Interval neutrosophic stochastic multiple attribute decision-making method based on cumulative prospect theory and generalized Shapley function

Abstract

Concerning the discrete stochastic multi-attribute decision making (SMADM) problems in which attribute values are interval neutrosophic numbers and the attribute weight is incompletely known, a novel SMADM method based on the cumulative prospect theory (CPT) and generalized Shapley function is proposed. Firstly, the value prospect of each attribute under interval neutrosophic environment is calculated as well as subjective probability weights respectively, and the prospect function values are obtained according to the formulation of prospect function. Then, following the maximum deviation principle, the optimization model with respect to the incompletely known attribute weight information is constructed to obtain the optimal fuzzy measure, and the optimal weights of each attribute based on the formulation of generalized Shapley function are generated. Further, by aggregating the above prospect function values and the optimal weights, the values of the comprehensive prospect function are obtained and then alternatives are ranked. Finally, an illustrate example is presented to demonstrate the validity and feasibility of the proposed method.

Keywords

Stochastic multiple-attribute decision making interval neutrosophic set cumulative prospect theory generalized shapley function

1 Introduction

Owning to the complexity of objective things and the limitations of human cognitive ability, as well as the indeterminacy of the problems in the real practice of decision making, more and more people have paid attention to the researches on SMADM, which is a kind of finite alternative selection problem with multiple attributes as stochastic variables. SMADM is an important part of modern decision theory, and has a wide range of practical background in economic management and engineering systems, e.g., project investment decision, project evaluation and production decision etc. From the point of view of the existing literatures on the SMADM, the researches can be roughly classified into two aspects: one is about the psychological behavior of the decision makers, and the other is about the correlation among the attributes.

For the first aspect, most of the SMADM problems are solved by the expected utility theory (EUT) based on the assumption that the decision maker’s behavior is completely rational [1]. However, it is impossible for decision makers to make a decision by complete rationality. Furtherly, Kahneman and Tversky [2, 3] proposed a new theory of bounded rationality called as “prospect theory” (PT), which makes decision-makers rely on the risk attitude and sensitivity according to the reference point selected in the face of gains and losses. PT thought that decision maker’s decision-making process is affected by many factors, and it is prone to produce systematic bias of emotional and cognitive aspects. Obviously, PT is more reasonable to solve the real decision making problems than EUT. To this end, the multi-attribute decision-making problems considering the bounded rational behavior of decision-makers based on PT have been proposed by many scholars [4, 5]. Levy [6] proposed the prospective stochastic dominance criterion based on the PT. This criterion took into account two aspects, that is, the decision maker’s attitude on the selection of reference points and the psychological behavior characteristics such as the alienation risk attitude for the gains and losses. Li et al. [7] presented a SMADM method in which attribute weights are unknown and the attribute values are intuitionistic fuzzy numbers (IFNs) based on prospect theory. Hu and Yang [8] studied the dynamic SMADM problem with discrete dynamic variables in attribute values based on the CPT and the set pair analysis method. Wang et al. [9] proposed a multi-criterion decision making method in which the weight is not fully defined and the criterion value is trapezoidal fuzzy numbers based on PT. Fan et al. [10, 11] solved a kind of multi-attribute decision-making problems in which the attribute values take the form of crisp numbers, interval numbers and linguistic variables based on the cumulative prospect theory. Li et al. [12] proposed an uncertain dynamic multi-attribute risk decision-making method by considering three reference points based on PT in which stage weights were incomplete known and attribute values were interval values. Zhang and Fan [13] proposed a decision method to solve multiple attribute risk decision making problems with different attribute aspiration-levels given by decision makers based on CPT. Zhang and Fan [14] considered the behavioral factors of decision makers for SMADM and transformed the decision matrix with stochastic variables into the matrix of gains and losses relative to the reference points. From above researches, we can draw the conclusions that (1) the above methods for SMADM problems are only for crisp numbers, interval numbers, IFNs and linguistic variables; (2) the above methods for SMADM problems ignore the interaction among attributes.

For the second aspect, there are some correlations among attributes in real SMADM problems [15, 16]. For example, there is a certain correlation among the market share of products and the short-term and long-term profits of enterprises. Therefore, how to solve the SMADM problem existing correlations among attributes has an important theoretical and practical significance. Lahdelma et al. [17] proposed a decision method for the discrete SMADM problem with attribute correlations based on Monte Carlo simulation calculation method. Considering the interactions among attributes, Liu et al. [18] calculated the overall evaluation values in the form of normal stochastic variables by aggregating the normalization of attribute values based on Choquet integral formulas. However, currently, there are few researches on the SMADM problems with the correlations among attributes, and the existing literature only processes the real number, and do not consider the influence of the decision maker’s bounded rational behavior on the decision results.

In addition, it should be noted that attribute values in the above SMADM methods are crisp numbers, fuzzy numbers, linguistic variables, interval values and intuitionistic fuzzy numbers which can only deal with incomplete information, and can’t deal with uncertain, inconsistent information [19]. In order to deal with all kinds of complex information more comprehensively, Smarandache [20, 21] proposed Neutrosophic Set (NS) which has an independent hesitate degree, so NS is more suitable for dealing with uncertain information than the above fuzzy sets. Now, NS has been widely used in image processing, pattern recognition and medical diagnosis, and has achieved remarkable applications [22 , 30]. Further, Interval Neutrosophic Set (INS), as a general form of NS, is produced by extending membership, uncertainty and non-membership of NS to interval, which further enhances the ability to express complex fuzzy information [24, 37]. However, until now, there are no researches for SMADM with interval Neutrosophic information.

Above all, from the above analysis, we can see that there are some gaps in the existing literatures on the SMADM problems. (1) Some methods for the SMADM problems consider the psychological behavior of decision makers based on PT or CPT, however, they don’t consider the interaction among attributes, and attribute values are only for crisp numbers, interval numbers, IFNs and linguistic variables, and they don’t deal with the INS. (2) Some methods for the SMADM problems consider the interaction among attributes, however, they cannot consider psychological behavior of decision makers, and the attribute values are only for crisp numbers and they don’t deal with the INS. Only Liu et al. [25] and Chang et al. [26] proposed the risk MADM method based on CPT, Choquet integral and the generalized Shapley value respectively. However, these two methods have a shortcoming: the fuzzy measure values are given directly by the experts when the fuzzy measure is adopted, which is too subjective to reflect the objectivity. (3) There are no existing methods for SMADM problems with INS.

Consequently, it is necessary to propose a new SMADM method under INS environment, which considers the following three situations: (1) attribute values take the form of INS; (2) the correlations between attributes are considered based on generalized Shapley function; (3) the bounded rationality of decision makers are considered based on PT. In this method, the prospect function values of each alternative are calculated by using the CPT and the algorithm of INS. Furthermore, the optimal attribute weight is obtained by the fuzzy measure and generalized Shapley function. Then, the comprehensive prospect function value is determined by calculating the weighted value to determine the ordered results of alternatives. The goals and motivation of this paper are: (1) to propose a novel decision making method for SMADM problems which considers the bounded rationality of decision makers and the correlations among attributes; (2) to develop a maximum deviation model to calculate the optimal fuzzy measure values; (3) to show the advantages of the proposed method compared with the existing methods by some examples.

The remainder of this paper is constructed as follows. We briefly review some basic concepts in regard to the NS, CPT, the fuzzy measure and the general Shapely function in Section 2. In Section 3, we establish a method based on the CPT and Shapley function with respect to the discrete SMADM problem in which the probability values are crisps and attribute values are INNs respectively, and the attribute weight is incompletely known. In Section 4, a practical example is given to show the effectiveness of the developed method. The conclusions are summarized, and the future work is specified in last Section.

2 Preliminaries

2.1 The neutrosophic set

Before introducing the notions of single valued neutrosophic set and interval neutrosophic set, we recall the definition of neutrosophic set.

Definition 1. [21] Let Y be a universe of discourse, with a generic element in Y represented by y. A neutrosophic set in Y is expressed by $S = {〈 y, {TM}_{S} (y), {IM}_{S} (y), {FM}_{S} (y) 〉 | y \in Y}$ (1)

Where TM_S (y), IM_S (y) and FM_S (y) are the truth-membership function, indeterminacy-membership function and falsity-membership function, respectively. TM_S (y), IM_S (y) and FM_S (y) are all real standard or nonstandard subsets of ] 0⁻, 1⁺ [.

There is no restriction on the sum of TM_S (y), IM_S (y) and FM_S (y), so we could get the following results. $0^{- -} ⩽ T M_{S} (y) + I M_{S} (y) + F M_{S} (y) ⩽ 3^{+}$ (2)

Since it is hard to apply the neutrosophic sets in practical applications, then the single valued neutrosophic set as a subclass of neutrosophic set was further proposed, which is defined as follows.

