Int J Adv Manuf Technol (2008) 36:401–418
DOI 10.1007/s00170-006-0842-6
ORIGINAL ARTICLE
A supply chain distribution network design model:
An interactive fuzzy goal programming-based
solution approach
Hasan Selim & Irem Ozkarahan
Received: 20 April 2006 / Accepted: 12 October 2006 / Published online: 12 December 2006
# Springer-Verlag London Limited 2006
Abstract A supply chain (SC) distribution network design
model is developed in this paper. The goal of the model is
to select the optimum numbers, locations and capacity
levels of plants and warehouses to deliver products to
retailers at the least cost while satisfying desired service
level to retailers. A maximal covering approach is used in
statement of the service level. The model distinguishes
itself from other models in this field in the modeling
approach used. Because of somewhat imprecise nature of
retailers’ demands and decision makers’ (DM) aspiration
levels for the goals, a fuzzy modeling approach is used.
Additionally, a novel and generic interactive fuzzy goal
programming (IFGP)-based solution approach is proposed
to determine the preferred compromise solution. To explore
the viability of the proposed model and the solution
approach, computational experiments are performed on
realistic scale case problems.
Keywords Supply chain . Distribution network design .
Interactive fuzzy goal programming . Maximal covering
Abbreviations
DM
decision maker
FGP
fuzzy goal programming
FST
fuzzy set theory
GP
goal programming
IFGP
interactive fuzzy goal programming
SC
supply chain
TCOST
total cost
INV
investment
TSERVL total service level
H. Selim (*) : I. Ozkarahan
Department of Industrial Engineering, Dokuz Eylul University,
35100 Izmir, Turkey
e-mail: hasan.selim@deu.edu.tr
I. Ozkarahan
e-mail: irem.ozkarahan@deu.edu.tr
1 Introduction
The network design problem is one of the most comprehensive strategic decision issues that need to be optimized
for the long-term efficient operation of whole supply chain
(SC). The decisions made for network design determine the
number and locations of raw material suppliers, manufacturing plants, intermediate warehouses and distribution
centers as well as select the distribution channel from
suppliers to customers and identify the transportation
volume among distributed facilities for an extended time
horizon. Effective design and management of SC networks
assists in production and delivery of a variety of products at
low cost, high quality, and short lead times. It is obvious
that an effective SC network structure is crucial in gaining a
competitive operational performance.
To cope with the complexity of the SC network design
problem, the network has been divided into several stages
in many previous studies. The number of stages is
determined based on the tradeoffs between fragmented
network problem complexity and the network integrality
[1]. For the SC network design problem, Jang et al. [2]
decompose the entire network into three sub-networks,
inbound network, distribution network and outbound
network.
A common objective in designing a distribution network
is to determine the least cost system design such that the
retailers’ demand is satisfied without exceeding the capacities of the warehouses and plants. This usually involves
making tradeoffs inherent among the cost components of
the system that include cost of opening and operating the
plants and warehouses and the inbound and outbound
transportation costs.
There are two key decisions when designing a distribution network [3]: 1) Will products be delivered to the
customer location or picked up from a pre-ordained site? 2)
Will products flow through an intermediary? Based on the
402
choices for the two decisions, there are various types of
distribution network designs that may be used to move
products from manufacturing plants to customer. The
readers may refer to Chopra and Meindl [3] for the detailed
description of each distribution option and the discussion
on its strengths and weaknesses.
In this paper, we deal with a SC distribution network
comprised of a set of manufacturing plants, warehouses and
retailers. The network is illustrated in Fig. 1.
In the design option considered, inventory is stored
locally at retail stores and distributor warehouses. The main
advantage of such a network structure is that it can lower
the delivery cost and provide a faster response than other
networks. The major disadvantage is the increased inventory and facility costs. As emphasized by Chopra and
Meindl [3], such a network is best suited for fast-moving
items where customers value the rapid response.
Cost or profit-based optimization is the most widely
used method for SC distribution network design problems.
However, more customer oriented approaches are required
in order to provide a sustainable competitive advantage in
today’s business environment. Nowadays, there is a trend to
consider customer service level as more critical. Customer
service level can be measured by various measures such as
customer response time, consistency of order cycle time,
accuracy of order fulfillment rate, delivery lead time,
flexibility in order quantity.
In this paper, we develop a multiobjective SC distribution network design model. The goal is to select the
optimum numbers, locations and capacity levels of plants
and warehouses to deliver the products to the retailers at the
Fig. 1 Structure of the SC
distribution network
Int J Adv Manuf Technol (2008) 36:401–418
least cost while satisfying the desired service level to the
retailers. Since the maximal covering approach [4] has
proved to be one of the most useful facility location models
from both theoretical and practical points of view, we use it
in statement of the service level. In the proposed model, a
coverage function which may differ among the retailers
according to their service standard request is defined for
each retailer.
Much of the decision making in the real world takes
place in an environment in which the goals, the constraints,
and the consequences of the possible actions are not known
precisely. As in the same manner, SCs operate in a
somehow uncertain environment. Uncertainty may be
associated with target values of objectives, external supply
and customer demand etc. Supply chain distribution
network design models developed so far either ignored
uncertainty or consider it approximately through the use of
probability concepts. However, when there is lack of
evidence available or lack of certainty in evidence, the
standard probabilistic reasoning methods are not appropriate. In this case, uncertain parameters can be specified
based on the experience and managerial subjective judgment. Fuzzy set theory (FST) [5] provides the appropriate
framework to describe and treat uncertainty [6]. In decision
sciences, fuzzy sets have had a great impact in preference
modeling and multi-criteria evaluation and have helped
bringing optimization techniques closer to the users needs.
As emphasized previously, SC distribution network
design problem is a strategic decision problem, the
optimization of which is crucial for the long-term efficient
operation of whole SC. The decisions to be made have
Int J Adv Manuf Technol (2008) 36:401–418
long-lasting effects and are costly, or sometimes impossible
to reverse. Because data are often incomplete and imprecise, SC distribution network design decisions generally use
forecasts based on aggregated data. As the environment or
system changes, such impreciseness also propagates and
can exert serious effects on the operation and management
of the system. Another source of impreciseness may be due
to vagueness in DMs’ interpretation or intent. This situation
occurs when exactness is not required or is not possible to
specify, or when exactness would unnecessarily limit
options. As stated by Reznik and Pham [7], exact or crisp
models when used in such cases would not be able to
faithfully simulate the richness and subtlety of the real work
space. Furthermore, in the worst case, misleading or
incorrect outcomes may result. Thus, there is a need to
articulate the problems arising from such impreciseness
with the view to construct appropriate models for their
representation, as well as suitable methods for processing
and manipulating them within the environment. Considering this need, in this study, we use FST in handling SC
distribution network design problem. More specifically, two
kind of impreciseness that may be faced in the problem,
i.e., imprecision in retailers’ demand and DMs’ aspiration
levels for the goals, are treated. Additionally, to provide
the DMs with a flexible and robust multi-objective
decision making technique, we propose a novel and
generic interactive fuzzy goal programming (IFGP)-based
solution approach. Through the solution approach, DMs
determine the preferred compromise solution.
