Fuzzy stochastic multi-criteria decision-making methods with interval neutrosophic probability based on regret theory

Abstract

Stock evaluation is a significant decision-making activity for investors. Due to the complexity of stock exchange market, evaluation information may be fuzzy and stochastic in the meantime. Therefore, it is essential to conduct the research on fuzzy stochastic multi-criteria decision-making (FSMCDM) methods. In this paper, at the beginning, we propose interval neutrosophic probability from the definition of neutrosophic probability. Then, a novel MCDM method is proposed with interval neutrosophic probability based on regret theory, in which criteria values are interval neutrosophic numbers (INNs). The method proposed with interval neutrosophic probability may be much better than methods using classic probability or fuzzy probability. Next, we apply the proposed method to stock selection problems and discuss the influence of parameters on ranking results. Finally, we compare regret theory with prospect theory, which is also an important theory of bounded rationality just like regret theory, to emphasize the characteristics the regret theory. Moreover, a comparative analysis is also conducted between the proposed method and existing methods under interval neutrosophic environment to demonstrate efficiency and applicability of the proposed method.

Keywords

Multi-criteria decision-making interval neutrosophic sets interval neutrosophic probability bounded rationality regret theory

1 Introduction

The capital market especially stock market has developed rapidly in recent years. more and more investors invest a large amount of money into the stock market [1]. However, the stock market is often complex and variable. Therefore, stock evaluation is an important problem for investors [2]. Many multi-criteria decision-making (MCDM) methods [3 –5] were utilized to deal with stock assessment problems [1, 6]. Due to complexity and variability of stock market, stock evaluation problem may have both fuzziness and randomness. To solve this type of problems, many researches on fuzzy stochastic multi-criteria decision-making (FSMCDM) methods has been studied [7 –14], but there are some certain limitations in these methods. The limitations are outlined below.

Peng and Dai [7], Peng and Yang [8], and Peng et al. [9] proposed FSMCDM methods where criteria values are pythagorean fuzzy numbers, interval-valued fuzzy numbers, and trapezoidal fuzzy numbers, respectively. However, these fuzzy numbers only can express the fuzzy, uncertain, and incomplete information in the decision-making processes [15 –18] while they overolook the neutrality in evalation information. Therefore, intuitionistic fuzzy numbers were used by Chen et al. [10] and Yuan and Li [11] to depict neutral information. Nevertheless, In some practical cases, there is much indeterminate and inconsistent information which intuitionistic fuzzy numbers cannot depict [19, 20]. Hence, the existing methods with fuzzy numbers or intuitionistic fuzzy numbers may not handle problems effectively and efficiently.

Peng and Dai [7] and Peng and Yang [8] utilized point probability to represent the probability distribution of criteria values. Moreover, Liu et al. [12] and Peng et al. [9] proposed FSMCDM methods with interval probability and trapezoidal fuzzy probability, separately. Nevertheless, the point probability, interval fuzzy probability, and trapezoidal fuzzy probability only consider the fuzziness of events’ results while the indeterminacy of events’ results is not taken into accounts [21].

Yuan and Li [11] transformed an intuitionistic trapezoidal fuzzy random matrix into a score function matrix. And a hybrid method, integrating Mahalanobis-Taguchi Gram-Schmidt with improved evidence theory, was utilized to cope with the score function matrix. Espino et al. [13] proposed a multi-criteria model which combined Integrated Value Model for Sustainable Assessments (MIVES) method with some auxiliary complements, including Monte Carlo Simulations, fuzzy sets, and the AHP model. Similarly, Monte Carlo Simulation was combined with VIKOR technique in Ref. [14] to address FSMCDM problems. Obviously, the above methods are based on complete rationality. DMs’ psychological behaviors are not taken into considerations. The results derived from the above methods on the basis of complete rationality may be unreasonable [22].

In order to cover the deficiencies in the existing FSMCDM methods, we propose a FSMCDM method with interval neutrosophic probability based on regret theory within interval neutrosophic environment. Next, the literature reviews related to interval neutrosophic sets (INSs), interval neutrosophic probability, and regret theory, are introduced.

INSs can be regarded as the particular cases of neutrosophic sets (NSs) [20, 23]. To deal with the indeterminate and inconsistent information in practical problems, Smarandache [19, 20] proposed NSs in 1998. However, without a specific presentation, NSs cannot be applied in many real-life cases. Therefore, single-valued neutrosophic sets (SVNSs) were put forward as variations of NSs [20, 23]. In SVNSs, the truth-membership function, indeterminacy-membership function, and falsity-membership function are a single value in interval [0, 1]. Nevertheless, there is an issue when we use SVNSs, that is, experts are difficult to describe alternatives with a certain value in some practical circumstances. To solve this issue, Wang et al. [24] developed INSs where the degrees of truth, indeterminacy, and falsity are an interval value, respectively. If there is only one element in an INS, this element is called as an interval neutrosophic number (INN). The NSs and their particular cases have been applied in many practical problems [25, 26], including evaluation problem of e-commerce websites [27], third party logistics selection problem [28], and multi-criteria decision-making problem [29].

INSs can describe the uncertain information found in stock selection problems appropriately. For instance, an expert is invited to evaluate the prospects of stocks. The stocks will go up or down, and at the same time, there will be great uncertainty. Therefore, the expert may evaluate stocks from three perspectives that are the degree of good future development, the degree of bad prospects, and the uncertainty of stocks. After thinking of the fundamental and technological sides of a stock comprehensively, the expert may consider the degree of future development is between 0.6 and 0.9. Simultaneously, he or she may think the degree of bad prospects is ranging from 0.3 to 0.4. Further, the expert may consider the degree of indeterminacy is fall in the interval [0.2, 0.3]. In this case, the truth-membership degree, indeterminacy-membership degree, and falsity-membership degree of INSs can perfectly describe the degree of good future development, the degree of indeterminacy, and the degree of bad prospects, respectively. The evaluation information can be represented by INSs as

$A = {〈 x, [0.6, 0.9], [0.2, 0.3], [0.3, 0.4] 〉 | x \in X |} .$

Many researchers have investigated MCDM methods under interval neutrosophic environment. The researches can be roughly grouped into three categories. The first group is based on aggregation operators, such as interval neutrosophic weighted averaging operator and weighted geometric operator [30], interval neutrosophic ordered weighted averaging operator and ordered weighted geometric operator [31], interval neutrosophic prioritized ordered weighted averaging operator [32] and some power generalized aggregation operators [33]. The second group is based on measures, including distance measure [34], similarity measures [35, 36], correlation coefficients [37, 38], and cross-entropies [39, 40]. The third group is on the basis of outranking relations [41].

Neutrosophic probability (NP) was originally proposed by Smarandache [21]. Then NP has been applied in many domains, such as quantum physics [42], data classification [43], target identification [44], but there is little research on NP in handling MCDM problems. NP is composed with three parts. That is, the chance that an event occurs, the chance that an event does not occur, and the indeterminate chance related to an event. Compared to point probability, interval probability, and fuzzy probability, it can depict the results of events perfectly. For example, there are two candidates No.1 and No.2 for presidency, and the probability that No.1 wins is 0.46, the probability that No.2 wins is 0.45, then 0.09 would be the probability of blank (the voters are not choosing any candidate) and black (the voters reject both candidates) votes together. Subsequently, point probability and fuzzy probability cannot describe three results at the same time. Neutrosophic probability that the candidate No.1 wins can be denoted by NP = (0.46, 0.09, 0.45) [21]. In some circumstances, experts give the probability with a range rather than a specific number. Therefore, interval neutrosophic probability, which is derived from NP, is developed in this study.

In order to make decisions more reasonable and effective, many studies have conducted based on bounded rationality rather than complete rationality. In the MCDM methods based on the assumptions of complete rationality, DMs make decisions with complete information, sufficient knowledge, and a large amount of time. However, in some practical decision-making cases, DMs are hard to select the alternative rationally due to their limitations of professional knowledge, information processing capability, and available time [22]. Thus, Simon [45] introduced the concept of bounded rationality in 1947. Since then, more and more MCDM methods have been studied in the perspective of bounded rationality [46, 47]. One of the most important decision-making theories is regret theory [22, 48].