Definition 2. [27 , 38] Let Y be a universe of discourse, with a generic element in Y represented by y. A single valued neutrosophic set $\bar{S}$ in Y is expressed by $\bar{S} = {〈 y, {TM}_{\bar{S}} (y), {IM}_{\bar{S}} (y), {FM}_{\bar{S}} (y) 〉 | y \in Y}$ (3) where the three functions TM_S (y), IM_S (y) and FM_S (y) satisfy ${TM}_{\bar{S}} (y),$ ${IM}_{\bar{S}} (y)$ , ${FM}_{\bar{S}} (y) \in [0, 1]$ and $0 ⩽ {TM}_{\bar{S}} (x) + {IM}_{\bar{S}} (x) + {FM}_{\bar{S}} (x) ⩽ 3$ , denoting the truth-membership function, indeterminacy-membership function and falsity-membership function, respectively.

Definition 3. [39, 40] Let Y be a universe of discourse, with a generic element in Y represented by y. An interval neutrosophic set in Y is expressed by $\begin{matrix} \hat{S} = {〈 y, {TM}_{\hat{S}} (y), {IM}_{\hat{S}} (y), {FM}_{\hat{S}} (y) 〉 | y \in Y} \\ = {y, 〈 \begin{matrix} [{TM}_{\hat{S}}^{L} (y), {TM}_{\hat{S}}^{U} (y)], [{IM}_{\hat{S}}^{L} (y), {IM}_{\hat{S}}^{U} (y)], \\ [{FM}_{\hat{S}}^{L} (y), {FM}_{\hat{S}}^{U} (y)] \end{matrix} 〉 | y \in Y} \end{matrix}$ (4) where the functions TM_S (y), IM_S (y) and FM_S (y) satisfy ${TM}_{\hat{S}} = [{TM}_{\hat{S}}^{L}, {TM}_{\hat{S}}^{U}] \subseteq [0, 1],$ ${IM}_{\hat{S}} = [{IM}_{\hat{S}}^{L}, {IM}_{\hat{S}}^{U}] \subseteq [0, 1]$ , ${FM}_{S} = [{FM}_{\hat{S}}^{L}, {FM}_{\hat{S}}^{U}] \subseteq [0, 1]$ and $0 ⩽ {TM}_{\hat{S}}^{U} + {IM}_{\hat{S}}^{U} + {FM}_{\hat{S}}^{U} ⩽ 3$ , representing the truth-membership function, indeterminacy-membership function and falsity-membership function, respectively.

For convenience $\hat{e} = 〈 TM (\hat{e}), IM (\hat{e}), FM (\hat{e}) 〉 = 〈 [{TM}^{L} (\hat{e}), {TM}^{U} (\hat{e})], [{IM}^{L} (\hat{e}), {IM}^{U} (\hat{e})], [{FM}^{L} (\hat{e}),$ ${FM}^{U} (\hat{e})] 〉$ is simply used to represent an element y in interval neutrosophic numbers (INNs).

Definition 4. [29] Let $\hat{e} = 〈 TM (\hat{e}), IM (\hat{e}), FM (\hat{e}) 〉 = 〈 \begin{matrix} [{TM}^{L} (\hat{e}), {TM}^{U} (\hat{e})], \\ [{IM}^{L} (\hat{e}), {IM}^{U} (\hat{e})], \\ [{FM}^{L} (\hat{e}), {FM}^{U} (\hat{e})] \end{matrix} 〉$ be an INN, the complement of $\hat{e}$ is defined as follows. $\begin{matrix} {\hat{e}}^{c} & = & 〈 {TM}^{c} (\hat{e}), {IM}^{c} (\hat{e}), {FM}^{c} (\hat{e}) 〉 \\ = & 〈 \begin{matrix} [{FM}^{L} (\hat{e}), {FM}^{U} (\hat{e})], \\ [1 - {IM}^{U} (\hat{e}), 1 - {IM}^{L} (\hat{e})], \\ [{TM}^{L} (\hat{e}), {TM}^{U} (\hat{e})] \end{matrix} 〉 \end{matrix}$ (5)

Definition 5. [31] Let $\hat{e} = 〈 TM (\hat{e}), IM (\hat{e}), FM (\hat{e}) 〉$ and $\hat{f} = 〈 TM (\hat{f}), IM (\hat{f}), FM (\hat{f}) 〉$ be any two INNs, then the Hamming distance between $\hat{a}$ and $\hat{b}$ is formulated as follows. $\begin{matrix} d (\hat{e}, \hat{f}) = \frac{1}{6} (| {TM}^{L} (\hat{e}) - {TM}^{L} (\hat{f}) | + | {TM}^{U} (\hat{e}) - {TM}^{U} (\hat{f}) | + \\ | {IM}^{L} (\hat{e}) - {IM}^{L} (\hat{f}) | + | {IM}^{U} (\hat{e}) - {IM}^{U} (\hat{f}) | + \\ | {FM}^{L} (\hat{e}) - {FM}^{L} (\hat{f}) | + | {FM}^{U} (\hat{e}) - {FM}^{U} (\hat{f}) | \end{matrix}$ (6)

Definition 6. [32] Let $\hat{e} = 〈 TM (\hat{e}), IM (\hat{e}), FM (\hat{e}) 〉$ and $\hat{f} = 〈 TM (\hat{f}), IM (\hat{f}), FM (\hat{f}) 〉$ be any two INNs, then the cosine of included angle between $\hat{a}$ and $\hat{b}$ is $\begin{matrix} cos (\hat{e}, \hat{f}) = \frac{1}{∥ \hat{e} ∥ • ∥ \hat{f} ∥} [{TM}^{L} (\hat{e}) {TM}^{L} (\hat{f}) + {TM}^{U} (\hat{e}) {TM}^{U} (\hat{f}) + \\ {IM}^{L} (\hat{e}) {IM}^{L} (\hat{f}) + {IM}^{U} (\hat{e}) {IM}^{U} (\hat{f}) + \\ {FM}^{L} (\hat{e}) {FM}^{L} (\hat{f}) + {FM}^{U} (\hat{e}) {FM}^{U} (\hat{f})] \end{matrix}$ (7)

Obviously, $0 < cos (\hat{e}, \hat{f}) ⩽ 1$ , when $\hat{f}$ is the ideal solution I = ([1.0, 1.0], [0.0, 0.0], [0.0, 0.0]), the bigger the $cos (\hat{f}, I)$ between $\hat{f}$ and I is, the more consistent the direction between $\hat{f}$ and I is. In this condition, ${cos}^{*} (\hat{e}, \hat{f}) = \frac{{TM}^{L} (\hat{e}) + {TM}^{U} (\hat{f})}{\sqrt{2 (\begin{matrix} ({TM}^{L} (\hat{e}))^{2} + ({TM}^{U} (\hat{f}))^{2} + \\ ({IM}^{L} (\hat{e}))^{2} + ({IM}^{U} (\hat{f}))^{2} + \\ ({FM}^{L} (\hat{e}))^{2} + ({FM}^{U} (\hat{f}))^{2} \end{matrix})}}$ (8)

In order to compare two INNs, we can give the following definition.

Definition 7. [32] Let $\hat{e} = 〈 TM (\hat{e}), IM (\hat{e}), FM (\hat{e}) 〉$ and $\hat{f} = 〈 TM (\hat{f}), IM (\hat{f}), FM (\hat{f}) 〉$ be any two INNs, if ${cos}^{*} (\hat{e}, I) < {cos}^{*} (\hat{f}, I)$ then $\hat{e} < \hat{f}$ .

2.2 The cumulative prospect theory (CPT)

Uncertain prospect f is a function of status set S to result set X, i.e. f: S → X, for any S_i ∈ S, there is f (S_i) = x_i, x_i ∈ X, in the CPT, the result of the prospect is incremented, i.e., the prospect f is expressed as a sequence of the order (x_i, A_i).

The comprehensive prospect value of prospect f = (x_i, A_i) includes two parts: the value function v and decision weight function π. The specific concept and operation rules are shown as follows:

(1) The value function

Kahneman and Tversky [2] give the value function in the form of a power function to reflect the risk aversion of the decision maker in the face of gains and the risk preference in the face of the losses, which is shown as follows: $v (Δ x_{i}) = {\begin{matrix} (Δ x_{i})^{α}, (Δ x_{i}) ⩾ 0 \\ - λ (Δ x_{i})^{β}, (Δ x_{i}) < 0 \end{matrix}$ (9)

Δx_i is the difference between the decision-making status S_i and the reference point, and its positive and negative represent the relative gains and losses based on the risk perception of the decision maker. When Δx_i is positive, it means gains, and whe Δx_i is negative, it means losses. α, β (0 < α, β < 1) are risk aversion and risk preference coefficients respectively, which represent the degrees of concavity and convexity of value functions in the area of the gains and losses. The greater the α, β are, the more likely the decision maker is to take risks; λ is the loss aversion coefficient, which represents the steep characteristic of the value function curve in the losses area than the gains region, and it indicates that decision makers are more sensitive to losses than gains. (2) Weight function

Tversky and Kahneman [3] thought that the probability weight is the subjective judgment made by the decision makers based on the probability p of the result, which is expressed as: $π_{i}^{+} = w^{+} (\sum_{j = i}^{n} p_{j}) - w^{+} (\sum_{j = i + 1}^{n} p_{j})$ (10) $π_{i}^{-} = w^{-} (\sum_{j = 1}^{i} p_{j}) - w^{-} (\sum_{j = 1}^{i - 1} p_{j})$ (11) and are non-linear functions, $w^{+} (p_{j}) = \frac{p_{j}^{γ}}{(p_{j}^{γ} + (1 - p_{j})^{γ})^{1 / γ}}$ (12) $w^{-} (p_{j}) = \frac{p_{j}^{δ}}{(p_{j}^{δ} + (1 - p_{j})^{δ})^{1 / δ}}$ (13)

The coefficients γ, δ represent the attitudes of risk decisions to gains and losses, respectively, and 0 < γ < δ < 1 indicates that the decision maker overestimated small probability events and underestimated the larger probability events. Tversky and Kahneman [3] gave the results γ = 0.61, δ = 0.69 α = β = 0.88, λ = 2.25.

