Keywords

1 Introduction

Multistage risk decision making (MSRDM) methods aim to rank alternatives or select the best alternative(s) by the aggregation of multistage risk decision-making (DM) information. MSRDM problems include risk, uncertainty and dynamics. Psychological factors of the decision-makers need to be taken into consideration to solve the risk DM problems. Decision-makers often consider the gain and loss under a reference point due to the bounded rationality. The fairness and satisfaction of DM is significantly influenced by a single reference point. In dynamic and uncertain conditions, using a single reference point will lead to the loss of some of the information about the distribution of the results. In the DM process with risk and dynamics, the psychological behavior of decision makers is inconsistent. In this context, the consideration of multiple reference points helps decision-makers to uncover the dynamic and risk characteristics of MSRDM problems, thus making a reasonable and comprehensive assessment of results.

Work on MSRDM methods have been increasing recently. In a multistage DM problem of finite-state automaton, a new optimization method stochastically develops a solution step-by-step in combination with a simulated annealing [1]. In an optimal investment problem with several projects, a new methodology is proposed based on experts’ evaluations. It consists of three stages [2]. The multistage one-shot DM problems under uncertainty are studied based on scenario [3]. A multi-stage technical screening and evaluation tool is proposed to determine the optimal technique scheme under fuzzy environment [4]. A multistage assignment model is presented for rescue teams to dynamically respond to the disaster chain [5].

With the increasing complexity of DM problems, more effective methods are developed to support decision makers’ judgments. DM methods considering reference points are one kind of resourceful methods. TOPSIS [6, 7] is widely used in MCDM problems. The idea of TOPSIS is to compare each solution with the positive ideal solution and the negative ideal solution, which are actually the two reference points. VIKOR methods [8] are also dependent on the positive and negative ideal solution, which are similar to the TOPSIS methods. Kahneman and Tversky [9, 10] presented the Prospect Theory to solve the risky DM problems.

The actual utility is obtained from comparison with a reference point. Due to the limited rationality of decision-makers’ thinking, it is difficult to judge by the evaluation value. The consideration of reference points can provide the basis for decision makers, and lead to better informed and well-reasoned decisions. DM methods considering the reference point have been gradually enriched. A prospect theory-based interval dynamic reference point method has been proposed for emergency DM [11]. A risk DM method has been proposed considering the dynamic reference point, the external reference point and the internal reference point [12]. A new method based on the concept of ideal solution has been presented as a possible variant of TOPSIS and VIKOR methods [13]. The newsvendor’s pricing and stocking decisions have been studied considering the impact of reference point effects [14].

Decision information often shows different forms, such as fuzzy numbers [2], interval numbers [12], linguistic sets [15], cloud models and so on. Due to the dynamic continuity of MSRDM process and the risky DM environment, information often shows fuzziness and randomness at the same time. The transformation between qualitative concepts and quantitative concepts is often needed to be dealt with. Linguistic set is usually used to express the decision maker’s judgment. However, linguistic sets are often ambiguous and uncertain, and very difficult to form accurate information [15]. In this context, Li presented cloud models to propose conversion between qualitative concept and quantitative representation [16]. Many new approaches to cloud models have been proposed to solve existing problems. Cloud Hierarchical Analysis (CHA) is an extension of AHP [17]. The Cloud Delphi hierarchical analysis has been presented for practical multi-criteria group DM problems [18]. DM methods combining linguistic sets and cloud model have been investigated [19]. Cloud model has been widely used in many problems, such as water quality assessment [20], image segmentation [21], and clustering problems [22].

This paper makes contributes to the MSRDM problems for normal cloud model with three reference points.

  1. (1)

    The new setting method for three reference points is proposed based on the data in different states and stages.

  2. (2)

    The aggregation method for different prospect values is proposed based on the synthetic degree of grey incidence.

  3. (3)

    A two-stage weight optimization method is proposed to solve the attribute weights and stage weights.

The rest of this paper is organized as follows. In Sect. 2, some related concepts and definitions are reviewed. Section 3 presents the MSRDM method for normal cloud model with three reference points. Section 4 provides a case followed by its analysis. Section 5 concludes the paper.

2 Preliminaries

This section briefly reviews the basic concepts and definitions associated with normal cloud model and prospect theory, and describes the problem addressed in this paper.

2.1 Basic Concepts and Definitions

Due to the complexity and uncertainty of DM problems, DM information often shows fuzziness and uncertainty. Normal cloud model provides an important background to represent fuzziness and randomness at the same time. Many scholars [18, 20, 23, 24] have carried out studies using normal cloud model.