In regret theory, DMs conduct comparison analyses between the results of selected alternative and results of other alternatives, so that their psychological behaviors will have regretful and joyful emotions. If the outcomes of selected alternative are worse than other alternatives, DMs may feel regretful during the decision-making processes. Otherwise, that is, if selected alternative is the best one, DMs may have rejoicing in their mind. Bell [49] and Loomes and Sugden [50] initially proposed regret theory, then many researchers have paid attention to regret theory [7, 8, 51]. Rai et al. [51] studied a compromise MCDM method in view of regret theory, subsequently, the method was utilized to deal with material selection problems. Moreover, Interval-valued fuzzy soft methods and pythagorean Fuzzy methods were proposed based on regret theory by Peng and Yang [8] and Peng and Dai [7], respectively.

Based on the above illustrations, a number of contributions are offered in this paper:

Interval neutrosophic probability is proposed to represent the fuzziness and indeterminacy of the result of one event, which is superior to classical probability and fuzzy probability in description.

The method based on regret theory is firstly proposed under interval neutrosophic environment, and this method is significant in theoretical innovation.

The proposed method is applied in Shanghai Stock Exchange and help investors to select stocks, which broaden the application environment of regret theory. Thus, this paper not only has theoretical innovation, but also has application contribution.

The rest of this paper is arranged below. In Section 2, we introduce the basic concepts of INSs and NP at the beginning. And the regret theory is also reviewed. In Section 3, we defined interval neutrosophic probability derived from NP. In Section 4, an MCDM method based on regret theory is proposed. Then, Section 5 provides an example of stock selection and compares the proposed method with the existing methods under interval neutrosophic environment. Similarly, a comparative analysis between prospect theory and regret theory is investigated. Section 6 offers a conclusion.

2 Preliminaries

This section presents the details of interval neutrosophic numbers and neutrosophic probability. Regret theory is also introduced. These definitions will be used in subsequent discussions.

2.1 Interval neutrosophic numbers

Definition 1. [24] Let X be a non-empty fixed set, with generic elements in X denoted by x. An INS A is characterized by a truth-membership degree T_A (x), an indeterminacy-membership degree I_A (x), and a falsity-membership degree F_A (x). For each element x in X, we have that $T_{A} (x) = [T_{A}^{L}, T_{A}^{U}]$ , $I_{A} (x) = [I_{A}^{L}, I_{A}^{U}]$ , $F_{A} (x) = [F_{A}^{L}, F_{A}^{U}] \subseteq [0, 1]$ , and $0 \leq T_{A}^{U} + I_{A}^{U} + F_{A}^{U} \leq 3$ . Therefore, an INS is denoted by $A = {〈 x, [T_{A}^{L}, T_{A}^{U}], [I_{A}^{L}, I_{A}^{U}], [F_{A}^{L}, F_{A}^{U}] 〉 | x \in X |} .$ (1)

A is called an interval neutrosophic number (INN) if X has only one element, denoted by $A = 〈 [T_{A}^{L}, T_{A}^{U}], [I_{A}^{L}, I_{A}^{U}], [F_{A}^{L}, F_{A}^{U}] 〉$ .

Definition 2. [24, 34] Let A be an INN, then the complement of A can be defined as: $A^{c} = 〈 [F_{A}^{L}, F_{A}^{U}], [1 - I_{A}^{U}, 1 - I_{A}^{L}], [T_{A}^{L}, T_{A}^{U}] 〉 .$ (2)

Definition 3. [34, 36] Let A and B be two INNs, then Euclidean distance between A and B is defined below: $\begin{matrix} d (A, B) = \\ \frac{1}{6} [{(T_{A}^{L} - T_{B}^{L})}^{2} + {(T_{A}^{U} - T_{B}^{U})}^{2} + {(I_{A}^{L} - I_{B}^{L})}^{2} \\ {+ {(I_{A}^{U} - I_{B}^{U})}^{2} + {(F_{A}^{L} - F_{B}^{L})}^{2} + {(F_{A}^{U} - F_{B}^{U})}^{2}]}^{\frac{1}{2}} \end{matrix} .$ (3)

Further, the operation laws and comparison method employed in subsequent discussion are referenced from Ref. [30].

2.2 Neutrosophic probability

Some notions related to NP are introduced in this subsection, and an example is given to illustrate the property of NP.

Definition 4. [21] Let φ be an event. The NP is composed of the chance that φ occurs ch (φ), the indeterminate chance related to φ ch (indeterm_φ), and the chance that φ does not occur $ch (\bar{φ})$ . A NP is denoted by $NP (φ) = (ch (φ), ch ({indeterm}_{φ}), ch (\bar{φ})),$ (4) where ch (φ), ch (indeterm_φ), and $ch (\bar{φ})$ are standard or nonstandard subsets of the nonstandard unitary interval ] ^-0, 1⁺]. In particular, the NP is regarded as crisp neutrosophic probability if ch (φ), ch (indeterm_φ), and $ch (\bar{φ})$ are a number ranging from 0 to 1, respectively. For the crisp neutrosophic probability, one has $^{-} 0 \leq c h (\bar{φ}) + c h (i n d e t e r m_{φ}) + c h (\bar{φ}) \leq 3^{+}$ .

It should be noted that the sum of neutrosophic probability components may be greater than 1 when we forecast an event from different criteria. And the following example is illustrated.

Example 1. [21] There will be a game between two handball teams No.3 and No.4 next week. According to the history of their previous disputes, team No.3 have 60% chance to win. On the basis of their last games with other handball teams in the actual season, team No.4 has a better performance than team No.3, and experts think that team No.4 has 70% chance to win upon this criterion. Further, others believe the chance that their game will be tie is 10% . Therefore, the NP that team No.3 wins over team No.4 is NP = (0.6, 0.1, 0.7). That is, the sum of three components is 0.6 + 0.1 + 0.7 > 1.

2.3 Regret theory

Bell [49] and Loomes and Sugden [50] initially proposed regret theory which is an important reasoning method. The regret theory reflects preferences by a utility function which considers regret and rejoicing. Regret represents the difference in DMs’ position resulting from choosing one of the unselected alternatives instead of the selected alternative. Rejoicing represents the extra pleasure gained from knowing that the optimal alternative wasselected [22].

Let x_i be the a criteria value of alternative A_i for i = 1, 2, …, m, respectively. The modified utility function of obtaining x_i is defined as [22]: $u_{i} = V (x_{i}) + R (V (x_{i}) - V (x^{*})),$ (5) where V (x) denotes the utility value; R (V (x_i) - V (x^*)) represents the regret-rejoice value; x^* is the maximum among x_i for i = 1, 2, …, m.

(1) The utility value

Let X = [X⁻, X⁺] be a criteria value, then the utility value of X is denoted by the following equation [52]: $V (X) = \int_{X^{-}}^{X^{+}} v (x) f (x) dx,$ (6) where v (x) is utility function and f (x) is the probability density function.

Bell [49] defined utility function as: $v (x) = \frac{1 - e^{- λ x}}{λ}, 0 < λ < 1,$ (7) where λ is risk aversion coefficient of DMs. The value of λ is larger, DMs are more prone to risk aversion. It should be noted that utility function v (x) is a monotonically increasing concave function, satisfying v′ (x) > 0 and v″ (x) < 0. Considering the parameter λ, Refs. [7, 8] gaveλ = 0.02.

In this paper, for probability density function, we take the normal distribution into consideration. The probability density function is defined as[53]: $f (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(x - μ)}^{2}}{2 σ^{2}}},$ (8) where μ = (X⁻ + X⁺)/ - 2 is the mean value, σ = (X⁺ - X⁻)/ - 6 is the standard deviation.

