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Supplier selection using revised multi-segment
goal programming model
Article in International Journal of Advanced Manufacturing Technology · February 2014
DOI: 10.1007/s00170-013-5368-0
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Supplier selection using revised multisegment goal programming model
Hossein Karimi & Alireza Rezaeinia
The International Journal of
Advanced Manufacturing Technology
ISSN 0268-3768
Volume 70
Combined 5-8
Int J Adv Manuf Technol (2014)
70:1227-1234
DOI 10.1007/s00170-013-5368-0
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Author's personal copy
Int J Adv Manuf Technol (2014) 70:1227–1234
DOI 10.1007/s00170-013-5368-0
ORIGINAL ARTICLE
Supplier selection using revised multi-segment
goal programming model
Hossein Karimi & Alireza Rezaeinia
Received: 5 November 2011 / Accepted: 27 September 2013 / Published online: 18 October 2013
# Springer-Verlag London 2013
Abstract Selecting the best supplier has been turned into a
strategic subject in the competitive market, in the recent years.
Supplier selection is one of the significant multiple criteria
decision-making problems, which is studied in this paper. This
research applies the multi-segment goal programming formulation, which considers different positions in the goal programming. This new kind of goal programming is revised in this
investigation. The advantage of the proposed method is that it
allows decision makers to set multiple aspiration levels for the
coefficient of variables. Hence, the decision makers obtain more
revenue and match with the reality of their problems. Unlike the
multi-segment goal programming proposed in the literature, the
revised formulation is linearized in the current study. The supplier
selection problem is solved by this new formulation in a special
circumstance. Finally, some concluding notes are remarked.
Keywords Supplier selection . Goal programming . Multiple
aspiration levels . Revised multi-segment goal programming
achievement of goals and their aspiration levels. In a characteristic manner, GP can be expressed as follows:
min
N
X
j f i ðxÞ−gi j
ð1Þ
i¼1
s:t:
ð2Þ
x∈ F
where f i (x) and g i are the linear function and the goal of the
ith objective, respectively, and F is a feasible set. Also, N is
the number of criteria.
Based on this formulation, researchers proposed many
formulations for GP, which had different attitudes. We introduced some of these models, which have been frequently
applied by different researchers.
Ignizio proposed weighted GP (WGP) and lexicographic
GP (LGP) [7]. These formulations are the following in order:
min
N
X
αi d iþ þ β i d i−
i¼1
ð3Þ
1 Introduction
s:t :
Goal programming (GP) is a kind of multiple objective programming, which is the progeny of multi-criteria decision
making (MCDM). In this research, the methodology known
as GP has been inspired from [1]. Further developments are
provided for instance by [2–6]. GP lets decision makers
(DMs) fix their aspiration levels for each goal. The objective
of GP is to minimize the undesirable deviations between the
f i ðxÞ−d iþ þ d i− ¼ gi ; i ¼ 1; 2…; N
ð4Þ
d iþ ; d i− ≥ 0;
ð5Þ
H. Karimi (*)
Department of Industrial Engineering, K. N. Toosi University of
Technology, No. 17, Pardis St., Molla Sadra St.,
Vanak Sq., Tehran, Iran
e-mail: [email protected]
A. Rezaeinia
Project Planning and Control, National Iranian Oil Company,
Pars Economic Special Zone, Assaluyeh, Iran
e-mail: [email protected]
i ¼ 1; 2; …; N
and Eq. (2) where d +i and d −i are the positive and negative
deviations between the ith objective and its goal, respectively;
α i and β i are the positive weights to the deviations.
2
3
n
n
n
X
X
X
þ
þ
þ
−
−
− 5
4
min
αi d i þ βi d i ; …;
αi d i þ β i d i ; …;
αi d i þ β i d i
i¼hi
i¼hr
i¼hQ
s: t :
ð6Þ
−
f i ðxÞ−d þ
i þ d i ¼ gi ;
i ¼ 1; 2; …; N ;
i∈hr
r ¼ 1; 2; …Q
ð7Þ
and Eqs. (2 and 5) where h r stands for the index set of goals
placed in the rth level. The other variables are defined in
WGP.
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Flavell presented min–max or Chebyshev GP (MGP), minimizing the maximum deviation [8]. This method is provided
as follows:
minD
s:t:
ð8Þ
−
D ≥ αi d þ
i þ βi d i ;
ð9Þ
f i ðxÞ−d þ
i
þ
d i−
¼ gi ; i ¼ 1; 2; …; N
N
X
ð2d i − f i ðxÞ þ gi Þ
ð10Þ
ð11Þ
i¼1
s: t:
d i −f i ðxÞ þ gi ≥ 0; i ¼ 1; 2; …; N
ð12Þ
d i ≥ 0; i ¼ 1; 2; …; N
ð13Þ
and Eq. (2).
In this formulation, d i is the positive deviation, and −f i (x)+
g i is the negative deviation.
Romero et al. provided a combinatorial formulation relied
on WGP and MGP [9]. This formulation structure can be
presented as follows:
minλ
N
X
i¼1
αi d iþ þ βi d i− þ ð1−λÞD
ð14Þ
s:t:
D ≥ αi d iþ þ β i d i− ;
ð15Þ
and Eqs. (2, 5, and 10) where λ and 1−λ are the weight of
unwanted deviation variables and maximum deviation, respectively. For λ =0, we can achieve the MGP; for λ =1, the
WGP is achieved; and for the other values of this parameter
belonging to the interval (0, 1), intermediate solutions are
provided by the weighted combination of WGP and MGP.
Chang proposed mixed binary GP (MBGP), which involved the achievement of goals [10]. This formulation expresses as follows:
min
N
X
i¼1
N
X
min
and Eqs. (2 and 5) where D is an extra variable to show the
maximum value of the deviation. The aforementioned formulations have numerous d i+ and d i− variables. Hence, Li proposed a new approach as follows [4]:
min
and Eqs. (2 and 5) where b i is a binary variable of ith goal.
Chang declared in some cases that there might be a situation where the DMs would like to make a decision on the
problem, with the goal that can be achieved from some specific aspiration levels [11]. Therefore, the multi-choice GP
(MCGP) problem can be expressed as follows:
i¼1
ð16Þ
s:t:
ð f i ðxÞ−gi Þbi −d iþ þ d i− ¼ 0; i ¼ 1; 2; …; N
ð17Þ
bi ∈f0; 1g
ð18Þ
i ¼ 1; 2; …; N
ð19Þ
s:t:
f i ðxÞ −d iþ þ d i− ¼
M
X
gij S ij ðBÞ ; i ¼ 1; 2; …; N
ð20Þ
j¼1
and Eqs. (2 and 5) where S ij (B ) represents a function of the
binary serial number. Other variables have been defined in the
preceding paragraphs. The revised MCGP is presented in [12].
