A multi-criteria decision-making with regret theory-based MULTIMOORA method under interval neutrosophic environment

Abstract

The MULTIMOORA (multiple multi-objective optimization by ratio analysis) method is useful for multiple criteria decision-making method. It is based on expected utility theory and assumes that decision makers are completely rational. However, some studies show that human beings are usually bounded rational, and their regret aversion behaviors play an important role in the decision-making process. Interval neutrosophic sets can more flexibly depict uncertain, incomplete and inconsistent information than single-valued neutrosophic sets. Therefore, this paper improves the traditional MULTIMOORA method by combining the regret theory under interval neutrosophic sets. Firstly, the regret theory is used to calculate the utility value and regret-rejoice value of each alternatives. Secondly, the criteria weights optimization model based on the maximizing deviation is constructed to obtain the weight vector. Then, the MULTIMOORA method is used to determine the order of the alternatives. Finally, an illustrative example about school selection is provided to demonstrate the feasibility of the proposed method. Sensitivity analysis shows the validity of the regret theory in the proposed method, and the ranking order change with different regret avoidance parameter. Comparisons are made with existing approaches to illustrate the advantage of the proposed method in reflecting decision makers’ psychological preference.

Keywords

Interval neutrosophic set regret theory multiple criteria decision making MULTIMOORA

1 Introduction

Multi-criteria decision-making (MCDM) refers to the use of a certain method to gather known decision-making information and to sort and select the best alternative in a certain way while considering multiple criteria. It has a wide range of applications in the fields of politics [1], economy [2], management [3] and engineering [4].

Due to the fuzziness and limitations from decision maker’s (DM) and the complexity of the problems, majority information of the decision-making in real situation is fuzzy and cannot be completely determined. Therefore, decision-making information cannot be directly expressed by crisp set. To solve this problem, Zadeh [5] proposed the fuzzy set theory which can process such kind of information effectively. But the fuzzy set only has the truth-membership and cannot express falsity-membership. Therefore, Atanassov [6, 7] proposed the intuitionistic fuzzy set by adding a falsity-membership function into fuzzy set. Intuitionistic fuzzy set is easier to express fuzzy information than fuzzy set, so it has been widely used in MCDM problems since it was proposed. However, intuitionistic fuzzy sets cannot deal with all types of uncertainties, such as indeterminate and inconsistent information. In real situation, the evaluation information given by DMs are also indeterminate and inconsistent. Therefore, in order to make up for this shortcoming, Smarandache [8] proposed the neutrosophic set (NS) by adding indeterminacy-membership on the basis of intuitionistic fuzzy set. With the adding of indeterminacy-membership, NS can express the incomplete, indeterminate and inconsistent information more accurately than intuitionistic fuzzy set, so it has been developed rapidly. In order to apply NS to the actual decision-making process, Wang [9] first proposed single-value neutrosophic set (SVNS) based on NS. He defined SVNS and its settlement sub, and proposed a MCDM method based on SVNS. However, sometimes the degree of truth-membership, indeterminacy-membership, and falsity-membership represented by single value cannot accurately reflect the fuzzy information of the DMs. Therefore, Wang et al. [10] defined interval neutrosophic set (INS) and interval neutrosophic number (INN) and INS-based set theory operators. INS can reflect the information of DMs more flexibly than SVNS.

The current research on INS can be divided into two aspects: information aggregation operators and information evaluation methods. For aggregation operators, which aggregate all the presumptive individual arguments into a holistic argument, there has been a wealth of research. Zhang, Wang and Chen [11] proposed two operators under INSs named interval neutrosophic number weighted averaging operator (INNWA) and interval neutrosophic number weighted geometric operator (INNWG), and established a MCDM method based on the proposed operators. Liu and Tang [12] combined power average operator and the generalized weighted aggregation operator to INS, and thus proposed three operators named interval neutrosophic power generalized aggregation operator, interval neutrosophic power generalized weighted aggregation operator and interval neutrosophic power generalized ordered weighted aggregation operator, respectively. Khan et al. [13] extended the power Bonferroni mean operator based on Dombi operations to INS and developed four operators, including interval-neutrosophic Dombi power Bonferroni mean operator, interval-neutrosophic weighted Dombi power Bonferroni mean operator, interval-neutrosophic Dombi power geometric Bonferroni mean operator and interval-neutrosophic weighted Dombi power geometric Bonferroni mean operator. A MCDM strategy was also developed based on these four operators. [14 –19] also studies aggregation operators in the INN environment.

There are also many studies on information evaluation methods. The traditional MCDM approaches like TODIM (tomada de decisao iterativa multicriterio) [20], TOPSIS (technique for order preference by similarity to an ideal solution) [21], VIKOR (vlsekriterijumska optimizacija i kompromisnoresenje) [22], AHP (analytichierarchy process) [23], EDAS (evaluation based on distance from average solution) [24, 25] and MULTIMOORA (multiple multi-objective optimization by ratio analysis) [26] have been extended to the INS environment. Among them, MULTIMOORA method consists of three subordinate methods and each of them has its advantages and disadvantages. There is evidence shown that the result derived by aggregation performs better than the results of any single method [27], which means that MULTIMOORA has strong robustness. Besides, the mathematical calculation process of the three methods is relatively simple, and the calculation time is small [27]. Therefore, to some extent, MULTIMOORA method is superior to other traditional methods in dealing with quantitative information. Thus, Zavadskas et al. [26] extended MULTIMOORA method to INS and combined it with a novel heuristic evaluation methodology to solve MCDM problems based on INS. But this method doesn’t take the DM’s psychological preference into consideration. Like most of the traditional MCDM methods, MULTIMOORA method is based on the expected utility theory which assumed that the DMs are completely rational. However, in real situation, limited by knowledge, time and capability, DMs cannot make completely rational decisions and are usually bounded rational [28, 29]. Therefore, in practical decision-making cases, the final result made by the DMs may deviate from the expected utility theory. In order to reduce the deviation, it is meaningful to combine the MULTIMOORA method with the bounded rationality theory.

The regret theory can reflect the bounded rationality of DMs. Its main idea is that the DMs will not only pay attention to the results of the chosen option, but also the expected regret or rejoice caused by the assumption of choosing other options. This kind of expected regret or rejoice will affect the psychology of the DMs and thus affect the decision. The regret theory can explain some events that cannot be explained by expected utility theory, such as Ellsberg’s paradox, Allias’s paradox, and so on, so it is widly used in decicion-making problems [30 –35]. Unfortunately, all these efforts are not focused on improvement of MULTIMOORA method under INS environment.

Based on the above analysis, we can find out the following:

The MULTIMOORA method is an excellent decision-making method with strong robustness and simple calculation. However, there are few researches on the MULTIMOORA method in INS environment. Moreover, the current research does not consider the bounded rationality of DMs.

DMs are always bounded rational when they make decisions, while most traditional MCDM methods and studies assume that DMs are completely rational. The traditional MCDM method may lead to deviations between the decision results and the actual results.

Regret theory can reflect the regret avoidance of DMs in decision-making process. But there is no application of regret theory to MCDM problem in INS environment at present.

Motivated by these gaps in the current research, a novel MCDM method based on MULTIMOORA method and regret theory under INS environment is developed. The contributions of this paper can be summarized as follows:

Regret theory is employed to process the initial decision data, so that the DMs’ regret-aversion attitude is reflected in the decision information, so as to obtain more realistic decision results.

The MULTIMOORA method is combined with regret theory and extended to the INS environment. This hybrid method not only has the advantages of MULTIMOORA, but also considers the bounded rationality of DMs.

For decision-making problems with completely unknown criteria weights, decision information is really good as a basis of objective weights. A novel optimization method based on the maximizing deviation is proposed to determining objective criteria weights.

