A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY
CHAIN WHERE LOGISTICS SERVICE
PROVIDERS TAKE PART
Ernesto Del R. Santibanez-Gonzalez
Department of Computer Science, Federal University of Ouro Preto (UFOP), Ouro Preto, M.G., Brazil
Geraldo Robson Mateus
Department of Computer Science, Federal University of Minas Gerais (UFMG), Belo Horizonte, M.G., Brazil
Henrique Pacca Luna
Computer Science Institute, Federal University of Alagoas (UFAL), Maceió, A.L., Brazil
Keywords: Logistics service providers, Supply chain design, Non-cooperative supply chain, Optimal network design,
Mathematical programming.
Abstract: This paper presents a mathematical model for the problem of designing not cooperative supply chain where
the logistic service providers take part. In this problem, manufacturers are not collaborating or having any
type of bargaining strategy among them, they compete for supplying products to retailers seeking to
maximize their profit. Logistics service providers also compete among them for providing logistics services
to manufacturers and delivering products to retailers. It is considered that manufacturers, logistics service
providers and retailers collaborate to maximize services. Normally this problem can not be modeled as an
optimization problem and we use a variational inequality approach to formulate it. The model determines
the optimal level of production for each manufacturer, the flows of products between manufacturers and
retailers, the flow of products each logistics service provider is going to move and the price retailers are
willing to pay to manufacturer in a non-cooperative environment. We demonstrate and discuss theory results
regarding existence and uniqueness of the solution for the model. An example is presented to illustrate some
properties of the problem.
1 INTRODUCTION
As mentioned by Frankel, Bolumole, Eltantawy,
Paulraj and Gundlach (2008), Giannakis and Croom
(2004), Gibson, Mentzer and Cook (2005),
Giunipero, Hooker, Joseph-Matthews, Yoon and
Brudvig (2008), Lambert, García-Dastugue, and
Croxton (2005), Larson, Poist and Halldórsson
(2007), Supply chain management (SCM) has
become a fertile field for the application of a wide
variety of disciplines, including finance, logistics,
operations management, operations research, and
information technology among others. For some
authors, the philosophy of SCM is to combine some
or all of these disciplines to produce a
comprehensive strategy for improving the
performance of the company (Giunipero et al.,
2008). The large number of articles published till
today, especially in the last twenty years, reflects the
enormous interest shown in SCM by the academic
and business world. Even when there is an enormous
number of publications, different authors agree that
despite the importance of SCM to gain competitive
advantages and improve the performance of
organizations (Cooper, Lambert & Pagh, 1997;
Croom, Romano & Giannakis, 2000; Elmuti, 2002;
Lambert, Cooper & Pagh, 1998; Gunasekaran, Patel
& Tirtiroglu, 2001; Sanders, 2009), there is still no
consensus on its definition, the limits for practical
application and the relationship between SCM and
409
Santibanez Gonzalez E., Robson Mateus G. and Pacca Luna H..
A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY CHAIN WHERE LOGISTICS SERVICE PROVIDERS TAKE PART.
DOI: 10.5220/0003617904090417
In Proceedings of the 13th International Conference on Enterprise Information Systems (SSSCM-2011), pages 409-417
ISBN: 978-989-8425-54-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
other disciplines or concepts (Frankel et al., 2008;
Gibson et al., 2005; Lambert et al., 2005; Larson et
al., 2007). For Wisner and Tan (2000) "The concept
is still evolving. There is no generally accepted
definition of SCM or a general understanding of
how SCM affects the characteristics and
organizational practices". Handfield and Bechtel
(2004) noted that "what we are seeing in the field of
SCM, which is that for years was defined in some
way, has now become larger with different
fragments of theory". Certainly the work developed
after year 2000 to define SCM, has helped to close
the gap of uncertainty and lack of agreement, but
still remain an open question.
