Supply network formation as a biform game

Supply network formation as a biform game
Jean-Claude Hennet*. Sonia Mahjoub*,**
* LSIS, CNRS-UMR 6168, Université Paul Cézanne, Faculté Saint Jérôme, Avenue Escadrille Normandie Niémen, 13397
Marseille Cedex 20, France (Tel: +33(4)91056016, e-mail: [email protected]).
** FIESTA , ISG Tunis, 41 rue de la liberté, 2000 Le Bardo, Tunisia (e-mail:[email protected])
Abstract: In the context of fixed market prices for the selected set of goods to be manufactured, supply
network formation problems have been previously analyzed as cooperative linear production games. In
particular, the profit sharing problem among partners of the winning coalition has been solved by a
perfectly competitive solution, called the Owen set. Now, if an enterprise network decides to organize
itself as a supply chain and imposes the wholesale price of its manufactured goods, then the supply chain
design problem under a price elastic random demand from the market can be formulated as a biform game,
combining a strategic subgame with a cooperative subgame. The decision variables of the strategic
subgame are the wholesale prices and the retail prices of the goods, while the results of the cooperative
subgame are the winning coalition and the payoff profile associated with it. The optimal global value
function is then computed as the solution of a quadratic programming problem. In this scheme, the
enterprise network plays the role of the Stackelberg leader and the retailer the role of the follower. The
paper studies this type of biform games. In particular, it shows the existence of a payoff policy that is fair,
efficient and individually rational. Copyright © 2010 IFAC
1
INTRODUCTION
This paper proposes a new supply chain model based on
game theory. In the context of international projects such as
the European coordinated action CODESNET (2009), it has
been observed that some networks of manufacturers have
now organized themselves both internally, in a cooperative
manner, by sharing their products and resources, and
externally, as dominant strategic actors relatively to their
suppliers and customers. The concept of co-opetition, coined
by Brandenburger and Nalebuff (1996), can be useful to
analyze such new structures of power and trade. However, in
this paper, competition does not only emerge from the
cooperative game between manufacturers. It is also the
leading trend of the profit sharing mechanism between
manufacturers and retailers.
Biform games have been introduces by Brandenburger and
Stuart (2007) to describe situations in which a supply chain
agent needs to make strategic decisions in a competitive
environment. This hybrid model has been adopted in several
SCM literatures. In particular, Anupindi et al. (2001)
analyzed a decentralized distribution system composed of
independent retailers. In the first stage, before demand
realization, each retailer makes its own decision on how
much to order. In the second stage, after observing the
demands, the retailers can cooperate by reallocating their
inventories and allocating the corresponding additional profit.
The authors have shown that this biform game has a nonempty core and have constructed an allocation mechanism
based on dual solution and contained in the core of the game.
Plambeck and Taylor (2005) studied a model with two
original equipment manufacturers (OEMs) who sell their
capacity to the contract manufacturer (CM). In the first stage,
the OEM non-cooperatively choose their capacity and
innovation levels. In the second cooperative stage, the
manufacturers pool their capacity and negotiate the allocation
of the additional profit obtained from capacity pooling.
In Chatain and Zemsky (2007), a biform game approach is
used to model a buyer-supplier relationship. First, suppliers
make initial proposals and take organizational decisions. This
stage is analyzed using a non-cooperative game theory
approach. Then, suppliers negotiate with buyers who seek to
outsource two tasks. In this stage, a cooperative game theory
is applied to characterize the outcome of bargaining among
the player over how to distribute the total surplus. Each
supplier’s share of the total surplus is the product of its added
value and its relative bargaining power.
The quadratic production game of this paper is defined as a
biform game that combines a strategic game between a
