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Decision-making from multiple uncertain experts: case of distribution center location selection

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The location selection of distribution center covers one of the important strategic decision issues for the logistics system managers. In view of the inherent uncertainty and inaccuracy of human decision-making, the future behavior of the market and companies, this paper adopts the improved multi-attribute and multi-Actor decision-making (MAADM) method as a fuzzy multi-attribute and multi-actor decision-making (FMAADM) method for solving the selection problem under an uncertain environment. The great strengths of our proposed method are: first, the integration of the decision-makers group preferences into the decision-making process, second, the consideration of the informations related to the alternatives and the criteria weights which are inaccurate, uncertain or incomplete, third, the verification of the obtained solution by both tests of concordance and non-discordance. To validate the FMAADM method, a decision support system was developed. Different experiments were provided based on comparative analysis of results and the sensitivity analysis. These experiments demonstrate the efficiency of our proposed method and its superiority over another existing methods.
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Soft Computing
https://doi.org/10.1007/s00500-020-05461-y
METHODOLOGIES AND APPLICATION
Decision-making from multiple uncertain experts: case of distribution
center location selection
Maroi Agrebi1·Mourad Abed1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
The location selection of distribution center covers one of the important strategic decision issues for the logistics system
managers. In view of the inherent uncertainty and inaccuracy of human decision-making, the future behavior of the market
and companies, this paper adopts the improved multi-attribute and multi-Actor decision-making (MAADM) method as
a fuzzy multi-attribute and multi-actor decision-making (FMAADM) method for solving the selection problem under an
uncertain environment. The great strengths of our proposed method are: first, the integration of the decision-makers group
preferences into the decision-making process, second, the consideration of the informations related to the alternatives and
the criteria weights which are inaccurate, uncertain or incomplete, third, the verification of the obtained solution by both
tests of concordance and non-discordance. To validate the FMAADM method, a decision support system was developed.
Different experiments were provided based on comparative analysis of results and the sensitivity analysis. These experiments
demonstrate the efficiency of our proposed method and its superiority over another existing methods.
Keywords Distribution center location selection ·Multi-attribute decision-making ·Multiple uncertain experts ·Uncertain
preferences ·Fuzzy set theory
1 Introduction
In an urban context, the city functioning means an important
rate of freight transportation. This functioning relies on var-
ious physical facilities, as shown in Fig. 1. These facilities
are considered as real drivers of logistics systems. There-
fore, the location selection of logistics facilities is viewed as
one of the main strategic issues of distribution system for
the logistics systems managers (Agrebi et al. 2016;Klose
and Drexl 2005). Indeed, the best location of logistics facil-
ities contributes to the logistics service quality and plays a
key role for operation in the future (Lee 2014). In terms of
logistical system design, location selection decisions are of
high priority, since such decisions involve long-term com-
Communicated by V. Loia.
BMaroi Agrebi
maroi.agrebi@gmail.com
Mourad Abed
mourad.abed@uphf.fr
1LAMIH UMR CNRS 8201, Université Polytechnique
Hauts-de-France, Jonas Building, Le Mont Houy, 59313
Valenciennes Cedex 9, France
mitment of resources and generally represent a substantial
investment, which may affect the long-term profitability and
sustainability of the firm. Besides, these decisions are usually
irreversible (Cagri Tolga et al. 2013).
The distribution center location, as a special case of logis-
tics facilities location, plays an important role not only
in minimizing traffic congestion and pollution, but also in
decreasing transport cost and maximizing of profit (Eldemir
and Onden 2016). Moreover, a good location of distribution
center may contribute in maximizing customers’ satisfaction,
as well as maximizing the acceptability by inhabitants, who
live near the logistics platforms and are impacted by vehicle
movements (Agrebi et al. 2016,2017). In addition, know-
ing that higher load factor in the city can decrease harmful
effects associated with city logistics (van Duin et al. 2012), a
good location of distribution center allows reductions in the
number of kilometers vehicle and better utilization rates for
vehicles (Huschebeck and Allen 2005).
In fact, select the best distribution center location from a
set of potential locations (alternatives) is difficult, especially
in a context of decision-making from multiple uncertain
experts (He et al. 2017; Sopha et al. 2018). In this context,
the aim of this paper is to treat this problem as a fuzzy multi-
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M.Agrebi,M.Abed
Fig. 1 Generic supply chain network (Melo et al. 2009)
criteria decision-making problem (FMCDM). The expected
decision must respect decision-makers group preferences and
existing evaluation criteria (e.g., the investment cost, the pos-
sibility of expansion, the availability of acquisition hardware,
the human resources, the proximity to suppliers, etc.).
In the literature, a number of studies have been conducted
on location selection problem of distribution center under
uncertain environment (Pamucar et al. 2020; Deveci et al.
2020). However, the existing methods represent main limi-
tations (Wolf 2011; Manav et al. 2013; Agrebi et al. 2017;
Chen et al. 2018; Agrebi 2018):
The multi-criteria aspect for the selection of the location
of distribution centers is important. However, most of the
existing methods seek to convert qualitative criteria and
sometimes even quantitative criteria to cost.
The majority of methods do not consider the impact of
the overall strategy of the company on decisions taking
into account qualitative criteria.
The selection of distribution centers location requires the
participation of several company departments, including
distribution, quality and sustainable development, etc.
However, most of the methods developed rely on a single
decision-maker and therefore, only on their preferences.
The location of distribution centers is based on maxi-
mizing certain criteria and minimizing others. However,
many methods do not consider these two aspects at the
same time, for example maximizing quality of service
and minimizing congestion at the same time.
– Metaheuristics for the fuzzy multi-objective decision-
making methods and fuzzy multi-objective decision-
making methods can deal with only quantitative criteria.
The consequence is that qualitative criteria are neglected
in the decision-making process.
In order to overcome the limitations and satisfy more
the distribution center location decision requirements as
well as the expectations of decision-makers, in this paper, a
new Fuzzy Multi-attribute and Multi-actor Decision-Making
(FMAADM) method is proposed. FMAADM method com-
bines the Multi-Attribute and Multi-Actor Decision-Making
(MAADM) method (Agrebi et al. 2017) based on ELECTRE
I method and the multiple criteria decision-making method
(Chen 2001) based on the fuzzy set theory (Yager 1996;
Zadeh 1975a,b,c; Zadeh et al. 1965). Although the MAADM
method has proved its strengths and robustness in the issue
of the distribution location selection (Agrebi et al. 2017),
it is not able to treat this problem under an uncertain envi-
ronment. Thus, we combine the MAADM method with the
fuzzy set theory. Indeed, the fuzzy set theory is an important
paradigm given its ability to treat uncertainty and impreci-
sion associated with information (Alias et al. 2019; Garg and
Rani 2019) and reduce its complexity (Awasthi et al. 2018).
The strengths of the FMAADM method are as follows:
First, the integration of the decision-makers group prefer-
ences into the decision-making process, knowing that the
human preferences are often ambiguous and uncertain,
Second, the consideration of the informations related to
the alternatives and the criteria weights which are inac-
curate, uncertain or incomplete (Arora and Garg 2018),
Third, the verification of the obtained solution by both
tests of concordance and non-discordance.
The main contributions of this paper are threefold:
We propose a novel method, FMAADM method, and
a novel decision support system to treat the decision-
making problem from multiple uncertain experts;
We experimentally valid that our proposed method pro-
vides the expected results by using real data and that our
method when confronted with the decision process acts
as expected within distribution center location selection
process;
We demonstrate not only the stability and the robustness
of the FMAADM method, but also its superiority over
two other existing methods, by applying the Sensitivity
analysis based on the simulation of scenarios.
The rest of paper is organized as follows. The literature
about the location selection of distribution center is presented
in Sect. 2. In Sect. 3, the FMAADM method is developed.
Sect. 4presents the experimental and operational validation
of our proposed method. Finally, Sect. 5concludes the paper
and presents the future research directions.
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Decision-making from multiple uncertain experts: case of distribution center location selection
2 Literature review
To arrange the survey of the problem of distribution center
location selection in various aspects, we will divide it into
the following parts: nature of the problem, existing methods,
and discussion.