This article argues that a SC is a set of
interacting organizations (among themselves) under
a common goal and are involved in the flow of
goods, services, resources and information. This is
characterized by the following basic elements:
a. Organizations and / or individuals can be
grouped under a common goal. The common goal
does not necessarily mean that all the organizations
share the same goal or objective.
b. The interactions between the organizations
can take many forms, such as exchange of raw
materials, exchange of goods, services, sale or
purchase of various resources, information
exchange, etc.;
c. The limit - or range- defining which
organizations are part of the supply chain, i.e. the
boundaries (scope) of the SC, is determined by the
type of problem that will be addressed and the
capabilities of the tools of analysis that are used.
For the purposes of this article, it is used the
concept of root manufacturer (provider) to describe a
provider that has no other provider, i.e. for which
there is no organization that provides to it products,
services or resources.
In this paper it is addressed the problem of
designing a supply chain involving the operation of
logistics service
providers (LSP) under a non-
cooperative environment. In particular it is worked
with triad structures for the supply chain which is
composed by manufacturers, retailers and LSP´
layers. In each layer, members of the supply chain
compete with other similar agents and work in a
non-cooperative scenario. Firms belonging to a
different layer of the supply chain work in a
collaborative environment. There are no firms with a
dominant position able to influence in the decisions
of the other members of the supply chain. Each
manufacturer wants to maximize its profitability and
the same is valid for retailers and LSP as well.
Manufacturers are located at the top layer of the
supply chain and are concerned with the production
of products and shipments to the retailers. The
manufacturers compete among them for delivering
products of equivalent quality to retailers through
LSP, whom also compete to attract the
manufacturers and deliver the products to retailers.
In this supply chain, the LSPs are located in the
middle layer. Each LSP is faced with handling and
delivering the products sold by manufacturers to
retailers, conducting transactions with both types of
agents - whom purchase the LSP services- and
retailers -whom purchase products from the
manufacturers-. Retailers are located at the bottom
layer of the supply chain. They demand a certain
quantity of products (single commodity) and agree
to purchase them from any manufacturer at a finite
price. Also transactions and prices per transaction
between manufacturers and LSP must be
determined. Till now authors are not aware of any
other paper working in this problem.
Supply chain design has been extensively
studied so far. For a discussion about the design
problem underlying it is recommended the book by
Simchi-Levi, Kaminsky and Simchi-Levi (2003)
besides the concept of supply chain assumed in this
paper is a bit different from the one discussed in the
book. For a detailed review of supply chain network
design problems and modeling approaches it is
recommended the work by Melo, Nickel and
Saldanha-Da-Gama (2009). Melo et al. (2009)
conducted a detailed literature review of facility
location models in the context of supply chain
management and particularly their applications to
supply chain network design. All the literature
reviewed in these works follows the traditional
supply chain network design models where there is
no competition among the agents and the models
and solution methods are focused on facility
location. For concepts and applications related to
network design in a broader context it is recommend
the book by Ahuja, Magnanti and Orlin (1999). For
additional background on supply chain, see also the
books by Bramel and Simchi-Levi (1997), Pardalos
and Tsitsiringos (2002), and the volume edited by
Simchi-Levi, Wu and Shen (2004).
This paper follows the work by Beckmann,
McGuire and Winsten (1956) who first identified the
applications of networks to conceptualize decision-
making of an organization and particularly in
manufacturing processes and product flows linking
also to the theory of firm. Related to this work is
also the work by Nash (1950, 1951) on game theory.
Nagurney Dong and Zhang (2002) addressed the
problem of a three tier supply chain network design
modeling where firms are located at the nodes of the
network, each firm have their individual profit-
maximization objective functions, and they seek to
determine the optimal flows between tiers of nodes
and also the prices of the product at the various tiers.
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
410
In the same line, Dong Zhang and Nagurney (2004)
developed a supply chain network model where a
finite-dimensional variational inequality was
formulated for the behavior of various decision
makers. Chiou (2008) considered a multi-tiered
supply chain network which contains manufacturers,
distributors and consumers and involving two-level
of decision makers. He proposed a new solution
scheme for the supply chain network design problem
formulated as a (non-convex and non-differentiable)
mathematical program with equilibrium constraints.
