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Abstract

In multiple criteria decision making (MCDM) with interval-valued belief distributions (IVBDs), individual IVBDs on multiple criteria are combined explicitly or implicitly to generate the expected utilities of alternatives, which can be used to make decisions with the aid of decision rules. To analyze an MCDM problem with a large number of criteria and grades used to profile IVBDs, effective algorithms are required to find the solutions to the optimization models within a large feasible region. An important issue is to identify an algorithm suitable for finding accurate solutions within a limited or acceptable time. To address this issue, four representative evolutionary algorithms, including genetic algorithm, differential evolution algorithm, particle swarm optimization algorithm, and gravitational search algorithm, are selected to combine individual IVBDs of alternatives and generate the minimum and maximum expected utilities of alternatives. By performing experiments with different numbers of criteria and grades, a comparative analysis of the four algorithms is provided with the aid of two indicators: accuracy and efficiency. Experimental results indicate that particle swarm optimization algorithm is the best among the four algorithms for combining individual IVBDs and generating the minimum and maximum expected utilities of alternatives.

Keywords

Multiple criteria decision making interval-valued belief distribution evolutionary algorithm accuracy efficiency

1. Introduction

In the Internet and Big Data era, human lifestyles have undergone an unprecedented revolution. An individual’s life is filled with a lot of data, which makes people more informed than ever before. At the same time, people must choose between deriving useful information and knowledge from data for use in practice and abandoning their attempt to employ data in real problems. Because of the availability of information, people usually choose to find effective information and knowledge from various types of data and use them in practical cases. Such a choice improves their capability to handle complex problems. The choice also results in more uncertain environments associated with real problems than before due to the randomness, unavailability, noise, sparsity, and variety of data. In this environment, people may have difficulty directly finding overall solutions to real problems. A feasible way to overcome the difficulty is to analyze real problems from multiple different perspectives, and then combine the relevant analyses to generate overall problem solutions. This method is called multiple criteria decision making (MCDM) in an uncertain environment.

To effectively model and analyze uncertain MCDM problems, many attempts have been made with the help of different uncertain expressions. Representative expressions include intuitionistic fuzzy sets [1], hesitant fuzzy linguistic sets [2], hesitant fuzzy sets [3], probabilistic linguistic sets [4], belief distributions [5], interval-valued fuzzy sets [6], interval-valued hesitant fuzzy linguistic sets [7], interval-valued intuitionistic fuzzy sets [8], interval-valued hesitant fuzzy sets [9], interval type-2 fuzzy sets [10], and interval-valued belief distributions [11]. In theory, MCDM methods with these expressions are sufficient for analyzing all real problems. From real cases or numerical examples in these studies, few methods are found which aim to solve large-scale problems with many alternatives and criteria. In addition to this, when interval-valued assessments are adopted, such as interval-valued hesitant fuzzy elements or interval-valued belief distributions, the search space for finding solutions increases exponentially with the increase in the number of interval-valued hesitant fuzzy element values or the number of interval-valued belief distribution grades.

Evolutionary computation provides a feasible and effective way to find acceptable or satisfactory solutions within a limited time. When MCDM problems are regarded as multi-objective optimization (MOO) problems constructed on a common set of variables, many evolutionary MOO approaches have been developed to find the optimum trade-off among criteria which is the most consistent with the preference of a decision maker [12, 13, 14, 15, 16, 17]. Three methods (priori, interactive, and posteriori) are usually applied to combine the preferences of a decision maker with the MOO process [15, 16]. If the preferences of a decision maker are not considered in the MOO process, the results of the MOO may not be satisfactory.

In practice, individual assessments on different criteria may not be always constructed on a common set of variables. For example, a radiologist determines whether a nodule of a patient is malignant from the perspectives of contour, echogenicity, calcification, and vascularity. It cannot be said that the judgments on the nodule with the consideration of contour and those with the consideration of any other perspective are made by the radiologist through the same set of variables (or features). As another example, when the same discipline at different universities is compared, many criteria are considered, such as research projects, publications, awards, patents, social services, and excellent alumni. Individual assessments on different criteria are generated from different data rather than common data. When encountering these situations, a decision maker takes into account the individual assessments on all criteria synthetically rather than improving the values of most objectives and balancing them to generate solutions. MCDM problems with large search spaces in these situations can also be solved by using evolutionary algorithms. For example, Javanbarg et al. [18] used particle swarm optimization (PSO) algorithm [19] to solve MCDM problems modeled by a fuzzy analytic hierarchy process, and Chen and Huang [20] used PSO algorithm to solve MCDM problems modeled by interval-valued intuitionistic fuzzy numbers.

Existing studies show that less attention has been paid to the application of evolutionary algorithms to MCDM with different sets of variables used on different criteria. This makes it questionable whether MCDM methods with different ways to characterize different types of uncertain nature (e.g., [3, 7, 8, 9, 10, 11]) can be applied to solve MCDM problems with large search spaces. There is a gap between the solution requirements of large-scale MCDM problems and relevant studies on effective solution approaches. Although there are few studies on the combination of evolutionary algorithms and MCDM with different sets of variables (e.g., [18, 20]), some important issues deserve investigation. The issues include: (1) why PSO algorithm has been selected for application in MCDM; (2) whether PSO algorithm can be applied to MCDM with different types of uncertain expressions other than interval-valued intuitionistic fuzzy numbers and fuzzy triangular numbers; and (3) which evolutionary algorithm has better performance when applied to MCDM with a specific type of uncertain expression. In fact, different evolutionary algorithms can be applied to solve the same real problem. For example, when determining the near-optimal scheme for recharging batteries at a battery swapping station, Wu et al. [21] used three representative evolutionary algorithms including genetic algorithm (GA) [22, 23], differential evolution (DE) algorithm [24], and PSO algorithm to find the minimum running cost. Their experimental results show that GA and DE algorithms achieve higher accuracies and lower efficiencies than PSO algorithm; specifically, PSO algorithm fails to obtain the objective. Inspired by this, much attention should be paid to a key issue, which is comparing the accuracies [21, 25] and efficiencies [14, 26] of different representative evolutionary algorithms for solving large-scale MCDM problems with specific types of uncertain expressions and different sets of variables.

In this paper, to address this key issue, we aim to compare the accuracies and efficiencies of four representative evolutionary algorithms, which are GA, DE algorithm, PSO algorithm, and gravitational search algorithm (GSA) [27], for analyzing MCDM problems modeled by interval-valued belief distributions (see Section 2). Because combining individual interval-valued belief distributions is an important and necessary MCDM sub-process of MCDM, the combination processes using the four algorithms are presented. To fairly compare the four algorithms, their original versions (instead of their extensions) are used in the processes. With the aid of the processes, experiments with different numbers of criteria and grades used to profile interval-valued belief distributions are performed to compare the accuracies and efficiencies of the four algorithms for combining interval-valued belief distributions and generating the expected utilities. The comparative analysis of experimental results helps select the appropriate evolutionary algorithm to find satisfactory solutions to MCDM problems with interval-valued belief distributions within a limited or acceptable time. A sensitivity analysis of the accuracies and efficiencies of the four algorithms is provided to highlight the conclusion drawn from the comparative analysis.

The rest of this paper is organized as follows. Section 2 recalls the modeling of MCDM problems by using belief distributions and interval-valued belief distributions. Section 3 presents the processes of four evolutionary algorithms for combining interval-valued belief distributions. Section 4 compares the accuracies and efficiencies of the four algorithms for combining interval-valued belief distributions. The results of the four algorithms for generating the expected utilities are compared in Section 5. A sensitivity analysis of the performance of the four algorithms is provided in Section 6. Finally, this paper is concluded in Section 7.

2. Preliminaries

2.1 Modeling of MCDM problems using belief distributions

In the evidential reasoning (ER) approach [28, 29, 30], which is a type of multiple criteria utility function method, belief distribution is used to characterize the uncertain preferences of a decision maker. Because belief distribution is a special case of interval-valued belief distribution, the method for modeling MCDM problems using belief distribution is reviewed first.

Suppose that alternative $a_{l}∼{}(l=1,\ldots,M)$ is evaluated on criterion $e_{i}$ ( $i=1,\ldots,L$ ) by using a set of grades $\Omega=\{H_{1},H_{2},\ldots,H_{N}\}$ , which is ordered increasingly from worst to best. The utilities of grades $u(H_{n})$ ( $n=1,\ldots,N$ ) satisfy the constraint $0=u(H_{1})<u(H_{2})<\ldots<u(H_{N})=1$ to reflect the difference among grades. Under the conditions, a belief distribution $B(e_{i}(a_{l}))=\{(H_{n},$ $\beta_{n,i}(a_{l})),$ $n=1,\ldots,N;$ $(\Omega,\beta_{\Omega,i}(a_{l}))\}$ describes the evaluation, where $\beta_{n,i}(a_{l})$ with $\beta_{n,i}(a_{l})\geqslant 0$ and $\sum_{n=1}^{N}{\beta_{n,i}(a_{l})}\leqslant 1$ denotes the belief degree assigned to grade $H_{n}$ , and $\beta_{\Omega,i}(a_{l})=1-\sum_{n=1}^{N}{\beta_{n,i}(a_{l})}$ represents the degree of global ignorance [31]. If $\beta_{\Omega,i}(a_{l})=$ 0, the assessment is complete; otherwise, it is incomplete. When $B(e_{i}(a_{l}))(i=1,\ldots,L,l=1,\ldots,M)$ is given, the belief decision matrix $S_{L\times M}$ is obtained.

Assume that criteria weights are represented by $w=(w_{1},w_{2},\ldots,w_{L})$ such that $0\leqslant w_{i}\leqslant 1$ and $\sum_{i=1}^{L}{w_{i}}=1$ . By combining individual belief distributions $B(e_{i}(a_{l}))(i=1,\ldots,L,l=1,\ldots,M)$ by using criteria weights and the ER rule [32], the overall belief distribution is obtained as $B(a_{l})=\{(H_{n},\beta_{n}(a_{l})),n=1,\ldots,N;(\Omega,\beta_{\Omega}(a_{l}))\}$ . Similar to the individual belief distribution, $\beta_{\Omega}(a_{l})$ represents the degree of aggregated global ignorance. It is not easy to directly compare the aggregated belief distributions of different alternatives in most cases. To facilitate comparison, $B(a_{l})(l=1,\ldots,M)$ is transformed by using the utilities of grades $u(H_{n})(n=1,\ldots,N)$ to the minimum and maximum expected utilities of alternative $a_{l}$ , which are $u^{-}(a_{l})=\sum_{n=2}^{N}{\beta_{n}(a_{l})u(H_{n})}+(\beta_{1}(a_{l})+\beta_% {\Omega}(a_{l}))u(H_{1})$ and $u^{+}(a_{l})=\sum_{n=1}^{N-1}{\beta_{n}(a_{l})u(H_{n})}+(\beta_{N}(a_{l})+% \beta_{\Omega}(a_{l}))u(H_{N}).$ From $u^{-}(a_{l})$ and $u^{+}(a_{l})$ , a decision rule, such as the Hurwicz rule [33], can be used to aid in generating solutions.

2.2 Combination of belief distributions

The contents in the above section show that the ER rule [32] is the key to find solutions to MCDM problems modeled by belief distributions, which is simply presented as follows.

.

