Download Toward the fair sharing of profit in a supply network formation

Toward the fair sharing of profit in a supply network formation
Jean-Claude Henneta,∗, Sonia Mahjouba,b
a
LSIS, CNRS-UMR 6168, Université Paul Cézanne, Faculté Saint Jérôme, Avenue
Escadrille Normandie Niémen, 13397 Marseille Cedex 20, France
b
FIESTA , ISG Tunis, 41 rue de la liberté, 2000 Le Bardo, Tunisia
Abstract
The design of a supply chain network can be interpreted as a coalition formation problem in cooperative
game theory and formulated as a linear production game (LPG). The companies which are members of the
optimal coalition share their manufacturing assets and resources to produce a set of end-products and globally
maximize their profits by selling them in a market. This paper investigates the possibility of combining the
requirement of coalition stability with a fair allocation of profits to the participants. It is shown that, in general,
the purely competitive allocation mechanism does not exhibit the property of fairness. A technique is proposed
to construct a stable and fair allocation system when the core of the game does not exclusively contain a set of
competitive allocations.
Keywords: supply chain, cooperative game theory, coalitions, linear programming, duality
*
Corresponding author. Tel. : +33 491 05 6016 ; fax: +33 491 05 6033,
E-mail address: [email protected]
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1. Introduction
In the context of modern information and communication networks, many firms are open to
finding new partnerships and penetrating new markets. In the manufacturing sector, several classical
paradigms and tools must be revisited in the light of more adaptability, flexibility and network
reconfigurability. This paper describes a supply network formation process as a network of
enterprises attempting to collaborate and capture a market share for a particular set of products.
The result of such a collaboration may be called an emergent supply network. The basic and
necessary ingredients to enable collaboration to prevail are the existence of a targeted market, and
resources for manufacturing, communication and logistics. This study focuses on the latter success
factor: existing resources could bring more profit if they were used jointly rather than separately. As
a consequence, many enterprises are ready to use their own resources and those of other firms to
produce services and products in the most advantageous quantity and at the lowest cost.
Supply networks are thus characterized by a basic common goal and a highly integrated
operational system, combined with a complex decentralized decisional organization. Such
characteristics are of major concern in game theory, under the classical decomposition into
cooperative (or coalitional) games and non-cooperative (or strategic) games. Models referred to as
‘non-cooperative’ are composed of players with different preference relations or utility functions.
Their actions obey a strategy that takes into account information (generally imperfect) on the other
players’ actions and preferences. As noted in Cachon and Netessine (2004), most supply chain
models based on game theory use a non-cooperative approach. However, cooperative game theory
seems more appropriate to analyze a supply network in its design stage, as it is characterized by
many possibilities for enterprise coalitions and allocation patterns for tasks and rewards. In such a
strategic stage, all the actors are ready to cooperate to create competitive advantages in the market.
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Partner enterprises share a common goal that is clearly identified: penetrate a market segment to
obtain the maximal global expected profit.
A general characteristic of cooperative games is the understanding of the players that they can
obtain a larger global benefit from pooling their resources than by acting separately. This
characteristic is particularly important when resources are scarce, as in the case of spare parts for
aircrafts studied by Kilpi et al. (2009), and when important economies of scale in the availability
service can be achieved by cooperation. Several authors have used cooperative game theory to
represent alliances between retailers in their relation to the market and/or to suppliers. Nagarajan
and Sošić (2008) have studied the benefits expected by retailers from price setting, pooling their
market share and sharing risks. Profit can also be increased by centralizing inventories. The works of
Granot and Sošić (2003), Cachon and Netessine (2004) and Reinhardt and Dada (2005) provide
convincing interpretations of supply chain design problems as cooperative games by focusing on
manufacturing capacity and inventory pooling. In the light of cooperative game theory, a supply
network can be modelled as a coalition of partners pooling their resources and sharing the same
utility function (profit). The partnership building problem can then be modelled as a cooperative
game with transferable utilities (TU-game). Such a game is firstly characterized by global optimization
of the supply chain value (total expected profit). A TU-game can thus be seen as a target model on
which the partners can agree to estimate the maximal value of the chain and the shares of the global
profit acceptable to all of them. Another practical advantage of this model is that it evaluates and
compares different possible coalitions. The results can thus be used as arguments in the supply chain
design stage, to convince the partners to be part of the best possible coalition and set up the joint
venture.
Once the comparative advantage of cooperation has been demonstrated, the key remaining
issue is the choice of the scheme for sharing benefits between the members. Expected profit is
interpreted as the transferable utility that should be allocated to players in a manner that is
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acceptable to all of them and guarantees coalitional stability. In cooperative game theory, several
concepts have been introduced for approaching the stability issue. A necessary condition for the
stability of a coalition is that no set of players is able to increase its members’ profits by forming a
different coalition. The set of payoff profiles that verifies this property is known as the core of the
TU-game (Gillies, 1959). It is the set of non-dominated feasible payoff profiles (also called
imputations) covering all the possible coalitions. Non-emptiness of the core has been shown if the
problem is convex (Shapley and Shubik, 1969). By introducing some restrictions on the possible
deviations from a coalition, several sets have been defined to characterize stable and/or balanced
imputations among the partners of the optimal coalition: stable set, bargaining set, nucleolus, kernel