2.1 Nature of the problem
In the literature, several studies have been conducted by
researchers on location selection problem of distribution cen-
ter under not only a certain environment but also an uncertain
environment. This depends on the nature of the used param-
eters in the decision-making process, notably the criteria
values, the desired value and importance weight of criteria
and the rating of each alternative location.
Regarding the certain environment, the used parameters
are known and fixed in advance and the problem in question
is characterized as static and deterministic problem (Agrebi
et al. 2017). On the contrary case, under an uncertain envi-
ronment, the real data and the information pertaining, mainly
the informations related to the alternatives and the criteria
weights, are imprecisely because of the inherent vagueness
of human preferences.
2.2 Existing methods
According to the application environment, the existing meth-
ods, for dealing with the location selection problem of
distribution center under uncertainty, can be mainly classi-
fied, as indicated in Table 1, into:
– Metaheuristics for the fuzzy multi-objective decision-
making (FMMODM) methods,
Fuzzy multi-criteria decision-making (FMCDM) meth-
ods categorized into Fuzzy multi-objective decision-
making (FMODM) methods and Fuzzy multi-attribute
decision-making (FMADM) methods.
The common point between the FMCDM and the
FMMODM methods is the development and the application
of fuzzy theory. Table 2presents the most proposed methods
in this issue.
2.2.1 Existing metaheuristics for the fuzzy multi-objective
decision-making methods
Wang et al. (2005) developed a method combining quan-
titative heuristic arithmetic, qualitative analytic hierarchy
process and fuzzy comprehensive evaluation. Their objective
is to realize the minimum expenses of distribution center and
system. Yang et al. (2007) proposed a hybrid method com-
bining Tabu search algorithm, genetic algorithm and fuzzy
simulation algorithm to seek the approximate best solution of
the model. Their objective is to minimize the total relevant
cost. Xu et al. (2011) developed a method based on Tabu
search algorithm, genetic algorithm and fuzzy simulation
algorithm. The objective is to minimize the total relevant cost
comprising of fixed costs of the distribution center and trans-
port costs and minimize the transportation time. Liu et al.
(2011) proposed a hybrid heuristic algorithm to deal with the
problem, which combines rough set methods and fuzzy logic.
The objective is to optimize the cost and combined earn-
ings. Zhou et al. (2015) proposed a solution model termed
as rough multi-objective synthesis effect model. This consti-
tutes a series of crisp multi-objective programming models
that reflect different decision consciousness for each decision
maker. The optimal solutions of the proposed model can be
obtained by using the genetic algorithm. Zhuge et al. (2016)
proposed a stochastic programming model on locating dis-
tribution centers determining their suitable scales, as well as
adjusting distribution centers so as to adapt to the dynami-
cally changing demands at the sites of retailers. Xiyang et al.
(2018) established a fuzzy multiobjective model to solve the
location problem. This model is based on two-stage supply
chain, taking into account the inventory status of the tight
front and rear nodes involved in the location of the distribu-
tion center. The core of this model is the impact of inventory
fluctuations on the supply chain.
In this first category, the metaheuristics for the fuzzy multi-
objective decision-making methods, the proposed methods
attempt to convert the multi-objective problem into a single
objective problem and then optimize this new single objec-
tive problem (Chen et al. 2018; Manav et al. 2013). In fact,
optimizing this single objective problem yields a single solu-
tion. But, the decision-makers need diverse options in the
real condition (Wolf 2011). However, there are some clas-
sical methods that require knowing the optimal solution of
each objective but acquiring this information is expensive
and time-consuming (Wolf 2011). In addition, it is difficult,
especially in the case of the non-deterministic situation, to
choose weights for which these methods are dependent.
2.2.2 Existing fuzzy multi-criteria decision-making methods
Chen (2001) proposed a multi-attribute decision-making
method based on fuzzy set theory. In this method, the
weights of the criteria and the evaluations of the alternatives
are described by linguistic variables expressed in triangular
fuzzy numbers. The final evaluation value of each distribution
center location is also expressed in a triangular fuzzy number.
Lee (2005) addressed the problem of selecting the location
of distribution centers in a fuzzy environment by proposing a
fuzzy multi-criteria decision-making method based on fuzzy
set theory and SWOT analysis. The goal is to refine the impre-
cision of decision data by using alternative scores and criteria
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M.Agrebi,M.Abed
Table 1 Comparison of some
characteristics between existing
methods
FMCDM FMMODM
FMADM FMODM
Alternatives Limited Limited Unlimited
Solution(s) One or more One or more One
Criteria Qualitative and/or quantitative Quantitative Quantitative
Table 2 Existing methods
Methods Proposed by
FMCDM methods
FMCDM method based on Fuzzy Set Theory (FST) Chen (2001)
FMCDM method based on fuzzy logic and SWOT (Strengths, Weaknesses, Opportunities,
Threats) analysis
Lee (2005)
Improved FMCDM approach ChuandLai(2005)
Fuzzy Analytic Hierarchy Process (AHP) Fan et al. (2006)
Multidimensional FMCDM method based on fuzzy quality function deployment and fuzzy
Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)
Zhang et al. (2006)
Fuzzy integrated hierarchical decision-making approach Chan et al. (2007)
FMCDM method based on fuzzy Analytic Network Process (ANP) Wei and Wang (2009)
Weighted fuzzy factor rating system Ou and Chou (2009)
FMCDM model Chou and Chang (2009)
Model of multi-level fuzzy optimization based on information entropy and AHP Zhang et al. (2009)
Fuzzy decision-making model based on engineering FST Yu et al. (2009)
Fuzzy TOPSIS method based on FST, the factor rating system and simple additive weighting Hu et al. (2009)
FMCDM method based on fuzzy Decision Making Trial and Evaluation Laboratory
(DEMATEL) and AHP/ANP
Kuo (2011)
FMCDM method based on fuzzy TOPSIS Awasthi et al. (2011)
Approach based on FST, fuzzy integration method, axiomatic fuzzy set approach based
fuzzy clustering algorithm and fuzzy TOPSIS
Wang et al. (2012)
Hybrid FMCDM method based on fuzzy Entropy Weight (EW), fuzzy AHP and fuzzy
TOPSIS
He et al. (2017)
Framework of hybrid spatial-fuzzy multi-criteria decision-making based on weighted
Geographical Information System data and fuzzy TOPSIS
Sopha et al. (2018)
Metaheuristics for FMODM
Method combines the quantitative heuristic arithmetic, the qualitative AHP and the fuzzy
comprehensive evaluation
Wang et al. (2005)
Method based on fuzzy chance-constrained programming, tabu search algorithm + genetic
algorithm + fuzzy simulation algorithm
Yang et al. (2007)
Multi-objective approach based on fuzzy AHP method and LP-metric method Jafari et al. (2010)
Heuristic Algorithm combines rough set methods and fuzzy logic Liu et al. (2011)
Multi-objective programming model with random fuzzy coefficients, chance-constrained
programming, the spanning tree-based genetic algorithm
Xu et al. (2011)
Fuzzy neural network method Li and Liu (2013)
Method based on cluster analysis, Floyd algorithm and fuzzy AHP Guo-qin and Hong-yan (2014)
Rough model based on crisp multi-objective programming models and genetic algorithm Zhou et al. (2015)
Method based on AHP and fuzzy comprehensive evaluation method Cheng and Zhou (2016)
Stochastic programming model Zhuge et al. (2016)
Fuzzy multi-objective model Xiyang et al. (2018)
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Decision-making from multiple uncertain experts: case of distribution center location selection
weights which are assigned as linguistic variables repre-
sented by fuzzy triangular numbers. Chan et al. (2007)devel-
oped fuzzy integrated hierarchical decision-making approach
which is the combination of the hierarchical decision-making
technique and the fuzzy decision making technique. Wei and
Wang (2009) applied fuzzy-ANP methodology to select the
location of distribution center. This methodology is a method
combined ANP, fuzzy theory, and fuzzy AHP. The rela-
tionship between the criteria is established using the ANP,
while the criteria weights as well as the evaluations of the
alternatives according to the various criteria are determined
using the fuzzy ANP. Ou and Chou (2009) are interested to
treat the problem of international distribution center selection
under uncertain environment. On this subject, they developed
weighted fuzzy factor rating system. Chou and Chang (2009)
proposed a fuzzy multiple criteria decision making model.