The author improved the model and linearized it. In addition,
it is applied for multi-period, multi-stage inventory controlled
supply chain problem [13]. MCGP with utility function, considered for goals, is also introduced lately [14]. Additionally, a
new GP called multi-coefficient GP is introduced recently
[15]. It can be used for group pricing discrimination problems.
The model of this type is mixed integer nonlinear programming. Lately, a new multi-aspiration goal programming
(MAGP) model is proposed to join in the multi-segment GP
(MSGP) and MCGP in order to solve the problems [16].
Also, Liao proposed a new GP model called as MSGP [17],
which is considered in this paper. A manager's objective of an
organization may include some of the following: intensifying
market share, obtaining difference profits, portioning the
product price, and so on, in different markets. In fact, the
incompleteness of the attainable information and conflicts of
organizational resource force DMs to build a reliable mathematical model for portraying their preference. The objectives
are so different in organizations, which they cannot be combined into a single proper goal, but the DMs attempt to set a
goal to get the acceptable solutions in solving the problem of a
multi-segment aspiration level. DMs would be interested to
minimize the deviations between the achievements of the goal
and their aspiration levels of variable coefficients of decision.
However, this formulation can be described as follows:
min
N
X
i¼1
d iþ þ d i− bi
d iþ þ d i− wi
d iþ þ d i−
ð21Þ
s: t:
N
X
0
S ij x i − d iþ þ d i− ¼ g i ;
i ¼ 1; 2; …; N
j ¼ 1; 2; …; M ð22Þ
i¼1
and Eqs. (2 and 5) where S ′ij is a variable coefficient of
decision representing the multi-segment aspiration levels of
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jth segment of ith goal in ith decision variable. Therefore, in
this model, the aspiration contribution level should be defined
only for one decision variable in each goal. In addition, other
binary variables should be added for solving MSGP problem
that if there have been many segments in each market, multiplicity of extra binary variables would be difficult to solve the
problem. Therefore, we propose a conceptual model considering more than one coefficient for decision variables in each
goal for segmentation and linearize the formulation with a
new idea.
The rest of the paper is arranged as follows: Section 2
introduces the revised multi-segment goal programming
(RMSGP) formulation. In Section 3, we apply the displayed
method to the supplier selection in an Iranian company. Finally, Section 4 presents the conclusion and future researches.
2 Revised multi-segment goal programming framework
Based on the MSGP method introduced by [17], an aspiration
contribution level should be defined only for one decision
variable in each goal. Unlike the mentioned MSGP, the revised
MSGP allows the decision maker to set multi-aspiration levels
for each decision variable in each goal. RMSGP suggests a
conceptual model for multi-choice in variables of aspiration
levels that it analyzes coefficients of decision variables. Hence,
decision maker can decide more effectively with achieving a
set of compromising solutions and minimizing the deviations
between the achievement of goals and their aspiration levels.
The general model is as follows:
N
X
min
i¼1
wi d iþ þ d i−
ð21Þ
s:t:
m X
hij
X
j¼1
hij
X
S ijk X ijk þ d i− − d iþ ¼ gi
∀i
ð23Þ
k¼1
X ijk ¼ Y j
∀i; j
ð24Þ
k¼1
X ijk ; Y j ≥ 0
Table 1 Differences between
MSGP and RMSGP
∀i ;j; k
and Eq. (5) where d +i and d −i are the positive and negative
deviations correlated to ith goal; w i is the weight for normalization of the deviations (e.g., wi ¼ g1 ). S ijk represents the kth
i
segment of jth variable in ith goal of the coefficients. X ijk is
their decision variable; Y j is decision variable of jth factor in
deciding (e.g., supplier in the supplier selection problem); g i
is the aspiration level of the goal; h ij is the number of segments for jth variable in ith goal.
The differences between this revised model and the first
model of MSGP are summarized in Table 1.
The following cases express the advantages of RMSGP:
(1) considering segmentation of all decision variables; (2) the
segmentation is distributed to one or more than one coefficient, not only to one of coefficients; (3) this model can be
solved for more than four segmentations, while MSGP cannot
be formulated easily; (4) changing the mixed integer programming to the linear programming; and (5) selecting one or more
than one coefficient for each decision variable with considering their percentage. Hence, using RMSGP, the formulation
can significantly describe the real-world problem.
In the next section, we apply our proposed formulation to
an important MCDM problem. In addition, the model is
described fluently to introduce RMSGP further.
3 Application of the MSGP model to the supplier selection
problem
In this section, we present the literature review and our proposed
method for supplier selection, respectively. Supplier selection
belongs to the multi-criteria decision-making problem. Therefore, decision-making techniques and criteria are two significant
elements in this problem. We review the literature about this
problem and its solution methods. One of the first researchers in
this field was Dickson [18]. He recognized 23 different criteria
for this problem focused on a questionnaire, which was sent to
some companies in North America. These criteria contain delivery time, production capacity, warranty situation, product quality,
performance, claims policy, production facilities, net price, and
technical capabilities. Moore and Fearon [19] provided a review,
which concentrated on industrial applications of computerassisted supplier selection models. Evans proposed that price,
quality, and delivery are the most important criteria for evaluating suppliers in the industrial market [20]. Shipley proposed
Model
Number of variables
segmented in each goal
Extra
variables
Extra
constraints
Class
Selecting coefficient
MSGP
RMSGP
One
One or more than one
Binary
Positive
0
M ×N
Nonlinear
Linear
One
One or more than one with
considering their proportion
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three criteria including quality, price, and delivery lead time for
supplier selection [21]. Ellram mentioned that the firm needed to
consider the product-offering price, quality, delivery time, and
service quality in supplier selection problem [22]. Weber et al.
arranged the literature on supplier selection in some groups by
reviewing many articles [23]. Moreover, de Boer et al. reviewed
the decision methods published in the literature for supporting
the supplier selection process [24]. Pi and Low proposed quality,
on-time delivery, price, and service for supplier evaluation using
Taguchi loss functions [25]. Aissaoui et al. [26] have presented
another literature review according to the purchasing process.