The rest of paper is organized as follows. Section 2 reviews kinds of literature related to MULTIMOORA method and regret theory. Section 3 presents some concepts and definitions of NSs, SVNSs, INSs and regret theory. In Section 4, the traditional MULTIMOORA method is introduced. And a novel multi-criteria decision making approach with INN is proposed in Section 5. Furthermore, a case study of school selection is provided to demonstrate the applicability of the proposed approach in Section 6. Comparison and sensitive analyses are performed to verify the effectiveness and superiority of the proposed method. Implications of this work are explained in Section 7. Finally, Section 8 presents some meaningful conclusions of this paper.

2 Literature review

This section provides a brief literature review on the MULTIMOORA method and regret theory, respectively.

2.1 Development of MULTIMOORA

The development of MULTIMOORA method can be traced back to the proposal of MOORA (Multi-objective optimization by ratio analysis) method. MOORA method was proposed by Brauers and Zavadskas [36] on the basis of previous researches [37], and it consists of two core part named Ratio System and Reference Point Approach. Later, Brauers and Zavadskas [38] developed MOORA to MULTIMOORA by adding Full Multiplicative Form and employing dominance theory to obtain the final integrative ranking. As a relatively new approach to solve MCDM problems, MULTIMOORA uses three subordinate methods to sort the alternatives. Each of the three subordinate methods has its advantages and disadvantages. Ratio System is a fully compensatory model and it works well when criteria are "independent". Conversely, Full Multiplicative Form is an incompletely-compensatory model and it performs better when faced with "dependent" criteria. Reference Point Approach is a non-compensatory model, which is more conservative than Ratio System and Full Multiplicative Form. Since these three methods can complement each other, the result derived by aggregation of them performs better than the results of any single method, which was proved by Brauers and Zavadskas [27].

With strong robustness, a lot of extensions and applications for MULTIMOORA method have emerged since it is proposed. For instance, Streimikiene et al. [39] proposed a new method which was combined MULTIMOORA method with TOPSIS method to analyze sustainable electricity production technologies. Baleɘentis et al. [40] extended MULTIMOORA method to fuzzy environment and applied it to solve a personal selection group decision making problem. And later, Baleɘentis and Zeng [41] used MULTIMOORA method to solve MCGDM problems under interval-valued fuzzy numbers environment. Balezentiene et al. [42] applied the fuzzy MULTIMOORA method to solve MCDM problems on energy crops selection. Zavadskas et al. [43] proposed a new method by combining MULTIMOORA method with ELECTRE IV to select effective technological system in construction. Subsequently, Zavadskas et al. [44] extended MULTIMOORA method to interval-valued intuitionistic fuzzy environment and applied it to solve group decision-making problems in engineering. And Zhao et al. [45] used MULTIMOORA method and continuous weighted entropy to analysis the failure mode and effect under interval-valued intuitionistic fuzzy environment. Dragisa et al. [46] proposed a MULTIMOORA method suitable for SVNNs, which is the first time that the MULTIMOORA method has been extended to the neutrosophic environment. Dong et al. [47] assessed the innovative ability of universities by an improved MULTIMOORA method with combined weights. Chen et al. [48] presented a Cloud-based ERP system selection model by integrating probabilistic linguistic MULTIMOORA method and Choquet integral operator. Liu and Li [49] proposed a probabilistic linguistic-dependent weighted average (PLDWA) operator and improved MULTIMOORA method by proposing a new final value determination method based on prospect theory under probabilistic linguistic environment.

In addition to the expansion of MUITOMOORA’s application environment, there are also many researches on the aggregation method of MULTIMOORA’s subordinate rankings. The most commonly used aggregation technique is dominance theory, which is also the original aggregation technique for MULTIMOORA proposed by Brauers and Zavadskas [38]. This method is based on three principles named Dominance, Equality and Transitiveness [27], and uses the three sub-rankings of MULTIMOORA method to get the final ranking. But it also means that the dominance theory only considers ordinal values and neglects the cardinal values of alternatives. With this in mind, in recent years, some methods of integrating cardinal and ordinal values have emerged as its substitutes. An excellent method is Improved Borda Rule. The application of the Borda Rule in MULTIMOORA was first proposed by Altuntas and Dereli [50], and then Wu et al. [51] proposed an Improved Borda Rule to MULTIMOORA method on the basis of the former research. The improved Borda rule considers both the order relations and the cardinal relations of the three subordinate methods of MULTIMOORA method. In this sense, the Improved Borda Rule is superior to dominance theory. Besides, dominance theory is based on manual comparison while Improved Borda Rule doesn’t need any manual comparison. Considering those, this paper employed Improved Borda Rule as the aggregation method of MULTIMOORA.

2.2 The regret theory in decision-making

The regret theory was proposed by Bell [52] and Loomes and Sugden [53] in 1982. Its main idea is the DMs will not only pay attention to what they got, but also what they could have got. The most primitive regret theory can only be used for the choice of two options. In 1994, Quiggin [54] modified the original utility function and extended it to the choice of multiple options. Since then, scholars have conducted a lot of discussions on regret theory. Zeelenberg [55] believes that in the decision-making process, the DM’s expected regret or rejoice when comparing the selected option with the rejected alternative will affect the decision. Connolly and Zeelenberg [56] proposed the decision-making judgment theory of regret, pointing out that regret in decision-making consists of two core elements: one is regret obtained by comparison with other results, and the other is self-blame when the choice is poor. Zeelenberg et al. [57] demonstrated through four sets of experiments that for the same consequences, inaction produces more regrets than actions. Kr $\ddot{a}$ hmer and Stone [58] extended the regret theory to a dynamic decision-making environment and proposed a dynamic regret theory model. Besides, regret theory has been applied in many areas including asset pricing [30], investment selection [31, 32], route choice [33], personnel selection [34], commercial recommendation [34], and new energy investment [35].

Based on a review of the above literature, we can conclude that MULTIMOORA method is a promising and robust approach in decision-making. And the regret theory can reflect the regret avoidance psychology of DMs. However, the MULTIMOORA does not take into account the regret avoidance psychology of DMs, and there is no study of regret theory combined with MNULTIMOORA under the environment of INN. In this study, the MULTIMOORA method is combined with the regret theory, and the hybrid approach is extended to the INN environment to solve MCDM problems.

3 Preliminaries

In order to better understand the content of this paper, some concepts and definitions of NSs, SVNSs, INSs and regret theory are introduced in this section.

3.1 Neutrosophic Sets (NSs)

Definition 1. [8] Let X be a space of points (objects), with a generic element in X denoted by x. A neutrosophic set A in X is characterized by a truth-membership function T_A (x), a indeterminacy-membership function I_A (x), and a falsity-membership function F_A (x). T_A (x) , I_A (x) , F_A (x) are real standard or non-standard subsets of ] 0^-, 1⁺ [. That is T_A (x) : X →]0^-, 1⁺ [, I_A (x) : X →]0^-, 1⁺ [, F_A (x) : X →]0^-, 1⁺ [.

There is no restriction on the sum of T_A (x), I_A (x) and F_A (x), so 0^- ≤ sup T_A (x) + sup I_A (x) + sup F_A (x) ≤3⁺.

Definition 2. [8] The complement of a neutrosophic set A is denoted by A^C and is defined as T_{A
^C} = {1⁺} - T_A (x), I_{A
^C} = {1⁺} - I_A (x), F_{A
^C} = {1⁺} - F_A (x) for every x in X.

Definition 3. [8] A neutrosophic set A is contained in the other neutrosophic set B, A ⊆ B if and only if inf T_A (x) ≤ inf T_B (x), sup T_A (x) ≤ sup T_B (x), inf I_A (x) ≥ inf I_B (x), sup I_A (x) ≥ sup I_B (x), inf F_A (x) ≥ inf F_B (x), and sup F_A (x) ≥ sup F_B (x) for every x in X.