The paper is further organized as follows: A review of
the related literature is presented in the next section.
Thereafter, basic concepts and the framework of fuzzy goal
programming (FGP), and the proposed IFGP-based solution approach are presented in Sects. 3 and 4, respectively.
Section 5 is devoted to the presentation of the crisp
formulation of the proposed SC distribution network design
model. Computational experiments are presented in Sect. 6.
Finally, conclusions are presented in Sect. 7.
2 Literature review
Numerous researchers have extensively studied facility and
demand allocation problems. Interested readers may refer to
detailed survey results, e.g., Francis et al. [8], Aikens [9],
Brandeau and Chiu [10], Beamon [11], Avella et al. [12]
and Pontrandolfo and Okogbaa [13].
Beamon [11] provides a focused review of literature in
the area of multi-stage SC design and analysis and classify
the models in the area into four categories: deterministic
analytic models, stochastic analytic models, economic
models, and simulation models. Avella et al. [12] present
their views on the state of the art and the future trends in
403
location analysis. Pontrandolfo and Okogbaa [13] review
the literature on the configuration as well as the coordination of the network of global facilities problems. They
define a framework that systematically addresses the global
manufacturing planning problem by identifying and classifying the variables involved therein.
The last decades of the twentieth century witnessed a
considerable expansion of SCs into international locations.
This growth in globalization, and the additional management challenges it brings, has motivated both practitioner
and academic interest in global SC management [14]. In
this paper, we do not consider the global issues in the
proposed model.
After mentioning some important reviews on SC
network design problem, we present a review of the several
relevant papers in the following. In our review, we focus
especially on the papers that develop or consider linear
deterministic SC network design models.
Geoffrion and Graves [15] presents a new method for the
solution of the problem addresses the optimal location of
distribution centers between plants and customers. They
develop an algorithm based on Benders’ decomposition for
solving multi-commodity distribution network design problem. Brown et al. [16] present a mixed integer model for a
multi-commodity production-distribution system. The objective of the model is to minimize the variable production
and shipping cost, fixed cost of equipment assignment and
plant operations. They apply to the model a primal
decomposition technique similar to Geoffrion and Graves’
[15] algorithm. Cohen and Lee [17] present a strategic
model structure and a hierarchical decomposition approach.
The scope of their work is to analyze interactions between
functions in a complete SC network. To model these
interactions they consider four sub modules where each
represents a part of the overall SC: (1) material control, (2)
production control, (3) finished goods stockpile, and (4)
distribution network control.
Pirkul and Jayaraman [18] consider a tri-echelon, multicommodity system concerning production, distribution and
transportation planning. The authors use a Lagrangean
relaxation-based heuristic to provide effective an effective
feasible solution. Jayaraman [19] studies the capacitated
warehouse location problem that involves locating a given
number of warehouses to satisfy customer demands for
different products. Pirkul and Jayaraman [1] extend the
previous problem by considering locating also a given
number of plants. They present a model for multicommodity, multi-plant, capacitated facility location problem, and develop a Lagrangean-based heuristic solution
procedure. Dogan and Goetschalckx [20] develop a mixed
integer linear programming model for the integrated design
of multi-period production-distribution systems. Their
paper contributes to the literature by developing an
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integrated design methodology for strategic production and
distribution systems using primal decomposition theory. It
also provides an acceleration methodology for solving the
problem. Tragantalerngsak et al. [21] consider a twoechelon facility location problem in which the facilities in
the first echelon are incapacitated and the facilities in the
second echelon are capacitated. The goal in their model is
to determine the number and locations of facilities in both
echelons in order to satisfy customer demand of the
product. They develop a Lagrangean relaxation-based
branch and bound algorithm to solve the problem. Lee et
al. [22] develop a multi-product mixed integer nonlinear
programming model to develop a capacity expansion of an
integrated production and distribution system. The system
comprises the multi-site batch plants and warehouses.
Melachrinodis and Min [23] design a multi-objective,
multi-period mixed integer programming model that determine the optimal relocation site and phase out schedule of a
combined manufacturing and distribution facility from SC
perspectives. Their research differentiates from the literature by considering both dynamic aspects and multiechelon network design. Sabri and Beamon [24] develop
a SC model that considers simultaneous strategic and
operational SC planning. The main contribution of the
work is the incorporation of production, delivery, and
demand uncertainty into one model. Pirkul and Jayaraman
[25] present an integrated logistic model, and develop an
efficient solution procedure for multi-commodity production-distribution problem.
Tsiakis et al. [26] develop a strategic planning model for
SC networks. The paper takes into consideration flexible
production facilities in which a number of products are
produced making use of shared resources, the economies of
scale in transportation, and uncertainty in product demand.
Jang et al. [2] propose a supply network with a global bill
of material. They model design and planning problems of a
supply network in a single combined system. Talluri and
Baker [27] propose a multi-phase mathematical programming approach for effective SC network design. Cakravastia
et al. [28] develop a mixed integer programming model of
the supplier (any manufacturer playing a lower-level
supporting role) selection process in designing a SC
network. The assumed objective of the SC is to minimize
the level of customer dissatisfaction, which is evaluated by
two criteria: (i) price and (ii) delivery lead time.
More recently, Beamon and Fernandes [29] study a
closed-loop SC in which manufacturers produce new
products and remanufacture used products. The multiperiod integer programming model uses the present worth
method to jointly analyze investment and operational costs.
Yeh [30] presents a hybrid heuristic algorithm to solve the
multi-stage SC network design problem. The algorithm
combines a greedy method, the linear programming
Int J Adv Manuf Technol (2008) 36:401–418
technique and three local search methods, the pair-wise
exchange procedure, the insert procedure and the remove
procedure. Eskigun et al. [31] deal with the design of a SC
distribution network considering lead time, location of
distribution facilities and choice of transportation mode.
They present a Lagrangian heuristic that gives good
solution quality in reasonable computational time. In a
recent paper, Amiri [32] addresses the distribution network
design problem in a SC system. The author develops a
mixed integer programming model and provides a heuristic
solution procedure.
The contribution of our current paper to the literature is
twofold: First, a fuzzy multi-objective model has been
developed for SC distribution network design problem.