They have focused on the classification of single and multiple
items and periods. Saen emphasized on the ordinal data [27].
However, with the widespread use of manufacturing philosophies such as just in time (JIT), the emphasis has shifted to the
simultaneous consideration of cardinal and ordinal data in the
supplier selection process. He proposed a new pair of assurance
region-imprecise data envelopment analysis for selecting the best
suppliers in the presence of both weight restrictions and imprecise data. Liao and Kao integrated the Taguchi loss function,
analytical hierarchy process, and multi-choice GP model for
solving the supplier selection problem [28]. Their method
allowed decision makers to fix multiple aspiration levels for
the decision criteria. Ho et al. [29] published another review
paper in supplier selection method with multiple criteria
decision-making approaches. Amin et al. applied quantified
SWOT in the context of supplier selection for the first time
[30]. They proposed a fuzzy linear programming model to
determine how much should be purchased from each supplier.
Vinodh et al. used fuzzy analytic network process approach for
the supplier selection process [31]. Yücenur et al. proposed a
model for choosing the global supplier by analytical hierarchy
process and analytical network process depending upon linguistic variable weight [32].
Our approach is relevant to represent a method in the previous section. Supplier selection is a multi-criteria problem, which
consists of both tangible and intangible factors. In order to select
the best suppliers, it is essential to make a trade-off between
some of conflicting tangible and intangible factors. Therefore,
making a dependable mathematical model can be useful for
better decision making on the management of problems.
An Iranian thermal system production company wants to
select a suitable supplier to purchase the input components.
Table 2 Segmentations of supplier's coefficients in each goal
Goal
Warranty degree
Price
Delivery time
Service satisfaction
The company has to evaluate suppliers with their different
aspiration levels in order to make optimal options. Their
suppliers cause some segmentation for the company goals.
Therefore, to cope with this problem, we proposed RMSGP,
which is the unique method that can consider this problem
situation. The following four goals have been defined for
supplier selection and segmentations of supplier's coefficients
in each goal (presented in Table 2):
Goal 1.
Goal 2.
Goal 3.
Goal 4.
Warranty degree goal with satisfaction level 80,
Price goal with satisfaction level 2200,
Delivery time goal with satisfaction level 2, and
Service satisfaction goal with satisfaction level 50.
In this study, suppliers suggested two or three aspiration
levels for different goals. For example, (Y 1, Y 2) suppliers
exposed (60, 63, 67; 70, 74, 78) aspiration levels for warranty
degree goal, respectively.
To deal with this problem, a new constraint should be
added to the RMSGP model indicating the selection concept.
This constraint is as follows:
M
X
Y j ¼ 1:
j¼1
In this constraint, it should be considered to be M =4,
because there are four suppliers. This constraint ensures that
the model selects only one supplier.
In basis of RMSGP model, this problem can be solved
using LINGO 8 to achieve the best supplier. The optimal
solutions are as (Y 1, Y 2, Y 3, Y 4)=(1, 0, 0, 0). The problem
results display that the supplier Y 1 is the best selection. In
addition, X i1k values are the following:
X 111 ¼ 0; X 112 ¼ 0; X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1 ; X 312 ¼ 0 ; X 411 ¼ 0 ; X 412 ¼ 0 ; X 413 ¼ 1 :
The segmentation is distributed to one coefficient in different
aspiration levels like the first, third, and fourth goals and is
divided into more than one coefficient for X 21k values between
0 and 1. In this optimal solution, the aspiration levels for the
second decision variable (i.e., first supplier) on the first, second,
third, and fourth goals are 67, (1407+792), 2, and 27, respectively. To normalize the deviation, each deviation is divided by
Suppliers' coefficients in each goal
Y1
Y2
Y3
Y4
60, 63, 67
2100, 2400
2, 5
23, 25, 27
70, 74, 78
2000, 2200, 2300
3, 4, 9
17, 19, 22
25, 29
1500, 1800
1, 2
25, 27, 29
56, 61
1600, 1900, 2200
3, 5
27, 29, 31
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Table 3 RMSGP model formulation
Model
Goal
h þ − þ − þ − þ − i
d þd
d þd
d 1 þd 1
d 2 þd 2
min
þ 2200
þ 3 2 3 þ 4 50 4
80
Minimize the deviations
ð60X 111 þ 63X 112 þ 67X 113 Þ þ ð70X 121 þ 74X 122 þ 78X 123 Þþ
ð25X 131 þ 29X 132 Þ132 þ ð56X 141 þ 61X 142 Þ þ d −1 −d þ
1 ¼ 80;
ð2100X 211 þ 2400X 212 Þ þ ð2000X 221 þ 2200X 222 þ 2300X 223 Þþ
ð1500X 231 þ 1800X 232 Þ þ ð1600X 241 þ 1900X 242 þ 2200X 243 Þþ
d −2 −d þ
2 ¼ 2200;
ð2X 311 þ 5X 312 Þ þ ð3X 321 þ 4X 322 þ 9X 323 Þ þ ðX 331 þ 2X 332 Þþ
ð3X 341 þ 5X 342 Þ þ d −3 −d þ
3 ¼ 2;
ð23X 411 þ 25X 412 þ 27X 413 Þ þ ð17X 421 þ 19X 422 þ 22X 423 Þþ
ð25X 431 þ 27X 432 þ 29X 433 Þ þ ð27X 441 þ 29X 442 þ 31X 443 Þþ
d −4 −d þ
4 ¼ 50;
Y 1 +Y 2 +Y 3 +Y 4 =1
X 311 +X 312 =Y 1
X 211 +X 212 =Y 1
X 411 þ X 412 þ
X 111 þ X 112 þ
X 413 ¼ Y 1
X 113 ¼ Y 1
X 121 þ X 122 þ
X 221 þ X 222 þ
X 321 þ X 322 þ
X 421 þ X 422 þ
X 123 ¼ Y 2
X 223 ¼ Y 2
X 323 ¼ Y 2
X 423 ¼ Y 2
X 231 +X 232 =Y 3
X 331 +X 332 =Y 3
X 131 +X 132 =Y 3
X 431 þ X 432 þ
X 433 ¼ Y 3
X 141 +X 142 =Y 4
X 341 +X 342 =Y 4
X 241 þ X 242 þ
X 441 þ X 442 þ
X 243 ¼ Y 4
X 443 ¼ Y 4
Achieve to warranty degree
its goal value. Therefore, we can add together the proportion of
the deviations. The minimum proportion of deviation from the
goals is 0.62 in this case. In respect to this result, we suggest
that this thermal system production company that selects the
Table 4 Sensitivity analysis
results for aspiration level
changes
Range change of
aspiration levels (%)
+10
Achieve to price
Achieve to delivery time
Achieve to service satisfaction
Select a supplier
Balance segmentation decision variable for the first supplier
Balance segmentation decision variable for the second supplier
Balance segmentation decision variable for the third supplier
Balance segmentation decision variable for the fourth supplier
first supplier can be suitable to achieve its goals. It is better for
the company to choose the third aspiration level of this supplier
to abut onto warranty degree. To achieve the price goal, the
company should accept 67 and 33 % of the first and second
Results
Absolute deviation
from each goal (%)
Selected
supplier
Selected aspiration level
Total
deviation