3.2 Single-value Neutrosophic Sets (SVNSs)

Definition 4. [9] Let X be a space of points (objects), a SVNSs A in x is characterized as the following: $A = {(x, T_{A} (x), I_{A} (x), F_{A} (x)) ∣ x \in X}$ where the truth-menbership function T_A (x), indetermi-nacy-menbership I_A (x) and falsity-membership function F_A (x), T_A (x) : X → [0, 1] , I_A (x) : X → [0, 1] and F_A (x) : X → [0, 1] with the condition 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3.

For convenience, a SVNN can be expressen to be A = (T_A, I_A, F_A), T_A ∈ [0, 1], I_A ∈ [0, 1], F_A ∈ [0, 1], and 0 ≤ T_A + I_A + F_A ≤ 3.

Definition 5. [59] Let A = 〈T_A, I_A, F_A〉 be a SVNN, then the Score Function S (A) of A is defined: $S (A) = \frac{(2 + T_{A} - I_{A} - F_{A})}{3}$ (1)

Definition 6. [59] Let A = 〈T_A, I_A, F_A〉 and B = 〈T_B, I_B, F_B〉 be two SVNNs. Then the normalized Hamming distance between A and B is $d_{H} (A, B) = \frac{1}{3} (| T_{A} - T_{B} | + | I_{A} - I_{B} | + | F_{A} - F_{B} |)$ (2)

Definition 7. [60] Let x_j (j = 1, 2, ⋯ , n) be a series of SVNNs, and w = (w₁, w₂, ⋯, w_n) be the weight vector of x_j (j = 1, 2, ⋯ , n), then the Single-valued Neutrosophic Weighted Average (SVNWA) operator and Single-valued Neutrosophic Weighted Geometric (SVNWG) operator are: $\begin{matrix} SVNWA (x_{1}, x_{2}, \dots, x_{n}) = \oplus_{j = 1}^{n} w_{j} x_{j} = \\ (1 - \prod_{j = 1}^{n} {(1 - T_{ij})}^{w_{j}}, \prod_{j = 1}^{n} {(I_{ij})}^{w_{j}}, \prod_{j = 1}^{n} {(F_{ij})}^{w_{j}}), \end{matrix}$ (3) $\begin{matrix} SVNWG (x_{1}, x_{2}, \dots, x_{n}) = \otimes_{j = 1}^{n} x_{j}^{w_{j}} = \\ (\prod_{j = 1}^{n} {(T_{ij})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - I_{ij})}^{w_{j}}, 1 - \prod_{j = 1}^{n} {(1 - F_{ij})}^{w_{j}}) . \end{matrix}$ (4)

3.3 Interval Neutrosophic Sets (INSs)

Definition 8. [10] Let X be a space of points (objects), with a generic element in X denoted by x. INS A in X is characterized by a truth-membership function T_A (x), an indeterminacy-membership function I_A (x) and a falsity-membership function F_A (x), then A can be denoted by A = {〈x, T_A (x) , I_A (x) , F_A (x) 〉|x ∈ X}, where $T_{A} (x) = [T_{A}^{L} (x), T_{A}^{U} (x)]$ , I_A (x) = $[I_{A}^{L} (x), I_{A}^{U} (x)]$ , $F_{A} (x) = [F_{A}^{L} (x),$ $F_{A}^{U} (x)] \subseteq [0, 1]$ for every x in X, and 0 ≤ sup T_A (x) + sup I_A (x) + sup F_A (x) ≤3.

For convenience, we refer to A = 〈x, T_A, $I_{A}, F_{A} 〉 = 〈 [T_{A}^{L}, T_{A}^{U}], [I_{A}^{L}, I_{A}^{U}], [F_{A}^{L}, F_{A}^{U}] 〉$ as interval neutrosophic number (INN), which is a basic unit of INS.

Definition 9. [10] The complement of A is denoted by A^C and is defined as $A^{C} = 〈 [F_{A}^{L}, F_{A}^{U}], [1 - T_{A}^{L}, 1 - T_{A}^{U}], [I_{A}^{L}, I_{A}^{U}] 〉$ .

Fig. 1

Utility function v (x).

Definition 10. [10] Let $A = 〈 [T_{A}^{L}, T_{A}^{U}], [I_{A}^{L},$ $I_{A}^{U}], [F_{A}^{L}, F_{A}^{U}] 〉$ and $B = 〈 [T_{B}^{L}, T_{B}^{U}],$ $[I_{B}^{L}, I_{B}^{U}],$ $[F_{B}^{L}, F_{B}^{U}] 〉$ be two INNs. The operations for INNs are defined as follows:

$\begin{matrix} (1) A \oplus B = 〈 [T_{A}^{L} + T_{B}^{L} - T_{A}^{L} • T_{B}^{L}, T_{A}^{U} + T_{B}^{U} - T_{A}^{U} • \\ T_{B}^{U}], [I_{A}^{L} • I_{B}^{L}, I_{A}^{U} • I_{B}^{U}], [F_{A}^{L} • F_{B}^{L}, F_{A}^{U} • F_{B}^{U}] 〉, \end{matrix}$

$\begin{matrix} (2) A \otimes B = 〈 [T_{A}^{L} • T_{B}^{L}, T_{A}^{U} • T_{B}^{U}], [I_{A}^{L} + I_{B}^{L} - \\ I_{A}^{L} • I_{B}^{L}, I_{A}^{U} + I_{B}^{U} - I_{A}^{U} • I_{B}^{U}], [F_{A}^{L} + F_{B}^{L} - \\ F_{A}^{L} • F_{B}^{L}, F_{A}^{U} + F_{B}^{U} - F_{A}^{U} • F_{B}^{U}] 〉, \end{matrix}$

$\begin{matrix} (3) λ A = 〈 [1 - {(1 - T_{A}^{L})}^{λ} 1 - {(1 - T_{A}^{U})}^{λ}], [{(I_{A}^{L})}^{λ}, \\ {(I_{A}^{U})}^{λ}], [{(F_{A}^{L})}^{λ}, {(F_{A}^{U})}^{λ}] 〉, \end{matrix}$

$\begin{matrix} (4) {(A)}^{λ} = 〈 [{(T_{A}^{L})}^{λ}, {(T_{A}^{U})}^{λ}], [1 - {(1 - I_{A}^{L})}^{λ}, 1 - \\ {(1 - I_{A}^{U})}^{λ}], [1 - {(1 - F_{A}^{L})}^{λ}, 1 - {(1 - F_{A}^{U})}^{λ}] 〉 . \end{matrix}$

3.4 Regret theory

The regret theory was first put forward independently by Bell [52], Loomes and Sugden [53]. They believe that in the decision-making process, the DMs will compare the results of their own choices with the possible results of other alternatives. If they find that they can get better results by choosing other alternatives, they will feel regret, otherwise, they will feel happy. Therefore, in the decision-making process, the DMs will anticipate the regret or joy that may arise from the decision-making, and try to avoid choosing the plan that will make them regret. Utility function, regret-rejoice function and perceived utility function are three important components of regret theory. Among them, the perceived utility function consists of utility function and regret-joy function. In practical case, the DMs are often risk-averse, so the utility function v (x) is usually a monotonically increasing concave function, satisfying v (x) ′ > 0 and v (x) ^′′ < 0. Currently, there are two kinds of utility functions: power utility function and exponential utility function. This paper chooses the exponential utility function which is expressed as $v (x) = \frac{1 - e^{- β x}}{β}$ (5) where, x indicates the evaluation information of DMs, v (x) represents the corresponding utility value of the evaluation information, as shown in Fig. 1. β is risk aversion coefficient, and 0 < β < 1. The bigger β is, the greater the degree of risk aversion of DM is.

Fig. 2

Regret-rejoice function R(Δv).

