition between
manufacturers with suppliers is more effective than that between
retailers and manufacturers.
The above discussed situation and assumptions made in this paper
are the keys for future researches. Taking into account more levels in
the supply chain through distribution centres, warehouses or final
stores will lead researchers to a comprehensive model for coordinating
the supply chain in the future. For more accuracy and to provide the
proposed models performable in reality and industrial situations, the
case studies based on industrial information from the existing supply
chains should be gathered and examined. In conclusion, as the competency
of information and complete information shared at different levels of
the supply chain under real circumstances seem to be impossible, using
incomplete or imperfect game theory approaches such as the signalling
game or Nash Bayesian game will solve this problem and reach more
realistic options in the future.
doi:10.3846/16111699.2014.926289
Caption: Fig. 1. The three-echelon supply network
Appendixes
Appendix 1. Best retailer responses
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Appendix 2. Best manufacturer responses
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Appendix 3. Best supplier responses
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
References
Adida, E.; Demiguel, V. 2011. Supply chain competition with
multiple manufacturers and retailers, Operations Research 59(1):
156-172. http://dx.doi.org/10.1287/opre.1100.0863
Arda, Y.; Hennet, J. C. 2005. Supply chain coordination through
contract negotiation, in Conference on Decision and Control, 12-15
October, 2005, Seville, Spain, 4658-4684.
Bai, S.; Wang, D. 2008. Research on inventory game of supply chain
based on credit coordination mechanism, in IEEE International Conference
on Service Operations and Logistics, and Informatics, 12-15 October,
2008, Beijing, China, 3037-3042.
Bazaraa, M. S.; Sherali, H. D.; Shetty, C. M. 1993. Nonlinear
programming: theory and algorithms. New Jersey: John Wiley & Sons.
Cachon, G. P.; Kok, G. A. 2010. Competing manufacturers in a retail
supply chain: on contractual form and coordination, Management Science
56(3): 571-589. http://dx.doi.org/10.1287/mnsc.1090.1122
Cachon, G. P.; Lariviere, M. A. 1999. Capacity choice and
allocation: strategic behavior and supply chain performance, Management
Science 45(8): 1091-1108. http://dx.doi.org/10.1287/mnsc.45.8.1091
Chakrabortty, S.; Pal, M.; Nayak, P. K. 2010. Solution of
interval-valued manufacturing inventory models with shortage,
International Journal of Engineering and Applied Sciences 4(2): 357-365.
Chandra, C.; Kamrani, A. K. 2004. Mass customization: a supply
chain approach. New York: Kluwer Academic Publishers.
http://dx.doi.org/10.1007/978-1-4419-9015-0
Chang, H. C. 2008. A note on the EPQ model with shortages and
variable lead time, Information and Management Sciences 15(1): 61-67.
Chen, S. S. 2009. Comparing Cournat output and Bertrand price
doupoly game. Taiwan: Dahan Institute of Technology.
Chopra, S.; Meindel, P. 2007. Supply chain management. New York:
Prenticehall.
Esmaeili, M.; Aryanejad, M.; Zeephongsekul, P. 2008. A game theory
approach in seller-buyer supply chain, European Journal Of Operation
Research 195(2): 442-448. http://dx.doi.org/10.1016/-j.ejor.2008.02.026
Feng, S. X. 2008. Coordination of pricing decisions in multiple
product supply chains, in 4th International Conference on Wireless
Communications, Networking and Mobile Computing, 12-14 October, 2008,
Dalian, China, 16-22.
Feng, J. S.; Jia, L. M.; Jiao, H. L. 2007. The three stage supply
chain coordination by revenue sharing contracts, in IEEE International
Conference on Grey Systems and Intelligent Services, 18-20 November,
2007, Nanjing, China, 1216-1220.
http://dx.doi.org/10.1109/GSIS.2007.4443466
Forrester, J. W. 1958. Industrial dynamics--a major breakthrough
for decision makers, Harvard Business Review 36(4): 37-66.
Gibbons, R. 2002. A primer in game theory. Essex: Pearson Education
Limited.
Govindan, K.; Nicoleta, M. 2011. Overview and classification of
coordination contracts within forward and reverse supply chains,
Discussion Papers on Business and Economics 7: 5-32.