Definition 1.

[23] Let U be the universe of discourse and \( \tilde{A} \) be a qualitative concept in U. If \( x \in U \) is a random instantiation of concept \( \tilde{A} \) that satisfies \( x\sim N\left( {Ex,En^{\prime 2} } \right),En^{\prime } \sim N\left( {En,He^{2} } \right), \) and the certainty degree of x belonging to \( \tilde{A} \) satisfies

$$ y = e^{{ - \frac{{\left( {x - Ex} \right)^{2} }}{{2(En^{\prime } )^{2} }}}} $$
(1)

The distribution of x in the universe U is called the normal cloud model and x can be called a cloud drop. The normal cloud model can effectively integrate the randomness and fuzziness of a concept through three parameters: Expectation Ex, Entropy En and Hyper Entropy He. Expectation Ex is the mathematical expectation of the cloud drops belonging to a concept in the universe. It can best represent the qualitative concept. Entropy En represents the uncertainty measurement of a qualitative concept. It is the measurement of randomness and fuzziness of the concept. Hyper Entropy He is the uncertain degree of entropy En [23].

Given two normal cloud models \( C_{i} (Ex_{i} ,En_{i} ,He_{i} ) \) and \( C_{j} (Ex_{j} ,En_{j} ,He_{j} ) \). Certain operation rules between two normal cloud models have been included in [18, 20].

  1. (1)

    \( C_{i} + C_{j} = \left( {Ex_{i} + Ex_{j} ,\sqrt {En_{i}^{2} + En_{j}^{2} } ,\sqrt {He_{i}^{2} + He^{2} } } \right). \)

  2. (2)

    \( C_{i} - C_{j} = \left( {Ex_{i} - Ex_{j} ,\sqrt {En_{i}^{2} + En_{j}^{2} } ,\sqrt {He_{i}^{2} + He^{2} } } \right). \)

  3. (3)

    \( C_{i} \times C_{j} = \left( {Ex_{i} \times Ex_{j} ,\left| {Ex_{i} Ex_{j} } \right|\sqrt {\left( {\frac{{En_{i} }}{{Ex_{i} }}} \right)^{2} + \left( {\frac{{En_{j} }}{{Ex_{j} }}} \right)^{2} } ,\left| {Ex_{i} Ex_{j} } \right|\sqrt {\left( {\frac{{He_{i} }}{{Ex_{i} }}} \right)^{2} + \left( {\frac{{He_{j} }}{{Ex_{j} }}} \right)^{2} } } \right). \)

  4. (4)

    \( C_{i} \div C_{j} = \left( {\frac{{Ex_{i} }}{{Ex_{j} }},\left| {\frac{{Ex_{i} }}{{Ex_{j} }}} \right|\sqrt {\left( {\frac{{En_{i} }}{{Ex_{i} }}} \right)^{2} + \left( {\frac{{En_{j} }}{{Ex_{j} }}} \right)^{2} } ,\left| {\frac{{Ex_{i} }}{{Ex_{j} }}} \right|\sqrt {\left( {\frac{{He_{i} }}{{Ex_{i} }}} \right)^{2} + \left( {\frac{{He_{j} }}{{Ex_{j} }}} \right)^{2} } } \right). \)

  5. (5)

    \( \lambda C_{i} = \left( {\lambda Ex_{i} ,\sqrt \lambda En_{i} ,\sqrt \lambda He_{i} } \right). \)

  6. (6)

    \( (C_{i} )^{\lambda } = \left( {Ex_{i}^{\lambda } ,\sqrt \lambda Ex_{i}^{\lambda - 1} En_{i} ,\sqrt \lambda Ex_{i}^{\lambda - 1} He_{i} } \right). \)

When applying normal cloud model, the similarity between two normal cloud models is commonly used.

Definition 2.

[24] Let \( C_{i} (Ex_{i} ,En_{i} ,He_{i} ) \) and \( C_{j} (Ex_{j} ,En_{j} ,He_{j} ) \) be two normal cloud models. The similarity between two normal cloud models based on shape and distance is defined as:

$$ sim_{c} (C_{i} ,C_{j} ) = sim_{d} (C_{i} ,C_{j} ) \times sim_{s} (C_{i} ,C_{j} ) $$
(2)

Where \( sim_{d} \left( {C_{i} , C_{j} } \right) \) represents the similarity between two normal cloud models based on distance, \( sim_{s} (C_{i} ,C_{j} ) \) represents the similarity between two normal cloud models based on shape.