Assuming that under the same criteria, the decision information of the A₁ and A₂ alternatives are x₁ and x₂ respectively, then the regret-rejoice function generated by the DMs choosing A₁ instead of A₂ is $R (x_{1}, x_{2}) = 1 - e^{- δ (v (x_{1}) - v (x_{2}))}$ (6) where δ is regret avoidance coefficient, meeting δ > 0. The bigger the δ is, the greater the regret aversion of DMs is. R (x₁, x₂) denotes the regret-rejoice value caused by DMs choosing A₁ instead of A₂. If R (x₁, x₂) > 0, R (x₁, x₂) is a joy value, indicating that the DM is pleased to choose alternative A₁ instead of A₂. Conversely, if R (x₁, x₂) < 0, it is a regret value, which means that the DM is regret to choose option A₁ rather than A₂. In addition, R (Δv) is a monotonically increasing concave function, as shown in Fig. 2, which means that the DMs are more sensitive to losses than to gains in the decision-making process. In other words, the DMs have unequal attitudes towards losses and gains. Therefore, taking A₂ as the standard, the DM’s perceived utility function for option A₁ is $u (x_{1}, x_{2}) = v (x) + R (x_{1} - x_{2})$ (7)

4 The traditional MULTIMOORA method

The traditional MULTIMOORA method was proposed by Brauers and Zavadskas in 2010 [61], and includes three subordinate methods named ratio system, reference point approach and full multiplicative form. Assume that X = (x_ij) _m×n is a decision matrix where x_ij denotes the value of alternative i with respect to criterion j. The three subordinate approaches are as follows.

Ratio system is started with the normalization of the original response matrix by the following formula. $x_{ij}^{*} = \frac{x_{ij}}{\sqrt{\sum_{i = 1}^{m} x_{ij}^{2}}}$ (8)

Then, the evaluation value of alternative A₍i) under the ratio system method is defined by $y_{i}^{*} = \sum_{j = 1}^{g} x_{ij}^{*} - \sum_{j = g + 1}^{n} x_{ij}^{*}$ (9) where j = 1, 2, ⋯ , g are the objectives to be maximized, and j = g + 1, g + 2, ⋯ , n are the objectives to be minimized. The final preference can get from the ordinal ranking of the $y_{i}^{*}$ .

Reference point approach aims to find the maximum distance between the maximum value of each objective and the associate assessment of the alternative. $min_{j} {max_{j} | r_{j} - x_{ij}^{*} |}$ (10) where $r_{j} = max x_{ij}^{*}$ . Then, the alternatives can be ranked in ascending d_i.

Full multiplicative form is the main improvement of MULTIMOORA over MOORA, which embodying maximization as well as minimization of purely multiplicative utility function. The utility of alternative A_i with criteria to be maximized and criteria to be minimized can be calculated as $U_{i} = \frac{A_{i}}{B_{i}}$ (11) where $A_{i} = \prod_{j = 1}^{g} x_{ij}$ and $B_{i} = \prod_{j = g + 1}^{n} x_{ij}$ denotes the product of criteria of the alternative A_i that needs to be maximized and minimized, respectively. The ranking of alternatives is obtained from the ordinal ranking of U_i.

Dominance theory [38] is usually used to aggregate these three rankings above into a single one.

Above all, the steps of traditional MULTIMOORA method are as follows:

Step 1. Normalize the original matrix by Equation (8);

Step 2. Obtain the rankings of ratio system by the ordinal ranking of the $y_{i}^{*}$ , which is calculated by Equation (9);

Step 3. Obtain the reference point approach ranking by Equation (10);

Step 4. Calculate and sort the utility values by the full multiple form model shown as Equation (11);

Step 5. Use the dominance theory to integrate the three sub-rankings obtained in step 2-4 to get the final ranking.

5 The proposed RT-MULTIMOORA method with INS

In order to reflect the regret and rejoice of DMs in the MCDM process under the INN environment, this section combined regret theory with the traditional MULTIMOORA method, and proposes an optimization model to determine the objective criteria weight, thus developing a new approach to solve the MCDM problem in the INN environment. The approach is divided into four phases named acquisition of decision-making information, data processing by regret theory, determining of criteria weights, and ranking of alternatives. In phase I, the decision-making background and original decision-making information are obtained. Next, in phase II, the regret theory is used to process the original decision and transform the original decision matrix into comprehensive perceive utility value matrix. Then in phase III, an optimization model is developed to obtain the objective criteria weights. Finally, in phase IV, MULTIMOORA method is used to obtain the final ranking of alternatives. Details of the four phases will be presented in the following subsections.

5.1 Phase I: Acquisition of decision-making information

Suppose in a MCDM problem, there are m alternatives A = (A₁, A₂, ⋯ , A_m) and n criteria C = (C₁, C₂, ⋯ , C_n) (including benefit-type criteria and cost-type criteria) with unknown criteria weights. Assume that the vector of the weight w = (w₁, w₂, ⋯ , w_n), where w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . $X = {(x_{ij})}_{m \times n} = 〈 [T_{ij}^{L}, T_{ij}^{U}], [I_{ij}^{L}, I_{ij}^{U}],$ ${[F_{ij}^{L}, F_{ij}^{U}] 〉}_{m \times n}$ is the decision matrix where x_ij takes the form of INNs for alternative A_i with respect to criteria C_j.

5.2 Phase II: Data processing by regret theory

Because the criteria value $x_{ij} = 〈 [T_{ij}^{L}, T_{ij}^{U}], [I_{ij}^{L},$ $I_{ij}^{U}],$ $[F_{ij}^{L}, F_{ij}^{U}] 〉$ is interval neutrosophic number, its truth-membership, indeterminacy-membership, and falsity-membership are all interval numbers. Since the interval number comes from the calculation of random sampling, the attribute value can be regarded as a random value in the interval, and it obeys a certain distribution. In fact, normal distribution and uniform distribution are the most common distributions. Therefore, considering the complexity and feasibility of calculation, this article assumes that the membership agrees of attribute values obey uniform distribution in the interval. Therefore, the probability density function of the membership degrees of attribute value can be respectively expressed as $f_{ij}^{S} (x) = {\begin{matrix} \frac{1}{S_{ij}^{U} - S_{ij}^{L}}, S_{ij}^{L} \leq x \leq S_{ij}^{U} \\ 0, otherwise \end{matrix}$ (12) where S = T, I, F, and $f_{ij}^{T} (x), f_{ij}^{I} (x)$ and $f_{ij}^{F} (x)$ refers to the probability density function of the truth-membership degree, indeterminacy-membership degree and the falsity-membership degree respectively. Then its corresponding utility values are obtained by $v_{ij}^{S} = \int_{S^{L}}^{S^{U}} v (x) f_{ij}^{S} (x) dx$ (13) where S = T, I, F, and $v (x) = \frac{1 - e^{- β x}}{β}$ . β is risk aversion coefficient, and 0 < β < 1. The bigger β is, the greater the risk aversion of DM is. $v_{ij}^{T}$ refers to the truth-membership degree utility value and the bigger the better, while $v_{ij}^{I}$ and $v_{ij}^{F}$ are the indeterminacy-membership degree utility value and falsity-membership degree utility value respectively, and the smaller the better. The utility matrix is denoted by V = (v_ij) m × n where .

In the traditional regret theory, the regret-rejoice function is calculate by comparing the size of each alternative relative to the reference points, which are the points with the maximum and minimum utility value. But in this condition, the reference points are fixed. In order to more accurately capture the regret or rejoice mood that DMs may have in choosing each alternative, this paper selects dynamic reference point, that is, compares each alternative with the other alternatives in turn, and finds the average value of regret or rejoice as the regret-rejoice value of the alternative relative to the other alternatives under the criteria C_j. Thus, the regret or rejoice value of truth-membership, indeterminacy-membership and falsity-membership for each alternative can be calculated as $R_{ikj}^{S} = {\begin{matrix} 1 - e^{- δ (v_{ij} - v_{kj})}, S = T \\ 1 - e^{- δ (v_{kj} - v_{ij})}, S = I, F \end{matrix}$ (14) $R_{ij}^{S} = \sum_{k} R_{ikj}^{S}$ (15) in which, S = T, I, F, k = 1, 2, ⋯ , m, k ≠ i, j = 1, 2, ⋯ , n. δ is regret avoidance coefficient, meeting δ > 0. The bigger the δ is, the greater the regret aversion of DMs is. Then, the regret-rejoice matrix can be denoted by , where R_ij represents the regret value or rejoice value of alternative A_i under criteria C_j. According to the regret theory, the comprehensive perceive utility value of x_ij can be calculated as $U_{ij}^{S} = v_{ij}^{S} + R_{ij}^{S}$ (16) in which S = T, I, F. Then the comprehensive perceive utility matrix can be denoted by .

