A multi-stage uncertain risk decision-making method with reference point based on extended LINMAP method

Abstract

The multi-stage uncertain risk decision-making problems are very common in production and life, such as emergency decision-making problems, batch product sampling inspection, etc. It’s very complex and interesting in applications. The multi-stage uncertain risk decision-making process often involves multiple reference points owing to decision-making environment and decision-makers’ mentalities. This paper aims to solve the multi-stage uncertain risk decision-making problems without probability information. Decision makers’ psychological behaviors are considered by setting the thresholds of possibility degree in different states. The attribute weight optimization model is established based on the extended LINMAP method through a two-stage model, considering the principle of maximizing the ranking consistency of alternatives in all states. The stage weight optimization model is established in the viewpoint of minimizing the opportunity loss. The optimal alternative is selected according to the compromise ranking values, which consider the satisfaction of decision makers. A case is presented to illustrate the feasibility and validity of the proposed method.

Keywords

Multi-stage uncertain risk decision-making weight optimization LINMAP relative entropy reference point

1 Introduction

Multi-stage uncertain risk decision-making (MSURDM) problems exist widely in daily life. For example, emergency assessment, quality inspection, disaster prediction and so on. There are several special issues that have to be concerned.

There may be several states without probability information in the future due to the uncontrollable natural factors and human factors. That may lead to certain risks and uncertainties to decision-making (DM)problems [8].

When facing DM risks, decision-makers (DMs) often have some risk appetite [5 –7]. Due to the limited rationality of humanity’s thinking, there is psychological perceptual deviation.

DM environment is always in a dynamic change [21 , 33]. The development levels at different stages are required to be considered.

Multiple states often exist in DM problems owing to the uncertain environment. DMs often rely on the expected value to solve this kind of problem, if the probability is known [1 –3]. Probability information is a prediction of future which needs a long-term observation. It’s hard to obtain the probability information. In this context, scholars have made a lot of research taking into account the multiple impacts on DM environment, DM information and DMs’ preferences. An evidence reasoning approach is proposed to solve the multi-criteria ranking problem in uncertain environment [10]. Immediate probabilities are modified based on the objective probability with the DMs’ attitudinal characters [1]. Interval probabilities are proposed to solve the multi-attribute DM problems with uncertain linguistic information [4].

It’s hard to get accurate information under complex and uncertain environment. Fuzzy set [11, 12], linguistic set [20], interval grey number [13] and so on are often used to express the DMs’ opinions. Interval grey number is an effective form to represent uncertain information. Julong Deng first proposed interval grey number to measure complex issues [14]. Later, many scholars represented by Sifeng Liu conducted an in-depth study and discussion on interval grey number [13]. A novel dominance relation between interval grey numbers is proposed for multiple attribute DM problems [9]. The degree of greyness and choquet integral is presented to solve the multiple attribute DM problems based on interval grey number [17].

The perceived deviation from DMs is ubiquitous in DM process [6]. People often make decisions by comparing the values of alternatives to the value of the reference point in order to reduce the perceived bias. Reference points are considered in the dynamic stochastic multi-criteria DM problems [18]. The target space closest to the reference point is presented to select the optimal alternative [19]. Multiple reference points are set in multi-attribute RDM problems [3]. The difference between the comprehensive evaluation point and the reference point is minimized to solve project evaluation problems [20].

DM problems are always in dynamic development. Multi-stage evaluation problems become a hot research topic. A multi-stage investment DM method is proposed based on expert evaluation [21]. A multi-objective DEA method is suggested to solve the multi-stage efficiency evaluation problem [22]. A multi-stage stochastic programming method is proposed to solve the water resource planning problem [23]. A new consensus reach model is proposed and used in a dynamic DM problem [24]. An interval dynamic reference point method based on prospect theory is suggested to solve the emergency DM problem [25].

Multiple attribute DM method is an effective way to settle the MSURDM problems. Linear programming technique for multidimensional analysis of preference (LINMAP) is a famous multiple attribute DM method to analyze the individual differences depending on a set of stimuli predefined. LINMAP method is first proposed by Srinivasan and Shocker [27]. Extensions of LINMAP can be divided into two classes. The first class is applied the decision data to uncertain forms, such as fuzzy sets [34], intuitionistic fuzzy sets [35], linguistic variables [36] and so on. The second class is the comprehensive application of LINMAP and other methods to solve different kinds of problems. LINMAP method is utilized to solve multiple criteria group DM problems [37]. LINMAP and TOPSIS are integrated to solve the hesitant fuzzy DM problems [38]. The extended LINMAP method is designed to solve the interval type-2 fuzzy DM problems [39].

This study aims to solve the MSURDM problems with reference points in which the probability information of states is unknown. Firstly, the threshold of possibility degree is presented to construct a new value function. The operator uses interval grey number, the possibility degree and the prospect theory. This makes it possible to consider DMs’ risk preferences. Secondly, the attribute weight optimization model is established based on the extended LINMAP method. The model helps to improve the ranking consistency of alternatives in different states. Thirdly, the stage weight optimization model is set up based on the relative entropy. The model helps to minimize the opportunity loss. Finally the best alternative is chosen according to the compromise ranking values, which consider DMs’ satisfaction.

This paper is structured as follows. Section 2 introduces the basic concepts. Section 3 presents the new MSURDM method. Section 4 makes a case study. Section 5 concludes the paper.

2 Preliminaries

DMs do not often know the distribution of DM information. They can only give the possible range of DM information. Interval grey number is an effective method to represent the uncertain information [13].

Definition 1. [13] A grey number with both a lower limit $\bar{a}$ and an upper limit $\underset{0.3 em -}{a}$ is called an interval grey number, denoted as $\tilde{a} (\otimes) \in [\underset{0.3 em -}{a}, \bar{a}]$ .

Operation rules of interval grey number are omitted [13]. There are both similarities and differences between interval grey number and interval number [13 –16]. The main difference between the two kinds of numbers is the range of number. An interval grey number is the only one number which belongs to the range. However, an interval number is the whole range [13].

Definition 2. [16] Let $\tilde{a} (\otimes) \in [\underset{0.3 em -}{a}, \bar{a}]$ and $\tilde{b} (\otimes) \in [\underset{0.3 em -}{b}, \bar{b}]$ be two interval grey numbers, and $l_{a} = \bar{a} - \underset{0.3 em -}{a}$ , $l_{b} = \bar{b} - \underset{0.3 em -}{b}$ , the possibility degree that $\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)$ can be defined as follows.

$\begin{matrix} p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) \\ = \frac{\min {l_{a} + l_{b}, \max (\bar{a} - \underset{0.3 em -}{b}, 0)}}{l_{a} + l_{b}} \end{matrix}$ (1)

The formula is derived from interval number [40]. It’s found that the possibility degree of interval number has also been used in the calculation of interval grey number [2 , 42], if two interval grey numbers are not identical.

In prospect theory, the value function defines the deviation of a value relative to its reference point.

Definition 3. [6] The value function is expressed in the form of a power law.