and the ‘Shapley value’ (see e.g. Osborne and Rubinstein, 1994). In recent years, several imputation
mechanisms have been proposed in the literature on supply chain analysis by cooperative game
theory.
Several authors have proposed the use of the ‘Shapley value’ approach in various contexts
related to supply chain design (Nagarajan and Sošić, 2008). In particular, Reinhardt and Dada (2005)
have studied the case of n firms cooperating by pooling their critical resources. Due to the difficulty
of computing the Shapley value for a large number of players, these authors have proposed a
pseudo-polynomial algorithm to compute the Shapley value allocation of benefits for particular
games, called ‘coalition symmetric’. In addition to its computational complexity, the Shapley value
also has the drawback of not necessarily belonging to the core, as pointed out by Granot and Sošić
(2003). Bartholdi and Kemahlıoglu-Ziya (2005) have also used the Shapley value to allocate profit in a
network composed of two retailers and one supplier. They have shown, in particular, that such an
allocation coordinates the supply chain but may appear unfair to the supplier.
Another example of an alliance between retailers can be found in Guardiola et al. (2007). The
authors analyze the case of a supply chain with one supplier and n retailers selling the same product
in separate markets. Through cooperation, the retailers increase their profit by obtaining a lower
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wholesale price from the supplier. In counterpart, the supplier receives equal side payments from all
the retailers. The maximal value of these payments is computed to guarantee that the allocation
mechanism belongs to the core of the retailers–supplier game. In an almost symmetrical manner, Jin
and Wu (2006) have shown that in online reverse auctions there is one unique strongly stable
suppliers’ coalition that allows for maximizing profit. Several studies have concentrated on the
costing aspects rather than on profit sharing. In particular, Charles and Hansen (2008) have applied
cooperative game theory to global cost minimization and cost allocation in an enterprise network.
They have shown that under classical concave cost functions for all members, the cost allocation
obtained by the activity based costing (ABC) technique is rational and belongs to the core of the
game.
In many practical situations, a supply network can be viewed as a multistage production
system in which the different production stages are performed by different enterprises. Material
requirement planning (MRP) theory (Grübbström, 1999; Grubbström and Huynh, 2006) can then be
used to define and distribute responsibilities and manufacturing orders among the partners. In this
view, the product structure supports the enterprise network organization, especially under an
extended view of the bill of materials (BOM) such as the generic BOM (Lamothe et al., 2005).
Additionally, multistage production by several producers highly differs from multistage production by
a single producer because of the need for coordinating requirements and contracts negotiation. It
also carries new possibilities for increasing the effectiveness of ordering and inventory policies (Arda
and Hennet, 2006). In the design stage, the largest possible set of firms should be investigated for
selecting partners and sharing resource costs and/or rewards in the most efficient manner.
This paper analyzes the problem of optimally choosing the partners to constitute a multistage
production system. By assumption, the firms who own manufacturing resources are ready to make
them available to the supply chain if they obtain sufficient rewards. In such a scheme, some
enterprises may be complementary (if they own complementary resources) and some may compete
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(if they own resources of the same type). In addition, they all compete with each other to obtain the
largest possible share of the global profit. Using MRP theory, this paper shows that the supply
network formation problem can be formulated as a linear production game (LPG). Since the pioneer
works of Shapley and Shubik (1972) and Owen (1975), many results have been obtained for this
particular class of cooperative game. LPGs are characterized by a linear value function to be
optimized by linear programming (LP) under linear constraints relating production to resources under
capacity constraints.
Since the work of Owen (1975), it has been well-known that the core of an LPG is generally not
empty and that a purely competitive allocation scheme, called the Owen set, is contained in the core.
This set is constructed from the optimal solution of the dual linear program, which defines the
shadow prices of the resources. In the Owen set, each player receives the reward corresponding to
the shadow prices of the resources that he owns. Then, if the dual problem has a unique solution,
the Owen set consists of a single point, called the purely competitive allocation (Van Gellekom et al.,
2000). These results have been extended to semi-infinite LP situations, under the assumption of a
finite bound on the maximal profit, in two basic cases. Fragnelli et al. (1999) have considered the
case of a potentially infinite number of end products, and Tijs et al. (2001) have studied the
possibility of an infinite number of transformation techniques.
Based on the previous works of Owen (1975) and Van Gellekom et al. (2000), this paper
explicitly constructs the Owen set of the multistage LPG under study. Then, the quality of the
solution is discussed in terms of stability and fairness. In particular, the existence of partners
obtaining a null profit is considered as an unfair property that decreases the robustness of the
coalition’s stability. The main issue of this paper is to study the existence of fair allocations in the
core of LPGs. Theorem 1 in section 4 gives a sufficient condition for existence and a technique to
construct such allocations. To solve this problem, some side payments are introduced from the
partners with a positive payoff to their associates with a null payoff in the Owen allocation. The
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maximal value of these side payments is computed under the condition of maintaining the allocation
in the core of the game. If this value is strictly positive, as may occur if the competition is not too
severe, then a new set of fair imputations lying in the core is constructed.
Section 2 presents some preliminaries on cooperative game theory. Then, section 3 represents
the multistage supply chain design problem as a cooperative linear production game. The problem is
solved in terms of maximizing profit and forming a coalition with the smallest number of partners.
The Owen set of this game is constructed and analyzed. It is shown via an example that, under the