(2) The regret-rejoice function

The regret-rejoice function is established to obtain the regret values of all alternatives. Bell [49] developed the regret-rejoice function by the following equation: $R (Δ v_{i}) = 1 - e^{- η Δ v_{i}},$ (9) where η is risk aversion coefficient ranging from 0 to 1. The value of η is larger, DMs are more prone to risk aversion. Δv denotes the utility value difference between V (x_i) and V (x^*) for i = 1, 2, …, m.

3 Interval neutrosophic probability

As the derivation of NP, interval neutrosophic probability is defined in this section. Next, an example is provided to illustrate the usage of interval neutrosophic probability.

In some real-life situations, NP cannot be used effectively to cope with MCDM problems. Due to the limitations of knowledge and complexity of environments, experts are hard to give specific numbers to describe the probabilities of events’ results. They may provide interval values because they cannot give the probability exactly. Therefore, interval neutrosophic probability is developed from NP. Since ch (φ), ch (indeterm_φ), and $ch (\bar{φ})$ are standard or nonstandard subsets of the nonstandard unitary interval ] ^-0, 1⁺] in a NP, we give the definitions of interval neutrosophic probability when the three components are an interval value included in [0, 1], respectively.

Definition 5. Let $INP = ([p_{i}^{TL}, p_{i}^{TU}], [p_{i}^{IL}, p_{i}^{IU}],$ $[p_{i}^{FL}, p_{i}^{FU}])$ (i = 1, 2, …, n) be n-dimensional interval neutrosophic probabilities depicting the probability of fundamental events in event set Ω. And the n-dimensional interval neutrosophic probabilities should satisfy $0 \leq p_{i}^{TL} \leq p_{i}^{TU} \leq 1$ , $0 \leq p_{i}^{IL} \leq p_{i}^{IU} \leq 1$ , and $0 \leq p_{i}^{FL} \leq p_{i}^{FU} \leq 1$ for i = 1, 2, …, n.

Example 2. The trends of China’s economy in the next few years have three kinds of status, good, fair, and bad. Economists think the chance of good status occurs is 0.5 at least and 0.9 at most according to the China’s economy over the past few years. On the basis of the current foreign and domestic economic conditions, the chance of good status does not occur will be fall within the range [0.3, 0.6]. In addition, some experts consider the China’s economy is uncertain, and the indeterminate chance is between 0.1 and 0.2. Therefore, the interval neutrosophic probability that the goodstatus occur is ([0.5, 0.9] , [0.1, 0.2] , [0.3, 0.6]). Similarly, experts can give interval neutrosophic probabilities of fair and bad status, ([0.7, 0.8] , [0.2, 0.3] , [0.4, 0.7]) and ([0.4, 0.8] , [0.1, 0.2] , [0.5, 0.8]), respectively.

From Definition 5, we can easily obtain $0 \leq p_{i}^{TU} + p_{i}^{IU} + p_{i}^{FU} \leq 3$ for i = 1, 2, …, n. Therefore, similar to NP, the sum of upper bound of three components in every interval neutrosophic probability does not need to be equal to 1. In Example 2, 0.9 + 0.2 + 0.6 ≠ 1, 0.8 + 0.3 + 0.7 ≠ 1, and 0.8 + 0.2 + 0.8 ≠ 1. Meanwhile, what should be paid attention to is $0 \leq \sum_{i = 1}^{n} p_{i}^{TU} \leq 3$ , that is, the sum of upper bound of the possibility of chance in all status does not need to be equal to 1. In Example 2,0.9 + 0.8 + 0.8 ≠ 1.

4 The novel MCDM method with interval neutrosophic probability

In this section, we describe decision-making problems with interval neutrosophic probability, and then a novel MCDM method based on regret theory is proposed to solve them.

4.1 Description of the decision-making problems

For a MCDM problem with interval neutrosophic information, assume that there are m alternatives A_i (i = 1, 2, …, m) to be evaluated with respect to n criteria C_j (j = 1, 2, …, n). Suppose that the weight vector of decision criteria is ω = (ω₁, ω₂, …, ω_n) where 0 ≤ ω_j ≤ 1 and $\sum_{j = 1}^{n} ω_{j} = 1$ . Let θ_t (t = 1, 2, …, q) be q kinds of possible nature status. For each kind of status, let ${INP}_{t} = ([p_{t}^{TL}, p_{t}^{TU}], [p_{t}^{IL}, p_{t}^{IU}], [p_{t}^{FL}, p_{t}^{FU}])$ be interval neutrosophic probability of the status θ_t for t = 1, 2, …, q, respectively, where $\sum_{t = 1}^{q} p_{t}^{TL} \leq 1 \leq \sum_{t = 1}^{q} p_{t}^{TU}$ and $p_{t}^{TL} + p_{t}^{IL} + p_{t}^{FL} \leq 1 \leq p_{t}^{TU} + p_{t}^{IU} + p_{t}^{FU}$ under each status θ_t. Let $B^{t} = {[b_{ij}^{t}]}_{m \times n}$ be the decision matrix, and $b_{ij}^{t} = 〈 [T_{ij}^{Lt}, T_{ij}^{Ut}], [I_{ij}^{Lt}, I_{ij}^{Ut}], [F_{ij}^{Lt}, F_{ij}^{Ut}] 〉$ is evaluation information for the alternative A_i with respect to C_j under the status θ_t.

4.2 Decision-making procedures based on regret theory

The main procedures of novel MCDM method based on regret theory are studied in this subsection. The proposed method based on regret theory focus on regret aversion of DMs.

Step 1. Standardizing the decision matrix.

Since there are both benefit and cost criteria in an MCDM problem, the decision matrix should be standardized so as to transform various criteria value into comparable value. If criterion C_j is a benefit criterion, interval neutrosophic number $b_{ij}^{t}$ does not need to be standardized, otherwise, $b_{ij}^{t}$ should be transformed to the complement of $b_{ij}^{t}$ . By Definition 2, the equation of the standardized transformation is defined as: $u_{ij}^{t} = {\begin{matrix} b_{ij}^{t}, & if C_{j} is a benefit criterion, \\ (b_{ij}^{t})^{c}, & if C_{j} is a cost criterion . \end{matrix}$ (10)

Step 2. Determining the ideal point.

Let $u_{j}^{*} = 〈 [T_{j}^{L *}, T_{j}^{U *}], [I_{j}^{L *}, I_{j}^{U *}], [F_{j}^{L *}, F_{j}^{U *}] 〉$ be the ideal point regarding criterion C_j. It can be calculated according to the formula below:

$\begin{matrix} u_{j}^{*} & = & max {u_{ij}^{t} | 1 < i < m, 1 < t < q}, \\ j = 1, 2, \dots, n . \end{matrix}$ (11)

Step 3. Computing the utility values.