$v (x) = {\begin{matrix} x^{α}, & x ⩾ 0 \\ - θ {(- x)}^{β}, & x < 0 \end{matrix}$ (2)

Where x denotes the gain or loss of a value relative to its reference point. x ⩾ 0 represents a gain, and x < 0 represents a loss. α is the risk-seeking coefficient, and β is the risk-averse coefficient. θ indicates the loss-averse coefficient. Tversky and Kahneman [7] used a nonlinear regression procedure to estimate that when α = β = 0.88 and θ = 2.25, the experimental results are more consistent with the empirical results. The set of values have already been used in much research [3 , 32]. So, we take α = β = 0.88, θ = 2.25.

Definition 4. [26] The relative entropy between two probabilities mass functions p (i) and q (i) (∑_ip (i) = 1, ∑_iq (i) = 1) is defined as follows. $D (p | | q) = - \sum_{i} p (i) \log_{2} \frac{p (i)}{q (i)}$ (3)

Where $0 \log_{2} \frac{0}{q (i)} = 0, p (i) \log_{2} \frac{p (i)}{0} = \infty$ . In general, p (i) represents the true distribution of data, q (i) represents the approximate distribution of p (i). The relative entropy satisfies the following properties.

D (p||q) ⩾ 0, if and only if p = q, D (p||q) = 0.

D (p||q) ≠ D (q||p).

3 Main problem and method

3.1 Problem description

This paper aims to rank alternatives or select the desirable alternative(s) from a set of feasible alternatives. The MSURDM problem is illustrated in Fig. 1. There are two kinds of information in the MSURDM problem: the values of alternatives and the values of reference points. There are several states in the future without the probability information. Decisions are made by comparing the gain or loss between the values of attributes and the reference points.

Fig.1

The MSURDM problem with reference points.

Let A ={ a₁, a₂, ⋯, a_I } be the set of I alternatives. Let C ={ c₁, c₂, ⋯, c_J } be the set of J attributes. Let $X^{t} = {({\tilde{x}}_{ij}^{t} (\otimes))}_{I \times J}$ be the DM information of T stages, where ${\tilde{x}}_{i j}^{t} (\otimes) \in [\underset{0.3 e m - i j t}{x}, {\bar{x}}_{i j}^{t}]$ denotes the performance of alternative a_i with respect to attribute c_j at the tth stage. Let $X^{t *} = ({\tilde{x}}_{1}^{t *} (\otimes), {\tilde{x}}_{2}^{t *} (\otimes), \dots, {\tilde{x}}_{J}^{t *} (\otimes))$ be the set of the reference points in the tth stage, where ${\tilde{x}}_{j}^{t *} (\otimes)$ denotes the value of the reference point with respect to the attribute c_j at the tth stage. Let λ = (λ¹, λ², ⋯, λ^T) be the weighting vector of T stages, where λ^t denotes the tth stage weight. Let $W^{t} = (w_{1}^{t}, w_{2}^{t}, \dots, w_{J}^{t})$ be the weighting vector of J attributes at the tth stage, where $w_{j}^{t}$ denotes the jth attribute-weight. Let S ={ s₁, s₂, ⋯, s_N } be the set of N states in future, where s_n denotes the nth state.

3.2 The definition of possible value function for interval grey number with reference points

Interval grey number is commonly used to express fuzzy and uncertain information [13]. But the distribution in a given interval range is unknown. The possibility degree is introduced to measure the relationship between two interval grey numbers.

In the existing methods [16], when $p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) \in [0.5, 1]$ , we have $\tilde{a} (\otimes) < \tilde{b} (\otimes)$ . When $p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) \in [0, 0.5)$ , we have $\tilde{a} (\otimes) < \tilde{b} (\otimes)$ .

Example 1. There are three interval grey numbers ${\tilde{a}}_{1} (\otimes) \in [1, 4], {\tilde{a}}_{2} (\otimes) \in [2, 3], {\tilde{a}}_{3} (\otimes) \in [4, 5]$ . We have $p ({\tilde{a}}_{1} (\otimes) ⩾ {\tilde{a}}_{2} (\otimes)) = 0.5$ , $p ({\tilde{a}}_{3} (\otimes) ⩾ {\tilde{a}}_{2} (\otimes)) = 1$ . In any case, we have ${\tilde{a}}_{3} (\otimes) > {\tilde{a}}_{2} (\otimes)$ . But ${\tilde{a}}_{1} (\otimes) ⩾ {\tilde{a}}_{2} (\otimes)$ may not be established, if ${\tilde{a}}_{1} (\otimes)$ falls in [1,2, 1,2].

Because the distributions of interval grey numbers are unknown, comparison of two interval grey numbers cannot only depend on the possibility degree. DM’s risk preferences in different states affect the relationships between interval grey numbers. It is necessary to make a comprehensive judgment on DMs’ psychological factors [6, 7].

Definition 5. Let ɛ (0 ⩽ ɛ ⩽ 1) be a threshold of possibility degree. The relationship between two interval grey numbers is defined as follows. $p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) {\begin{matrix} \in (ɛ, 1], \tilde{a} (\otimes) ≻ \tilde{b} (\otimes) \\ = ɛ, \tilde{a} (\otimes) \approx \tilde{b} (\otimes) \\ \in [0, ɛ), \tilde{a} (\otimes) ≺ \tilde{b} (\otimes) \end{matrix}$ (4)

In practice, ɛ can be obtained by interviews with experts. In a low-risk environment, DMs may set a threshold of possibility degree closer to 0, which means a smaller threshold of possibility degree can make $\tilde{a} (\otimes) ≻ \tilde{b} (\otimes)$ . In a high-risk environment, DMs may set a threshold of possibility degree closer to 1, which means a bigger threshold of possibility degree is more likely to reach $\tilde{a} (\otimes) ≻ \tilde{b} (\otimes)$ .

Definition 6. Let $\tilde{a} (\otimes)$ and $\tilde{b} (\otimes)$ be two interval grey numbers. $\tilde{a} (\otimes)$ is the value of an attribute, and $\tilde{b} (\otimes)$ is the value of a reference point. For a given threshold of possibility degree ɛ (0 ⩽ ɛ ⩽ 1), the possible value function to measure deviations of $\tilde{a} (\otimes)$ relative to $\tilde{b} (\otimes)$ is defined as follows. $v_{(a, b)} = {\begin{matrix} (d_{p (a, b)})^{α} & \tilde{a} (\otimes) ⩾ \tilde{b} (\otimes) \\ - θ {(d_{p (a, b)})}^{β} & \tilde{a} (\otimes) < \tilde{b} (\otimes) \end{matrix}$ (5)

Where $d_{p (a, b)} = | p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) - ɛ | \cdot d_{(a, b)}$ . d_p(a,b) denotes the gain or loss of $\tilde{a} (\otimes)$ relative to $\tilde{b} (\otimes)$ . $d_{(a, b)} = \sqrt{{(\underset{0.3 em -}{a} - \bar{b})}^{2} + {(\bar{a} - \underset{0.3 em -}{b})}^{2}}$ denotes the distance between $\tilde{a} (\otimes)$ and $\tilde{b} (\otimes)$ [13]. The greater the value of $| p (\tilde{a} (\otimes) ⩾ \tilde{b} (\otimes)) - ɛ |$ is, the greater the probability of a gain or loss.