Therefore, the regret theory transforms the original decision matrix from INNs to SVNNs through a series of calculations. Subsequent calculations will be based on this comprehensive perception utility matrix.

5.3 Phase III: Determining of criteria weights

After data processing, we need to determine the criteria weights for they are completely unknown. Weight calculation is an old topic with many methods to measure. When the weights are completely unknown, the most used objective weight method is maximizing deviation method [62, 63]. Maximizing deviation method was proposed by Wang [62], which hold the view that the greater the deviation of different alternatives under the same criteria, the more information the criteria covers, and then the more important the criteria is. Therefore, based on this method, this paper built a criteria weight optimization model as $M 1 {\begin{matrix} max Z (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj}) w_{j}, \\ s . t . \sum_{j = 1}^{n} w_{j}^{2} = 1, w_{j} \in [0, 1] . \end{matrix}$ (17)

Then, the Lagrangian is constructed to solve M1. $\begin{matrix} L (w, λ) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj}) w_{j} + λ (\sum_{j = 1}^{n} w_{j}^{2} - 1) \end{matrix}$ (18) where λ is the Lagrange multiplier. Let the derivative of L (w, λ) with respect to w_j and λ be equal to the value 0, the following equations can be obtained. ${\begin{matrix} \frac{\partial L (w, λ)}{w_{j}} = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj}) + 2 λ w_{j} = 0, \\ \frac{\partial L (w, λ)}{λ} = \sum_{j = 1}^{n} w_{j}^{2} - 1 = 0 . \end{matrix}$ (19)

By solving Equation (19), we can obtain the criteria weights as $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj})}{\sqrt{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} {d (U_{ij}, U_{kj})}^{2}}} .$ (20)

Equation (20) can be normalized as Equation (21). In other words, the normalized criteria weight based on M1 is denote as Equation (21). $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (U_{ij}, U_{kj})} .$ (21)

5.4 Phase IV: Ranking of alternatives

5.4.1 Ratio system

Based on the traditional MULTIMOORA method introduced in Section 4, we can use the SVNWA operator (Equation (3)) to integrate the above information. $\begin{matrix} Y_{i}^{+} = (1 - {\prod_{j \in J_{1}} (1 - U_{ij}^{T})}^{w_{j}}, \prod_{j \in J_{1}} {(U_{ij}^{I})}^{w_{j}}, \prod_{j \in J_{1}} {(U_{ij}^{F})}^{w_{j}}) \end{matrix}$ (22) $Y_{i}^{-} = (1 - {\prod_{j \in J_{2}} (1 - U_{ij}^{T})}^{w_{j}}, \prod_{j \in J_{2}} {(U_{ij}^{I})}^{w_{j}}, \prod_{j \in J_{2}} {(U_{ij}^{F})}^{w_{j}})$ (23)

Next, use Equation (1) to calculate the Score function of $y_{i}^{+} = S (Y_{i}^{+})$ and $y_{i}^{-} = S (Y_{i}^{-})$ . Then, we can obtain the final utility of ratio system. $y_{i} = y_{i}^{+} - y_{i}^{-}, i = 1, 2, \dots, n .$ (24)

The bigger the y_i is, the better the A_i is.

5.4.2 Reference point approach

In general, the criteria can be divided into two types named efficient criteria and cost criteria. For efficient criteria, the bigger the criteria value is, the better the alternative is. On the contrary, for cost criteria, the bigger the criteria value is, the worse the alternative is. In order to eliminate the influence of different types, we need to calculate the reference point $U_{j}^{+}$ of efficient factor and cost factor on the basis of Section 5.2. $U_{j}^{+} = {\begin{matrix} (max_{i} U_{ij}^{T}, min_{i} U_{ij}^{I}, min_{i} U_{ij}^{F}), j \in J_{1}, \\ (min_{i} U_{ij}^{T}, max_{i} U_{ij}^{I}, max_{i} U_{ij}^{F}), j \in J_{2} . \end{matrix}$ (25)

According to Equation (25), we can pick up the maximal objective reference point vector $U_{j}^{+}$ . The closer to the reference point, the better the alternative is. So, we can calculate the normalized Hamming distance between alternatives and the reference point vector according to Equation (2). Then the utility of reference point approach is defined as follows. $\begin{matrix} d_{i} = max_{j} d (U_{ij}, U_{j}^{+}) w_{j}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n \end{matrix}$ (26)

The smaller the d_i is, the better the alternative A_i is. So the alternatives are ranked in ascending order of d_i.

5.4.3 Full multiple form

Based on the SVNWG operator (Equation (4)), we can obtain $\begin{matrix} a_{i} = \prod_{j \in J_{1}} U_{ij} = ({\prod_{j \in J_{1}} U_{ij}^{T}}^{w_{j}}, 1 - \prod_{j \in J_{1}} {(1 - U_{ij}^{I})}^{w_{j}}, \\ 1 - \prod_{j \in J_{1}} {(1 - U_{ij}^{F})}^{w_{j}}), \end{matrix}$ (27) $\begin{matrix} b_{i} = \prod_{j \in J_{2}} U_{ij} = ({\prod_{j \in J_{2}} U_{ij}^{T}}^{w_{j}}, 1 - \prod_{j \in J_{2}} {(1 - U_{ij}^{I})}^{w_{j}}, \\ 1 - \prod_{j \in J_{2}} {(1 - U_{ij}^{F})}^{w_{j}}) . \end{matrix}$ (28)

Calculate the Score function S (a_i) and S (b_i) of a_i and b_i, then, we can get the final ranking by $u_{i} = \frac{S (a_{i})}{S (b_{i})} .$ (29) The alternatives are ranked in descending order of u_i.

5.4.4 Determination of final ranking result

According to the three subordinate methods of MULTIMOORA method, three rankings can be obtained. In this step, we need to integrate the above three rankings to obtain the final decision. The traditional MULTIMOORA method used dominance theory to obtain the final result. But it only considers ordinal values and neglects the relative importance of alternatives. Improved Borda rule [51] considers both order relations and utility values. In this sense, the improved Borda rule is superior to dominance theory. So, this paper uses the improved Borda rule to obtain the final decision. The improved Borda score IBS (A_i) of each alternative can be calculated as Equation (30), and the alternatives are ranked based on descending IBS (A_i). $\begin{matrix} IBS (A_{i}) & = y_{i}^{*} \frac{m - r (y_{i}) + 1}{m (m + 1) / 2} - d_{i}^{*} \frac{r (d_{i})}{m (m + 1) / 2} \\ + u_{i}^{*} \frac{m - r (u_{i}) + 1}{m (m + 1) / 2}, \end{matrix}$ (30) where $y_{i}^{*} = \frac{y_{i}}{\sqrt{\sum_{i = 1}^{m} {(y_{i})}^{2}}}, d_{i}^{*} = \frac{d_{i}}{\sqrt{\sum_{i = 1}^{m} {(d_{i})}^{2}}}$ and $u_{i}^{*} = \frac{u_{i}}{\sqrt{\sum_{i = 1}^{m} {(u_{i})}^{2}}}$ , r (y_i), r (d_i) and r (u_i) are the ranking order of ratio system, reference point approach and full multiple form respectively.