Owen allocation, some participants of the optimal coalition may receive no payoff. To correct this
drawback as much as possible, the side-payments mechanism is proposed and studied in section 4.
The main result of the paper is stated in Theorem 1. It provides a test of existence and a technique to
construct an efficient, rational and fair allocation. Some conclusions on the applicability of the
method are finally exposed.
2. Some preliminaries on TU-games
Classically, a game involves a finite set of N 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 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. Each player i ∈ N seeks to maximize
his utility function, which is the payoff that he can obtain from belonging to a coalition. By
convention, all the utilities are nonnegative. The minimal utility that a player may obtain is zero. In
this model, the players’ actions within a coalition are implicit. In order to maximize his utility
function, the only possible decision a player can take is to belong to a coalition or not. If, as the result
of the game, coalition S , S ⊆ N prevails, then each player i ∈ S obtains a share vi ( S ) ≥ 0 such
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that:
∑ vi (S ) = v(S ) , and each player j of N not belonging to
S ( j ∈ N \ S ) has a null payoff:
i∈S
v j ( S ) = 0 . Notation N \ S represents the set of players that belong to N but not to S . All the
utilities are transferable (TU-game) in the sense that they are all shares of the global payoff.
A TU-game is thus simply noted (N, v) . It raises two basic problems:
•
global utility maximization: determination of the maximal value function and the coalitions S
for which this value is obtained,
•
allocation problem: determination of the endowments of the agents by distributing the global
payoff among them.
As in Osborne and Rubinstein (1994), an S-feasible payoff profile is defined as a vector (u i ) i∈S
such that
∑ ui = v(S ) , and a feasible payoff profile as a vector (u i ) i∈N
i∈S
such that
∑ ui = v(N ) .
i∈N
Let v * be the maximal global payoff of the TU-game (N, v) :
v* = max v( S ) .
S ∈P (N )
(1)
Consider a feasible payoff profile (u i ) i∈N . With every coalition S we associate a payoff u (S )
defined by expression (2):
u(S ) =
∑ ui .
(2)
i∈S
Several properties will be now defined. These properties appear necessary for a payoff profile
to solve a coalitional game.
Property 1: Efficiency (Pareto optimality)
The feasible payoff profile (ui )i∈N is said to be efficient (or Pareto optimal) if and only if
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N
u (N ) = ∑ ui = v * .
(3)
i =1
Another condition for a feasible payoff profile (ui )i∈N to be accepted by all the players is
(coalitional) rationality, as defined below.
Property 2: 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 ) .
(4)
The core of the game (N, v) is a game-theoretic concept introduced in Gillies (1959). It is
defined as follows.
Definition 1: Core
The core of a TU-game is the set of feasible payoff profiles (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.
A major issue in the analysis of a TU cooperative game consists in characterizing its core. It can
be noted that equations (2) and (3) and linear inequalities (4) define a closed and convex set.
However, due to the combinatorial growth of the number of coalitions (cardinal of P (N ) ) with the
number of players, N, the core is generally difficult to construct and fully explore. The approach that
will be followed in this study for the LPG will rather be to construct a core solution by a known
algorithm and then to locally modify this solution without escaping the core.
An interesting index to characterize the value of a subset S' ⊆ S ⊆ N
is its marginal
contribution in S , denoted ∆ S (S ' ) and defined as follows.
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Definition 2: Relative marginal contribution
The marginal contribution of a subset S ' of a coalition S is:
∆ S ( S ' ) = v( S ) − v( S \ S ' )
(5)
where notation S \ S ' represents the set of players that belong to S but not to S ' .
In a classical manner, marginal contributions relative to the so-called ‘grand coalition’ N are
simply called ‘marginal contributions’.
Property 3: Marginal contribution principle
An allocation that lies in the core satisfies the marginal contribution principle:
u(S ) ≤ ∆ N (S ) ∀S; S ∈ P (N ) .
(6)
The proof of this property is straightforward: a feasible payoff profile (u i ) i∈N
the
core
satisfies
u (N \ S ) ≥ v(N \ S )
for
every
coalition
S.
that lies in
Then,
from
u ( S ) = u (N ) − u (N \ S ) = v(N ) − u (N \ S ) , one derives u ( S ) ≤ v(N ) − v(N \ S ) .
Hence, a core allocation is such that any coalition cannot obtain a payoff greater than its
marginal contribution. With the objective of studying the core of a TU-game, inequalities (6) are
particularly useful since they provide upper bounds to coalition payoffs, while rationality conditions
(4) provide lower bounds.
The maximal global payoff of the TU-game (N, v) , v * , has been defined by equation (1). As
this value can be obtained for several coalitions, it is possible to introduce a secondary criterion to
define the optimal coalitions.
Definition 3: Optimal coalition
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The optimal cardinality the TU-game (N, v) is: s* = min{card ( S ) v( S ) = v *}. An optimal
coalition of the TU-game (N, v) is a coalition S * ⊆ N that satisfies v( S * ) = v * and card ( S ) = s * .
The most critical situation for a player in a TU-game is when his utility remains at its minimal
value, zero. In other words, his share of the global payoff, v * , is null. For a given feasible payoff
profile (ui )i∈N , a player who does not receive any payoff is called a ‘null payoff player’ (NPP),
according to the following definition.
Definition 4: Null payoff player (NPP)
An NPP with respect to a feasible payoff profile (u i ) i∈N is a player i ∈ N such that ui = 0 .
For every optimal coalition S * , two types of NPPs may be distinguished: players outside and
players inside the optimal coalition S * .
The following properties derive from the definitions above.
Property 4
Under a core allocation profile, any player who does not belong to every optimal coalition is
an NPP.
Proof. Suppose j ∈ N \ S * with S * as an optimal coalition. Then, by the efficiency property
of the payoff profile, u (N ) = u (N \ S *) + u ( S *) = u ( S *) = v * , therefore
∑u
i∈N \ S *
i
= 0 and, by the
nonnegativity of utilities, u j = 0 ∀j ∈ N \ S * .
Property 5
Under a core allocation profile, any NPP which belongs to an optimal coalition, i ∈ S * satisfies
the two following conditions: v({}
i ) >0 .
i ) = 0 and ∆ S* ( {}
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Proof. Condition ui ≥ v({}
i ) ∀i ∈ N is a well-known property, called ‘individual rationality’. For