Theorem 1. For two given thresholds of possibi-lity degree ɛ1 and ɛ2, when ɛ1 < ɛ2, there is $v {(\tilde{a} (\otimes), \tilde{b} (\otimes))}_{ɛ 1} > v {(\tilde{a} (\otimes), \tilde{b} (\otimes))}_{ɛ 2}$ .

Proof. (Omitted.)

DMs’ risk preferences have an important impact on DM values. The introduction of the threshold of possibility degree makes it possible to consider DMs’ risk preferences. Using different thresholds of possibility degree to measure DM values is more suitable for risk and uncertain information.

3.3 The two-stage attribute weight optimization model based on the extended LINMAP method

DM goals change over time, so attribute weights are very different from different stages. Traditional approaches (AHP method [24], the maximum entropy method [27], LINMAP method [28]) can solve DM problems with no risk. They cannot adapt to the DM environment under multiple states without probability information [2, 4].

DMs can make scientific decisions easier if the ranking results in different states are more consistent. TOPSIS [38], LINMAP [28] are value-based methods that rank alternatives according to the closeness to the ideal alternative. The traditional idea of LINMAP [28] is to construct the consistency and non-consistency of the alternative to the ideal alternative. LINMAP can realize the comparisons without a given rank. Therefore, the traditional LINMAP method is extended to the uncertain RDM environment to reduce the ranking conflict in different states.

In the tth stage, $V_{ij}^{t} (n)$ represents the value of alternative a_i with respect to attribute c_j in state s_n, which can be obtained by Equation (5). $V_{i}^{t} (n) = \sum_{j = 1}^{J} w_{j}^{t} V_{ij}^{t} (n) (i = 1, 2, \dots, I)$ represents the value of alternative a_i in state s_n. For two alternatives a_k and a_l (k, l = 1, 2, ⋯, I ; k ≠ l), the consistency and non-consistency of $V_{k}^{t} (n) ⩾ V_{l}^{t} (n)$ based on LINMAP is denoted as V (k, l) ^t+ (n), V (k, l) ^t- (n). $V {(k, l)}^{t +} (n) = \max {0, V_{k}^{t} (n) - V_{l}^{t} (n)}$ (6) $V {(k, l)}^{t -} (n) = \max {0, V_{l}^{t} (n) - V_{k}^{t} (n)}$ (7)

The pair of alternatives (k, l) is ordered, which means alternative a_k is better than a_l.

Definition 7. The total ranking consistency of (k, l) in all states is defined as ΔV^t (k, l). $Δ V^{t} (k, l) = \sum_{n = 1}^{N} (V {(k, l)}^{t +} (n) - V {(k, l)}^{t -} (n))$ (8)

The bigger the value of ΔV^t (k, l), the higher the ranking consistency of (k, l). The smaller the value of ΔV^t (k, l), the higher the ranking consistency of (l, k). The bigger the value of (ΔV^t (k, l)) ², the higher the ranking consistency of (k, l) or (l, k).

Attribute weight shows the relative importance of the attributes with respect to the DM goal. It is difficult to determine the exact attribute weight, due to the fuzziness of human thinking and the increasing complexity of the DM environment.

The following settings are considered when establishing the attribute weight optimization model. (1) In the viewpoint of ranking alternatives more easily, it is feasible to set the objective function with $f (w_{j}^{t}) = \sum_{k = 1, l > k}^{I} {(Δ V^{t} (k, l))}^{2}$ to be maximum, which indicates the total ranking consistency of all pairs of alternatives to be maximum. (2) Attribute weight meets the prior conditions H₁, which can be expressed in 5 forms [29]. (3) The entropy of attribute weights can be set to a range [γ₁lnJ, γ₂lnJ], taking into account the differences between attribute weight and the full utilization of DM information. The greater the value of γ₂lnJ, the greater the amount of information available to the problem [26]. The lower the value of γ₁lnJ, the bigger the difference between attribute weights. This setting ( $- \sum_{j = 1}^{J} w_{j}^{t} * \ln w_{j}^{t} in [γ_{1} \ln J, γ_{2} \ln J]$ ) cannot only make full use of information but also maintain the differences between attribute weight.

Following consideration of the above principles, model M1 is established. $\begin{matrix} maxf (w_{j}^{t}) = \\ \sum_{\begin{matrix} k = 1, \\ l > k \end{matrix}}^{I} (\sum_{n = 1}^{N} \max {0, \sum_{j = 1}^{J} (w_{j}^{t} V_{kj}^{t} (n) - w_{j}^{t} V_{lj}^{t} (n))} \\ {- \sum_{n = 1}^{N} \max {0, \sum_{j = 1}^{J} (w_{j}^{t} V_{lj}^{t} (n) - w_{j}^{t} V_{kj}^{t} (n))})}^{2} \\ s . t . {\begin{matrix} \sum_{j = 1}^{J} w_{j}^{t} = 1 \\ w^{t} = (w_{1}^{t}, w_{2}^{t}, \dots, w_{J}^{t}) \in H_{1}^{t} \\ - \sum_{j = 1}^{J} w_{j}^{t} * \ln w_{j}^{t} \in [γ_{1} lnJ, γ_{2} lnJ] \end{matrix} M 1 \end{matrix}$

In model M1, $f (w_{j}^{t})$ represents the total ranking consistency of all pairs of alternatives in all states. Constraints are as follows. The sum of attribute weights is equal to 1. Attribute weight meets the prior conditions $H_{1}^{t}$ . The entropy of attribute weights is setting in [γ₁lnJ, γ₂lnJ] (0 ⩽ γ₁ ⩽ γ₂ ⩽ 1). The higher the value of γ₂, the more fully utilized DM information can be. The lower the value of γ₁, the greater the differences between attribute weights are.

It’s hard to determine the exact values of γ₁, γ₂. γ₁lnJ indicates the minimal entropy of attribute weights. γ₂lnJ indicates the maximal entropy of attribute weights. By establishing model M2, the minimum value of γ₁ and the maximum value of γ₂ can be calculated under the prior conditions.

$\begin{matrix} \max / minH (w_{j}^{t}) = - \sum_{j = 1}^{J} w_{j}^{t} * {lnw}_{j}^{t} \\ s . t . {\begin{matrix} \sum_{j = 1}^{J} w_{j}^{t} = 1 \\ w^{t} = (w_{1}^{t}, w_{2}^{t}, \dots, w_{J}^{t}) \in H_{1}^{t} \end{matrix} M 2 \end{matrix}$

The objective function represents the maximum entropy or the minimum entropy. The constraint conditions are the same with those in model M1. Then calculate $γ_{2}^{*} = maxH (w_{j}^{t}) / lnJ$ and $γ_{1}^{*} = minH (w_{j}^{t}) / lnJ$ . Thus, DMs can set $H (w_{j}^{t}) \in [γ_{1} lnJ, γ_{2} lnJ]$ $(γ_{1}^{*} ⩽ γ_{1} ⩽ γ_{2} ⩽ γ_{2}^{*})$ according to the actual situation.