Zhang et al. (2009) proposed model of multilevel fuzzy
optimization into location decision on distribution center of
emergency logistics for emergency event and used informa-
tion entropy and analytical hierarchy process to determine the
combined weight of the indexes. Yu et al. (2009) presented
the new optimal selection for alternative programs of logistics
center using the fuzzy decision-making model based on engi-
neering fuzzy set theory. Hu et al. (2009) proposed a fuzzy
TOPSIS method integrating fuzzy set theory, the factor rat-
ing system and simple additive weighting to evaluate facility
locations alternatives. Jafari et al. (2010) treat the problem of
distribution center location as a multiobjective problem. Cost
minimization is the primary objective in this area. Then, they
proposed a method for the uncapacitated single stage facility
location problem in which a fuzzy AHP method is used to
achieving these importances. Awasthi et al. (2011) presented
a multi-criteria decision making approach for location plan-
ning for urban distribution centers under uncertainty. Their
proposed approach involves identification of potential loca-
tions, selection of evaluation criteria, using of fuzzy theory
to quantify criteria values under uncertainty and application
of fuzzy TOPSIS to evaluate and select the best location for
implementing an urban distribution center. Kuo (2011)pro-
posed an hybrid method combining the concepts of fuzzy
DEMATEL and a method of fuzzy multiple criteria decision-
making in a fuzzy environment. The fuzzy DEMATEL is
proposed to arrange a suitable structure between criteria, and
the analytic hierarchy/network process (AHP/ANP) is used
to construct weights of all criteria. The linguistic terms char-
acterized by triangular fuzzy numbers are used to denote the
evaluation values of all alternatives versus various criteria.
Finally, the aggregation fuzzy assessments of different alter-
natives are ranked to determine the best selection. Wang et al.
(2012) proposed fuzzy integration and clustering approach
using the improved axiomatic fuzzy set theory developed for
location clustering based on multiple hierarchical evaluation
criteria. Then, they applied the technique for order preference
by similarity to ideal solution for evaluating and selecting
the best candidate for each cluster. Guo-qin and Hong-yan
(2014) employed the cluster analysis method and Floyd algo-
rithm to achieve minimization of all paths to get the Shanghai
agricultural product logistics distribution center alternatives.
Combined with the Fuzzy AHP, an analysis of Shanghai agri-
cultural products logistics distribution center alternatives is
performed. Cheng and Zhou (2016), to improve the efficiency
of decision-making, proposed a method combining AHP and
fuzzy and comprehensive evaluation method. He et al. (2017)
proposed an hybrid fuzzy multiple-criteria decision-making
method, in order to achieve operational efficiency and reduce
operational cost. Their method combines fuzzy AHP method,
fuzzy-entropy method, and fuzzy TOPSIS method. Sopha
et al. (2018) proposed a framework of hybrid spatial fuzzy
multi-criteria decision-making based on weighted Geograph-
ical Information System data and fuzzy TOPSIS.
In this second category, the fuzzy multicriteria decision-
making methods, existing methods cited above are used in
the case where the number of predetermined alternatives is
limited. This set of alternatives satisfies each objective in a
specified level (Wolf 2011). Then, the best solution is selected
according to the priority of each objective and the interaction
between them. However, these methods neglect the aspects
of concordance and non-concordance to verify that a suffi-
cient majority of criteria, represented by their weight, are in
favoroftheassertionAi outclass Ak, and, to make it possi-
ble to refuse the outclass of an alternative over another when
there is too much opposition on at least one criterion. Indeed,
most existing methods not consider qualitative criteria in the
process decision. Besides, many methods consider only one
objective, maximization or minimization, for example max-
imizing quality of service or minimizing congestion but not
the both.
2.3 Discussion
In order to satisfy more distribution center location deci-
sion requirements, this paper presents a new Fuzzy Multi-
Attribute and Multi-Actor Decision-Making (FMAADM)
method. The proposed method combines the Multi-Attribute
and Multi-Actor Decision-Making (MAADM) method
(Agrebi et al. 2017) based on ELECTRE I method (Milosavl-
jevi´cetal.2018; Collette and Siarry 2011;Roy1968) and
the multiple criteria decision-making method (Chen 2001)
based on the fuzzy set theory (Si et al. 2019; Yager 1996;
Zadeh 1975a,b,c; Zadeh et al. 1965). Although the MAADM
method has proved its strengths and robustness in the issue
of the distribution location selection (Agrebi et al. 2017), it is
not able to treat this problem under an uncertain environment.
Thus, we combine the MAADM method with the fuzzy set
theory. We argue below the choice of MAADM method and
fuzzy set theory.
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M.Agrebi,M.Abed
Table 3 Advantages/disadvantages of a number of MADM methods (Ayadi 2010;Agrebietal.2017;Agrebi2018)
Method Advantages Disadvantages
AHP The hierarchical structure of the decision problem The explosion of the number of pairwise
comparisons in the case of processing a large
number of elements
The semantic scale used to express the preferences
of the decision-maker
The reversal of the alternatives rank following the
deletion/addition of one or more alternatives
The association of a numerical scale with the
semantic scale is restrictive
TOPSIS The introduction of the notions of ideal and anti-ideal The requirement that attributes must be cardinal in
nature
Easy to apply In the event of the alternatives are bad, TOPSIS
offers the best alternative among the bad ones
The arbitrariness of the distance choice to the ideal
point and to the anti-ideal point
ELECTRE I The introduction of the kernel notion makes it
possible to narrow the field of study to focus only
on the best alternatives
The requirement to translate the performance of
alternatives into scores can lead to loss of control
over data
The non-consideration of group decision-makers
MAADM The non-need to translate the performance of
alternatives into scores
The absence of fuzzy notion in the choices of the
decision maker
ELECTRE II The alternatives’ outranking from the best to the
worst
The requirement of cardinal evaluations and the a
priori articulation of preferences
ELECTRE III The alternatives’ outranking from the best to the
worst
The need for a large number of technical parameters
The admission of the fuzzy notion in the choices of
the decision maker and the introduction of the veto
threshold
Complex and sometimes difficult to interpret
ELECTRE IV The association with each criterion of the preference
thresholds and the overpressure of the criteria
weighting
The need for a large number of technical parameters
ELECTRE TRI The ability to process a large number of alternatives The need for a large number of technical parameters
The inability in some cases to compare each
alternative with the alternatives limiting the
different categories
The definition of categories linked to the choice of
benchmark alternatives
PROMETHEE I The construction of the valued outranking
relationship reflecting an intensity of preference
Indifference is in practice very rare given the
numerous calculations to obtain the flows
PROMETHEE II The construction of a total preorder excluding
incomparability and greatly reducing indifference
Pairwise comparison are only used to hide the
calculation of the final score for each alternative
PROMETHEE III The introduction of indifference thresholds, which
minimizes numerous calculations
The indifference thresholds have no concrete
interpretation for the decision maker These
thresholds are the subject of statistical calculations
which make POMETHEE III less accessible
2.3.1 ELECTRE method
ELECTRE method and its different derivatives (ELECTRE
I, ELECTRE II, ELECTRE III, ELECTRE IV, and ELEC-
TRE TRI) are considered to be the most preferred methods
among several outranking methods like PROMETHEE and
its derivatives (PROMETHEE I and II), ORESTE, QUAL-
IFLES, MELCHIOR, MAPPACC, PRAGMA, and TACTIC
(Zandi and Roghanian 2013; Agrebi et al. 2017)asshown
in Table 3. Furthermore, ELECTRE method is considered as
one of the best methods which take into account both desir-
able directions (Min and Max) (Farahani and Asgari 2007;
Agrebi 2018)
ELECTRE I method, as a derivative of ELECTRE method,
is suitable for solving the location selection problem distri-
bution centers as a multi-criteria selection problem under
certain environment. This method allows to select the best
locations among a limited set of alternatives by respect-
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Decision-making from multiple uncertain experts: case of distribution center location selection
ing of qualitative and quantitative criteria (Kumar et al.