Second, a novel and generic IFGP-based solution approach is proposed to determine the preferred compromise
solution.
3 Fuzzy goal programming
Goal programming (GP) is one of the most powerful, multiobjective decision making approaches in practical decision
making. In a standard GP formulation, goals are defined
precisely. However, application of GP to the real life problems may be faced with two important difficulties. One of
which is expressing the DMs’ vague goals mathematically
and the second is the need to optimize all goals simultaneously. In such situations, the use of FST comes in handy.
Applying FST into goal programming (GP) has the
advantage of allowing for the vague aspirations of a DM,
which can then be qualified by some natural language
terms. The FST in GP was first considered by Narasimhan
[33]. Goal programming in fuzzy environment is further
developed by Hannan [34], Ignizio [35], Narasimhan and
Rubin [36], Tiwari et al. [37, 38] and others.
A fuzzy set A can be characterized by a membership
function, usually denoted by μ, which assigns to each
object of a domain its grade of membership in A. The
nearer the value of membership function to unity, the higher
the grade of membership of element or object in a fuzzy set
A. Various types of membership functions can be used to
represent the fuzzy set.
A typical FGP problem formulation can be stated as
follows:
Find
xi ;
i ¼ 1; :::; n
9
>
>
>
>
>
>
=
Zm ðxi Þ Z m
m ¼ 1; :::; M
Zk ðxi Þ Z k
gj ð x i Þ bj
k ¼ M þ 1; :::; K
>
>
>
>
j ¼ 1; :::; J
>
>
;
i ¼ 1; :::; n
xi 0
ð1Þ
Int J Adv Manuf Technol (2008) 36:401–418
405
where, Zm(xi) is the mth goal constraint, Zk (xi) is the kth
goal constraint, Z m ðxi Þ is the target value of the mth goal,
Z k ðxi Þ is the target value of the kth goal, gj (xi) is the jth
inequality constraint and bj is the available resource of
inequality constraint j.
In formulation (1), the symbols “≺ and ≻” denote the
fuzzified versions of “≤ and ≥” and can be read as
approximately less / greater than or equal to. These two
types of linguistic terms have different meanings. Under
approximately less than or equal to situation, the goal m is
allowed to be spread to the right-hand-side of Z m (Z m ¼ lm
where lm denote the lower bound for the mth objective)
with a certain range of rm(Z m þ rm ¼ um , where um denote
the upper bound for the mth objective). Similarly, with
approximatelygreater than or equal to, pk is the allowed
left side of Z k Z k pk ¼ lk ; and Z k ¼ uk .
As can be seen, GP and FGP have some similarities.
Both of them need an aspiration level for each objective.
These aspiration levels are determined by DMs. In addition
to the aspiration levels of the goals, FGP needs max-min
limits (uk,lk) for each goal. While the DMs decide the maxmin limits, the linear programming results are starting
points and the intervals are covered by these results.
Generally, the DMs find estimates of the upper (u) and
lower (l) values for each goal using payoff table (see
Table 1). Therefore, the feasibility of each fuzzy goal is
guaranteed.
Here, Zm(X) denotes the mth objective function, and X(m)
is the optimal solution of the mth single objective problem.
Solving the problem with X(m) (m=1,..., M) for each
objective, a payoff matrix with entries Zpm ¼ Zm X ðpÞ , m,
p=1,..., M can be formulated as presented in Table 1. Here,
um ¼ max ðZ1m ; Z2m ; . . . ; ZMm Þ and lm =Zmm, m=1,..., M.
After constructing fuzzified aspiration levels with respect
to the linguistic terms of approximately less than or equal
to, and approximately greater than or equal to, the
membership functions can be developed for each goal.
Using Belman & Zadeh’s [39] min operator approach,
one can obtain the feasible fuzzy solution set by the
intersection of all membership functions representing the
fuzzy goals. This solution set is then characterized by its
membership μF(x) which is:
mF ð xÞ ¼ mZ1 ð xÞ \ mZ2 ð xÞ:::: \ mZk ð xÞ
¼ min mZ1 ð xÞ; mZ2 ð xÞ; ::::; mZk ð xÞ
ð2Þ
Then the optimum decision can be determined to be the
maximum degree of membership for the fuzzy decision:
max mF ð xÞ ¼ max min mZ1 ð xÞ; mZ2 ð xÞ; :::; mZk ð xÞ
x2F
x2F
ð3Þ
By introducing the auxiliary variable λ, which is the
overall satisfactory level of compromise, formulation (2)
Table 1 The payoff table
X (1)
X (2)
..
.
X (M)
Z1(X)
Z2(X)
...
ZM(X)
Z11
Z21
...
ZM1
Z12
Z22
...
ZM2
...
...
...
...
Z1M
Z2M
...
ZMM
can be transformed to the following conventional linear
programming problem [40]:
9
maximize 1
>
>
>
>
>
>
subject to
>
>
>
=
1 μ Zk
k ¼ 1; :::; K
ð4Þ
gj ðxi Þ bj i ¼ 1; :::; n; j ¼ 1; :::; J >
>
>
>
>
>
xi 0
i ¼ 1; :::; n
>
>
>
;
1 2 ½0; 1:
4 The proposed IFGP-based solution approach
By use of the interactive paradigm, interactive fuzzy
multiobjective decision making approaches have been
investigated to improve the flexibility and robustness of
multiobjective decision making techniques. They provide
learning process about the system, whereby the DM can
learn to recognize good solutions and relative importance of
factors in the system [41]. The main advantage of
interactive approaches is that the DM controls the search
direction during the solution procedure and, as a result, the
efficient solution achieves his/her preferences [42]. Literature in the class of fuzzy interactive programming includes
Werners [43, 44], Leung [45], Fabian et al. [46], Sasaki et
al. [47] and Baptistella and Ollero [48].
Belman and Zadeh’s [39] min operator focuses only on
the maximization of the minimum membership grade. It is
not a compensatory operator. That is, goals with a high
degree of membership are not traded off against goals with
a low degree of membership. Therefore, some computationally efficient compensatory operators (see [41]) can be
used in setting the objective function in fuzzy programming
to investigate better results.
One criterion used to evaluate the performance of
compensatory operators in fuzzy optimization is monotonicity. Among the compensatory operators which are well
suited in solving multiobjective programming problems,
Werners’ [49] ‘fuzzy and’ operator has an advantage of
being a strongly monotonically increasing function. That is,
it is positively related with the compensation rate. Furthermore, it is easy to handle, and has generated reasonable
consistent results in applications. For those reasons, we
employ Werners’s ‘fuzzy and’ operator in the proposed
IFGP-based solution approach.