In some way, the problem addressed in this
paper could be considered as a Strategic Planning
problem involving a long-term planning horizon and
the selection of some mix of manufacturer, LSP and
retailers in order to satisfy customer´s demand.
Eventually could also involve a facility location
problem and capacity planning of
manufacturing/servicing units to supply retailers, as
well as the transportation capacity required among
manufacturers and retailers. Simchi-Levi et al.
(2003) stated that “the strategic level deals with
decisions that have a long-lasting effect on the firm.
These include decisions regarding the number,
location and capacities of warehouses and
manufacturing plants, or the flow of material
through the logistics network”. This statement
establishes a clear link between location models and
strategic SCM. Problems of this type are mostly
modeled as mixed integer linear programming
problems, for example, see the paper by Chauhan
Nagi and Proth (2004) and the annotated
bibliography for a discussion. However, in the
general case, the kind of problem we work on this
paper can not be modeled as an optimization
problem, and then we use a variational inequality
approach to formulate it. One of the central points in
this type of problem is to demonstrate the existence
of the solution and demonstrate its uniqueness.
Normally, the complexity of this problem derives
from considering that the cost functions for members
of the supply chain depend on each other, then the
Jacobian of the cost functions is no longer
symmetric.
The paper by Meixell and Gargeya (2005)
reviewed the literature on models to support the
design of global supply chain and focused on the
logistics aspects of the supply chain, i.e., the
movement of goods from the point of origin to the
point of consumption. Meixell and Gargeya (2005)
studied a supply chain design problem comprising
the decisions regarding the number and location of
production facilities, the amount of capacity at each
facility, the assignment of each market region to one
or more locations, and manufacturer selection for
sub-assemblies, components and materials. Global
supply chain design extends this definition to
include selection of facilities at international
locations, and the special globalization factors this
involves.
We do not make any distinction among global or
domestic supply chain design. Cohen, Fisher and
Jaikumar (1989) “present the main features that
differentiate an international supply chain from a
single-country model” (as cited in Vidal and
Goetschalkx, 1997). In this paper, the selection of
locations for production (manufacturers) and/or
distribution facilities in global supply chains scale is
modeled/considered in implicit way. The model
focuses on solving the problem of how the material
flows from manufacturers to retailers and the
definition of how many products should be
manufactured and delivered when a LSP take part.
This paper is organized as follows. In section 2 is
discussed the formulation of the supply chain design
problem involving the operation of logistics services
providers, whom attend the demand of
manufacturers for delivering products to their
customers. In Section 3 is derived and analyzed the
existence and uniqueness of the solution for the
problem. Finally in Section 4 conclusions are
presented.
2 PROBLEM FORMULATION
We address the problem of producing and delivering
homogeneous products from manufacturers to
retailers through logistics service providers
contracted by the manufacturers. The model consists
of n profit-maximizing manufacturers, with a typical
manufacturer denoted by i; m profit-maximizing
retailers, with retailers denoted by j and with a
typical demand denoted by
, and q profit-
maximizing logistics service providers (LSP), with a
typical LSP denoted by k. The manufacturers are
involved in the production of homogeneous
products, which can be purchased by the retailers,
who, in turn, make the product available to
consumers at the demand markets. The links in the
supply chain network denote
transportation/transaction operations. In every layer,
manufacturers, LSP and retailer compete in the
sense of Nash (1950, 1951). Each firm acts in his
own benefit and will determine his optimal choice
given the optimal choices of the competitors, in such
a way that the whole system gets an equilibrium
state.
A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY CHAIN WHERE LOGISTICS SERVICE PROVIDERS
TAKE PART
411
2.1 The Manufacturers Optimization
Problem
Each manufacturer i seeks to maximize its profit by
setting the total quantity
that he must produce
and determining for each retailer j, the sales price p
and the quantity of product x
to deliver to retailer j.
