Int. J. Production Economics 131 (2011) 139–145
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Int. J. Production Economics
journal homepage: www.elsevier.com/locate/ijpe
A weighted max–min model for fuzzy multi-objective supplier selection in a
supply chain
A. Amid a,n, S.H. Ghodsypour b, C. O’Brien c
a
Islamic Azad University, Tehran North Branch, Engineering School, Industrial Engineering Department, Tehran, Iran
Industrial Engineering Department, Amirkabir University of Technology, Tehran, Iran
c
Nottingham University Business School, Jubilee Campus, NG8 1BB, UK
b
a r t i c l e in fo
abstract
Article history:
Received 6 July 2008
Accepted 28 April 2010
Available online 4 May 2010
Supplier selection is one of the most important activities of purchasing departments. This importance is
increased even more by new strategies in a supply chain. Supplier selection is a multi-criteria decision
making problem in which criteria have different relative importance. In practice, for supplier selection
problems, many input information are not known precisely. The fuzzy set theories can be employed due
to the presence of vagueness and imprecision of information. A weighted max–min fuzzy model is
developed to handle effectively the vagueness of input data and different weights of criteria in this
problem. Due to this model, the achievement level of objective functions matches the relative
importance of the objective functions. In this paper, an analytic hierarchy process (AHP) is used to
determine the weights of criteria. The proposed model can help the decision maker (DM) to find out the
appropriate order to each supplier, and allows the purchasing manager(s) to manage supply chain
performance on cost, quality and service. The model is explained by an illustrative example.
& 2010 Elsevier B.V. All rights reserved.
Keywords:
Supplier selection
Fuzzy multi-objective decision making
Weighted max–min model
1. Introduction
Within new strategies for purchasing and manufacturing,
suppliers play a key role in achieving corporate competitiveness.
Hence, selecting the right suppliers is a vital component of these
strategies. In most industries the cost of raw materials and
component parts constitutes the major cost of a product, such
that in some cases it can account for up to 70% (Ghobadian et al.,
1993). Thus the purchasing department can play a key role in an
organization’s efficiency and effectiveness because of the contribution of supplier performance on cost, quality, delivery and
service in achieving the objectives of a supply chain. Supplier
selection is a multiple criteria problem that includes both
qualitative and quantitative factors. The relative importance of
the criteria and sub-criteria are determined by top management
and purchasing managers based upon supply chain strategies.
In a real case, decision makers do not have exact and complete
information related to decision criteria and constraints. In these
cases the theory of fuzzy sets is one of the best tools to handle
uncertainty. Fuzzy set theories are employed in the supplier
selection problem due to the presence of vagueness and
imprecision of information. Amid et al. (2006) developed a
n
Corresponding author. Tel./fax: + 98 21 88787204.
E-mail addresses: a_amid@iau-tnb.ac.ir (A. Amid), ghodsypo@aut.ac.ir
(S.H. Ghodsypour), chris.obrien@nottingham.ac.uk (C. O’Brien).
0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijpe.2010.04.044
weighted additive fuzzy model for supplier selection problems
to deal with: imprecise inputs and the basic problem of
determining the weights of quantitative/qualitative criteria under
conditions of multiple sourcing and capacity constraints. In a
weighted additive model, there is no guarantee that the achievement levels of fuzzy goals are consistent with desirable relative
weights or the DM’s expectations. When the DM provides the
weight of the objective functions, the ratio of membership
functions achievement level should be as close as possible to
the ratio of objective weights in order to reflect the relative
importance of the criteria. However in the weighted additive
model, the ratio of achievement levels is not necessarily the same
as that of the objective weights.
In this paper, a weighted max–min fuzzy multi-objective
model has been developed to enable the purchasing managers to
assign the order quantities to each supplier based on supply chain
strategies.