2017). Moreover, it includes the decision-makers and their
preferences into the decision-making process. Besides, the
obtained results are validated using concordance and non-
discordance tests. However, it does not consider neither
several decision makers, nor the information uncertainty
and imprecision. MAADM method (Agrebi et al. 2017;
Agrebi 2018) as an extension of ELECTRE I takes into
account several decision-makers into the decision process.
Thus, the fuzzy set theory is used to address this limita-
tion.
2.3.2 Fuzzy set theory
Fuzzy set theory is an extension of ordinary set theory was
introduced by Zadeh et al. (1965) for dealing with uncer-
tainty and imprecision which are inherent to human judgment
in decision making processes through the use of linguis-
tic terms and degrees of membership (Alias et al. 2019;
Garg and Rani 2019; Tayal et al. 2014; Zouggari and Beny-
oucef 2012). Indeed, a fuzzy set is a class of objects with
grades of membership. These grades present the degree of
stability to which certain element belongs to a fuzzy set
(Zadeh et al. 1965). Therefore, it is economically sensible
for an enterprise decision maker to use fuzzy set theory,
one of the artificial intelligence techniques (Simi´cetal.
2017).
In the multi-criteria environment, fuzzy set theory had an
impact on classification techniques and contributed to the
proposal of new decision-making methods (Awasthi et al.
2018; Bashiri and Hosseininezhad 2009; Chen 2000,2001;
Chu and Lai 2005;Ertu˘grul 2011; Hwang and Thill 2005;
Kahraman et al. 2006; Kaya and Çinar 2006;Lietal.2011;
Rebaa 2003; Takaˇci et al. 2012; Trivedi and Singh 2017; Zhou
and Liu 2007). These methods make it possible to treat uncer-
tainty based on the idea of order. In addition, they are based on
a methodology of representation and use of vague and uncer-
tain knowledge, called the theory of approximate reasoning,
better known as fuzzy logic. In addition, they consider classes
of objects whose boundaries are not clearly defined by intro-
ducing a membership function taking values between zero
and one.
In short, fuzzy set theory offers a mathematically pre-
cise way of modeling vague preferences, for example setting
weights of performance scores on criteria. Simply stated,
fuzzy set theory makes it possible to mathematically describe
statements like: “criterion X should have a weight of around
0.8”. Besides, fuzzy set theory can be combined with other
techniques to improve the quality of results (Simi´cetal.2017)
and improve the decision-making process by making it more
realistic.
Fig. 2 Flowchart of FMAADM method
3 The multi-attribute and multi-actor
decision-making (FMAADM) method
The Multi-Attribute and Multi-Actor Decision-Making
(FMAADM) method is described in this section. First,
Sect. 3.1 details the procedure of the FMAADM, second,
Sect. 3.2 represents its architecture, and third, Sect. 3.3 shows
the proposed decision support system based on FMAADM
method.
3.1 FMAADM method procedure
The FMAADM method comprises, essentially, the ten steps
described as follows. The flowchart of this method is pre-
sented as shown in Fig. 2.
Step 1. Constitution of decision-makers’ committee:This
step consists in forming a committee of decision-makers (K)
from various departments (distribution, quality, sustainable
development, etc.) in order to defend the departments inter-
ests that represents each decision-maker. The goal is to reflect
the general case of the request and to treat the selection prob-
lem in a broad perspective by including different points of
view (Turkoglu and Genevois 2017).
Step 2. Identification of potential locations: This step con-
sists in identifying a set of potential locations (alternatives)
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M.Agrebi,M.Abed
of distribution centers (Ai=1,...,m) based on sustain-
able freight regulations, decision-makers’ preferences and
knowledge conditions of freight transportation. The potential
locations are those that cater to the interest of all city stake-
holders, that is, city residents, logistics operators, municipal
administrations, and so forth (Awasthi et al. 2011).
Step 3. Selection of evaluation criteria: This step consists
in selecting ncriteria (Cj, where j=1,...,n) such as con-
nectivity to multimodal transport, proximity to customers and
transportation cost, etc. Compared with the selected criteria,
the alternatives will be evaluated.
Step 4. Determination of the fuzzy weight of criteria:This
step consists in assigning the importance of ncriteria by K
decision-makers. The goal is to establish the matrix criteria’s
importance (W) based on Eq. (1) expressed as follows:
W=w1w2... w
n.
W=1
Kw1
j+w2
j+···+wK
j,(1)
where wj(j=1,2,3,...,n) is the weight of criterion (Cj).
Step 5. Evaluations of alternatives and determination of
the fuzzy decision matrix: This step consists, first, in evalu-
ating malternatives by Kdecision-makers with respect to the
criteria (Cj, where j=1,...,n) and then in constructing
the fuzzy decision matrix (D) based on Eq. (2) and expressed
as follows:
D=
x11 x12 ... x1n
x21 x22 ... x2n
.. .
.. .
. . ... .
xm1xm2... xmn
,
where xij,i,jis the rating of alternative Ai(i=1,2,…,m)
with respect to criterion Cj.
xij =1
Kx1
ij +x2
ij +···+xK
ij .(2)
Step 6: Construction of the normalized fuzzy decision
matrix: This step is based on the normalization of the fuzzy
decision matrix (D) by using the linear scale transforma-
tion. The aim is to ensure that the evaluations above preserve
the property that the ranges of normalized fuzzy numbers
belong to [0, 1]. Then, the normalized fuzzy decision matrix
is obtained and denoted by R.
R=rijmn,
rij =aij
c
j
,bij
c
j
,cij
c
j,jB,
rij =a
cij ,a
bij ,a
aij ,jC,
c
j=max
icijifj B,
aj=min
iaijifj C,(3)
where rij is the normalized value of xij,Band Care the set
of benefit criteria and cost criteria, respectively.
Considering the different importance of each criterion, the
final fuzzy evaluation value of each alternative is calculated
as:
Pi=rij.wj,i=1,2,...,m,(4)
where Piis thefinal fuzzy evaluation value of alternative Ai.
Step 7. Establishment relations between alternatives with
respect to each criterion: By respecting to each criterion, the
pairwise comparisons of the alternatives Aiand Ak(where
k[i,...,m] and k= i) are established as follows:
J+(Ai,Ak)={j|Cj(Ai)>Cj(Ak)},(5)
where J+(Ai,Ak)the set of criteria for which the alternative
Aiis preferred over Ak.
J=(Ai,Ak)={j|Cj(Ai)=Cj(Ak)},(6)
where J=(Ai,Ak)the set of criteria for which the alternative
Aiis equal in preference to alternative Ak.
J(Ai,Ak)={j|Cj(Ai)<Cj(Ak)},(7)
where J(Ai,Ak)the set of criteria for which the alternative
Akis preferred over Ai.
Step 8. Conversion of relations between alternatives in
numerical values: In this step, the sum of the criteria weights
is determined in each set of criteria as follows:
P+(Ai,Ak)=
j
wjjJ+(Ai,Ak).(8)
P=(Ai,Ak)=
j
wjjJ=(Ai,Ak).(9)
P(Ai,Ak)=
j
wjjJ(Ai,Ak).(10)
Step 9. Merger the numerical values: This step consists
in merging the numerical values by calculating the Concor-
dance Index (CI) and the Disconcordance Index (DI).
123
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Decision-making from multiple uncertain experts: case of distribution center location selection
Concordance index (CI): This index expresses how much
the hypothesis (Aioutclasses Ak) is consistent with the
reality represented by the evaluations of alternatives.
CIik =P+(Ai,Ak)+P=(Ai,Ak)
P(Ai,Ak),(11)
where P(Ai,Ak)=P+(Ai,Ak)+P=(Ai,Ak)+P(Ai,Ak).
Set of concordance:
J(Ai,Ak)=J+(Ai,Ak)J=(Ai,Ak). (12)
Disconcordance index (DI):
DIik =(0,0,0)if J(Ai,Ak)=∅
(1,1,1)
j×max(Cj(Ak)Cj(Ai)) where jJ(Ai,Ak), otherwise
(13)
where jis the amplitude of the scale associated with crite-
rion j.
Step 10. Filtration of alternatives: This step allows to
extract from the set of the potential alternatives Ai(where
i=1,...,m) the set of alternatives which respect Eq. (14).
From this set, one alternative will finally be retained. It is the
alternative that outclasses more alternatives.