a core allocation, this property is verified as a particular instance of rationality property (4), for sets
S ⊂ N with cardinality 1. For an NPP, condition ui = 0 implies v({}
i ) = 0 . To show ∆ S * ( {}
i ) > 0 , it
suffices to apply definition 5 to note that the reverse condition, ∆ S *( {}
i ) = 0 , would contradict the
minimality of coalition S * .
With this background, it is now possible to propose a definition of fairness for payoff profiles.
Definition 5: Fairness
A feasible payoff profile (u i ) i∈N is fair if all the members of an optimal coalition S * obtain a
strictly positive payoff : ui > 0 ∀i ∈ S * .
This definition raises the problem of the existence of fair solutions belonging to the core of a
TU-game.
To show that some TU-games cannot admit any fair solution, it suffices to construct games
that possess several different optimal coalitions, optimality meaning that the coalition creates the
maximal utility value, v* , and has minimal cardinality in the set of coalitions S verifying v( S ) = v* . It
will now be shown that games with this property can be constructed, as inspired by Aumann (1964),
by replicating one or several players of an optimal coalition.
Consider a production game (N, v) in which the players own resources that they put into
common use to manufacture products that create value. Consider a player i who belongs to an
optimal coalition S * and suppose that he owns resources that are useful but not critical, in the sense
that he possesses all of them in excess of what is required for the optimal production mix. Other
resources that he does not own are critical. Let us now define a new game, (N ∪ {i' }, v) , by
introducing a new player, denoted i’, exactly identical to player i.
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Because of the non-criticality of the resources owned by players i and i’, the act of increasing
their quantity does not change the optimal production mix. Hence, the optimal value of the new
(
)
game is the same as the optimal value, v*, of the original game. Coalitions S * and S * \ {}
i ∪ {i '} are
both optimal for the new game. Therefore, Property 4 implies ui = ui ' = 0 for any core allocation u of
the game (N ∪ {i' }, v) . And yet player i belongs to the optimal coalition S * , and player i’ to the
(
)
optimal coalition S * \ {}
i ∪ {i '} . Therefore, player i is an NPP in S * and player i’ an NPP in
(S * \ {}i )∪ {i'}.
As a consequence, a necessary condition for the existence of a fair solution in the core of a TUgame is the uniqueness of the optimal coalition S * . Note that this is an intrinsic property of the
game, independent from the core allocation considered.
3. Coalition formation for profit maximization
3.1. A multistage model
Consider a network of N firms represented by numbers in the set N = {1,..., N } . These firms
are willing to cooperate to produce commodities and sell them in a market. These commodities
belong to a set of g manufactured products (or families of products) i=1,…,g. Typically, manufacturing
recipes are fixed and well defined. The gozinto graph describes the product structure and has no
cycle. It can then be decomposed into levels: level 0 products are the g final products. Then,
intermediate and primary products are numbered in the increasing order of their level. The level of
product i, for i=g+1,...,n is the maximal number of stages to transform product i into a final product.
Each production stage is supposed to have several input products but only one output product. The
BOM technical matrix, G, is defined as follows: according to the given manufacturing recipe,
production of one unit of product i requires the combination of components j=1,...,n in
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quantities G ji . It can be noted that under a level-consistent ordering of products, matrix G then has
a lower triangular structure (Hennet, 2003).
In the context of MRP theory (Grübbström, 1999; Grubbström and Huynh, 2006), matrix G is
the input matrix. When combined with the lead-time (output) matrix, it can be used to define the
generalized Leontieff inverse of the system. However, the scope here will be restricted to stationary
balance equations under a simple production structure. In this case, the output matrix reduces to the
identity matrix.
Let x = (( xij ) be the matrix of the quantities of product i produced (or purchased) at firm j and
y = ( y1  yn )T be the output vector of products during a reference period. The components of this
matrix and vector are the variables of the design problem. For simplicity, quantities per period (or
throughputs) are supposed to be continuous: x ∈ ℜ n+× N , y ∈ ℜ n+ . Due to the sharing of resources, the
problem can be formulated in terms of the global throughput vector, denoted ω and related to
matrix x through the elementary summation relation (7):
ω = x1 with 1 being the unit vector of dimension N.
(7)
The output vector can be computed from the global throughput vector by the following
relation:
y = ( I − G )ω
(8)
with I the identity matrix of dimension n × n .
Matrix G being componentwise nonnegative and lower triangular (with 0s on the diagonal),
matrix ( I − G ) is a regular –M-matrix and ( I − G ) −1 is lower-triangular (with 1s on the diagonal)
and nonnegative (Berman and Plemmons, 1979). Then the BOM can be expressed as follows:
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ω = ( I − G )−1 y .
(9)
There are N enterprises which are candidates to be part of the supply chain to be created.
Each candidate enterprise is characterized by its production resources: manufacturing plants,
machines, work teams, robots, pallets, storage areas, etc.
As in Van Gellekom et al. (2000), a coalition S is defined as a subset of the set N of N
enterprises with characteristic vector e S ∈ {0,1}N such that:
(e S ) j = 1
(e S ) j = 0
if j ∈ S
.
if j ∉ S
(10)
For the R types of resources considered (r=1,…,R), let crj be the amount of resource r available
at enterprise j, C = ((crj )) ∈ ℜ
R× N
, and mri the amount of resource r necessary to produce one unit
of product i, M = ((mri )) ∈ ℜ R×n .
Capacity constraints for coalition S are written:
n
∑ mriωi ≤ ∑ crj or equivalently
i =1
j ∈S
n
∑ mriωi ≤ ∑ crj (e S ) j .
i =1
(11)
j∈N
It can be noted that the left-hand side quantities in inequalities (11) fully characterize coalition S.
One originality of the proposed model is that it combines the coalition decision problem
through the choice of vector eS , with the multistage manufacturing model represented by
constraints (9) and (11). The second originality is related to the choice of the objective function that
aggregates the benefits expected from manufacturing activities. The value chain concept introduces
transfer prices for intermediate goods so that each manufacturing stage should be profitable.
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3.2. The Value chain