2.3 Fuzzy measures and the generalized Shapely function

Definition 8. [33] A fuzzy measure on a finite set M = {1, 2, …, m} is a set function μ: P (M) → [0, 1]. It satisfies the following conditions:

Boundary: μ (φ) =0, μ (M) =1,

Monotonicity: if E, F ∈ P (M) and E ⊂ F, then μ (E) ⩽ μ (F),

where P (M) is the power set of M, Then μ is the fuzzy measure on M.

In multi-criteria decision-making, μ (E) can be regarded as the importance of the attribute E.

The formulation of the generalized Shapely function was provided by Shapely [34], which is shown as follows. $\begin{matrix} φ_{i} (μ, M) = \sum_{T \subseteq N ∖ i} \frac{(m - t - 1)! t!}{m!} (μ (i \cup T) - μ (T)) \\ \forall i \in M \end{matrix}$ (14) where μ is a fuzzy measure on M = {1, 2, …, m}, m and t represent the cardinalities of M and T, respectively. From the formulation of generalized Shapely function, we know that it is an expectation value of the marginal contribution between the element i ∈ M and every combination in M ∖ i. From the formula (14) and the definition 8, it is easy to know that {φ_i (μ, M)} _i∈M is a weight vector, because φ_i (μ, M) ⩾0 for any element i ∈ M, and $\sum_{i = 1}^{m} φ_{i} (μ, M) = 1$ . It should be noted that if there are no interactions between elements, then the generalized Shapely function values are equal to their own importance.

3 Interval Neutrosophic stochastic MADM Method Based on CPT and generalized Shapley function

3.1 Problem description

For a SMADM problem with interval neutrosophic information in which the criteria are interactive. Let A = {a₁, a₂, ⋯ a_m} be an alternative set and C = {c₁, c₂, ⋯ c_n} be the set of attributes. For each attribute c_j, there are t_j possible statuses Θ_k = (θ₁, θ₂, ⋯, θ_{t
_j}) and $p_{ij}^{k}$ is the probability occurred under the status θ_k (1 ⩽ k ⩽ t_j), where $0 ⩽ p_{j}^{k} ⩽ 1$ and $\sum_{k = 1}^{t_{j}} p_{j}^{k} = 1$ . Let $R = [r_{ij}^{k}]_{m \times n}$ be a decision matrix, where $r_{ij}^{k} = < [T_{ij}^{Lk}, T_{ij}^{Uk}],$ $[I_{ij}^{Lk}, I_{ij}^{Uk}], [F_{ij}^{Lk}, F_{ij}^{Uk}]) > (i = 1, 2, \dots m; j = 1, 2, \dots, n, k = 1, 2, \dots t_{j})$ is the attribute value in the form of the interval neutrosophic information for alternative A_i with respect to attribute c_j under the status θ_k, in which $[T_{ij}^{L}, T_{ij}^{U}] \subseteq [0, 1], [I_{ij}^{L}, I_{ij}^{U}] \subseteq [0, 1], [F_{ij}^{L}, F_{ij}^{U}] \subseteq [0, 1]$ , and $0 ⩽ T_{ij}^{U} + I_{ij}^{U} + F_{ij}^{U} ⩽ 3$ . Assume that the importance of attributes is given as $U_{c_{1}} = [w_{1}^{l}, w_{1}^{u}], U_{c_{2}} = [w_{2}^{l}, w_{2}^{u}], \dots, U_{c_{n}} = [w_{n}^{l}, w_{n}^{u}]$ . Based on these conditions, the goal is to rank the alternatives for the SMADM problem.

3.2 The calculation of the prospect function value

The CPT is developed on the basis of the PT. Some unreasonable assumptions about the descriptive aspects of the expected utility theory are relaxed. Based on the CPT, the steps are shown as follows.

Interval neutrosophic prospect is constructed in order to calculate the prospect value function with respect to INS. In this method, interval neutrosophic prospect can be expressed as $f_{ij} = (r_{ij}^{1}, p_{ij}^{1}; r_{ij}^{2}, p_{ij}^{2}; \dots r_{ij}^{t_{j}}, p_{ij}^{t_{j}})$ . Where, $r_{ij}^{k}$ is the kth occurring result of interval neutrosophic prospect f_ij. $p_{ij}^{k}$ is the corresponding probability of the result $r_{ij}^{k}$ . $p_{ij}^{k}$ is the exact number, and $r_{ij}^{k}$ is in the form of INS, i = 1, 2, …, m; j = 1, 2, …, n, k = 1, 2, … t_j.

The determination of the reference point. The reference point is generally determined by the decision maker according to his/her risk preference. Commonly, the current status, mean value, minimum value and maximum value can be selected as the reference point in the real decision making. In the INS environment, we can choose the middle point of the interval neutrosophic prospect as a decision reference point. The determination of gains and loss values. Calculate the distance $d (r_{ij}^{k}, \tilde{r})$ about the result $r_{ij}^{k}$ with respect to the reference point $\tilde{r}$ according to the distance formula of INN in the Definition 5, and then calculate the change value $Δ r_{ij}^{k}$ of the gains and losses values for the result $r_{ij}^{k}$ relative to the reference point $\tilde{r}$ .

Δ r_{ij}^{k} = {\begin{matrix} \begin{matrix} d (r_{ij}^{k}, \tilde{r}), & r_{ij}^{k} > \tilde{r} \end{matrix} \\ \begin{matrix} 0 & r_{ij}^{k} = \tilde{r} \end{matrix} \\ \begin{matrix} - d (r_{ij}^{k}, \tilde{r}), & r_{ij}^{k} < \tilde{r} \end{matrix} \end{matrix}

(15)

4. The Determination of the interval neutrosophic value function. On the basis of Step 3, we can calculate the interval neutrosophic value function by the following formula. $v (Δ r_{ij}^{k}) = {\begin{matrix} (Δ r_{ij}^{k})^{α}, (Δ r_{ij}^{k}) ⩾ 0 \\ - λ (- Δ r_{ij}^{k})^{β}, (Δ r_{ij}^{k}) < 0 \end{matrix}$ (16)

Where, α, β can take the value 0.88, λ = 2.25.

5. Construct an ordered prospect corresponding to the t_j prospects. The permutation (•) is defined, which makes change value $Δ r_{ij}^{k}$ of the gains and losses values rearranged in order of magnitude, that is, Δr_ij(1) ⩾ Δr_ij(2) ⩾ ⋯ ⩾ Δr_ij(l) ⩾ 0 ⩾ Δr_ij(l+1) ⩾ ⋯ ⩾ Δr_{ij(t_j)}. In this way, the rearrangement of the interval neutrosophic prospect ${\tilde{f}}_{ij} = (r_{ij (1)}, p_{ij (1)}; r_{ij (2)}, p_{ij (2)}; \dots r_{ij (t_{j})}, p_{ij (t_{j})})$ is an orderly prospect.

6. Calculate the weighting functions $π_{ij (z)}^{+}$ and $π_{ij (z)}^{-}$ according to the following formulas. $\begin{matrix} π_{i j (1)}^{+} = w^{+} (p^{i j (z)}) \\ π_{i j (z)}^{+} = w^{+} (\sum_{z = 1}^{l} p^{i j (z)}) - w^{+} (\sum_{z = 1}^{l - 1} p^{i j (z)}) , 2 ⩽ z ⩽ l \\ π_{i j (z)}^{-} = w^{-} (\sum_{z = l}^{t_{j}} p^{i j (z)}) - w^{-} (\sum_{z = l + 1}^{t_{j}} p_{i j (z)}), \\ l + 1 ⩽ z ⩽ t_{j} - 1 \\ π_{i j (t_{j})}^{-} = w^{-} (p^{i j (z)}) \end{matrix}$ (17) where, w⁺ (•) and w⁻ (•) can be obtained according to the definition of interval neutrosophic probability weight function.

