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

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.

1 Introduction

In an urban context, the city functioning means an important rate of freight transportation. This functioning relies on various physical facilities, as shown in Fig. 1. These facilities are considered as real drivers of logistics systems. Therefore, 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 facilities 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 commitment 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 logistics 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, knowing 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 multicriteria decision-making problem (FMCDM). The expected decision must respect decision-makers group preferences and existing evaluation criteria (e.g., the investment cost, the possibility of expansion, the availability of acquisition hardware, the human resources, the proximity to suppliers, etc.).

Fig. 1

Generic supply chain network (Melo et al. 2009)

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 limitations (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 maximizing 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 combines 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 environment. 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 imprecision 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 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 (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 provides 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. 4 presents the experimental and operational validation of our proposed method. Finally, Sect. 5 concludes the paper and presents the future research directions.

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 center under not only a certain environment but also an uncertain environment. This depends on the nature of the used parameters 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 environment, 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 methods, for dealing with the location selection problem of distribution center under uncertainty, can be mainly classified, as indicated in Table 1, into:

• Metaheuristics for the fuzzy multi-objective decision-making (FMMODM) methods,

• Fuzzy multi-criteria decision-making (FMCDM) methods categorized into Fuzzy multi-objective decision-making (FMODM) methods and Fuzzy multi-attribute decision-making (FMADM) methods.

Table 1 Comparison of some characteristics between existing methods

The common point between the FMCDM and the FMMODM methods is the development and the application of fuzzy theory. Table 2 presents the most proposed methods in this issue.

Table 2 Existing methods

2.2.1 Existing metaheuristics for the fuzzy multi-objective decision-making methods

Wang et al. (2005) developed a method combining quantitative 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 combining 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 transport 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 earnings. Zhou et al. (2015) proposed a solution model termed as rough multi-objective synthesis effect model. This constitutes 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 distribution centers determining their suitable scales, as well as adjusting distribution centers so as to adapt to the dynamically 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 distribution 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 objective problem (Chen et al. 2018; Manav et al. 2013). In fact, optimizing this single objective problem yields a single solution. But, the decision-makers need diverse options in the real condition (Wolf 2011). However, there are some classical 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 imprecision of decision data by using alternative scores and criteria weights which are assigned as linguistic variables represented by fuzzy triangular numbers. Chan et al. (2007) developed 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 relationship 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 information 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 engineering fuzzy set theory. Hu et al. (2009) proposed a fuzzy TOPSIS method integrating fuzzy set theory, the factor rating 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 planning for urban distribution centers under uncertainty. Their proposed approach involves identification of potential locations, 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) proposed 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 characterized by triangular fuzzy numbers are used to denote the evaluation values of all alternatives versus various criteria. Finally, the aggregation fuzzy assessments of different alternatives 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 algorithm 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 agricultural 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 Geographical 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 sufficient majority of criteria, represented by their weight, are in favor of the assertion Ai outclass Ak, and, to make it possible 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 maximizing quality of service or minimizing congestion but not the both.

2.3 Discussion

In order to satisfy more distribution center location decision 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 (Milosavljević et al. 2018; Collette and Siarry 2011; Roy 1968) 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.

Table 3 Advantages/disadvantages of a number of MADM methods (Ayadi 2010; Agrebi et al. 2017; Agrebi 2018)

2.3.1 ELECTRE method

ELECTRE method and its different derivatives (ELECTRE I, ELECTRE II, ELECTRE III, ELECTRE IV, and ELECTRE TRI) are considered to be the most preferred methods among several outranking methods like PROMETHEE and its derivatives (PROMETHEE I and II), ORESTE, QUALIFLES, MELCHIOR, MAPPACC, PRAGMA, and TACTIC (Zandi and Roghanian 2013; Agrebi et al. 2017) as shown in Table 3. Furthermore, ELECTRE method is considered as one of the best methods which take into account both desirable 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 distribution 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 respecting 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 limitation.

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 uncertainty and imprecision which are inherent to human judgment in decision making processes through the use of linguistic terms and degrees of membership (Alias et al. 2019; Garg and Rani 2019; Tayal et al. 2014; Zouggari and Benyoucef 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 (Simić et al. 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; Ertuğrul 2011; Hwang and Thill 2005; Kahraman et al. 2006; Kaya and Çinar 2006; Li et al. 2011; Rebaa 2003; Takači et al. 2012; Trivedi and Singh 2017; Zhou and Liu 2007). These methods make it possible to treat uncertainty based on the idea of order. In addition, they are based on a methodology of representation and use of vague and uncertain 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 introducing a membership function taking values between zero and one.