[32] Given individual assessments $B(e_{i}(a_{l}))(i=1,\ldots,L)$ and their weights $w_{i}$ , the combined result of the first $i$ assessments is defined as

$\displaystyle\{(H_{n},\beta_{n,b(i)}(a_{l})),n\!=\!1,\ldots,N;(\Omega,\beta_{% \Omega,b(i)}(a_{l}))\},$ (1)

where

$\displaystyle\beta_{n,b(i)}(a_{l})=\frac{\hat{\beta}_{n,b(i)}(a_{l})}{\sum_{n=% 1}^{N}\hat{\beta}_{n,b(i)}(a_{l})+\hat{\beta}_{\Omega,b(i)}(a_{l})},$ (2) $\displaystyle\beta_{\Omega,b(i)}(a_{l})=\frac{\hat{\beta}_{\Omega,b(i)}(a_{l})% }{\sum_{n=1}^{N}{\hat{\beta}_{n,b(i)}(a_{l})}+\hat{\beta}_{\Omega,b(i)}(a_{l})},$ (3) $\displaystyle\vec{\beta}_{n,b(i)}(a_{l})=$ $\displaystyle\frac{\hat{\beta}_{n,b(i)}(a_{l})}{\sum_{n=1}^{N}{\hat{\beta}_{n,% b(i)}(a_{l})}+\hat{\beta}_{\Omega,b(i)}(a_{l})+\hat{\beta}_{P(\Omega),b(i)}(a_% {l})},$ (4) $\displaystyle\vec{\beta}_{\Omega,b(i)}(a_{l})=$ $\displaystyle\frac{\hat{\beta}_{\Omega,b(i)}(a_{l})}{\sum_{n=1}^{N}{\hat{\beta% }_{n,b(i)}(a_{l})}+\hat{\beta}_{\Omega,b(i)}(a_{l})+\hat{\beta}_{P(\Omega),b(i% )}(a_{l})},$ (5) $\displaystyle\vec{\beta}_{P(\Omega),b(i)}(a_{l})=$ $\displaystyle\frac{\hat{\beta}_{P(\Omega),b(i)}(a_{l})}{\sum_{n=1}^{N}{\hat{% \beta}_{n,b(i)}(a_{l})}+\hat{\beta}_{\Omega,b(i)}(a_{l})+\hat{\beta}_{P(\Omega% ),b(i)}(a_{l})},$ (6) $\displaystyle\hat{\beta}_{n,b(i)}(a_{l})=[(1-w_{i})\cdot\vec{\beta}_{n,b(i-1)}% (a_{l})$ $\displaystyle\quad+\vec{\beta}_{P(\Omega),b(i-1)}(a_{l})\cdot w_{i}\cdot\beta_% {n,i}(a_{l})]$ $\displaystyle\quad+\vec{\beta}_{n,b(i-1)}(a_{l})\cdot w_{i}\beta_{n,i}(a_{l})$ $\displaystyle\quad+\vec{\beta}_{n,b(i-1)}(a_{l})\cdot w_{i}\beta_{\Omega,i}(a_% {l})$ $\displaystyle\quad+\vec{\beta}_{\Omega,b(i-1)}(a_{l})\cdot w_{i}\beta_{n,i}(a_% {l}),$ (7) $\displaystyle\hat{\beta}_{\Omega,b(i)}(a_{l})=[(1-w_{i})\cdot\vec{\beta}_{% \Omega,b(i-1)}(a_{l})$ $\displaystyle\quad+\vec{\beta}_{P(\Omega),b(i-1)}(a_{l})\cdot w_{i}\cdot\beta_% {\Omega,i}(a_{l})]$ $\displaystyle\quad+\vec{\beta}_{\Omega,b(i-1)}(a_{l})\cdot w_{i}\beta_{\Omega,% i}(a_{l}),$ (8)

and

$\displaystyle\hat{\beta}_{P(\Omega),b(i)}(a_{l})=(1-w_{i})\cdot\vec{\beta}_{P(% \Omega),b(i-1)}(a_{l}).$ (9)

In Definition 1, $P(\Omega)$ represents the power set of $\Omega$ , and it is satisfied that $0\leqslant\beta_{n,b(i)}(a_{l})$ , $\beta_{\Omega,b(i)}(a_{l})$ , $\vec{\beta}_{n,b(i)}(a_{l})$ , $\vec{\beta}_{\Omega,b(i)}(a_{l})\leqslant 1$ , $0\leqslant\vec{\beta}_{P(\Omega),b(i)}(a_{l})\leqslant 1$ , and $\sum^{N}_{n-1}\vec{\beta}_{n,b(i)}(a_{l})+\vec{\beta}_{\Omega,b(i)}(a_{l})+% \vec{\beta}_{P(\Omega),b(i)}(a_{l})=$ 1 for $i=2,\ldots,L$ recursively. Specifically, in Eqs (1)–(9), $(\hat{\beta}_{n,b(i)}(a_{l})$ , $n=1,\ldots,N$ , $\hat{\beta}_{\Omega,b(i)}(a_{l})$ , $\hat{\beta}_{P(\Omega),b(i)}(a_{l}))$ means the unnormalized combination of the first $i-1$ iterative assessments $(\vec{\beta}_{n,b(i-1)}(a_{l}),n=1,\ldots,N,\vec{\beta}_{\Omega,b(i-1)}(a_{l})% ,\vec{\beta}_{P(\Omega),b(i-1)}(a_{l}))$ and the $i$ th assessment $(\beta_{n,i}(a_{l}),n=1,\ldots,N,\beta_{\Omega,i}(a_{l}))$ . Noted that the assessment $(\vec{\beta}_{n,b(2)}(a_{l})$ , $n=1,\ldots,N$ , $\vec{\beta}_{\Omega,b(2)}(a_{l})$ , $\vec{\beta}_{P(\Omega),b(2)}(a_{l}))$ is obtained by combining $(\beta_{n,1}(a_{l})$ , $n=1,\ldots,N,\beta_{\Omega,1}(a_{l}))$ and $(\beta_{n,2}(a_{l})$ , $n=1,\ldots,N,\beta_{\Omega,2}(a_{l}))$ . In Eqs (1)–(1), $(\vec{\beta}_{n,b(i)}(a_{l}),n=1,\ldots,N,\vec{\beta}_{\Omega,b(i)}(a_{l}),% \vec{\beta}_{P(\Omega),b(i)}(a_{l}))$ means the normalization of $(\hat{\beta}_{n,b(i)}(a_{l}),n=1,\ldots,N$ , $\hat{\beta}_{\Omega,b(i)}(a_{l})$ , $\hat{\beta}_{P(\Omega),b(i)}(a_{l}))$ , which considers both the ignorance on the set of $\Omega$ and that on the power set of $\Omega$ . In Eqs (2)–(3), $(\beta_{n,b(i)}(a_{l}),n=1,\ldots,N,\beta_{\Omega,b(i)}(a_{l}))$ means the normalization of $(\hat{\beta}_{n,b(i)}(a_{l}),n=1,\ldots,N,\hat{\beta}_{\Omega,b(i)}$ $(a_{l}))$ , which considers the ignorance on the set of $\Omega$ .

2.3 Modeling of MCDM problems using interval- valued belief distributions

Due to the lack of sufficient data and knowledge or the nature of the decision problems under consideration, in some situations, a decision maker can only provide interval-valued belief distributions (IVBDs) as the evaluations of alternatives. For example, when a radiologist provides the diagnostic category in thyroid imaging reporting and data system published by Horvath et al. [34] for the thyroid nodule of a patient, he or she only reports the interval-valued cancer risk rather than the precise cancer risk of the patient.

In this situation, individual IVBDs are represented by $B(e_{i}(a_{l}))=\{(H_{n},[\beta_{n,i}^{-}(a_{l}),\beta_{n,i}^{+}(a_{l})]),n=1,$ $\ldots,N;(\Omega,[\beta_{\Omega,i}^{-}(a_{l}),\beta_{\Omega,i}^{+}(a_{l})])\}$ [11]. If it is satisfied that $\sum_{n=1}^{N}{\beta_{n,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}(a_{l})\leqslant 1$ and $\sum_{n=1}^{N}\beta_{n,i}^{+}$ $(a_{l})+\beta_{\Omega,i}^{+}(a_{l})\geqslant 1$ , the IVBDs are called valid [35]. Otherwise, they are invalid and cannot be used to generate valid belief distributions. Valid IVBDs are said to be normalized [35] only when it is satisfied that

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)-(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))$ $\displaystyle\quad\geqslant 1,∼{}∼{}n=1,\ldots,N,$ (10) $\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)-(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))$ $\displaystyle\quad\geqslant 1,$ (11) $\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)+(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))$ $\displaystyle\quad\leqslant 1,∼{}∼{}n=1,\ldots,N,\text{ and}$ (12) $\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)+(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))$ $\displaystyle\quad\leqslant 1.$ (13)

Normalized IVBDs are valid but valid IVBDs may be unnormalized [35].

From normalized individual IVBDs, a pair of optimization problems is constructed by using the ER rule to generate the aggregated IVBD $B(a_{l})=\{(H_{n}$ , $[\beta_{n}^{-}(a_{l}),\beta_{n}^{+}(a_{l})]),n=1,\ldots,N;(\Omega,[\beta_{% \Omega}^{-}(a_{l})$ , $\beta_{\Omega}^{+}(a_{l})])\}$ [36, 37].

$\displaystyle\textrm{MIN/MAX}∼{}\beta_{n}(a_{l})$ (14) $\displaystyle\text{s.t. }\beta_{n,i}^{-}(a_{l})\leqslant\beta_{n,i}^{*}(a_{l})% \leqslant\beta_{n,i}^{+}(a_{l}),$ (15) $\displaystyle\beta_{\Omega,i}^{-}(a_{l})\leqslant\beta_{\Omega,i}^{*}(a_{l})% \leqslant\beta_{\Omega,i}^{+}(a_{l}),$ (16) $\displaystyle\sum_{n=1}^{N}{\beta_{n,i}^{*}(a_{l})}+\beta_{\Omega,i}^{*}(a_{l}% )=1.$ (17)

In the pair of optimization problems, $\beta_{n,i}^{*}(a_{l})$ and $\beta_{\Omega,i}^{*}(a_{l})$ represent decision variables, which form belief distributions limited to IVBDs. When the objective of the pair of optimization problems is changed to $\beta_{\Omega}(a_{l})$ , $\beta_{\Omega}^{-}(a_{l})$ and $\beta_{\Omega}^{+}(a_{l})$ can be obtained. From the aggregated IVBD and the utilities of grades $u(H_{n})(n=1,\ldots,N)$ , the following optimization model is constructed to determine the minimum and maximum expected utilities $u^{-}(a_{l})$ and $u^{+}(a_{l})$ [36, 37].

$\displaystyle\text{MIN}\sum_{n=2}^{N}{\beta_{n}^{*}(a_{l})u(H_{n})}+(\beta_{1}% ^{*}(a_{l})+\beta_{\Omega}^{*}(a_{l}))u(H_{1})$ (18) $\displaystyle\text{s.t.}\beta_{n}^{-}(a_{l})\leqslant\beta_{n}^{*}(a_{l})% \leqslant\beta_{n}^{+}(a_{l}),\beta_{\Omega}^{-}(a_{l})\leqslant\beta_{\Omega}% ^{*}(a_{l})\leqslant\beta_{\Omega,i}^{+}(a_{l}),\sum_{n=1}^{N}{\beta_{n}^{*}(a% _{l})}+\beta_{\Omega}^{*}(a_{l})=1.$ (21)

Solving this model, in which $\beta_{n}^{*}(a_{l})$ and $\beta_{\Omega}^{*}(a_{l})$ represent decision variables, generates the optimal $u^{-}(a_{l})$ . When the objective of this model is changed to “MAX $\sum_{n=1}^{N-1}{\beta_{n}^{*}(a_{l})u(H_{n})}+(\beta_{N}^{*}(a_{l})+\beta_{% \Omega}^{*}(a_{l}))u(H_{N})$ ”, the optimal $u^{+}(a_{l})$ can be obtained. If the aggregated IVBD is not required to analyze the decision problem under consideration, the optimization model shown in Eqs (18)–(21) can be modified as follows to determine $u^{-}(a_{l})$ and $u^{+}(a_{l})$ [36, 37].

$\displaystyle\text{MIN}\sum_{n=2}^{N}{\beta_{n}(a_{l})u(H_{n})}+(\beta_{1}(a_{% l})+\beta_{\Omega}(a_{l}))u(H_{1})$ (22) $\displaystyle\text{s.t.}\beta_{n,i}^{-}(a_{l})\leqslant\beta_{n,i}^{*}(a_{l})% \leqslant\beta_{n,i}^{+}(a_{l}),\beta_{\Omega,i}^{*}(a_{l})\leqslant\beta_{% \Omega,i}^{+}(a_{l}),\sum_{n=1}^{N}{\beta_{n,i}^{*}(a_{l})}+\beta_{\Omega,i}^{% *}(a_{l})=1.$ (25)

The combination of individual belief distributions by using the ER rule to generate the aggregated belief distribution $B(a_{l})=\{(H_{n},\beta_{n}(a_{l})),n=1,\ldots,$ $N;(\Omega,\beta_{\Omega}(a_{l}))\}$ is implicitly involved in this optimization model. Similarly, the optimal $u^{-}(a_{l})$ can be obtained from solving this model and the optimal $u^{+}(a_{l})$ from solving the model with the objective of “MAX $\sum_{n=1}^{N-1}{\beta_{n}(a_{l})u(H_{n})}+(\beta_{N}(a_{l})+\beta_{\Omega}(a_% {l}))u(H_{N})$ ”.

