Linguistic assessment information risky multi-criteria decision-making about wind power investment

Abstract

In this paper, we give a risky decision-making approach based on prospect theory and cloud theory to solve the wind power investment problem. In this problem, the criteria value of alternative is linguistic assessment information and the criteria’s weights are partially known. Firstly, we use the cloud theory to transfer the linguistic variables into one-dimensional normal cloud model. On the basis of defining the normal cloud model comparison rule and viewing all other alternatives as the reference point, a cloud prospect value function can be defined. Then, the cloud prospect decision-making matrix can be attained. Secondly, to get the optimal criteria weights, an optimization programming model which satisfies the algorithm of maximizing deviation and the decision makers’ subjective information is enacted. After that, we aggregate the clouds of each alternative as a comprehensive cloud. Then the order of alternatives can be listed by comparing comprehensive cloud of each alternative. Finally, an illustrative example about wind power investment is given and we verify the effectiveness and feasibility of this approach which can be valuable in the wind power investment decision-making.

Keywords

Wind power investment risky decision making linguistic assessment maximizing deviation

1 Introduction

With the increasing severity of energy and environment problems, renewable energy is competitively developed all over the world. Especially wind energy which is clean and abundant, is attached great importance and becomes the fastest developing renewable energy. However, the uncertainty and volatility of wind power may influence the operation and stability of grid network dramatically [1]. Recently, multi-criteria decision-making has been used in many fields such as power, energy and military. And the multi-criteria decision-making theory in which the criteria value of alternative is precise number has been relatively perfect. Yet, in reality decision-making process, the maker often uses language to describe the objects [2, 3]. Thus, the research on how to deal with linguistic assessment information risky multi-criteria decision-making about wind power investment has an important practical significance.

About the linguistic assessment information risky multi-criteria decision-making, the literature almost centers on how to deal with linguistic assessment. Literature [2 –4] transfer the linguistic assessment information into some kind of fuzzy number and get the solution order based on the fuzzy numerical calculation. Literature [5, 6] build the two-tuple semantic to transfer the language. Literature [7] uses uncertain linguistic hybrid geometric mean operator to solve the linguistic assessment information in group decision-making. In literature [8 –10], the language are transferred into one-dimensional normal cloud model, and by defining the calculation rule about cloud model, they give the optimal solution. In these methods above, if we build model with linguistic variables directly, we would fund that it is easily to ignore some hidden information. If we use traditional fuzzy sets to transfer the linguistic information into fuzzy mathematics to make decision, the calculation result of membership function would be exact numerical. In this condition, the fuzzy mathematics would be forced to precise mathematical category which violates the basic theory of fuzzy mathematics. Besides, the general fuzzy membership functions cannot reflect the relationship between fuzziness and randomness. Generally speaking, in cloud model, the basic principle of it is that it merges the probability theory and fuzzy set theory together, and transfer qualitative concept into numerical description via a certain algorithm. What’ more, it can deal with the hidden information to some extent. Because the cloud model can reflect the randomness and fuzziness of linguistic assessment information comprehensively [11], the research on linguistic variable decision-making based on cloud model has more and more attention [12, 13].

However, in the actual decision process, the external factors which are hard to be predicted will affect the decision maker and result to the different ordering consequence. As for this problem, researchers use the Expected Utility Theory to solve it, but the theory cannot explain the Allias Paradox and Ellsberg Paradox because it just can be efficacious on the assuming that people are totally rational which is impossible in the reality society. In 1979, Kahneman and Tversky proposed the Prospect Theory to make up for the disadvantage of Expected Utility Theory [14]. In this theory, it points out that people are limited rational and there exists system perception deviation when people make risky decision-making. This kind of system perception deviation is overestimate the low probability events and underestimate the high probability event [15]. From the description, the Prospect Theory is more coincidence with the reality.