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.63
Goal (1)=7.87
Goal (2)=5
Goal (3)=10
+5
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.61
Goal (4)=40.60
Goal (1)=12.06
Goal (2)=0.23
Goal (3)=5
0
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.62
Goal (4)=43.30
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
−5
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:28; X 212 ¼ 0:72;
X 311 ¼ 0:96; X 312 ¼ 0:04; X 413 ¼ 1
0.69
Goal (4)=46
Goal (1)=20.44
Goal (2)=0
Goal (3)=0
−10
Y 1 =1
X 113 ¼ 1; X 212 ¼ 1; X 311 ¼ 0:93;
X 312 ¼ 0:07; X 413 ¼ 1
0.78
Goal (4)=48.70
Goal (1)=24.63
Goal (2)=1.82
Goal (3)=0
Goal (4)=51.40
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Table 5 Sensitivity analysis results in goal value changes
Goal
Goal (1)
Goal (2)
Goal (3)
Goal (4)
Goal range change
Results
Absolute deviation
from each goal (%)
Supplier
Aspiration level
Total deviation
+10 % (88)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.70
−10 % (72)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.53
+10 % (2420)
Y 1 =1
X 113 ¼ 1; X 212 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.63
−10 % (1980)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.68
+10 % (2.2)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 0:93; X 312 ¼ 0:07; X 413 ¼ 1
0.62
−10 % (1.8)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.73
+10 % (55)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.67
−10 % (45)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.56
aspiration levels of this supplier. In addition, this company had better select the first and third aspiration levels of
Fig. 1 Results of sensitivity
analysis by varying the goal value
Goal (1)=23.86
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=6.94
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0.83
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=6.06
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=11.11
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=50.09
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=46
the first supplier to obtain delivery time and service satisfaction, respectively.
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The whole formulation of this problem is described in
Table 3.
In order to know how the change in value from +10 to −10 %
in the all aspiration levels affects the deviation from the goals
and the total deviation, sensitivity analysis of aspiration level is
run. In Table 4, the sensitivity analysis results are shown. As it
can be seen from this table, in all aspiration levels, an increase of
10 % will cause an 8.36 % lower change in warranty degree,
5 % higher change in price, 10 % higher change in delivery
time, and 6.60 % lower change in service satisfaction. In
contrast, in all aspiration levels, a decrease of 10 % will cause
an 8.38, 1.82, and 5.40 % higher change in warranty degree,
price, and service satisfaction, respectively. No change in the
delivery time will be happened by a 10 % decrease of all
aspiration levels. An increase of 5 % of all aspiration levels will
lead to a 4.19 % lower change in warranty degree, 0.23 %
higher change, 5 % higher change in delivery time, and 2.70 %
lower change in service satisfaction. In contrast, in all aspiration
levels, a decrease of 5 % will lead to a 4.19 and 2.70 % higher
change in warranty degree and service satisfaction, respectively.
This reduction will cause no change in price and delivery time.
We see in Table 4 that in the all aspiration levels, increasing
causes lower total deviation, but the supplier selected in all of
them is the same.
In order to analyze how the different value of each goal affects
the deviation from the goals and the total deviation. We change
each goal from +10 to −10 % for sensitivity analysis of goal
value. In Table 5, the sensitivity analysis results are summarized.
As can be seen in Table 5, the warranty degree increase of
10 % will lead to a 0.08 higher change in the total deviation. In
contrast, this goal decrease of 10 % will cause a 0.09 lower
change in the total deviation. The prices grow, and a decrease
of 10 % will cause a 0.01 and 0.06 higher changes in the
objective function of the RMSGP, respectively. The 10 %
increase of delivery time will lead to no change in the total
deviation. This goal reduction of 10 % will cause a 0.11 higher
change in the total deviation. The service satisfaction growth
and reduction of 10 % will cause a 0.05 and 0.0.6 lower and
higher change in the total deviation, respectively. The results
of sensitivity analysis of goal change are shown in Fig. 1.
It has been inferred from this figure that a 10 % reduction of
the warranty degree does not create any impact on the decision
made, which has been revealed by the model without change
on this goal. However, this reduction causes the lowest change
in the total deviation among all studied changes. The 10 %
reduction of warranty degree leads to a 9.31 % lower change
in warranty degree and no change in the other goals.
4 Conclusion
Drawing and optimizing efficient supply chain network make
available competitive benefits to buyers and companies.
1233
Moreover, supplier selection, the procedure of selecting suitable suppliers who are proficient to provide the buyer with the
accurate quality products or services at the appropriate price
and at the optimal time and quantities, is one of the most
significant activities for determining an efficient supply chain.
On the other hand, this is a hard problem since supplier
selection is considered as a multi-objective decision-making
problem. Hence, in this paper, with concentration to the GP
techniques, a revised MSGP model is proposed to improve
MSGP model. This approach is being different from MSGP
model to consider multi-aspiration levels for each decision
variable in each goal. Four different suppliers that make
thermal products are selected by using RMSGP method. This
result represents that selective supplier provides better performance levels at the suppliers' different inputs. As for the future
researches, we recommend a new solution methodology based
on multi-choice goal programming which can be developed
for the different segments, and the efficiency of the model
according to this solution methodology can be examined.
Acknowledgments The authors would like to thank the anonymous
referees for their valuable contributions, which led to improvements in
this paper. Moreover, the second author has been partially supported by
the National Iranian Oil Company (NIOC).