7. Obtain the prospect function value V (f_ij) by the following formula. $\begin{matrix} V (f^{i j}) = \sum_{z = 1}^{l} π_{i j (z)}^{+} v (Δ r^{i j (z)}) + \\ \sum_{z = l + 1}^{t_{j}} π_{i j (z)}^{-} v (Δ r^{i j (z)}) \end{matrix}$ (18)

3.3 The calculation of the comprehensive prospect value

From the formula (14), the generalized Shapley function value can be regarded as the weighted average value of the marginal contribution of a single attribute c_j, that is, the weight of the attribute c_j. Therefore, the weight can be obtained by calculating the generalized Shapley function value of the interrelated attributes.

The maximum deviation method has been widely used in the MADM method. The basic idea of this method is: if the difference between the attribute values of each alternative is small for a certain attribute, the importance of this attribute is relatively small, that is, the its weight should be small; Likewise, if for a certain attribute, the difference between the attribute values of each alternative is large, the importance of this attribute is relatively large, that is, the its weight should be large.

Assuming that $D_{ij} (φ_{c_{i}} (μ, c)) = \sum_{k = 1}^{m} d (v_{ij}, v_{kj})$ φ_{c
_i} (μ, c) represents the deviation of the alternative a_i from the other alternatives under the attribute c_j, $D_{j} (φ_{c_{i}} (μ, c)) = \sum_{i = 1}^{m} D_{ij} (φ_{c_{i}} (μ, c)) = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (v_{ij}, v_{kj}) φ_{c_{i}} (μ, c)$ represents the total deviation of every alternative from the other alternatives under the attribute c_j, and $D (φ_{c_{i}} (μ, c)) = \sum_{j = 1}^{n} D_{j} (φ_{c_{i}} (μ, c)) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (v_{ij}, v_{kj}) φ_{c_{i}} (μ, c)$ represents the total deviation of all the attributes for each alternative. If the attribute weight information is partly known, the model shown in Equation (19) can be constructed to solve the optimal fuzzy measure μ (c_i). $\begin{matrix} max \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{n} d (v_{ij}, v_{kj}) φ_{c_{j}} (μ, c) \\ s . t . {\begin{matrix} μ (Φ) = 0, μ (C) = 1 \\ μ (S) ⩽ μ (T) \forall S, T \subseteq C, S \subseteq T \\ μ (c_{j}) \in {Uc}_{j}, j = 1, 2, \dots, n \end{matrix} \end{matrix}$ (19) where, φ = (φ_{c
₁} (μ, c), φ_{c
₂} (μ, c), ⋯ φ_{c
_n} (μ, c)) is the Shapley value vector of attribute c_j (j = 1, 2, …, n), μ is the fuzzy measure of Index c_j (j = 1, 2, …, n), d (v_ij, v_kj) represents the distance between the two prospect functional values.

After getting the value of V (f_ij) and φ_{c
_j} (μ, c), the comprehensive prospect values for each alternative can be calculated by the following formula: $V_{a_{i}} = \sum_{j = 1}^{n} V (f_{ij}) φ_{c_{j}} (μ, c)$ (20)

Obviously, the greater the comprehensive prospect value is, the better the alternative a_i is. Therefore, we can determine the ranking of the alternatives according to the comprehensive prospect values.

Remark 1. According to the definition of the fuzzy measure, μ (c_i) is a function defined on the power set, that is, under the n attributes there will be 2ⁿ fuzzy measure (including the null and universal set). Furthermore the optimal fuzzy measure of μ (c_i) is obtained by the formula (19). Then, according to the definition of the generalized Shapley value, the weight value φ_{c
_j} (μ, c) of each attribute is calculated.

3.4 The procedure of the interval neutrosophic SMADM based on CPT

Based on the above analysis, we give a procedure of the interval neutrosophic SMADM based on CPT showed in Fig. 1. A brief description of the procedure is given as follows:

Fig. 1

The procedure of the interval neutrosophic SMADM based on CPT.

The procedure can be divided into four parts. In the first part, the evaluation matrix $R = {[r_{ij}^{k}]}_{m \times n}$ and the prospect probability $p_{ij}^{k}$ are obtained from the given data. In the second part, the reference point $\tilde{r}$ is determined, the distance $d_{ij}^{k}$ and $Δ r_{ij}^{k}$ of the attribute value are calculated, then the prospect value function value $v_{ij}^{k}$ and the subjective probability weights $π_{ij (z)}^{+}$ and $π_{ij (z)}^{-}$ are obtained, and prospect value V (f_ij) can be gotten. In the third part, in order to obtain the comprehensive prospect value and take into account the interaction between attributes, the optimal fuzzy measure μ (c_i) is obtained by constructing the model by using the maximum deviation method and the generalized Shapley value. In the fourth part, based on V (f_ij) and φ_{c
_j} (μ, c), the comprehensive prospect value V_{a
_i} of each alternative is gotten by weighted calculation.

In summary, the steps of the interval neuttrosophic SMADM based on CPT are shown as follows.

Step 1. Definite reference point $\tilde{r}$ ;

Step 2. Calculate the distance $d (r_{ij}^{k}, \tilde{r})$ between $r_{ij}^{k}$ and the reference point $\tilde{r}$ by definition 5, denoted as $d_{ij}^{k}$ .

Step 3. Calculate the change value $Δ r_{ij}^{k}$ of gains and losses values for the result $r_{ij}^{k}$ relative to the reference point $\tilde{r}$ according to the formula (15).

Step 4. Calculate the value function of the INNs which is obtained by formula (16), denoted as ${\tilde{v}}_{ij}^{k}$ .

Step 5. Calculate the subjective probability weight functions $π_{ij (l)}^{+}$ and $π_{ij (l)}^{-}$

Step 6. Calculate the values of prospect function V (f_ij) according to the prospect value function and the probability weight function.

Step 7. Construct linear programming for the optimal fuzzy measure on attribute set C and calculate the φ_{c
_j} (μ, c) by formulas (14) and (19).

Step 8. Calculate the weighted prospect values and rank alternative according to their values.

4 An illustrative example

In order to verify the effectiveness of this method, an example is used to show the application of the proposed method. The company is planning to set up a new factory. There are three alternatives (a₁, a₂, a₃), and four attributes are considered by decision makers which are economic benefits c₁, policy benefits c₂, social benefits c₃, and green benefits c₄. The attribute weight is incompletely known. In accordance with the market survey, there are four natural statuses in attribute c₁ and c₂, including very good θ₁, good θ₂, fair θ₃, and poor θ₄, and there are three natural statuses in attribute c₃ and c₄, including very good θ₁, good θ₂, and fair θ₃. The evaluated value of every natural status θ_k (k = 1, 2, …, t_j) with respect to each attributes c_j (j = 1, 2, 3, 4) for the alternative A_i (i = 1, 2, 3) can be represented by interval neutrosophic number

r_{ij}^{k} = < [{TM}_{ij}^{kL}, {TM}_{ij}^{kU}], [{IM}_{ij}^{kL}, {IM}_{ij}^{kU}], [{FM}_{ij}^{kL}, {FM}_{ij}^{kU}]) >

(i = 1, 2, …, m; j = 1, 2, …, n; k = 1, 2, …, t_j). The interval neutrosophic matrix

R = [r_{ij}^{k}]_{m \times n}

is listed in Table 1. The probability of every natural status θ_k (k = 1, 2, …, t_j) is represented as the crisp numbers. Suppose that the importance of attributes is given as U_{c
₁} = [0.3, 0.35], U_{c
₂} = [0.35, 0.45], U_{c
₃} = [0.2, 0.25] and U_{c
₄} = [0.25, 0.3].

Table 1
The evaluation matrix of three possible alternatives With respected to the four criteria