5.5 Solution framework of the INN-RT-MULTIMOO RA method

Fig. 3

Flowchart of RT-MULTIMOORA method for INS.

To clarify the RT-MULTIMOORA method based on INNs, the framework for the proposed method is visualized in Fig. 3, and the specific procedure is organized as follows:

Step 1. Obtain the original decision matrix X;

Step 2. Calculate the utility matrix V according to Equations (13);

Step 3. Compute the regret-rejoice matrix R by Equations (15);

Step 4. Obtain the comprehensive perceive utility matrix U according to Equation (16);

Step 5. Use M1 proposed in Section 5.3 to determine the objective weight vector w of criteria;

Step 6. Obtain utilities and rankings of MULTIMOORA by Equations (29);

Step 7. Aggregate the three sub-rankings of MULTIMOORA method by the improved Borda rule, and the final ranking is obtained.

6 Illustrative example

In order to demonstrate the feasibility of the proposed method, this section introduced an illustrate example that applies the proposed INN-RT-MULTIMOO RA method to solve a MCDM problem about school selection. The example demonstrates the validity, application, and effectiveness of the proposed method.

6.1 Decision context and procedures

School selection is an important problem that every student will experience. A good school can provide students with excellent educational resources and lay a solid foundation for their future development. There are many standards for a good school, such as teaching quality, teacher resources, school tuition, etc. Students need to consider these criteria comprehensively and choose the most suitable school. This can be regarded as a MCDM problem. Due to the complexity of the reality, its criteria values are often not directly represented by crisp set. In this condition, INS is suitable for expressing the criterion value.

Suppose a high school student has just graduated from middle school and faces the problem of choosing a university. He needs to choose a university based on his college entrance examination scores. After preliminary screening, there are four universities that matched with his college entrance examination scores, respectively denoted as A₁, A₂, A₃, A₄. Next, he needs to choose the most suitable university based on three criteria named

C₁: teaching quality,

C₂: teacher resources,

C₃: distance from home,

where C₁ and C₂ are benefit criteria, and C₃ is a cost criterion. Assume that the objective weight vector of the criteria w = (w₁, w₂, w₃) is unknown and the four possible alternatives are to be evaluated under the above three criteria by the form of INNs. The MCDM problem can be solved by the proposed method involving the following steps.

Step 1. Obtain the original decision matrix X.

According to the evaluation of four universities based on three criteria, the following decision matrix expressed in INNs can be obtained as Table 1.

Table 1
Original decision matrix X with INN

C ₁ C ₂ C ₃

A ₁ 〈[0.4, 0.5] , [0.2, 0.3] , [0.3, 0.4] 〉〈[0.4, 0.6] , [0.1, 0.3] , [0.2, 0.4] 〉〈[0.7, 0.9] , [0.2, 0.3] , [0.4, 0.5] 〉

A ₂ 〈[0.6, 0.7] , [0.1, 0.2] , [0.2, 0.3] 〉〈[0.6, 0.7] , [0.1, 0.2] , [0.2, 0.3] 〉〈[0.3, 0.6] , [0.3, 0.5] , [0.8, 0.9] 〉

A ₃ 〈[0.3, 0.6] , [0.2, 0.3] , [0.3, 0.4] 〉〈[0.5, 0.6] , [0.2, 0.3] , [0.3, 0.4] 〉〈[0.4, 0.5] , [0.2, 0.4] , [0.7, 0.9] 〉

A ₄ 〈[0.7, 0.8] , [0.0, 0.1] , [0.1, 0.2] 〉〈[0.6, 0.7] , [0.1, 0.2] , [0.1, 0.3] 〉〈[0.6, 0.7] , [0.3, 0.4] , [0.8, 0.9] 〉

C ₁	C ₂	C ₃
A ₁	〈[0.4, 0.5] , [0.2, 0.3] , [0.3, 0.4] 〉	〈[0.4, 0.6] , [0.1, 0.3] , [0.2, 0.4] 〉	〈[0.7, 0.9] , [0.2, 0.3] , [0.4, 0.5] 〉
A ₂	〈[0.6, 0.7] , [0.1, 0.2] , [0.2, 0.3] 〉	〈[0.6, 0.7] , [0.1, 0.2] , [0.2, 0.3] 〉	〈[0.3, 0.6] , [0.3, 0.5] , [0.8, 0.9] 〉
A ₃	〈[0.3, 0.6] , [0.2, 0.3] , [0.3, 0.4] 〉	〈[0.5, 0.6] , [0.2, 0.3] , [0.3, 0.4] 〉	〈[0.4, 0.5] , [0.2, 0.4] , [0.7, 0.9] 〉
A ₄	〈[0.7, 0.8] , [0.0, 0.1] , [0.1, 0.2] 〉	〈[0.6, 0.7] , [0.1, 0.2] , [0.1, 0.3] 〉	〈[0.6, 0.7] , [0.3, 0.4] , [0.8, 0.9] 〉

Step 2. Calculate the utility matrix V.

Let β = 0.88, which is verified experimentally by Tversky and Kahneman [65] in 1992. Then, according to Equations (13), the utility matrix V = (v_ij) _m × n can be calculated as Table 2.

Table 2

Utility matrix V obtain by regret theory

C ₁	C ₂	C ₃
A ₁	〈0.3713, 0.2241, 0.3010〉	〈0.4036, 0.1822, 0.2625〉	〈0.5736, 0.2241, 0.3713〉
A ₂	〈0.4948, 0.1402, 0.2241〉	〈0.4948, 0.1402, 0.2241〉	〈0.3694, 0.3361, 0.5983〉
A ₃	〈0.3694, 0.2241, 0.3010〉	〈0.4358, 0.2241, 0.3010〉	〈0.3713, 0.2625, 0.5736〉
A ₄	〈0.5488, 0.0486, 0.1402〉	〈0.4948, 0.1402, 0.1822〉	〈0.4948, 0.3010, 0.5983〉

Step 3. Compute the regret-rejoice matrix R by Equations (15).

Let the regret avoidance coefficient δ = 0.3 which is the commonly used value in literatures, and the regret-rejoice matrix can be calculated as Table 3.

Table 3

Regret-rejoice matrix R obtained by regret theory

C ₁	C ₂	C ₃
A ₁	〈 - 0.0306, - 0.0265, - 0.0242〉	〈 - 0.0217, - 0.0043, - 0.0082〉	〈 0.0472, 0.0224, 0.0635 〉
A ₂	〈 0.0190, 0.0073, 0.0067 〉	〈 0.0148, 0.0125, 0.0072 〉	〈 - 0.0340, - 0.0224, - 0.0260〉
A ₃	〈 - 0.0314, - 0.0265, - 0.0242〉	〈 - 0.0087, - 0.0212, - 0.0237〉	〈 - 0.0332, 0.0072, - 0.0159 〉
A ₄	〈 0.0401, 0.0432, 0.0397 〉	〈 0.0148, 0.0125, 0.0238 〉	〈 0.0165, - 0.0081, - 0.0260 〉

Step 4. Obtain the comprehensive perceive utility matrix U according to Equation (16).

Then, we can obtain the comprehensive perceive utility matrix as Table 4.

Table 4

Comprehensive perceive utility matrix U obtained by regret theory

C ₁	C ₂	C ₃
A ₁	〈0.3407, 0.1976, 0.2767〉	〈0.3818, 0.1779, 0.2544〉	〈0.6208, 0.2466, 0.4348〉
A ₂	〈0.5138, 0.1475, 0.2308〉	〈0.5096, 0.1527, 0.2313〉	〈0.3353, 0.3138, 0.5724〉
A ₃	〈0.3379, 0.1976, 0.2767〉	〈0.4271, 0.2029, 0.2772〉	〈0.3381, 0.2698, 0.5577〉
A ₄	〈0.5890, 0.0918, 0.1799〉	〈0.5096, 0.1527, 0.2059〉	〈0.5113, 0.2928, 0.5724〉

Step 5. Use the optimization model M1 proposed in Section 5.3 to determine the objective weight vector of criteria.