2. Literature review
Dickson (1966) first identified and analyzed the importance of
23 criteria for supplier selection based on a survey of purchasing
managers. Weber et al. (1991) reviewed 74 articles discussing
supplier selection criteria. They also concluded that supplier
selection is a multi-criteria problem and the priority of criteria
depends on each purchasing situation. Roa and Kiser (1980) and
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A. Amid et al. / Int. J. Production Economics 131 (2011) 139–145
Bache et al. (1987) identified, respectively, 60 and 51 criteria for
supplier selection.
Weber and Current (1993) used a multi-objective approach to
systematically analyze the trade-offs between conflicting criteria in
supplier selection problems. Ghodsypour and O’Brien (1997, 1998)
developed an integrated AHP and linear programming model to
consider both qualitative and quantitative factors in a purchasing
activity. Karpak et al. (1999) used a goal programming model to
minimize costs and maximize delivery reliability and quality in
supplier selection when assigning the order quantities to each
supplier. Degraeve and Roodhooft (2000) developed a total cost
approach with mathematical programming to treat supplier
selection using activity based cost information. Ghodsypour and
O’Brien (2001) developed a mixed-integer non-linear programming
approach to minimize total cost of logistics, including net price,
storage, ordering costs and transportation in supplier selection.
Cebi and Bayraktar (2003) proposed an integrated model for
supplier selection. They used an integrated lexicographic goal
programming (LGP) and AHP to consider both quantitative and
qualitative conflicting factors. Barla (2003) proposed the supplier
selection and evaluation for a manufacturing company under lean
philosophy. The supplier selection and evaluation process was
conducted by multi-attribute selection model (MSM) in five basic
steps. Demirtas and Ustün (2008) proposed an integrated approach
using analytic network process (ANP) with multi-objective mixed
integer linear programming (MOMILP) to consider both tangible
and intangible factors in choosing the best suppliers and define the
optimum quantities among selected suppliers to maximize the
total value of purchasing and minimize the budget and defect rate.
Ng (2008) proposed a weighted linear model and a transformation
technique to solve a multi-criteria supplier selection problem.
Some authors have used fuzzy set theory (FST) to deal with
uncertainty. In fuzzy programming, the decision-maker (DM) is no
longer forced to formulate the problem in a precise and rigid form.
Based on fuzzy logic approaches, Erol and Ferrel (2003) proposed a
methodology that assists decision-makers to use qualitative and
quantitative data in a multi-objective mathematical programming
model. The methodology uses fuzzy QFD to convert qualitative
information into quantitative parameters. They used this methodology for selecting the best software system for a particular
application. Kwang et al. (2002) introduced a combined scoring
method with fuzzy expert systems approach for determination of
best supplier. Kahraman et al. (2003) developed a fuzzy AHP model
to select the best supplier firm providing the most satisfaction for
the criteria determined. Dogan and Sahin (2003) proposed a
supplier selection model for multi-periods under uncertainty
conditions. The supplier selection process is performed by choosing
the supplier that minimizes the present total additional costs
associated with the purchase decision. The activity-based cost is
used in their model.
These papers deal with single sourcing supplier selection in
which one supplier can satisfy all buyers’ needs. However, our
model discusses multiple sourcing (Ghodsypour and O’Brien,
1998). Kumar et al. (2004) proposed fuzzy goal programming for
the supplier selection problem with multiple sourcing that
includes three primary goals: minimizing the net cost, minimizing
the net rejections and minimizing the net late deliveries, subject
to realistic constraints regarding buyer’s demand and vendors’
capacity. In their proposed model, a weightless technique is used
in which there is no difference between objective functions. In
other words, the objectives are assumed equally important in this
approach and there is no possibility for the DM to emphasize
objectives with heavy weights. In the real situation for supplier
selection problems, the weights of criteria are different and
depend on purchasing strategies in a supply chain (Wang et al.,
2004).
As stated above, Amid et al. (2006, 2009) developed a weighted
additive fuzzy model for supplier selection problems to deal with:
imprecise inputs and the basic problem of determining weights of
quantitative/qualitative criteria under conditions of multiple
sourcing and capacity constraints. In their weighted additive
model, there is no guarantee that the achievement levels of fuzzy
goals are consistent with desirable relative weights or the DM’s
expectations (Chen and Tasi, 2001; Amid et al., 2006).