CI
ik ct
DIik dtAiSA
k,(14)
where:
ct is the threshold of concordance beyond which the
hypothesis AiSA
kis considered as valid.
dt is the threshold of discordance below which the
hypothesis AiSA
kis no longer valid.
We remind that Sis the outranking relation ( AiSA
kmeans
that Aiis at least as good as Ak).
3.2 FMAADM method architecture
The architecture of FMAADM method consists of three lev-
els notably users level, user-application interface level and
application level. Fig. 3gives a general view of the interac-
tions between levels.
User level: The users are the involved decision-makers
into the decision-making process. They are invited to
express their preferences by regarding the importance
of each criterion and each alternative. From these pref-
erences, the matrix criteria’s importance (W) and the
decision matrix (D) are determined. The goal is to store
them into the knowledge base in order to use them for
selecting the best alternative.
User–application interface: This level regroups the com-
munication interfaces between the decision-makers and
the application.
Application level: Configuration module: This module
ensures the configuration of the decision-making process
in accordance with the selection policy of the enterprise
such as the number of alternatives to choose, and the
decision-makers.
Simulator fuzzy multi-attribute : This simulator is based
on FMAADM method. It is developed for generating
the decision-makers preferences based on linguistic vari-
ables represented by triangular fuzzy number. The aim is
to find the best alternative among the set of potential alter-
natives.
Fuzzy module : The role of this module is to accommo-
date the uncertain parameters of the selection process.
The decision-maker communicates the linguistic vari-
ables in order to express his point of view about the
criteria importance and alternatives. Subsequently, the
fuzzy module translates the equivalence in triangular
fuzzy number.
3.3 Proposed decision support system
In order to find an appropriate solution to users’ needs and
specificities, we developed the decision support system based
on FMAADM method. The interface and the functionality of
our system are implemented in Java 8. Netbeans1has been
selected as the appropriate development environment. Also,
the system uses XML2format for information transmission
and storage (saving performed studies or projects). In addi-
tion, we made use of some APIs such as Apache POI,3JDBC4
in order to manage data, which may be extracted from excel
files. Users can generate data automatically based on a ran-
dom generator or existing data source and manually. We note
that the random generator is basically used for testing pur-
pose. Fig. 4presents the architecture of our S-SSD.
1https://netbeans.org/.
2https://www.w3.org/XML/.
3https://poi.apache.org/.
4http://www.oracle.com/technetwork/java/javase/jdbc/index.html.
123
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M.Agrebi,M.Abed
Fig. 3 FMAADM method
architecture
Fig. 4 Architecture of the proposed decision support system
123
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Decision-making from multiple uncertain experts: case of distribution center location selection
Table 4 Evaluation criteria for
location selection Criteria Illustrations Units Types
Accessibility (C1)Access by public and private transport modes
to the location
Quantitative Benefit
Security (C2)Security of the location from accidents Qualitative Benefit
Connectivity to multimodal
transport (C3)
Connectivity of the distribution center with
other modes of transport
Quantitative Benefit
Costs (C4)Costs combining land cost, vehicle resources
cost, policy cost and taxes
Quantitative Cost
Environmental impact (C5)Impact of the implementation of the
distribution center on the environment
Quantitative Cost
Proximity to customers
(C6)
Distance of location to customer Quantitative Benefit
Proximity to suppliers (C7)Distance of location to suppliers Quantitative Benefit
Resource availability (C8)Availability of resources to various uses Quantitative Benefit
Conformance to sustainable
freight regulations (C9)
Ability to conform to sustainable freight
restriction imposed by public authorities.
Qualitative Benefit
Possibility of expansion
(C10)
Capability to augment the size to
accommodate increasing demands
Quantitative Benefit
Quality of service (C11)Ability to assure timely and reliable service to
clients
Qualitative Benefit
4 Experimental validation
In order to establish the necessary consensus between the
decision-making problem and a proposed method, Bisdorff
et al. (2015) present four kinds of validation:
Conceptual validation: verify on what each precise con-
cept represents and how this is useful for the decision-
making’s problem.
Logical validation: verify whether the concepts are logi-
cally consistent and meaningful.
Experimental validation: test the method using experi-
mental data in order to show that the method provides
the expected results and possibly check formal require-
ments such as convergence of an algorithm, accuracy of
a classification, and sensitivity to small variations of the
parameters.
Operational validation: show that the method when con-
fronted with the decision process acts as expected within
such a decision-making process.
To this end, in this section, performance of our proposed
method is validated by a case of an accompany, which is
interested in selecting a new distribution center location. The
selection of the best location is done by a committee of three
decision-makers D1,D2and D3. The aim of which is to select
a best location among three alternatives A1,A2and A3.The
selection decision is made based on eleven main evaluation
criteria C1,...,C11. As shown in Table 4,C4and C5are cost
criteria and the remaining of criteria are the benefit criteria.
The hierarchical structure of this case study is illustrated as
shown in Fig. 5.
4.1 Application of the FMAADM method
The computational procedure of the FMAADM method is
summarized as the following steps.
Foremost, using the linguistic variables (Awasthi et al.
2016,2011;Heetal.2017) presented in Tables 5and 6,the
criteria and the alternatives are evaluated by the decision-
makers. Table 7presents the importance of criteria and the
weight of each criterion calculated using Eq. (1). Table 8sum-
marizes the evaluations of the alternatives. Then, the fuzzy
decision matrix (D) is constructed using Eq. (2)asshownin
Table 9. The normalized fuzzy decision matrix (R) is deter-
mined using Eq. (3) and presented in Table 10.
Afterward, considering the criteria, the final fuzzy eval-
uation value (P) of each alternative is determined using
Eq. (4) as shown in Table 11. Therefore, the relationship
(J+,J=,J) between the alternatives is established as
shown in Table 12, by calculating the difference between two
final fuzzy evaluation value of each alternative using Eqs. (5),
(6) and (7). These relations are converted subsequently, using
Eqs. (8), (9) and (10) , in numerical values (P+,P=,P)by
calculating the set of concordance Jas shown in Tables 12
and 13 . The merge of the numerical values is obtained, using
Eqs. (11), (12) and (13) , by calculating of the coefficients of
concordance Cik and the coefficients of disconcordance Dik
as shown in Table 14.
Finally, the test of concordance and the test of non-
disconcordance are done using Eq. (14) in order to filter the
123
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M.Agrebi,M.Abed
Fig. 5 Hierarchical structure of
the distribution center’s location
selection
Table 5 Linguistic variables for the importance weight of criteria
Linguistic term Membership function
Very low (VL) (1, 1, 3)
Low (L) (1, 3, 5)
Medium (M) (3, 5, 7)
High (H) (5, 7, 9)
Very high (H) (7, 9, 9)
Table 6 Linguistic variables for the ratings
Linguistic term Membership function
Very poor (VP) (1, 1, 3)
Poor (P) (1, 3, 5)
Fair (F) (3, 5, 7)
Good (G) (5, 7, 9)
Very good (VG) (7, 9, 9)
alternatives. For that, the threshold ct is fixedto (0.5, 0.7, 0.9).
This test is satisfied if CI
ik (0.5, 0.7, 0.9). The threshold dt
is fixed to (0.3, 0.5, 0.7). Then, the test is satisfied if DIik
(0.3, 0.5, 0.7).
The CI
ik which satisfied the test of concordance are CI
21,
CI
31 and CI
32.TheDIik which satisfied the test of non-
disconcordance are DI12,DI13 and DI23. Therefore, based
on both tests, we found that : the alternative A1outclasses the
alternatives A2and A3. Then, we can infer that the alternative
A1is the best alternative.
To validate experimentally our proposed method, an appli-
cation of FMAADM by using real data was presented. The
obtained results show that FMAADM method when con-
fronted with the decision process acts as expected within
distribution center location selection process. In the follow-
ing Sect. 4.2, a comparative analysis of the obtained results
by our method and two existing methods will be detailed.