In short, fuzzy set theory offers a mathematically precise 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 (Simić et al. 2017) and improve the decision-making process by making it more realistic.

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 presented as shown in Fig. 2.

Fig. 2

Flowchart of FMAADM method

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 interests that represents each decision-maker. The goal is to reflect the general case of the request and to treat the selection problem in a broad perspective by including different points of view (Turkoglu and Genevois 2017).

Step 2. Identification of potential locations: This step consists in identifying a set of potential locations (alternatives) of distribution centers ($A_i = 1,\ldots , m$) based on sustainable freight regulations, decision-makers’ preferences and knowledge conditions of freight transportation. The potential locations are those that cater to the interest of all city stakeholders, 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 n criteria ($C_j$, where $j = 1, \ldots , n$) such as connectivity 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 n criteria by K decision-makers. The goal is to establish the matrix criteria’s importance (W) based on Eq. (1) expressed as follows:

$$\begin{aligned} {W}= & {} \begin{bmatrix} {w}_{1} &{} {w}_{2} &{}\ldots &{} {w}_{n}\\ \end{bmatrix}.\nonumber \\ {W}= & {} \frac{1}{K} \begin{bmatrix} {w}_{j}^1 + {w}_{j}^2 + \cdots + {w}_{j}^K\\ \end{bmatrix}, \end{aligned}$$

(1)

where $w_j$ ($j = 1, 2, 3, \ldots , n$) is the weight of criterion ($C_j$).

Step 5. Evaluations of alternatives and determination of the fuzzy decision matrix: This step consists, first, in evaluating m alternatives by K decision-makers with respect to the criteria ($C_j$, where $j = 1, \ldots , n$) and then in constructing the fuzzy decision matrix (D) based on Eq. (2) and expressed as follows:

$$\begin{aligned} {D}= \begin{bmatrix} {x}_{11} &{} {x}_{12} &{}\ldots &{} {x}_{1n}\\ {x}_{21} &{} {x}_{22} &{}\ldots &{} {x}_{2n}\\ . &{} . &{}&{} .\\ . &{} . &{}&{} .\\ . &{} . &{}\ldots &{} .\\ {x}_{m1} &{} {x}_{m2} &{}\ldots &{} {x}_{mn}\\ \end{bmatrix}, \end{aligned}$$

where ${x}_{ij}$, $\forall$ i,j is the rating of alternative $A_i$ (i=1,2,...,m) with respect to criterion $C_j$.

$$\begin{aligned} {x}_{ij}= \frac{1}{K} \begin{bmatrix} {x}_{ij}^1 + {x}_{ij}^2 + \cdots + {x}_{ij}^K \end{bmatrix}. \end{aligned}$$

(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 transformation. 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.

(3)

where $r_{ij}$ is the normalized value of $x_{ij}$, B and C are 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:

$$\begin{aligned} {P}_{i}={r}_{ij} . {w}_{j}, i=1, 2, \ldots , m, \end{aligned}$$

(4)

where ${P}_{i}$ is the final fuzzy evaluation value of alternative $A_i$.

Step 7. Establishment relations between alternatives with respect to each criterion: By respecting to each criterion, the pairwise comparisons of the alternatives $A_i$ and $A_k$ (where k $\in$ [$i, \ldots ,m$] and $k \ne i$) are established as follows:

$$\begin{aligned} {J^+}_{(A_i, A_k)} = \{j | C_j(A_i) > C_j( A_k)\}, \end{aligned}$$

(5)

where ${J^+}_{(A_i, A_k)}$ the set of criteria for which the alternative $A_i$ is preferred over $A_k$.

$$\begin{aligned} {J^=}_{(A_i, A_k)} = \{j | C_j(A_i) = C_j( A_k)\}, \end{aligned}$$

(6)

where ${J^=}_{(A_i, A_k)}$ the set of criteria for which the alternative $A_i$ is equal in preference to alternative $A_k$.

$$\begin{aligned} {J^-}_{(A_i, A_k)} = \{j | C_j(A_i) < C_j( A_k)\}, \end{aligned}$$

(7)

where ${J^-}_{(A_i, A_k)}$ the set of criteria for which the alternative $A_k$ is preferred over $A_i$.