References
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Supplier selection using revised multi-segment
goal programming model
Article in International Journal of Advanced Manufacturing Technology · February 2014
DOI: 10.1007/s00170-013-5368-0
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Supplier selection using revised multisegment goal programming model
Hossein Karimi & Alireza Rezaeinia
The International Journal of
Advanced Manufacturing Technology
ISSN 0268-3768
Volume 70
Combined 5-8
Int J Adv Manuf Technol (2014)
70:1227-1234
DOI 10.1007/s00170-013-5368-0
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Int J Adv Manuf Technol (2014) 70:1227–1234
DOI 10.1007/s00170-013-5368-0
ORIGINAL ARTICLE
Supplier selection using revised multi-segment
goal programming model
Hossein Karimi & Alireza Rezaeinia
Received: 5 November 2011 / Accepted: 27 September 2013 / Published online: 18 October 2013
# Springer-Verlag London 2013
Abstract Selecting the best supplier has been turned into a
strategic subject in the competitive market, in the recent years.
Supplier selection is one of the significant multiple criteria
decision-making problems, which is studied in this paper. This
research applies the multi-segment goal programming formulation, which considers different positions in the goal programming. This new kind of goal programming is revised in this
investigation. The advantage of the proposed method is that it
allows decision makers to set multiple aspiration levels for the
coefficient of variables. Hence, the decision makers obtain more
revenue and match with the reality of their problems. Unlike the
multi-segment goal programming proposed in the literature, the
revised formulation is linearized in the current study. The supplier
selection problem is solved by this new formulation in a special
circumstance. Finally, some concluding notes are remarked.
Keywords Supplier selection . Goal programming . Multiple
aspiration levels . Revised multi-segment goal programming
achievement of goals and their aspiration levels. In a characteristic manner, GP can be expressed as follows:
min
N
X
j f i ðxÞ−gi j
ð1Þ
i¼1
s:t:
ð2Þ
x∈ F
where f i (x) and g i are the linear function and the goal of the
ith objective, respectively, and F is a feasible set. Also, N is
the number of criteria.
Based on this formulation, researchers proposed many
formulations for GP, which had different attitudes. We introduced some of these models, which have been frequently
applied by different researchers.
Ignizio proposed weighted GP (WGP) and lexicographic
GP (LGP) [7]. These formulations are the following in order:
min
N
X
αi d iþ þ β i d i−
i¼1
ð3Þ
1 Introduction
s:t :
Goal programming (GP) is a kind of multiple objective programming, which is the progeny of multi-criteria decision
making (MCDM). In this research, the methodology known
as GP has been inspired from [1]. Further developments are
provided for instance by [2–6]. GP lets decision makers
(DMs) fix their aspiration levels for each goal. The objective
of GP is to minimize the undesirable deviations between the
f i ðxÞ−d iþ þ d i− ¼ gi ; i ¼ 1; 2…; N
ð4Þ
d iþ ; d i− ≥ 0;
ð5Þ
H. Karimi (*)
Department of Industrial Engineering, K. N. Toosi University of
Technology, No. 17, Pardis St., Molla Sadra St.,
Vanak Sq., Tehran, Iran
e-mail: [email protected]
A. Rezaeinia
Project Planning and Control, National Iranian Oil Company,
Pars Economic Special Zone, Assaluyeh, Iran
e-mail: [email protected]
i ¼ 1; 2; …; N
and Eq. (2) where d +i and d −i are the positive and negative
deviations between the ith objective and its goal, respectively;
α i and β i are the positive weights to the deviations.
2
3
n
n
n
X
X
X
þ
þ
þ
−
−
− 5
4
min
αi d i þ βi d i ; …;
αi d i þ β i d i ; …;
αi d i þ β i d i
i¼hi
i¼hr
i¼hQ
s: t :
ð6Þ
−
f i ðxÞ−d þ
i þ d i ¼ gi ;
i ¼ 1; 2; …; N ;
i∈hr
r ¼ 1; 2; …Q
ð7Þ
and Eqs. (2 and 5) where h r stands for the index set of goals
placed in the rth level. The other variables are defined in
WGP.
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Flavell presented min–max or Chebyshev GP (MGP), minimizing the maximum deviation [8]. This method is provided
as follows:
minD
s:t:
ð8Þ
−
D ≥ αi d þ
i þ βi d i ;
ð9Þ
f i ðxÞ−d þ
i
þ
d i−
¼ gi ; i ¼ 1; 2; …; N
N
X
ð2d i − f i ðxÞ þ gi Þ
ð10Þ
ð11Þ
i¼1
s: t:
d i −f i ðxÞ þ gi ≥ 0; i ¼ 1; 2; …; N
ð12Þ
d i ≥ 0; i ¼ 1; 2; …; N
ð13Þ
and Eq. (2).
In this formulation, d i is the positive deviation, and −f i (x)+
g i is the negative deviation.
Romero et al. provided a combinatorial formulation relied
on WGP and MGP [9]. This formulation structure can be
presented as follows:
minλ
N
X
i¼1
αi d iþ þ βi d i− þ ð1−λÞD
ð14Þ
s:t:
D ≥ αi d iþ þ β i d i− ;
ð15Þ
and Eqs. (2, 5, and 10) where λ and 1−λ are the weight of
unwanted deviation variables and maximum deviation, respectively. For λ =0, we can achieve the MGP; for λ =1, the
WGP is achieved; and for the other values of this parameter
belonging to the interval (0, 1), intermediate solutions are
provided by the weighted combination of WGP and MGP.
Chang proposed mixed binary GP (MBGP), which involved the achievement of goals [10]. This formulation expresses as follows:
min
N
X
i¼1
N
X
min
and Eqs. (2 and 5) where D is an extra variable to show the
maximum value of the deviation. The aforementioned formulations have numerous d i+ and d i− variables. Hence, Li proposed a new approach as follows [4]:
min
and Eqs. (2 and 5) where b i is a binary variable of ith goal.
Chang declared in some cases that there might be a situation where the DMs would like to make a decision on the
problem, with the goal that can be achieved from some specific aspiration levels [11]. Therefore, the multi-choice GP
(MCGP) problem can be expressed as follows:
i¼1
ð16Þ
s:t:
ð f i ðxÞ−gi Þbi −d iþ þ d i− ¼ 0; i ¼ 1; 2; …; N
ð17Þ
bi ∈f0; 1g
ð18Þ
i ¼ 1; 2; …; N
ð19Þ
s:t:
f i ðxÞ −d iþ þ d i− ¼
M
X
gij S ij ðBÞ ; i ¼ 1; 2; …; N
ð20Þ
j¼1
and Eqs. (2 and 5) where S ij (B ) represents a function of the
binary serial number. Other variables have been defined in the
preceding paragraphs. The revised MCGP is presented in [12].