Theorem 2. If the feasible region exists and is bounded, model M2 must get the optimal solution.

Proof. Model M2 is a single objective programming problem. If the feasible region exists and is bounded, model M2 must have the optimal solution depending on the existence theorem of the optimal solution [30]. $H_{1}^{t}$ is a set of prior conditions of the attribute weights in the tth stage. If the prior conditions are not reasonable, the feasible region of model M2 may not exist, which means the model has no optimal solution. DMs should change the prior conditions considering the actual situation.

Theorem 3. If model M2 has the optimal solution and $γ_{1}^{*} ⩽ γ_{1} ⩽ γ_{2} ⩽ γ_{2}^{*}$ , model M1 must get the optimal solution.

Proof. (omitted).

Then the value of alternative a_i of state s_n in stage t $V_{i}^{t} (n)$ can be calculated through the following formula. $V_{i}^{t} (n) = \sum_{j = 1}^{J} w_{j}^{t} V_{ij}^{t} (n)$ (9)

From the viewpoint of maximizing the ranking consistency in different states, the attribute weight optimization model is established based on the idea of LINMAP. In the constraint conditions, the full utilization of information and the differences between attribute weight are also considered. In MSURDM environment, stage weights have a great influence on the comprehensive ranking values. The method to optimize stage weight is researched in the next part.

3.4 The stage weight optimization model based on relative entropy

If probability information is unknown, multiple DM principles according to DMs’ risk preferences are usually used, such as the Optimistic Principle (OP), the Pessimistic Principle (PP), the Mean Value Principle (MVP), the Minimum Regret Principle (MRP).

Example 2. RDM information is showed in Table 1 with three states (S1, S2, S3) and three alternatives (A1, A2, A3). The results obtained by the above DM principles are showed in Table 1.

Table 1
An example of RDM problems

S1 S2 S3 OP PP MVP MRP

A1 40 15 20 40 15 25 0

A2 20 15 40 40 15 25 0

A3 40 20 20 40 20 26.7 0

	S1	S2	S3	OP	PP	MVP
A1	40	15	20	40	15	25
A2	20	15	40	40	15	25
A3	40	20	20	40	20	26.7

It is unable to find the best alternative or an effective rank according to OP and MRP. The best alternative can be obtained according to PP and MVP. But the three alternatives cannot be ranked effectively. There are no effective comparisons between states and alternatives when using the above DM principles. The relative entropy method [26] cannot only compare the differences between two distributions, but also compare the differences between each alternative. Here, the relative entropy method is extended to the MSURDM problem without probability information.

The relative entropy D (p||q) represents the information loss when using the possible distribution q (i) to fit the real distribution p (i) [26]. DMs always want the information loss as small as possible, which means D (p||q) as small as possible.

Definition 8. Let s_nr be a real state. Let s_ne be an error state. The corresponding value of each state is $V_{i}^{t} (nr)$ and $V_{i}^{t} (ne)$ . When using the data in state s_ne to fit the data in state s_nr, the opportunity loss caused by DM errors can be defined as D (s_nr||s_ne):

$D (s_{nr} | | s_{ne}) = \sum_{i} \frac{V_{i}^{t} (nr)}{\sum_{i = 1}^{I} V_{i}^{t} (nr)} \log_{2} \frac{\frac{V_{i}^{t} (nr)}{\sum_{i = 1}^{I} V_{i}^{t} (nr)}}{\frac{V_{i}^{t} (ne)}{\sum_{i = 1}^{I} V_{i}^{t} (ne)}}$ (10)

The thresholds of possibility degree in real state s_nr and error state s_ne is ɛ_nr and ɛ_ne. If ɛ_nr > ɛ_ne, $V_{ij}^{t} (nr) < V_{ij}^{t} (ne)$ gets to work according to Theorem 1., there is no opportunity loss. If ɛ_nr < ɛ_ne, $V_{ij}^{t} (nr) > V_{ij}^{t} (ne)$ gets to work, there is an opportunity loss.

Definition 9. The total opportunity loss in all states can be defined as D. $\begin{array}{l} D = \sum_{n 1 = 1, n 2 > n 1}^{N} D (S_{n 1} ∥ S_{n 1}) \\ \sum_{\begin{array}{l} n 1 = 1, \\ n 2 > n 1 \end{array}}^{N} \sum_{i} \frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)} \\ \begin{array}{l} \times \log_{2} \end{array} \frac{\frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)}}{\frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)}} \end{array}$ (11)

For any two states s_n1 ∈ S and s_n2 ∈ S, if n2 > n1 then ɛ_n1 < ɛ_n2.

In MSURDM process, the following factors are taken into consideration. (1) Due to the uncertain and fuzzy DM information, the opportunity loss caused by DM mistakes often exists objectively. The lower the value of D, the smaller the opportunity loss is. So, the objective function is set to minimize the total opportunity loss. (2) Stage weight meets the prior conditions H₂, which can be expressed in 5 forms [29]. (3) Orness measure can reflect DM’s preference for recent and long-term data [31]. (4) Things always develop from tiny, non-significant quantitative changes to significant and fundamental changes in nature. In the time of observation, those changes tend to be quantitative changes which are relatively stable within a certain range. So, the changes of adjacent stages are relatively stable, which should be set in a certain range. There is this setting $\frac{λ^{t + 1} V_{i}^{t + 1} (n) - λ^{t} V_{i}^{t} (n)}{λ^{t + 1} V_{i}^{t + 1} (n)} \in [- η, η]$ . The larger the value of η, the greater the allowable range of the change rate.

Following consideration of the above principles, model M3 is established. $\begin{array}{l} \sum_{\begin{array}{l} n 1 = 1, \\ n 2 > n 1 \end{array}}^{N} \sum_{i} \frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)} \\ \begin{array}{l} \times \log_{2} \end{array} \frac{\frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)}}{\frac{\sum_{t = 1}^{T} λ^{t} V_{i}^{t} (n 1)}{\sum_{i = 1}^{I} \sum_{i = 1}^{T} λ^{t} V_{i}^{t} (n 1)}} \\ s . t . {\begin{matrix} \sum_{t = 1}^{T} λ^{t} = 1 \\ λ = (λ^{1}, λ^{2}, \dots, λ^{T}) \in H_{2} \\ o r n e s s (λ_{t}) = δ = \frac{\sum_{t = 1}^{T} (T - t) λ^{t}}{T - 1} \\ \frac{λ^{t + 1} V_{i}^{t + 1} (n) - λ^{t} V_{i}^{t} (n)}{λ^{t + 1} V_{i}^{t + 1} (n)} \in [- η, η], η \geq 0 \end{matrix} M 3 \end{array}$

The objective function D (λ^t) represents the total opportunity loss in all states. Constraints are the following conditions. The sum of stage weight is equal to 1. Stage weight meets the prior conditions H₂. Stage weights satisfy the orness measure. δ indicates the DMs’ preference for recent and long-term data (0 ⩽ δ ⩽ 1). The smaller the value of δ, the more attention DMs pay to the recent data. Otherwise DMs pay more attention to the long-term data. The last condition indicates that the change rate of adjacent stages is in the interval range of [- η, η]. Generally, η is usually given by DMs depending upon practical experience.