	c ₁
	θ ₁	θ ₂	θ ₃	θ ₄
p	0.3	0.4	0.1	0.2
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])	([0.2,0.66],[0.2,0.3],[0.1,0.6])	([0.7,0.8],[0.0,0.1],[0.1,0.2])
a ₂	([0.6,0.8],[0.1,0.2],[0.2,0.3])	([0.6,0.7],[0.1,0.2],[0.2,0.3])	([0.1,0.6],[0.1,0.2],[0.2,0.3])	([0.6,0.7],[0.1,0.2],[0.1,0.3])
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])	([0.1,0.3],[0.2,0.3],[0.3,0.4])	([0.6,0.7],[0.1,0.2],[0.1,0.3])
	c ₂
θ ₁	θ ₂	θ ₃	θ ₄
p	0.1	0.3	0.4	0.2
a ₁	([0.2,0.5],[0.1,0.3],[0.3,0.5])	([0.2,0.6],[0.1,0.4],[0.1,0.4])	([0.2,0.6],[0.2,0.3],[0.3,0.4])	([0.3,0.8],[0.0,0.1],[0.1,0.4])
a ₂	([0.4,0.5],[0.1,0.4],[0.1,0.3])	([0.3,0.7],[0.1,0.5],[0.1,0.3])	([0.3,0.7],[0.1,0.9],[0.1,0.5])	([0.2,0.7],[0.1,0.4],[0.1,0.5])
a ₃	([0.4,0.6],[0.2,0.3],[0.3,0.4])	([0.3,0.6],[0.1,0.3],[0.3,0.6])	([0.2,0.6],[0.1,0.3],[0.2,0.4])	([0.4,0.7],[0.1,0.5],[0.1,0.6])
	c ₃
	θ ₁	θ ₂	θ ₃
p	0.3	0.5	0.2
a ₁	([0.2,0.4],[0.1,0.3],[0.3,0.9])	([0.1,0.2],[0.2,0.8],[0.1,0.2])	([0.1,0.6],[0.1,0.4],[0.3,0.4])
a ₂	([0.1,0.9],[0.2,0.7],[0.1,0.9])	([0.2,0.7],[0.1,0.3],[0.1,0.3])	([0.2,0.7],[0.1,0.3],[0.2,0.5])
a ₃	([0.2,0.3],[0.1,0.4],[0.3,0.4])	([0.4,0.6],[0.2,0.4],[0.3,0.5])	([0.4,0.6],[0.2,0.7],[0.1,0.4])
	c ₄
	θ ₁	θ ₂	θ ₃
p	0.2	0.4	0.4
a ₁	([0.1,0.5],[0.2,0.3],[0.2,0.4])	([0.2,0.3],[0.1,0.4],[0.2,0.3])	([0.1,0.2],[0.1,0.3],[0.2,0.9])
a ₂	([0.2,0.7],[0.1,0.4],[0.2,0.9])	([0.1,0.3],[0.1,0.8],[0.1,0.3)	(0.6,0.7],[0.1,0.3],[0.2,0.6])
a ₃	([0.3,0.8],[0.2,0.6],[0.3,0.7])	([0.3,0.6],[0.2,0.4],[0.3,0.6])	([0.3,0.6],[0.2,0.3],[0.1,0.4])

4.1 The steps of the example

We utilize the proposed interval neutrosophic SMADM method to sort the alternatives. In order to obtain the optimal alternative(s), the specific steps are as follows:

Step 1. Determine the reference point $\tilde{r}$ . Decision makers choose the reference points according to their risk preferences and psychological status. In this example, we can set: $\tilde{r} = ([0.5, 0.5], [0.5, 0.5], [0.5, 0.5])$

Step 2. By Definition 5, calculate the distance $d (r_{ij}^{k}, \tilde{r})$ between $r_{ij}^{k}$ and the reference point $\tilde{r}$ , denoted as $d_{ij}^{k}$ , i.e., $\begin{matrix} d_{11}^{1} = 0 . 150, d_{11}^{2} = 0 . 200, d_{11}^{3} = 0 . 233, d_{11}^{4} = 0 . 350, \\ d_{21}^{1} = 0 . 267, d_{21}^{2} = 0 . 250, d_{31}^{3} = 0 . 283, d_{11}^{4} = 0 . 217, \\ d_{31}^{1} = 0 . 183, d_{31}^{2} = 0 . 150, d_{31}^{3} = 0 . 233, d_{31}^{4} = 0 . 267, \\ d_{12}^{1} = 0 . 183, d_{12}^{2} = 0 . 233, d_{12}^{3} = 0 . 200, d_{12}^{4} = 0 . 317, \\ d_{22}^{1} = 0 . 200, d_{22}^{2} = 0 . 233, d_{22}^{3} = 0 . 267, d_{22}^{4} = 0 . 283, \\ d_{32}^{1} = 0 . 167, d_{32}^{2} = 0 . 200, d_{32}^{3} = 0 . 233, d_{32}^{4} = 0 . 270, \\ d_{13}^{1} = 0 . 267, d_{13}^{2} = 0 . 333, d_{13}^{3} = 0 . 217, d_{23}^{1} = 0 . 350, \\ d_{23}^{2} = 0 . 283, d_{23}^{3} = 0 . 233, d_{33}^{1} = 0 . 217, d_{33}^{2} = 0 . 133, \\ d_{33}^{3} = 0 . 200, d_{14}^{1} = 0 . 217, d_{14}^{2} = 0 . 250, d_{14}^{3} = 0 . 330, \\ d_{24}^{1} = 0 . 283, d_{24}^{2} = 0 . 317, d_{24}^{3} = 0 . 217, d_{34}^{1} = 0 . 217, \\ d_{34}^{2} = 0 . 167, d_{34}^{3} = 0 . 283 \end{matrix}$

Step 3. Calculate changes $Δ {\tilde{r}}_{ij}^{k}$ which are shown as follows. $\begin{matrix} Δ {\tilde{r}}_{11}^{1} = 0.15, Δ {\tilde{r}}_{11}^{2} = 0.2, Δ {\tilde{r}}_{11}^{3} = - 0.233, Δ {\tilde{r}}_{11}^{4} = 0.35, \\ Δ {\tilde{r}}_{21}^{1} = 0.267, Δ {\tilde{r}}_{21}^{2} = 0.25, Δ {\tilde{r}}_{21}^{3} = 0.283, Δ {\tilde{r}}_{21}^{4} = 0.217, \\ Δ {\tilde{r}}_{31}^{1} = 0.183, Δ {\tilde{r}}_{31}^{2} = 0.15, Δ {\tilde{r}}_{31}^{4} = 0.267, Δ {\tilde{r}}_{12}^{1} = 0.183, \\ Δ {\tilde{r}}_{31}^{3} = - 0.233, Δ {\tilde{r}}_{12}^{2} = 0.233, Δ {\tilde{r}}_{12}^{3} = - 0.2, \\ Δ {\tilde{r}}_{12}^{4} = 0.317, Δ {\tilde{r}}_{22}^{1} = 0.2, Δ {\tilde{r}}_{22}^{2} = 0.233, \\ Δ {\tilde{r}}_{22}^{3} = - 0.267, Δ {\tilde{r}}_{22}^{4} = - 0.283, Δ {\tilde{r}}_{32}^{1} = 0.167, \\ Δ {\tilde{r}}_{32}^{2} = 0.2, Δ {\tilde{r}}_{32}^{3} = - 0.233, Δ {\tilde{r}}_{32}^{4} = 0.217, \end{matrix}$ $\begin{matrix} Δ {\tilde{r}}_{13}^{1} = - 0.267, Δ {\tilde{r}}_{13}^{2} = 0.333, Δ {\tilde{r}}_{13}^{3} = - 0.217, \\ Δ {\tilde{r}}_{23}^{1} = - 0.35, Δ {\tilde{r}}_{23}^{2} = 0.283, Δ {\tilde{r}}_{23}^{3} = 0.233, \\ Δ {\tilde{r}}_{33}^{1} = - 0.217, Δ {\tilde{r}}_{33}^{2} = 0.133, Δ {\tilde{r}}_{33}^{3} = - 0.2, \\ Δ {\tilde{r}}_{14}^{1} = - 0.217, Δ {\tilde{r}}_{14}^{2} = - 0.25, Δ {\tilde{r}}_{14}^{3} = - 0.333, \\ Δ {\tilde{r}}_{24}^{1} = - 0.283, Δ {\tilde{r}}_{24}^{2} = - 0.317, Δ {\tilde{r}}_{24}^{3} = 0.217, \\ Δ {\tilde{r}}_{34}^{1} = 0.217, Δ {\tilde{r}}_{34}^{2} = 0.167, Δ {\tilde{r}}_{34}^{3} = - 0.283 . \end{matrix}$

Step 4. Calculate the value function ${\tilde{v}}_{ij}^{k}$ of the INNs according to the formula (28), i.e., $\begin{matrix} {\tilde{v}}_{11}^{1} = 0.188, {\tilde{v}}_{11}^{2} = 0.243, {\tilde{v}}_{11}^{3} = - 0.624, {\tilde{v}}_{11}^{4} = 0.397, \\ {\tilde{v}}_{21}^{1} = 0 . 313, {\tilde{v}}_{21}^{2} = 0 . 295, {\tilde{v}}_{21}^{3} = - 0 . 329, {\tilde{v}}_{21}^{4} = 0.261, \\ {\tilde{v}}_{31}^{1} = 0.224, {\tilde{v}}_{31}^{2} = 0.188, {\tilde{v}}_{31}^{3} = - 0.624, {\tilde{v}}_{31}^{4} = 0.313, \\ {\tilde{v}}_{12}^{1} = 0 . 224, {\tilde{v}}_{12}^{2} = 0 . 278, {\tilde{v}}_{12}^{3} = - 0 . 243, {\tilde{v}}_{12}^{4} = 0 . 364, \\ {\tilde{v}}_{22}^{1} = 0.243, {\tilde{v}}_{22}^{2} = 0.278, {\tilde{v}}_{22}^{3} = - 0.704, {\tilde{v}}_{22}^{4} = - 0.741, \\ {\tilde{v}}_{22}^{4} = - 0.741, {\tilde{v}}_{32}^{1} = 0.207, {\tilde{v}}_{32}^{2} = - 0.243, \\ {\tilde{v}}_{32}^{3} = - 0.278, {\tilde{v}}_{32}^{4} = - 0.361, {\tilde{v}}_{13}^{1} = - 0.0.704, \\ {\tilde{v}}_{13}^{2} = - 0.855, {\tilde{v}}_{13}^{3} = 0.587, {\tilde{v}}_{23}^{2} = 0.329, \\ {\tilde{v}}_{23}^{3} = - 0.278, {\tilde{v}}_{33}^{1} = - 0.587, {\tilde{v}}_{33}^{2} = 0.169, {\tilde{v}}_{33}^{3} = - 0.243, \\ {\tilde{v}}_{14}^{1} = - 0.587, {\tilde{v}}_{14}^{2} = - 0.664, {\tilde{v}}_{14}^{3} = - 0.855, \\ {\tilde{v}}_{24}^{1} = - 0.741, {\tilde{v}}_{24}^{2} = - 0.819, {\tilde{v}}_{24}^{3} = 0.261, {\tilde{v}}_{34}^{1} = 0.261, \\ {\tilde{v}}_{34}^{2} = 0.207, {\tilde{v}}_{34}^{3} = 0.329 . \end{matrix}$

