A novel model for multi-criteria assessment based on BWM and possibilistic chance-constrained programming

Highlights

• A new approach was developed based on BWM and PCCP.

• Possibility, necessity and credibility concepts were used to obtain weights of criteria.

• Three perspectives (optimistic, pessimistic and intermediate) were used.

• The proposed models were validated by analyzing two numerical examples.

• Trapezoidal fuzzy numbers were used to develop the proposed models.

Abstract

The best-worst method (BWM) is one of the most important methods for determining the weights of criteria or options in multi-criteria decision-making (MCDM) and has attracted the attention of many researchers due to its advantages such as fewer numbers of comparisons and higher consistency rate. Given that real-world decision making is often associated with uncertainty, in this study several fuzzy linear programming models have been developed using trapezoidal fuzzy numbers for BWM that can calculate the optimal weights of criteria. The proposed models are based on three measures of possibility, necessity, and credibility, which are parts of possibilistic chance-constrained programming (PCCP). Development of BWM based on possibilistic distribution allows the decision-maker (DM) to take into account uncertainties in the calculation of weights as well as include his optimistic, pessimistic, and intermediate attitudes in determining the weight of decision criteria. The possibility approach reflects the DM's optimistic view of the issue. The necessity approach is used in situations where the DM prefers a pessimistic view, and the credibility approach indicates that DM has an intermediate view between optimistic and pessimistic views, or in other words, considers an intermediate approach between possibility and necessity approaches. Finally, the feasibility and effectiveness of the proposed approaches were tested using two numerical examples and the sensitivity of the results was analyzed for different values of uncertainty (alpha parameter). Also, by analyzing the coefficient of variation, it was found that the results of the proposed models had very little dispersion for different values of uncertainty, and this confirmed the validity of the results. Examining the results of the proposed models revealed that the possibility approach provides more robust results than other approaches when the levels of confidence of the decision-makers are changed.

Introduction

Often, decision-making scenarios involve choosing an alternative from among the available alternatives. Existence of various criteria and large number of alternatives makes it necessary to solve the problem with multi-criteria decision-making (MCDM) methods (Chen, 2000). MCDM is an important branch of decision theory. Based on the solution space, MCDM problems are divided into two categories: continuous and discrete. To solve continuous problems, multi-objective decision-making (MODM) methods are used, and on the other hand, to solve discrete problems, multi-attribute decision-making (MADM) methods are used (Zavadskas et al., 2014). The main focus of this study is on the latter category. According to the recent literature, MCDM is used to describe discrete MCDM (Rezaei, 2015). The MCDM approach helps decision-makers (DMs) to choose the right option from among other alternatives using mathematical relations and computational methods (Sharaf, 2018). Due to its ability to evaluate different alternatives using certain criteria, MCDM is a very useful tool for analyzing complex problems that occur on a daily basis (Stević et al., 2020). In recent years, various MCDM methods have been developed and used to solve many complex problems in the real world (Ilieva et al., 2018). Here are some of the most commonly used MCDM methods: Analytic hierarchy process (AHP) (Saaty, 1990) ،analytic network process (Saaty, 1996); simple additive weighting (SAW) (MacCrimmon, 1968), technique for order preference by similarity to ideal solution (TOPSIS) (Tzeng and Huang, 2011), Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) (Opricovic, 1998); elimination and choice expressing reality (ELECTRE) (Roy, 1971), the preference ranking organization method for enrichment evaluation (PROMETEE) (Mareschal et al., 1984), data envelopment analysis (DEA) (Charnes et al., 1978), decision making trial and evaluation laboratory (DEMATEL) (Fontela and Gabus, 1976), Evaluation based on Distance from Average Solution (EDAS) (Keshavarz Ghorabaee et al., 2015), COmbinative Distance-based ASsessment (CODAS) (Keshavarz Ghorabaee et al., 2016) and Measurement Alternatives and Ranking according to COmpromise Solution (MARCOS) (Stević et al., 2020). Preferences of individuals or DMs are used as the primary data for MCDM and are analyzed by various methods.

The best-worst method (BWM) is one of the newest MCDM methods, which was proposed by Rezaei (2015). In this method, the decision-maker (DM) selects the best and worst decision criteria from among the available criteria; then, using paired comparisons, determines the priority of the best criterion over each of other criteria as well as the priority of each criterion over the worst criterion. Then, a programming model is formed and the optimal weights of the criteria are obtained by solving the model. This method requires fewer pair comparisons than previous methods such as AHP and also offers more consistent comparisons.