In prospect theory, alternatives are selected based on the prospect value.

Definition 3.

[9] The prospect value is defined by the value function and the probability weight function:

$$ V(x) = \sum\nolimits_{k} {\pi (p_{k} ) \cdot v(x_{k} )} $$
(3)

Definition 4.

[10] The value function is expressed in the form of a power law according to the following formula:

$$ v(x_{k} ) = \left\{ {\begin{array}{*{20}l} {(x_{k} )^{\alpha } ,} \hfill & {x_{k} \ge 0} \hfill \\ { - \theta ( - x_{k} )^{\beta } } \hfill & {x_{k} < 0} \hfill \\ \end{array} } \right. $$
(4)

Where xk denotes the gain or loss of the value when comparing an alternative to its reference point. When xk ≥ 0, it represents a gain. When xk < 0, it represents a loss. α and β represent the concave and convex degree of the value power function v(xk) in the region of gain and loss respectively. \( \theta \) indicates the loss-averse coefficient.

Definition 5.

[10] The probability weight function is defined as

$$ \pi (p_{k} ) = \left\{ {\begin{array}{*{20}c} {\frac{{(p_{k} )^{\gamma } }}{{\left( {\left( {p_{k} } \right)^{\gamma } + \left( {1 - p_{k} } \right)^{\gamma } } \right)^{1/\gamma } }}} & {x_{k} \ge 0} \\ {\frac{{(p_{k} )^{\delta } }}{{\left( {(p_{k} )^{\delta } + (1 - p_{k} )^{\delta } } \right)^{1/\delta } }}} & {x_{k} < 0} \\ \end{array} } \right. $$
(5)

where γ and δ are the risk gain and loss attitude coefficients respectively.

Tversky and Kahneman [10] found that when \( \alpha \, = \,\beta \, = \,0. 8 8, \theta \, = \, 2. 2 5, \gamma \, = \,0. 6 1, \delta \, = \,0. 7 2 \), the experimental results are more consistent with the empirical results. To simplify calculation, we also take the above values in the paper.

2.2 Problem Description

The MSRDM process involves multiple stages and multiple states, which is usually risk, uncertainty and dynamic. Considering multiple reference points is helpful to make decisions results from multiple perspectives. This paper aims to select the desirable alternative(s) from a set of feasible alternatives according to the MSRDM problems.

In the MSRDM problem, let \( A = \left\{ {a_{i} \left| {i = 1,2, \cdots ,I} \right.} \right\} \) be the set of I alternatives. Let \( C = \left\{ {c_{j} \left| {j = 1,2} \right., \ldots ,J} \right\} \) be the set of J attributes. Let \( M = \left\{ {m^{t} \left| {t = 1,2, \ldots ,T} \right.} \right\} \) be the set of T stages. Let \( S = \left\{ {s^{n} \left| {n = 1,2, \ldots ,N} \right.} \right\} \) be the set of N natural states. Let \( WC = \left\{ {wc_{1} ,wc_{2} , \ldots ,wc_{J} } \right\} \) be the weighting set of J attributes \( \left( {wc_{j} \in [0,1],\sum\nolimits_{j = 1}^{J} {wc_{j} = 1} } \right),WM = \left\{ {wm^{1} ,wm^{2} , \ldots ,wm^{T} } \right\} \) be the weighting set of M stages \( \left( {wm^{t} \in [0,1],\sum\nolimits_{t = 1}^{T} {wm^{t} = 1} } \right),P = \left\{ {p(s^{1} ),p(s^{2} ), \ldots ,p(s^{N} )} \right\} \) be the probability set of N states \( \left( {p(s^{n} ) \in [0,1],\sum\nolimits_{n = 1}^{N} {p(s^{n} ) = 1} } \right) \). Let \( X^{tn} = (x_{ij}^{tn} )_{I \times J} \)) be the DM matrix in the stage mt in the state sn, which is showed in Table 1. \( x_{ij}^{tn} = \left( {Ex_{ij}^{tn} ,En_{ij}^{tn} , He_{ij}^{tn} } \right) \) is the decision value of alternative ai with respect to attribute cj in stage mt in state sn.

Table 1. MSRDM evaluation information.
Full size table

3 A MSRDM Method for Normal Cloud Model with Three Reference Points

3.1 The Setting Method of Three Reference Points

In MSRDM problems, the performance of an alternative will change dynamically as time goes on. Therefore, current situation, development trend and decision goal fluctuate with the change of stage. Considering a single reference point is difficult to evaluate the current situation, dynamic and inspiring nature comprehensively. Thus, it is difficult to systematically describe the development trend and characteristics.