3. Four evolutionary algorithms for MCDM with IVBDs

When the number of criteria $L$ and the number of grades $N$ are large, solving the optimization problems shown in Eqs (14)–(17) and (22)–(25) becomes difficult. Evolutionary algorithms are helpful for finding solutions to the optimization problems with large $L$ and $N$ . A key issue is to find the evolutionary algorithm with higher accuracy and efficiency among feasible algorithms. To address this issue, four evolutionary algorithms (GA, DE algorithm, PSO algorithm, and GSA) are compared. These algorithms are selected because many of their extensions have been developed to handle real problems in different fields. For GA, its chromosome coding [38, 39] and structure [40] were improved and it was used to conduct combinational dispatching decision [41], ischemic beat classification [42], and the generation of trading strategies for stock markets [43]. As to DE algorithm, its neighborhood-based mutation operator [44], dynamic parameter selection [45], self-adapting control parameters [46], and hybrid cross-generation mutation operation [47] have been developed, and it was used to solve permutation flow shop scheduling problems [48] and periodic railway timetable scheduling problems [49]. With respect to PSO algorithm, its stability [50] and impacts of coefficients on movement patterns [51] were analyzed, and it was used to conduct cancer classification [52] and population classification in fire evacuation [53], and model the gene regulatory networks [54]. Concerning about GSA, its nearest neighbor scheme [55] was developed, and it was used to conduct feature selection for face recognition [56], unit commitment in power system operation [57], and parameter identification for a water turbine regulation system [58].

In the following, the original processes of the four evolutionary algorithms for solving the pair of optimization problems shown in Eqs (14)–(17) are presented to facilitate comparing the accuracies and efficiencies of the four algorithms. When the objective in Eq. (14) is changed to that in Eq. (22), the similar processes of the four algorithms can be used to determine the minimum and maximum expected utilities, which are omitted to avoid repetition. To guarantee a fair comparison, extensions of the four algorithms are not adopted.

3.1 GA process for combining individual IVBDs

The GA process for combining individual IVBDs is presented as follows.

Step 1: Initialization

For the pair of optimization problems in Eqs (14)–(17), the $j$ th chromosome in the $t$ th ( $t=$ 1) iteration is represented by

$\displaystyle C^{j,t}(a_{l})=\{(\beta_{1,1}^{j,t}(a_{l}),\ldots,\beta_{N,1}^{j% ,t}(a_{l}),\beta_{\Omega,1}^{j,t}(a_{l})),\ldots,$ $\displaystyle\qquad(\beta_{1,L}^{j,t}(a_{l}),\ldots,\beta_{N,L}^{j,t}(a_{l}),% \beta_{\Omega,L}^{j,t}(a_{l}))\}.$ (26)

Randomly generating $N_{G}$ chromosomes completes the initialization of the GA process. Assume that the maximum number of iterations is $N_{t}$ . The crossover probability threshold $C_{GA}$ and the mutation probability threshold $M_{GA}$ are set as 0.6 and 0.1, respectively.

Step 2: Performance evaluation

When the lower bound of $\beta_{n}(a_{l})$ is optimized, the fitness value of the chromosome $C^{j,t}(a_{l})$ is set as $F^{j,t}(a_{l})=\beta_{n}(a_{l})$ . Conversely, it is set as $F^{j,t}(a_{l})=-\beta_{n}(a_{l})$ to optimize the upper bound of $\beta_{n}(a_{l})$ . As indicated in Section 2.1, $\beta_{n}(a_{l})$ is limited to $[0,1]$ , which means that $F^{j,t}(a_{l})$ is limited to $[-1,1]$ .

Step 3: Selection

The selection probability of the chromosome $C^{j,t}(a_{l})$ is defined as

$\displaystyle p^{j,t}(a_{l})=\frac{1-(F^{j,t}(a_{l})-(-1))/2}{\sum_{k=1}^{N_{G% }}{(1-(F^{k,t}(a_{l})-(-1))/2)}},$ (27)

where $1-(F^{j,t}(a_{l})-(-1))/2$ indicates that the smaller the fitness value $F^{j,t}(a_{l})$ , the larger the possibility of selecting the chromosome $C^{j,t}(a_{l})$ . Given a random selection threshold $\delta^{t}(a_{l})$ , the chromosome with a selection probability larger than $\delta^{t}(a_{l})$ is retained to perform crossover and mutation operations.

Step 4: Crossover

After selection, two chromosomes $C^{j,t}(a_{l})$ and $C^{k,t}(a_{l})$ are randomly selected first. The belief distributions $(\beta_{1,i1}^{j,t}(a_{l}),\ldots,\beta_{N,i1}^{j,t}(a_{l}),\beta_{\Omega,i1}^% {j,t}(a_{l}))$ and $(\beta_{1,i1}^{k,t}$ $(a_{l}),\ldots,\beta_{N,i1}^{k,t}(a_{l}),\beta_{\Omega,i1}^{k,t}(a_{l}))$ in the two chromosomes are randomly selected to perform the crossover operation. Given a random indicator of crossover $C_{I}$ , if $C_{I}>C_{GA}$ , the crossover operation continues; otherwise, it ends. When $C_{I}>C_{GA}$ , given a random crossover coefficient $\gamma^{t}(a_{l})$ , the crossed belief distributions are obtained as

$\displaystyle\hat{\beta}_{n,i1}^{j,t}(a_{l})=\gamma^{t}(a_{l})\cdot\beta_{n,i1% }^{k,t}(a_{l})\quad+(1-\gamma^{t}(a_{l}))\cdot\beta_{n,i1}^{j,t}(a_{l}),$ (28) $\displaystyle\hat{\beta}_{n,i1}^{k,t}(a_{l})=\gamma^{t}(a_{l})\cdot\beta_{n,i1% }^{j,t}(a_{l})\quad+(1-\gamma^{t}(a_{l}))\cdot\beta_{n,i1}^{k,t}(a_{l}),$ (29) $\displaystyle\hat{\beta}_{\Omega,i1}^{j,t}(a_{l})=\gamma^{t}(a_{l})\cdot\beta_% {\Omega,i1}^{k,t}(a_{l})\quad+(1-\gamma^{t}(a_{l}))\cdot\beta_{\Omega,i1}^{j,t% }(a_{l}),\text{ and}$ (30) $\displaystyle\hat{\beta}_{n,i1}^{k,t}(a_{l})=\gamma^{t}(a_{l})\cdot\beta_{% \Omega,i1}^{j,t}(a_{l})\quad+(1-\gamma^{t}(a_{l}))\cdot\beta_{\Omega,i1}^{k,t}% (a_{l}).$ (31)

Step 5: Mutation

After the selection and crossover operations, a chromosome $C^{j,t}(a_{l})$ and one belief distribution of the chromosome $(\beta_{1,i1}^{j,t}(a_{l}),\ldots,\beta_{N,i1}^{j,t}(a_{l}),\beta_{\Omega,i1}^% {j,t}(a_{l}))$ are randomly selected to perform the mutation operation. Given a random indicator of mutation $M_{I}$ , if $M_{I}>M_{GA}$ , the crossover operation continues; otherwise, it ends. When $M_{I}>M_{GA}$ , given a random mutation probability $\eta^{t}(a_{l})$ , when $\eta^{t}(a_{l})>0.5$ , the belief degree $\beta_{n,i1}^{j,t}(a_{l})$ and the ignorance $\beta_{\Omega,i1}^{j,t}(a_{l})$ are mutated to be increased to

$\displaystyle\vec{\beta}^{j,t}_{n,i1}=\beta_{n,i1}^{j,t}(a_{l})+\Delta\beta_{n% ,i1}^{j,t}(a_{l})\text{ and}$ (32) $\displaystyle\vec{\beta}^{j,t}_{\Omega,i1}=\beta_{\Omega,i1}^{j,t}(a_{l})+% \Delta\beta_{\Omega,i1}^{j,t}(a_{l}),$ (33)

where

$\displaystyle\Delta\beta_{n,i1}^{j,t}(a_{l})=(\beta_{n,i1}^{+}(a_{l})-\beta_{n% ,i1}^{j,t}(a_{l}))\quad\cdot({1-\eta^{t}(a_{l})^{(1-t/N_{t})^{2}}})\text{ and}$ (34) $\displaystyle\Delta\beta_{\Omega,i1}^{j,t}(a_{l})=(\beta_{\Omega,i1}^{+}(a_{l}% )-\beta_{\Omega,i1}^{j,t}(a_{l}))\quad\cdot({1-\eta^{t}(a_{l})^{(1-t/N_{t})^{2% }}}).$ (35)

As presented in Section 2.1, $\sum_{n=1}^{N}{\beta_{n,i1}^{j,t}(a_{l})}+\beta_{\Omega,i1}^{j,t}(a_{l})=1$ must be satisfied for the normalized belief distribution, so the increased quantity $\sum_{n=1}^{N}{\Delta\beta_{n,i1}^{j,t}(a_{l})}+\Delta\beta_{\Omega,i1}^{j,t}(% a_{l})$ must be removed from the belief distribution $(\vec{\beta}^{j,t}_{1,i1}(a_{l}),\ldots,\vec{\beta}^{j,t}_{N,i1}(a_{l}),\vec{% \beta}^{j,t}_{\Omega,i1}$ $(a_{l}))$ . By following the rule that the larger $\beta_{n,i1}^{j,t}(a_{l})-\beta_{n,i1}^{-}(a_{l})$ (or $\beta_{\Omega,i1}^{j,t}(a_{l})-\beta_{\Omega,i1}^{-}(a_{l}))$ , the more the decrease in $\vec{\beta}^{j,t}_{n,i1}(a_{l})$ (or $\vec{\beta}^{j,t}_{\Omega,i1}(a_{l})$ ) is, the normalized belief distribution is obtained as

$\displaystyle\tilde{\beta}^{j,t}_{n,i1}(a_{l})=\vec{\beta}^{j,t}_{n,i1}(a_{l})$ (36) $\displaystyle-\left({\sum_{n=1}^{N}{\Delta\beta_{n,i1}^{j,t}(a_{l})}+\Delta% \beta_{\Omega,i1}^{j,t}(a_{l})}\right)\cdot$

and

(37)

\displaystyle\tilde{\beta}^{j,t}_{\Omega,i1}(a_{l})=\vec{\beta}^{j,t}_{\Omega,% i1}(a_{l})

\displaystyle-\left({\sum_{n=1}^{N}{\Delta\beta_{n,i1}^{j,t}(a_{l})}+\Delta% \beta_{\Omega,i1}^{j,t}(a_{l})}\right)\cdot

When $\eta^{t}(a_{l})\leqslant$ 0.5, the belief degree $\beta_{n,i1}^{j,t}(a_{l})$ and the ignorance $\beta_{\Omega,i1}^{j,t}(a_{l})$ are mutated to be decreased to

(38)

\displaystyle\vec{\beta}^{j,t}_{n,i1}(a_{l})=\beta_{n,i1}^{j,t}(a_{l})-\Delta% \beta_{n,i1}^{j,t}(a_{l})\text{ and}

\displaystyle\vec{\beta}^{j,t}_{\Omega,i1}(a_{l})=\beta_{\Omega,i1}^{j,t}(a_{l% })-\Delta\beta_{\Omega,i1}^{j,t}(a_{l}),

(39)

where

$\displaystyle\Delta\beta_{n,i1}^{j,t}(a_{l})=(\beta_{n,i1}^{j,t}(a_{l})-\beta_% {n,i1}^{-}(a_{l}))\cdot({1-\eta^{t}(a_{l})^{(1-t/N_{t})^{2}}})$ (40)

and

$\displaystyle\Delta\beta_{\Omega,i1}^{j,t}(a_{l})=(\beta_{\Omega,i1}^{j,t}(a_{% l})-\beta_{\Omega,i1}^{-}(a_{l}))\cdot({1-\eta^{t}(a_{l})^{(1-t/N_{t})^{2}}}).$ (41)

By following a similar rule, the normalized belief distribution is obtained as

$\displaystyle\tilde{\beta}^{j,t}_{n,i1}(a_{l})=\vec{\beta}^{j,t}_{n,i1}(a_{l})$ (42) $\displaystyle+\left({\sum_{n=1}^{N}{\Delta\beta_{n,i1}^{j,t}(a_{l})}+\Delta% \beta_{\Omega,i1}^{j,t}(a_{l})}\right)\cdot$

and

(43)

\displaystyle\tilde{\beta}^{j,t}_{\Omega,i1}(a_{l})=\vec{\beta}^{j,t}_{\Omega,% i1}(a_{l})

\displaystyle+\left({\sum_{n=1}^{N}{\Delta\beta_{n,i1}^{j,t}(a_{l})}+\Delta% \beta_{\Omega,i1}^{j,t}(a_{l})}\right)\cdot

Note that as $1-\eta^{t}(a_{l})^{(1-t/N_{t})^{2}}$ is certainly limited to $[0,1]$ when $0\leqslant\eta^{t}(a_{l})\leqslant$ 1, $\vec{\beta}^{j,t}_{n,i1}(a_{l})$ in Eqs (32) and (38) and $\vec{\beta}^{j,t}_{\Omega,i1}(a_{l})$ in Eqs (33) and (39) are certainly limited to $[\beta_{n,i1}^{-}(a_{l}),\beta_{n,i1}^{+}(a_{l})]$ and $[\beta_{\Omega,i1}^{-}(a_{l}),\beta_{\Omega,i1}^{+}(a_{l})]$ , respectively.