The utility values of $u_{ij}^{t}$ for alternative A_i regarding criterion C_j under status θ_t can be calculated as follows: $\begin{matrix} V (u_{ij}^{t}) = \\ \frac{1}{3} \int_{T_{ij}^{Lt}}^{T_{ij}^{Ut}} \frac{1 - e^{- 0.02 x}}{0.02} \frac{1}{\sqrt{2 π} σ_{ij}^{Tt}} e^{- \frac{{(x - μ_{ij}^{Tt})}^{2}}{2 {(σ_{ij}^{Tt})}^{2}}} dx \\ + \frac{2}{3} \times \int_{δ}^{1} \frac{1 - e^{- 0.02 x}}{0.02} \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(x - μ)}^{2}}{2 {(σ)}^{2}}} dx \\ - \frac{1}{3} \int_{I_{ij}^{Lt}}^{I_{ij}^{Ut}} \frac{1 - e^{- 0.02 x}}{0.02} \frac{1}{\sqrt{2 π} σ_{ij}^{It}} e^{- \frac{{(x - μ_{ij}^{It})}^{2}}{2 {(σ_{ij}^{It})}^{2}}} dx \\ - \frac{1}{3} \int_{F_{ij}^{Lt}}^{F_{ij}^{Ut}} \frac{1 - e^{- 0.02 x}}{0.02} \frac{1}{\sqrt{2 π} σ_{ij}^{Ft}} e^{- \frac{{(x - μ_{ij}^{Ft})}^{2}}{2 {(σ_{ij}^{Ft})}^{2}}} dx, \end{matrix}$ (12) where $μ_{ij}^{Tt} = (T_{ij}^{Lt} + T_{ij}^{Ut}) / - 2$ , $σ_{ij}^{Tt} = (T_{ij}^{Ut} -$ $T_{ij}^{Lt}) / - 6$ , $μ_{ij}^{It} = (I_{ij}^{Lt} + I_{ij}^{Ut}) / - 2$ , $σ_{ij}^{It} = (I_{ij}^{Ut} -$ $I_{ij}^{Lt}) / - 6$ , $μ_{ij}^{Ft} = (F_{ij}^{Lt} + F_{ij}^{Ut}) / - 2$ , and $σ_{ij}^{Ft} =$ $(F_{ij}^{Ut} - F_{ij}^{Lt}) / - 6$ . In particular, δ = 1 - ξ and ξ is a very tiny positive number. Meanwhile, μ = (δ + 1)/ - 2 and σ = (1 - δ)/ - 6.

Step 4. Computing the regret-rejoice function.

The regret values $R_{ij}^{t}$ for alternative A_i with respect to criterion C_j under status θ_t can be determined by the regret-rejoice function. And the regret-rejoice function can be represented by following equation: $R_{ij}^{t} = R (V (u_{ij}^{t}) - V (u_{j}^{*})) = 1 - e^{- η (V (u_{ij}^{t}) - V (u_{j}^{*}))},$ (13)

Step 5. Computing the comprehensive perceived utility values.

Let E_ij be the comprehensive perceived utility value for alternative A_i in terms of criterion C_j, then E_ij can be calculated as follows: $E_{ij} = \sum_{t = 1}^{q} {INP}_{t} \times (V (u_{ij}^{t}) + R_{ij}^{t}),$ (14)

The comprehensive perceived utility matrix is denoted by EX = (E_ij) _m×n and E_ij is an interval neutrosophic number.

Step 6. Determining the criteria weights.

The criterion weight ω_j with respect to criterion C_j can be calculated as follows: $ω_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (E_{ij}, E_{kj})}{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (E_{ij}, E_{kj})},$ (15) where d (E_ij, E_kj) is Euclidean distance between the comprehensive perceived utility value E_ij and E_kj which is defined in Definition 3.

Step 7. Identifying the positive and negative ideal solution.

Let $f^{*} = 〈 f_{1}^{*}, f_{2}^{*}, \dots, f_{n}^{*} 〉$ and $f^{-} = 〈 f_{1}^{-}, f_{2}^{-}, \dots, f_{n}^{-} 〉$ be the positive and negative ideal solution, respectively. Utilizing the comparison method in Ref. [30], the ideal solution $f_{j}^{*}$ and $f_{j}^{-}$ regarding C_j can be obtained by following formulas: $f_{j}^{*} = max {E_{ij} | 1 < i < m}, for j = 1, 2, \dots, n,$ (16) $f_{j}^{-} = min {E_{ij} | 1 < i < m}, for j = 1, 2, \dots, n .$ (17)

Step 8. Calculating the relative closeness degree.

We can obtain the relative closeness degree D (A_i) for the alternative A_i by the equation below: $D (A_{i}) = \frac{d^{-} (E_{ij}, f^{-})}{d^{*} (E_{ij}, f^{*}) + d^{-} (E_{ij}, f^{-})},$ (18) where $d^{*} (E_{ij}, f^{*}) = \sum_{j = 1}^{n} ω_{j} d (E_{ij}, f^{*})$ and $d^{-} (E_{ij}, f^{-}) = \sum_{j = 1}^{n} ω_{j} d (E_{ij}, f^{-})$ .

Step 9. Ranking all alternatives.

We can use relative closeness degree to rank all the alternatives. The greater the value of D (A_i) is, the better the alternative is.

5 Illustration example of stock assessment process

In this section, the proposed method based on regret theory in this paper is applied in stock assessment process to demonstrate its effectiveness and applicability.

Shanghai Stock Exchange (SSE) is one of two stock exchanges in mainland China. Since the establishment of SSE in 1990, a great deal of companies was listed on the SSE. As of March 2017, there are 1236 companies come into SSE, of which the total market capitalization has reached ¥ 298 trillion. The SSE incorporates many industries included banking business, real-estate industry, manufacturing industry, construction business, and new energy industry. New energy is the resource beyond traditional energy which composes solar energy, wind energy, biomass energy, and geothermal energy. With the decreases of traditional energy on earth, the new energy leads to more and moreattentions.

An investment company wants to select a new energy stock in SSE to invest. Assume that there are four alternatives stocks to select. That is, Aerospace Electrical A₁, Great Wall Electrical A₂, Jingneng Thermal Power A₃, and Minjiang Hydropower A₄. To make decisions objectively and scientifically, this investment company invites many new energy experts to evaluate the stocks with respect to three criteria. That is, liquidity C₁, profitability C₂, and stability C₃. The detailed descriptions of these criteria are shown in Table 1 and the criteria weights information is completely unknown.

Table 1
Decision criteria and their descriptions

Criterion Description

Liquidity Liquidity represents the ability of stock cash flow, denoted by transaction numbers, cash inflows, and payments of the stock in a period of time.

Profitability Profitability represents the ability of company payoff, denoted by the earnings per share and dividends per share.

Stability Stability represents the ability of company sustainable development, denoted by R&D expenditure and marketing expenditure.

Criterion	Description
Liquidity	Liquidity represents the ability of stock cash flow, denoted by transaction numbers, cash inflows, and payments of the stock in a period of time.
Profitability	Profitability represents the ability of company payoff, denoted by the earnings per share and dividends per share.
Stability	Stability represents the ability of company sustainable development, denoted by R&D expenditure and marketing expenditure.

During the process of stock selection, we should evaluate future trend of the stocks in corresponding stock markets. Since uncertainty is existing in stock exchange market, it has three possible status: good θ₁, fair θ₂, and poor θ₃. The probability of each kind of status is an interval neutrosophic probability, denoted by INP₁ = ([0.2, 0.5] , [0.1, 0.3] , [0.5, 0.6]), INP₂ = ([0.3, 0.8] , [0.2, 0.3] , [0.4, 0.7]), and INP₃ = ([0.4, 0.7] , [0.1, 0.2] , [0.4, 0.5]). Interval neutrosophic probability can fully depict the indeterminacy of three status, and the proposed methods with interval neutrosophic probability may be reasonable and feasible.