manufacturers’ network and the market, and a cooperative
game within the manufacturing network. In the strategic
game, the manufacturing network is supposed to dominate
the market, who acts as a Stackelberg follower. The
consumers’ optimization problem determines the market
prices on the basis of the wholesale prices imposed by
manufacturing network.
2
SOME PRELIMINARIES ON GAME THEORY
Biform games combine strategic games with cooperative
games. Some preliminaries on both types of games are useful
to understand this study.
2.1
Strategic games.
Classically, a non-cooperative static game is a collection
N , ,
where
N  1,..., N  is a finite set of players
with N = card( N ), X is a set of decisions available for each
represent the
, ,….,
player i,
1, … … , N and
utility (or payoff) received by each player. The payoff of each
player depends on the strategy chosen by all the players.
Given an N-players game, player i aims to choose a strategy
that maximizes his utility function π x , x , given
that the other players’ strategy is summarized by decisions
…
…
. Then, the best strategy of
.
,
player i is defined by:
A set of actions x , x , … . , x
game if:
is a Nash equilibrium of the
x x
i
1, … . , N.
(1)
use the locally optimal strategy x . From this definition, a
Nash equilibrium is a set of actions from which no player can
improve the value of his utility function by unilaterally
deviating from it.
Stackelberg games are strategic games with 2 players. They
are also called leader-follower games. They are not in the
normal form since they are dynamic with 2 steps. The leader
plays first, anticipating the decision of the follower, and the
follower has no other choice than to act optimally as
anticipated by the leader. Such games generally reach a
compromise situation, called the Stackelberg equilibrium.
The leader’s optimal decision, denoted x , is computed
recursively from the knowledge of the follower’s optimal
response function x x :
argmax π x , x x
players, denoted N  1,..., N  , with N = Card( N ). A
coalition S is a subset of N : S  N . The set P (N ) is the
set of all the subsets of N .
In a TU (Transferable Utility) cooperative (or coalitional)
game in the sense of Von Neumann and Morgenstern (1944),
each coalition S  P (N ) is characterized by a value function
v( S )  0 . The value v(S ) is the maximal utility (or payoff)
that can be obtained by coalition S. All the utilities are
transferable (TU-game) in the sense that they are all shares of
the global payoff. Each player i  N seeks to maximize his
utility function, which is the payoff that he can obtain from
belonging to a coalition S  N . Notation N \ S represents
the set of players that belong to N but not to S . If S is the
winning coalition, then any player
j  N \ S has a null
payoff.
Let v * be the maximal global payoff of the TU-game (N, v) :
v*  max v( S ) .
S P (N )
(3)
 ui  v * .
With every coalition S we associate a payoff
iN
u(S ) defined by:
In the normal form of the game, each player i selects his
optimal strategy x assuming that all the other players also
x
Classically, a cooperative game involves a finite set of N
A feasible payoff profile is a vector (ui ) iN such that
Definition 2.1 Nash equilibrium
argmax π x , x
2.2 TU-cooperative games.
and x
x x . (2)
u(S ) 
 ui .
(4)
iS
Several properties will now be defined.
Definition 2.2: Efficiency (Pareto optimality)
The feasible payoff profile (u i ) iN is said to be efficient (or
Pareto optimal) if and only if
N
u (N )   u i  v * .
(5)
i 1
Definition 2.3: Rationality
A feasible payoff profile (ui )iN is said to be rational if the
payoff of every coalition S is larger than its value v(S ) :
u ( S )  v( S ) S; S  P (N ) .
(6)
Definition 2.4: Core
Let y  ( y1  y n )T be the output vector of products during a
The core of a TU-game is the set of feasible payoff profiles
reference period. Equation (8) is called the demand curve. As
in Lariviere and Porteus (2001), the retailer faces the inverse
demand curve obtained from the optimality conditions of the
market game. The products being assumed independent, the
inverse demand curve for each product i=1,…,n is:
(u i ) iN that satisfy conditions (3) and (4). Namely, it is the
set of feasible payoff profiles that are both efficient (Pareto
optimal) and rational.
As in Gillies (1959), the core is defined as:”the set of
feasible outcomes that cannot be improved upon by any
coalition of players”.
pi  
1
i