Recently, BWM has received the attention of many researchers and numerous articles have been published on this topic, including in the area of supplier selection (Gupta and Barua, 2017, Rezaei et al., 2016, Badi and Ballem, 2018, Cheraghalipour et al., 2018), performance evaluation (You et al., 2017, Gupta, 2018), site selection (Maghsoodi et al., 2019), performance measurement (Salimi and Rezaei, 2016, Gupta et al., 2017), etc. A number of other researchers have extended BWM and added new capabilities to its model. In the following we will review some of these studies. A summary of the articles that have developed BWM can also be seen in Table 1.

Rezaei (2016) extended BWM by adding a linear programming model and some details to BWM. A new BWM-based group decision making model was introduced for uncertain conditions. They developed the intuitionistic fuzzy multiplicative best-worst method (IFMBWM) using intuitionistic fuzzy multiplicative preference relations (IFMPRs) for multi-criteria group decision making, and proposed several max–min programming models based on it. They also proposed new relationships to calculate consistency rates in order to observe the degree of consistency of comparisons and the reliability of the results. Finally, they used the proposed approach to solve health management problems (Mou et al., 2016). Also, in another study, Guo and Zhao (2017) investigated BWM in fuzzy environment. They modeled BWM under uncertainty using graded mean integration representation (GMIR) and nonlinear programming (NLP) and analyzed the proposed method using case studies. A new approach was introduced based on interval-valued fuzzy-rough numbers (IVFRN) to deal with uncertainty. In this approach, original multi-criteria model was extended using the IVFRN approach. The model extended the traditional stages of BWM and MABAC (Multi-Attributive Border Approximation Area Comparison). This model was tested and validated through studying the optimal selection of firefighting helicopters. The results showed that the proposed method enables better criteria evaluation compared to the traditional fuzzy and rough approaches (Pamučar et al., 2018).

Aboutorab et al. (2018) introduced the z-number BWM version. They applied their proposed method in uncertain environment and evaluated the model using a case study. Tabatabaei et al. (2019) proposed BWM as an integrated model for group decision making. The proposed model helped the decision-making team when it was necessary to simultaneously incorporate the manager's opinion and expert opinions.

The Bayesian BWM was introduced in order to calculate weights of decision-making criteria for solving group decision-making problems. They applied the Bayesian hierarchical model for this purpose. A new ranking method was also used and the proposed model was analyzed using a numerical example (Mohammadi and Rezaei, 2019).

BWM was applied using two mathematical models for group decision making and the main parameters of the models were analyzed. The results showed that the proposed method was suitable for group decision making (Safarzadeh et al., 2018). In another study, integrated model based on BWM and Weighted Aggregates Sum Product Assessment (WASPAS), and MABAC with interval rough numbers (IRN) were used to evaluate third-party logistics (3PL) providers (Pamucar et al., 2019).

Tabatabaei et al. (2019) proposed a hierarchical BWM for decision making in situations in which calculation of the weights of decision criteria and sub-criteria is required, then evaluated the benefits and efficiency of the model using two numerical examples.

BWM was used to evaluate the performance of hospitals using hesitant fuzzy linguistic information. Also, a method was proposed for correcting inconsistencies between paired comparisons and finally a comparative analysis was performed to assess validity and applicability of the model (Liao et al., 2019).

In another study, BWM was applied to solve MCDM problems in a fuzzy environment. In the proposed method, performing all pairwise comparisons was not necessary and only reference comparisons were required. Reference comparisons included determining the fuzzy priority of the best alternative over each of other alternatives as well as the priority of each alternative over the worst alternative. Then a fully fuzzy linear mathematical model was formulated to obtain weights of the criteria. The model was also used to calculate scores of the alternatives. The advantages of their proposed method include: less data needed for decision making, high capability to obtain a reliable solution, and a suitable method to combine with other methods. Finally, hospital maintenance evaluation was performed to evaluate the proposed method (Karimi et al., 2020)

In another study, hierarchical group decision making algorithm based on axiomatic design principles was used in fuzzy environment. Then, as a case study, the proposed method was applied for conceptual design of a loudspeaker and the capabilities of the model were evaluated (Maghsoodi et al., 2019).