Theorem 4. If the feasible region of model M3 is nonempty and bounded, model M3 must have the optimal solution.

Proof. The relative entropy function is a convex function [26]. The objective function D (λ_t) is the sum of convex functions, which is also a convex function. If the feasible region of model M3 is nonempty and bounded, there must be a minimum value of D (λ_t), according to the properties of convex functions. So, model M3 must have the optimal solution.

There are different absolute change rates with data in different stages. It is difficult to determine the absolute change rates. DMs can only give the probably range, such as $| \frac{λ^{t + 1} V_{i}^{t + 1} (n) - λ^{t} V_{i}^{t} (n)}{λ^{t + 1} V_{i}^{t + 1} (n)} | ⩽ η$ . If the constraint is too harsh, it may cause the feasible region to be empty. Then it will lead to no optimal solution for model M3. At this time, DMs should change η according to actual data.

By solving model M3, the stage weights λ = (λ¹, λ², ⋯, λ^T) can be obtained. The comprehensive values V_i (n) can be calculated according to Equation (12).

$V_{i} (n) = \sum_{t = 1}^{T} \sum_{j = 1}^{J} λ^{t} w_{j}^{t} V_{ij}^{t} (n)$ (12)

In RDM environment, DMs may consider both the possible benefit and the possible regret value. Take into account this, the compromise ranking value V_i is designed as follows. $\begin{matrix} V_{i} = μ \frac{\max_{n} V_{i} (n)}{\max_{n} V_{i} (n) - \min_{n} V_{i} (n)} + \\ (1 - μ) \frac{\max_{n} \max_{i} V_{i} (n) - \max_{n} V_{i} (n)}{\max_{n} \max_{i} V_{i} (n) - \min_{n} \min_{i} V_{i} (n)} \end{matrix}$ (13)

Where max_nV_i (n) - min_nV_i (n) represents the opportunity cost by selecting alternative a_i. $\frac{\max_{n} V_{i} (n)}{\max_{n} V_{i} (n) - \min_{n} V_{i} (n)}$ represents the maximum value of per unit opportunity cost. max_nmax_iV_i (n) - min_nmin_iV_i (n) represents the global maximum opportunity cost. $\frac{\max_{n} \max_{i} V_{i} (n) - \max_{n} V_{i} (n)}{\max_{n} \max_{i} V_{i} (n) - \min_{n} \min_{i} V_{i} (n)}$ represents the minimum regret value of per unit global maximum opportunity cost. μ is the preference coefficient. μ > 0.5 indicates that DMs pay more attention to the maximum value. μ > 0.5 indicates that DMs pay more attention to the minimum regret value. μ = 0.5 indicates that DMs pay equal attention to the maximum value and the minimum regret value.

Steps to solve the MSURDM problems are as follows.

Step 1. Input and normalize DM information.

To simplify the expression, the normalized data expression is unchanged. For benefit data ${\tilde{x}}_{i j}^{t} (\otimes) \in [\underset{0.3 e m - i j t, {\bar{x}}_{i j}^{t}}{x}]$ , the normalized data is ${\tilde{x}}_{i j}^{t} (\otimes) \in [\frac{\underset{0.3 e m - i j t}{x} \max_{i} {\bar{x}}_{i j}^{t}}{, \frac{{\bar{x}}_{i j}^{t}}{\max_{i} {\bar{x}}_{i j}^{t}}]}$ . For cost data ${\tilde{x}}_{i j}^{t} (\otimes) \in [\underset{0.3 e m - i j t}{x}, {\bar{x}}_{i j}^{t}]$ , the normalized data is ${\tilde{x}}_{i j}^{t} (\otimes) \in [\frac{\min_{i} \underset{0.3 e m - i j t}{x}}{{\bar{x}}_{i j}^{t}}, \frac{\min_{i} \underset{0.3 e m - i j t}{x}}{\underset{0.3 e m - i j t}{x}}]$ .

Step 2. Calculate the possibility degrees and the possible values.

The possibility degrees $p ({\tilde{x}}_{ij}^{t} (\otimes) ⩾ {\tilde{x}}_{j}^{t *} (\otimes))$ can be calculated by Equation (1). The possible values $V_{ij}^{t} (n)$ can be calculated by Equation (5), according to the given thresholds of possibility degree ɛ ={ ɛ₁, ɛ₂, ⋯, ɛ_N }.

Step 3. Calculate the attribute weights and the values of alternatives of different states.

The range of entropy for attribute weights $[γ_{1}^{*} lnJ, γ_{2}^{*} lnJ]$ can be obtained by model M2. DMs set the entropy range as [γ₁lnJ, γ₂lnJ]. Attribute weights $w^{t} = {(w_{1}^{t}, w_{2}^{t}, \dots, w_{J}^{t})}^{T}$ can be obtained by model M1. $V_{i}^{t} (n)$ can be obtained by Equation (9).

Step 4. Calculate the stage weights and the comprehensive values of alternatives of different states.

The change rate [- η, η] and the orness measure δ are given according to the knowledge, experience and preference of DMs. By solving model M3, the stage weights λ = (λ¹, λ², ⋯, λ^T) can be obtained. The comprehensive value V_i (n) in all stages can be calculated by Equation (12).

Step 5. Calculate the compromise ranking values.

V_i can be calculated by Equation (13) with the given preference coefficient μ. The best alternative can be selected by max_iV_i.

These steps can be implemented by Excel, Visual Basic 6.0, Lingo, MATLAB programming and so on. Users can package the algorithm programmatically. In practical applications, these programs can be used to solve problems easily and quickly.

From the viewpoint of minimizing the opportunity loss in all states, the stage weight optimization model is set up based on the relative entropy. In the constraint conditions, the DMs’ preference for recent and long-term data and the stability of changes between adjacent stages are considered. Finally, the best alternative can be selected by the compromise ranking values.