The idea of setting the three reference points is showed in Fig. 1. The developmental reference point (DRP) is set by the performances of the previous stage. Compared with the DRP, the progress from the previous stage to the present stage can be obtained. The state reference point (SRP) is set by the expected performance of multiple states in the present stage. Unlike with the SRP, the extent to which one alternative is better than the expected performance of multiple states can be obtained. The promoting reference point (PRP) is set by the potential and the performance of the previous stage, which can be seen as the goal. The PRP can be used to adjust the degree and direction of the effort. Compared with the PRP, the degree of effort in the present stage can be obtained. In order to fully compare the advantages and disadvantages of MSRDM information, this paper sets up three reference points, i.e., development, multistate and promotion.

Fig. 1.
figure 1

The idea of setting the three reference points

Full size image

State Reference Point 1. In multiple natural states, the expected value is the probability-weighted average of all possible values. It represents the central tendency of the values in multiple states. The expected value is what one expects to happen on average. If the value of an alternative is higher than the expected value, it will be a gain for the alternative to the expected value. The SRP is set by the average value of the expected values of alternatives in one stage.

Definition 6.

The SRP of the attribute cj at stage mt is defined as

$$ r_{j}^{ts} = \frac{1}{I}\sum\nolimits_{i = 1}^{I} {\sum\nolimits_{n = 1}^{N} {p(s^{n} ) \cdot x_{ij}^{tn} = \left( {Ex_{j}^{ts} ,En_{j}^{ts} , He_{j}^{ts} } \right)} } $$
(6)

Developmental Reference Point 2. From the viewpoint of development, the actual development level of an alternative at the present stage can be obtained by comparing the data with the data of the previous stage. The DRP is set by the performance of the previous stage. Thus, the progress of the present stage can be obtained by comparing it with the DRP. Compared with the previous stage, the greater the gain at the present stage, the better the development level of the present stage.

Definition 7.

The DRP of the attribute cj at stage mt in natural state sn is defined as

$$ r_{j}^{tnd} = \frac{1}{I}\sum\nolimits_{i = 1}^{I} {x_{ij}^{t - 1,n} = \left( {Ex_{j}^{tnd} ,En_{j}^{tnd} , He_{j}^{tnd} } \right)} $$
(7)

Promoting Reference Point 3. From the viewpoint of promotion, reasonable goals should be set up to motivate people’s subjective initiative. The PRP is set by the potential and performance of the previous stage. It actually is an estimate of the present stage based on the resources and historical foundations. The degree of realization of the PRPs reflects the degree of effort and the potential. Compared with the PRP, the greater the gain of the alternative, the better.

Definition 8.

The maximum growth potential of the alternative ai with respect to the attribute cj at stage mt−1 in natural state sn is defined as

$$ \tau_{ij}^{t - 1,n} = \frac{{max_{j} x_{ij}^{t - 1,n} }}{{x_{ij}^{t - 1,n} }} $$
(8)

Definition 9.

The average maximum growth potential of the attribute cj at stage mt−1 in natural state \( s^{n} \) is defined as

$$ \tau_{j}^{t - 1,n} = \frac{1}{I}\sum\nolimits_{i = 1}^{I} {\left( {\frac{{\max_{j} x_{ij}^{t - 1,n} }}{{x_{ij}^{t - 1,n} }}} \right)} $$
(9)

Definition 10.

The PRP of the attribute cj at stage mt in natural state \( s^{n} \) is defined as

$$ r_{j}^{tnp} = \tau_{j}^{t - 1,n} \frac{1}{I}\sum\nolimits_{i = 1}^{I} {x_{ij}^{t - 1,n} = \left( {Ex_{j}^{tnp} ,En_{j}^{tnp} , He_{j}^{tnp} } \right)} $$
(10)

3.2 The Value Function for Normal Cloud Model Under Three Reference Points

Comparing with three reference points can measure the performance of alternatives from different perspectives. The performance of one alternative at the current stage can be measured by comparing with the SRP. The performance of one alternative at the current stage can be measured by comparing with the SRP. The development performance of one alternative from the previous stage to the present stage can be measured by comparing with the DRP. Whether one alternative reaches the expected potential level can be measured by comparing with the GRP. When compared with the three reference points, the better the gain is, the better the alternative is. Then a value function is defined to obtain the gain or loss from a reference point. Take the DRP as an example.

Definition 11.