Haeri and Rezaei (2019) extended BWM using gray (interval) numbers theory. The proposed model was able to identify the interrelationships between the criteria. Using their proposed model, they solved the green supplier selection problem with regard to economic and environmental criteria, and also ranked the alternatives using interval analysis.

Brunelli and Rezaei (2019) examined BWM from a mathematical perspective and added a new metric to the overall BWM framework. Using this metric, the original idea of BWM did not change but ultimately converted to an optimization problem that could be linearized and solved.

Mi and Liao (2019) developed a BWM using hesitant fuzzy information in three models with different objectives. Using the normalized weights of criteria obtained by their method, they extended the evaluation based on distance from average solution (EDAS) method for hesitant uncertain fuzzy environment. They conducted a case study for deciding on choosing commercial endowment insurance products. Finally, they evaluate validity and reliability of their method by comparing it with other available methods.

Wu et al. (2019) developed a BWM based on interval type-2 fuzzy sets (IT2FSs) for multi-criteria group decision-making (MCGDM). They then proposed an integrated approach using BWM and VIKOR methods in the interval type-2 fuzzy environment, which had advantages over the basic BWM and fuzzy BWM. They evaluated their approach by applying it for green supplier selection.

Amiri and Emamat (2020) proposed two models based on nonlinear and linear BWM. In their proposed models, the number of constraints was reduced to n-2, where n represented the number of criteria. Using a numerical example, they showed the capabilities and the total deviations of the proposed models and compared them with the total deviations obtained from nonlinear and linear BWM. In addition to reducing the complexity of the calculations, the proposed models also had acceptable total deviations.

Recently, some integrated fuzzy models based on BWM technique and fuzzy preference programming were proposed in another study for weighting and evaluating decision criteria. The proposed models were applicable for individual and group decision making scenarios. One of the advantages of the proposed models was that there was no need to calculate the consistency rate (CR) of comparisons separately; the consistency of comparisons in their models was determined by the model. They evaluated hospitals performance as a case study to validate their method (Amiri et al., 2020).

Rezaei (2020) introduced concentration ratio for measuring optimal intervals in nonlinear BWM. He showed the relationship between consistency ratio and concentration ratio and stated that these two indicators together give DMs more insights about the nonlinear BWM results. Concentration ratio shows how much the weights obtained from nonlinear BWM tend to a single point.

Since human judgments and decisions are ambiguous and are usually not spelled out in a clear way, they cannot be entered with Crisp exact numbers. Hence the fuzzy theory presented by Zadeh (1965), is used to aid the decision-making process. It uses human preferences and judgments as fuzzy numbers in an uncertain environment. This uncertainty may exist in data, constraints, resources, etc. Therefore, we need to use decision-making methods in uncertain environments to make decisions in such cases.

Dubois et al. (2003) generally categorize uncertainty as follows: (1) input data uncertainty; (2) flexibility in constraints and goals. For the first case, commonly referred to as epistemic uncertainty, possibilistic programming (PP) methods are used. In the second case, there is flexibility in the values of the objective function as well as the flexibility in the sources or the right hand side (RHS) numbers, and we need to use flexible mathematical programming (Bellman and Zadeh, 1970, Mula et al., 2006). We need to distinguish between epistemic uncertainty and flexibility in data constraints and goals and lack of data. Flexibility is modeled using fuzzy constraints while epistemic uncertainty is modeled using fuzzy coefficients and PP (Mula et al., 2007).

On the other hand, two main categories of uncertainty in data are randomness uncertainty and epistemic uncertainty (Mula et al., 2006, Mula et al., 2007). Randomness uncertainty is used when the parameters are naturally random and their distribution is known. In such cases, stochastic programming is used to deal with uncertainty. Epistemic uncertainty is used when data are ambiguous and inaccurate, and DMs face a lack of information. PP is used to deal this type of uncertainty (Dehghan et al., 2018). PP is used when there is a lack of knowledge (epistemic uncertainty) about the exact values of input data (parameters) due to the inaccessibility of the required data; in this situation suitable possibilistic distributions are derived as fuzzy numbers using available information and data and preferences of DMs (Mousazadeh et al., 2014).