Step 5. The subjective probability weight functions $π_{ij (l)}^{+}$ and $π_{ij (l)}^{-}$ are calculated, i.e., $\begin{matrix} π_{11 (1)}^{+} = 0.261, π_{11 (2)}^{+} = 0.213, π_{11 (3)}^{+} = 0.238, \\ π_{11 (4)}^{-} = 0.17, π_{21 (1)}^{+} = 0.186, π_{21 (2)}^{+} = 0.184, \\ π_{21 (3)}^{+} = 0 . 237, π_{21 (4)}^{+} = 0 . 393, π_{31 (1)}^{+} = 0 . 261, \\ π_{31 (2)}^{+} = 0.16, π_{31 (3)}^{+} = 0.291, π_{31 (4)}^{-} = 0.17, \\ π_{12 (1)}^{+} = 0 . 261, π_{12 (2)}^{+} = 0 . 16, π_{12 (3)}^{+} = 0.291, \\ π_{12 (4)}^{+} = 0.288, π_{22 (1)}^{+} = 0.318, π_{22 (2)}^{+} = 0.052, \\ π_{22 (3)}^{-} = 0.261, π_{22 (4)}^{-} = 0.257, π_{32 (1)}^{+} = 0.37, \\ π_{32 (2)}^{+} = 0.474, π_{32 (3)}^{+} = 0.238, π_{32 (4)}^{+} = 0.288, \\ π_{13 (1)}^{-} = 0.412, π_{13 (2)}^{-} = 0.134, π_{13 (3)}^{-} = 0.454, \\ π_{23 (1)}^{+} = 0.421, π_{23 (2)}^{+} = 0.113, π_{23 (3)}^{-} = 0.328, \\ π_{33 (1)}^{+} = 0.261, π_{33 (2)}^{+} = 0.273, π_{33 (3)}^{-} = 0.328, \\ π_{14 (1)}^{-} = 0.331, π_{14 (2)}^{-} = 0.277, π_{14 (3)}^{-} = 0.391, \\ π_{24 (1)}^{+} = 0.37, π_{24 (2)}^{-} = 0.126, π_{24 (3)}^{-} = 0.391, \\ π_{34 (1)}^{+} = 0.37, π_{34 (2)}^{+} = 0.104, π_{34 (3)}^{+} = 0.526 . \end{matrix}$

Step 6. Calculate the values of prospect function V (f_ij) by formula (18), i.e., $\begin{matrix} V (f_{11}) = 0.094, V (f_{21}) = 0.291, V (f_{31}) = 0.066, \\ V (f_{12}) = 0 . 275, V (f_{22}) = - 0.273, \end{matrix}$ $\begin{matrix} V (f_{32}) = - 0.083, V (f_{13}) = - 0.757, \\ V (f_{23}) = - 0.123, V (f_{33}) = - 0.083, \\ V (f_{14}) = - 0.712, V (f_{24}) = - 0.317, V (f_{34}) = 0.258 . \end{matrix}$

Step 7. According to the formula (19), we get the following linear programming for the optimal fuzzy measure on attribute set C. $\begin{matrix} max = - 0 . 529 (μ (c_{1}) - μ (c_{2}, c_{3}, c_{4})) - 0.006 (μ (c_{2}) \\ - μ (c_{1}, c_{3}, c_{4})) + 0 . 07 (μ (c_{3}) - μ (c_{1}, c_{2}, c_{4})) + \\ 0 . 465 (μ (c_{4}) - μ (c_{1}, c_{2}, c_{3})) - 0.267 (μ (c_{1}, c_{2}) - \\ μ (c_{3}, c_{4})) - 0 . 229 (μ (c_{1}, c_{3}) - μ (c_{2}, c_{4})) \\ - 0 . 032 (μ (c_{1}, c_{4}) - μ (c_{2}, c_{3})) + 2 . 486 \\ {\begin{matrix} μ (c_{1}, c_{2}, c_{3}, c_{4}) = 1 μ (Φ) = 0 \\ υ (S) ⩽ υ (T) \forall S, T \subseteq {c_{1}, c_{2}, c_{3}, c_{4}} S \subseteq T \\ μ (c_{1}) \in [0 . 3, 0 . 35], μ (c_{2}) \in [0 . 35, 0 . 45], \\ μ (c_{3}) \in [0 . 2, 0 . 25], μ (c_{4}) \in [0 . 25, 0 . 3] \end{matrix} \end{matrix}$

Solving this above model by Lingo software, we obtain $\begin{matrix} μ (c_{1}) = μ (c_{4}) = μ (c_{1}, c_{2}) = μ (c_{1}, c_{3}) \\ = μ (c_{1}, c_{4}) = μ (c_{2}, c_{3}) = μ (c_{1}, c_{2}, c_{3}) = 0.3 \\ μ (c_{2}) = 0.15, μ (c_{3}) = 0.25, μ (c_{2}, c_{4}) \\ = μ (c_{3}, c_{4}) = μ (c_{1}, c_{2}, c_{4}) = μ (c_{2}, c_{3}, c_{4}) \\ = μ (c_{1}, c_{3}, c_{4}) = μ (c_{1}, c_{2}, c_{3}, c_{4}) = 1 \end{matrix}$

By formula (14), we get the following attribute Shapely values $\begin{matrix} φ_{c_{1}} (ν, C) = 0.092, φ_{c_{2}} (ν, C) = 0.158, \\ φ_{c_{3}} (ν, C) = 0.192, φ_{c_{4}} (ν, C) = 0.558 . \end{matrix}$

Step 8. The weighted prospect values are calculated and sorted according to their values, i.e., $\begin{matrix} V_{a 1} = 0.092 \times 0.094 + 0.158 \times 0.275 + \\ 0.192 \times (- 0.757) + 0.558 \times (- 0.712) = - 0.491 \end{matrix}$ $\begin{matrix} V_{a 2} = 0.092 \times 0.291 + 0.158 \times (- 0.273) + \\ 0.192 \times (- 0.123) + 0.558 \times (- 0.317) = 0.271 \end{matrix}$ $\begin{matrix} V_{a 3} = 0.092 \times 0.066 + 0.158 \times 0.344 + \\ 0.192 \times (- 0.083) + 0.558 \times (- 0.258) = - 0.146 \end{matrix}$

So the final ranking is result a₂ ≻ a₃ ≻ a₁. Obviously, a₂ is the optimal alternative.

4.2 The steps of the example

From the formula of the prospect value function, we can know that the reference point of the prospect value function has a direct impact on the prospect value of the result. The sensitivity analysis of the decision making method is made by changing the decision reference points. In this paper, some new reference points are assigned to

{\tilde{r}}_{1} = ([1, 1], [0, 0], [0, 0])

{\tilde{r}}_{2} = ([0, 0], [0, 0], [1, 1])

{\tilde{r}}_{3} = ([0, 0], [1, 1], [0, 0])

. According to the above method, the sorted results are involved in Table 2.

Table 2
The ranking results of alternatives with different referent points

Different referent points	Ranking orders of alternatives	The optimal alternative
([1,1],[0,0],[0,0])	a₂ ≻ a₃ ≻ a₁	a ₂
([0,0],[0,0],[1,1])	a₂ ≻ a₁ ≻ a₃	a ₂
([0,0],[1,1],[0,0])	a₂ ≻ a₁ ≻ a₃	a ₂
([0.5,0.5],[0.5,0.5],[0.5,0.5])	a₂ ≻ a₃ ≻ a₁	a ₂

As shown from Table 2, with the difference of the reference points, the final ranking results have been changed. More specifically, when the reference point is selected ([1, 1], [0, 0], [0, 0]), ([0.5, 0.5], [0.5, 0.5], [0.5, 0.5]), the ranking result is a₂ ≻ a₃ ≻ a₁; when the reference point is selected ([0, 0], [0, 0], [1, 1]), ([0, 0], [1, 1], [0, 0]), the ranking result is a₂ ≻ a₁ ≻ a₃.