Let χ = ( χ 1 ,  , χ R ) T be the vector of unit costs for resources. The unit cost for product i, γ i , is
assumed to be the same for all the firms possessing the required resources. It is determined by the
cost resource requirement for producing one unit of product i:
R
γ i = ∑ χ r mri .
(12)
r =1
Let γ = (γ 1 ,, γ n )T be the vector of unit production costs for the products. The set of relations
(12) for all the products can be written in vector form:
γ = MTχ .
(13)
Prices may be partly or totally exogenous. Let p = ( p1  pn )T be the vector of market prices for
the final products.
A necessary and sufficient condition for the global profitability of the supply chain is:
pT y − γ T ω ≥ 0 ,
(14)
or, using relation (9),
[ pT − γ T ( I − G ) −1 ] y ≥ 0 .
(15)
The prices of final products can be supposed fixed and exogenous. The global profitability
condition (15) can be considered sufficient for the supply chain to be viable.
More restrictive conditions will now be established to allow for a total decomposability of the
multistage manufacturing process. It is now assumed that the prices of intermediate products are
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negotiated in the network so that each manufacturing stage is profitable. Under this more restrictive
requirement, the following inequality should be verified for any product i:
n
pi ≥ γ i + ∑ Gij p j
j =1
(16)
.
Let π = (π1,, π n )T be the vector of unit profits for products. The unit profit π i associated with
product i is:
n
π i = pi − γ i − ∑ Gij p j
j =1
(17)
.
Condition (16) corresponds to the profitability conditions π i ≥ 0 ∀i ∈ {1,, n} that characterize
the value chain. These conditions can be gathered into the following condition in vector form:
π = (I − G) p − γ ≥ 0 .
(18)
Condition (18) may appear unnecessarily restrictive. However, it can be used to fix the transfer
prices between products in the network so that each manufacturing stage is intrinsically profitable.
3.3. Global profit maximization
The maximal total payoff, v * , is obtained from the solution of the mixed variables’ Linear
Programming problem (P):
T
N
Maximize v =< ( I − G ) p − M χ , y >= ∑ π i yi
i =1
subject to M ( I − G ) −1 y ≤ Qe
(P)
y ∈ ℜn+ , e ∈ {0,1}N
with the cardinality of the coalition as the complementary objective to be minimized:
17
N
s = ∑ (e) j .
(19)
j =1
The secondary objective can be solved together with the global profit maximization problem by
adding a ‘small’ term − ε (
N
∑ (e
j =1
S
) j ) with 0 <ε <<1 to v in the objective function of (P). The modified
problem, (P'), is formulated as follows:
n
N
Maximize ϕ = ∑ π i y i − ε (∑ e j )
i =1
j =1
−1
subject to M ( I − G ) y ≤ QeS
y ∈ ℜ +n , e ∈ {0,1}
N
The term − ε (
N
∑ (e)
j =1
j
(P')
.
) is said to be ‘small enough’ if the optimal solutions of (P) and (P’) are
identical with respect to the optimal vector y * . This property is achieved for any value of ε smaller
than a threshold value ε 0 > 0. The optimal coalition of lowest cardinality S* is then obtained by
resolution of problem (P'):
S * = {j ; j ∈ {1,  , N }, e j * = 1}.
This coalition is supposed not empty. Accordingly, the maximal value function of the TU-game is
supposed strictly positive. It is exactly computed from the solution of (P’) by: v* = ϕ * +ε (
N
∑ e *) .
j =1
j
In problem (P), vector e is a vector of binary variables. By definition, problem ( PS ) relates to a
particular coalition S ⊆ N . It is defined by the same constraints as (P), but with the vector of
variables e replaced by the known vector eS given by (10):
(e S ) j = 1
(e S ) j = 0
if j ∈ S
.
if j ∉ S
18
N
Maximize vS = ∑ π i yi
i =1
subject to M ( I − G ) −1 y ≤ QeS
(PS)
y ∈ ℜ +n .
The TU-game (N, v) defined in this section is an LPG as defined in Owen (1975). It is an N-persons
game in which the value v(S ) of a coalition S is obtained as the solution of an LP problem, (PS),
defined by the nonnegative production matrix M ( I − G ) −1 , the nonnegative resource matrix Q and
the nonnegative unit profit vector π .
Several properties can be derived from this definition. In particular, it is super-additive:
v( S ) + v(T ) ≥ v( S ∪ T ) for all disjoint coalitions S and T , and it has a non-empty core (Owen,
1975). Superadditivity implies, in particular, that if S * is an optimal coalition, then v( S ) = v* for any
coalition S such that S * ⊆ S . In particular, the maximal total payoff, v* can simply be obtained for
the grand coalition, N , by solving problem PN .
3.4. The core of the linear production game
The core of a TU-game is often difficult to determine. Some subsets are of particular importance.
It is a classical result that in exchange economies (markets with transferable payoffs), every
competitive allocation is in the core (Shubik, 1959). For LPGs, the set of competitive allocations has
been characterized by Owen (1975) and called the Owen set by Van Gellekom et al. (2000).
Conversely, for coalitions with a limited number of players, as in the supply chains example, the
core may be substantially larger. On the other hand, the set of competitive allocations often reduces
to a single point. It is thus interesting, in particular regarding improving negotiation convergence and
robustness, to consider payoff allocations that lie in the core but possibly outside the set of
competitive allocations.
19
3.5. The Owen set
The numerical resolution of problem (P’) solves the global utility maximization problem presented
in section 3.3.: it defines the maximal total payoff, v*, and an optimal output vector y*. Problem (P)
can then be replaced by problem ( PS * ), in which all the variables are real numbers.
At this stage, however, the allocation problem, which determines the share of the total payoff to
be distributed to each coalition partner, still remains open.
Consider the dual of ( PS ), denoted ( DS ):
Minimize wS =
R
∑ qr ( S ) z r
r =1
−T
subject to ( I − G )
MTz ≥π
z = (z1, ,z R )T ∈ ℜ+R
( DS )
.
The coefficient of variable zr in the objective function is the quantity of resource r available for
production if coalition S is selected:
N
qr ( S ) = ∑ (eS ) j crj = ∑ crj .
j =1
(20)
j ∈S
It can be noted that the set of constraints of ( DS ) is the same for any coalition S. Further, since
the optimal dual variables zr* ( S ) can be interpreted as shadow prices for resources, they determine
a vector of payoffs, the so-called ‘Owen set’ for this TU-game, which is optimal in the context of a
purely competitive economy (Van Gellekom et al., 2000). Owen (1975) has shown that the Owen set
of an LPG is contained in the core of the LPG.
20
Consider the optimal solution (S*, y*) of problem (P). The purely competitive payoff profile
u* = u ( S *) is also called the Owen set or the Owen point, since it is generally a single point. It is
obtained from the solution (wS*, z*(S*)) of ( DS * ) through the following relation:
R