PP is a common approach in fuzzy programming models and has many applications in this field. Many articles have been published about this approach: In the field of reverse logistics and green logistics (Pishvaee and Torabi, 2010, Qin and Ji, 2010, Pishvaee and Razmi, 2012, Vahdani et al., 2013, Pishvaee et al., 2012), allocation models (Xu and Zhou, 2013, Önüt et al., 2008, Zahiri et al., 2014), portfolio selection (Mehlawat and Gupta, 2014, Li and Xu, 2007), data envelopment analysis (DEA) (Lertworasirikul et al., 2003, Peykani et al., 2018), etc. In these studies, DMs faced uncertainty and used PP to deal with it. Uncertainty may be due to lack of access to information or lack of knowledge.

The purpose of this study is to propose a fuzzy MCDM method based on BWM using PP approach. In this approach, a best-worst linear programming model is developed based on three approaches of possibility, necessity, and credibility, and new fuzzy linear programming models are introduced for decision making under uncertainty. In this way, DMs can have an optimistic view, pessimistic view, or a view between the two about the decision-making problem. They can use the proposed approach as a supportive decision-making framework in cases where decision-makers' preferences are close to each other and the criteria cannot be separated well. Of course, this is not the first time that MCDM methods (as supportive systems) have helped DMs. For example, in a study on water security sustainability evaluation, an integrated MCDM approach was used, which provided a multi-stage decision support framework by integrating the MCDM scientific and strategic methods, including BWM, DEMATEL, and TOPSIS and assisted DMs in assessing the sustainability of water security (Nie et al., 2018).

In a recent study to deal with incomplete, uncertain, or imprecise data, an integrated decision-making approach including the improved AHP method and picture fuzzy PROMETHEE II was developed, which uses picture fuzzy numbers (PFNs) and it is able to reduce errors made by human and deal with incomplete data or lack of decision data. They used the improved AHP method (which was based on expert mean assessment) and the extended picture fuzzy PROMETHEE II method (which also included psychological behaviors of DMs) to solve a decision-making problem about the effect of environmental issues on tourism attraction and analyzed the results (Tian et al., 2020). In another study, an integrated MCDM framework based on picture fuzzy information including picture fuzzy exponential entropy and extended VIKOR method (in order to avoid problems caused by lack of information) was proposed to select a sustainable supplier (Peng et al., 2020).

In the field of fuzzy systems and sets and in the fuzzy mathematical programming field, the most common problems and studies are related to fuzzy linear programming (Verdegay, 2015). The fuzzy models developed in this study are linear and can be used to calculate optimal weights of decision criteria. Many methods have been used in the decision making process to process and synthesize fuzzy information. These methods incorporate fuzzy information into the linear programming model so that DMs can obtain optimal solutions by solving the model (Luhandjula, 1986). In the proposed approach in this paper, new fuzzy linear programming models have been developed to deal with uncertainty or lack of information in decision data. These models can calculate optimal weights of criteria. This is an important issue in the development of MCDM methods.

With regard to the reviewed literature and the need to develop MCDM methods to deal with information shortages and to reduce decision errors, and also given the advantages of BWM over other decision making methods, combining BWM with PP approach can provide a suitable tool for DMs to deal with the lack of information in decision data. Therefore, according to previous studies on the development of BWM in uncertain environments, the main features of this study include the following:

• Development of BWM method in fuzzy environment using the advantages of the possibilistic chance-constrained programming (PCCP) to deal with epistemic uncertainty (which includes lack of information in input data).

• Proposing the fuzzy linear programming models based on the possibility, necessity, and credibility measures which enable decision-makers to consider appropriate levels of uncertainty in their decisions using optimistic, pessimistic and intermediate views, and also enable them to calculate optimal weights of criteria by careful evaluation.

• The use of trapezoidal fuzzy numbers to develop new models which facilitate the decision-making process since in the trapezoidal fuzzy number membership function, there is more than one number with membership degree 1. In other words, the trapezoidal fuzzy numbers are more extended and more flexible than the triangular fuzzy numbers.

The rest of the paper will be organized as follows: summary of the BWM and its steps is provided in Section 2. In Section 3 possibilistic programming and approaches of possibility, necessity, and credibility will be discussed. The research methodology involved in developing PP-based BWM will be described in Section 4 and proposed models will be defined. In Section 5, capabilities of the proposed models will be examined using two numerical examples. The paper is concluded in Section 6.