In this paper, a weighted max–min fuzzy multi-objective
model has been developed for the supplier selection problem to
overcome the above problem. This fuzzy model enables the
purchasing managers not only to consider the imprecision of
information but also to take the limitations of buyer and supplier
into account in calculating the order quantities from each
supplier. The analytic hierarchy process (AHP) approach is often
suggested for solving a complex problem and it has been applied
in a wide variety of decision making contexts (Saaty, 1978, 1990).
It also provides a structured approach for determining the
weights of criteria. AHP is used to determine the weights of
criteria in the model presented.
The paper is organized as follows: Section 3 presents the fuzzy
multi-objective model and its crisp formulation for the supplier
selection problem in which the objectives are not equally
important and have different weights. First, a general linear
multi-objective formulation for this problem is considered and
then some definitions and appropriate approach for solving this
decision making problem are discussed. Section 4 gives the
numerical example and explains the results. Finally, the concluding remarks are presented in Section 5.
3. The multi-objective supplier selection model
A general multi-objective model for the supplier selection
problem can be stated as follows (Weber and Current, 1993):
Min Z1 ,Z2 ,. . .,Zk
ð1Þ
Max Zk þ 1 ,Zk þ 2 ,. . .,Zp
ð2Þ
subject to:
x A Xd ,
Xd ¼ fx=gs ðxÞ rbs ,
s ¼ 1,2,. . .,mg
ð3Þ
in which the Z1, Z2,y,Zk are the negative objectives or criteria for
minimization like cost, late delivery, etc. and Zk + 1, Zk + 2,y,Zp are
the positive objectives or criteria for maximization such as
quality, on time delivery, after sale service and so on. Xd is the
set of feasible solutions that satisfy the set of system and policy
constraints.
It is clear that the supplier selection problem is an optimization problem, which requires that formulation of objective
functions. Not every criterion in this problem is quantitative.
This problem is recognized by Ghodsypour and O’Brien (1998).
They proposed an integrated method that uses AHP to deal with
both qualitative and quantitative criteria. A comprehensive
review of criteria for supplier selection is presented in Ghodsypour and O’Brien (1996). He concluded that the number and the
weights of criteria depend on purchasing strategies.
To have a typical model, the purchasing criteria are assumed to
be quality, net price and delivery in this paper. These objectives
were cited most often in ordering decision (Roa and Kiser, 1980;
Ghodsypour and O’Brien, 1998).
In order to formulate this model, the following notations are
defined:
D
xi
demand over period
the number of units purchased from the ith supplier
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A. Amid et al. / Int. J. Production Economics 131 (2011) 139–145
goals (Zk) and maximization goals (Zl) are given as follows:
per unit net purchase cost from supplier i
capacity of ith supplier
percentage of quality level of ith supplier
percentage of service level of ith supplier
number of suppliers
Pi
Ci
Fi
Si
n
8
1
>
<
mzk ðxÞ ¼ fmzk ¼ ðZkþ Zk ðxÞÞ=ðZkþ Zk Þ
>
:0
A typical model for purchasing a single item in multiple
sourcing networks is as follows:
n
X
Min Z1 ¼
Pi xi
ð4Þ
8
>
<1
mzl ðxÞ ¼ fmzl ¼ ðZl ðxÞZl Þ=ðZlþ Zl Þ
>
:0
for Zk r Zk
for Zk rZk ðxÞ r Zkþ
ðk ¼ 1,2,. . .,pÞ
for Zk Z Zkþ
ð14Þ
for Zl Z Zlþ
for Zl r Zl ðxÞ r Zlþ
for Zl r Zl ðl ¼ p þ1,p þ 2,. . .,qÞ
ð15Þ
i¼1
Max Z2 ¼
n
X
Fi xi
ð5Þ
n
X
Si xi
ð6Þ
i¼1
Max Z3 ¼
i¼1
subject to:
n
X
xi ZD
ð7Þ
i ¼ 1,2,. . .,n
i ¼ 1,2,. . .,n
xi Z0,
With Zimmermann’s approach (1978, 1993), using max–min
as the operator, the above fuzzy models (10)–(13) is equivalent to
solving the following crisp model:
ð16Þ
ð8Þ
subject to:
ð9Þ
l rfmZj ðxÞ, J ¼ 1,. . .,q ðfor all objective functionsÞ
The capacity constraint of a supplier is also considered in the
model. Three objective functions – net price (4), quality (5) and
service (6) – are formulated to minimize total monetary cost, and
maximize total quality and service level of purchased items.