Step 6: Update

After the selection, crossover, and mutation operations are performed, the fitness values of all chromosomes are recalculated to update the best objective with the corresponding solution.

Step 7: Termination

If $N_{t}$ iterations have been completed, the best objective with the corresponding solution is obtained as the lower bound or upper bound of $\beta_{n}(a_{l})$ . Otherwise, go to Step 3.

3.2

The DE algorithm process for combining individual IVBDs is presented as follows.

Step 1: Initialization

For the pair of optimization problems in Eqs (14)–(17), the $j$ th individual in the $t$ th iteration ( $t=$ 1) is represented by

(44)

\displaystyle I^{j,t}(a_{l})=\{(\beta_{1,1}^{j,t}(a_{l}),\ldots,\beta_{N,1}^{j% ,t}(a_{l}),\beta_{\Omega,1}^{j,t}(a_{l})),\ldots,(\beta_{1,L}^{j,t}(a_{l}),% \ldots,\beta_{N,L}^{j,t}(a_{l}),\beta_{\Omega,L}^{j,t}(a_{l}))\}.

The $N_{D}$ individuals are randomly generated and the maximum number of iterations $N_{t}$ is set. The coefficient of mutation operation $m_{d}$ and the crossover probability threshold $C_{DE}$ are set as 0.9 and 0.6, respectively.

Step 2: Performance evaluation

Through the same process as Section 3.1, the fitness value of the individual $I^{j,t}(a_{l})$ is obtained.

Step 3: Mutation

For the individual $I^{j,t}(a_{l})$ , three individuals are randomly selected, which are $I^{k1,t}(a_{l})$ , $I^{k2,t}(a_{l})$ , and $I^{k3,t}(a_{l})$ . From the three individuals, the individual $I^{j,t}(a_{l})$ is mutated to be

$\displaystyle\vec{\beta}^{j,t}_{n,i}(a_{l})=\beta_{n,i}^{k1,t}(a_{l})+m_{d}% \cdot(\beta_{n,i}^{k2,t}(a_{l})-\beta_{n,i}^{k3,t}(a_{l}))$

and

$\displaystyle\vec{\beta}^{j,t}_{\Omega,i}(a_{l})=\beta_{\Omega,i}^{k1,t}(a_{l}% )+m_{d}\cdot(\beta_{\Omega,i}^{k2,t}(a_{l})-\beta_{\Omega,i}^{k3,t}(a_{l})),$ $\displaystyle i=1,\ldots,L$ (46)

To guarantee that the mutated belief distribution is feasible, $\vec{\beta}^{j,t}_{n,i}(a_{l})$ and $\vec{\beta}^{j,t}_{\Omega,i}(a_{l})$ are strictly limited to $[\beta_{n,i}^{-}(a_{l}),\beta_{n,i}^{+}(a_{l})]$ and $[\beta_{\Omega,i}^{-}(a_{l}),\beta_{\Omega,i}^{+}(a_{l})]$ . This strict requirement, however, does not guarantee that $\sum^{N}_{n=1}\vec{\beta}^{j,t}_{n,i1}(a_{l})+\vec{\beta}^{j,t}_{\Omega,i1}(a_% {l})\!=\!1$ . When $\sum^{N}_{n=1}{\vec{\beta}^{j,t}_{n,i1}(a_{l})}$ $+\vec{\beta}^{j,t}_{\Omega,i1}(a_{l})>1$ or $\sum^{N}_{n=1}{\vec{\beta}^{j,t}_{n,i1}(a_{l})}+\vec{\beta}^{j,t}_{\Omega,i1}(% a_{l})<1$ , Eqs (36)–(37) or Eqs (42)–(43) can be similarly used to generate the normalized belief distribution $(\tilde{\beta}^{j,t}_{1,i}(a_{l}),\ldots,\tilde{\beta}^{j,t}_{N,i}(a_{l}),% \tilde{\beta}^{j,t}_{\Omega,i}(a_{l}))$ satisfying $\sum^{N}_{n=1}{\vec{\beta}^{j,t}_{n,i1}(a_{l})}+\vec{\beta}_{\Omega,i1}^{j,t}(% a_{l})=1$ .

Step 4: Crossover

Given a random indicator of crossover $C_{I}$ , the crossed belief distribution derived from the mutated one and the original one is obtained as

$\displaystyle\hat{\beta}_{n,i}^{j,t}(a_{l})=\begin{cases}\tilde{\beta}_{n,i}^{% j,t}(a_{l})&\text{$C_{I}\leqslant C_{DE}$}\\ \beta_{n,i1}^{j,t}(a_{l})&\text{others}\end{cases}$ (47)

and

$\displaystyle\hat{\beta}_{\Omega,i}^{j,t}(a_{l})=\begin{cases}\tilde{\beta}_{% \Omega,i}^{j,t}(a_{l})&\text{$C_{I}\leqslant C_{DE}$}\\ \beta_{\Omega,i1}^{j,t}(a_{l})&\text{others}\end{cases}$ (48)

Then, the crossed individual $\hat{I}^{j,t}(a_{l})$ is formed by the crossed belief distributions $(\hat{\beta}_{1,i}^{j,t}(a_{l}),\ldots,\hat{\beta}_{N,i}^{j,t}(a_{l}),$ $\hat{\beta}_{\Omega,i}^{j,t}(a_{l}))(i=1,\ldots,L)$ .

Step 5: Selection

If the fitness value of the crossed individual $\hat{I}^{j,t}(a_{l})$ is better than that of $I^{j,t}(a_{l})$ , $I^{j,t}(a_{l})$ is set as $\hat{I}^{j,t}(a_{l})$ . Otherwise, $I^{j,t}(a_{l})$ remains unchanged.

Step 6: Update

After the mutation, crossover, and selection operations are performed, the fitness values of all individuals are recalculated to update the best objective with the corresponding solution.

Step 7: Termination

If $N_{t}$ iterations are completed, the best objective with the corresponding solution is obtained as the lower bound or upper bound of $\beta_{n}(a_{l})$ . Otherwise, go to Step 3.

3.3 PSO algorithm process for combining individual IVBDs

The PSO algorithm process for combining individual IVBDs is presented as follows.

Step 1: Initialization

For the pair of optimization problems in Eqs (14)–(17), the position and the velocity of the $j$ th particle in the $t$ th ( $t=$ 1) iteration are represented respectively as

$\displaystyle P^{j,t}(a_{l})=\{(\beta_{1,1}^{j,t}(a_{l}),\ldots,\beta_{N,1}^{j% ,t}(a_{l}),\beta_{\Omega,1}^{j,t}(a_{l})),\ldots,$ $\displaystyle(\beta_{1,L}^{j,t}(a_{l}),\ldots,\beta_{N,L}^{j,t}(a_{l}),\beta_{% \Omega,L}^{j,t}(a_{l}))\}$ (49)

and

$\displaystyle V^{j,t}(a_{l})=\{(v_{1,1}^{j,t}(a_{l}),\ldots,v_{N,1}^{j,t}(a_{l% }),v_{\Omega,1}^{j,t}(a_{l})),\ldots,$ $\displaystyle(v_{1,L}^{j,t}(a_{l}),\ldots,v_{N,L}^{j,t}(a_{l}),v_{\Omega,L}^{j% ,t}(a_{l}))\}.$ (50)

The $N_{P}$ particles are randomly generated with positions and velocities, and the maximum number of iterations $N_{t}$ is set. Meanwhile, the inertia coefficient $w_{p}$ , the particle coefficient to track its historical best position $c_{1}$ , and the particle coefficient to track the historical best position of all particles $c_{2}$ are set as 1, 2, and 2, respectively.

Step 2: Performance evaluation

Through the same process as Section 3.1, the fitness value of each particle is obtained. From the fitness values of all particles, the initial value of the best position of each particle and that of the best position of all particles are also obtained as $\bar{P}^{j,t}(a_{l})$ and $\bar{P}^{t}(a_{l})$ , respectively. Here, we have from Eq. (3.3) that

$\displaystyle\bar{P}^{j,t}(a_{l})=\{(\bar{\beta}_{1,1}^{j,t}(a_{l}),\ldots,% \bar{\beta}_{N,1}^{j,t}(a_{l}),\bar{\beta}_{\Omega,1}^{j,t}(a_{l})),\ldots,(% \bar{\beta}_{1,L}^{j,t}(a_{l}),\ldots,\bar{\beta}_{N,L}^{j,t}(a_{l}),\bar{% \beta}_{\Omega,L}^{j,t}(a_{l}))\}$

and

$\displaystyle\bar{P}^{t}(a_{l})=\{(\bar{\beta}_{1,1}^{t}(a_{l}),\ldots,\bar{% \beta}_{N,1}^{t}(a_{l}),\bar{\beta}_{\Omega,1}^{t}(a_{l})),\ldots,(\bar{\beta}% _{1,L}^{t}(a_{l}),\ldots,\bar{\beta}_{N,L}^{t}(a_{l}),\bar{\beta}_{\Omega,L}^{% t}(a_{l}))\}.$

Step 3: Position update

Two random real numbers $r_{1}$ and $r_{2}$ limited to $[0,1]$ are generated to constrain $c_{1}$ and $c_{2}$ , respectively. From $w_{p},c_{1},c_{2},r_{1},r_{2},\bar{P}^{j,t}(a_{l})$ , and $\bar{P}^{t}(a_{l})$ , the velocity of the particle $P^{j,t}(a_{l})$ , $V^{j,t}(a_{l})$ is updated to be

$\displaystyle v_{n,i}^{j,t+1}(a_{l})=w_{p}\cdot v_{n,i}^{j,t}(a_{l})+c_{1}% \cdot r_{1}\cdot(\bar{\beta}_{n,i}^{j,t}(a_{l})-$ $\displaystyle\beta_{n,i}^{j,t}(a_{l}))+c_{2}\cdot r_{2}\cdot(\bar{\beta}_{n,i}% ^{t}(a_{l})-\beta_{n,i}^{j,t}(a_{l}))$ (51)

and

$\displaystyle v_{\Omega,i}^{j,t+1}(a_{l})=w_{p}\cdot v_{\Omega,i}^{j,t}(a_{l})% +c_{1}\cdot r_{1}\cdot(\bar{\beta}_{\Omega,i}^{j,t}(a_{l})-$ $\displaystyle\beta_{\Omega,i}^{j,t}(a_{l}))+c_{2}\cdot r_{2}\cdot(\bar{\beta}_% {\Omega,i}^{t}(a_{l})-\beta_{\Omega,i}^{j,t}(a_{l})).$ (52)

To avoid extreme velocity, $v_{n,i}^{j,t+1}(a_{l})$ is required to be limited to $[-0.25\cdot(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l})),0.25\cdot(\beta_{n,% i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))]$ .

From the updated velocity of the particle $P^{j,t}(a_{l})$ , the position of the $j$ th particle is updated to be

$\displaystyle\beta_{n,i}^{j,t+1}(a_{l})=\beta_{n,i}^{j,t}(a_{l})+v_{n,i}^{j,t+% 1}(a_{l})\text{ and}$ (53) $\displaystyle\beta_{\Omega,i}^{j,t+1}(a_{l})=\beta_{\Omega,i}^{j,t}(a_{l})+v_{% \Omega,i}^{j,t+1}(a_{l}).$ (54)

To guarantee the feasibility of the updated position of the $j$ th particle, $\beta_{n,i}^{j,t+1}(a_{l})$ and $\beta_{\Omega,i}^{j,t+1}(a_{l})$ are required to be limited to $[\beta_{n,i}^{-}(a_{l}),\beta_{n,i}^{+}(a_{l})]$ and $[\beta_{\Omega,i}^{-}(a_{l}),$ $\beta_{\Omega,i}^{+}(a_{l})]$ , respectively.

The feasible updated position of the $j$ th particle may not guarantee that $\sum_{n=1}^{N}{\beta_{n,i}^{j,t+1}(a_{l})}+\beta_{\Omega,i}^{j,t+1}(a_{l})=1$ . On the condition that $\sum_{n=1}^{N}{\beta_{n,i}^{j,t+1}(a_{l})}+\beta_{\Omega,i}^{j,t+1}(a_{l})>1$ or $\sum_{n=1}^{N}{\beta_{n,i}^{j,t+1}(a_{l})}+\beta_{\Omega,i}^{j,t+1}(a_{l})<1$ , Eqs (36)–(37) or Eqs (42)–(43) can be similarly used to generate the normalized position $\{(\beta_{1,i}^{j,t+1}(a_{l}),\ldots,\beta_{N,i}^{j,t+1}(a_{l})$ , $\beta_{\Omega,i}^{j,t+1}(a_{l})),i=1,\ldots,L\}$ .