In this problem all the delivery services from the n
producers (i=1,…n) to the m retailers are contracted
to logistics service providers (LSP) k (k=1,…q) at a
price p
. A manufacturer i can contract one or more
LSP k to deliver the products purchased by the
retailer j. Manufacturer i have a service cost function
(
) to make available the products to the
retailer j through LSP k. Consider that
=
∑
x
∀=1…,
. There are not production costs
at this stage of the problem:
=
−
−
(
)
(1)
s.t.
≥0∀,
,
(2)
Notice that the total production of manufacturer i
must satisfy that.
=
∀=1,…, (3)
That is, all the products sold by manufacturer i to
all the retailers j, j=1,…,m is equal to all the
products delivered by manufacturer i to all the
retailers j through LSP k, k=1,…,q.
In this problem, the service costs functions
(
) for each manufacturer i are continuously
differentiable and convex. Assuming that the
manufacturers compete in a non-cooperative
fashion in the sense of Nash (1950, 1951), which
states, in this context, that each manufacturer will
determine his optimal production quantity and
shipments, given the optimal ones of the
competitors, the optimality conditions for all
manufacturers i simultaneously are as follow:
Determine
∗
∗
є R satisfying:
(
∗
+
(
∗
)
)
∗
−
∗
−
∗
∗
−
∗
≥0,
(4)
≥0,∀,
,
(4a)
Where
∗
∗
are the optimal values
for the corresponding variables.
2.2 The Logistics Service Providers
Profit Maximizing Problem
The LSPs are also profit-maximizing agents. They
seek to maximize the profit resulting from selling
their services to manufacturers and the costs of
servicing the retailers - the costumers of the
manufacturers-. Remember that all the LSPs
compete in the sense of Nash. For each LSP k, the
problem is the following:
=
−
(
)
(5)
s.t.
=
∀=1,…,
(6)
≥0,∀,
,
(6a)
LSPs incur in operating costs
(.) to take
products from manufacturer i and deliver to retailer
j. The cost function
(.) is continuous
differentiable and convex. The profit for LSP k in
(5) is given by the prices
charged to
manufacturer i to deliver the products to retailer j
minus the corresponding costs
(.). It is
reasonable consider that the LSP deliver all the
products arrived from the manufacturers, what is
modeled by (6).
As in the manufacturer case, the LSPs act in a
non-cooperative way and there are not bargaining or
any type of collaboration between them. The
optimality conditions for all LSPs simultaneously
are as follows, determine
∗
є R such that:
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
412
(
(
)
−
∗
)
∗
−
∗
≥0,
(7)
≥0,∀,
,
(7a)
Where
∗
is the optimal flow of products
coming from manufacturer i that LSP k will deliver
to retailer j.
2.3 The Retailers Problem
The retailers j purchase products to manufacturer i at
sale price
∗
, but they also consider in their
decision the unit cost
(
) of making the
transaction with this manufacturer i. These costs are
continuous and depend on the quantity of products
purchased by others retailers, then could be
interpreted as the procurement cost incurred by the
retailer in a non-cooperative scenario. The retailers
are willing to pay a demand price
∗
for the
products. Then the equilibrium conditions for all
retailers j=1,…,m are the following:
∗
+
=
∗
≥0
≥
∗
=0
(8)
=
∀
=1,…,
(9)
In the same way as the other agents in the supply
chain, considering that (8) must hold for all market
in equilibrium, the variational inequality problem
can be expressed as follows:
∗
+
∗
−
∗
≥0
(10)
∗
≥0∀,
(10a)
2.4 Supply Chain Management
Perspective
In a non-cooperative scenario with manufacturer,
retailers and logistics service providers seeking to
maximize their profit hold equilibrium conditions
for the supply chain in which the total quantity of
products that manufacturer produces and ships to
retailers through logistics service providers must be
equal to the amount of product purchased by a
retailer, as well as the quantity of products the
logistics service provider receive from the
manufacturer to be delivered to retailers, must be
equal to the shipment of the logistics service
providers to the retailers. Finally the shipments and
price pattern must satisfy the sum of inequalities (4),
(7) and (10) described previously. Formally this is
stated as follows:
2.4.1 Definition 1
The equilibrium state of the given supply chain
design problem in the presence of logistics service
providers is one where the flows of products
between manufacturers, logistics service providers
and retailers coincide and the product shipments and
prices satisfy the sum of the optimality conditions
(4), (7), and the conditions (10).