The value function with the DRP is defined as

$$ v_{ij}^{tnd} = \left\{ {\begin{array}{*{20}c} {\left( {1 - sim\left( {x_{ij}^{tn} ,r_{j}^{tnd} } \right)} \right)^{\alpha } } & {Ex_{ij}^{tn} \ge Ex_{j}^{tnd} } \\ { - \theta \left( {1 - sim\left( {x_{ij}^{tn} ,r_{j}^{tnd} } \right)} \right)^{\beta } } & {Ex_{ij}^{tn} < Ex_{j}^{tnd} } \\ \end{array} } \right. $$
(11)

where \( sim\left( {x_{ij}^{tn} ,r_{j}^{tnd} } \right) \) is the similarity between the attribute value and the DRP, which can be calculated by (2). When \( Ex_{ij}^{tn} \ge Ex_{j}^{tnd} \), the value is a gain. The bigger the similarity between the attribute value and the DRP, the smaller the value of \( x_{ij}^{tn} \) as compared with the DRP \( r_{j}^{tnp} \). When \( Ex_{ij}^{tn} < Ex_{j}^{tnd} \), the value is a loss. The bigger the similarity between the attribute value and the DRP, the bigger the value of \( x_{ij}^{tn} \) compared with the DRP \( r_{j}^{tnp} \).

The value functions for the SRP and the PRP are the same as the DRP.

3.3 The Aggregation Method for Three Kinds of Prospect Values

The prospect values with respect to each reference point can be calculated by Eq. (3). Then we can get \( v_{ij}^{ts} ,v_{ij}^{td} ,v_{ij}^{tp} \). The comprehensive prospect values based on multiple reference points can be obtained by:

$$ v_{ij}^{t} = \lambda_{j}^{ts} \cdot v_{ij}^{ts} + \lambda_{j}^{td} \cdot v_{ij}^{td} + \lambda_{j}^{tp} \cdot v_{ij}^{tp} $$
(12)

Where \( \lambda_{j}^{ts} ,\lambda_{j}^{td} ,\lambda_{j}^{tp} \left( {\lambda_{j}^{ts} + \lambda_{j}^{td} + \lambda_{j}^{tp} = 1, \lambda_{j}^{ts} ,\lambda_{j}^{td} ,\lambda_{j}^{tp} \in [0,1]} \right) \) represent the preference coefficients of different reference points at stage \( m^{t} \). The preference coefficients can be given by decision makers in accordance with the actual situation. The coefficient of preference can also be determined according to the connections between the three kinds of prospect values.

The synthetic degree of grey incidence can describe the overall relationship of closeness between sequences [25]. So, we take the synthetic degree of grey incidence to obtain the preference coefficients of different reference points.

Let \( X_{j}^{ts} = \left( {v_{1j}^{ts} ,v_{2j}^{ts} , \ldots ,v_{Ij}^{ts} } \right),X_{j}^{td} = \left( {v_{1j}^{td} ,v_{2j}^{td} , \ldots ,v_{Ij}^{td} } \right),X_{j}^{tp} = \left( {v_{1j}^{tp} ,v_{2j}^{tp} , \ldots ,v_{Ij}^{tp} } \right) \) be the sequences of the attribute cj at stage \( m^{t} \) with respect to the three reference points.

The synthetic degree of grey incidence between each of the two-reference points can be calculated as \( \rho_{j}^{tsd} ,\rho_{j}^{tsp} ,\rho_{j}^{tdp} \) [25]. Thus, the preference coefficients of different reference points can be obtained by:

$$ \lambda_{j}^{ts} = \frac{1}{2} \cdot \frac{{\rho_{j}^{tsd} + \rho_{j}^{tsp} }}{{\rho_{j}^{tsd} + \rho_{j}^{tsp} + \rho_{j}^{tdp} }} $$
(13)
$$ \lambda_{j}^{td} = \frac{1}{2} \cdot \frac{{\rho_{j}^{tsd} + \rho_{j}^{tdp} }}{{\rho_{j}^{tsd} + \rho_{j}^{tsp} + \rho_{j}^{tdp} }} $$
(14)
$$ \lambda_{j}^{tp} = \frac{1}{2} \cdot \frac{{\rho_{j}^{tsp} + \rho_{j}^{tdp} }}{{\rho_{j}^{tsd} + \rho_{j}^{tsp} + \rho_{j}^{tdp} }} $$
(15)

The prospect values of alternatives at stage mt can be obtained by

$$ v_{i}^{t} = \sum\nolimits_{j = 1}^{J} {wc_{j} \cdot v_{ij}^{t} } $$
(16)