Step 4: Best position update

After the positions of all particles are updated, the fitness values of all particles are recalculated to update the historical best position of each particle and the historical best position of all particles to be $\bar{P}^{j,t+1}(a_{l})$ and $\bar{P}^{t+1}(a_{l})$ , respectively.

Step 5: Termination

If $N_{t}$ iterations have been completed, the best position of each particle and the best position of all particles are obtained, in which the best position of all particles is used to generate the lower bound or upper bound of $\beta_{n}(a_{l})$ . Otherwise, go to Step 3.

3.4 GSA process for combining individual IVBDs

The GSA process for combining individual IVBDs is presented as follows.

Step 1: Initialization

For the pair of optimization problems in Eqs (14)–(17), the position and velocity of the $j$ th agent (mass) in the $t$ th ( $t=1$ ) iteration are represented by $P^{j,t}(a_{l})$ and $V^{j,t}(a_{l})$ , respectively, with the help of Eqs (3.3)–(3.3).

The $N_{S}$ agents are randomly generated and the maximum number of iterations $N_{t}$ is set. The small constant $\varepsilon$ involved in the calculation of the force acting on one mass from another is set as 0.001.

Step 2: Performance evaluation

Through the same process as Section 3.1, the fitness value of each agent is obtained.

Step 3: Update of gravitational coefficient and inertial mass

In the $t$ th iteration, the gravitational coefficient is calculated by $G^{t}=100\cdot e^{-20\cdot t/N_{t}}$ . From the fitness values of all masses $F^{j,t}(a_{l})$ ( $j=1,\ldots,N_{S})$ , the best and worst fitness values are obtained as

$\displaystyle F^{t+}(a_{l})=\min\limits_{j\in\{1,\ldots,N_{S}\}}F^{j,t}(a_{l})$ (55)

and

$\displaystyle F^{t-}(a_{l})=\max\limits_{j\in\{1,\ldots,N_{S}\}}F^{j,t}(a_{l}),$ (56)

respectively. By using $F^{t+}(a_{l})$ and $F^{t-}(a_{l})$ , the inertial mass of the $j$ th agent is calculated as

$\displaystyle M^{j,t}(a_{l})=\frac{m^{j,t}(a_{l})}{\sum_{k=1}^{N_{S}}{m^{k,t}(% a_{l})}},$ (57)

where

$\displaystyle m^{j,t}(a_{l})=\frac{F^{j,t}(a_{l})-F^{t-}(a_{l})}{F^{t+}(a_{l})% -F^{t-}(a_{l})}.$

The passive and active gravitational masses $M^{pj,t}(a_{l})$ and $M^{aj,t}(a_{l})$ are equal to $M^{j,t}(a_{l})$ .

Step 4: Calculation of the total force acting on the $j$ th agent

In the $t$ th iteration, the force acting on the $j$ th agent from the $k$ th agent is calculated as

$\displaystyle F_{n,i}^{jk,t}(a_{l})=G^{t}\cdot\frac{M^{pj,t}(a_{l})\cdot M^{ak% ,t}(a_{l})}{R^{jk,t}(a_{l})+\varepsilon}\cdot(\beta_{n,i}^{k,t}(a_{l})-\beta_{% n,i}^{j,t}(a_{l}))$ (58)

and

$\displaystyle F_{\Omega,i}^{jk,t}(a_{l})=G^{t}\cdot\frac{M^{pj,t}(a_{l})\cdot M% ^{ak,t}(a_{l})}{R^{jk,t}(a_{l})+\varepsilon}\cdot(\beta_{\Omega,i}^{k,t}(a_{l}% )-\beta_{\Omega,i}^{j,t}(a_{l})),$ (59)

where $R^{jk,t}(a_{l})=\|{A^{j,t}(a_{l}),A^{k,t}(a_{l})}\|_{2}$ represents the Euclidean distance between the $j$ th and $k$ th agents.

Assume that $K_{B}$ is the set of first $K$ agents with the biggest inertial mass. As $K$ is linearly decreased to 1 in the last iteration, it is set as the integer part of $(N_{S}-1)-(N_{S}-1)\cdot(t-1)/N_{t}$ . Then, the total force acting on the $j$ th agent from the agents in $K_{B}$ is calculated as

$\displaystyle F_{n,i}^{j,t}(a_{l})=\sum_{k\in K_{B},k\neq j}{r_{n,i}^{jk,t}(a_% {l})\cdot F_{n,i}^{jk,t}(a_{l})}\text{ and}$ (60) $\displaystyle F_{\Omega,i}^{jk,t}(a_{l})=\sum_{k\in K_{B},k\neq j}{r_{\Omega,i% }^{jk,t}(a_{l})\cdot F_{\Omega,i}^{jk,t}(a_{l})},$ (61)

where $r_{n,i}^{jk,t}(a_{l})$ and $r_{\Omega,i}^{jk,t}(a_{l})$ are random real numbers limited to [0, 1].

Step 5: Calculation of acceleration and velocity

The acceleration of the $j$ th agent in the $t$ th iteration is calculated as

$\displaystyle a_{n,i}^{j,t}(a_{l})=\frac{F_{n,i}^{j,t}(a_{l})}{M^{j,t}(a_{l})}% \text{ and}$ (62) $\displaystyle a_{\Omega,i}^{j,t}(a_{l})=\frac{F_{\Omega,i}^{j,t}(a_{l})}{M^{j,% t}(a_{l})}.$ (63)

Step 6: Update of the agent’s position

From the acceleration of the $j$ th agent, the velocity of the $j$ th agent is updated to be

$\displaystyle v_{n,i}^{j,t+1}(a_{l})=r_{n,i}^{j,t}(a_{l})\cdot v_{n,i}^{j,t}(a% _{l})+a_{n,i}^{j,t}(a_{l})\text{ and}$ (64) $\displaystyle v_{\Omega,i}^{j,t+1}(a_{l})=r_{\Omega,i}^{j,t}(a_{l})\cdot v_{% \Omega,i}^{j,t}(a_{l})+a_{\Omega,i}^{j,t}(a_{l}),$ (65)

where $r_{n,i}^{j,t}(a_{l})$ and $r_{\Omega,i}^{j,t}(a_{l})$ are random real numbers limited to $[0,1]$ .

The resulting velocity is then used to calculate the updated position of the $j$ th agent as

$\displaystyle\beta_{n,i}^{j,t+1}(a_{l})=\beta_{n,i}^{j,t}(a_{l})+v_{n,i}^{j,t+% 1}(a_{l})\text{ and}$ (66) $\displaystyle\beta_{\Omega,i}^{j,t+1}(a_{l})=\beta_{\Omega,i}^{j,t}(a_{l})+v_{% \Omega,i}^{j,t+1}(a_{l}).$ (67)

Through the same process as Step 3 of Section 3.3, the normalized position of the $j$ th agent is obtained, which is ensured to be within the feasible region of the optimization problems in Eqs (14)–(17).

Step 7: Termination

If $N_{t}$ iterations have been completed, the best position of each agent is obtained, which is used to calculate the fitness value of each agent and then generate the lower bound or upper bound of $\beta_{n}(a_{l})$ . Otherwise, go to Step 3.

4. Comparison for generating aggregated IVBDs

For a MCDM problem modeled by IVBDs, the aggregated IVBD of each alternative is usually used to analyze the problem. Although the final solution cannot be directly generated from the aggregated IVBDs of alternatives in most cases, beneficial analyses can be generally obtained. More importantly, the aggregated IVBDs of alternatives can be combined with the utilities of grades $u(H_{n})∼{}(n=1,\ldots,N)$ , to generate the minimum and maximum expected utilities, which are used to generate a solution to the problem with the help of a decision rule. In view of the importance of generating the aggregated IVBDs of alternatives, some problems with different numbers of criteria and grades are used to compare the four evolutionary algorithms presented in Section 3.

Table 1
The optimal $\beta_{1}^{-}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.0782, 0.555)	(0.0625, 2.891)	(0.0376, 0.801)	(0.0662, 20.822)
(0.3, 14, 7)	(0.0512, 0.735)	(0.0415, 5.765)	(0.0328, 1.439)	(0.0594, 38.282)
(0.3, 18, 9)	(0.0590, 0.987)	(0.0320, 9.674)	(0.0270, 2.230)	(0.0473, 60.450)
(0.3, 22, 11)	(0.0291, 1.268)	(0.0262, 15.459)	(0.0224, 3.133)	(0.0482, 87.444)
(0.3, 26, 13)	(0.0374, 1.520)	(0.0217, 31.423)	(0.0193, 4.333)	(0.0339, 121.479)
(0.3, 30, 15)	(0.0400, 1.897)	(0.0184, 52.084)	(0.0175, 5.574)	(0.0405, 159.405)
(0.8, 10, 5)	(0.1431, 0.531)	(0.1327, 2.938)	(0.1287, 0.903)	(0.1308, 21.081)
(0.8, 14, 7)	(0.1067, 0.722)	(0.1008, 5.812)	(0.0979, 1.513)	(0.1051, 37.769)
(0.8, 18, 9)	(0.0822, 0.978)	(0.0796, 9.740)	(0.0786, 2.232)	(0.0842, 60.748)
(0.8, 22, 11)	(0.0733, 1.238)	(0.0666, 15.399)	(0.0657, 3.254)	(0.0730, 87.954)
(0.8, 26, 13)	(0.0631, 1.568)	(0.0569, 24.340)	(0.0565, 4.325)	(0.0629, 121.029)
(0.8, 30, 15)	(0.0546, 1.815)	(0.0497, 55.450)	(0.0495, 5.520)	(0.0529, 160.301)

4.1 Aggregation comparison of the four evolutionary algorithms by using specified IVBDs

Suppose that all individual IVBDs are normalized in a MCDM problem, which makes Eqs (2.3)–(2.3) hold. There are two types of situations in which IVBDs are normalized. One is that

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))=1$

and

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))=1,$

$\displaystyle\left(\sum_{m=1}^{N}\beta_{m,i}^{-}(a_{l})+\beta_{\Omega,i}^{-}(a% _{l})\right)$ $\displaystyle+(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))=1$

and

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)$ $\displaystyle+(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))=1,$

while the other is that

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))>1,$ $\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))>1,$ $\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)$ $\displaystyle+(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))<1,\text{∼{}and}$ $\displaystyle\left(\sum_{m=1}^{N}\beta_{m,i}^{-}(a_{l})+\beta_{\Omega,i}^{-}(a% _{l})\right)$ $\displaystyle+(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))<1.$

The IVBDs in the first type of situation are regarded as specified and those in the second type of situation as general. In the following, specified IVBDs are first used as foundations to compare the accuracies [21] and efficiencies [14] of the four evolutionary algorithms presented in Section 3 for combining individual IVBDs.

We focus on the aggregation of specified IVBDs satisfying

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)$ $\displaystyle+(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))=1$

and

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{-}% (a_{l})\right)$ $\displaystyle+(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))=1.$

For simplicity, assume that

$\displaystyle[\beta_{n,i}^{-}(a_{l}),\beta_{n,i}^{+}(a_{l})]=[\beta_{\Omega,i}% ^{-}(a_{l}),\beta_{\Omega,i}^{+}(a_{l})]$ $\displaystyle=\left[\frac{x}{N+1},1-\frac{N\cdot x}{N+1}\right]$

with $0\leqslant x\leqslant 1$ , which indicates that

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))=N-(N-1)\cdot x>1$

and

$\displaystyle\left(\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{+}% (a_{l})\right)$ $\displaystyle-(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))=N-(N-1% )\cdot x>1.$

With this assumption, without loss of generality, four evolutionary algorithms are used to determine $\beta_{1}^{-}(a_{l})$ and $\beta_{1}^{+}(a_{l})$ by solving the pair of optimization problems shown in Eqs (14)–(17) with different values of $x$ , $L$ , and $N$ . The optimization results generated using the four algorithms can be directly used to compare their accuracies, and the solution time of the four algorithms can be used to compare their efficiencies.