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:

$$\begin{aligned} {P^+}_{(A_i, A_k)} = \sum _{j} w_j~\forall j \in {J^+}_{(A_i, A_k)}. \end{aligned}$$

(8)

$$\begin{aligned} {P^=}_{(A_i, A_k)}= \sum _{j} w_j~\forall j \in {J^=}_{(A_i, A_k)}. \end{aligned}$$

(9)

$$\begin{aligned} {P^-}_{(A_i, A_k)} = \sum _{j} w_j~\forall j \in {J^-}_{(A_i, A_k)}. \end{aligned}$$

(10)

Step 9. Merger the numerical values: This step consists in merging the numerical values by calculating the Concordance Index (CI) and the Disconcordance Index (DI).

• Concordance index (CI): This index expresses how much the hypothesis ($A_i$ outclasses $A_k$) is consistent with the reality represented by the evaluations of alternatives.

$$\begin{aligned} {CI}_{ik}= \frac{P^+(A_i, A_k)+P^=(A_i, A_k)}{P(A_i, A_k)}, \end{aligned}$$

(11)

where ${P}_{(A_i, A_k)}$ = ${P^+}_{(A_i, A_k)}$ + ${P^=}_{(A_i, A_k)}$ + ${P^-}_{(A_i, A_k)}$.

• Set of concordance:

$$\begin{aligned} J(A_i, A_k) = J^+(A_i, A_k) \cup J^=(A_i, A_k). \end{aligned}$$

(12)

• Disconcordance index (DI):

$$\begin{aligned} DI_{ik}= {\left\{ \begin{array}{ll} (0 , 0 , 0) \quad if \, J^-(A_i, A_k)=\emptyset \\ \frac{(1 , 1 , 1)}{\partial _j}\times \max (C_j(A_k)-C_j(A_i)) \quad \text {where}\, j \in J^-(A_i, A_k), \,\text {otherwise}\, \end{array}\right. }\nonumber \\ \end{aligned}$$

(13)

where $\partial _j$ is the amplitude of the scale associated with criterion j.

Step 10. Filtration of alternatives: This step allows to extract from the set of the potential alternatives $A_i$ (where $i = 1,\ldots , m$) 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.

$$\begin{aligned} \left. \begin{aligned} CI_{ik} \ge ct \\ DI_{ik} \le dt \\ \end{aligned}\right\} \Leftrightarrow A_i \, S \, A_k, \end{aligned}$$

(14)

where:

• ct is the threshold of concordance beyond which the hypothesis $A_i \, S \, A_k$ is considered as valid.

• dt is the threshold of discordance below which the hypothesis $A_i \, S \, A_k$ is no longer valid.

We remind that S is the outranking relation ($A_i \,S \, A_k$ means that $A_i$ is at least as good as $A_k$).

3.2 FMAADM method architecture

The architecture of FMAADM method consists of three levels notably users level, user-application interface level and application level. Fig. 3 gives a general view of the interactions between levels.

Fig. 3

FMAADM method architecture

• 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 preferences, 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 communication 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 variables represented by triangular fuzzy number. The aim is to find the best alternative among the set of potential alternatives.

  Fuzzy module : The role of this module is to accommodate the uncertain parameters of the selection process. The decision-maker communicates the linguistic variables 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. Netbeans¹ has been selected as the appropriate development environment. Also, the system uses XML² format for information transmission and storage (saving performed studies or projects). In addition, we made use of some APIs such as Apache POI,³ JDBC⁴ in order to manage data, which may be extracted from excel files. Users can generate data automatically based on a random generator or existing data source and manually. We note that the random generator is basically used for testing purpose. Fig. 4 presents the architecture of our S-SSD.

Fig. 4

Architecture of the proposed decision support system

Table 4 Evaluation criteria for location selection

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 concept represents and how this is useful for the decision-making’s problem.

• Logical validation: verify whether the concepts are logically consistent and meaningful.

• Experimental validation: test the method using experimental data in order to show that the method provides the expected results and possibly check formal requirements 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 confronted 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 $D_1$, $D_2$ and $D_3$. The aim of which is to select a best location among three alternatives $A_1$, $A_2$ and $A_3$. The selection decision is made based on eleven main evaluation criteria $C_1, \ldots , C_{11}$. As shown in Table 4, $C_4$ and $C_5$ are 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.

Fig. 5

Hierarchical structure of the distribution center’s location selection

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; He et al. 2017) presented in Tables 5 and 6 , the criteria and the alternatives are evaluated by the decision-makers. Table 7 presents the importance of criteria and the weight of each criterion calculated using Eq. (1). Table 8 summarizes the evaluations of the alternatives. Then, the fuzzy decision matrix (D) is constructed using Eq. (2) as shown in Table 9. The normalized fuzzy decision matrix (R) is determined using Eq. (3) and presented in Table 10.