406
Int J Adv Manuf Technol (2008) 36:401–418
Werners [49] formulates the ‘fuzzy and’ operator as
follows:
(
)
X
μD ð xÞ ¼ Max + min ðμk ð xÞÞ þ ð1 + Þð1=K Þ
μk ð xÞ
k
k
ð5Þ
where K is the total number of fuzzy objectives and
constraints, μk(x) is the membership function of fuzzy goal
k, and + is the coefficient of compensation defined within
the interval [0,1]. By adopting min operator into Eq. (5),
the following linear programming problem can be formed:
maximize
+ 1 þ ð1 + Þð1=K Þ
subject to
μ k ð xÞ 1 þ 1 k ;
X
k
1 ; 1 k ; + 2 ½0; 1:
1k
9
>
>
>
=
8k 2 K; 8x 2 X >
>
>
;
ð6Þ
In real life decision problems, the relative importance of
the objectives assigned by the DMs may not be equal, and
may change over time. Different from ‘fuzzy and’ operator,
the proposed IFGP approach consider the relative importance of the objectives, and consequently provides a more
realistic structure.
To reflect the relative importance of λks to the objective
function, we develop and use a modified version of the
‘fuzzy and’ operator in the proposed solution approach.
More specifically, we use the following formulation:
P
maximize gl þ ð1 g Þ wk lk
ð7Þ
k
subject to the constraints in formulation (6), where
P
wk ¼ 1.
k
In order to determine the weights, there are some good
approaches in the literature, such as analytic hierarchy
process, weighted least square method and the entropy
method etc. Also, there are some fuzzy approaches for
finding crisp weights in fuzzy environment. However,
determination of the weights is not the focus of this study.
We think that the coefficient of compensation (γ) can be
treated as the degree of willingness of the SC partners to
sacrifice the aspiration levels for their goals to some extent
in the short run to provide the loyalty of their partners and/
or to strengthen their competitive position in the long run.
We also think that the coefficient of compensation can be
determined through a consensus decision making process.
In this process, complete unanimity is not the goal - that is
rarely possible. However, it is possible for each SC member
to have had the opportunity to express their opinion, be
listened to, and accept a group decision based on its logic
and feasibility considering all relevant factors. This requires
the mutual trust and respect of each member.
Fig. 2 Flowchart of the proposed IFGP-based solution approach
We assert that uncertainties of the input data and the
DMs’ aspiration levels for the goals in multiobjective linear
programming problems can be treated through the proposed
IFGP-based solution approach, and consequently, the
preferred compromise solution can be determined. Before
introducing the proposed solution approach, we think that
presentation of some definitions and theoretic explanations
related to the topic may be useful for clarity.
Definitions of the compromise solution and the preferred
compromise solution are presented in the following [42].
mij
LBi
Fig. 3 Illustration of linear coverage function
UBi
dtij
Int J Adv Manuf Technol (2008) 36:401–418
407
Table 2 Formulation of the
objective functions
Compromise solution: A feasible vector X* Z S is called a
compromise solution of the problem iff X* Z E and
Z ðX Þ ^X 2S Z ð X Þ where Z(X) is the objective function, S
is the feasible region, $ stands for “minimum” and E is the
set of efficient solutions.
This definition imposes two conditions on the solution
for it to be a compromise solution. First, the solution should
be efficient. Second, the compromise solution is the closest
solution to the ideal one that maximizes the underlying
utility function of the DM. In real-world cases, knowledge
of the set of efficient solutions E is not always necessary.
On the other hand, the DMs’ preferences are to be
considered in determining the final compromise solution.
Preferred compromise solution: If the compromise solution
satisfies the DMs’ preferences, then it is called the preferred
compromise solution.
The proposed IFGP-based solution approach can be
summarized in the following steps:
–
Step 1: Develop the conventional (crisp) linear
programming formulation of the problem.
Table 3 Expected demand of the retailers
Demand for product I (ai l)
992
423
659
806
107
500
741
257
956
129
408
573
873
222
377
439
783
936
317
887
556
388
467
501
777
759
516
327
664
929
831
683
653
669
701
264
507
174
878
609
Demand for product II (ai2)
352
378
649
483
709
449
455
528
484
514
229
898
844
306
845
758
508
916
171
403
282
968
960
789
195
736
198
630
955
186
894
845
935
968
610
976
496
813
424
505
976
885
847
334
776
923
927
320
658
700
880
988
629
971
743
339
811
466
720
724
–
–
–
–
–
Step 2: Obtain efficient extreme solutions (payoff
values) used for constructing the membership functions
of the objectives. If the DM selects one of them as a
preferred compromise solution go to Step 7. Otherwise
go to Step 3.
Step 3: Define the membership function of each fuzzy
objective using upper and lower bounds of the
objectives.
Step 4: Considering the membership functions defined
in Step 3 and γ (fix the value of γ to 1 in the first
iteration) develop the formulation of the problem using
proposed ‘modified fuzzy and’ operator.
Step 5: Obtain a compromise solution and present the
solution to the DM. If the DM accepts it, go to Step 7.
Otherwise, go to Step 6.
Step 6: Ask the DM if he want to modify the
coefficient of compensation (γ), and membership
functions of the objectives, and go to Step 3. Actually,
definition of a unique rule, e.g., selection of the initial
value, change direction or rate of variation in each step,
by which the value of γ is varied is difficult since it
depends on DMs’ preferences. For instance, if the
problem under concern is so sensitive to γ, the rate of
variation should be sufficiently slow. When the DM
tries to modify membership functions of the objectives
and constraints, only the following variations are
acceptable [43]: a) the increase of lower bound (lk)
for the maximization objectives, b) the decrease of
upper bound (um) for the minimization objective, c) the
Table 4 Lower and upper bounds for the distance
Retailers
Lower bound (LBi)
Upper bound (UBi)
1–12
13–27
28–50
500
600
700
650
750
850
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Int J Adv Manuf Technol (2008) 36:401–418
Table 5 The payoff table
TCOST
INV
TSERVL
–
Table 7 Compromise solution results with min operator
TCOST
INV
TSERVL
γ λ
2,720,667
3,250,000
3,250,000
2,663,710
2,229,724
2,994,668
52,046
50,000
61,532
1 0.5317 2,968,563 2,583,509 56,131
decrease of maximum tolerance (dj) is an acceptable
modification which can guarantee an efficient solution
in the recalculated compromise solution step. In order
to avoid the possibility of getting into infeasible
solution sets because of excess increase of lk or excess
decrease of um, we should increase lk and decrease um
as few requirements as possible in each iteration.
Step 7: Stop.
The flow chart of these steps is shown in Fig. 2.