4 Case study

4.1 Background of case

To purchase a certain electronic component, Commercial Aircraft Corporation of China carries out the supplier evaluation. Products from 4 suppliers comprise the set of alternatives A = { a₁, a₂, ⋯, a_I } (I = 4). Evaluation attributes comprise the set of attributes C = { c₁, c₂, ⋯, c_J } (J = 5). DM information from T (T = 4) stages comprises the DM matrix ${({\tilde{x}}_{ij}^{t} (\otimes))}_{I \times J}$ . Technical experts propose the reference point $X^{t *} = ({\tilde{x}}_{1}^{t *} (\otimes), {\tilde{x}}_{2}^{t *} (\otimes), \dots, {\tilde{x}}_{J}^{t *} (\otimes))$ for each stage. Using environment of the electronic component is divided into 3 severity levels, which form the set of states S = { s₁, s₂, ⋯, s_N } (N = 3). DM information ${({\tilde{x}}_{ij}^{t} (\otimes))}_{I \times J}$ and the values of reference points ${\tilde{x}}_{j}^{t *} (\otimes)$ are showed in Table 2. The set of thresholds of possibility degree is ɛ ={ ɛ₁ = 0.35, ɛ₂ = 0.55, ɛ₃ = 0.75 }. The set of priori conditions for attribute weights is $H_{1}^{t} = {w_{j}^{t} \in [0.05, 0.3]; \sum_{j = 1}^{J} w_{j}^{t} = 1}$ . The set of priori conditions for stage weights is $H_{2} = {λ^{t} \in [0.05, 0.4]; \sum_{t = 1}^{T} λ^{t} = 1}$ .

Table 2
Decision data in different stages

Stage Alternative c ₁ c ₂ c ₃ c ₄ c ₅

t = 1 a ₁ [1840,1884] [3.759,3.865] [10.00,16.00] [0.30,0.51] [26.98,27.02]

a ₂ [1825,1877] [3.855,3.870] [12.00,15.00] [0.20,0.69] [26.86,26.90]

a ₃ [1835,1850] [3.860,3.865] [13.00,15.00] [0.30,0.55] [26.97,27.13]

a ₄ [1832,1860] [3.840,3.880] [10.50,15.50] [0.02,0.57] [26.94,27.06]

X ^1* [1833,1890] [3.850,3.880] [11.00,17.00] [0.01,1.00] [26.86,27.20]

t = 2 a ₁ [1835,1885] [3.810,3.865] [10.50,16.00] [0.15,0.66] [27.03,27.12]

a ₂ [1830,1880] [3.820,3.875] [11.00,14.00] [0.25,0.75] [26.96,27.02]

a ₃ [1840,1860] [3.585,3.870] [11.00,15.50] [0.33,0.55] [26.94,27.15]

a ₄ [1833,1870] [3.650,3.890] [10.50,15.50] [0.15,0.63] [26.93,27.00]

X ^2* [1825,1890] [3.800,3.885] [10.00,17.00] [0.01,1.00] [26.97,27.20]

t = 3 a ₁ [1835,1880] [3.825,3.880] [11.00,16.00] [0.20,0.65] [27.00,27.13]

a ₂ [1825,1885] [3.825,3.875] [12.00,16.50] [0.30,0.70] [27.06,27.09]

a ₃ [1835,1875] [3.835,3.865] [12.00,16.50] [0.40,0.60] [26.99,27.05]

a ₄ [1833,1870] [3.650,3.900] [10.50,16.00] [0.23,0.58] [27.01,27.13]

X ^3* [1825,1895] [3.755,3.905] [10.00,16.00] [0.05,1.00] [26.95,27.20]

t = 4 a ₁ [1840,1880] [3.825,3.900] [11.00,16.00] [0.35,0.65] [26.99,27.11]

a ₂ [1830,1885] [3.825,3.875] [12.00,15.50] [0.30,0.70] [26.96,27.01]

a ₃ [1835,1880] [3.835,3.880] [12.00,16.00] [0.43,0.63] [26.89,27.00]

a ₄ [1833,1865] [3.715,3.895] [10.50,16.50] [0.33,0.65] [26.93,27.05]

X ^4* [1825,1895] [3.755,3.905] [10.00,16.00] [0.05,1.00] [26.98,27.20]

Stage	Alternative	c ₁	c ₂	c ₃	c ₄	c ₅
t = 1	a ₁	[1840,1884]	[3.759,3.865]	[10.00,16.00]	[0.30,0.51]	[26.98,27.02]
	a ₂	[1825,1877]	[3.855,3.870]	[12.00,15.00]	[0.20,0.69]	[26.86,26.90]
	a ₃	[1835,1850]	[3.860,3.865]	[13.00,15.00]	[0.30,0.55]	[26.97,27.13]
	a ₄	[1832,1860]	[3.840,3.880]	[10.50,15.50]	[0.02,0.57]	[26.94,27.06]
X ^1*	[1833,1890]	[3.850,3.880]	[11.00,17.00]	[0.01,1.00]	[26.86,27.20]
t = 2	a ₁	[1835,1885]	[3.810,3.865]	[10.50,16.00]	[0.15,0.66]	[27.03,27.12]
	a ₂	[1830,1880]	[3.820,3.875]	[11.00,14.00]	[0.25,0.75]	[26.96,27.02]
	a ₃	[1840,1860]	[3.585,3.870]	[11.00,15.50]	[0.33,0.55]	[26.94,27.15]
	a ₄	[1833,1870]	[3.650,3.890]	[10.50,15.50]	[0.15,0.63]	[26.93,27.00]
X ^2*	[1825,1890]	[3.800,3.885]	[10.00,17.00]	[0.01,1.00]	[26.97,27.20]
	t = 3	a ₁	[1835,1880]	[3.825,3.880]	[11.00,16.00]	[0.20,0.65]	[27.00,27.13]
	a ₂	[1825,1885]	[3.825,3.875]	[12.00,16.50]	[0.30,0.70]	[27.06,27.09]
	a ₃	[1835,1875]	[3.835,3.865]	[12.00,16.50]	[0.40,0.60]	[26.99,27.05]
	a ₄	[1833,1870]	[3.650,3.900]	[10.50,16.00]	[0.23,0.58]	[27.01,27.13]
X ^3*	[1825,1895]	[3.755,3.905]	[10.00,16.00]	[0.05,1.00]	[26.95,27.20]
t = 4	a ₁	[1840,1880]	[3.825,3.900]	[11.00,16.00]	[0.35,0.65]	[26.99,27.11]
	a ₂	[1830,1885]	[3.825,3.875]	[12.00,15.50]	[0.30,0.70]	[26.96,27.01]
	a ₃	[1835,1880]	[3.835,3.880]	[12.00,16.00]	[0.43,0.63]	[26.89,27.00]
	a ₄	[1833,1865]	[3.715,3.895]	[10.50,16.50]	[0.33,0.65]	[26.93,27.05]
X ^4*	[1825,1895]	[3.755,3.905]	[10.00,16.00]	[0.05,1.00]	[26.98,27.20]

The following attributes are considered. c₁ represents the battery capacity. c₂ represents the ultimate strength before external damage. c₃ represents the time that can be used continuously. c₄ represents the ability to resist corrosion damage to the surrounding medium. c₅ represents the ability to perform functions without error. The using environment of the electronic component is displayed as follows. s₁ represents low interference. s₂ represents moderate interference. s₃ represents high interference.