2.4.2 Theorem 1
A product shipment(
∗
,
∗
) is an equilibrium
pattern of the supply chain design model according
to Definition 1 if and only if it satisfies the
variational inequality problem:
(
+
∗
)∗(
−
∗
)
+
∗(
−
∗
)≥0
(11)
,
≥0∀,
,
(11a)
Proof
Consider the definition 1. Sum up the
inequalities (4), (7) and (10). After algebraic
operations it gets the inequality (11).
Now, consider the inequality (11). To the first
term in bracket of inequality (11) adds
−
,
then follows:
(
+
∗
+
−
)∗(
−
∗
)
(12)
To the second term in bracket of inequality (11)
add
−
then
A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY CHAIN WHERE LOGISTICS SERVICE PROVIDERS
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413
(
)+
−
∗(
−
∗
)
(13)
After the above additions (12) and (13),
inequality (11) can be rewritten as follows:
(
(
∗
)
+
)∗(
−
∗
)−
∗
∗
−
∗
+(
(
)
−
∗
)
∗
−
∗
+
∗
+
∗
−
∗
≥0
(14)
In the above inequality (14), the first two terms
of (14) are the same of (4), the third term is equal to
(7) and the last term is identical to (10). Hence
inequality (11) is the sum of conditions (4), (7) and
(10) according to Definition 1. So the proof is
complete.
3 THEORY RESULTS
In this Section, it is presented some qualitative
properties regarding inequalities (11). In particular it
derived the existence and uniqueness of the solution
to (11).
3.1 Theorem 2: Existence of the
Solution
Assuming that the feasible set is nonempty, then
variational inequality (11) admits a solution.
Proof
Since there is a finite demand for the products in
the market, i.e., d ≤ u for some μεR
. Then each
retailer j =1,…,m also demands a finite amount
≤
of product from the manufacturers, for some
μ
εR
.
By the side of manufacturers, each of them, i
=1,…,n, has a finite capacity of production
. Then
for each pair i, j, there is a finite capacity of
shipments
εR
for the retailers. That is
≤
.
Since
≤
, then there is
εR
such that
≤
for each i, j and k=1,…,q . This is, by the
side of LSP, they deliver a finite amount of products
to retailers already sold by manufacturers and
demanded in a finite amount by the retailers.
Suppose we define
as the quantity of
products retailer j is receiving from LSP k. Since
≤
, and
≤
for each i, j and k, then
≤
for some j,k.
So it can be said that the set
=
,
,
≥0
≤
,
≤
,
(15)
≤
,∀,
,
is bounded, closed and convex, then X is a
compact subset .
Now, let H be a real Hilbert space, whose inner
product is denoted by <,>. Let X be a nonempty
closed convex subset of H and A:X→ H a nonlinear
map. Then (11) can be written as the problem of
finding x
∗
in standard variational inequality format
such as,
∗
|
−
∗
≥0,∀
(16)
Where mapping A has a correspondence with the
terms in (11).
Assuming the mapping A is continuous, from
(15) and (16) there is a solution for (11).
3.2 Theorem 3: Uniqueness of the
Solution
Assume the conditions in Theorem 2 and that the
map A(X) is strictly monotone on X, that is:
−
−>0,∀,
(17)
Then the solution x* to variational inequality
(16) is unique.
Proof
Given that inequality (11) can be re-written as
inequality (16), follows from standard theory of
inequality the Theorem 3.