The weight vector of criteria can be calculated as w = (0.3890, 0.2097, 0.4013).

Step 6. Obtain the RT-MULTIMOORA rankings.

According to Equations (29), the utilities and rankings of alternatives under ratio system, reference point approach and full multiplicative form can be obtained and shown in Table 5.

Step 7. Aggregate the three sub-rankings of RT-MULTIMOORA method by the improved Borda rule.

After the above steps, three sorts of alternatives are obtained. Now we need to integrate the results into one sort by using the improved Bolda rule. According to the improved Borda rule, the three kinds of sub-utilities y_i, d_i, u_i can be aggregate into one value IBS (A_i). By sorting this value, the final preference, as shown in Table 5, is obtained. The final result explains that the alternative A₂ is the best choice and A₄ is the suboptimal one, while A₁ is the worst option.

Table 5

The results derived from INN-PT-MULTIMOORA

RS		RPA		FMF		Final Rank
y _i	r (y_i)	d _i	r (d_i)	u _i	r (u_i)	IBS (A_i)	r (A_i)
A ₁	0.1231	4	0.0656	4	0.8941	4	-0.2074	4
A ₂	0.2965	2	0.0236	1	1.0976	1	0.3721	1
A ₃	0.2091	3	0.0588	3	0.9991	3	-0.0026	3
A ₄	0.3055	1	0.0263	2	1.0741	2	0.3519	2

6.2 Sensitive analysis

Regret theory involves two parameters, risk aversion coefficient β and regret avoidance coefficient δ. Among them, the risk aversion coefficient β reflects the degree of risk aversion of the DMs. The larger the β is, the more risk-averse DMs are. The regret avoidance coefficient δ reflects the degree of regret avoidance of the DMs. The larger the value of δ means that DM has a stronger perception of regret. We combined the regret theory with MULTIMOORA method in our proposed method, so the change of β and δ can reflect DMs behavioral preferences. This paper mainly discuss the influence of DMs’ regret psychology on decision-making results, so we let β = 0.88, which is the result of Tversky and Kahneman’s experimental verification and change the value of δ of the example in Section 6 to discuss the influence of DMs’ regret avoidance degree on decision-making results.

Let β = 0.88, and δ varies from 0.01 to 1.00 with a step size of 0.01, the final results are shown in Table 6. In order to capture the changes in the decision-making results more intuitively, we expressed the numerical results in the form of a graph, as illustrated in Fig. 4. As we can see from the Table 4, the final result slightly changes with the varies value of δ. When 0.01 ≤ δ ≤ 0.23, the final result is A₄ ≻ A₂ ≻ A₃ ≻ A₁. In this case, the δ is at a low level, and the DMs’ perception of regret is relatively weak, and them are more inclined to make a rational decision. So, A₄ is the optimal choice and A₁ is the worst one. As δ gradually increases, the ranking result is changed. When changing from 0.24 to 0.91, the order of A₂ and A₄ has changed. The optimal choice becomes to A₂, and the worst choice is still A₁. In this case, the regret aversion of DMs increases, and they are prone to choose the alternative with the least regret as the optimal choice. When 0.92 ≤ δ ≤ 1.00, the final result becomes to A₂ ≻ A₃ ≻ A₄ ≻ A₁, and the best option has been maintained at A₂, which shows that when the DMs have greater regret aversion, A₂ is the alternative with the least regret. In other words, when the DMs have a low level of regret-aversion, them are prone to choose A₄ as the optimal choice, but when they have a deep regret-aversion, they will regret choosing A₄ and more willing to choose A₂ as the best one. Besides, A₁ and A₃ have never been the optimal choice and A₁ is almost the last one whatever the δ is. That is because the decision value of A₁ and A₃ are really small and DMs will not regret not choosing these two alternatives.

Table 6
Final results with varies δ

δ Final ranking Optimal choice

0.01-0.23 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

0.24-0.91 A₂ ≻ A₄ ≻ A₃ ≻ A₁ A ₂

0.92-1.00 A₂ ≻ A₃ ≻ A₄ ≻ A₁ A ₂

δ	Final ranking	Optimal choice
0.01-0.23	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A ₄
0.24-0.91	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A ₂
0.92-1.00	A₂ ≻ A₃ ≻ A₄ ≻ A₁	A ₂

As we can see in Fig. 4, the numerical result IBS (A_i) is also changing with varied δ. IBS (A₁), IBS (A₂), and IBS (A₃) are increase with the increase of δ, while IBS (A₄) decreases with the increase of δ. This is because the value of δ affects the DM’s feelings of regret, and at the same time affects the comprehensive utility value, thereby affecting the criteria weights, and ultimately affecting the ranking results.

Fig. 4

The numerical results with changing δ.

In summary, the final ranking and numerical results changed with the change of regret aversion coefficient, which shows that the proposed method can really reflect the psychological attitude of DMs and also proves the effectiveness of adding regret theory to MCDM problem.

6.3 Comparative discussion

In order to further verify the superiority of the proposed INN-RT-MULTIMOORA method, this section conducts a comparative analysis with the results based on the same MCDM problem. Different existing methods from Ye [64], Chi and Liu [63] and Zhang et al. [66, 67] are employed to solve the same MCDM problem in Section 6.

In Ye’s methods [64], the similarity measures between each alternative and the most desirable alternative(s) are employed to determine the final ranking. The similarity measures are calculated based on weighted Hamming distance and weighted Euclidean distance. In order to make the results comparable, we uniformly use the criteria weights obtained in this paper for calculation.

The method from Chi and Liu [63] are based on the ideal of TOPSIS method. Firstly, as in this paper, maximizing deviation method is utilized to derive objective weights. And then, the relative closeness coefficient is used to determine the ranking of alternatives.

In the procedure of Zhang et al.’s methods [66], the INNWA and INNWG operators are used to calculate the comprehensive INNs, and the likelihood matrices based on the score function are constructed to determine the final ranking. Similarly, we set the criteria weights obtained in this paper as the weight vector for the comparison convenience. Besides, Zhang et al. [67] defined some outranking relations for INNs based on ELECTRE IV method, and used it to obtain the MCDM results.

Conduct these procedures, the comparison results can be achieved and shown in Table 7.

Table 7
The final rankings with different method

Method Ranking Optimal one

Ye [64]

Similarity Measures based on Hamming distance A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

Similarity Measures based on Euclidean distance A₂ ≻ A₄ ≻ A₃ ≻ A₁ A ₂

Chi and Liu [63] A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

Zhang et al. [66]

Method based on INNWA operator A₄ ≻ A₁ ≻ A₂ ≻ A₃ A ₄

Method based on INNWG operator A₁ ≻ A₄ ≻ A₂ ≻ A₃ A ₁

Zhang et al. (p = 0.2, q = 0.1) [67] A₄ ≻ A₁ ≻ A₂ ≻ A₃ A ₄

The proposed method

(β = 0.88, δ = 0.1) A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

(β = 0.88, δ = 0.3) A₂ ≻ A₄ ≻ A₃ ≻ A₁ A ₂

Method	Ranking	Optimal one
Ye [64]
Similarity Measures based on Hamming distance	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A ₄
Similarity Measures based on Euclidean distance	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A ₂
Chi and Liu [63]	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A ₄
Zhang et al. [66]
Method based on INNWA operator	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A ₄
Method based on INNWG operator	A₁ ≻ A₄ ≻ A₂ ≻ A₃	A ₁
Zhang et al. (p = 0.2, q = 0.1) [67]	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A ₄
The proposed method
(β = 0.88, δ = 0.1)	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A ₄
(β = 0.88, δ = 0.3)	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A ₂

It can be seen from Table 7 that the final results are varied for different methods. When β = 0.88 and δ = 0.1, the result of the proposed INN-RT-MULTIMOORA method is the same as those of Ye [64] (based on Hamming distance) and Chi and Liu [63], while slightly different from those of Zhang et al. [66, 67]. And when β = 0.88 and δ = 0.1, the result of the proposed method is the same as the result of the similarity measures based on Euclidean distance proposed by Ye [64], and different from those of others. The reasons for the consistency or inconsistency of the rankings can be explained as follows.