Many stocks under different status have different performance. Some stocks may have good performance under good status but have very bad performance under bad status. Also, some stocks may not have pretty good performance under good status but have not bad performance under bad status. Therefore, we should assess these stocks under three status, respectively. Fuzzy set theory is applied widely in the process of stock assessment [1, 6] to depict the evaluation information. And the unique properties of INNs introduced in Section 1 allow them to express fuzzy information perfectly and conveniently. As a result, we suggest experts utilize INNs to evaluate these four alternatives with respect to three criteria under three statuses. For example, when experts evaluate alternative A₁ regarding criterion C₁ under status θ₁, they may consider the degree that A₁ has good future development to fall within the range [0.7, 0.8]. Simultaneously, the experts think the degree that A₁ has bad prospects is ranging from 0.2 to 0.4. Further, the degree of indeterminacy is 0.2 at least and 0.3 at most. In this case, the evaluation information is depicted as 〈 [0.7, 0.8] , [0.2, 0.3] , [0.2, 0.4]〉. All the assessment information of experts is presented as B¹, B² and B³. $\begin{matrix} B^{1} = {[\begin{matrix} 〈 [0.7, 0.8], [0.2, 0.3], [0.2, 0.4] 〉 & 〈 [0.7, 0.9], [0.1, 0.3], [0.1, 0.4] 〉 & 〈 [0.8, 0.9], [0.1, 0.2], [0.4, 0.5] 〉 \\ 〈 [0.8, 0.9], [0.2, 0.3], [0.1, 0.3] 〉 & 〈 [0.5, 0.7], [0.3, 0.4], [0.1, 0.2] 〉 & 〈 [0.6, 0.8], [0.2, 0.4], [0.4, 0.5] 〉 \\ 〈 [0.6, 0.9], [0.1, 0.3], [0.1, 0.2] 〉 & 〈 [0.5, 0.8], [0.1, 0.2], [0.2, 0.4] 〉 & 〈 [0.7, 0.9], [0.1, 0.2], [0.2, 0.3] 〉 \\ 〈 [0.7, 0.8], [0.2, 0.3], [0.1, 0.2] 〉 & 〈 [0.6, 0.7], [0.2, 0.4], [0.1, 0.4] 〉 & 〈 [0.5, 0.7], [0.2, 0.3], [0.2, 0.3] 〉 \end{matrix}]}_{4 x 3}, \\ B_{ij}^{2} = {[\begin{matrix} 〈 [0.8, 0.9], [0.3, 0.4], [0.2, 0.3] 〉 & 〈 [0.8, 0.9], [0.1, 0.2], [0.4, 0.5] 〉 & 〈 [0.5, 0.6], [0.2, 0.3], [0.1, 0.3] 〉 \\ 〈 [0.6, 0.8], [0.3, 0.4], [0.4, 0.5] 〉 & 〈 [0.8, 0.9], [0.3, 0.4], [0.2, 0.3] 〉 & 〈 [0.6, 0.7], [0.3, 0.4], [0.1, 0.2] 〉 \\ 〈 [0.7, 0.8], [0.2, 0.3], [0.2, 0.3] 〉 & 〈 [0.7, 0.9], [0.1, 0.2], [0.1, 0.3] 〉 & 〈 [0.8, 0.9], [0.1, 0.2], [0.4, 0.5] 〉 \\ 〈 [0.7, 0.9], [0.1, 0.2], [0.1, 0.2] 〉 & 〈 [0.5, 0.7], [0.3, 0.4], [0.2, 0.4] 〉 & 〈 [0.6, 0.7], [0.2, 0.4], [0.4, 0.5] 〉 \end{matrix}]}_{4 x 3}, \\ B_{ij}^{3} = {[\begin{matrix} 〈 [0.8, 0.9], [0.1, 0.3], [0.3, 0.5] 〉 & 〈 [0.5, 0.6], [0.3, 0.4], [0.4, 0.5] 〉 & 〈 [0.6, 0.9], [0.1, 0.2], [0.3, 0.4] 〉 \\ 〈 [0.5, 0.7], [0.3, 0.4], [0.1, 0.3] 〉 & 〈 [0.6, 0.7], [0.2, 0.3], [0.1, 0.2] 〉 & 〈 [0.8, 0.9], [0.2, 0.3], [0.3, 0.4] 〉 \\ 〈 [0.8, 0.9], [0.1, 0.4], [0.1, 0.2] 〉 & 〈 [0.7, 0.8], [0.2, 0.3], [0.1, 0.3] 〉 & 〈 [0.7, 0.9], [0.2, 0.4], [0.3, 0.5] 〉 \\ 〈 [0.6, 0.8], [0.1, 0.3], [0.2, 0.3] 〉 & 〈 [0.6, 0.7], [0.1, 0.3], [0.1, 0.3] 〉 & 〈 [0.7, 0.8], [0.2, 0.4], [0.2, 0.3] 〉 \end{matrix}]}_{4 x 3} . \end{matrix}$ (19)

DMs may have strong psychological activities in the process of stock selection. That is, DMs may have a feeling of regret aversion when choosing stocks. Therefore, the proposed method based on regret theory can be utilized to evaluate stocks effectively and efficiently.

5.1 Illustration of the proposed method based on regret theory

In this subsection, we utilize the proposed method based on regret theory to evaluate the stock alternatives. All the alternatives are ranked and the best alternative is selected.

Step 1. Standardizing the decision matrix.

The decision matrix do not need to standardize since all the corresponding criteria for alternatives are benefit criteria.

Step 2. Determining the ideal point.

According to comparison of criteria values, the ideal point can be identified by Equation (11). The ideal point is: $\begin{matrix} u^{*} = {〈 [0.7, 0.9], [0.1, 0.2], [0.1, 0.2] 〉, \\ 〈 [0.7, 0.9], [0.1, 0.2], [0.1, 0.3] 〉, \\ 〈 [0.7, 0.9], [0.1, 0.2], [0.2, 0.3] 〉} . \end{matrix}$ (20)

Step 3. Computing the utility values.

The utility values are calculated by Equation (12) and shown in Table 2.

Table 2

The utility values

	Status θ₁			Status θ₂			Status θ₃
	C ₁	C ₂	C ₃	C ₁	C ₂	C ₃	C ₁	C ₂	C ₃
A ₁	0.7234	0.7729	0.7397	0.7396	0.7397	0.6909	0.7397	0.5752	0.7400
A ₂	0.7892	0.6908	0.6410	0.6245	0.7396	0.7072	0.6742	0.7403	0.7396
A ₃	0.7896	0.7237	0.7894	0.7399	0.8060	0.7397	0.8058	0.7565	0.6902
A ₄	0.7730	0.6906	0.6907	0.8225	0.6412	0.6246	0.7401	0.7403	0.7234

Table 3

The regret values

	Status θ₁			Status θ₂			Status θ₃
	C ₁	C ₂	C ₃	C ₁	C ₂	C ₃	C ₁	C ₂	C ₃
A ₁	−0.0302	−0.0100	0.0150	−0.0252	−0.0201	−0.0300	−0.0252	−0.0717	−0.0149
A ₂	−0.0100	−0.0352	−0.0455	−0.0612	−0.0201	−0.0250	−0.0455	−0.0199	−0.0151
A ₃	−0.0099	−0.0250	0	−0.0251	0	−0.0150	−0.0050	−0.0150	−0.0302
A ₄	−0.0150	−0.0352	−0.0300	0	−0.0507	−0.0507	−0.0250	−0.0199	−0.0200

Table 4

The effect of parameter η on the ranking results

	A ₁	A ₂	A ₃	A ₄	Ranking results
η = 0.1	0.3667	0.3000	1.0000	0.5130	A₃ ≻ A₄ ≻ A₁ ≻ A₂
η = 0.3	0.3709	0.3020	1.0000	0.5150	A₃ ≻ A₄ ≻ A₁ ≻ A₂
η = 0.5	0.3756	0.3048	1.0000	0.5177	A₃ ≻ A₄ ≻ A₁ ≻ A₂
η = 0.7	0.3807	0.3081	1.0000	0.5210	A₃ ≻ A₄ ≻ A₁ ≻ A₂

Step 4. Computing the regret-rejoice function.

Giving parameter η = 0.3. Using Equation (13), the regret values can be computed in Table 3.

Step 5. Computing the comprehensive perceived utility values.