4 SOME NUMERICAL
EXAMPLES
In this section we provide a numerical example of
the model presented in previous section and also
discuss the results and some interesting issues. The
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
414
example was figure out in order to illustrate the
problem and be simple to solve it. We use some data
from the literature (Braess, Nagurney and
Wakolbinger, 2005) and adapt them to our problem.
The example was solved analytically, and algorithms
for solving general cases could be discussed in a
future paper.
Consider a problem as represented by the graph
depicted in Figure 1, with two manufacturers (m=2),
two logistics service providers (q=2) and one retailer
(n=1),
Figure 1: Example supply chain.
The transaction cost functions
faced by
manufacturers i and LSP k to meet demand of
customer j are given by:
=0,5
+50
;
=5
=0,5
+100
;
=10
The operating cost functions
of LSP k, to
deliver products from the manufacturer i to the
customer j are the following:
=5
;
=0,5
+50
=10
;
=0,5
+100
The transaction cost functions
associated
with the customer j in obtaining products from
manufacturer i are given by:
=5
;
=10
The demand is set to D=6 units.
Analyzing the data and after some algebraic
operations it is obtained the following values for the
variables
and
:
=
=3
=
=0
The demand price customer is willing to pay is
113 for the products sold by manufacturer 1. There
are no products purchased to manufacturer 2. Using
(8) we can obtain the price charged by manufacturer
1, that is equal to 83.
From this example and regarding the model, we
can observe the following issues:
i.- The demand price retailer(s) are willing to pay
to manufacturers make no difference whether the
retailer is serviced by LSP 1 or LSP 2. But in
practice, if LSP (anyone) has a value-added service
offering to customers, and the customers perceive
this difference among the LSPs then, they could be
willing to pay more for the same product but making
the difference by the service they receive. So this
fact, proved in practice, it is considered in the model
by the operating cost functions c
of LSP k, and
also by the price ρ
charged to manufacturer i by
LSP k to service customer j. Nevertheless, could be
also interested to include in the model a service cost
perceived by the customers depending of the LSP.
ii.- The demand price retailer(s) are willing to
pay is exactly the sum of the “operating costs” of
manufacturers and LSPs plus the cost of obtaining
the products. This last cost in fact, could include the
benefits perceived by retailers in making the
purchasing, and a kind of procurement costs,
including all the internal costs incurred by the
retailer in making the purchasing to manufacturers.
5 CONCLUSIONS
Supply chain management sometimes confounded
with logistics, is a multidisciplinary approach for
effective and efficient management of the supply
chain.. In turn, supply chain is a set of interacting
organizations (among themselves) under a common
goal and they are involved in the flow of goods,
services, resources and / or information. In this
paper is used this approach to model a non-
cooperative problem in a supply chain composed of
manufacturers and retailers and where there are
logistics service providers servicing the demand of
retailers. The model considers that manufacturers,
retailers as well as LSP act on their own advantage,
seeking to maximize their profit individually. Also,
the model considers that the agents located in
different layers -manufacturer, retailers and LSP-
collaborate to get the best available service level. In
the optimal solution, the model determines the flow
of products going from the manufacturers to retailers
and passing across the LSP. Hence the model
permits to handle the amount of products shipped
from each manufacturer to each retailer and
specifying the LSP servicing both the manufacturer
A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY CHAIN WHERE LOGISTICS SERVICE PROVIDERS
TAKE PART
415
and the retailer. The model also determines the price
the retailer agrees to pay for the products sold by
manufacturers. Some theory results are also
analyzed in term of existence and uniqueness of the
solution to the problem. An example is discussed to
illustrate the model.
ACKNOWLEDGEMENTS
This research was partially supported by the
Fundação de Amparo á Pesquisa do Estado de Minas
Gerais – FAPEMIG, CNPq e Capes - Brazil.
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A MODEL FOR DESIGNING NON COOPERATIVE SUPPLY CHAIN WHERE LOGISTICS SERVICE PROVIDERS
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