To fairly compare the four algorithms, it is specified that $N_{G}=N_{D}=N_{P}=N_{S}=100$ and $N_{t}=400$ . Meanwhile, the values of $x$ are set as 0.3 and 0.8, the value of $N$ is changed from 5 to 15 with a step of 2, and the value of $L$ is set as $2\cdot N$ . The criteria weights can be determined by various methods, such as direct rating [59], point allocation [60], the eigenvector method [61], and the best-worst method [62]. For convenience, the criteria weights are set as $\{w_{i}=1/L,i=1,\ldots,L\}$ in the comparison of the four algorithms. Under the conditions, the generation of the optimal $\beta_{1}^{-}(a_{l})$ and $\beta_{1}^{+}(a_{l})$ from the four algorithms is implemented using C of VS. Net 2017 on a personal computer with an Intel Core (TM) Duo i7-6700 CPU running at 3.4 GHz with 16 GB of RAM and a 64-bits Windows 7 operating system. The optimal $\beta_{1}^{-}(a_{l})$ and the solution time using the four algorithms with 12 sets of $(x,L,N)$ are presented in Table 1; the optimal $\beta_{1}^{+}(a_{l})$ and the solution time using the four algorithms with 12 sets of $(x,L,N)$ are presented in Table 2. The cell (0.0782, 0.555) in Table 1 means that the optimal $\beta_{1}^{-}(a_{l})$ is obtained as 0.0782 by using GA with a solution time of 0.555 seconds on the condition that $(x,L,N)$ is set as (0.3, 10, 5). Other cells in Tables 1 and 2 can be similarly understood.

Table 2
The optimal $\beta_{1}^{+}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.3850, 0.504)	(0.4790, 2.999)	(0.8115, 0.877)	(0.4714, 24.298)
(0.3, 14, 7)	(0.2941, 0.774)	(0.3896, 5.786)	(0.8018, 1.480)	(0.3740, 38.316)
(0.3, 18, 9)	(0.2166, 0.979)	(0.2923, 9.584)	(0.7204, 2.239)	(0.3326, 60.755)
(0.3, 22, 11)	(0.2081, 1.223)	(0.2789, 15.188)	(0.7305, 3.210)	(0.3090, 87.262)
(0.3, 26, 13)	(0.2098, 1.566)	(0.3248, 32.706)	(0.6827, 4.340)	(0.2828, 121.762)
(0.3, 30, 15)	(0.1593, 1.890)	(0.3135, 45.742)	(0.5694, 5.483)	(0.2701, 160.274)
(0.8, 10, 5)	(0.2181, 0.525)	(0.2451, 3.033)	(0.3564, 0.872)	(0.3062, 21.235)
(0.8, 14, 7)	(0.1621, 0.746)	(0.1805, 5.750)	(0.3052, 1.450)	(0.2447, 38.103)
(0.8, 18, 9)	(0.1404, 0.994)	(0.1591, 9.789)	(0.2752, 2.258)	(0.2009, 60.821)
(0.8, 22, 11)	(0.1123, 1.236)	(0.1416, 15.172)	(0.2555, 3.216)	(0.1790, 88.943)
(0.8, 26, 13)	(0.0982, 1.553)	(0.1308, 23.824)	(0.2504, 4.281)	(0.1661, 120.884)
(0.8, 30, 15)	(0.0907, 1.884)	(0.1305, 52.407)	(0.2388, 5.463)	(0.1570, 157.929)

Suppose that the optimal values of $\beta_{1}^{-}(a_{l})$ and $\beta_{1}^{-}(a_{l})$ generated using the four evolutionary algorithms to combine specified IVBDs are represented by $\beta_{1}^{G-}(a_{l}),\beta_{1}^{D-}(a_{l}),\beta_{1}^{P-}(a_{l}),\beta_{1}^{S% -}(a_{l}),\beta_{1}^{G+}(a_{l}),\beta_{1}^{D+}$ $(a_{l}),\beta_{1}^{P+}(a_{l})$ , and $\beta_{1}^{S+}(a_{l})$ . Table 1 shows that $\beta_{1}^{P-}$ $(a_{l})<\max\{\beta_{1}^{G-}(a_{l}),\beta_{1}^{G-}(a_{l}),\beta_{1}^{S-}(a_{l})\}$ always holds and $\beta_{1}^{D-}(a_{l})<\beta_{1}^{S-}(a_{l})<\beta_{1}^{G-}(a_{l})$ holds in most cases. Given this fact, $A_{P}>A_{D}>A_{S}>A_{G}$ , where $A_{P},A_{G},A_{D}$ , and $A_{S}$ represent the accuracies of PSO, GA, DE, and GSA for generating the optimal $\beta_{1}^{-}(a_{l})$ . On the other hand, Table 1 also shows that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , where $T_{G}$ , $T_{P}$ , $T_{D}$ , and $T_{S}$ represent the solution time for generating the optimal $\beta_{1}^{-}(a_{l})$ using the four algorithms (the notation ‘ $\ll$ ’ denotes ‘is greatly less than’). This indicates that $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ , where $E_{G},E_{P},E_{D}$ , and $E_{S}$ represent the efficiencies of the four algorithms for generating the optimal $\beta_{1}^{-}(a_{l})$ (the notation ‘ $\gg$ ’ denotes ‘is greatly better than’). In considering the accuracy and efficiency results, PSO algorithm is the best for generating the optimal $\beta_{1}^{-}(a_{l})$ among the four algorithms.

As for the optimal $\beta_{1}^{+}(a_{l})$ , Table 2 shows that when $x=0.3$ , $\beta_{1}^{P+}(a_{l})$ is clearly larger than $\beta_{1}^{G+}(a_{l})$ , $\beta_{1}^{D+}(a_{l})$ , and $\beta_{1}^{D+}(a_{l})$ , and $\beta_{1}^{D+}(a_{l})>\beta_{1}^{S+}(a_{l})>\beta_{1}^{G+}(a_{l})$ holds in most cases. This indicates that $A_{P}>A_{D}>A_{S}>A_{G}$ . The solution time for generating the optimal $\beta_{1}^{+}(a_{l})$ using the four algorithms also satisfies that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , which means that $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ . When $x=$ 0.8, the relationship between the optimal values of $\beta_{1}^{+}(a_{l})$ generated using the four algorithms changes to $\beta_{1}^{P+}(a_{l})>\beta_{1}^{S+}(a_{l})>\beta_{1}^{D+}(a_{l})>\beta_{1}^{G% +}(a_{l})$ , which means that $A_{P}>A_{S}>A_{D}>A_{G}$ . The relationship between the efficiencies of the four algorithms is the same as that in the situation where $x=0.3$ . To focus on a balance between accuracy and efficiency, PSO algorithm is the best among the four algorithms for generating the optimal $\beta_{1}^{+}(a_{l})$ . As a whole, the above experiments indicate that PSO algorithm is more suitable to combine specified IVBDs than the other three algorithms.

Table 3

The optimal $\beta_{\Omega}^{-}(a_{l})$ and the solution time using the four algorithms to combine general IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.0701, 0.558)	(0.0700, 2.928)	(0.0678, 0.885)	(0.0701, 21.005)
(0.3, 14, 7)	(0.0536, 0.813)	(0.0536, 6.107)	(0.0531, 1.598)	(0.0537, 38.347)
(0.3, 18, 9)	(0.0434, 1.055)	(0.0434, 9.704)	(0.0434, 2.418)	(0.0434, 60.438)
(0.3, 22, 11)	(0.0364, 1.322)	(0.0364, 15.297)	(0.0364, 3.351)	(0.0364, 86.852)
(0.3, 26, 13)	(0.0313, 1.656)	(0.0313, 24.618)	(0.0313, 4.522)	(0.0313, 119.780)
(0.3, 30, 15)	(0.0275, 2.040)	(0.0275, 55.714)	(0.0275, 5.893)	(0.0275, 161.294)
(0.8, 10, 5)	(0.2016, 0.535)	(0.2016, 2.879)	(0.2007, 0.926)	(0.2015, 20.944)
(0.8, 14, 7)	(0.1510, 0.795)	(0.1509, 5.607)	(0.1505, 1.570)	(0.1509, 37.760)
(0.8, 18, 9)	(0.1206, 1.055)	(0.1206, 9.783)	(0.1206, 2.378)	(0.1206, 59.566)
(0.8, 22, 11)	(0.1004, 1.324)	(0.1004, 15.172)	(0.1004, 3.281)	(0.1005, 86.264)
(0.8, 26, 13)	(0.0860, 1.696)	(0.0860, 32.307)	(0.0860, 4.556)	(0.0860, 119.355)
(0.8, 30, 15)	(0.0753, 2.025)	(0.0753, 59.966)	(0.0753, 5.882)	(0.0753, 157.933)

Table 4

The optimal $\beta_{\Omega}^{+}(a_{l})$ and the solution time using the four algorithms to combine general IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.1040, 0.554)	(0.3582, 2.907)	(0.465, 0.886)	(0.1003, 20.895)
(0.3, 14, 7)	(0.0668, 0.791)	(0.2703, 5.799)	(0.3426, 1.570)	(0.0701, 38.243)
(0.3, 18, 9)	(0.0603, 1.027)	(0.2177, 9.751)	(0.2711, 2.368)	(0.0532, 59.594)
(0.3, 22, 11)	(0.0456, 1.333)	(0.1895, 15.388)	(0.2228, 3.334)	(0.0437, 86.064)
(0.3, 26, 13)	(0.0370, 1.682)	(0.1618, 25.732)	(0.1844, 4.489)	(0.0372, 119.700)
(0.3, 30, 15)	(0.0382, 2.015)	(0.1505, 59.092)	(0.1653, 5.793)	(0.0322, 159.731)
(0.8, 10, 5)	(0.2188, 0.524)	(0.2821, 2.916)	(0.3166, 0.899)	(0.2336, 21.023)
(0.8, 14, 7)	(0.1606, 0.815)	(0.2123, 5.704)	(0.2343, 1.57)	(0.1673, 37.875)
(0.8, 18, 9)	(0.1259, 1.025)	(0.1747, 9.826)	(0.1861, 2.371)	(0.1315, 59.97)
(0.8, 22, 11)	(0.1046, 1.304)	(0.1457, 15.341)	(0.1544, 3.344)	(0.1077, 86.177)
(0.8, 26, 13)	(0.0881, 1.666)	(0.1243, 22.675)	(0.1316, 4.469)	(0.0912, 118.923)
(0.8, 30, 15)	(0.0785, 2.028)	(0.1115, 58.346)	(0.1135, 5.811)	(0.0797, 158.347)

4.2 Aggregation comparison of the four evolutionary algorithms using general IVBDs

For a MCDM problem with general IVBDs, suppose that

$\displaystyle[\beta_{1}^{+}(a_{l}),\beta_{n,i}^{+}(a_{l})]=\bigg{[}\frac{0.5% \cdot x}{N+1}+\frac{(n-1)\cdot x}{N\cdot(N+1)},$ $\displaystyle\frac{0.5\cdot(2-x)}{N+1}+\frac{(n-1)\cdot(2-x)}{N\cdot(N+1)}% \bigg{]}$

and

$\displaystyle[\beta_{\Omega,i}^{-}(a_{l}),\beta_{\Omega,i}^{+}(a_{l})]=\bigg{[% }\frac{1.5\cdot x}{N+1},\frac{1.5\cdot(2-x)}{N+1}\bigg{]}$

with $0\leqslant x\leqslant 1$ . In this situation it can be obtained that

$\displaystyle\Bigg{(}\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{% -}(a_{l})\Bigg{)}$ $\displaystyle+(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))$ $\displaystyle=x+\frac{3N-2}{N(N+1)}\cdot(1-x)<1,$ $\displaystyle\Bigg{(}\sum_{m=1}^{N}{\beta_{m,i}^{-}(a_{l})}+\beta_{\Omega,i}^{% -}(a_{l})\Bigg{)}$ $\displaystyle+(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))$ $\displaystyle=x+\frac{3\cdot(1-x)}{N+1}<1,$ $\displaystyle\Bigg{(}\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{% +}(a_{l})\Bigg{)}$ $\displaystyle-(\beta_{n,i}^{+}(a_{l})-\beta_{n,i}^{-}(a_{l}))$ $\displaystyle=(2-x)-\frac{3N-2}{N(N+1)}\cdot(1-x)>1,$

and

$\displaystyle\Bigg{(}\sum_{m=1}^{N}{\beta_{m,i}^{+}(a_{l})}+\beta_{\Omega,i}^{% +}(a_{l})\Bigg{)}$ $\displaystyle-(\beta_{\Omega,i}^{+}(a_{l})-\beta_{\Omega,i}^{-}(a_{l}))$ $\displaystyle=(2-x)+\frac{3\cdot(1-x)}{N+1}>1.$

On this assumption, normalized IVBDs are combined using the pair of optimization problems shown in Eqs (14)–(17). With 12 sets of $(x,L,N)$ , the optimal values of $\beta_{\Omega}^{-}(a_{l})$ and $\beta_{\Omega}^{+}(a_{l})$ are obtained using the four evolutionary algorithms. The results and the corresponding solution time are presented in Tables 3 and 4, respectively.