The prospect values of alternatives at all stages can be obtained by

$$ v_{i} = \sum\nolimits_{t = 1}^{T} {\sum\nolimits_{j = 1}^{J} {wm^{t} \cdot wc_{j} \cdot v_{ij}^{t} } } $$
(17)

3.4 The Two-Stage Optimization Model

In MSRDM problems, the weights of attributes and stages can be given by decision makers. In some situations, it is difficult to determine the exact weights for decision-makers. Inappropriate weight setting may lead to errors in DM results. In this paper, we add the decision maker’s judgment of weights to the priori information set. In this way, the weight optimization model can be more objective to determine the attribute weight, and we also take into account the influence of the subjective weights by DMs.

According to the idea of minimax reference point optimization [26], we designed the two-stage optimization model to solve the weights. The idea of the two-stage optimization model is showed in Fig. 2. The abscissa represents the weights of attributes, and the ordinate represents the weights of stages.

Fig. 2.
figure 2

The idea of the two-stage optimization method

Full size image

In the first stage of the two-stage optimization model, the maximum value of each alternative at each stage is obtained. In the second stage of the two-stage optimization model, the biggest distance between the maximum value and the actual value is minimized step by step. For example, the biggest distance in Fig. 2 is ɛ3 at first. After optimization by modeling, the maximum distance is changed to ɛ2. Finally, the biggest distance between the maximum value and the actual value is minimized. All the values of alternatives are as close as possible to the maximum value.

For the convenience of calculation, the prospect values of alternatives are standardized, the annotation is unchanged. Then we have \( v_{ij}^{t} \in [0,1] \).

The first stage model M1 is used to calculate the maximum value of each alternative at each stage. M1 is defined as

$$ max(v_{i}^{t} ) = max\sum\nolimits_{j = 1}^{J} {wc_{j} \cdot v_{ij}^{t} } $$
(18)
$$ \left\{ {\begin{array}{*{20}c} {\sum\nolimits_{j = 1}^{J} {wc_{j} \cdot v_{ij}^{t} \le 1,\,i = 1,2, \ldots ,I} } \\ {\,\,wc_{j} \in H_{1} ,\,j = 1,2, \ldots ,J} \\ \end{array} } \right. $$
(19)

The maximum value of each alternative at each stage can be calculated as \( v_{i}^{t*} \). Each alternative can get its maximum value if model M1 is bounded. Because \( v_{ij}^{t} \in \left[ {0,1} \right] \) and \( wc_{j} \in \left[ {0,{ 1}} \right] \), the value of \( \sum\nolimits_{j = 1}^{J} {wc_{j} \cdot v_{ij}^{t} } \) must be in the range [0, 1]. Attribute weights meet the prior set H1, which can be expressed in 5 forms [27]. The prior set about attribute weights is usually decided by multiple decision makers.

The second stage model M2 aims to minimize the biggest distance between the maximum value and the actual value. M2 is defined as:

$$ min(\varepsilon ) $$
(20)
$$ \left\{ {\begin{array}{*{20}c} {\sum\nolimits_{t = 1}^{T} {wm^{t} \cdot \left( {v_{i}^{t*} - \sum\nolimits_{j = 1}^{J} {wc_{j} \cdot v_{ij}^{t} } } \right) \le \varepsilon ,i = 1,2, \ldots ,I} } \\ {\sum\nolimits_{t = 1}^{T} {\sum\nolimits_{j = 1}^{J} {wm^{t} \cdot wc_{j} \cdot v_{ij}^{t} \le 1} } } \\ {\,\,wc_{j} \in H_{1} ,j = 1,2, \ldots ,J} \\ {wm^{t} \in H_{2} ,t = 1,2, \ldots ,T} \\ \end{array} } \right. $$
(21)

The minimum value of ɛ can be obtained from solving model M2. Constraints are the following conditions. The biggest distance between the maximum value and the actual value is less than or equal to ɛ. Each alternative is effective. Because \( v_{ij}^{t} \in [0,1],wc_{j} \in [0,1] \) and \( wm^{t} \in \left[ {0,{ 1}} \right] \), the value of \( \sum\nolimits_{t = 1}^{T} {\sum\nolimits_{j = 1}^{J} {wm^{t} \cdot wc_{j} \cdot v_{ij}^{t} } } \) must in the range [0, 1]. The attribute weight and the stage weight satisfy the prior set H1 and H2 respectively. The prior sets about attribute and stage weights are usually decided by multiple decision makers.