The author improved the model and linearized it. In addition,
it is applied for multi-period, multi-stage inventory controlled
supply chain problem [13]. MCGP with utility function, considered for goals, is also introduced lately [14]. Additionally, a
new GP called multi-coefficient GP is introduced recently
[15]. It can be used for group pricing discrimination problems.
The model of this type is mixed integer nonlinear programming. Lately, a new multi-aspiration goal programming
(MAGP) model is proposed to join in the multi-segment GP
(MSGP) and MCGP in order to solve the problems [16].
Also, Liao proposed a new GP model called as MSGP [17],
which is considered in this paper. A manager's objective of an
organization may include some of the following: intensifying
market share, obtaining difference profits, portioning the
product price, and so on, in different markets. In fact, the
incompleteness of the attainable information and conflicts of
organizational resource force DMs to build a reliable mathematical model for portraying their preference. The objectives
are so different in organizations, which they cannot be combined into a single proper goal, but the DMs attempt to set a
goal to get the acceptable solutions in solving the problem of a
multi-segment aspiration level. DMs would be interested to
minimize the deviations between the achievements of the goal
and their aspiration levels of variable coefficients of decision.
However, this formulation can be described as follows:
min
N
X
i¼1
d iþ þ d i− bi
d iþ þ d i− wi
d iþ þ d i−
ð21Þ
s: t:
N
X
0
S ij x i − d iþ þ d i− ¼ g i ;
i ¼ 1; 2; …; N
j ¼ 1; 2; …; M ð22Þ
i¼1
and Eqs. (2 and 5) where S ′ij is a variable coefficient of
decision representing the multi-segment aspiration levels of
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jth segment of ith goal in ith decision variable. Therefore, in
this model, the aspiration contribution level should be defined
only for one decision variable in each goal. In addition, other
binary variables should be added for solving MSGP problem
that if there have been many segments in each market, multiplicity of extra binary variables would be difficult to solve the
problem. Therefore, we propose a conceptual model considering more than one coefficient for decision variables in each
goal for segmentation and linearize the formulation with a
new idea.
The rest of the paper is arranged as follows: Section 2
introduces the revised multi-segment goal programming
(RMSGP) formulation. In Section 3, we apply the displayed
method to the supplier selection in an Iranian company. Finally, Section 4 presents the conclusion and future researches.
2 Revised multi-segment goal programming framework
Based on the MSGP method introduced by [17], an aspiration
contribution level should be defined only for one decision
variable in each goal. Unlike the mentioned MSGP, the revised
MSGP allows the decision maker to set multi-aspiration levels
for each decision variable in each goal. RMSGP suggests a
conceptual model for multi-choice in variables of aspiration
levels that it analyzes coefficients of decision variables. Hence,
decision maker can decide more effectively with achieving a
set of compromising solutions and minimizing the deviations
between the achievement of goals and their aspiration levels.
The general model is as follows:
N
X
min
i¼1
wi d iþ þ d i−
ð21Þ
s:t:
m X
hij
X
j¼1
hij
X
S ijk X ijk þ d i− − d iþ ¼ gi
∀i
ð23Þ
k¼1
X ijk ¼ Y j
∀i; j
ð24Þ
k¼1
X ijk ; Y j ≥ 0
Table 1 Differences between
MSGP and RMSGP
∀i ;j; k
and Eq. (5) where d +i and d −i are the positive and negative
deviations correlated to ith goal; w i is the weight for normalization of the deviations (e.g., wi ¼ g1 ). S ijk represents the kth
i
segment of jth variable in ith goal of the coefficients. X ijk is
their decision variable; Y j is decision variable of jth factor in
deciding (e.g., supplier in the supplier selection problem); g i
is the aspiration level of the goal; h ij is the number of segments for jth variable in ith goal.
The differences between this revised model and the first
model of MSGP are summarized in Table 1.
The following cases express the advantages of RMSGP:
(1) considering segmentation of all decision variables; (2) the
segmentation is distributed to one or more than one coefficient, not only to one of coefficients; (3) this model can be
solved for more than four segmentations, while MSGP cannot
be formulated easily; (4) changing the mixed integer programming to the linear programming; and (5) selecting one or more
than one coefficient for each decision variable with considering their percentage. Hence, using RMSGP, the formulation
can significantly describe the real-world problem.
In the next section, we apply our proposed formulation to
an important MCDM problem. In addition, the model is
described fluently to introduce RMSGP further.
3 Application of the MSGP model to the supplier selection
problem
In this section, we present the literature review and our proposed
method for supplier selection, respectively. Supplier selection
belongs to the multi-criteria decision-making problem. Therefore, decision-making techniques and criteria are two significant
elements in this problem. We review the literature about this
problem and its solution methods. One of the first researchers in
this field was Dickson [18]. He recognized 23 different criteria
for this problem focused on a questionnaire, which was sent to
some companies in North America. These criteria contain delivery time, production capacity, warranty situation, product quality,
performance, claims policy, production facilities, net price, and
technical capabilities. Moore and Fearon [19] provided a review,
which concentrated on industrial applications of computerassisted supplier selection models. Evans proposed that price,
quality, and delivery are the most important criteria for evaluating suppliers in the industrial market [20]. Shipley proposed
Model
Number of variables
segmented in each goal
Extra
variables
Extra
constraints
Class
Selecting coefficient
MSGP
RMSGP
One
One or more than one
Binary
Positive
0
M ×N
Nonlinear
Linear
One
One or more than one with
considering their proportion
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three criteria including quality, price, and delivery lead time for
supplier selection [21]. Ellram mentioned that the firm needed to
consider the product-offering price, quality, delivery time, and
service quality in supplier selection problem [22]. Weber et al.
arranged the literature on supplier selection in some groups by
reviewing many articles [23]. Moreover, de Boer et al. reviewed
the decision methods published in the literature for supporting
the supplier selection process [24]. Pi and Low proposed quality,
on-time delivery, price, and service for supplier evaluation using
Taguchi loss functions [25]. Aissaoui et al. [26] have presented
another literature review according to the purchasing process.
They have focused on the classification of single and multiple
items and periods. Saen emphasized on the ordinal data [27].