Constraint (4) ensures that demand is satisfied. Constraint set (8)
means that order quantities to each supplier should be equal or less
than its capacity and constraint set (9) prohibits negative orders.
In a real situation for a supplier selection problem, all
objectives might not be achieved simultaneously under the
system constraints; the decision maker may define a tolerance
limit and membership function m (Zj(x)) for the jth fuzzy goals.
It was shown that linear programming problems (1)–(3) with
fuzzy goal may be presented as follows:
Find a vector xT ¼[x1, xn] to satisfy:
Z~ k ¼
and
obtained through solving the multi-objective
problem as a single objective using, each time, only one objective.
Zkþ is the maximum value (worst solution) of negative objective Zk
and Zl is the minimum value (worst solution) of the positive
objective function Zl (Lai and Hwang, 1994; Hwang and Yoon,
1981). Linear membership functions mzj ðxÞ are shown in Fig. 1.
Max l
i¼1
xi rCi ,
Zlþ are
Zk
r ¼ 1,. . .,m
gr ðxÞ r br ,
xi Z 0,
ð17Þ
i ¼ 1,. . .,n,
ð18Þ
l A ½0,1
ð19Þ
The max–min’s approach does not consider the relative
importance of objective functions. In this solution, objectives are
equally important.
One model that takes into account the objectives’ weights is
the additive model of Tiwari et al. (1987), which is formulated as
follows:
Max
q
X
wj mzj ðxÞ
ð20Þ
j¼1
k ¼ 1,2,. . .,p
ð10Þ
subject to:
gr ðxÞ r br , r ¼ 1,. . .,m
ð21Þ
l ¼ p þ 1, p þ 2,. . .,q
ð11Þ
xi Z 0,
ð22Þ
n
X
cki xi ¼ r Zk0 ,
n
X
cli xi Z Zl0 ,
i¼1
Z~ l ¼
i¼1
q
X
subject to:
gs ðxÞ ¼
n
X
wj ¼ 1,
wi Z 0
ð23Þ
j¼1
asi xi rbs ,
s ¼ 1,. . .,m
ð12Þ
i¼1
xi Z0,
i ¼ 1,. . .,n
i ¼ 1,. . .,n
ð13Þ
where cki, cli, asi and bs are crisp values. In this model, the sign indicates the fuzzy environment. The symbol r in the
constraints set denotes the fuzzified version of r and has
linguistic interpretation ‘‘essentially smaller than or equal to’’
and the symbol Z has linguistic interpretation ‘‘essentially
greater than or equal to’’. Zk0 and Zl0 are the aspiration levels that
the decision-maker wants to reach.
Zimmermann (1978) extended his fuzzy linear programming
approach to the fuzzy multi-objective linear programming problems
(10)–(13). He expressed objective functions Zj, j¼1,y,q, by fuzzy
sets whose membership functions increase linearly from 0 to 1. In
this approach, the membership function of objectives is formulated
by separating every objective function into its maximum and
minimum values. The linear membership function for minimization
where wj is the coefficient weighting that presents the relative
importance among the fuzzy goals. The following crisp single
objective programming is equivalent to the above fuzzy model:
Max
q
X
wj lj
ð24Þ
j¼1
1
µ l (zl)
µ k (zk)
Zk
-
Zk
+
Zl
-
Zl
+
Fig. 1. Objective function as fuzzy number: (a) min Zk and (b) max Zl.