Gumus, T. A.; Guneri, F. A. 2007. Multi-echelon inventory
management in supply chains with uncertain demand and lead times:
literature review from an operational research perspective, in
Proceedings of the Institution of Mechanical Engineers, Part B: Journal
of Engineering Manufacture 221(10): 1553-1570.
http://dx.doi.org/10.1243/09544054JEM889
Gupta, D.; Weerawat, W. 2006. Supplier manufacturer coordination in
capacitated two stage supply chains, Elsevier: European Journal of
Operational Research 175(1): 67-89.
http://dx.doi.org/10.1016/j.ejor.2005.04.021
Hennet, J. C.; Arda, Y. 2008. Supply chain coordination: a game
theory approach, Engineering Applications of Artificial Intelligence
21(3): 399-405. http://dx.doi.org/10.1016/j.engappai.2007.10.003
Jaber, M. Y.; Osman, I. H.; Guiffrida, A. L. 2006. Coordinating a
three level supply chain with price discounts, price dependent demand,
and profit sharing, International Journal of Integrated Supply Chain
2(1): 28-49. http://dx.doi.org/10.1504/IJISM.2006.008337
Jaafarnejad, A.; Amoozad Mahdiraji, H.; Mohaghar, A.;
Modarresyazdi, M. 2012. Retailers leadership mathematical modeling in
unlimited three echelon supply chain: noncooperative game theory
approach, Archives Des Science 65(6): 81-90.
Jia, P.; Amoozad Mahdiraji, H.; Govindan, K.; Meidute, I. 2013.
Leadership selection in an unlimited three-echelon supply chain, Journal
of Business Economics and Management 14(3): 616-637.
http://dx.doi.org/10.3846/16111699.2012.761648
Jiazhen, H.; Qin, L. 2008. Revenue coordination contract based on
Stackelberg game in upsteam supply chain, in 4th International
Conference on Wireless Communications, Networking and Mobile Computing,
12-14 October, 2008, Dalian, China, 1-5.
Kaplinski, O.; Tamosaitiene, J. 2010. Game theory applications in
construction engineering and management, Technological and Economic
Development of Economy 16(2): 348-363.
http://dx.doi.org/10.3846/tede.2010.22
Kaviani, M. R.; Chaharsooghi, K.; Naimi Saidgh, A. 2011.
Stackelberg game theory approach for manufacturer--retailer supply chain
coordination using imperialist competitive algorithm, in 2nd
International Conference on Logistics & Supply Chain Conference,
22-23 November, 2011, Tehran, Iran, 130-145.
Lee, J. W. 1993. Determining order quantity and selling price by
geomettric programming: optimal solution, bounds, and sensitivity,
Decesion Science 24(1): 76-87.
http://dx.doi.org/10.1111/j.1540-5915.1993.tb00463.x
Lee, H. L.; Padmanabhan, V.; Seungjin, W. 1997. The bullwhip effect
in supply chains, Sloan Management Review Spring: 93-102.
Leng, M. M.; Parlar, M. 2010. Game theoretic analysis of
decentralized assembly supply chains: non cooperative equilibria vs.
coordination with cost sharing contracts, European Journal of
Operational Research 204(1): 96-104.
http://dx.doi.org/10.1016/j.ejor.2009.10.011
Leng, M. M.; Zhu, A. 2009. Side payments contracts in two person
nonzero sum supply chain games: review, discussion and applications,
European Journal of Operational Research 196(2): 600-618.
http://dx.doi.org/10.1016/j.ejor.2008.03.029
Liu, Y.; Zhang, H. 2006. Supply chain coordination with contracts
for online game industry, in IEEE International Conference on Innovation
and Technology, June, 2006, Singapore, China, 867-871.
Oganezov, K. N. 2006. Inventory models for production systems with
constant linear demand, time value of money and perishable or
nonperiashable items. Virginia: West Virginia University.
Osborne, M. J. 2004. An introduction to game theory. Newyork:
Oxford University Press.
Peldschus, F. 2008. Experience of the game theory application in
construction management, Technological and Economic Development of
Economy 14(4): 531-545.
Peldschus, F.; Zavadskas, E. K. 2005. Fuzzy matrix games
multi-criteria model for decisionmaking in engineering, Informatica
16(1): 107-120. http://dx.doi.org/10.3846/1392-8619.2008.14.531-545
Peldschus, F.; Zavadskas, E. K. 2012. Equilibrium approaches for
construction processes- multi-objective decision making for construction
projects, Bauingenieur 87(5): 210-215.
Peldschus, F.; Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J.
2010. Sustainable assessment of construction site by applying game
theory, Inzinerine Ekonomika--Engineering Economics 21(3): 223-236.