According to the utility values and regret values calculated before, the comprehensive perceived utility values can be obtained by Equation (14) and shown as follows: $\begin{matrix} E_{ij} = [\begin{matrix} 〈 [0.5391, 0.9171], [0.0124, 0.0582], [0.1670, 0.3315] 〉 \\ 〈 [0.5014, 0.8896], [0.0158, 0.0722], [0.1955, 0.3553] 〉 \\ 〈 [0.5674, 0.9297], [0.0083, 0.0456], [0.1453, 0.2987] 〉 \\ 〈 [0.5630, 0.9335], [0.0090, 0.0472], [0.1445, 0.3084] 〉 \end{matrix} \\ \begin{matrix} 〈 [0.4955, 0.8991], [0.0170, 0.0746], [0.1921, 0.3696] 〉 \\ 〈 [0.5374, 0.9162], [0.0132, 0.0599], [0.1697, 0.3359] 〉 \\ 〈 [0.5605, 0.9310], [0.0099, 0.0495], [0.1492, 0.3140] 〉 \\ 〈 [0.5156, 0.8969], [0.0163, 0.0700], [0.1910, 0.3518] 〉 \end{matrix} . \\ {\begin{matrix} 〈 [0.5360, 0.9127], [0.0123, 0.0587], [0.1700, 0.3301] 〉 \\ 〈 [0.5259, 0.9077], [0.0160, 0.0669], [0.1824, 0.3500] 〉 \\ 〈 [0.5378, 0.9186], [0.0111, 0.0558], [0.1627, 0.3265] 〉 \\ 〈 [0.5091, 0.8923], [0.0172, 0.0729], [0.1963, 0.3571] 〉 \end{matrix}]}_{4 x 3} \end{matrix}$ (21)

Step 6. Determining the criteria weights.

Using Equation (15), the criteria weights are obtained as ω = (0.4008, 0.3727, 0.2264).

Step 7. Identifying the positive and negative ideal solutions.

The positive and negative ideal solutions are determined by Equations (16, 17), and the ideal solutions are represented in the following paragraphs: $\begin{matrix} f^{*} = {〈 [0.5674, 0.9297], [0.0083, 0.0456], [0.1453, 0.2987] 〉, \\ 〈 [0.5605, 0.9310], [0.0099, 0.0495], [0.1492, 0.3140] 〉, \\ 〈 [0.5378, 0.9186], [0.0111, 0.0558], [0.1627, 0.3265] 〉}, \end{matrix}$ (22) $\begin{matrix} f^{-} = {〈 [0.5014, 0.8896], [0.0158, 0.0722], [0.1955, 0.3553] 〉, \\ 〈 [0.4955, 0.8896], [0.0158, 0.0722], [0.1955, 0.3553] 〉, \\ 〈 [0.5091, 0.8923], [0.0171, 0.0729], [0.1963, 0.3571] 〉} . \end{matrix}$ (23)

Step 8. Calculating the relative closeness degree.

Using Equation (18), the relative closeness degrees are calculated as follows:

D (A₁) = 0.3709, D (A₂) = 0.3020,

D (A₃) = 1.0000, D (A₄) = 0.5150.

Step 9. Ranking all alternatives.

We can use relative closeness degrees to rank all the alternatives, That is, A₃ ≻ A₄ ≻ A₁ ≻ A₂. Further, the best alternative is A₃.

5.2 Analysis and discussion

This subsection investigates how the parameters influence the results of proposed method at the beginning. Then it conduct a comparative analysis between the method based on prospect theory and method based on regret theory regarding the features of each method. Finally, we compare the proposed method in this paper with the methods proposed under interval neutrosophic environment in other papers and the advantages of proposed method areconcluded.

(1) The effect of parameters on ranking results

For the parameters λ and η of regret theory, parameter λ was provided by scholars [7, 8]. However, the parameter η is variable. Therefore, we only need to change the parameter η to observe variation of ranking results, which is in Table 4.

When we change the parameter η, the ranking orders of all alternatives are not changed. The alternative A₃ is the best and A₂ is the worst all the time. Therefore, the proposed method based on regret theory has good stability in decision-makingprocess.

(2) Comparative analysis between prospect theory and regret theory

The prospect theory is also a significant theory based in bounded rationality just like regret theory. Thus, a comparative analysis is studied between the method based on prospect theory and method based on regret theory.

According to Refs. [54, 55], the method on the basis of prospect theory is developed under interval neutrosophic environment to compare with the method based on regret theory. In the method based on prospect theory, the utility values are denoted by evaluation difference between each alternative and reference point, and the probability weight function is utilized to make a fuzzy stochastic MCDM problem reduce to a fuzzy MCDM problem.

The ranking results of the methods based on prospect theory and regret theory which are shown in Table 5.

It can be seen that ranking results using two methods are identical. Hence, to some extent, the proposed method in this paper is effective and applicable. However, features of these two methods are different and the differences are explained below.

Table 5
The ranking results of two methods

Methods Ranking results

The method based on prospect theory A₃ ≻ A₄ ≻ A₁ ≻ A₂

α = β = 0.88, θ = 2.25, γ = 0.61 and δ = 0.72

The proposed method A₃ ≻ A₄ ≻ A₁ ≻ A₂

λ = 0.02 and η = 0.3

Methods	Ranking results
The method based on prospect theory	A₃ ≻ A₄ ≻ A₁ ≻ A₂
α = β = 0.88, θ = 2.25, γ = 0.61 and δ = 0.72
The proposed method	A₃ ≻ A₄ ≻ A₁ ≻ A₂
λ = 0.02 and η = 0.3

Though the decision-making psychology activities are taken into consideration by both of the method based on prospect theory and method based on regret theory, the focus of each method is different. The prospect theory emphasizes loss aversion while the regret theory considers regret aversion. That is, the prospect theory takes the distinction between individual expectation and selected alternative into account. Meanwhile, the regret theory focuses on comparison between the selected alternative and other alternatives to prevent from regret when choosing a bad alternative. In addition, the computational complexity of each method is different. Namely, the number of parameters in each method is not equal. The method based on prospect theory has five parameters while the method based on regret theory has only two parameters. In other word, the regret theory is easier than prospecttheory.

(3) Comparative analysis with extant methods under interval neutrosophic environment

A comparative analysis is conducted between proposed method and fuzzy MCDM methods with interval neutrosophic information.

Since there is no existing research on fuzzy stochastic decision-making problems under interval neutrosophic environment, for the convenience of comparison, we use expected decision matrix to transform fuzzy stochastic MCDM problems into fuzzy MCDM problems. The expected evaluate value edm_ij for the alternative A_i regarding criterion C_j is ${edm}_{ij} = \sum_{t = 1}^{q} {INP}_{t} b_{ij}^{t}$ . And the expected decision matrix is shown below: $\begin{matrix} E_{ij} = [\begin{matrix} 〈 [0.5556, 0.9378], [0.0234, 0.1302], [0.1810, 0.4503] 〉 \\ 〈 [0.4490, 0.8990], [0.0456, 0.1538], [0.1619, 0.3978] 〉 \\ 〈 [0.5273, 0.9267], [0.0130, 0.1353], [0.1316, 0.3223] 〉 \\ 〈 [0.4837, 0.9261], [0.0149, 0.0987], [0.1316, 0.3359] 〉 \end{matrix} \\ \begin{matrix} 〈 [0.4771, 0.9107], [0.0197, 0.1167], [0.2253, 0.4845] 〉 \\ 〈 [0.4802, 0.9072], [0.0456, 0.1480], [0.1316, 0.3223] 〉 \\ 〈 [0.4881, 0.9261], [0.0149, 0.0852], [0.1270, 0.3903] 〉 \\ 〈 [0.4315, 0.8541], [0.0234, 0.1480], [0.1316, 0.4051] 〉 \end{matrix} . \\ {\begin{matrix} 〈 [0.4574, 0.8942], [0.0130, 0.0808], [0.1868, 0.4424] 〉 \\ 〈 [0.5093, 0.9023], [0.0345, 0.1480], [0.1868, 0.4256] 〉 \\ 〈 [0.5294, 0.9430], [0.0149, 0.1007], [0.2227, 0.4590] 〉 \\ 〈 [0.4686, 0.8742], [0.0282, 0.1538], [0.1997, 0.3978] 〉 \end{matrix}]}_{4 x 3} \end{matrix}$ (24)

Using the expected decision matrix as the evaluation information in fuzzy MCDM problems, and the fuzzy MCDM methods under interval neutrosophic environment are listed below:

In Ye’s method [36], similarity measures between INSs are proposed based on Hamming and Euclidean distances. And the similarity measures between each alternative and ideal alternative are used to rank all the alternatives and obtain the best one.