yi  i
(9)
i
Quantities and prices being nonnegative, a necessary
condition for equations (9) to be valid is :
Definition 2.5: Optimal coalition
The optimal cardinality of the TU-game (N, v) is:
s*  mincard ( S ) v( S )  v *. An optimal coalition of the

p i  p MAX , with p MAX  i .
(10)
i
i
i
TU-game (N, v) is a coalition S *  N that satisfies
By convention, condition (10) is always satisfied if p i is not
v ( S * )  v * and card ( S )  s * .
the actual retailer price for product i, but is obtained from the
Definition 2.6. Convexity
actual retailer price for product I, denoted p a through the
A TU cooperative game is convex if and only if:
following relation (11).
i
v( S  T )  v( S )  v(T )  v( S  T ) S  N, T  N (7)
3
3.1
THE SUPPLY CHAIN MODEL
p i  min( p a , p MAX )
i
3.2
The market game
Consider a retailer selling on a market a set of products
numbered i=1,…,n. In the market game between the retailer
and the set of customers, the retailer plays first, by proposing
a price vector p=(p1…pn)T and the market reacts by buying a
quantity that depends on this price and on its habits and
requirements. The supply-demand negotiation game can be
represented as an iterative process. The current price p i (t ) is
the decision variable fixed by the retailer and the currently
purchased quantity, y i (t ) ,is the decision variable of the
present and past quantities and prices purchased by customers
at periods t,t-1, t-2,…t-h+1, with h the system memory,
supposed finite.
In a generic manner, we write: y i (t )  f i ( Y i (t  1), Pi (t) ) . For
each product i=1,…,n, the market game is supposed to reach
a stable equilibrium for which the expected quantity y i sold
over a reference period, satisfies
y i    i p i  i
retailer over the reference period is :
under conditions: wi  p i .
∏
The price vector,
p   Diag (
1
i
is obtained from (9) in the form:
) y  Diag (
i
)1
i
(13)
diagonal terms mi , and 1 is the vector with all the
components equal to 1, and the appropriate dimension (n in
this case).
The objective is to find the optimal vector
∏ , with:
Diag (
1
i
)y
1 Diag (
that maximize
i
)y
i
The optimality condition takes the following form:
(12)
where Diag ( m i ) denotes a diagonal matrix with generic
∏
(8)
The retailer’s problem
For each final product sold on the market, the retailer faces a
stochastic demand. Considering the price-dependent expected
quantity sold, y i for i=1,…,n, the expected profit of the
market. Different models of the market reaction function can
be investigated. Let Y i (t) and P i (t) be the vectors of
(11)
i
(14)
∏
2
And since
(15)
0, the criterion is strictly concave and admits
a single optimal solution. For each product, the optimal
expected demand is:
 
yi  i  i wi
2
2
(16)
The non negativity of this quantity derives from inequalities
in (10) and (12). Accordingly, the proposed retail price is
derived from (9):
pi 
w
i
 i
2 i
2
.
(17)
It is assumed that the vector of wholesale prices, w, is
determined by the manufacturer’s network, who acts as a
Stackelberg leader. It is related to the output vector y, by:
(18)
N
vector eS  0,1 such that:
(eS ) j  1
if j  S
(eS ) j  0
if j  S
.
(20)
be the amount of resource r available at enterprise j,
B  (( Brj ))   RN , and Ari the amount of resource r
Then, as a Stackelberg follower, the retailer reacts by
choosing the retail prices (17) that maximize his expected
profit. From (11), (17), (18), the retailer expected profit is:
∑
As in Van Gellekom et al. (2000), a coalition S is defined as a
subset of the set N of N enterprises with characteristic
For the R types of resources considered (r=1,…,R), let Brj