*
u j * = ∑ crj zr if j ∈ S *
.
u j * = ∑ (eS * ) j crj zr * or equivalently, 
r =1
r =1

=
∉
u
j
S
*
0
if
*
j

R
(21)
Expression (21) shows that the payoff of each player equals the value of his resource bundle
under the marginal price. Moreover, this vector of payoffs forms a subset of the core in this
production game. However, it will be shown that the Owen set allocation has some drawbacks, in
particular a lack of fairness that may induce a corresponding lack of stability robustness against the
objections to coalition S*.
3.6. Example
Consider the BOM of the example in Hennet (2003) with two final products (1 and 2), three
intermediate products (3, 4 and 5), the unit profit vector π = (22 25 0 0 0)T and the technical
0
0

matrix
G = 2

0
1
0
0
2
0
2
0
0
0
3
2
0
0
0
0
0
0
0
. Four resources are necessary for the five products at the different
0

0
0
1
0
manufacturing stages, with the following requirement matrix M = 
0

2
1
0
0
2
0
1
1
1
0
1
1
1
0
0 .
2

1
Ten enterprises are candidates for partnership in the supply chain. The amounts of the four
resources owned by the ten firms are represented in the following matrix:
21
0
0
C=
0

3
0
0
0
0
1
0
0
3
0
0 5 5 9 10 0 0 0 2 .
0 12 10 7 13 0 9 19 0

3 0 5 5 0 27 20 0 2
The optimal total payoff and optimal coalition are obtained from the solution of the LP (P)
(with the additional term to obtain a coalition of lowest cardinality). The maximal total payoff
is v* = 87.5 , obtained for y* = [0 3.5 0 0 0 ]T . It is obtained by the minimal coalition
S * = {3 4 5 6 7 8 9} and also, indeed, by any coalition containing S*.
The associated purely competitive payoff profile (Owen set) is obtained by formula (21) for
z * = [0 0 1.25 0] :
u* = [0 0 15.00 12.50 8.75 16.25 0.00 11.25 23.75 0] .
In this solution, the endowment of partner 7 is null. Yet, partner 7 is important to the coalition
since for coalition S * −{7} , the total payoff drops down to 46.875! Its individual marginal
contribution in S* is:
∆ S * ({7}) = v(S *) − v(S * \{7}) ≅ 40.625 .
The reason for a null endowment is that, with partner 7 in the coalition, resource 4 is in excess
and its shadow price falls to 0. The individual marginal contribution of partner 7 in the grand coalition
N is also strictly positive:
∆ N ({7}) = v(N ) − v(N \ {7}) = 87.5 − 59.375 = 28.125 .
From the marginal contribution principle (6), non nullity of ∆ N ({7}) is a necessary condition
for the existence of a core payoff profile (ui )i∈N with u7 > 0 .
22
4. Improving payoff functions with side payments
4.1. Some criticisms of the Owen set allocation
The property of not allocating any payoff to the players with resources in excess is inherent in
the Owen set since this solution rule derives from the duality principle in linear programming. In
duality theory, a shadow price indicates the value of one additional unit of the resource associated
with the corresponding primal constraint. Thus, if a dual variable is equal to zero (zr* = 0), this means
that the addition of one unit of resource r has no effect on the optimal objective function. Hence the
allocations based on the values of the optimal dual variables exhibit this property: the players with
scarce resources share the total worth of the coalition between them, and the players with excess
resources get a null payoff.
In light of these shortcomings, the Owen set solution may become critically stable since the
players with excess resources are indifferent as to whether they participate. Such a lack of incentive
toward NPPs is not desirable in supply chain design because it does not provide stability robustness
against objections.
Therefore, even if the allocation mechanism may better reward the players with scarce
resources, allocating a null imputation to the players with all resources in excess may not be the best
solution. The next section explores the possibility of constructing a core solution with a strictly