In Chi and Liu’s method [34], maximizing deviation method is utilized to determine the criteria weights. An extended TOPSIS method based on Hamming distance is developed to rank alternatives.

Since criteria weights are completely known in Ye’s method, we use maximizing deviation method to determine the criteria weights in Ye’s method. Further, for comparing conveniently, we utilize Euclidean distance-based similarity in Ye’s method and Euclidean distance in Chi and Liu’s method. The ranking results obtained by different methods are shown in Table 6.

Table 6

The ranking results of four methods

Methods	Ranking results
Ye’s method [36]	A₃ ≻ A₂ ≻ A₄ ≻ A₁
Chi and Liu’s method [34]	A₃ ≻ A₄ ≻ A₂ ≻ A₁
The method based on regret theory	A₃ ≻ A₄ ≻ A₁ ≻ A₂

It can be seen that there are some differences in Table 6. Though alternative A₃ is the best one in all of the methods, the ranking orders are inconsistent. The reasons why ranking results are different are presented as follows.

The difference between Ye’s method [36] and the proposed method based on regret theory is the orders of A₁, A₂, and A₄. This is due to Ye’s method [36] only can solve MCDM problems on the premise of complete rationality. Psychological behavior of DMs is not taken into account. By contrast, the proposed method can consider regret aversion ofDMs.

The positions of A₁ and A₂ are different between Chi and Liu’s method [34] and the proposed method based on regret theory. Though Chi and Liu’s method [34] take both positive and negative ideal point into consideration, it does not consider bounded rationality. Therefore, the ranking results calculated by Chi and Liu’s method [34] are different from those obtained by the proposed method based on regret theory even both of two methods use maximizing deviation method and TOPSIS.

According to the analyses illustrated above, the proposed method in this paper based on regret theory has a lot of advantages which are representedbelow:

The probability in MCDM problems is in a form of interval neutrosophic probability. There are three possible status in the stock exchange market. That is, good, fair, or poor. Interval neutrosophic probability can fully depict the three possible results while classic probability or the fuzzy probability only can describe one result every time. Hence, the proposed method with interval neutrosophic probability is more reasonable and effective.

The method proposed in this paper take DMs’ psychology activities into consideration in the decision-making process. Compared with the traditional methods based on complete rationality, the proposed method based on bounded rationally is more likely applied in real-life situations.

In MCDM problems, the criteria values are denoted by INSs. As particular cases of neutrosophic sets, INSs can represent the evaluation information perfectly and conveniently in the decision-making process.

The maximizing deviation method under interval neutrosophic environment can be used to calculate criteria weights when weight information is completely unknown.

6 Conclusion

Both fuzzy and stochastic information are existing in real-life decision-making environment. Therefore, FSMCDM methods are studied necessarily to deal with the hybrid information. In this paper, we first proposed interval neutrosophic probability which is derived from NP. Then a novel MCDM method is developed with interval neutrosophic probability based on bounded rationality, in which criteria values are INNs. In the proposed method, maximizing deviation method is used to calculate the criteria weights when weights information is completely unknown. The proposed method is based on regret theory which incorporates TOPSIS to rank the alternatives. Finally, we applied the proposed method in the process of stock selection to verify the availability and applicability of them. Further, we discuss the influences of parameters on ranking results and compare prospect theory with regret theory. The comparative analyses between the proposed method and extant methods with INSs are also conducted.

The proposed method can deal with the evaluation information efficiently and perfectly in stock selection problems with INNs and interval neutrosophic probability. Moreover, interval neutrosophic probability is initially proposed and applied to address MCDM problems in this paper. Subsequently, regret theory are incorporated in proposed method to consider DMs’ psychological preference. Thus, the proposed method have greater advantages than methods based on complete rationality in real-life applications.

In the future, we expect to apply interval neutrosophic probability in some problems which are full of indeterminacy, including the forecast of China’s economic environment, medical diagnosis, and pattern recognition. What’s more, we may use uncertain linguistic variables to represent the probability instead of a specific number or a neutrosophic number. Simultaneously, we could expand the scope of application of regret theory.

Conflict of interests

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

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 71571193).

References

Hatami-Marbini

and Kangi

, An extension of fuzzy TOPSIS for a group decision making with an application to tehran stock exchange, Applied Soft Computing 52 (2017), 1084–1097.

Shen

K.-Y.

and Tzeng

G.-H.

, Combined soft computing model for value stock selection based on fundamental analysis, Applied Soft Computing 37 (2015), 142–155.

Zhou

, Wang

J.Q.

and Zhang

H.Y.

, Stochastic multicriteria decision-making approach based on SMAA-ELECTRE with extended gray numbers, International Transactions in Operational Research (2017). doi: 10.1111/itor.12380, 2016

S.-M.

, Wang

J.-Q.

and Li

, A multi-criteria decision-making model for hotel selection with linguistic distribution assessments, Applied Soft Computing 67 (2018), 741–755.

S.-M.

, Zhang

H.-Y.

and Wang

J.-Q.

, Hesitant fuzzy linguistic maclaurin symmetric mean operators and their applications to multi-criteria decision-making problem, International Journal of Intelligent Systems 33 (2018), 953–982.

Ghobadi

and Torabi

, Investment rankings based on technical analysis by Fuzzy MCDM in Tehran stock exchange, Ekonomika 61 (2015), 117–129.

Peng

and Dai

, Approaches to pythagorean fuzzy stochastic multi-criteria decision making based on prospect theory and regret theory with new distance measure and score function, International Journal of Intelligent Systems 32 (2017), 1187–1214.

Peng

and Yang

, Algorithms for interval-valued fuzzy soft sets in stochastic multi-criteria decision making based on regret theory and prospect theory with combined weight, Applied Soft Computing 54 (2017), 415–430.

Peng

, Liu

and Su

, Research on the random multi-attribute decision-making methods with trapezoidal fuzzy probability based on prospect theory, Journal of Intelligent & Fuzzy Systems 26 (2014), 2131–2141.

10.

Chen

Z.-S.

, Chin

K.-S.

, Ding

and Li

Y.-L.

, Triangular intuitionistic fuzzy random decision making based on combination of parametric estimation, score functions, and prospect theory, Journal of Intelligent & Fuzzy Systems 30 (2016), 3567–3581.

11.

Yuan

and Li

, A new method for multi-attribute decision making with intuitionistic trapezoidal fuzzy random variable, International Journal of Fuzzy Systems 19 (2017), 15–26.

12.

Liu

, Jin

, Zhang

, Su

and Wang

, Research on the multi-attribute decision-making under risk with interval probability based on prospect theory and the uncertain linguistic variables, Knowledge-Based Systems 24 (2011), 554–561.

13.

Jato-Espino

, Rodriguez-Hernandez

, Andrés-Valeri

V.C.

and Ballester-Muñoz

, A fuzzy stochastic multi-criteria model for the selection of urban pervious pavements, Expert Systems with Applications 41 (2014), 6807–6817.

14.

Mousavi

S.M.

, Jolai

and Tavakkoli-Moghaddam

, A fuzzy stochastic multi-attribute group decision-making approach for selection problems, Group Decision and Negotiation 22 (2013), 207–233.

15.

Büyüközkan

and Göçer

, Application of a new combined intuitionistic fuzzy MCDM approach based on axiomatic design methodology for the supplier selection problem, Applied Soft Computing 52 (2017), 1222–1238.

16.

Blanco-Mesa

, Merigó

J.M.

and Gil-Lafuente

A.M.

, Fuzzy decision making: A bibliometric-based review, Journal of Intelligent & Fuzzy Systems 32 (2017), 2033–2050.

17.

Merigó

J.M.

, Fuzzy decision making with immediate probabilities, Computers & Industrial Engineering 58 (2010), 651–657.