The results of the proposed approach are the same as that of Ye’s methods [64] to some extent. But those two methods are based on different ideas. Ye’s method [64] focuses on the similarity measurement based on distance calculation, while the proposed method focuses on the evaluation of information by the subordinate method. Besides, the proposed method considers the DM’s regret avoidance behaviour while Ye’s method [64] does not. Moreover, the criteria weight in Ye’s method [64] is directly assumed, but in practical case, the weight is usually not unknown. This article gives a weight calculation model when the decision weight is completely unknown.

When β = 0.88 and δ = 0.1, the result of the proposed approach is the same as that of Chi and Liu [63]. But the Chi and Liu’s method [63] has disadvantages. This method based on TOPSIS method, and the relative closeness coefficients are calculated based on the relative ideal solutions. However, the calculation of the positive ideal solution (PIS) and negative ideal solution (NIS) is carried out separately for the degree of truth-membership, indeterminacy-membership, and falsity-membership which may lead to the unreal existence of PIS or NIS. Besides, TOPSIS method also have not consider the bounded rationality of DMs.

The difference between the ranking results of Zhang et al.’s methods [66, 67] and that of the proposed approach is mainly reflected in the position of A₁. In the proposed method, A₁ is always the worst option whatever the δ changes, while in methods [66, 67], A₁ is superior to A₂ and A₃, and even been the best choice when using INNWG operator in [66]. This may caused by the different ideas and the inherent characteristics of aggregation operators. The proposed method is focuses on the evaluation of information while the method in [66] focuses on the integration of information. Besides, the methods in [66] does not give the calculation method of the criterion weight, and the method in [67] does not take the criterion weight into consideration but the proposed method does. Moreover, the outranking method of Zhang et al. [67] has to convert the INNs into real numbers which may cause information loss or distortion. Furthermore, Zhang et al.’s methods are based on expected utility theory while the proposed method is based on bounded rational theory.

From above discussion, the advantages of the proposed INN-PT-MULTIMOORA method over the other six approaches can be simply summarized into the following points.

The addition of regret theory enables the proposed method to effectively reflect the regret mental state of DMs in the decision-making process. The parameter setting of regret avoidance coefficient δ and risk aversion coefficient β is based on the risk preference of DMs, which makes the final result more in line with the real situation.

Three subordinate methods of MULTIMOORA method are used in the proposed method simultaneously, which enhances the robustness of the decision result. Moreover, the calculations required for the three subordinate methods are relatively straightforward and effective.

In real situation, the criterion weight is usually unknown. The proposed approach can easily handle with this situation by using the proposed optimization model based on maximizing derivation method.

7 Implication

In this study, an RT based MULTIMOORA method is constructed to solve the neutrosophic MCDM problem. The proposed research framework provides valuable implications for theory and applications. The following paragraphs provide the implications through the current study.

Theoretically, the current research is a new contribution to the existing body of knowledge in the field of MCDM in NS. By combining RT with the traditional MULTIMOORA method, a neutrosophic MCDM method considering the bounded rationality of the DMs is developed, which fills the research gap in the relevant literature. This method enriches and develops the research of RT in neutrosophic MCDM, and expands the theory and method of MCDM in INN environment. Overall, the contribution of the current study to the field of operations research and management is a comprehensive and quantifiable framework for selecting alternatives considering multiple criteria.

In application, this study successfully demonstrates the practicality of the proposed INN-RT-MULTIMOORA method in a practical case of school selection. The new method can help DMs choose the school that is most suitable and the least likely to make them regret based on their scores and the attributes of different schools. In addition, the proposed general framework can also be applied to different fields such as supplier management, project management, medical option selection and sustainable development to solve different MCDM problems. The methodology of this study will motivate and guide managers and DMs to choose the most suitable option by analyzing the evaluation of different alternatives in multiple dimensions and the psychological state (regret or joy) generated by choosing each option. Thus, the current work promises significant application implications for management practitioners and general DMs.

8 Conclusion

Due to INS’s flexibility in expressing uncertain, incomplete and inconsistent information and the ubiquity of MCDM problem in daily life, the study of MCDM methods with INS is highly significant. The MULTIMOORA method is an excellent fuzzy MCDM approach. It is based on the expected utility theory, while researches prove that DMs are bounded rationally when making decisions. Therefore, in order to make the decision more in line with the real situation, this paper combined the regret theory and the MULTIMOORA method based on INS environment, thus constructing a new MCDM method named INN-RT-MULTIMOORA method. Specifically, after some concepts of NSs and regret theory and the traditional MULTIMOORA method were introduced, the INN-PT-MULTIMOORA method for MCDM was developed. The regret theory is used to process the original decision information, thereby transforming the decision information represented by INNs into a comprehensive perception regret decision matrix represented by SVNNs. Then, an optimization model based on maximizing deviation is constructed to obtain the objective weight of criteria. MULTIMOORA approach is used to handle decision matrix, and improved Borda rule is used to obtain the final result. Finally, an illustrated application of school selection proves the feasibility of the proposed method. Sensitivity analysis with different regret avoidance parameter values was performed to show the validity of the regret theory in the proposed method. Furthermore, comparative analysis with six available MCDM approaches also illustrate the superiority of our method.

In comparison with six available MCDM approaches, the advantages of the proposed INN-RT-MULTIMO ORA method can be summarized as: (1) Regret theory can describe the regret psychology of DMs in the decision-making process. DMs can adjust the risk aversion coefficient according to their regret aversion levels. So the result will be more convincing. (2) MULTIMOORA method is a promising MCDM method consists of three subordinate methods. Each of the subordinate method is both convenient and effective. So the proposed method is also simple and effective; (3) When the criteria weights are completely unknown, the optimization model proposed is a good solution to obtain objective weights. The proposed method not only makes full use of objective decision-making information, but also considers the DMs’ subjective risk preference, which makes the decision-making process more practical.

A shortcoming of the proposed approach is that it only considers the case where the criteria weights are completely unknown, and does not discuss other complicated cases. Accordingly, the one future research direction is to further improve the weight determine of complex situations. Besides, it is meaningful to apply the proposed method to solve complex MCDM problems, and extended it to more domains like venture capital, supplier selection and personnel selection, etc.

Footnotes

Acknowledgments

This work was supported by the Humanities and Social Sciences Foundation of Ministry of Education of the Peoples Republic of China under Grant 17YJA630115.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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A multi-criteria decision-making with regret theory-based MULTIMOORA method under interval neutrosophic environment

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Development of MULTIMOORA

2.2 The regret theory in decision-making

3 Preliminaries

3.1 Neutrosophic Sets (NSs)

3.2 Single-value Neutrosophic Sets (SVNSs)

5.1 Phase I: Acquisition of decision-making information

5.2 Phase II: Data processing by regret theory

5.4.1 Ratio system

6.1 Decision context and procedures

Table 6 Final results with varies δ δ Final ranking Optimal choice 0.01-0.23 A4 ≻ A2 ≻ A3 ≻ A1 A 4 0.24-0.91 A2 ≻ A4 ≻ A3 ≻ A1 A 2 0.92-1.00 A2 ≻ A3 ≻ A4 ≻ A1 A 2

8 Conclusion

Footnotes

Acknowledgments

Conflict of interest

References

Table 6
Final results with varies δ

δ Final ranking Optimal choice

0.01-0.23 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

0.24-0.91 A₂ ≻ A₄ ≻ A₃ ≻ A₁ A ₂

0.92-1.00 A₂ ≻ A₃ ≻ A₄ ≻ A₁ A ₂