In this section, we present the crisp formulation of the
proposed SC distribution network design model. The goal
of the model is to select the optimum numbers, locations
and capacity levels of plants and warehouses to deliver the
products to the retailers at the least cost while satisfying the
desired service level to the retailers. Maximal covering
approach is used in statement of the service level, and a
coverage function which may differ among the retailers
according to their service standard request is defined for
each retailer. The proposed model distinguishes itself from
other models in this field in the modeling approach used.
Because of somehow imprecise nature of retailers’ demand
and DMs’ aspiration levels for the goals, fuzzy modeling
approach is used. Additionally, a novel and generic IFGPbased solution approach is proposed to determine the
preferred compromise solution.
The mathematical model is developed on the basis of the
following assumptions:
–
–
–
INV
TSERVL μTCOST μINV
μTSERVL
0.5317 0.5317 0.5317
The retailers have demand for a multitude of products,
and the warehouses are responsible for right-time
delivery of a right amount of products.
Decision makers of the plants, warehouses and retailers
share information and collaborate with each other to
design an effective distribution network.
Decisions are made within a single period.
5.1 Notations and definitions
Definitions of sets, parameters and decision variables of the
proposed model are presented in the following.
5 Crisp formulation of the proposed model
–
–
TCOST
The network considered encompasses a set of retailers
with known locations, and possible discrete set of
location zones/sites where warehouses and plants are
located.
Different capacity levels are available to both the
potential plants and warehouses.
Table 6 Lower and upper bounds for the objectives
–
Sets
I
J
K
L
R
H
set of zones where retailers are located,
potential warehouse locations,
potential plant locations,
set of products,
set of capacity levels available for warehouses,
set of capacity levels available for plants.
–
Parameters
Tjkl
Cijl
fkh
gjr
OPkh
OWjr
ail
sl
Wjr
Objectives
Lower bound
Upper bound
TCOST
INV
TSERVL
2,720,667
2,229,724
50,000
3,250,000
2,994,668
61,532
ql
Dkh
dtij
variable cost to transport one unit of product
l from the plant in zone k to the warehouse in
zone j
variable cost to transport one unit of product
l from the warehouse in zone j to the retailer in
zone i
fixed portion of the operating cost of the plant in
zone k with capacity level h
fixed portion of the annual possession and
operating costs of the warehouse in zone j with
capacity level r
opening cost of the plant in zone k with capacity
level h
opening cost of the warehouse in zone j with
capacity level r
demand for product l by the retailer in zone i
required throughput capacity of a warehouses for
product l
throughput capacity of the warehouse in zone j
with capacity level r
required production capacity of a plant for
product l
capacity of the plant in zone k with capacity level h
distance between zone i and zone j
Int J Adv Manuf Technol (2008) 36:401–418
Table 8 Solution results of the
model by the proposed solution
approach
a
409
γ
λ
TCOST
INV
TSERVL
μTCOST
μINV
μTSERVL
1a
0.9
0.8-0.7
0.6-0.5
0.4-0.3
0.2-0
0.5317
0.5317
0.5280
0.5234
0.4374
0.4231
2,968,563
2,947,188
2,924,110
2,909,366
2,827,727
2,823,297
2,583,509
2,583,509
2,586,257
2,589,729
2,654,768
2,665,540
56,131
56,131
56,229
56,185
57,216
57,310
0.5317
0.5721
0.6157
0.6435
0.7977
0.8061
0.5317
0.5317
0.5280
0.5234
0.4374
0.4231
0.5317
0.5317
0.5402
0.5364
0.6257
0.6339
min operator
mij
LBi,
UBi
–
Yjkl
Xijl
Zjr
Pkh
coverage parameter that denotes the coverage level
of the retailer in zone i by the warehouse in zone j
the parameters that are used in defining the
coverage parameter of the retailer in zone i, LBi
and UBi denote lower and upper bound for the
distance, respectively (see Fig. 3).
Decision variables
amount of product l transported to the warehouse in
zone j from the plant in zone k,
amount of product l transported to the retailer in zone
i from the warehouse in zone j,
binary variable that indicates whether a warehouse
with capacity level r is constructed in zone j,
binary variable that indicates whether a plant with
capacity level h is constructed in zone k.
5.2 The objective functions
houses, the third one maximizes the total service level
provided to the retailers.
5.3 The constraints
The constraints of the proposed model and their definitions
are presented in the following.
X
Xijl ¼ ail for all i 2 I and l 2 L;
ð11Þ
j
Constraint set (11) ensures that all demand from retailers
is satisfied by warehouses.
XX
X
sl Xijl Wjr Zjr for all j 2 J ;
ð12Þ
i
r
l
Constraint set (12) limits the distribution quantities that
are shipped from warehouses to retailers to the throughput
limits of warehouses.
X
Zjr 1 for all j 2 J ;
ð13Þ
r
Objective functions of the model are formulated as follows:
As can be seen from Table 2, the first objective function
minimizes total cost made of: the transportation costs of
products from plants to warehouses and from warehouses to
retailers, and the fixed costs associated with the plants and
the warehouses. While the second objective function
minimizes the investment in opening plants and wareFig. 4 Solution results by the
proposed solution approach
Constraint set (13) ensures that a warehouse can be
assigned at most one capacity level.
X
X
Xijl Yjkl for all j 2 J and l 2 L;
ð14Þ
i
k
Constraint set (14) guarantees that all demand from
retailer in zone i for product l is balanced by the total units
410
Int J Adv Manuf Technol (2008) 36:401–418
Table 9 Solution results obtained by the proposed solution approach
in terms of utilization of the plants and warehouses
γ
1
0.9
0.8-0.7
0.6-0.5
0.4-0.3
0.2-0
Utilization of the plants
Utilization of the warehouses
Min
Max
Average
Min
Max
Average
0.87
0.87
0.89
0.87
0.87
0.88
1
1
1
1
1
1
0.97
0.97
0.97
0.97
0.97
0.96
0.90
0.91
0.82
0.90
0.86
0.75
1.00
1.00
1.00
1.00
1.00
1.00
0.97
0.97
0.97
0.97
0.97
0.93
of product l available at warehouse in zone j that has been
supplied from open plants.
XX
X
qkl Yjkl Dkh Pkh for all k 2 K;
ð15Þ
j
l
h
Constraints in set (15) represent the capacity restrictions
of the plants in terms of their total shipments to the
warehouses.
X
Pkh 1 for all k 2 K;
ð16Þ
h
Constraint set (16) ensures that a plant can be assigned
at most one capacity level.
Zjr 2 f0; 1g for all j 2 J ; r 2 R; Pkh 2 f0; 1g
for all k 2 K; h 2 H
ð17Þ
Finally, constraint set (17) enforces the binary and nonnegativity restrictions on the decision variables.