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A. Amid et al. / Int. J. Production Economics 131 (2011) 139–145
subject to:
lj r fmzj ðxÞ, J ¼ 1,. . .,q
gr ðxÞ r br ,
ðfor all objective functionsÞ
r ¼ 1,. . .,m
ð25Þ
ð26Þ
lj A ½0,1, J ¼ 1,. . .,q
ð27Þ
q
X
ð28Þ
wj ¼ 1,
wj Z0
3.2. Model algorithm
j¼1
xi Z0,
i ¼ 1,. . .,n
ð29Þ
When the DM provides the weight of objective functions, the
ratio of achievement level membership functions should be as
close as possible to the ratio of objective weights in order to
reflect the relative importance of criteria. However in this model,
the ratio of achievement levels is not necessarily the same as that
of the objective weights.
Lin (2004) proposed a weighted max–min model such that the
ratio of the achievement level objective functions is as close to the
ratio of weight or importance of objectives. This model is
formulated as follows:
Max l
ð30Þ
subject to:
wj l rfmzj ðxÞ,
gr ðxÞ r br ,
J ¼ 1,. . .,q
ðfor all objective functionsÞ
r ¼ 1,. . .,m
ð31Þ
ð32Þ
l A ½0,1
q
X
management. Then by applying Saaty’s 1–9 scales, these preferences should be quantified and then a pairwise comparison
matrix can be structured. After that, the weights of criteria and a
consistency ratio (CR) will be calculated. For more details of
calculating priority in AHP see Anderson et al. (1994).
The proposed weighted max–min model for fuzzy multiobjective supplier selection problem is stated in the following
steps:
ð33Þ
wj ¼ 1,
wi Z 0
i ¼ 1,. . .,n
ð35Þ
This model is equivalent to solving (1)–(3) with new membership functions as follows:
8
1=wk
for Zk r Zk
>
<
ð36Þ
m0zk ðxÞ ¼ fmzk ðxÞ=wk for Zk r Zk ðxÞ rZkþ ðk ¼ 1,2,. . .,pÞ
>
:0
for Zk Z Zkþ
8
1=wl
>
<
m0zl ðxÞ ¼ fmzl ðxÞ=wl
>
:0
bound and upper bound for the jth objective (Zj).
Step 5: For the objective functions find the membership
function according to (14) and (15).
Step 6: Calculate the weight of the criteria.
Step 7: Formulate the equivalent crisp model of the weighted
max–min for fuzzy multi-objective problem according to
Eqs. (30)–(35).
Step 8: Find the optimal solution vector xn, where xn is the
efficient solution of the original multi-objective supplier
selection problem with the DM’s preferences.
ð34Þ
j¼1
xi Z0,
Step 1: Construct the supplier selection model according to the
criteria and constraints of the buyer and suppliers.
Step 2: Solve the multi-objective supplier selection problem as
a single objective supplier selection problem using each time
only one objective. This value is the best value for this
objective as other objectives are absent.
Step 3: From the results of step 2 determine the corresponding
values for every objective at each solution derived.
Step 4: From steps 2 and 3, for each objective function find a
lower bound and an upper bound corresponding to the set of
solutions for each objective. Let Zj and Zjþ denote the lower
for Zl ZZlþ
for Zl rZl ðxÞ rZlþ
ðl ¼ p þ 1,. . .,qÞ
ð37Þ
for Zl rZl
The new membership function values and optimal achievement level (ln) can exceed unity since wj o1. Nevertheless, the
actual achievement level for each objective may never exceed
unity. This model finds an optimal solution within the feasible
area such that the ratio of the achievement levels is as close to the
ratio of objective weights as possible (see Lin, 2004 if necessary).
To elicit weight or priority among goals/objectives from a DM
is a very important initial process to solve this model. For
specifying the weight of goals in this article the analytic hierarchy
process (AHP) (Saaty, 1978) is utilized.