There are two kinds of interval grey numbers. One is the values of alternatives with respect to each attribute. It is mainly acquired by quantitative methods. The other one is the values of reference points with respect to each attribute. It is mainly acquired by} technical analysis and expert discussion. The information about the thresholds of possibility degree, the attribute weights and the stage weights are acquired by interviews with experts.

4.2 The MSURDM procedure

Step 1. Input and normalize DM information. (omitted). Step 2. Calculate the possibility degrees and the possible values (omitted). Step 3. Calculate the attribute weights and the values of alternatives of different states. By solving model M2, $γ_{1}^{*} = 0.86, γ_{2}^{*} = 1$ can be obtained. Considering the fully utilization of information and the differences between attribute weights, DMs set γ₁ = 0.95, γ₂ = 1. By solving model M1, the attribute weights in each stage are obtained as follows. $\begin{matrix} W^{1} = (0.1283, 0.1477, 0.2980, 0.3000, 0.1260), \\ W^{2} = (0.1669, 0.2803, 0.0904, 0.3000, 0.1624), \\ W^{3} = (0.1248, 0.1583, 0.2955, 0.3000, 0.1214), \\ W^{4} = (0.1317, 0.1678, 0.2912, 0.3000, 0.1093) . \end{matrix}$ (14) $V_{i}^{t} (n)$ can be calculated by Equation (9), which are presented in Fig. 2. The ordinate is $V_{i}^{t} (n)$ and the horizontal ordinate is alternatives.

Fig.2

The possible values of alternatives.

The comparison of the ranking of alternatives in different states is showed as follows. In the first, second and third stage, the rankings are the same in state s₂ and s₃, but the ranking of state s₁ is slightly different from the other states. In the fourth stage, the rankings are the same in three states. In most cases, alternative a₃ always ranks first. Alternative a₄ always ranks in the latter position. Alternatives a₁ and a₂ rank in the middle position. The results indicate that the ranking consistencies are high in different stages and different states. The conclusion is in line with the idea of model M1. Step 4. Calculate the stage weights and the comprehensive values of alternatives of different states. Since DMs are more interested in the recent data. The orness measure is set to δ = 0.4. According to the requirements of stage change rate, DMs set η = 1.25. By solving model M3, the stage weights λ = (0.1835, 0.1433, 0.3628, 0.3104) can be obtained. It indicates that the third stage is the most important, the second stage is the least important. Calculate the comprehensive value V_i (n) by Equation (12) (as showed in Fig. 3). The ordinate is V_i (n) and the horizontal ordinate is alternatives. In state s₂ and s₃, the ranking is a₃ > a₂ > a₁ > a₄. In state s₁, the ranking is a₂ > a₃ > a₁ > a₄. The comprehensive value of alternative a₃ is the highest, and the comprehensive value of alternative a₄ is the lowest.

Fig.3

The comprehensive values of different states.

Step 5. Calculate the compromise ranking values. The preference coefficient is set to μ = 0.5. By Equation (13), the compromise ranking values V₁ = 0.1973, V₂ = 0.2038, V₃ = 0.2066, V₄ = 0.1309, a₃ > a₂ > a₁ > a₄ can be obtained. The best alternative is a₃.

4.3 Sensitivity analysis of the preference coefficient

Individual preferences and judgments of value and other subjective factors affect the final DM results. The preference coefficient μ is different from DMs. Let μ change according to step (μ) = 0.1, the change of the compromise ranking values is showed in Fig. 4. The ordinate represents the change of μ from zero to one. The horizontal ordinate represents the compromise ranking values V_i.

Fig.4

The change of the compromise ranking values as μ changes.

If μ ⩽ 0.3, the ranking of alternatives is a₄ > a₁ > a₃ > a₂. If μ = 0.4, the ranking of alternatives is a₃ > a₁ > a₄ > a₂. If μ ⩾ 0.5, the ranking of alternatives is a₃ > a₂ > a₁ > a₄. The change of μ reflects DMs’ risk preference. If μ gets closer to 0, DMs prefer the minimum regret values, which shows that DMs are more conservative. If μ gets closer to 1, DMs prefer the maximum values, which shows that DMs are more adventurous. It is more scientific than the direct judgment.

4.4 Comparison and analysis

4.4.1 Method comparisons for the value function

Value function is used to measure the gain and loss when comparing with reference points. The existing methods [18 , 32] mainly study the expansion of value functions dealing with different data forms. The changing of DMs’ risk preferences is not considered. Considering this, the possible value function is proposed to measure the gain and loss of an attribute to its reference point in different states. The proposed method is compared with the method in literature [25]. Take the data in the first stage as anexample. The possible values $V_{ij}^{1} (n)$ obtained by the proposed method are showed in Fig. 5. The ordinate represents alternatives in different states. The horizontal ordinate represents the possible values $V_{ij}^{1} (n)$ . It is founded that the values of two attributes (c₃ and c₄) vary greatly in different states.

Fig.5

The possible values obtained by the proposed method.

The values

\bar{V_{ij}^{1}}

obtained by the method in literature [25] are showed in Table 3. It is founded that there is little difference between the values in different states. The values of these two attributes (c₄ and c₅) are zero with respect to differentalternatives.

Table 3

The values obtained by the method in literature [25]

	a ₁	a ₂	a ₄
c ₁	0.0000	– 0.0021	– 0.0003
c ₂	– 0.0117	0.0000	– 0.0013
c ₃	– 0.0294	0.0000	– 0.0147
c ₄	0.0000	0.0000	0.0000
c ₅	0.0000	0.0000	0.0000

The results show that. (1) The proposed method has a higher utilization rate of information. In a comparison between an attribute and a reference point, there are four endpoints in total. The four endpoints are fully used in the paper. Only two or three endpoints are used in literature [25]. (2) The consideration of DMs’ risk preferences in this paper makes the result more diversified under different states. The result of the method in literature [25] is relatively unitary. (3) The results of the proposed method are more obvious. Attribute data obtained by the proposed method are more distinct as can be observed from Fig. 5. The expression of values in literature [25] is too rough seen from Table 3. Multiple attribute values are zero. It is difficult to describe the differences between attributes.

4.4.2 Method comparisons for attribute weight calculation

From the viewpoint of maximizing the ranking consistency of alternatives, the attribute weight optimization model is built based on the extended LINMAP method. To illustrate the superiority of the proposed method in Section 3.3, the proposed method is compared with the method in literature [2]. Take the data in the first stage as an example. The result is displayed in Fig. 6. The ordinate represents the comprehensive values and the compromise ranking values. M1S1, M1S2 and M1S3 represent the comprehensive values of each state by the proposed method. M1R represents the compromise ranking values by the proposed method. M2S1, M2S2 and M2S3 represent the comprehensive values of each state calculated by the method in literature [2]. M2R represents the compromise ranking values by the method in literature [2].

Fig.6

The comparison of the results by the two methods.