4.3 Comparing with the existing methods

In order to verify the validity and to show the advantages of the method, we can make a comparison with the interval neutrosophic ELECTRE method, the method based on interval neutrosophic aggregation operator and interval neutrosophic TOPSIS method.

(1) Compared with interval neutrosophic ELECTRE method which was proposed by Liu and Li [ 35].

Because the method is not used to the SMADM under interval neutrosophic environment directly, we need to modify the ELECTRE method. Firstly, we use the INWA operator to aggregate the evaluated values with respect to varied state θ_i under each attribute c_j. Then the specific steps are involved:

Step 1. Compare the two pairs for any two alternatives under the same attribute c_j based on the aggregated matrix to establish the possibility degree matrix $P_{m \times m}^{k}$ : $p_{m \times m}^{k} = [\begin{matrix} 0.5 & p_{12}^{k} & \dots & p_{1 m}^{k} \\ p_{21}^{k} & 0.5 & \dots & p_{2 m}^{k} \\ \dots & \dots & \dots & ⋮ \\ p_{m 1}^{k} & p_{m 2}^{k} & \dots & 0.5 \end{matrix}]$

Step 2. Analyze the probability matrix $P_{m \times m}^{k}$ according to the sorting formula and then calculate the decision matrix Y_m×n: $Y_{m \times n} = [\begin{matrix} y_{11} & y_{12} & \dots & y_{1 n} \\ y_{21} & y_{22} & \dots & y_{2 n} \\ \dots & \dots & \dots & ⋮ \\ y_{m 1} & y_{m 2} & \dots & y_{mn} \end{matrix}]$

Step 3. Normalize the Y matrix to obtain the matrix R_m×n.

Step 4. Construct the priority relation and calculate the priority matrix CM.

Step 5. Construct the relative inferiority matrix DM.

Step 6. Calculate modified weighted matrix E, which comes as: $E = [\begin{matrix} e_{11} & e_{12} & \dots & e_{1 n} \\ e_{21} & e_{22} & \dots & e_{2 n} \\ \dots & \dots & \dots & ⋮ \\ e_{m 1} & e_{m 2} & \dots & e_{mn} \end{matrix}]$

Where e_ki = cm_ki • (1 - dm_ki).

Step 7. Calculate net dominance δ_k and rank the alternatives by this value δ_k. The formula of δ_k is shown as follows: $δ_{k} = \underset{i \neq k}{\sum_{i = 1}^{m}} e_{ki} - \underset{i \neq k}{\sum_{i = 1}^{m}} e_{ki}, k = 1, 2, \dots, m$ (21)

According to the steps 1–7, we get the modified weighted matrix E and net dominance δ_k respectively. Due to space constraints, the other steps are omitted. The results are shown as follows. $E = [\begin{matrix} 1.000 & 0.000 & 0.000 \\ 0.261 & 1.000 & 0.172 \\ 0.000 & 0.217 & 1.000 \end{matrix}]$ $δ_{k} = (- 0.261, 0.216, 0.045)$

From above values δ_k, we can obtain the ranking of the alternatives a₂ ≻ a₃ ≻ a₁, so the optimal alternatives is a₂.

Obviously, the extended ELECTRE method produced the same ranking results as the proposed method in this paper. This can verify the validity of our method.

(2) Compared with the method based on interval neutrosophic aggregation operator.

Based on INPGWA operator proposed by Liu and Tang [24] and based on ITFPOWA operator proposed by Zhang et al. [36], we propose a modified SMADM method which is more suitable to deal with the SMADM problem under interval neutrosophic environment. The specific method steps are as follows:

Step 1. Normalize the matrix, namely, Transformed cost type to benefit by the complement in formula (5).

Step 2. Calculate the supports. $\begin{matrix} Sup (r_{ij}^{k}, r_{ij}^{l}) = 1 - d (r_{ij}^{k}, r_{ij}^{l}), i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n; k, l = 1, 2 \dots, t_{j} \end{matrix}$ (22)

Where $d (r_{ij}^{k}, r_{ij}^{l})$ is Harmming distance between interval neutrosophic numbers $r_{ij}^{k}$ and $r_{ij}^{l}$ , which is defined by formula (18).

Step 3. Calculate $T (r_{ij}^{k})$ which is $\begin{matrix} T (r_{ij}^{k}) = \underset{l \neq k}{\sum_{l = 1}^{t_{j}}} Sup (r_{ij}^{k}, r_{ij}^{l}), i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n; k, l = 1, 2 \dots, t_{j} \end{matrix}$ (23)

Step 4. Calculate the weights ${\tilde{p}}_{ij}^{k}$ . $\begin{matrix} {\tilde{p}}_{ij}^{k} = \frac{p_{ij}^{k} (1 + T (r_{ij}^{k}))}{\sum_{k = 1}^{t_{j}} p_{ij}^{k} (1 + T (r_{ij}^{k}))}, i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n; k, l = 1, 2 \dots, t_{j} \end{matrix}$ (24)

Step 5. Aggregate the evaluation information of each attribute by IINPGWA operator. $\begin{matrix} r_{ij} = INPGWA (r_{ij}^{1}, r_{ij}^{2}, \dots, r_{ij}^{t_{j}}) = \\ ([\begin{matrix} (1 - \prod_{k = 1}^{t_{j}} (1 - (TM (r_{ij}^{kL}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ}, \\ (1 - \prod_{k = 1}^{t_{j}} (1 - (TM (r_{ij}^{kU}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ} \end{matrix}], \\ \begin{matrix} [\begin{matrix} 1 - (1 - \prod_{k = 1}^{t_{j}} (1 - (1 - IM (r_{ij}^{kL}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ}, \\ 1 - (1 - \prod_{k = 1}^{t_{j}} (1 - (1 - IM (r_{ij}^{kU}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ} \end{matrix}] \\ [\begin{matrix} 1 - (1 - \prod_{k = 1}^{t_{j}} (1 - (1 - FM (r_{ij}^{kL}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ}, \\ 1 - (1 - \prod_{k = 1}^{t_{j}} (1 - (1 - TM (r_{ij}^{kU}))^{λ})^{{\tilde{p}}_{ij}^{k}})^{1 / λ} \end{matrix}] \end{matrix}) \end{matrix}$ (25)

Step 6. Calculate T (r_ij) $\begin{matrix} T (r_{ij}) = \underset{l \neq k}{\sum_{l = 1}^{t_{j}}} Sup (r_{ij}, r_{il}), i = 1, 2, \dots, m; \\ j, l = 1, 2, \dots, n \end{matrix}$ (26)

Step 7. Calculate the weights ϖ_ij. $\begin{matrix} ϖ_{i j} = \frac{w_{j} (1 + T (r^{i j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (r^{i j}))}, i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n \end{matrix}$ (27)

Owning to the incompletely known attribute weights w_j, we can utilize the maximization deviation method to determine the attribute weights based on the aggregated matrix just like the Step1 in the extended TOPIS Method below.

Step 8. Calculate the comprehensive evaluation value of each alternative, which is shown as follows: $\begin{matrix} \begin{matrix} r_{i} = I N P G W A (r_{i 1}, r_{i 2}, \dots, r_{i n}) \\ = ([\begin{matrix} {(1 - \prod_{j = 1}^{n} {(1 - {(T M (r_{i j}^{L}))}^{λ})}^{ϖ}^{^{i j}})}^{1 / λ}, \\ {(1 - \prod_{j = 1}^{n} {(1 - {(T M (r_{i j}^{U}))}^{λ})}^{ϖ^{i j}})}^{1 / λ} \end{matrix}], \end{matrix} \end{matrix} \begin{matrix} [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - I M (r_{i j}^{L}))}^{λ})}^{ϖ^{i j}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - I M (r_{i j}^{U}))}^{λ})}^{ϖ^{i j}})}^{1 / λ} \end{matrix}] \\ [\begin{matrix} 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - F M (r_{i j}^{L}))}^{λ})}^{ϖ^{i j}})}^{1 / λ}, \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - F M (r_{i j}^{U}))}^{λ})}^{ϖ^{i j}})}^{1 / λ} \end{matrix}]) \end{matrix}$

Step 9. Calculate the cos(r_i, I), where I = ([1, 1], [0, 0], [0, 0]) by formula (8).

Step 10. Rank all the alternatives and select the optimal one(s) according to the value of cos(r_i, I)

According to the steps 1–10, we get the cos(r_i, I). Due to space constraints, the other steps are omitted. The results are shown as below: $\begin{matrix} cos (r_{1}, I) = 0.609, cos (r_{2}, I) = 0.831, \\ cos (r_{3}, I) = 0.815 \end{matrix}$

According to the value of cos(r_i, I), we can obtain the ranking of the alternatives, i.e., a₂ ≻ a₃ ≻ a₁, So the optimal alternatives is a₂.

Obviously, this method produced the same ranking results as the proposed method in this paper. This can verify the validity of our method.

(3) Compared with the interval neutrosophic TOPSIS method.