3.1. Calculate the weight of objective functions
In the AHP method, after a definition of supplier selection
objectives, the hierarchy structure of the problem can be built.
Once the hierarchy has been structured the weights of criteria
should be calculated. By using pairwise comparison, the preference of criteria will be asked from top and purchasing
The model algorithm is illustrated through a numerical
example.
4. Numerical example
For supplying a new product to a market assume that three
suppliers should be managed. The purchasing criteria are net
price, quality and service that are defined as:
Net purchase price including transportation costs
Quality includes defects and manufacturing capabilities and
continuous quality improvement
Service involves delivery speed and reliability, response to
changes, product development, financial and organizational
capabilities.
According to the defined criteria, the structure of the problem is
as shown in Fig. 2.
The capacity constraint of a supplier is also considered in the
model. It is assumed that the input data from suppliers’
performance on these criteria are not known precisely. The
estimated values of their cost, quality and service level and
constraints of suppliers are presented in Table 1. The demand is
predicted to be about 1000.
The crisp formulation of the numerical example is
presented as:
Z1 ¼ 13x1 þ 11:5x2 þ 15x3
Z2 ¼ 0:8x1 þ 0:7x2 þ 0:95x3
Z3 ¼ 0:85x1 þ 0:75x2 þ0:80x3
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A. Amid et al. / Int. J. Production Economics 131 (2011) 139–145
Supplier selection
Quality
Manufacturing
capabilities
Defects
Net cost
Total quality
management
Service
On-time
delivery
Response
to changes
Product
development
Financial &Org.
capability
Fig. 2. Supplier selection criteria.
Table 1
Suppliers’ quantitative information.
Supplier 1
Supplier 2
Supplier 3
Table 3
Solutions to numerical example by different approaches.
Cost
Quality (%)
Service (%)
Capacity
13
11.5
15
80
70
95
85
75
80
700
600
500
l1
l2
l3
Table 2
Data set for membership functions.
Z1(net cost)
Z2(quality level)
Z3(service level)
Z1
Z2
Z3
x1
x2
x3
m¼0
m¼1
m¼0
–
740
770
12,100
875
835
14,000
–
–
Weighted max–min
Additive weighted
Weightless
12,803
792
780
150
542
308
0.56
0.19
0.17
12,100
740
807
400
600
0
1
0.0
0.3
13,048
807
802
388
336
275
0.5
0.5
0.5
w3 ¼0.16. Based on the weighted max–min models (30)–(35), the
crisp single objective formulation for this problem is as follows:
Max l
subject to:
0:63 l r
14000ð13x1 þ 11:5x2 þ15x3 Þ
1900
0:21 l r
ð0:8x1 þ 0:7x2 þ0:95x3 Þ740
135
0:16 l r
ð0:85x1 þ 0:75x2 þ 0:80x3 Þ770
65
subject to:
x1 þx2 þx3 ¼ 1000
x1 r700,
xi Z0,
x2 r600,
x3 r 500
i ¼ 1,. . .,3
Three objective functions Z1, Z2 and Z3 are cost, quality and
service, respectively, and xi is the number of units purchased from
the ith supplier.
The linear membership function is used for fuzzifying the
objective functions for the above problem according to (14) and
(15). The data set for the values of the lower bounds and upper
bounds of the objective functions are given in Table 2.