6 Computational experiments
6.1 The problem and parameter structuring
To explore the viability of the proposed model and the
IFGP-based solution approach, computational experiments
are presented in this section. The experiments are classified
into two categories. Imprecision in the DMs’ aspiration
levels for the goals is treated in the first category, while
imprecision in both the retailers’ demand and DMs’
aspiration levels for the goals is treated simultaneously in
the second category. Solutions of the proposed model are
performed using Werners’ (1988) ‘fuzzy and’ operator and
the proposed solution approach, and the results are compared.
A hypothetically constructed SC distribution network
design problem with 50 retailer zones, 20 potential
warehouse sites and 15 potential plant sites is considered
in the computational experiments. It is assumed that two
different types of product are demanded by the retailers.
Coordinates of the retailer zones, potential warehouses and
plant sites are generated from a uniform distribution over a
square with side 3000. Euclidean distances are used in
defining the coverage parameters. CPLEX 9.1 optimization
software is used at the solution stage.
Before presenting the computational experiments, let us
explain the parameter structuring of the hypothetical SC
distribution network design problem under concern.
Expected demand of the retailers for two different
products is drawn from a uniform distribution between
100 and 1000 as given in Table 3.
Five capacity levels are used for the capacities available
to both the potential plants and warehouses. The opening
cost of the warehouse in zone j with capacity level 3 (OWj3)
are drawn from a uniform distribution between 90,000 and
120,000. The opening costs of the warehouses for the other
capacity levels are computed as follows: OWj1 =0.75*OWj3,
OWj2 =0.85*OWj3, OWj4 =1.15*OWj3, OWj5 =1.25*OWj3.
Cost coefficients of OPkh are computed in terms of the
warehouses costs as OPkh =4*OWkh. Fixed portion of the
annual possession and operating costs of the warehouse in
zone j with capacity level 3 (gj3) and the plant in zone k
with capacity level 3 (fk3) are drawn from a uniform
distribution between 18,000 and 25,000 and 75,000 and
100,000, respectively. Fixed portion of the annual possession
and operating costs of warehouses and plants for the other
capacity levels are computed as follows: gj1 =0.75*gj3, gj2 =
0.85*gj3, gj4 =1.15*gj3, gj5 =1.25*gj3 and fk1 =0.75*fk3, fk2 =
0.85*fk3, fk4 =1.15*fk3, fk5 =1.25*fk3.
Required throughput capacity of a warehouse for product
l and required production capacity of a plant for product l are
given as follows: s1 =1, s2 =1 and q1 =1, q2 =2. The cost
coefficients Cijl and Tjkl are computed as being proportional
to the Euclidean distance among the locations of warehouses
and retailers, and plants and warehouses, respectively.
Table 10 Solution results of the model by Werners’ ‘fuzzy and’ operator
γ
λ
TCOST
INV
TSERVL
μTCOST
μINV
μTSERVL
1
0.9
0.8-0.6
0.5
0.4-0.1
0
0.5317
0.5317
0.5280
0.5234
0.4448
0.0000
2,968,563
2,947,407
2,924,274
2,909,366
2,879,877
2,879,877
2,583,509
2,583,509
2,586,257
2,589,729
2,649,159
2,649,159
56,131
56,135
56,233
56,185
58,338
58,338
0.5317
0.5716
0.6154
0.6435
0.6992
0.6992
0.5317
0.5317
0.5280
0.5234
0.4448
0.4448
0.5317
0.5320
0.5405
0.5364
0.7231
0.7231
Int J Adv Manuf Technol (2008) 36:401–418
411
Fig. 5 Solution results of the
model by Werners’ ‘fuzzy and’
operator
Specifically, Cijl and Tjkl are drawn from a uniform
distribution between 0.025*dtij and 0.035*dtij and 0.045*dtjk
and 0.055*dtjk, respectively. The parameters that are used in
defining the coverage parameter of retailer in zone i (LBi,
UBi), are given in Table 4. Throughput limit of warehouse in
zone j with capacity level r (Wjr) and capacity of the plant in
zone k with capacity level h (Dkh) are taken as follows.
Wjr =4000, 6000, 8000, 10000, 12000, Dkh =15000,
20000, 30000, 35000, 40000.
6.2 Implementation category I: treatment of fuzzy
aspiration levels for the goals
In this category of experiments, the imprecision in goal
achievement is allowed through the specification of an
interval of acceptable achievement rather than a crisp
value.
µa
il
6.2.1 Solution by the proposed approach
Solution of the SC distribution network design problem by
the proposed approach is presented step by step in the
following.
–
Step 1:
The crisp formulation of the problem has been developed using Eqs. (8 to 17) in Sect. 5.
– Step 2:
Efficient extreme solutions of the problem are presented
in Table 5.
It is assumed here that the DMs don’t choose any of the
efficient extreme solutions as the preferred compromise
solution and proceed to Step 3.
– Step 3:
Considering the efficient extreme solutions given in
Table 5, the lower and upper bounds of the objectives can
be determined. In our case, the corresponding minimum
and maximum values of the efficient extreme solutions are
determined as the lower and upper bounds, respectively, as
presented in Table 6.
Table 11 The payoff table
0.8 a il
ail
1.2 a il
Fig. 6 Membership function of the retailers’ demand
ail
TCOST
INV
TSERVL
TCOST
INV
TSERVL
TCOST
INV
TSERVL
2,236,907
8,405,707
21,173,751
2,720,667
3,250,000
3,250,000
2,401,231
1,480,163
8,076,997
2,663,710
2,229,724
2,994,668
41,180
11,072
73,839
52,046
50,000
61,532
412
Int J Adv Manuf Technol (2008) 36:401–418
Membership functions of fuzzy objectives can be
formulated now using the upper and lower bounds as
follows:
8
0
if
TCOST > 3; 250; 000
>
>
>
< 3; 250; 000 TCOST
if
2; 720; 667 < TCOST 3; 250; 000
μTCOST ¼
> 3; 250; 000 2; 720; 667
>
>
:
1
if
TCOST 2; 720; 667:
ð18Þ
μINV ¼
8
0
>
>
>
<
2; 994; 668 INV
>
2;
994;
668 2; 229; 724
>
>
:
1
if
INV > 2; 994; 668
if
2; 229; 724 < INV 2; 994; 668
if
INV 2; 229; 724:
ð19Þ
μTSERVL ¼
8
1
>
< TSERVL 50; 000
>
: 61; 532 50; 000
0
if
TSERVL > 61; 532
if
50; 000 < TSERVL 61; 532
if
TSERVL 50; 000:
ð20Þ
–
Step 4:
Considering the membership functions in Step 3,
and using the ‘modified fuzzy and’ operator, mathematical formulation of the problem can be developed as
follows:
Table 12 Lower and upper bounds for the objectives
Objectives
Lower bound
Upper bound
TCOST
INV
TSERVL
2,236,907
1,480,163
50,000
3,250,000
2,994,668
73,839
9
maximize γλ þ ð1 γ Þ½0:45λ1 þ 0:20λ2 þ 0:35λ3 >
>
>
>
subject to
>
>
=
μTCOST λ þ λ1 ;
μINV λ þ λ2 ;
>
>
>
>
μTSERVL λ þ λ3 ;
>
>
;
μTCOST ; μINV ; μTSERVL ; λ; λk ðk ¼ 1; 2; 3Þ; γ 2 ½0; 1
ð21Þ
and other system constraints (11 to 17).