Afterward, considering the criteria, the final fuzzy evaluation 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 J as 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 $C_{ik}$ and the coefficients of disconcordance $D_{ik}$ as shown in Table 14.

Table 5 Linguistic variables for the importance weight of criteria

Table 6 Linguistic variables for the ratings

Table 7 Importance and weight of criteria

Finally, the test of concordance and the test of non-disconcordance are done using Eq. (14) in order to filter the alternatives. For that, the threshold ct is fixed to (0.5, 0.7, 0.9). This test is satisfied if $CI_{ik} \ge$ (0.5, 0.7, 0.9). The threshold dt is fixed to (0.3, 0.5, 0.7). Then, the test is satisfied if $DI_{ik} \le$ (0.3, 0.5, 0.7).

The $CI_{ik}$ which satisfied the test of concordance are $CI_{21}$, $CI_{31}$ and $CI_{32}$. The $DI_{ik}$ which satisfied the test of non-disconcordance are $DI_{12}$, $DI_{13}$ and $DI_{23}$. Therefore, based on both tests, we found that : the alternative $A_{1}$ outclasses the alternatives $A_{2}$ and $A_{3}$. Then, we can infer that the alternative $A_1$ is the best alternative.

Table 8 Evaluations of alternatives

Table 9 Fuzzy decision matrix

To validate experimentally our proposed method, an application of FMAADM by using real data was presented. The obtained results show that FMAADM method when confronted with the decision process acts as expected within distribution center location selection process. In the following Sect. 4.2, a comparative analysis of the obtained results by our method and two existing methods will be detailed.

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 environment 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 Information System data and fuzzy TOPSIS, is invented by Sopha et al. (2018). Table 15 recapitulates the obtained outranking and selected location by applying the different methods.

Table 10 Normalized fuzzy decision matrix

Table 11 Final fuzzy evaluation value of alternatives

Table 12 Summary of relations between alternatives

Table 13 Summary of converted relations between alternatives in numerical values

Table 14 Concordance and discordance index

Table 15 Comparative outranking

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. However, 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 impossible 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 characteristics:

• 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 $A_i$ S $A_k$, and, on the other hand, to make it possible to refuse the outclass of an alternative 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.

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 alternatives ($A_1$, $A_2$ and $A_3$) 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 TOPSIS (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 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 conducted. 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 ($C_4$) and ($C_5$) is the lowest weight equal to (1, 1, 3) and the weights of the benefit category criteria ($C_1$–$C_3$ and $C_6$–$C_{11}$) are set as the highest weight equal to (7, 9, 9).

Table 16 Experimental results under different experiments

Fig. 6

Sensitivity analysis results by using He et al.’s method

Fig. 7

Sensitivity analysis results by using Sopha et al.’s method

Fig. 8

Sensitivity analysis results by using FMAADM method

Among the 18 experiments by each method:

• By using He et al.’ method, as shown in Fig. 6, for 9 experiments (1–5, 11, 14, 15 and 17), the best location is $A_1$. The location $A_2$ has appeared as the winner for 3 experiments (6, 8 and 9). As for the location $A_3$ has 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, in 9 experiments (1–5, 8, 12, 15 and 18), the selected location is $A_1$. The alternative $A_2$ has 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 $A_3$.

• By using the FMAADM method, as shown in Fig. 8, for 13 experiments (1–7, 9, 10, 13, 14 and 16–18), the alternative $A_1$ has emerged as the best location. Contrariwise, in experiment 11, the alternative $A_3$ has appeared as the winner. In the rest of experiments (8, 12, and 15), both the alternatives $A_1$ and $A_3$ have emerged as the best locations. 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 $C_4$ and $C_5$ is set as the highest (experiments 9 and 10) or lowest (experiments 17 and 18), then the best solution is always the alternative $A_1$. In the opposite case, when the weight of benefit criteria $C_1$–$C_3$ and $C_6$–$C_{11}$ is set as the highest (experiments 6, 7, 9, 10, 13, 14 and 16), then the best solution is changed from the alternative $A_1$ to $A_3$ (experiment 11) and to both the alternatives $A_1$ and $A_3$ in 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 purpose 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 center among a set of potential locations. The expected decision must respect not only a set of criteria which are often contradictory 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 system 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 evaluated by three decision-makers according to eleven criteria. The obtained results by our FMAADM method were compared 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 superiority 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 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. Furthermore, 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.