However, with the widespread use of manufacturing philosophies such as just in time (JIT), the emphasis has shifted to the
simultaneous consideration of cardinal and ordinal data in the
supplier selection process. He proposed a new pair of assurance
region-imprecise data envelopment analysis for selecting the best
suppliers in the presence of both weight restrictions and imprecise data. Liao and Kao integrated the Taguchi loss function,
analytical hierarchy process, and multi-choice GP model for
solving the supplier selection problem [28]. Their method
allowed decision makers to fix multiple aspiration levels for
the decision criteria. Ho et al. [29] published another review
paper in supplier selection method with multiple criteria
decision-making approaches. Amin et al. applied quantified
SWOT in the context of supplier selection for the first time
[30]. They proposed a fuzzy linear programming model to
determine how much should be purchased from each supplier.
Vinodh et al. used fuzzy analytic network process approach for
the supplier selection process [31]. Yücenur et al. proposed a
model for choosing the global supplier by analytical hierarchy
process and analytical network process depending upon linguistic variable weight [32].
Our approach is relevant to represent a method in the previous section. Supplier selection is a multi-criteria problem, which
consists of both tangible and intangible factors. In order to select
the best suppliers, it is essential to make a trade-off between
some of conflicting tangible and intangible factors. Therefore,
making a dependable mathematical model can be useful for
better decision making on the management of problems.
An Iranian thermal system production company wants to
select a suitable supplier to purchase the input components.
Table 2 Segmentations of supplier's coefficients in each goal
Goal
Warranty degree
Price
Delivery time
Service satisfaction
The company has to evaluate suppliers with their different
aspiration levels in order to make optimal options. Their
suppliers cause some segmentation for the company goals.
Therefore, to cope with this problem, we proposed RMSGP,
which is the unique method that can consider this problem
situation. The following four goals have been defined for
supplier selection and segmentations of supplier's coefficients
in each goal (presented in Table 2):
Goal 1.
Goal 2.
Goal 3.
Goal 4.
Warranty degree goal with satisfaction level 80,
Price goal with satisfaction level 2200,
Delivery time goal with satisfaction level 2, and
Service satisfaction goal with satisfaction level 50.
In this study, suppliers suggested two or three aspiration
levels for different goals. For example, (Y 1, Y 2) suppliers
exposed (60, 63, 67; 70, 74, 78) aspiration levels for warranty
degree goal, respectively.
To deal with this problem, a new constraint should be
added to the RMSGP model indicating the selection concept.
This constraint is as follows:
M
X
Y j ¼ 1:
j¼1
In this constraint, it should be considered to be M =4,
because there are four suppliers. This constraint ensures that
the model selects only one supplier.
In basis of RMSGP model, this problem can be solved
using LINGO 8 to achieve the best supplier. The optimal
solutions are as (Y 1, Y 2, Y 3, Y 4)=(1, 0, 0, 0). The problem
results display that the supplier Y 1 is the best selection. In
addition, X i1k values are the following:
X 111 ¼ 0; X 112 ¼ 0; X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1 ; X 312 ¼ 0 ; X 411 ¼ 0 ; X 412 ¼ 0 ; X 413 ¼ 1 :
The segmentation is distributed to one coefficient in different
aspiration levels like the first, third, and fourth goals and is
divided into more than one coefficient for X 21k values between
0 and 1. In this optimal solution, the aspiration levels for the
second decision variable (i.e., first supplier) on the first, second,
third, and fourth goals are 67, (1407+792), 2, and 27, respectively. To normalize the deviation, each deviation is divided by
Suppliers' coefficients in each goal
Y1
Y2
Y3
Y4
60, 63, 67
2100, 2400
2, 5
23, 25, 27
70, 74, 78
2000, 2200, 2300
3, 4, 9
17, 19, 22
25, 29
1500, 1800
1, 2
25, 27, 29
56, 61
1600, 1900, 2200
3, 5
27, 29, 31
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Table 3 RMSGP model formulation
Model
Goal
h þ − þ − þ − þ − i
d þd
d þd
d 1 þd 1
d 2 þd 2
min
þ 2200
þ 3 2 3 þ 4 50 4
80
Minimize the deviations
ð60X 111 þ 63X 112 þ 67X 113 Þ þ ð70X 121 þ 74X 122 þ 78X 123 Þþ
ð25X 131 þ 29X 132 Þ132 þ ð56X 141 þ 61X 142 Þ þ d −1 −d þ
1 ¼ 80;
ð2100X 211 þ 2400X 212 Þ þ ð2000X 221 þ 2200X 222 þ 2300X 223 Þþ
ð1500X 231 þ 1800X 232 Þ þ ð1600X 241 þ 1900X 242 þ 2200X 243 Þþ
d −2 −d þ
2 ¼ 2200;
ð2X 311 þ 5X 312 Þ þ ð3X 321 þ 4X 322 þ 9X 323 Þ þ ðX 331 þ 2X 332 Þþ
ð3X 341 þ 5X 342 Þ þ d −3 −d þ
3 ¼ 2;
ð23X 411 þ 25X 412 þ 27X 413 Þ þ ð17X 421 þ 19X 422 þ 22X 423 Þþ
ð25X 431 þ 27X 432 þ 29X 433 Þ þ ð27X 441 þ 29X 442 þ 31X 443 Þþ
d −4 −d þ
4 ¼ 50;
Y 1 +Y 2 +Y 3 +Y 4 =1
X 311 +X 312 =Y 1
X 211 +X 212 =Y 1
X 411 þ X 412 þ
X 111 þ X 112 þ
X 413 ¼ Y 1
X 113 ¼ Y 1
X 121 þ X 122 þ
X 221 þ X 222 þ
X 321 þ X 322 þ
X 421 þ X 422 þ
X 123 ¼ Y 2
X 223 ¼ Y 2
X 323 ¼ Y 2
X 423 ¼ Y 2
X 231 +X 232 =Y 3
X 331 +X 332 =Y 3
X 131 +X 132 =Y 3
X 431 þ X 432 þ
X 433 ¼ Y 3
X 141 +X 142 =Y 4
X 341 +X 342 =Y 4
X 241 þ X 242 þ
X 441 þ X 442 þ
X 243 ¼ Y 4
X 443 ¼ Y 4
Achieve to warranty degree
its goal value. Therefore, we can add together the proportion of
the deviations. The minimum proportion of deviation from the
goals is 0.62 in this case. In respect to this result, we suggest
that this thermal system production company that selects the
Table 4 Sensitivity analysis
results for aspiration level
changes
Range change of
aspiration levels (%)
+10
Achieve to price
Achieve to delivery time
Achieve to service satisfaction
Select a supplier
Balance segmentation decision variable for the first supplier
Balance segmentation decision variable for the second supplier
Balance segmentation decision variable for the third supplier
Balance segmentation decision variable for the fourth supplier
first supplier can be suitable to achieve its goals. It is better for
the company to choose the third aspiration level of this supplier
to abut onto warranty degree. To achieve the price goal, the
company should accept 67 and 33 % of the first and second
Results
Absolute deviation
from each goal (%)
Selected
supplier
Selected aspiration level
Total
deviation
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.63
Goal (1)=7.87
Goal (2)=5
Goal (3)=10
+5
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.61