Table 5
The optimal $u^{-}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from specified IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.3244, 0.528)	(0.3077, 2.862)	(0.0994, 0.880)	(0.2452, 20.994)
(0.3, 14, 7)	(0.3376, 0.777)	(0.3118, 5.638)	(0.0991, 1.499)	(0.2567, 38.204)
(0.3, 18, 9)	(0.3418, 1.007)	(0.3305, 9.812)	(0.1378, 2.236)	(0.2401, 59.379)
(0.3, 22, 11)	(0.3378, 1.289)	(0.3321, 15.137)	(0.1567, 3.176)	(0.2324, 86.494)
(0.3, 26, 13)	(0.3645, 1.558)	(0.3292, 22.405)	(0.1735, 4.232)	(0.2242, 118.134)
(0.3, 30, 15)	(0.3878, 1.911)	(0.3073, 58.054)	(0.1784, 5.468)	(0.2089, 157.374)
(0.8, 10, 5)	(0.3926, 0.539)	(0.3755, 2.933)	(0.3232, 0.864)	(0.3502, 20.970)
(0.8, 14, 7)	(0.4129, 0.768)	(0.4027, 5.661)	(0.3469, 1.502)	(0.3725, 38.287)
(0.8, 18, 9)	(0.4240, 0.981)	(0.4151, 9.722)	(0.3586, 2.196)	(0.3818, 59.199)
(0.8, 22, 11)	(0.4286, 1.287)	(0.4221, 15.103)	(0.3779, 3.172)	(0.3883, 86.891)
(0.8, 26, 13)	(0.4452, 1.526)	(0.425, 22.214)	(0.3829, 4.236)	(0.3903, 119.757)
(0.8, 30, 15)	(0.4463, 1.920)	(0.4192, 56.365)	(0.3877, 5.705)	(0.3906, 156.803)

Table 6

The optimal $u^{+}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from specified IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.5966, 0.541)	(0.6597, 2.870)	(0.8239, 0.851)	(0.6074, 21.007)
(0.3, 14, 7)	(0.5655, 0.753)	(0.6215, 5.764)	(0.8369, 1.427)	(0.5360, 37.884)
(0.3, 18, 9)	(0.5565, 0.989)	(0.6159, 9.657)	(0.7688, 2.098)	(0.5257, 59.638)
(0.3, 22, 11)	(0.5635, 1.250)	(0.6033, 15.249)	(0.8156, 2.989)	(0.5677, 86.497)
(0.3, 26, 13)	(0.5499, 1.553)	(0.6110, 22.341)	(0.7716, 4.112)	(0.4957, 118.346)
(0.3, 30, 15)	(0.5408, 1.916)	(0.6492, 60.179)	(0.7735, 5.294)	(0.5161, 156.776)
(0.8, 10, 5)	(0.4591, 0.558)	(0.4809, 2.949)	(0.5371, 0.844)	(0.4705, 21.176)
(0.8, 14, 7)	(0.4617, 0.762)	(0.4875, 5.611)	(0.5413, 1.496)	(0.4659, 38.006)
(0.8, 18, 9)	(0.4780, 0.986)	(0.4951, 9.591)	(0.5462, 2.133)	(0.4770, 59.343)
(0.8, 22, 11)	(0.4788, 1.290)	(0.5041, 15.336)	(0.5423, 3.138)	(0.4799, 86.089)
(0.8, 26, 13)	(0.4769, 1.547)	(0.5064, 22.220)	(0.5505, 4.135)	(0.4788, 118.085)
(0.8, 30, 15)	(0.4824, 1.925)	(0.5193, 59.399)	(0.5554, 5.275)	(0.4818, 160.296)

Suppose that the optimal values of $\beta_{\Omega}^{-}(a_{l})$ and $\beta_{\Omega}^{+}(a_{l})$ generated using the four evolutionary algorithms to combine general IVBDs are represented by $\beta_{\Omega}^{G-}(a_{l}),\beta_{\Omega}^{D-}(a_{l}),\beta_{\Omega}^{P-}(a_{l% }),\beta_{\Omega}^{S-}(a_{l}),\beta_{\Omega}^{G+}(a_{l}),\beta_{\Omega}^{D+}$ $(a_{l}),\beta_{\Omega}^{P+}(a_{l})$ , and $\beta_{\Omega}^{S+}(a_{l})$ . Table 3 shows that $\beta_{\Omega}^{P-}$ $(a_{l})\leqslant\max\{\beta_{\Omega}^{G-}(a_{l}),\beta_{\Omega}^{D-}(a_{l}),% \beta_{\Omega}^{S-}(a_{l})\}$ always holds and $\beta_{\Omega}^{G-}(a_{l}),\beta_{\Omega}^{D-}(a_{l})$ , and $\beta_{\Omega}^{S-}(a_{l})$ are very close to each other. This fact indicates that $A_{P}>A_{D}\approx A_{S}\approx A_{G}$ (the notation ‘ $\approx$ ’ denotes ‘is almost equal to’). Table 3 shows that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , which reveals that $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ . With a balanced consideration of accuracy and efficiency, PSO algorithm is most suitable among the four algorithms to generate the optimal $\beta_{\Omega}^{-}(a_{l})$ .

From the observations shown in Table 4, $\beta_{\Omega}^{P+}(a_{l})>\beta_{\Omega}^{D+}(a_{l})>\max\{\beta_{\Omega}^{S+% }(a_{l}),\beta_{\Omega}^{G+}(a_{l})\}$ always holds and $\beta_{\Omega}^{S+}(a_{l})>\beta_{\Omega}^{G+}(a_{l})$ holds in most cases. This indicates that $A_{P}>A_{D}>A_{S}>A_{G}$ . Regarding solution time, Table 4 shows that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , and further, $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ . Similar to obtaining the optimal $\beta_{\Omega}^{-}(a_{l})$ , PSO algorithm remains the best among the four algorithms for generating the optimal $\beta_{\Omega}^{+}(a_{l})$ from the perspectives of accuracy and efficiency. As a result, PSO algorithm is the most applicable among the four algorithms for combining general IVBDs.

5. Comparison for generating expected utilities

When the aggregated IVBDs of alternatives are not needed, the optimal expected utilities of alternatives are required to generate a solution to an MCDM problem with the aid of a decision rule. In this situation, we compare the accuracies and efficiencies of the four evolutionary algorithms for generating the optimal expected utilities using specified and general IVBDs.

5.1 Utility comparison of the four evolutionary algorithms using specified IVBDs

In the same situation as specified in Section 4.1, GA, DE, PSO, and GSA are used to solve the optimization model shown in Eqs (22)–(25) to find the optimal $u^{-}(a_{l})$ . When the objective of the model is changed to “MAX $\sum_{n=1}^{N-1}{\beta_{n}(a_{l})u(H_{n})}+(\beta_{N}(a_{l})+\beta_{\Omega}(a_% {l}))\cdot u(H_{N})$ ”, the optimal $u^{+}(a_{l})$ can be found by solving the model. The relevant optimal results are shown in Tables 5 and 6. The cell $(0.3244,0.528)$ in Table 5 means that the optimal $u^{-}(a_{l})$ is obtained as 0.3244 by using GA with the solution time of 0.528 seconds on the condition that $(x,L,N)$ is set as $(0.3,10,5)$ . Other cells in Tables 5 and 6 can be similarly understood.

Table 7
The optimal $u^{-}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from general IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.5354, 0.531)	(0.4176, 2.898)	(0.2081, 0.901)	(0.4590, 20.960)
(0.3, 14, 7)	(0.5408, 0.817)	(0.4288, 5.725)	(0.2959, 1.554)	(0.4911, 37.531)
(0.3, 18, 9)	(0.5486, 1.042)	(0.4377, 9.655)	(0.3355, 2.346)	(0.5101, 59.419)
(0.3, 22, 11)	(0.5488, 1.350)	(0.4643, 15.380)	(0.3831, 3.328)	(0.5226, 86.591)
(0.3, 26, 13)	(0.5604, 1.668)	(0.4800, 22.574)	(0.3964, 4.428)	(0.5315, 118.212)
(0.3, 30, 15)	(0.5612, 2.014)	(0.488, 57.761)	(0.3970, 5.752)	(0.5381, 157.928)
(0.8, 10, 5)	(0.4827, 0.543)	(0.4378, 3.010)	(0.3870, 0.904)	(0.4590, 21.191)
(0.8, 14, 7)	(0.5069, 0.804)	(0.4723, 5.714)	(0.4237, 1.550)	(0.4911, 37.530)
(0.8, 18, 9)	(0.5229, 1.037)	(0.4871, 9.793)	(0.4514, 2.342)	(0.5101, 59.716)
(0.8, 22, 11)	(0.5328, 1.383)	(0.5060, 15.460)	(0.4750, 3.338)	(0.5226, 86.399)
(0.8, 26, 13)	(0.5400, 1.669)	(0.5166, 22.634)	(0.4910, 4.442)	(0.5315, 118.550)
(0.8, 30, 15)	(0.5448, 2.032)	(0.5234, 51.735)	(0.4967, 5.769)	(0.5381, 157.839)

Table 8

The optimal $u^{+}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from general IVBDs with 12 sets of $(x,L,N)$

Parameters	GA	DE	PSO	GSA
(0.3, 10, 5)	(0.6091, 0.576)	(0.6222, 2.935)	(0.7011, 0.893)	(0.6754, 20.848)
(0.3, 14, 7)	(0.6052, 0.782)	(0.6103, 5.646)	(0.7181, 1.534)	(0.6453, 37.990)
(0.3, 18, 9)	(0.5938, 1.040)	(0.6146, 9.728)	(0.7034, 2.347)	(0.6306, 59.004)
(0.3, 22, 11)	(0.5954, 1.333)	(0.6074, 15.511)	(0.6916, 3.332)	(0.6207, 86.464)
(0.3, 26, 13)	(0.5913, 1.660)	(0.6032, 22.362)	(0.6768, 4.532)	(0.6139, 118)
(0.3, 30, 15)	(0.5965, 2.081)	(0.6033, 57.356)	(0.6705, 5.811)	(0.6079, 156.517)
(0.8, 10, 5)	(0.5041, 0.544)	(0.5083, 2.879)	(0.5351, 0.953)	(0.5338, 20.929)
(0.8, 14, 7)	(0.5253, 0.779)	(0.5266, 5.737)	(0.5558, 1.561)	(0.5476, 37.757)
(0.8, 18, 9)	(0.5345, 1.047)	(0.5410, 9.747)	(0.5659, 2.325)	(0.5552, 59.468)
(0.8, 22, 11)	(0.5438, 1.340)	(0.5456, 15.238)	(0.5658, 3.369)	(0.5598, 86.487)
(0.8, 26, 13)	(0.5489, 1.701)	(0.5523, 22.778)	(0.5779, 4.450)	(0.5629, 118.599)
(0.8, 30, 15)	(0.5547, 2.009)	(0.5580, 57.085)	(0.5747, 5.734)	(0.5651, 156.441)

Suppose that the optimal $u^{-}(a_{l})$ and $u^{+}(a_{l})$ derived from GA, DE, PSO, and GSA are represented by $u_{G}^{-}(a_{l}),u_{D}^{-}(a_{l}),u_{P}^{-}(a_{l}),u_{S}^{-}(a_{l}),u_{G}^{+}(% a_{l}),u_{D}^{+}(a_{l}),u_{P}^{+}$ $(a_{l})$ , and $u_{S}^{+}(a_{l})$ . Table 5 shows that $u_{P}^{-}(a_{l})<u_{S}^{-}(a_{l})<u_{D}^{-}(a_{l})<u_{G}^{-}(a_{l})$ , which indicates that $A_{P}>A_{S}>A_{D}>A_{G}$ . The efficiencies of the four algorithms satisfy $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ in accordance with the fact that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ as shown in Table 5. By comprehensively comparing the accuracies and efficiencies of the four algorithms, PSO algorithm is considered the best algorithm for generating the optimal $u^{-}(a_{l})$ .

Table 6 shows that $u_{P}^{+}(a_{l})\!>\!u_{D}^{+}(a_{l})\!>\!\max\{u_{G}^{+}(a_{l})$ , $u_{S}^{+}(a_{l})\}$ always holds and $u_{G}^{+}(a_{l})>u_{S}^{+}(a_{l})$ holds in most cases, which means that $A_{P}>A_{D}>A_{G}>A_{S}$ . Table 6 also shows that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , which means that $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ . A preferable balance between accuracy and efficiency indicates that PSO algorithm is the most suitable to generate the optimal $u^{+}(a_{l})$ , compared to the other three algorithms. As a whole, PSO algorithm is the best choice among the four algorithms to generate the expected utilities from specified IVBDs.