positive payoff for each player of the optimal coalition.
4.2. Trying to combine fairness with rationality
Due to the criticisms directed at the Owen set as a competitive solution, other practical
allocation schemes will be explored to ensure that the maximal payoff is fairly allocated to the
players of the LPG game. So, the objective is to find an allocation that is fair and belongs to the core
of the LPG. These requirements can be formulated by the following set of constraints:
23
n
∑x
i =1
i
= v* ;
∑ x j ≥ v( S )
(22)
∀S ⊂ N ; and
(23)
j ∈S
x j > 0 ∀j ⊂ S *
.
(24)
As shown in the previous example, the optimal payoff may strongly decrease if a player who only
owns resources that are globally in excess decides not to participate in the cooperative game. This is
why condition (24), denoted ‘fairness property’, has been added to core requirements (22) and (23);
it provides a positive gain to every partner of the optimal coalition. However, the existence of
solutions to the set of conditions (22), (23) and (24) is not guaranteed in general. The purpose of this
section is precisely to characterize the cases when such solutions exist.
By construction, the core of an LPG is not empty if problem ( PN ) has a finite optimal solution
that is strictly positive, v(N ) > 0 . This is clear from the fact that an Owen solution exists whenever
the dual problem ( DN ) , or equivalently ( DS * ), has a solution. Then, in order to explore whether the
core of the game contains solutions that also satisfy (24), the following set of imputations can be
constructed.
The purely competitive payoff profile u* = u ( S *) has been defined by (21). The set N can be
decomposed into three disjoint subsets, as follows:
N = S 0 * ∪S 1 * ∪S 2 *
(25)
with S0 * = {i; i ∈ S * and ui* = 0} , S1 * = {i; i ∈ S * and ui* > 0} , S 2 * = N − S * .
24
The cardinals of these sets are respectively denoted s 0 *, s 1 *, s 2 * with, by assumption,
s 0 * ≥ 1, s 1 * ≥ 1 . This corresponds to the case when the total payoff is strictly positive and some of
the players in the optimal coalition are NPP: their Owen allocation is null.
A new set of imputations, denoted w , is defined as follows:
wi = ui * +α = α ∀i ∈ S0 *

 wi = ui * − β ∀i ∈ S1 * ,
 w = u * = 0 ∀i ∈ S *
i
i
2

(26)
α ≥ 0 , β ≥ 0 , α s0 * = β s1 * .
(27)
with
 ∑ u i * −v ( S ) 



 i∈S1
β ≤ min 
 with S1 = S ∩ S1 * and s 1 = card ( S1 )
s1
S ⊂N






(28)
β < min ui *
(29)
i∈S *
1
Theorem 1
A set of imputations w that satisfies conditions (26)-(29) belongs to the core of the LPG.
Furthermore, there exist strictly positive values of β that satisfy (28)-(29) and define fair imputations
if
and
only
if
s0 *
β = min~
S ⊂ S s 1 s 0 * −s 0 s 1




 ∑ u j * −v( S ∪ S 2 *)  > 0
*  j ∈S

1


with
~
S ⊂ S*
and
s1 s 0 * − s0 s 1 * > 0 .
Proof
25
•
Efficiency : From condition (26),
n
∑ wi = ∑ ui * + α s0 * − β s1 * . Then, Property (22) for
i =1
i∈N
imputation w is obtained from relation (27) and the efficiency property of the Owen
n
n
i =1
i =1
imputation (
•
∑ ui * = v * ): ∑ wi = v * .
Fairness:
 ∑ u i * −v ( S ) 


 i∈S1

By construction of S1 * , min u i > 0 . Then, If β = min 
 > 0 , it is possible to
s1
i∈S *
S ⊂N


1




select β > 0 such that (22) and (23) are satisfied.
•
Rationality:
The Owen imputation belongs to the core and thus satisfies the rationality property:
∑ u j * ≥ v( S )
j ∈S
∀S ⊂ N
(30)
.
Consider now the new imputation w and a coalition S ⊂ N . Then, define S0 = S ∩ S0 * , with
S1 = S ∩ S1 * , S2 = S ∩ S2 * and s 0 = card ( S 0 ) , s 1 = card ( S1 ) .
∑ w j = ∑ u j * + α s0 − β s1
j ∈S
j ∈S
If α s 0 − β s 1 ≥ 0 , then condition (25) implies
∑ w j ≥ v(S ) for any value of
β satisfying
j ∈S
(27), (28) and (29).
26
Consider now the case α s 0 − β s 1 < 0 . Using relation (18), this condition can be equivalently
replaced (with s 0 * ≠ 0 ) by s1 s 0 * − s0 s 1 * > 0 . Then, condition
∑ w j ≥ v(S ) is satisfied for any
j ∈S
value of β that verifies
s0 *
β≤
s 1 s 0 * −s 0 s 1