Goal (4)=40.60
Goal (1)=12.06
Goal (2)=0.23
Goal (3)=5
0
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.62
Goal (4)=43.30
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
−5
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:28; X 212 ¼ 0:72;
X 311 ¼ 0:96; X 312 ¼ 0:04; X 413 ¼ 1
0.69
Goal (4)=46
Goal (1)=20.44
Goal (2)=0
Goal (3)=0
−10
Y 1 =1
X 113 ¼ 1; X 212 ¼ 1; X 311 ¼ 0:93;
X 312 ¼ 0:07; X 413 ¼ 1
0.78
Goal (4)=48.70
Goal (1)=24.63
Goal (2)=1.82
Goal (3)=0
Goal (4)=51.40
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Table 5 Sensitivity analysis results in goal value changes
Goal
Goal (1)
Goal (2)
Goal (3)
Goal (4)
Goal range change
Results
Absolute deviation
from each goal (%)
Supplier
Aspiration level
Total deviation
+10 % (88)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.70
−10 % (72)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.53
+10 % (2420)
Y 1 =1
X 113 ¼ 1; X 212 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.63
−10 % (1980)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 1;
X 311 ¼ 1; X 413 ¼ 1
0.68
+10 % (2.2)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 0:93; X 312 ¼ 0:07; X 413 ¼ 1
0.62
−10 % (1.8)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.73
+10 % (55)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.67
−10 % (45)
Y 1 =1
X 113 ¼ 1; X 211 ¼ 0:67; X 212 ¼ 0:33;
X 311 ¼ 1; X 413 ¼ 1
0.56
aspiration levels of this supplier. In addition, this company had better select the first and third aspiration levels of
Fig. 1 Results of sensitivity
analysis by varying the goal value
Goal (1)=23.86
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=6.94
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0.83
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=6.06
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=11.11
Goal (4)=46
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=50.09
Goal (1)=16.25
Goal (2)=0
Goal (3)=0
Goal (4)=46
the first supplier to obtain delivery time and service satisfaction, respectively.
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The whole formulation of this problem is described in
Table 3.
In order to know how the change in value from +10 to −10 %
in the all aspiration levels affects the deviation from the goals
and the total deviation, sensitivity analysis of aspiration level is
run. In Table 4, the sensitivity analysis results are shown. As it
can be seen from this table, in all aspiration levels, an increase of
10 % will cause an 8.36 % lower change in warranty degree,
5 % higher change in price, 10 % higher change in delivery
time, and 6.60 % lower change in service satisfaction. In
contrast, in all aspiration levels, a decrease of 10 % will cause
an 8.38, 1.82, and 5.40 % higher change in warranty degree,
price, and service satisfaction, respectively. No change in the
delivery time will be happened by a 10 % decrease of all
aspiration levels. An increase of 5 % of all aspiration levels will
lead to a 4.19 % lower change in warranty degree, 0.23 %
higher change, 5 % higher change in delivery time, and 2.70 %
lower change in service satisfaction. In contrast, in all aspiration
levels, a decrease of 5 % will lead to a 4.19 and 2.70 % higher
change in warranty degree and service satisfaction, respectively.
This reduction will cause no change in price and delivery time.
We see in Table 4 that in the all aspiration levels, increasing
causes lower total deviation, but the supplier selected in all of
them is the same.
In order to analyze how the different value of each goal affects
the deviation from the goals and the total deviation. We change
each goal from +10 to −10 % for sensitivity analysis of goal
value. In Table 5, the sensitivity analysis results are summarized.
As can be seen in Table 5, the warranty degree increase of
10 % will lead to a 0.08 higher change in the total deviation. In
contrast, this goal decrease of 10 % will cause a 0.09 lower
change in the total deviation. The prices grow, and a decrease
of 10 % will cause a 0.01 and 0.06 higher changes in the
objective function of the RMSGP, respectively. The 10 %
increase of delivery time will lead to no change in the total
deviation. This goal reduction of 10 % will cause a 0.11 higher
change in the total deviation. The service satisfaction growth
and reduction of 10 % will cause a 0.05 and 0.0.6 lower and
higher change in the total deviation, respectively. The results
of sensitivity analysis of goal change are shown in Fig. 1.
It has been inferred from this figure that a 10 % reduction of
the warranty degree does not create any impact on the decision
made, which has been revealed by the model without change
on this goal. However, this reduction causes the lowest change
in the total deviation among all studied changes. The 10 %
reduction of warranty degree leads to a 9.31 % lower change
in warranty degree and no change in the other goals.
4 Conclusion
Drawing and optimizing efficient supply chain network make
available competitive benefits to buyers and companies.
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Moreover, supplier selection, the procedure of selecting suitable suppliers who are proficient to provide the buyer with the
accurate quality products or services at the appropriate price
and at the optimal time and quantities, is one of the most
significant activities for determining an efficient supply chain.
On the other hand, this is a hard problem since supplier
selection is considered as a multi-objective decision-making
problem. Hence, in this paper, with concentration to the GP
techniques, a revised MSGP model is proposed to improve
MSGP model. This approach is being different from MSGP
model to consider multi-aspiration levels for each decision
variable in each goal. Four different suppliers that make
thermal products are selected by using RMSGP method. This
result represents that selective supplier provides better performance levels at the suppliers' different inputs. As for the future
researches, we recommend a new solution methodology based
on multi-choice goal programming which can be developed
for the different segments, and the efficiency of the model
according to this solution methodology can be examined.
Acknowledgments The authors would like to thank the anonymous
referees for their valuable contributions, which led to improvements in
this paper. Moreover, the second author has been partially supported by
the National Iranian Oil Company (NIOC).
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