Pentico, D. W.; Drake, M. J.; Toews, C. 2009. The deterministic EPQ
with partial backordering: a new approach, Omega 37(3): 624-636.
http://dx.doi.org/10.1016/j.omega.2008.03.002
Rasmusen, E.; Blackwell, B. 2005. Games and information; an
introduction to game theory. 4th ed. Indiana: Indiana University Press.
Simchi-Levi, D.; Kaminsky, P.; Simchi-Levi, E. 2004. Managing the
supply chain: the definitive guide for the business professional. New
York: Mc-Graw Hill.
Stadtler, H.; Kilger, C. 2007. Supply chain management and advanced
planning: concepts, models, software, and Case Studies. New York:
Springer.
Stein, H. D.; Ginevicius, R. 2010. New co-opettion approach for
supply chain applications and the implementation a new allocation rule,
in 6th International Conference Business and Management, 13-14 May,
2010, Vilnius, Lithuania, 147-152.
http://dx.doi.org/10.3846/jbem.2010.21
Walker, W. 2005. Supply chain architecture: a blueprint for
networking the flow of material, Information and Cash. Florida: CRC
Press.
Wang, W. Y.; Michael, H. S.; Patrick, Y. 2007. Supply chain
management: issues in the new era of collaboration and competition.
Pensylvania: Idea Group Publishing.
Wang, L.; Sun, X.; Dang, F. 2009. Dynamic cooperation mechanism in
supply chain for perishable agricultural products under one to multi,
Advances in Neural Networks: Lecture Noes in Computer Science 5553:
1212-1221. http://dx.doi.org/10.1007/s11424-009-9166-6
Wang, X.; Tang, W. 2009. Fuzzy EPQ inventory models with backorder,
Journal of System and Science Complexity 22(2): 313-323.
Wang, S.; Zhao, L. 2009. Helper mode analysis in a supply chain:
based on supermodular game theory, International Conference on
Management and Service Science, 20-22 September, 2009, Wuhan, China,
978-983.
Wisner, J.; Tan, K.; Leong, G. 2008. Principles of supply chain
management: a balanced approach. Ohio: South-Western Cengage Learning.
Xu, Y.; Zhong, H. 2011. Benefit mechanism designing: for
coordinating three stages supply chain, in International Conference on
Management Science and Industrial Engineering, 811 January, 2011,
Harbin, China, 966-969.
Ying, H. L.; Qi, C. Y.; Sheng, J. Z. 2007. Research on the
coordination mechanism model of the three level supply chain, in
International Conference on management Science and Engineering, 20-22
August, 2007, Harbin, China, 20-25.
Zandi, F.; Tavana, M.; O'Connor, A. 2012. A strategic
cooperative game-theoretic model for market segmentation with
application to banking in emerging economies, Technological and Economic
Development of Economy 18(3): 389-423.
http://dx.doi.org/10.3846/20294913.2012.688072
Zavadskas , E. K.; Peldschus, F.; Ustinovicius, L. 2004. Game
theory in building technology and management. Vilnius: Technika.
Zavadskas, E. K.; Turskis, Z. 2008. A new logarithmic normalization
method in game theory, Informatica 19(2): 303-314.
Zavadskas, E. K.; Turskis, Z. 2011. Multiple criteria decision
making (MCDM) methods in economics: an overview, Technological and
Economic Development of Economy 17(2): 397-427.
http://dx.doi.org/10.3846/20294913.2011.593291
Received 08 August 2013; accepted 16 May 2014
Hannan Amoozad Mahdiraji (1), Kannan Govindan (2), Edmundas
Kazimieras Zavadskas (3), Seyed Hossein Razavi Hajiagha (4)
(1) Department of Management, Kashan Branch, Islamic Azad
University, Kashan, Iran
(2) Department of Business and Economics, University of Southern
Denmark, Odense, Denmark
(3) Faculty of Civil Engineering, Vilnius Gediminas Technical
University, Sauletekio al. 11, LT-10223Vilnius, Lithuania
(4) Department of Management, Kashan Branch, Islamic Azad
University, Kashan, Iran
E-mails: (1) h.amoozad@ut.ac.ir; (2) gov@sam.sdu.dk;
(3) edmundas.zavadskas@vgtu.lt (corresponding author) (4)
s.hossein.r@gmail.com
Zhao, Y; Wang, S.; Cheng, T. E.; Yang, X.; Huang, Z. 2010.
Coordination of supply chains by option contracts: a cooperative game
theory approach, European Journal of Operational Research 207(2):
668-675. http://dx.doi.org/10.1016/j.ejor.2010.05.017
Hannan AMOOZAD MAHDIRAJI. PhD in operation and manufacturing
management from the University of Teheran, BA in industrial engineering.