The proposed method can better distinguish the attributes than the method in literature [2]. The attribute weighting vector obtained by the proposed method is W¹. The attribute weighting vector obtained by the method in literature [2] is W^1′.

\begin{matrix} W^{1} = {(0.1283, 0.1477, 0.2980, 0.3000, 0.1260)}^{T}, \\ W^{1^{'}} = {(0.5, 0.35, 0.05, 0.05, 0.05)}^{T} . \end{matrix}

There is no difference between the three attribute weights (

w_{3}^{1^{'}}, w_{4}^{1^{'}}, w_{5}^{1^{'}}

). The difference between the attribute weight W¹ calculated by the proposed method is larger than the method in literature [2]. The difference between the comprehensive values (or the compromise ranking values) by the proposed method is more obvious than the method in literature [2] (as showed in Fig. 6). The ranking of alternatives in different states by the proposed method is more consistent than that by the method in literature [2]. The results show that. (1) Diversity of attribute weights by the proposed method can be bigger due to the consideration of the entropy range of attribute weights. (2) The ranking consistency by the proposed method is better, which is in line with the idea of model M1. The idea of the method in literature [2] is to make alternatives as close to the positive ideal alternative as possible, which is similar to the TOPSIS method. This idea may lead to a smaller gap between different ranking values.

4.4.3 Method comparisons for stage weight calculation

In order to compare the characteristics of model M3, DM data are calculated according to the proposed method and the method in literature [33]. The stage weights calculated by the proposed method is λ = (0.1835, 0.1433, 0.3628, 0.3104). The stage weights calculated by the method in literature [33] is λ′ = (0.1016, 0.1381, 0.2502, 0.5101). The method in literature [33] is based on the temporal characteristics of the data, and does not take into account the changes of data in different stages. The proposed method takes into account both the temporal characteristics and the stability of the data changes. These considerations are more in line with the actual DM environment. The comprehensive values and the compromise ranking values are showed in Table 4. According to the performances of the compromise ranking values, the values obtained by the method in literature [3] are bigger than the proposed method. From the final rank of alternatives, the two methods get the same rank a₃ > a₂ > a₁ > a₄, so the proposed method is desirable. According to the comparison results with different methods, the proposed method has advantages in the following aspects. (1) Considering the impact of DMs’ risk preferences is more in line with the actual DM process. (2) The proposed method makes full use of information, so the differences between attribute values are obvious. (3) The proposed method can improve the ranking consistency under different states. (4) Considering the minimization of the opportunity loss, the temporal characteristics and the stability of multistage data changes, the results can be both subjective and objective. The proposed method aims to solve the MSURDM problems without probability information. Based on the ideas of maximizing the ranking consistency and minimizing the opportunity loss, the proposed method can also be extended to solve the MSURDM problems with known probability information.

Table 4
The ranking result by different methods

The proposed method Method in literature [33]

s ₁ s ₂ s ₃ s ₁ s ₂ s ₃

a ₁ 0.0654 – 0.0433 – 0.1370 0.0708 – 0.0371 – 0.1306

a ₂ 0.0822 – 0.0240 – 0.1195 0.0833 – 0.0222 – 0.1174

a ₃ 0.0755 – 0.0201 – 0.1072 0.0793 – 0.0162 – 0.1040

a ₄ 0.0542 – 0.0573 – 0.1529 0.0622 – 0.0488 – 0.1444

ranking V₁ = 0.1973, V₂ = 0.2038 V₁ = 0.2033, V₂ = 0.2076

V₃ = 0.2066, V₄ = 0.1309 V₃ = 0.2251, V₄ = 0.1969

a₃ > a₂ > a₁ > a₄ a₃ > a₂ > a₁ > a₄

The proposed method	Method in literature [33]
a ₁	0.0654	– 0.0433	– 0.1370	0.0708	– 0.0371	– 0.1306
a ₂	0.0822	– 0.0240	– 0.1195	0.0833	– 0.0222	– 0.1174
a ₃	0.0755	– 0.0201	– 0.1072	0.0793	– 0.0162	– 0.1040
a ₄	0.0542	– 0.0573	– 0.1529	0.0622	– 0.0488	– 0.1444
ranking		V₁ = 0.1973, V₂ = 0.2038	V₁ = 0.2033, V₂ = 0.2076
	V₃ = 0.2066, V₄ = 0.1309	V₃ = 0.2251, V₄ = 0.1969
	a₃ > a₂ > a₁ > a₄		a₃ > a₂ > a₁ > a₄

5 Conclusions

This study is capable in dealing with the MSURDM problems without probability information and allows DMs to express their risk preferences under different states. First, considering the impact of DMs’ risk preferences, the threshold of possibility degree is proposed to design a new value function. Our results are more distinct and have a higher utilization rate. Next, based on the LINMAP method and the entropy-weighing method, a two-stage model is developed to handle with attribute weight. The comparison results show that the ranking consistency by the proposed method is better and the diversity of attribute weights is bigger. Then, to aggregate the values in different states, the relative entropy method is used to measure the opportunity loss. Thus, the stage weight optimization model is designed in the viewpoint of minimizing the opportunity loss. The comparison results show that the proposed method is feasible and desirable.

In future work, a more appropriate method to deal with the MSURDM problems with uncertain probability information are an interesting topic. Besides, the MSURDM problems with multiple reference points are also considered.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China [grant numbers 71171112, 71502073, 71601002]; Scientific Innovation Research of College Graduates in Jiangsu Province the Fundamental Research Funds for the Central Universities [grant numbers KYZZ16_0147]; Humanities and Social Sciences Foundation of Ministry of Education of China [grant numbers 16YJC630077]; Anhui Provincial Natural Science Foundation [grant numbers 1708085MG168].

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The proposed method			Method in literature [33]
	s ₁	s ₂	s ₃	s ₁	s ₂	s ₃
a ₁	0.0654	– 0.0433	– 0.1370	0.0708	– 0.0371	– 0.1306
a ₂	0.0822	– 0.0240	– 0.1195	0.0833	– 0.0222	– 0.1174
a ₃	0.0755	– 0.0201	– 0.1072	0.0793	– 0.0162	– 0.1040
a ₄	0.0542	– 0.0573	– 0.1529	0.0622	– 0.0488	– 0.1444
ranking		V₁ = 0.1973, V₂ = 0.2038			V₁ = 0.2033, V₂ = 0.2076
	V₃ = 0.2066, V₄ = 0.1309			V₃ = 0.2251, V₄ = 0.1969
	a₃ > a₂ > a₁ > a₄				a₃ > a₂ > a₁ > a₄

A multi-stage uncertain risk decision-making method with reference point based on extended LINMAP method

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 Problem description

Table 1 An example of RDM problems S1 S2 S3 OP PP MVP MRP A1 40 15 20 40 15 25 0 A2 20 15 40 40 15 25 0 A3 40 20 20 40 20 26.7 0

4.1 Background of case

4.4.1 Method comparisons for the value function

Footnotes

Acknowledgments

References

Table 1
An example of RDM problems

S1 S2 S3 OP PP MVP MRP

A1 40 15 20 40 15 25 0

A2 20 15 40 40 15 25 0

A3 40 20 20 40 20 26.7 0