Then the final weight can be calculated as wmt* and \( wc_{j}^{*} \), the final ranking value can be calculated as vi. The bigger the value of vi, the better the alternative ai is.

$$ v_{i} = \sum\nolimits_{t = 1}^{T} {\sum\nolimits_{j = 1}^{J} {wm^{t*} \cdot wc_{j}^{*} \cdot v_{ij}^{t} } } $$
(22)

3.5 The DM Procedure of the MSRDM Method

The DM procedure to solve the MSRDM problems with three reference points is demonstrated in the following steps.

Step 1. Set three reference points.

The three reference points are obtained from (6)–(10).

Step 2. Calculate the prospect values.

Use (2) to calculate the similarity between \( x_{ij}^{tn} \) and the reference points. Then the prospect values under each reference point can be obtained from (11).

Step 3. Calculate the preference coefficients and the comprehensive prospect values.

Calculate the synthetic degree of grey incidence between each of the two-reference points [25]. Then, use (13)–(15) to calculate the preference coefficients, and use Eq. (12) to calculate the comprehensive prospect values.

Step 4. Solve the attribute weights and the stage weights.

Build model M1 (18)–(19) to get the maximum values \( v_{i}^{t*} \) of each alternative at different stages. Build model M2 (20)–(21) to solve the attribute weights and the stage weights.

Step 5. Calculate the final ranking value.

The final ranking values can be calculated from (22). The best alternative is max (vi).

4 Numeral Example Analysis

4.1 Numeral Example Background

A pharmaceutical company carried out a risk assessment of the quality of products. There are 15 products to be evaluated, which comprise the set of alternatives A = {a1, a2, …, aI}(I = 15). Evaluation attributes comprise the set of attributes C = {c1, c2, …, cJ}(J = 4). Three natural states comprise the set of natural states S = {sn|n = 1, 2, …, N}(N = 3). The evaluation information from T = 4 stages comprises the decision-making matrix \( \left( {x_{ij}^{tn} } \right)_{I \times J} \). Attribute weights meet the prior set \( H_{1} = \left\{ {\sum\nolimits_{j = 1}^{J} {wc_{j} = 1;wc_{j} \ge 0.15;wc_{1} + wc_{3} \le 0.45} } \right\} \), Stage weights meet the prior set \( H_{2} = \left\{ {\sum\nolimits_{t = 1}^{T} {wm^{t} = 1;0 < wm^{1} \le 0.2;wm^{2} > 0.15;wm^{1} + wm^{2} \le 0.45;wm^{3} \ge 0.2;wm^{4} \ge 0.3} } \right\}. \)

There are four attributes to describe the products to be evaluated. c1 represents the management level of raw material. It can be evaluated by decision makers with the linguistic sets (very good, good, medium, poor, very poor). c2 represents the qualified rate of product quality. It can be obtained according to the data of product inspection. c3 represents technological level. It can be evaluated by decision makers with the linguistic sets (very good, good, medium, poor, very poor). c4 represents economic benefit. It can be obtained from the annual profit ratio.

There are three natural states: s1 = the low risks tate, \( s^{2} = the\, medium \,{\text{risk }}\,{\text{state}} \), \( s^{3} = the\, high \,{\text{risk}}\,{\text{state}} \). According to historical data, we have P = {p(s1) = 0.65, p(s2) = 0.25, p(s3) = 0.1}. The DM matrixes at different stages are showed in the appendix as Tables 2, 3 and 4.

Table 2. The evaluation information in stage m2
Full size table
Table 3. The evaluation information in stage m3
Full size table
Table 4. The evaluation information in stage m4
Full size table

4.2 The Calculation Process

Step 1. Set three reference points.

  1. (1)

    Use (6) to calculate the SRP in each stage.

Take the attribute c1 as an example; the SRPs of attribute c1 in four stages are showed in Fig. 3. The SRPs in stage m1, m2, m3, m4 are represented as \( r_{1}^{1s} = \left( {6.283,3.431,0.691} \right),r_{1}^{2s} = \left( {6.048,3.32,0.724} \right),r_{1}^{1s} = \left( {5.721,3.214,0.767} \right),r_{1}^{1s} = \left( {6.016,3.342,0.73} \right) \).The symbols (“.”, “+”, “∆”, “○”) in Fig. 3 represent the SRP in stage m1, m2, m3, m4. Comparing the values of Ex in SRPs \( (r_{1}^{1s} , r_{1}^{2s} ,r_{1}^{3s} ,r_{1}^{4s} ) \), we obtain that \( Ex_{1}^{1s} > Ex_{1}^{2s} > Ex_{1}^{4s} > Ex_{1}^{3s} \). The SRP in stage m1 is bigger than the other three reference points. The SRP in stage m3 is smaller than the other three reference points.