2
wi  i 
y
i i i
∏
the follower, the retailer can only accept or reject the
manufacturer’s proposal. It is assumed that the retailer agrees
to conclude any contract, provided that he obtains an
expected profit greater than his opportunity cost which is set
equal to zero by convention. After the manufacturers network
has set the vector of wholesale prices, w, the retailer
determines p (or equivalently ) to maximize his expected
profit. Having anticipated the retailer’s reaction function (13),
to maximize his expected profit.
the coalition determines
The pair of optimal vectors
can thus be determined
,
by the manufacturers’ network.
.
(19)
necessary
to
A  (( Ari ))  
produce
Rn
one
unit
The manufacturers’ network
produce commodities and sell them in a market. The N
manufacturers compete to be partners in a coalition
N .
Each candidate enterprise is characterized by its production
resources: manufacturing plants, machines, work teams,
robots, pallets, storage areas, etc. Mathematically, each firm
,
1 … … . . , of R
owns a vector
, ,…..,
,
types of resources. These resources can be used, directly or
indirectly to produce the vector
,…..,
of final
products. The coalition incurs manufacturing costs
,…..,
per unit of each final product and sells the
,…..,
to
products at the wholesale price vector
the retailer who acts as an intermediate party between the
manufacturers’ network and the final consumers.
Under a wholesale price contracts, the coalition of
manufacturers acts as the Stackelberg leader by fixing the
wholesale price vector w as a take-it-or-leave-it proposal. As
i,
Resource capacity constraints for coalition S are thus written:
3.4
Consider a network of N firms represented by numbers in the
set N  1,..., N . These firms are willing to cooperate to
product
.
.
3.3
of
(21)
The manufacturers’ game
At the manufacturing stage, two different problems must be
solved: the strategic problem of selecting the wholesale price
vector w, and the cooperative problem of optimizing the
production vector y and the coalition characteristic vector
eS . The profit optimization problem can be formulated as
follows:
Maximize
,
(22)
Subject to
0,1
For a given vector w, problem (22) characterizes a
cooperative game, namely the Linear Production Game
(LPG) studied in Owen (1975) and Hennet and Mahjoub
(2009). In the biform game studied in this paper, variables
are decision variables with optimal values related to the
through relations (18). Acting as the
optimal output values
Stackelberg leader in the strategic game with the retailer, the
manufacturing network anticipates the optimal reaction of the
retailer by substituting equations (18) into the objective
function of problem (22). The obtained set of quadratic
programming problems (P) defines a quadratic production
game denoted (QPG).
Maximize
∑
,
Subject to
(P)
0,1
,
4
THE QUADRATIC PRODUCTION GAME
By assuming exogenous prices imposed by the market, the
LPG describes a competitive economic situation. On the
contrary, the quadratic production game (QPG) described in
this paper is more appropriate to describe an oligopolistic
situation in which the manufacturing network imposes its
decisions to the retailer who himself has a dominant position
over customers and imposes the retail prices. In this context,
the QPG addresses the three following issues:



4.1
the profit maximization problem for the
manufacturing network considered as a whole,
the coalition decision problem through the choice of
vector e S ,
the problem of profit allocation to the members of
the optimal coalition.
Global profit maximization
Consider a coalition S, S  N . The maximal profit that can
be obtained by this coalition is obtained as the solution of the
following problem, denoted (PS):
Subject to
(PS)
,
,
0,1
solution of problem (PS), denoted v(S ) , is obtained for the
e N  1 . Note that matrices A and B in (PS) are
componentwise nonnegative. For any set S  N , e S  e N
and BeS BeN . Then, the optimal solution of (PS) is feasible
for (PN) and the maximal expected profit can be obtained as
the optimal solution of (PN).
The global profit maximization problem can thus be solved
through solving (PN) instead of (P), with the advantage of
solving a problem in which all the variables are continuous.
It can be noticed that property 4.1 does not imply optimality
of the grand coalition is the sense of definition. It may
happen that some coalitions with smaller cardinality than N
also yield the optimal expected profit.
4.2
Profit allocation in a coalition
From definition 2.4, any profit allocation policy in the core of
the QPG is efficient and rational. Other properties can
differentiate allocations. In particular, it is desirable to relate
profit allocations of players to their marginal contribution to
the value function. Classically (see e.g. Osborne and
Rubinstein, 1994), the marginal contribution of player i to
coalition S  N with i  S is defined by:
 i ( S )  v( S  
i )  v( S )
(23)
A particular allocation policy, introduced by Shapley (1953)
has been shown to possess the best properties in terms of
balance and fairness. It is called the Shapley value, and
defined by :
1
  i (Si (r ))
N ! r R
(24)
for each i in N where R is the set of all N! orderings of N ,
and Si (r ) is the set of players preceding i in the ordering r.
Furthermore, the following result applies (Shapley, 1971):
Formally, problem (PS) is similar to problem (P), except for
the fact that in problem (P),
is a vector of decision
variables, while in problem (PS), vector
is the fixed
characteristic vector of the investigated coalition, S. The
following result can be derived from the comparison of
problems (PS) for different coalitions S  N . The optimal
output vector denoted
Consider the grand coalition N . Its characteristic vector is
i (N , v) 
∑
Maximize
Property 4.1 The grand coalition generates the optimal profit
.
Property 4.2 If the QPG is convex, the Shapley value
allocation is in the core.
Unfortunately, convexity is not guaranteed in general for the
QPG, as it is illustrated in the example. It is then possible to
differentiate coalitional rationality (not verified in general)
from individual rationality. Finally, the manufacturers’ game
can be solved in a fair, efficient and individually rational
manner through the following steps:
1.
Solve problem ( PN ) to obtain the maximal profit
2.
3.
4.
and the optimal output vector y,
Set the wholesale price vector w computed by (18),
Set the market price vector p computed by (17),
Compute the Shapley value allocation (24) to
allocate the expected profit among the partners.
Computation of the Shapley value allocation requires
computing the solution of all the problems (PS) for S  N ,
and this, of course, can be very time consuming for large sets
of manufacturing partners.
4.3
A numerical example
A very simple numerical example is constructed to illustrate
 1 0.5
10 0 0 
the
approach:
A
, B