18.

Merigó

J.M.

, Gil-Lafuente

A.M.

and Yager

R.R.

, An overview of fuzzy research with bibliometric indicators, Applied Soft Computing 27 (2015), 420–433.

19.

Smarandache

, A unifying field in logics: Neutrosophic logic. Neutrosophy, neutrosophic set, probability, American Research Press, Rehoboth, 1999.

20.

Smarandache

,Neutrosophy: Neutrosophic probability, set, and logic, American Research Press, Rehoboth, 1998.

21.

Smarandache

,Introduction to neutrosophic measure, neutrosophic integral, and neutrosophic probability, Sitech & Education Publisher, Craiova-Columbus, 2013.

22.

Zhou

, Wang

J.-Q.

and Zhang

H.-Y.

, Grey stochastic multi-criteria decision-making based on regret theory and TOPSIS, International Journal of Machine Learning and Cybernetics 8 (2017), 651–664.

23.

Wang

H.-B.

, Smarandache

, Zhang

Y.-Q.

and Sunderraman

, Single valued neutrosophic sets, Muiltispace and Multistruncture 4 (2010), 410–413.

24.

Wang

, Smarandache

, Sunderraman

, Zhang

Y.-Q.

, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis, Phoenix, 2005.

25.

Liang

R.-X.

, Wang

J.-Q.

and Zhang

H.-Y.

, A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information, Neural Computing and Applications (2017). doi: 10.1007/s00521-017-2925-8

26.

Peng

H.-G.

, Zhang

H.-Y.

and Wang

J.-Q.

, Probability multi-valued neutrosophic sets and its application in multi-criteria group decision-making problems, Neural Computing and Applications 30 (2018), 563–583.

27.

Liang

R.-X.

, Wang

J.-Q.

and Zhang

H.-Y.

, Evaluation of e-commerce websites: An integrated approach under a single-valued trapezoidal neutrosophic environment, Knowledge-Based Systems 135 (2017), 44–59.

28.

, Wang

J.-Q.

and Zhang

H.-Y.

, Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers, Neural Computing and Applications (2016). doi: 10.1007/s00521-016-2660-6

29.

Liang

R.-X.

, Wang

J.-Q.

and Li

, Multi-criteria group decision-making method based on interdependent inputs of single-valued trapezoidal neutrosophic information, Neural Computing and Applications 30 (2018), 241–260.

30.

Zhang

H.-Y.

, Wang

J.-Q.

and Chen

X.-H.

, Interval neutrosophic sets and their application in multicriteria decision making problems, The Scientific World Journal (2014). doi: 10.1155/2014/645953

31.

, Multiple attribute decision-making method based on the possibility degree ranking method and ordered weighted aggregation operators of interval neutrosophic numbers, Journal of Intelligent & Fuzzy Systems 28 (2015), 1307–1317.

32.

Liu

and Wang

, Interval neutrosophic prioritized OWA operator and its application to multiple attribute decision making, Journal of Systems Science and Complexity 29 (2016), 681–697.

33.

Liu

and Tang

, Some power generalized aggregation operators based on the interval neutrosophic sets and their application to decision making, Journal of Intelligent & Fuzzy Systems 30 (2016), 2517–2528.

34.

Chi

and Liu

, An extended TOPSIS method for the multiple attribute decision making problems based on interval neutrosophic set, Neutrosophic Sets and Systems 1 (2013), 63–70.

35.

Broumi

and Smarandache

, Cosine similarity measure of interval valued neutrosophic sets, Neutrosophic Theory and Its Applications 5 (2014), 15–20.

36.

, Similarity measures between interval neutrosophic sets and their applications in multicriteria decision-making, Journal of Intelligent & Fuzzy Systems 26 (2014), 165–172.

37.

, Improved correlation coefficients of single valued neutrosophic sets and interval neutrosophic sets for multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 27 (2014), 2453–2462.

38.

Zhang

H.-Y.

, Ji

, Wang

J.-Q.

and Chen

X.-H.

, An improved weighted correlation coefficient based on integrated weight for interval neutrosophic sets and its application in multi-criteria decision-making problems, International Journal of Computational Intelligence Systems 8 (2015), 1027–1043.

39.

Tian

Z.P.

, Zhang

H.Y.

, Wang

J.Q.

and Chen

X.H.

, Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets, International Journal of Systems Science 47 (2016), 3598–3608.

40.

Şahin

, Cross-entropy measure on interval neutrosophic sets and its applications in multicriteria decision making, Neural Computing and Applications 28 (2015), 1177–1187.

41.

Zhang

H.-Y.

, Wang

J.-Q.

and Chen

X.-H.

, An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets, Neural Computing and Applications 27 (2016), 615–627.

42.

Smarandache

, An introduction to the neutrosophic probability applied in quantum physics, arXiv preprint math/0010088, 2000.

43.

Bhutani

, Kumar

, Aggarwal

,Multi-attribute data classification using Neutrosophic probability, in: India Conference (INDICON), IEEE, 2015.

44.

Smarandache

, Abbas

, Chibani

, Hadjadji

and Omar

, PCR5 and neutrosophic probability in target identification, Nonlinear Dynamics and Chaos Advances and Perspectives 4 (2016), 45–50.

45.

Simon

H.A.

, Effects of increased productivity upon the ratio of urban to rural population, Econometrica, Journal of the Econometric Society 15 (1947), 31–42.

46.

Ren

, Xu

and Gou

, Pythagorean fuzzy TODIM approach to multi-criteria decision making, Applied Soft Computing 42 (2016), 246–259.

47.

Zhang

, Zhu

, Liu

and Chen

, Regret theory-based group decision-making with multidimensional preference and incomplete weight information, Information Fusion 31 (2016), 1–13.

48.

Yang

and Wang

J.-Q.

, SMAA-based model for decision aiding using regret theory in discrete Z-number context, Applied Soft Computing 65 (2018), 590–602.

49.

Bell

D.E.

, Regret in decision making under uncertainty, Operations Research 30 (1982), 961–981.

50.

Loomes

and Sugden

, Regret theory: An alternative theory of rational choice under uncertainty, The Economic Journal 92 (1982), 805–824.

51.

Rai

, Jha

G.K.

, Chatterjee

and Chakraborty

, Material selection in manufacturing environment using compromise ranking and regret theory-based compromise ranking methods: A comparative study, Universal Journal of Materials Science 1 (2013), 69–77.

52.

Zhang

, Fan

and Chen

, Risky multiple attribute decision making with regret aversion, Journal of Systems & Management 23 (2014), 111–117.

53.

Zhou

, Xie

, Pan

, Probability theory and mathematical statistics, Higher Education Press, Beijing, 2001.

54.

Kahneman

and Tversky

, Prospect theory: An analysis of decision under risk, Econometrica: Journal of The Econometric Society (1979), 263–291.

55.

Tversky

and Kahneman

, Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992), 297–323.

Fuzzy stochastic multi-criteria decision-making methods with interval neutrosophic probability based on regret theory

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Interval neutrosophic numbers

4 The novel MCDM method with interval neutrosophic probability

4.1 Description of the decision-making problems

4.2 Decision-making procedures based on regret theory

Table 5 The ranking results of two methods Methods Ranking results The method based on prospect theory A3 ≻ A4 ≻ A1 ≻ A2 α = β = 0.88, θ = 2.25, γ = 0.61 and δ = 0.72 The proposed method A3 ≻ A4 ≻ A1 ≻ A2 λ = 0.02 and η = 0.3

Conflict of interests

Footnotes

Acknowledgments

References

Table 5
The ranking results of two methods

Methods Ranking results

The method based on prospect theory A₃ ≻ A₄ ≻ A₁ ≻ A₂

α = β = 0.88, θ = 2.25, γ = 0.61 and δ = 0.72

The proposed method A₃ ≻ A₄ ≻ A₁ ≻ A₂

λ = 0.02 and η = 0.3