As we stated previously, the relative weights for the
membership functions of the objectives can be determined
by the DMs using various methods. It is assumed here that,
the weights are determined by the DMs as presented in the
objective function of the model.
Table 13 Solution results of the model by the proposed solution approach
γ
λ
TCOST
INV
TSERVL
μTCOST
μINV
μTSERVL
1
0.9-0.6
0.5
0.4-0.3
0.2
0.1
0
0.3327
0.3327
0.2938
0.2669
0.2390
0.0932
0.0000
2,912,471
2,911,738
2,758,226
2,680,598
2,628,604
2,511,505
2,428,348
2,484,436
2,484,436
2,542,970
2,583,538
2,625,496
2,500,082
2,486,397
57,947
57,931
57,004
56,362
55,697
52,221
50,161
0.3327
0.3339
0.4854
0.5620
0.6134
0.7290
0.8110
0.3327
0.3327
0.2938
0.2669
0.2390
0.3223
0.3314
0.3327
0.3327
0.2938
0.2669
0.2390
0.0932
0.0068
Int J Adv Manuf Technol (2008) 36:401–418
–
Step 5:
By fixing the value of γ to 1, the solution given in
Table 7 is obtained.
It is assumed here that the DMs need more improvement
in the results, and want to consider the solution results of
the model with different coefficient of compensation (γ) to
make the final decision.
– Step 6:
In this step, a set of solutions corresponding to the
different values of γ are obtained and presented to the DMs.
The results are given in Table 8.
The results presented in Table 8 are illustrated graphically in Fig. 4.
Besides the cost, investment and service level, utilizations of the plants and warehouses are important indicators
which should be considered in designing a SC distribution
network. Therefore, we compare the solution results in
terms of utilizations of the plants and warehouses in
Table 9.
If the results of the model with γ=0.4 (or 0.3) are
compared to those of the model with γ=1, it can be realized
that a substantial increase (50.03%) can be provided in
achievement level of the membership function of total cost
objective (μTCOST) with a decrease by 17.74% in that of the
second (μINV) objective. Furthermore, achievement level of
the total service level objective is also improved by
17.68%. It can also be concluded that all solution results
seem reasonable from the utilization rates of the plants and
warehouses point of view. The average utilization rates are
considerably high in all of the solution alternatives.
Considering the results given in Tables 8 and 9 together,
let’s suppose that the DMs accept the results of the model
with γ=0.4, and that they consider this solution the
preferred compromise solution. Then, the procedure is
terminated at Step 7.
Fig. 7 Solution results of the
model with the proposed solution approach
413
Table 14 Solution results of the proposed model in terms of
utilization of the plants and warehouses
γ
1
0.9-0.6
0.5
0.4-0.3
0.2
0.1
0
Utilization of the plants
Utilization of the warehouses
Min
Max
Average
Min
Max
Average
0.97
0.92
1
0.97
0.95
0.97
0.87
1
1
1
1
1
1
1
0.99
0.97
1
0.99
0.98
0.99
0.97
0.67
0.61
0.79
0.63
0.56
0.67
0.67
1
1
1
1
1
1
1
0.95
0.92
0.97
0.91
0.93
0.95
0.95
6.2.2 Solution by Werners’ ‘fuzzy and’ operator
Using Werners’ ‘fuzzy and’ operator we can formulate the
SC network design problem under concern as follows:
9
maximize + 1 þ ð1 + Þ½ð1 1 þ 1 2 þ 1 3 Þ=3
>
>
>
>
subject to
>
>
=
μTCOST 1 þ 1 1
μINV 1 þ 1 2
>
>
>
>
μTSERVL 1 þ 1 3
>
>
;
μTCOST ; μINV ; μTSERVL ; 1 ; 1 k ðk ¼ 1; 2; 3Þ; + 2 ½0; 1
ð22Þ
and other system constraints (11 to 17).
In the same manner as the previous application, a set of
solutions corresponding to the different values of γ are
obtained in this application. The results are presented in
Table 10 and illustrated graphically in Fig. 5.
414
Int J Adv Manuf Technol (2008) 36:401–418
Table 15 Solution results of the problem with 'fuzzy and' operator
γ
λ
TCOST
INV
TSERVL
μTCOST
μINV
μTSERV
1
0.9-0.6
0.5
0.4
0.3
0.2
0.1
0
0.3327
0.3327
0.3313
0.3308
0.2948
0.2762
0.2443
0.0000
2,912,471
2,911,738
2,914,338
2,914,819
2,763,686
2,706,503
2,654,303
2,500,376
2,484,436
2,484,436
2,470,430
2,467,130
2,535,344
2,569,532
2,573,163
2,430,673
57,947
57,931
57,898
57,887
57,028
56,584
55,823
51,679
0.3327
0.3339
0.3313
0.3308
0.4800
0.5365
0.5880
0.7399
0.3327
0.3327
0.3420
0.3442
0.2989
0.2762
0.2738
0.3684
0.3327
0.3327
0.3313
0.3308
0.2948
0.2762
0.2443
0.0704
Considering the results given in Table 10, let’s
suppose that the DMs accept again the results of the model
with γ=0.4.
6.3 Implementation category II: treatment of fuzzy demand
and fuzzy aspiration levels for the goals
We assume here that the retailers’ demand pattern can be
affected to some extent using demand management techniques. That is, the retailers’ demand can be altered within a
range. Under this assumption, we state retailers’ demand as
fuzzy parameters using triangular membership function as
illustrated in Fig. 6. The lower and upper bounds for each of
the retailer’s demand are assumed 80% and 120% of their
corresponding expected demand (ail) values, respectively.
6.3.1 Solution by the proposed approach
Efficient extreme solutions of the problem are presented in
Table 11. The upper part of the table is constructed by
solving the problem considering the individual objective
functions subject to fuzzy constraint set while the crisp
constraint set is considered in the lower part.