Table 7 Importance and weight
of criteria CjDecision-makers Weight
D1D2D3
C1VH (7, 9, 9) VH (7, 9, 9) VH (7, 9, 9) (7, 9, 9)
C2VH (7, 9, 9) H (5, 7, 9) VH (7, 9, 9) (6.33, 8.33, 9)
C3VH (7, 9, 9) H (5, 7, 9) VH (7, 9, 9) (6.33, 8.33, 9)
C4H (5, 7, 9) VH (7, 9, 9) VH (7, 9, 9) (6.33, 8.33, 9)
C5M (3, 5, 7) H (5, 7, 9) M (3, 5, 7) (3.66, 5.66, 7.66)
C6VH (7, 9, 9) H (5, 7, 9) VH (7, 9, 9) (6.33, 8.33, 9)
C7H (5, 7, 9) VH (7, 9, 9) H (5, 7, 9) (5.66, 7.66, 9)
C8M (3, 5, 7) H (5, 7, 9) H (5, 7, 9) (4.33, 6.33, 8.33)
C9H (5, 7, 9) H (5, 7, 9) VH (7, 9, 9) (5.66, 7.66, 9)
C10 VH (7, 9, 9) H (5, 7, 9) VH (7, 9, 9) (6.33, 8.33, 9)
C11 VH (7, 9, 9) VH (7, 9, 9) VH (7, 9, 9) (7, 9, 9)
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Decision-making from multiple uncertain experts: case of distribution center location selection
Table 8 Evaluations of alternatives
CjAiDecision-makers
D1D2D3
C1A1VG(7,9,9) G(5,7,9) VG(7,9,9)
A2G (5, 7, 9) VG (7, 9, 9) VG (7, 9, 9)
A3VG(7,9,9) VG(7,9,9) VG(7,9,9)
C2A1G(5,7,9) VG(7,9,9) G(5,7,9)
A2F (3, 5, 7) G (5, 7, 9) F (3, 5, 7)
A3P (1, 3, 5) P (1, 3, 5) F (3, 5, 7)
C3A1F (3, 5, 7) G (5, 7, 9) F (3, 5, 7)
A2F (3, 5, 7) F (3, 5, 7) F (3, 5, 7)
A3F (3, 5, 7) G (5, 7, 9) G (5, 7, 9)
C4A1G (5, 7, 9) G (5, 7, 9) F (3, 5, 7)
A2F (3, 5, 7) G (5, 7, 9) F (3, 5, 7)
A3F (3, 5, 7) G (5, 7, 9) G (5, 7, 9)
C5A1G (5, 7,9) F (3, 5, 7) G (5, 7, 9)
A2G (5, 7, 9) F (3, 5, 7) F (3, 5, 7)
A3G (5, 7, 9) P (1, 3, 5) G (5, 7, 9)
C6A1G(5, 7, 9) VG (7, 9, 9) VG (7, 9, 9)
A2VG (7, 9, 9) VG (7, 9, 9) G (5, 7, 9)
A3VG (7, 9, 9) VG (7, 9, 9) G (5, 7, 9)
C7A1F(3,5,7) VG(7,9,9) F(3,5,7)
A2G (5, 7, 9) G (5, 7, 9) F (3, 5, 7)
A3G (5, 7, 9) G(5, 7, 9) (5, 7,9)
C8A1F (3, 5, 7) F (3, 5, 7) G (5, 7, 9)
A2G (5, 7, 9) F (3, 5, 7) F (3, 5, 7)
A3F (3, 5, 7) F (3, 5, 7) F (3, 5, 7)
C9A1F (3, 5, 7) F (3, 5, 7) F (3, 5, 7)
A2F (3, 5, 7) P (1, 3, 5) P (1, 3, 5)
A3VP(1,3,5) VP(1,1,3) VP(1,1,3)
C10 A1G (5, 7, 9) G (5, 7, 9) VG (7, 9, 9)
A2VG(7,9,9) G(5,7,9) VG(7,9,9)
A3G (5, 7, 9) G (5, 7, 9) G (5, 7, 9)
C11 A1VG(7,9,9) G(5,7,9) VG(7,9,9)
A2VG(7,9,9) G(5,7,9) G(5,7,9)
A3G (5, 7, 9) VG (7, 9, 9) VG (7, 9, 9)
4.2 Comparative analysis of the results
In this section, we compare the results of our proposed
method with two other existing methods under fuzzy envi-
ronment so that the consistency of the aforesaid results can
be justified: the first method, the hybrid FMCDM method
based on fuzzy Entropy Weight (EW), fuzzy AHP and fuzzy
TOPSIS, is proposed by He et al. (2017) and, the second
method, the framework of hybrid spatial-fuzzy multi-criteria
decision-making based on weighted Geographical Informa-
tion System data and fuzzy TOPSIS, is invented by Sopha
Table 9 Fuzzy decision matrix
A1A2A3
C1(6.33, 8.33, 9) (6.33, 8.33, 9) (7, 9, 9)
C2(5.66, 7.66, 9) (3.66, 5.66, 7.66) (1.66, 3.66, 5.66)
C3(3.66, 5.66, 7.66) (3, 5, 7) (4.33, 6.33, 8.33)
C4(4.33, 6.33, 8.33) (3.66, 5.66, 7.66) (4.33, 6.33, 8.33)
C5(4.33, 6.33, 8.33) (3.66, 5.66, 7.66) (3.66, 5.66, 7.66)
C6(6.33, 8.33, 9) (6.33, 8.33, 9) (6.33, 8.33, 9)
C7(4.33, 6.33, 7.66) (4.33, 6.33, 8.33) (5, 7, 9)
C8(3.66, 5.66, 7.66) (3.66, 5.66, 7.66) (3, 5, 7)
C9(3, 5, 7) (1.66, 3.66, 5.66) (1, 1.66, 3.66)
C10 (5.66, 7.66, 9) (6.33, 8.33, 9) (5, 7, 9)
C11 (6.33, 8.33, 9) (5.66, 7.66, 9) (6.33, 8.33, 9)
et al. (2018). Table 15 recapitulates the obtained outranking
and selected location by applying the different methods.
In fact, to evaluate the goodness of a result produced by
any method, we need to make a comparison with the true
result which is of course unknown (Munier et al. 2019). If
it was known, we would not need multi-criteria decision-
making, as we do when we use the Linear Programming,
because if there is an optimum solution it will find it. How-
ever, the Linear Programming works with only one objective
and with only quantitative criteria, and these conditions are
generally not present in real life scenarios. Since it is impos-
sible to realize if a result is good or bad, common sense says
that the practitioner has to consider a method that fits his
needs, and consequently it is expected that its result must be
better that a one that does not (Agrebi 2018; Munier 2011).
According to the afore-given discussion and analysis,
compared with the group decision-making methods from the
literature (He et al. 2017; Sopha et al. 2018), our proposed
method in this paper has the following three major charac-
teristics:
Our proposed method applies both tests of concordance
and non-discordance. On the one hand, to ensure that a
sufficient majority of criteria, represented by their weight,
are in favor of the assertion AiSAk, and, on the other
hand, to make it possible to refuse the outclass of an alter-
native over another when there is too much opposition on
at least one criterion,
The FMAADM method makes it possible to restrict the
field of study to focus only on the best alternatives based
in the kernel concept. Contrariwise, He et al.’s method
and Sopha et al.’s method, if all alternatives are bad, they
offer the best alternative among the bad ones,
And compared with the He et al.’s method our proposed
method possesses the ability to treat a large number of
alternatives.