 ∑ u j * −v( S ) .
*  j∈S

1


(31)
Then, by super-additivity of v , v( S 0 ∪ S1 ∪ S 2 ) ≤ v( S 0 ∪ S1 ∪ S 2 *) and the two sets have
the same parameters s 0 , s 1 . Therefore it suffices to test condition (31) for sets such that
s1 s 0 * − s0 s 1 * > 0 and s 2 = s 2 * . Conversely, if condition (31) for some set S implies β = 0 , then
imputation u * is the only one satisfying conditions (26)-(29).
4.3. Example
In the example of section 2.5, the optimal coalition is S * = {3 4 5 6
grand coalition N
7 8 9} and the
can be partitioned as follows: N = S0 * ∪S1 * ∪S 2 * with S0 = {7} ,
S1 = {3,4,5,6,8,9} and S 2 = {1,2,10}. The total number of coalitions in N is 210 − 1 = 1023 .
However, using theorem 1, only 2 6 − 1 = 63 coalitions have to be tested to determine the value of
β . In this example, s 0 * = 1, s 1 * = 6 and condition s1 s 0 * − s0 s 1 * > 0 requires s0 = 0 . Then,
the possible values of s1 ( s1 =1,…,6) generate the 2 6 − 1 sets for which condition (27) has to be
tested. The minimal bound, β = 1.25 is obtained for the set S * = {1,2,6,8,10} and since
β > min ui , the value β defines the following imputation which belongs to the core and satisfies
i∈ S
1
the condition of fairness:
27
w = [0.00 0.00 13.75 11.25 7.50 15.00 7.50 10.00 22.50 0.00] .
The proposed technique has actually constructed the set of core allocations w( β ) defined by
parameter β in the interval [0 1.25] and such that:
w( β ) = [0 0 15.00 - β 12.50 - β 8.75 - β 16.25 - β 6 β 11.25 - β 23.75 - β 0] .
For β > 0 , the solution w( β ) belongs to the core and has the property of fairness. The feasible
interval for β can be used as a negotiation space by the players of the optimal coalition.
4.4. Discussion
The situation described in the numerical example has only 10 players with resource capacities
that are not oversized. It has been shown that, in this particular case, the core is not reduced to a
single point and it is possible to construct imputations that are both fair and rational.
In contrast to this situation, other numerical experiments have been performed with more
players and several owners for each resource. These results show that, in this case, the core of an
LPG game is generally reduced to a single point, which is precisely the Owen competitive allocation.
This result is not surprising if one realizes that in an open market in which many players have similar
abilities and equipment, the core of an LPG tends toward a single point that characterizes the
perfectly competitive situation. Convergence of the core to the Owen set of an LPG has been shown
and characterized by Owen (1975) and Semet and Zamel (1984) for games in which players are
replicated when the number of replications tends to infinity. In a broader context, a well-known
result is that, for large numbers of players, the core of the game tends to be the set of competitive
allocations (Aumann, 1964).
Using the LPG framework, this study has provided a rational explanation for the difficulty in
conciliating economic efficiency with a fair repartition of profit. A related finding is that, in Linear
28
Production Games, cooperation and competition are complementarily entangled rather than
opposed. Many practical examples of cooperative competition, also called “co-opetition”
(Brandenburger and Nalebuff, 1997), can be found in today’s industrial world. A well-known example
is the short-term cooperation that took place between the competing companies IBM and Oracle to
develop ERPs for SMEs. On the one hand, coalition formation between retailers, manufacturers, and
suppliers can provide a competitive advantage on the market because of the increased
manufacturing possibilities generated by the sharing of competence and resources. On the other
hand, competition between owners of similar resources or between manufacturers of similar
products tends to decrease the value of these resources or products. As a consequence, it may
jeopardize the coalitional stability by decreasing the partners’ motivation to cooperate. If a resource
becomes potentially available beyond its necessary level, its marginal value decreases to zero, and so
does the profit allotted to its owners for possessing it. In this respect, a good business strategy for a
company could be to maintain the uniqueness of its products through technical progress and
innovation.
The search for cases when fair repartitions of profits exist also indicates that in order to
maintain good profit margins, an enterprise should become a partner of its complementors and
avoid associations with its competitors. The difficulty is that the same company may be both the
enterprise’s complementor and its competitor. In such a case, a possible path to reach a win–win
situation is not to enter the game as it is, but rather to change its rules, typically through negotiation
with the partners. The current industrial and commercial practices show that many agreements and
contracts are currently established to reach mutually profitable situations, with a particular attention
paid to the business legal regulations to be respected. In terms of practical significance, this study
has thus provided some clues for achieving economic profitability, by identifying some key properties
that generate positive profits: ownership or production of assets that are not easily available
elsewhere and cooperation with firms producing complements of the enterprise’s own products.
29
5. Conclusions
In the light of cooperative game theory, supply networks can be modelled as profit-maximizing
systems creating value in their socio-economic environment, considered as a market. Under the
assumption of a perfect sharing of resources and production capacities in their manufacturing
environment, supply network design problems can be represented as problems of optimal coalition
formation. Then, coalition stability requires efficiency and rationality in the distribution of profit
among the enterprises of the supply network. However, if the manufacturing environment is highly
competitive, only the owners of resources that are marginally scarce receive a strictly positive share
of the profit. A firm does not receive a share of the profit if all its resources are also possessed by
other firms in the manufacturing environment and are globally in excess. Its revenue simply covers its
costs. This result is consistent with the general equilibrium analysis in competitive economies.
However, such profit allocation rules are not motivating and may appear unfair to the firms that
belong to the optimal coalition without owning any scarce resource. This paper has shown that in
moderately competitive manufacturing environments, where resources are not very abundant, the
core of the game, which is the set of efficient and rational profit allocations, is not always restricted
to the purely competitive profit allocation rule. Under these conditions, a set of feasible allocation
rules has been constructed to guarantee a positive profit to all the enterprises in the supply network.
A possible extension of this work could be to use this feasible set as a negotiation set for the firms of
the supply network.
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