Assistant Professor of Kashan branch, Islamic Azad University and the
Chief of Planning and Systems Department of Iran Mercantile Exchange.
Has published nearly 15 papers on supply chains and MCDM models in
international journals and conferences.
Kannan GOVINDAN is currently an Associate Professor of operations
and supply chain management at the Department of Business and Economics,
University of Southern Denmark, Odense M, Denmark. Has published more
than 65 papers in the refereed international journals and more than 70
papers in conferences. Was awarded a gold medal for the best PhD thesis.
Research interests: logistics, supply chain management, green and
sustainable supply chain management, reverse logistics and maritime
logistics.
Edmundas Kazimieras ZAVADSKAS. PhD, DSc, h.c.multi. Prof., the Head
of the Department of Construction Technology and Management at Vilnius
Gediminas Technical University, Lithuania. Senior Research Fellow at the
Research Institute of Smart Building Technologies. PhD in Building
Structures (1973). Dr Sc in Building Technology and Management (1987). A
member of Lithuanian and several foreign Academies of Sciences. Doctore
Honoris Causa from Poznan, Saint Petersburg and Kiev universities.
Honorary International Chair Professor of the National Taipei University
of Technology. A member of international organizations; a member of
steering and programme committees at many international conferences; a
member of the editorial boards of several research journals; the author
and co-author of more than 400 papers and a number of monographs in
Lithuanian, English, German and Russian. The editor-in-chief of journals
Technological and Economic Development of Economy and Journal of Civil
Engineering and Management. Research interests: building technology and
management, decision-making theory, automation in design and decision
support systems.
Seyed Hossein RAZAVI HAJIAGHA. Assistant Professor at the
Department of Systemic and Productivity Studies, Institute for Trade
Studies and Research, Teheran, Iran. PhD in Production and Operation
Management (2012). The author and co-author of about 20 scientific
papers. Research interests: multiple criteria analysis, decision-making
theories, data envelopment analysis and mathematical modelling of
industrial problems.
Table 1. Coordination contracts
Contract Description
Wholesale The buyer pays a fixed and quantity-independent
price of each purchased unit for the seller
Discount Quantity-dependent unit prices
Buyback The seller promises to compensate the buyer for
unsold quantities
Revenue sharing The downstream agent commits to return a pre
negotiated portion of its realized profits to the
upstream agent
Rebate The upstream agent rewards the downstream agent for
every sold unit
Side payment Lump-sum monetary transfers among the contracting
agents that are independent of the amount of trade
and used as compensation and incentive alignment
mechanisms
Flexible In contrast with a rebate contract
Push & pull By increasing the amount of selling or buying, the
upstream agent proposes a lower price
Table 2. The notation of decision variables and parameters
Description Note Description
Product n and ordering cost [MATHEMATICAL Retailer's margin
of supplier s EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
Manufacturer's margin [G.sub.n] Selling price of
product n to
retailer r
Manufacturer's total revenue [TR.sub.n] Retailer's setup
cost
Total manufacturing cost [TC.sub.n] Retailer's
holding cost
coefficient
Supplier's margin [G.sub.S] Retailer's total
revenue
Supplier's total revenue [TR.sub.S] Retailer's total
cost
Supplier's total cost [TC.sub.S] Retailer's total
payoff
Supplier's unit cost of a [G.sub.S] Coefficient of a
raw material raw material in