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M.Agrebi,M.Abed
Table 10 Normalized fuzzy
decision matrix Cja
jc
jDecision-makers
A1A2A3
C16.33 9 (0.7, 0.75, 1) (0.7, 0.75, 1) (0.7, 0.7, 0.9)
C21.66 9 (0.18, 0.21, 0.29) (0.21, 0.29, 0.45) (0.29, 0.45, 1)
C33 8.33 (0.43, 0.67, 0.91) (0.36, 0.60, 0.84) (0.51, 0.75, 1)
C43.66 8.33 (0.51, 0.75, 1) (0.43, 0.67, 0.91) (0.51, 0.75, 1)
C53.66 8.33 (0.51, 0.75, 1) (0.43, 0.67, 0.91) (0.43, 0.67, 0.91)
C66.33 9 (0.7, 0.92, 1) (0.7, 0.92, 1) (0.7, 0.92, 1)
C74.33 9 (0.48, 0.7, 0.85) (0.48, 0.7, 0.92) (0.55, 0.77, 1)
C83 7.66 (0.52, 0.73, 1) (0.52, 0.73, 1) (0.39, 0.65, 0.91)
C91 7 (0.42, 0.71, 1) (0.23, 0.52, 0.8) (0.14, 0.23, 0.52)
C10 5 9 (0.62, 0.85, 1) (0.7, 0.92, 1) (0.55, 0.77 , 1)
C11 5.66 9 (0.7, 0.92, 1) (0.62, 0.85, 1) (0.7, 0.92, 1)
Table 11 Final fuzzy evaluation
value of alternatives A1A2A3
C1(4.9, 6.75, 9) (4.9, 6.75, 9) (4.9, 6.3, 8.1)
C2(1.13, 1.74, 2.61) (1.32, 2.41, 4.05) (1.83, 3.74, 9)
C3(2.72, 5.58, 8.19) (2.27, 4.99, 7.56) (3.22, 6.24, 9)
C4(3.22, 6.24, 9) (2.72, 5.58, 8.19) (3.22, 6.24, 9)
C5(1.86, 4.24, 7.66) (1.57, 3.79, 6.97) (1.57, 3.79, 6.97)
C6(4.43, 7.66, 9) (4.43, 7.66, 9) (4.43, 7.66, 9)
C7(2.71, 5.36, 7.65) (2.71, 5.36, 8.28) (3.11, 5.89, 9)
C8(2.25, 4.62, 8.33) (2.25, 4.62, 8.33) (1.68, 4.11, 7.58)
C9(2.37, 5.43, 9) (1.30, 3.98, 7.2) (0.79, 1.76, 4.68)
C10 (3.92, 7.08, 9) (4.43, 7.66, 9) (3.48, 6.41, 9)
C11 (4.9, 8.28, 9) (4.34, 7.65, 9) (3.48, 6.41, 9)
Table 12 Summary of relations
between alternatives AiA1A2A3
A1 {2, 7, 10} {2, 3, 7}
JA2{3, 4, 5, 9, 11} {2, 3, 4, 7}
A3{1, 5, 8, 9, 10, 11} {1, 8, 9, 10, 11}
A1 {1, 6, 8} {4, 6}
J=A2{1, 6, 8} {5, 6}
A3{4, 6} {5, 6}
A1 {3, 4, 5, 9, 11} {1, 5, 8, 9, 10, 11}
J+A2{2, 7, 10} {1, 8, 9, 10, 11}
A3{2, 3, 7} {2, 3, 4, 7}
In the following Sect. 4.3, the Sensitivity analysis will be
applied to verify the stability of outranking obtained by
FMAADM and two existing methods.
4.3 Sensitivity analysis
In order to verify the stability of the outranking of alter-
natives (A1,A2and A3) shown above in Sect. 4.1,the
Sensitivity analysis based on the simulation of scenarios was
applied by using: (1) the hybrid FMCDM method based on
fuzzy Entropy Weight (EW), fuzzy AHP and fuzzy TOP-
SIS (He et al.’s method He et al. 2017), (2) the framework of
hybrid spatial-fuzzy multi-criteria decision-making based on
weighted Geographical Information System data and fuzzy
TOPSIS (Sopha et al.’s method Sopha et al. 2018) and (3) the
FMAADM method. The objective is to test the stability of
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Decision-making from multiple uncertain experts: case of distribution center location selection
Table 13 Summary of
converted relations between
alternatives in numerical values
AiA1A2A3
P
ik A1 {18.32, 24.32, 27} {18.32, 24.32, 27}
A2{28.98, 38.98, 43.66} {24.65, 32.65, 36}
A3{33.98, 45.98, 51.99} {30.32, 40.32, 44.33}
P=
ik A1 {17.66, 23.66, 26.33} {12.66, 16.66, 18}
A2{17.66, 23.66, 26.33} {9.99, 13.99, 16.66}
A3{12.66, 16.66, 18} {9.99, 13.99, 16.66}
P+
ik A1 {28.98, 38.98, 43.66} {33.98, 45.98, 51.99}
A2{18.32, 24.32, 27} {30.32, 40.32, 44.33}
A3{18.32, 24.32, 27} {24.65, 32.65, 36}
Table 14 Concordance and
discordance index AiA1A2A3
CI
ik A1 (0.71, 0.72, 0.72) (0.71, 0.72, 0.72)
A2(0.55, 0.55, 0.55) (0.62, 0.62, 0.62)
A3(0.47, 0.47, 0.47) (0.53, 0.53, 0.54)
DIik A1–(0.47, 0, 0.44) (0.38, 0.14, 0.51)
A2(0.38, 0.14, 0.59) (0.38, 0.14, 0.59)
A3(0.47, 0.07, 0.51) (0.28, 0.22, 0.51)
Table 15 Comparative outranking
Literature Main method l Outranking Selected location
He et al. (2017) Hybrid FMCDM method based on fuzzy Entropy
Weight (EW), fuzzy AHP and fuzzy TOPSIS
A1SA2SA3A1
Sopha et al. (2018) Framework of hybrid spatial-fuzzy multi-criteria
decision-making based on weighted Geographical
Information System data and fuzzy TOPSIS
A1SA3SA2A1
This paper FMAADM method A1SA2and A1SA3A1
the obtained results vis-a-vis variations’ weight of the eleven
criteria used to evaluate the different potential alternatives,
since the criteria weight significantly affects the rank (Agrebi
et al. 2017; Lee and Chang 2018).
To this end, 18 experiments by each method were con-
ducted. Table 16 summarizes the obtained location in each
experiment. It can be seen, in the 5 first experiments, that the
weights of all criteria are set equal to (1, 1, 3), (1, 3, 5), (3,
5, 7), (5, 7, 9) and (7, 9, 9). In experiment 6 to 16, the weight
of one criterion is set as the highest weight (7, 9, 9) and the
remaining criteria are set to the lowest weight (1, 1, 3). In
experiment 17 and 18, the weight of the cost category criteria
(C4) and (C5) is the lowest weight equal to (1, 1, 3) and the
weights of the benefit category criteria (C1C3and C6C11)
are set as the highest weight equal to (7, 9, 9).
Among the 18 experiments by each method:
By using He et al.’ method, as shown in Fig. 6,for9
experiments (1–5, 11, 14, 15 and 17), the best location
is A1. The location A2has appeared as the winner for
3 experiments (6, 8 and 9). As for the location A3has
emerged as the winner for 6 experiments (7, 10, 12, 13,
16 and 18).
By using Sopha et al.’ method, as shown in Fig. 7,in9
experiments (1–5, 8, 12, 15 and 18), the selected location
is A1. The alternative A2has emerged as the best location
in 2 experiments (7 and 14) and in the rest of experiments
(6, 9, 10, 11, 13, 16 and 17) the best location is A3.