5.2 Utility comparison of the four evolutionary algorithms using general IVBDs

In the same situation as specified in Section 4.2, experiments similar to those in Section 5.1 are performed to generate the optimal $u^{-}(a_{l})$ and $u^{+}(a_{l})$ , which are shown in Tables 7 and 8.

Table 9
The optimal $\beta_{1}^{+}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with $(x,L,N)=(0.3,30,15)$

Parameters	GA	DE	PSO	GSA
(250, 25)	(0.1408, 0.354)	(0.1734, 5.383)	(0.581, 0.964)	(0.2978, 6.808)
(300, 50)	(0.1452, 0.816)	(0.2101, 12.744)	(0.4881, 2.137)	(0.2628, 30.54)
(350, 75)	(0.146, 1.371)	(0.2321, 22.348)	(0.6464, 3.662)	(0.2278, 78.957)
(400, 100)	(0.1593, 1.89)	(0.3135, 45.742)	(0.5694, 5.483)	(0.2701, 160.274)
(450, 125)	(0.1523, 2.852)	(0.2831, 49.215)	(0.6206, 7.84)	(0.3777, 278.827)
(500, 150)	(0.1792, 3.765)	(0.32, 82.107)	(0.7191, 10.375)	(0.3437, 447.453)
(550, 175)	(0.1775, 4.819)	(0.3357, 113.646)	(0.7422, 13.228)	(0.3172, 666.885)
(600, 200)	(0.1885, 6.02)	(0.3746, 164.941)	(0.7422, 16.543)	(0.2981, 951.078)
(650, 225)	(0.1786, 7.287)	(0.3715, 199.959)	(0.7869, 20.198)	(0.2864, 1317.75)
(700, 250)	(0.1704, 8.759)	(0.4025, 278.158)	(0.7191, 24.925)	(0.2881, 1727.87)
(750, 275)	(0.1725, 10.197)	(0.4409, 396.131)	(0.7191, 28.207)	(0.2683, 2253.34)
(800, 300)	(0.1807, 11.861)	(0.4247, 551.442)	(0.7422, 32.563)	(0.275, 2832.62)

Table 10

The optimal $u^{-}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from general IVBDs with $(x,L,N)=(0.3,30,15)$

Parameters	GA	DE	PSO	GSA
(250, 25)	(0.5664, 0.402)	(0.5231, 5.473)	(0.4322, 1.033)	(0.5381, 6.852)
(300, 50)	(0.5626, 0.889)	(0.5075, 13.066)	(0.4258, 2.293)	(0.5381, 30.726)
(350, 75)	(0.5627, 1.523)	(0.4930, 22.723)	(0.4167, 3.969)	(0.5381, 78.735)
(400, 100)	(0.5612, 2.014)	(0.4880, 57.761)	(0.3970, 5.752)	(0.5381, 157.928)
(450, 125)	(0.5627, 3.030)	(0.4867, 48.366)	(0.4133, 8.104)	(0.5381, 278.903)
(500, 150)	(0.5572, 3.965)	(0.4818, 80.606)	(0.4056, 10.898)	(0.5381, 444.781)
(550, 175)	(0.5594, 5.031)	(0.4794, 120.798)	(0.4109, 13.834)	(0.5381, 665.554)
(600, 200)	(0.5583, 6.287)	(0.4773, 175.615)	(0.4109, 17.338)	(0.5381, 951.278)
(650, 225)	(0.5615, 7.585)	(0.4762, 225.757)	(0.3876, 20.861)	(0.5381, 1294.31)
(700, 250)	(0.5601, 9.049)	(0.4743, 315.150)	(0.4044, 24.870)	(0.5381, 1723.89)
(750, 275)	(0.5583, 10.553)	(0.4746, 346.506)	(0.4088, 29.440)	(0.5381, 2232.51)
(800, 300)	(0.5552, 12.278)	(0.4715, 399.129)	(0.4153, 34.104)	(0.5381, 2852.88)

Table 7 shows that $u_{P}^{-}(a_{l})<u_{D}^{-}(a_{l})<u_{S}^{-}(a_{l})<u_{G}^{-}(a_{l})$ , which means that $A_{P}>$ $A_{D}>A_{S}>A_{G}$ . Additionally, Table 7 shows that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , which means that $E_{G}>E_{P}$ $\gg E_{D}\gg E_{S}$ . The accuracies and efficiencies of the four algorithms show that PSO algorithm is best suited for generating the optimal $u^{-}(a_{l})$ .

The conclusion of $A_{P}>A_{S}>A_{D}>A_{G}$ is drawn from the observation of $u_{P}^{+}(a_{l})>u_{S}^{+}(a_{l})>u_{D}^{+}(a_{l})>u_{G}^{+}(a_{l})$ shown in Table 8. The relationship between the efficiencies of the four algorithms is the same as that for optimizing $u^{-}(a_{l})$ . To reach a rational balance between accuracy and efficiency, PSO algorithm should be selected to generate the optimal $u^{+}(a_{l})$ . Considering what has been analyzed from Tables 7 and 8, we conclude that PSO algorithm should be considered the best among the four algorithms for generating the expected utilities from general IVBDs.

6. Sensitivity analysis

In Sections 4 and 5, experiments to compare the accuracies and efficiencies of the four evolutionary algorithms are conducted on the condition that $N_{G}=N_{D}=N_{P}=N_{S}=100$ and $N_{t}=400$ . An interesting area of investigation is finding the changes in the accuracies and efficiencies of the four algorithms with the variation in $N_{t}$ and $N_{r}$ on the assumption that $N_{r}=N_{G}=N_{D}=N_{P}=N_{S}$ .

To investigate this, when $(x,L,N)=(0.3,30,15)$ , we reperform the experiments to generate the optimal $\beta_{1}^{+}(a_{l})$ and $u^{-}(a_{l})$ using the four algorithms to combine specified IVBDs and generate the expected utilities from general IVBDs, respectively. In the experiments, $N_{t}$ is changed from 250 to 800 with a step size of 50 and $N_{r}$ from 25 to 300 with a step size of 25. The relevant results are shown in Tables 9 and 10. The cell (0.1408, 0.354) in Table 9 means that the optimal $\beta_{1}^{+}(a_{l})$ is obtained as 0.1408 using GA with the solution time of 0.354 seconds on the condition that $(x,L,N)$ is set as $(0.3,30,15)$ . Other cells in Tables 9 and 10 can be similarly understood.

To facilitate observing the movement of the optimal $\beta_{1}^{+}(a_{l})$ and $u^{-}(a_{l})$ with the variation in $N_{t}$ and $N_{r}$ , the relevant data shown in Tables 9 and 10 are plotted in Figs 1 and 2.

Figure 1.

Movement of the optimal $\beta_{1}^{+}(a_{l})$ with the variation in $N_{t}$ and $N_{r}$ .

Figure 2.

Movement of the optimal $u^{-}(a_{l})$ with the variation in $N_{t}$ and $N_{r}$ .

Figure 1 shows that with the increase in $N_{t}$ and $N_{r}$ , $\beta_{1}^{G+}(a_{l})$ and $\beta_{1}^{D+}(a_{l})$ increase slightly with small fluctuations, $\beta_{1}^{P+}(a_{l})$ increases significantly with fluctuations, and $\beta_{1}^{S+}(a_{l})$ fluctuates irregularly. Figure 2 shows that with the increase in $N_{t}$ and $N_{r}$ , $u_{G}^{-}(a_{l})$ , $u_{D}^{-}(a_{l})$ , and $u_{P}^{-}(a_{l})$ decrease with small fluctuations, and $u_{S}^{-}(a_{l})$ remains unchanged. These observations reveal that the increase in $N_{t}$ and $N_{r}$ cannot significantly improve the accuracies of GA, DE, and GSA for generating the optimal $\beta_{1}^{+}(a_{l})$ and $u^{-}(a_{l})$ . The same conclusion can be drawn when PSO algorithm is used to generate $u^{-}(a_{l})$ . In the situation in which the optimal $\beta_{1}^{+}(a_{l})$ is generated using PSO algorithm, the opposite conclusion is drawn.

As for the efficiencies of the four algorithms, Tables 9 and 10 show that $T_{G}<T_{P}\ll T_{D}\ll T_{S}$ , which means that $E_{G}>E_{P}\gg E_{D}\gg E_{S}$ . By comparing the accuracies and efficiencies of the four algorithms with the variation in $N_{t}$ and $N_{r}$ , PSO algorithm is considered the best among the four algorithms for combining IVBDs and generating the expected utilities.

7. Conclusions

In the process of solving MCDM problems with IVBDs, individual IVBDs are explicitly combined to generate an aggregated IVBD or implicitly combined in the optimization model of generating the minimum and maximum expected utilities. For an MCDM problem with a large number of criteria and grades, implementing the accurate combination of IVBDs within an acceptable time is difficult. Evolutionary algorithms provide effective methods for overcoming this difficulty. When these algorithms are selected to combine IVBDs or generate the expected utilities by implicitly combining IVBDs, a new issue occurs: which algorithm is suitable for accurately combining IVBDs within an acceptable time? To address this issue, four representative evolutionary algorithms with many extensions and applications, including GA, DE algorithm, PSO algorithm, and GSA, are selected to explicitly and implicitly combine IVBDs. By performing experiments on combining IVBDs and generating the expected utilities, a comparative analysis of the four algorithms is provided with the help of two indicators: accuracy and efficiency. The analysis indicates that PSO algorithm is more suitable than the other three algorithms to combine IVBDs and generate the expected utilities. This conclusion is highlighted by a sensitivity analysis of the four algorithms’ accuracies and efficiencies.

In this paper, we only compare the original versions of the four evolutionary algorithms for analyzing MCDM problems with IVBDs. Whether the extensions of the four algorithms can improve their accuracies and efficiencies for combining IVBDs deserves consideration, and will be investigated in the future. In addition, for MCDM problems with other types of uncertain preferences, such as linguistic term set, intuitionistic fuzzy set, hesitant fuzzy set, and interval type-2 fuzzy set, it would also be interesting to compare the accuracies and efficiencies of the four algorithms for analyzing the problems.

Footnotes

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant Nos. 71622003, 71571060).

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Comparative analysis of evolutionary algorithms for multiple criteria decision making with interval-valued belief distributions

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Modeling of MCDM problems using belief distributions

2.2 Combination of belief distributions

.

3.1 GA process for combining individual IVBDs

Step 1: Initialization

Step 2: Performance evaluation

Step 3: Selection

Step 4: Crossover

Step 5: Mutation

Step 6: Update

Step 7: Termination

Step 1: Initialization

Step 2: Performance evaluation

Step 3: Mutation

Step 4: Crossover

Step 5: Selection

Step 6: Update

Step 7: Termination

Step 1: Initialization

Step 2: Performance evaluation

Step 3: Position update

Step 4: Best position update

Step 5: Termination

Step 1: Initialization

Step 2: Performance evaluation

Step 3: Update of gravitational coefficient and inertial mass

Step 4: Calculation of the total force acting on the j th agent

Step 5: Calculation of acceleration and velocity

Step 6: Update of the agent’s position

Step 7: Termination

Table 1 The optimal β 1 - ⁢ ( a l ) and the solution time using the four algorithms to combine specified IVBDs with 12 sets of ( x , L , N )

Table 2 The optimal β 1 + ⁢ ( a l ) and the solution time using the four algorithms to combine specified IVBDs with 12 sets of ( x , L , N )

Table 5 The optimal u - ⁢ ( a l ) and the solution time using the four algorithms to generate the expected utilities from specified IVBDs with 12 sets of ( x , L , N )

5.1 Utility comparison of the four evolutionary algorithms using specified IVBDs

Table 7 The optimal u - ⁢ ( a l ) and the solution time using the four algorithms to generate the expected utilities from general IVBDs with 12 sets of ( x , L , N )

Table 9 The optimal β 1 + ⁢ ( a l ) and the solution time using the four algorithms to combine specified IVBDs with ( x , L , N ) = ( 0.3 , 30 , 15 )

Footnotes

Acknowledgments

References

Step 4: Calculation of the total force acting on the $j$ th agent

Table 1
The optimal $\beta_{1}^{-}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with 12 sets of $(x,L,N)$

Table 2
The optimal $\beta_{1}^{+}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with 12 sets of $(x,L,N)$

Table 5
The optimal $u^{-}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from specified IVBDs with 12 sets of $(x,L,N)$

Table 7
The optimal $u^{-}(a_{l})$ and the solution time using the four algorithms to generate the expected utilities from general IVBDs with 12 sets of $(x,L,N)$

Table 9
The optimal $\beta_{1}^{+}(a_{l})$ and the solution time using the four algorithms to combine specified IVBDs with $(x,L,N)=(0.3,30,15)$