product n
Supplier's holding cost [MATHEMATICAL Price of a raw
coefficient EXPRESSION NOT material from s
REPRODUCIBLE IN of product n
ASCII]
Supplier's ordering cost [MATHEMATICAL Variable
EXPRESSION NOT manufacturing
REPRODUCIBLE IN cost
ASCII]
Integrated payoff function of the manufacturer and retailer
Integrated payoff function of the manufacturer and supplier
Description Note
Product n and ordering cost [G.sub.r]
of supplier s
Manufacturer's margin [P.sub.n]
Manufacturer's total revenue [MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
Total manufacturing cost [k'.sub.n]
Supplier's margin [TR.sub.r]
Supplier's total revenue [TC.sub.r]
Supplier's total cost [Z.sub.r]
Supplier's unit cost of a [MATHEMATICAL
raw material EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
Supplier's holding cost [MATHEMATICAL
coefficient EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
Supplier's ordering cost [MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
Integrated payoff function [Z.sub.MR]
of the manufacturer and
retailer
Integrated payoff function [Z.sub.SM]
of the manufacturer and
supplier
Table 4. Initial data on the numerical example
Amount Par Amount Par
2 M 2 R
4 [MATHEMATICAL 2 S
EXPRESSION NOT
REPRODUCIBLE
IN ASCII]
0.15 [k'.sub.1] 5 [MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE
IN ASCII] (2)
1.1 [[phi]'.sub.1] 0.2 [k'.sub.2]
3 [k.sub.sn] (11) 1.15 [[phi]'.sub.2]
3 [k.sub.sn] (21) 4 [k.sub.sn] (12)
6 [Co.sub.sn] (11) 3 [k.sub.sn] (22)
4 [Co.sub.sn] (21) 5 [Co.sub.sn] (12)
1 [C.sub.B] (1) = 6 [Co.sub.sn] (22)
[C.sub.B] (2)
25 [MATHEMATICAL 0.5 [MATHEMATICAL
EXPRESSION NOT EXPRESSION NOT
REPRODUCIBLE REPRODUCIBLE
IN ASCII] (1) IN ASCII]
0.15 [MATHEMATICAL 24 [MATHEMATICAL
EXPRESSION NOT EXPRESSION NOT
REPRODUCIBLE REPRODUCIBLE
IN ASCII] (1) IN ASCII] (2)
2 [MATHEMATICAL 0.2 [MATHEMATICAL
EXPRESSION NOT EXPRESSION NOT
REPRODUCIBLE REPRODUCIBLE
IN ASCII] (1) IN ASCII] (2)
15 PC (1) = PC (2) 1.5 [MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE
IN ASCII] (2)
8 [MATHEMATICAL 7 [MATHEMATICAL
EXPRESSION NOT EXPRESSION NOT
REPRODUCIBLE REPRODUCIBLE
IN ASCII] (2) IN ASCII] (1)
1.1 [[phi].sub.2] 1.15 [[phi].sub.1]
Table 5. Key parameters for sensitivity analysis
Max Min Par
1.25 1.2 [alpha]
0.15 0.05 [beta]
0.1 0.01 [gamma]
4000 3000 k
4 2 u
Table 6. Design of experiment
[beta] [alpha] k u [gamma] Design
0.15 1.2 4000 4 0.01 1
0.05 1.2 3000 2 0.1 2
0.05 1.2 3000 4 0.01 3
0.05 1.25 3000 4 0.1 4
0.15 1.25 4000 4 0.1 5
0.15 1.2 3000 4 0.1 6
0.1 1.225 3500 3 0.055 7
0.15 1.25 3000 4 0.01 8
0.15 1.25 3000 2 0.1 9
0.15 1.2 3000 2 0.01 10
0.05 1.25 3000 2 0.01 11
0.05 1.2 4000 4 0.1 12
0.05 1.2 4000 2 0.01 13
0.15 1.2 4000 2 0.1 14
0.15 1.25 4000 2 0.01 15
0.05 1.25 4000 4 0.01 16
0.05 1.25 4000 2 0.1 17
Table 7. The overall profit of the supply chain
Design Decentralized MR game SM game
1 2288 1918 3406
2 1593 1939 1775
3 1552 1929 1737
4 1067 1849 1315
5 2442 1969 2407
6 2552 1866 2535
7 1995 1980 2073
8 1800 2462 1869
9 1819 1501 1839
10 2572 1493 2552
11 1088 1892 1320
12 2135 1451 2345
13 2168 1467 2380
14 2438 2503 3421
15 2498 1900 2474
16 1500 2441 1764
17 1507 2492 1794
Table 8. Comparison results from three coalition and
independent games
P-Value T-Value St Dev Mean Experiments
0.193 1.36 608 2177 17
355 1944 17
705 232 17
0.988 -0.01 498 1942 17
355 1944 17
672 -2 17
0.009 -2.96 498 1942 17
608 2177 17
327.4 -234.9 17
Table 9. The analysis of three proposed models for SC behaviour
Decentralized Game MR Game
P Effect Coefficient Effect Coefficient
Gama 10.7 5.4 8.4 4.2
U -43.1 -21.6 87.2 43.6
K 366.8 183.4 151.3 71.6
Alpha -447.1 -223.1 242.6 121.3
Beta 724.9 362.4 * *
SM Game
P Effect Coefficient
Gama -8.6 -4.3
U -21.9 -10.9
K 631.2 315.2
Alpha -671.2 -375.3
Beta 759.3 379.4