,
0.7 0.8
 0 10 20
 2
20
5
   ,    , , c   . The unconstrained optimum of
 3
40
5
the QPG is: v*  32.29 for y*  [2.5 6.25] . This solution is
feasible for coalitions 1,2,3, 1,2 , 1,3 , and v( S )  0 for
S  1, 2, 3, 2,3. For this problem, the core allocation is
unique: (v*, 0, 0) and the Shapley allocation is:
Brandenburger A.M. and Nalebuff B. (1996). Co-Opetition:
A revolution mindset that combines competition and
cooperation, Currency Doubleday, New York.
Brandenburger, A. and Stuart, H. (2007). Biform games.
Management science, 53(4), 537-549
Chatain, O. and Zemsky, P. (2007). The horizontal scope of
the firm: organizational tradeoffs vs. buyer-supplier
relationships. Management science, 53(4), 550-565.
CODESNET (2009), A European roadmap to SME networks
development, A. Villa, D. Antonelli Eds, Springer.
Gillies, D.B. (1959). Solutions to general non-zero-sum
games. In:. Contributions to the theory of Games vol. IV.
Annals of math studies, vol. 40. (Tucker, A.W., Luce,
R.D., (Eds.)), 47–85., Princeton, NJ: Princeton
University.
Hennet, J.C. and Mahjoub, S. (2009). A Cooperative
Approach to Supply Chain Network Design, Preprints of
the 13th IFAC Symposium on Information Control
Problems in Manufacturing (INCOM'09), 1545-1550.
does not apply, for instance, if S  1,2, T  1,3. The
Lariviere, M.A. and Porteus, E.L. (2001). Selling to the
newsvendor: an analysis of price-only contracts.
Manufacturing & service operations management, 3(4),
293-305.
vector of wholesale prices imposed by the manufacturing
network to the retailer is: w   7.5  . Then, the optimal
9.17 


Osborne, M.J. and Rubinstein, A. (1994). A course in game
theory. The MIT Press, Cambridge, Massachussetts,
U.S.A, London, England.
( 2 v*, 1 v*, 1 v * ). This QPG is not convex since property (7)
3
6
6
vector of retail prices is: p   8.75  .
11.25


5
Owen, G. (1975). On the core of linear production games.
Mathematical Programming, 9, 358–370.
CONCLUSIONS
In the study reported in this paper, the network of
manufacturers acts as a Stackelberg leader relatively to the
retailer. This situation has generated a QPG not yet studied in
the literature, but still relatively easy to solve. It has been
shown that in general, this game is not convex and therefore
that coalitional rationality and fairness of the allocation
policy are not always compatible. The reverse case, when the
retailer is the Stackleberg leader, gives rise to a different
biform game that seems to be more difficult, since the
reaction function of the manufacturers’ network cannot be
expressed analytically. This problem still seems to be open.
6
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