In Appendix A, the membership functions for three objectives
are provided by which to minimize the total monetary cost and
maximize the total quality and service level of the purchased
items. The fuzzy multi-objective formulation for the example
problem is as follows:
Find [x1, x2, x3] to satisfy:
H
Z~ 1 ¼ 13x1 þ 11:5x2 þ 15x3 r Z10
H
Z~ 2 ¼ 0:8x1 þ 0:7x2 þ0:95x3 Z Z20
H
Z~ 3 ¼ 0:85x1 þ 0:75x2 þ 0:80x3 Z Z30
subject to:
x1 þx2 þx3 ¼ 1000
x1 r700,
xi Z0,
x2 r600,
x1 þ x2 þ x3 ¼ 1000
x r1700,
x r 3500
x r2600,
x1 ,x2 ,x3 Z 0
The linear programming software LINDO/LINGO is used to
solve this problem. The optimal solution for the above formulation is as follows:
x1 ¼ 232,
x2 ¼ 600
Z1 ¼ 12550,
and
Z2 ¼ 770
x3 ¼ 167
and
Z3 ¼ 820
and achievement level objective functions are
mz1 ðxÞ ¼ 0:56, mz2 ðxÞ ¼ 0:19 and mz3 ðxÞ ¼ 0:17
Table 3 compares the solutions obtained by different
approaches. In Zimmermann’s weightless approach, there is no
difference between various importance of criteria and the
objectives are equally weighted; consequently, the achievement
level for all objective functions is
mz1 ðxÞ ¼ mz2 ðxÞ ¼ mz3 ðxÞ ¼ 0:5
x3 r 500
i ¼ 1,. . .,3
The weight of cost, quality and service obtained from example
Ghodsypour and O’Brien (1998), are w1 ¼0.63, w2 ¼0.21 and
Table 3 represents that the additive model is not acceptable
since the achievement levels are not corresponding to the weight
of objectives. The achieved level of the second objective is lower
than that of the third objective even though the weight of the
second objective is heavier than that of the third objective.
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A. Amid et al. / Int. J. Production Economics 131 (2011) 139–145
Comparing the solutions obtained by the weighted max–min
reveals that the proposed model manages to find an optimal
solution such that the ratio of the achieved levels is the same as
the weights of objectives and the solution is more consistent than
other solutions with the DM’s preferences or expectations. In
other word, (m1 4 m2 4 m3) agrees with (w1 4w2 4w3).
ðcÞ
mZ3 ðxÞ ¼
8
1
>
>
< Z 770
3
>
>
:
835
0
Z3 Z 835
770 o Z2 o 835
Z3 ¼ 0:85x1 þ 0:75x2 þ 0:8x3
Z2 r 770
References
5. Summary and conclusions
Supplier selection is a multiple criteria decision making
problem that includes both qualitative and quantitative criteria.
These tangible and intangible factors are not equally important. In
real cases, many input data are not known precisely for decision
making. Simultaneously, in this model, vagueness of input data
and varying importance of quantitative/qualitative criteria are
considered. The relative weights of criteria are obtained using
Saaty’s ANP method. In real cases, the proposed model can help a
DM to find out the appropriate order to each supplier, and allows
purchasing manager(s) to manage supply chain performance on
cost, quality, service, etc.
Moreover, the fuzzy multi-objective supplier selection problem is transformed into a weighted max–min fuzzy programming model and its equivalent crisp single objective LP
programming, in order that the achievement level of the objective
functions matches the relative importance of the objective
functions. This transformation reduces the dimension of the
system, giving less computational complexity, and makes the
application of fuzzy methodology more understandable. Finally,
the proposed model can be implemented in other multi-objective
optimization problems, in which the values of criteria are
expressed in vague terms and are not equally important.
Appendix A
Fig. 3 shows the membership functions:
ðaÞ
mZ1 ðxÞ ¼
8
1
>
>
< 14000Z
1
>
>
:
ðbÞ
mZ2 ðxÞ ¼
1900
0
12100 o Z1 o 14000
Z1 ¼ 13x1 þ 11:5x2 þ 15x3
Z1 Z14000
8
1
>
>
< Z 740
2
>
>
:
Z1 r12100
135
0
Z2 Z 875
740o Z2 o 875
Z2 ¼ 0:8x1 þ 0:7x2 þ 0:95x3
Z2 r 740
1
1
0
0
12100
net costs
14000
740
875
quality
1
0
770 835
service
Fig. 3. Membership functions: (a) net cost (Z1) objective function, (b) quality (Z2)
objective function and (c) service (Z3) objective function.
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