Fig. 3.
figure 3

The SRPs in different stages

Full size image
  1. (2)

    Use (7) to calculate the DRP in each state in each stage.

Take the attribute c2 in stage m1 as an example. The DRP in different states are represented as \( r_{2}^{11d} = \left( {0.923,0.069,0.018} \right),r_{2}^{12d} = \left( {0.923,0.042,0.21} \right),r_{2}^{13d} = \left( {0.899,0.065,0.019} \right) \). We can find that \( Ex_{2}^{11d} \cong Ex_{2}^{12d} > Ex_{2}^{13d} \). But the cloud drops of \( r_{2}^{12d} \) are more dispersed than the other two \( (He_{2}^{12d} > He_{2}^{13d} > He_{2}^{11d} ) \).

  1. (3)

    Calculate the PRP in each state in each stage by Eq. (10) (omitted).

Take the attribute c3 in stage m1 as an example. The PRP in different states are represented as \( r_{3}^{11p} = \left( {8.253,4.843,0.926} \right),r_{2}^{12p} = \left( {8.404,5.1,0.948} \right),r_{2}^{13p} = \left( {6.839,4.163,0.949} \right). \) We can find that \( Ex_{3}^{12p} > Ex_{3}^{11p} > Ex_{3}^{13p} . \)

Step 2. Calculate the prospect values.

Calculate the similarities and the prospect values (omitted). Take the prospect values in stage m1 as an example. The prospect values are showed in Fig. 4. “s, d, p” in Fig. 4 represent the state, development and PRP.

Fig. 4.
figure 4

The stacked column diagram of the prospect values in stage m1

Full size image

Compared with the SRPs, the prospect values of several alternatives with respect to attributes c1 and c2 are greater than 0, which means gains. This shows that the performances of these alternatives are higher than the expected performance of multiple states.

Compared with the DRPs, the prospect values of several alternatives with respect to attributes c1, c2 and c3 are greater than 0, which means gains. This shows that the performances of these alternatives are higher than the levels of the previous stage. This means that these alternatives are working harder at the present stage than in the previous stage.

Compared with the PRPs, the prospect values are always lower than 0, which means losses. This shows that the performances of alternatives do not reach their potential. This means that these alternatives have not fully exploited their potential.

Step 3. Calculate the preference coefficients and the comprehensive prospect values.

Calculate the preference coefficients by using (13)–(15). The comprehensive prospect values based on multiple reference points can be obtained from (12) based on the preference coefficients (omitted).

Step 4. Calculate the attribute weights and the stage weights.

Solve model M1 and M2. The attribute weights and the stage weights can be obtained as WC = {0.15, 0.345, 0.15, 0.355} and WM = {0.2, 0.25, 0.2, 0.35}. We can find that the weights of c2 and c4 are bigger than c1 and c3. The weights of stage m4 is bigger than the other three stages.

Step 5. Calculate the final ranking value.

The final ranking value can be calculated from (22) and we get the ranking values of alternatives (as showed in Fig. 5).

Fig. 5.
figure 5

The ranking values of alternatives

Full size image

Figure 5 shows that the best alternative is a15; the performances of alternative a15 are good under different reference points. The prospect values of alternative a15 are gains from the SRP and the DRP. This shows that the performance of alternative a15 is better than the expected performance of all alternatives in one stage and it is better than in the previous stages. The contribution of alternative a15 is greater, thus it is the best one.

5 Conclusions

The paper aims to propose a new MSRDM method for normal cloud model considering three reference points. The progress, current performance and the degree of effort are the criteria for measuring the multistage development of things. In this paper, the SRP is proposed to measure the current performance of an alternative under multiple states. The DRP is proposed to measure the progress in the previous stage. The PRP is proposed to measure the degree of effort and potential. Thus, a value function is defined to obtain the gain or loss from a reference point for normal cloud model. The prospect values under three reference points are aggregated by the synthetic degree of grey incidence. Then a two-stage weight optimization model is built to obtain the attribute weights and the stage weights based on the idea of minimax reference point optimization. The numeral example analysis shows its feasibility and validity for solving the MSRDM problems.

There are many interesting issues related to the problem with multiple reference points. The MCDM/GDM problems with multiple reference points will be researched in future. And a cloud model is a useful tool to deal with large quantity of data. We will develop a more appropriate method to deal with the cloud model.