By using the FMAADM method, as shown in Fig. 8,for
13 experiments (1–7, 9, 10, 13, 14 and 16–18), the alter-
native A1has emerged as the best location. Contrariwise,
in experiment 11, the alternative A3has appeared as the
winner. In the rest of experiments (8, 12, and 15), both
the alternatives A1and A3have emerged as the best loca-
tions. Therefore, we can say that the location decision is
relatively insensitive to cost criteria weight. It can be seen
where the weight of cost criteria C4and C5is set as the
highest (experiments 9 and 10) or lowest (experiments
17 and 18), then the best solution is always the alterna-
tive A1. In the opposite case, when the weight of benefit
123
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M.Agrebi,M.Abed
Table 16 Experimental results under different experiments
NDescription Selected location by three methods
He et al. (2017) Sopha et al. (2018) FMAADM
1 All criteria weights = (1, 1, 3) A1A1A1
2 All criteria weights = (1, 3, 5) A1A1A1
3 All criteria weights = (3, 5, 7) A1A1A1
4 All criteria weights = (5, 7, 9) A1A1A1
5 All criteria weights = (7, 9, 9) A1A1A1
6 Weight of criteria 1 = (7, 9, 9) A2A3A1
Weight of remaining criteria = (1, 1, 3)
7 Weight of criteria 2 = (7, 9, 9) A3A2A1
Weight of remaining criteria = (1, 1, 3)
8 Weight of criteria 3 = (7, 9 , 9) A2A1A1and A3
Weight of remaining criteria = (1, 1, 3)
9 Weight of criteria 4 = (7, 9, 9) A2A3A1
Weight of remaining criteria =(1, 1, 3)
10 Weight of criteria 5 = (7, 9, 9) A3A3A1
Weight of remaining criteria =(1, 1, 3)
11 Weight of criteria 6= (7, 9, 9) A1A3A3
Weight of remaining criteria = (1, 1, 3)
12 Weight of criteria 7 = (7, 9, 9) A3A1A1and A3
Weight of remaining criteria = (1, 1, 3)
13 Weight of criteria 8 = (7, 9, 9) A3A3A1
Weight of remaining criteria = (1, 1, 3)
14 Weight of criteria 9 =(7, 9, 9) A1A2A1
Weight of remaining criteria = (1, 1, 3)
15 Weight of criteria 10 = (7, 9, 9) A1A1A1and A3
Weight of remaining criteria= (1, 1, 3)
16 Weight of criteria 11 = (7, 9, 9) A3A3A1
Weight of remaining criteria = (1, 1, 3)
17 Weight of criteria 4 = (1, 1, 3) A1A3A1
Weight of remaining criteria = (7, 9, 9)
18 Weight of criteria 5 = (1, 1, 3) A3A1A1
Weight of remaining criteria = (7, 9, 9)
Fig. 6 Sensitivity analysis
results by using He et al.’s
method
123
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Decision-making from multiple uncertain experts: case of distribution center location selection
Fig. 7 Sensitivity analysis
results by using Sopha et al.’s
method
Fig. 8 Sensitivity analysis
results by using FMAADM
method
criteria C1C3and C6C11 is set as the highest (experi-
ments 6, 7, 9, 10, 13, 14 and 16), then the best solution is
changed from the alternative A1to A3(experiment 11)
and to both the alternatives A1and A3in experiments 8,
12, and 15.
In short, compared with He et al.’ method and Sopha et al.’
method, the FMAADM method, as proposed in this paper,
is rather stable and robust under an uncertain environment.
Thus, it can be recommended to decision makers for the pur-
pose of distribution center location selection.
5 Conclusion and future work
The aim of this paper is to help decision-makers group to
select, under uncertainty, the best location of distribution cen-
ter among a set of potential locations. The expected decision
must respect not only a set of criteria which are often con-
tradictory but also the decision-makers preferences.
For this purpose, the FMAADM method is proposed.
This fuzzy method possesses three great strengths: first,
the integration of the decision-makers group preferences
into the decision-making process, knowing that the human
preferences are often ambiguous and uncertain, second, the
consideration of the informations related to the alternatives
and the criteria weights which are inaccurate, uncertain or
incomplete, third, the verification of the obtained solution by
both tests of concordance and non-discordance.
In order to validate the FMAADM method, the S-SSD sys-
tem is developed. Then, we conducted a case study whose
objective is to select the best location among three potential
locations under uncertainty. These three alternatives are eval-
uated by three decision-makers according to eleven criteria.
The obtained results by our FMAADM method were com-
pared by the results obtained by two other recent methods.
This comparison proves that the FMAADM method meets
the desired objective and thus retained for the selection of
the best location of distribution center under an uncertain
context of the multi-attribute and the multi-actor. Moreover,
the sensitivity analysis was conducted in order to verify the
stability of our method. 54 experiments were provided. The
comparative analysis demonstrates not only the stability and
the robustness of the FMAADM method, but also its superi-
ority over the two other methods.
Based on the obtained results of FMAADM method and
its validation, our study advances the knowledge in the issue
of multicriteria decision making problem. This through the
treatment of the problem from multiple uncertain experts
while ensuring that a sufficient majority of criteria are in favor
of the outranking, and, on the other hand, to make it possible
to refuse the outclass of an alternative over an other when
there is too much opposition on at least one criterion. Besides,
based in the kernel concept, it is possible to restrict the field
of study to focus only on the best alternatives. Contrariwise
123
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M.Agrebi,M.Abed
of a number of methods, if all alternatives are bad, they offer
the best alternative among the bad ones.
Despite the case studies carried out in this paper, it would
be relevant to test our method on other real business issues
in order to validate its generalization, and if possible in
various fields: logistics, biomedical, tourism, etc. Further-
more, future researches may focus on the exploitation of
decision-makers preferences through similarity analysis to
build virtual experts communities. Moreover, we expect to
propose adaptations in the Big Data context by proposing
an approach to build ontologies from a large amount of data
and extend experiments to support the contribution of the
proposal. Besides, we count improve our system to be able
to extract the important criteria according to the studied case,
and this, automatically. Finally, we expect integrate a results
visualization module with an explanation sub-system.
Compliance with ethical standards
Conflict of interest The authors declare no potential conflict of interests
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... The methodology was then tested on a case study with 7 criteria and 9 alternatives. Agrebi and Abed [1] considered the DC location selection problem under an uncertain environment and proposed a methodology based on a fuzzy multi-attribute and multi-actor decision-making (FMAADM) method for solving the aforementioned problem. Kieu et al. [2] in their paper proposed a methodology based on a Spherical Fuzzy Analytic Hierarchy Process (SF-AHP) and Combined Compromise Solution (CoCoSo) methods for the selection of DC location in the agricultural supply chain. ...
... In Table 1 a detailed explanation of each criterion as well as references are provided. The price of the plot for the distribution center (DC) [4,7] Transportation time (C2) The time that elapses from the moment of receiving the order up to delivery to the customer [2,7] Distance to customers (C3) Distance between the plot and the customers [4,7] Proximity to suppliers (C4) Distance between the plot and the suppliers [4,7] Availability of labor resources (C5) Availability of labor resources in the proximity of the plot [1,4] Accessibility (C6) Plot's existing accessibility [3,7] Proximity to highway (C7) ...
... The plot's proximity to the highway [4,7] Expansion possibility (C8) Possibility to expend DC on the plot [1,7] In the first phase of the proposed methodology, a BWM was applied in order to determine the criteria weights that were then used in the second phase (MABAC method). When implementing BWM, and in accordance with the second step, the best and the worst criteria should be defined. ...
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... Instead, specific criteria are utilized for the distribution of agricultural materials and raw products to farmers. Connection between the location and the potential for using multimodal transport [17]; [34][35][36][37] Benefit ...
... The location's proximity to customers and suppliers [1]; [17]; [34]; [40] Cost C4 Land costs The site's monetary value per square metre [1]; [19]; [41] Cost C5 ...
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Thesis
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Le travail de recherche présenté dans cette thèse s’inscrit dans la continuité des travaux de l’aide à la décision multi-critère de groupe (décideurs), particulièrement dans le champ de sélection de la localisation des centres de distribution. Dans un environnement certain, si la décision de sélection de la localisation des centres de distribution a donné lieu à plusieurs travaux de recherche, elle n’a jamais été l’objet, à notre connaissance, d’une décision prise par plusieurs décideurs. À cet égard, le premier objectif de cette thèse est de proposer une méthode d’aide à la décision multi-attribut et multi-acteur (MAADM) pour résoudre le problème posé. Pour se faire, nous avons adapté et étendu la méthode ELECTRE I. Dans un environnement incertain, au vu de l’incertitude inhérente et l’imprécision du processus décisionnel humain ainsi que les comportements futurs du marché et des entreprises, le deuxième objectif de cette thèse est de développer une méthode floue d’aide à la décision multi-attribut et multi-acteur (FMAADM) pour traiter le problème en question. Pour cela, nous avons couplé la méthode MAADM avec la théorie des ensembles flous. Pour la validation des deux contributions, nous avons conçu un système d’aide à la décision (S-DSS) pour implémenter les algorithmes de la méthode MAADM et la méthode FMAADM. Sur la base du S-DSS, deux études expérimentales ont été menées. Nous avons, aussi, appliqué une analyse de sensibilité pour vérifier la sensibilité de la solution retenue vis-à-vis aux variations de poids des critères d’évaluation. Les résultats obtenus prouvent que les deux méthodes proposées répondent à l’objectif recherché et ainsi retenues pour la sélection de la meilleure localisation dans un contexte certain/incertain de multi-attribut et multi-acteur.
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