As in the same manner of our preceding experiment with
the proposed solution approach, it is assumed here that the
Fig. 8 Solution results of the
problem with Werners’ ‘fuzzy
and’ operator
DMs don’t select any of the efficient extreme solutions as
the preferred compromise solution and proceed to the next
step.
Considering the efficient extreme solutions given in
Table 11, the lower and upper bounds of the objectives are
determined as documented in Table 12.
Using the ‘modified fuzzy and’ operator, mathematical
formulation of the problem can be developed as follows:
max imize
γλ þ ð1 γ Þ½0:45λ1 þ 0:20λ2 þ 0:35λ3 subject to
μTCOST λ þ λ1 ;
μINV λ þ λ2 ;
μTSERVL λ þ λ3 ;
μDEMANDil λ;
μTCOST ; μINV ; μTSERVL ; μDEMANDil ; λ; λk ; γ 2 ½0; 1
9
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
;
ð23Þ
and other system constraints (11 to 17).
A set of solutions for the above problem with different
values of γ are obtained and presented in Table 13.
The solution results presented in Table 13 are illustrated
graphically in Fig. 7.
The solution results are also compared in terms of
utilization of the plants and warehouses in Table 14.
Int J Adv Manuf Technol (2008) 36:401–418
415
Table 16 Comparison of the
solution results
Solution approach
Implementation category
Objective function
Proposed
approacha
Werners’ 'fuzzy
and' operatorb
Category I: Fuzzy aspiration
levels for the goals
TCOST
INV
TSERVL
Utilization of plants
2,827,727
2,654,768
57,216
0.87
1
0.97
0.86
1
0.97
2,758,226
2,542,970
57,004
1
1
1
0.79
1
0.97
2,879,877
2,649,159
58,338
0.84
1
0.95
0.81
1
0.97
2,763,686
2,535,344
57,028
0.98
1
1
1
1
1
min
max
aver.
min
max
aver.
Utilization of warehouses
Category II: Fuzzy demand
and fuzzy aspiration levels for the
goals
TCOST
INV
TSERVL
Utilization of plants
γ=0.4 and γ=0.5 for the first
and second category,
respectively.
b
γ=0.4 and γ=0.3 for the first
and second category,
respectively.
min
max
aver.
min
max
aver.
a
Utilization of warehouses
6.3.2 Solution by Werners’ ‘fuzzy and’ operator
Comparing the results given in Table 13 together with
consideration of the utilization of the plants and warehouses, let’s suppose that the DMs accept the results of the
model with γ=0.5, and that they consider this solution the
preferred compromise solution. Then, the procedure is
terminated at Step 7.
If the results of the model with γ=0.5 are compared to
those of the model with γ=1, it can be realized that
substantial increase (45.9%) can be provided in achievement level of the total cost objective with a decrease by
11.7% in those of the second and third objective function.
In terms of utilization rates of the plants and warehouses,
the model with γ=0.5 provides better results compared to
the other solutions.
Fig. 9 Comparison of the
results in terms of TCOST and
INV objectives
Using Werners’ ‘fuzzy and’ operator, we obtain a set of
solutions corresponding to different values of γ. The results
are presented in Table 15 and illustrated graphically in
Fig. 8.
6.4 Comparison of the results
The solution results obtained by the proposed solution
approach and Werners’ ‘fuzzy and’ operator for the two
different implementation categories are compared in
Table 16.
The results presented in Table 16 are illustrated
graphically in Figs. 9, 10 and 11.
If the results presented in Table 16 are analyzed, it can
be seen that, for the first category of experiments, the
2,900,000
2,850,000
2,800,000
2,750,000
2,700,000
2,650,000
2,600,000
2,550,000
2,500,000
2,450,000
2,400,000
Proposed approach
Werners' approach
TCOST
INV
Category I
TCOST
INV
Category II
416
Int J Adv Manuf Technol (2008) 36:401–418
Fig. 10 Comparison of the
results in terms of TSERVL
60,000
TSERVL
58,000
Proposed approach
56,000
Werners' approach
54,000
52,000
50,000
Category I
relative importance levels assigned to the membership
functions of the objectives can be reflected to the results
to some extent by the proposed solution approach. More
specifically, by using the proposed approach with given
weighting structure, total cost objective can be improved by
1.81% while total service level objective is deteriorated by
1.92%, and investment objective remains almost the same
compared to the results by Werners’ approach. On the other
hand, the results obtained by the proposed and Werners’
approaches are almost the same for the second category of
experiments. That is, the relative importance of the
objectives doesn’t influence on the results of the problem
instance. It can be concluded here that, the proposed
solution approach may provide different and even more
preferable results compared to the Werners’ ‘fuzzy and’
approach. But, it should be emphasized that the weight
structure and the structure of the problem, e.g., fuzzy vs.
crisp parameters, influences the level of differences
between the two approaches. The proposed IFGP-based
solution approach can be employed by the DMs as a
flexible and robust multi-objective decision making approach to determine the preferred compromise solution.
Fig. 11 Comparison of the
results in terms of utilization of
the plants and warehouses
Category II
7 Conclusions
A multi-objective linear programming model is developed
in this paper to address the SC distribution network design
problem. The goal of the model is to select the optimum
numbers, locations and capacity levels of plants and
warehouses to deliver the products to the retailers at the
least cost while satisfying the desired service level. The
model distinguishes itself from other models in this field in
the modeling approach used. Decision makers’ imprecise
aspiration levels for the goals and retailers’ imprecise
demand are incorporated into the model using fuzzy
modeling approach, which is otherwise not possible by
conventional mathematical programming methods.
This paper also contributes to the literature by proposing
a novel and generic IFGP-based solution approach which
determines the preferred compromise solution for multiobjective decision problems.
Results of the computational experiments performed on
realistic scale case problems explore the viability of the
proposed solution approach and the SC distribution
network design model. The results also point out that SC
1
0.9
0.8
0.7
0.6
Proposed approach
0.5
Werners' approach
0.4
0.3
min
aver.
Util.of plants
min
aver.
Util. of
warehouses
Category I
min
aver.
Util.of plants
min
aver.
Util. of
warehouses
Category II
Int J Adv Manuf Technol (2008) 36:401–418
distribution network design problem can be handled in a
more flexible, robust and realistic way through the
proposed model and the IFGP-based solution approach.
The proposed model can be used in real life industrial
application in restructuring, i.e., expanding or narrowing, of
an existing SC distribution network besides the design of a
new network. Such a strategic model can be a part of a
decision support system developed for the collaborative SC
management practices.
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