Firstly, the prospect value matrix obtained by our method is used as the original matrix in the extended TOPSIS method. The steps for the TOPSIS method are as follows:

Step 1. Utilize the maximization deviation method to determine the attribute weights. The model is established as follows: $\begin{matrix} \begin{matrix} max = 0.9 w (c_{1}) + 2.468 w (c_{2}) + 2.696 w (c_{3}) \\ + 3.88 w (c_{4}) \\ {\begin{matrix} w (c_{1}) + w (c_{2}) + w (c_{3}) + w (c_{4}) = 1 \\ \begin{matrix} w (c_{1}) \in [0 . 3, 0 . 35], w (c_{2}) \in [0 . 35, 0 . 45], \\ w (c_{3}) \in [0 . 2, 0 . 25], w (c_{4}) \in [0 . 25, 0 . 3] \end{matrix} \end{matrix} \end{matrix} \end{matrix}$

Solving this above model by Lingo software, we obtain $\begin{matrix} w (c_{1}) = 0.3, w (c_{2}) = 0.15, w (c_{3}) = 0.25, \\ w (c_{4}) = 0.3 \end{matrix}$

Step 2. Construct weighting matrix shown as follows.

$[\begin{matrix} 0.028 & 0.041 & - 0.189 & - 0.214 \\ 0.087 & - 0.041 & - 0.031 & - 0.095 \\ 0.020 & 0.052 & - 0.021 & 0.078 \end{matrix}]$

Step 3. Identify the positive ideal solution and negative ideal solution, we get $\begin{matrix} {\hat{r}}^{+} = (\begin{matrix} 0.087, & \begin{matrix} 0.052, & - 0.021, & 0.078 \end{matrix} \end{matrix}) \\ {\hat{r}}^{-} = (\begin{matrix} 0.020, & - 0.041, & - 0.189, & - 0.214 \end{matrix}) \end{matrix}$

Step 4. Calculate the distance from the solution to the ideal solution and the negative ideal solution respectively: $\begin{matrix} D_{1}^{+} = 0.116, D_{2}^{+} = 0.038, D_{1}^{+} = 0.004, \\ D_{1}^{-} = 0.007, D_{2}^{-} = 0.044, D_{3}^{-} = 0.122 \end{matrix}$

Step 5. Calculate the relative closeness coefficient of each alternative (i.e., comprehensive evaluation index). $C_{1}^{*} = 0.055, C_{2}^{*} = 0.533, C_{3}^{*} = 0.964$

Step 6. Rank all the alternatives { a₁, a₂, a₃ } according to the relative closeness coefficient $C_{i}^{*}$ (i = 1, 2, 3), and get $a_{3} ≻ a_{2} ≻ a_{1}$

So, the optimal alternative is the a₃. Obviously, this ranking result is not the same as that produced by our method.

Because these two methods all use the prospect value matrix, i.e., they considered the bounded rationality. However, the proposed method adopt the generalized Shapley function, it takes into account the relevance of attributes while the interval neutrosophic TOPSIS method considered the attributes independent. We can analyze the fuzzy integrals obtained using the generalized Shapley function, that is, the attribute weights exist redundant and complementary relations which are shown as follows. $\begin{matrix} μ (c_{1}) + μ (c_{2}) = 0.45 > μ (c_{1}, c_{2}) = 0.3, \\ μ (c_{1}) + μ (c_{3}) = 0.55 > μ (c_{1}, c_{3}) = 0.3 \\ μ (c_{1}) + μ (c_{4}) = 0.6 > μ (c_{1}, c_{4}) = 0.3, \\ μ (c_{2}) + μ (c_{3}) = 0.4 > μ (c_{2}, c_{3}) = 0.3 \\ μ (c_{2}) + μ (c_{4}) = 0.45 < μ (c_{2}, c_{4}) = 1, \\ μ (c_{3}) + μ (c_{4}) = 0.55 < μ (c_{3}, c_{4}) = 1 \end{matrix}$

From this result, we can see complementary relationship exists in the attributes c₁ and c₂, c₁ and c₃, c₁ and c₄, c₂ and c₃, and redundant relationship exists in c₂ and c₄, c₃ and c₄. Thus the ranking result obtained by our method based on the generalized Shapley function is more accurate.

The comparisons of four methods are shown in Table 3.

Table 3

The comparisons of four methods

Methods	Rankings of alternatives
The proposed SMADM method based on CPT and Shapley function	a₂ ≻ a₃ ≻ a₁
Interval neutrosophic ELECTRE method [35]	a₂ ≻ a₃ ≻ a₁
Method based on interval neutrosophic aggregation operator [24, 36]	a₂ ≻ a₃ ≻ a₁
Interval neutrosophic TOPSIS method based on the CPT [42]	a₃ ≻ a₂ ≻ a₁

4.4 Further analysis

Based on the above comparative calculations and analysis, we can obviously see that the ranking result obtained by using the proposed SMADM method is the same as that obtained by using the method based on the interval neutrosophic aggregation operator and interval neutrosophic ELECTRE method. The main reason is that the proposed SMADM method takes into account not only the risk preferences/attitude of the decision maker, but also the interactive relation between multiple attributes of the SMADM problem, so it yields more reasonable and suitable decision result. Comparing with the method based on the interval neutrosophic aggregation operator, interval neutrosophic TOPSIS method and interval neutrosophic ELECTRE method, the proposed interval neutrosophic SMADM method based on the CPT and generalized Shapley function has some advantages, which are listed as follows:

Compared with the interval neutrosophic TOPSIS method based on CPT, the proposed interval neutrosophic SMADM method is based on CPT and the generalized Shapley function which considers not only the decision maker’s psychological behavior but also the interactive attribute relation. According to the above analysis, the ranking of alternatives produced by the interval neutrosophic TOPSIS method is not consistent with the proposed method. The reason is that the modified TOPSIS method has shortcoming, that is, it doesn’t consider the interrelation of the attribute values. In our method, we obtain the optimal fuzzy measure by maximum deviation principle, and therefore can calculate the optimal weights of each attribute value on the basis of the generalized Shapley function. The ranking of alternatives produced by the proposed method is more optimal than the extended TOPSIS method.

Compared with the method based on the interval neutrosophic aggregation operator, The proposed interval neutral SMADM method based on CPT, fully considering the psychological behavior preference of decision makers in the SMADM problem, can reflect the risk preferences and practical requirements as well to produce more accurate and flexible results, while the neutral interval SMADM method based on aggregation operator (i.e. INPGWA) is on the basis of the assumption that the decision maker is complete rationality, and therefore the decision-making behavior and psychological factors are not taken into account in the actual decision-making process. Moreover, The INPGWA can only deal with the relationship between the two attributes, but for the interaction among multiple attributes, it can’t do anything. The proposed interval neutrosophic SMADM method uses generalized Shapley function values to represent attribute weights, and it can well deal with the interaction between multiple attributes.

Compared with the interval neutrosophic ELECTRE method. Firstly, the solution of the proposed method is based on the quantitative calculation of each attribute value, and tries to construct the complete order on the feasible alternative set according to the result of this calculation. The interval neutrosophic ELECTRE method establishes the outranking relation based on the priority order of the attribute, which is a weaker order relation, that is, the partial order rather than the complete order, therefore the interval neutrosophic ELECTRE method cannot make full use of the information provided by decision matrix. Secondly, the interval neutrosophic ELECTRE method fails to consider the decision maker’s psychological behavior and the interactive attribute relation while our method can finish these functions.

In accordance with the above three group comparisons and analyses, it is clear that the proposed method is more effective and reasonable than the other three methods.

5 Conclusion

The SMADM method is widely used in practical decision-making. At the same time, multi-attribute decision-making method considering the bounded rationality of decision-makers and the interrelation of the attribute values has become the hotspot of decision-making science. In this paper, a new SMADM method is proposed for multi-attribute decision making problems with attributes as INNs based on the CPT and the generalized Shapley value function, which has the advantages: considering the influence of the decision maker’s subjective preference and the interrelated relationship between the multiple attributes. Therefore, the proposed method can give more reasonable ranking result than the existing methods. In the future, we will continue to study the applications of the proposed methods in other practical decision, e.g., economics, management and so on [41].

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos.71771140 and 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 15BGLJ06,16CGLJ31 and 16CKJJ27), the Humanities and Social Sciences Research Project of the Colleges and universities of Shandong province(J16YF06), the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057), and Key research and development program of Shandong Province (2016GNC110016).

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Interval neutrosophic stochastic multiple attribute decision-making method based on cumulative prospect theory and generalized Shapley function

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The neutrosophic set

3.1 Problem description

3.2 The calculation of the prospect function value

Table 1 The evaluation matrix of three possible alternatives With respected to the four criteria

4.2 The steps of the example

Table 2 The ranking results of alternatives with different referent points

5 Conclusion

Footnotes

Acknowledgments

References

Table 1
The evaluation matrix of three possible alternatives With respected to the four criteria

Table 2
The ranking results of alternatives with different referent points