Currently, the application of Prospect Theory in linguistic assessment information risky multi-criteria decision-making is little, there are some literatures researching the application [16 –18]. In the three literatures, the linguistic information is transferred into interval numbers, the phrase symbol and the triangular fuzzy number respectively, and by design the calculation rule, the prospect decision-making matrixes are obtained. Whereas, in these researches, the reference point is fixed, which is given by decision makers. The disadvantage of these applications is that we don’t know why it should use the fixed reference point and cannot get the comprehensive prospect value [19, 20].

Generally speaking, as for the decision-making in which the criteria value of alternative is linguistic assessment information and the criteria’s weights are partially known, we combine the cloud model and Prospect Theory together and give a method to make risky multi-criteria decision. In this method, the reference point is dynamic. And we build the cloud prospect value function on the base of cloud calculation rule. Besides, in order to increase the reliability of the decision-making result, we form an optimal model which can include the objective and subjective weights information. And at last, we compare the optimal solution in this paper and literature [9], we find that method in this paper is more feasible, and it can be valuable in wind power investment decision-making.

2 Cloud model and prospect theory

2.1 Cloud Model

2.1.1 Definition

In 1995, academician Deyi Li proposed the concept and mechanism of cloud model [21]. The cloud model is a quantitative model which can transfer qualitative concept into numerical description.

Definition 1. [22] Supposing U is the quantitative domain and C is a quality concept in U. And there exists an x which is a random realization of the quality concept C, besides, the membership μ (x) ∈ [0, 1] is a random number with stable tendency, that is to say, it satisfies that μ : U→ [0, 1], ∀x ∈ U, → μ (x). On this case, the distribution of x in U is called cloud C (U). Each x is called a cloud drop.

The Cloud Model can describe the randomness and fuzziness of qualitative concepts by three digital characteristics, namely, Ex(Expectation), En(Entropy), and He(Hyper entropy). Thus, the Cloud Model is usually expressed as C (Ex, En, He). Among these characteristics, Ex characterizes the quality concept best which represents the center position in the domain. En measures the fuzziness of a qualitative concept representing the range of values that could be accepted in the domain. Moreover, it also reflects the dispersing extent of the cloud drops. The larger En is, the larger the fuzziness and randomness will be. As for He, it represents the uncertainty of En in reflecting the dispersion of the Cloud drops.

2.1.2 calculation and comparison rules

Definition 2. [23] C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) are two normal clouds in domain U, the basic calculation rules are as follow.

$C_{1} + C_{2} = ({Ex}_{1} + {Ex}_{2}, \sqrt{{En}_{1}^{2} + {En}_{2}^{2}},$ $\sqrt{{He}_{1}^{2} + {He}_{2}^{2}})$

$C_{1} - C_{2} = ({Ex}_{1} - {Ex}_{2}, \sqrt{{En}_{1}^{2} + {En}_{2}^{2}},$ $\sqrt{{He}_{1}^{2} + {He}_{2}^{2}})$

λC₁ = (λEx₁, En₁, He₁)

$C_{1}^{α} = ({Ex}_{1}^{α}, \sqrt{α} \times {Ex}_{1}^{α - 1} \times {En}_{1}, \sqrt{α} \times {Ex}_{1}^{α - 1} \times {He}_{1})$

Definition 3. [24] C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) are two one-dimensional normal clouds in domain U, the Hamming distance between them is defined below. $d (C_{1}, C_{2}) = | (1 - ℓ_{1}) {Ex}_{1} - (1 - ℓ_{2}) {Ex}_{2} |$ (1)

Where, $ℓ_{1} = \frac{\sqrt{{En}_{1}^{2} + {He}_{1}^{2}}}{\sqrt{{En}_{1}^{2} + {He}_{1}^{2}} \sqrt{{En}_{2}^{2} + {He}_{2}^{2}}}$

$ℓ_{2} = \frac{\sqrt{{En}_{2}^{2} + {He}_{2}^{2}}}{\sqrt{{En}_{1}^{2} + {He}_{1}^{2}} \sqrt{{En}_{2}^{2} + {He}_{2}^{2}}}$ . Especially, when there exists En₁ = He₁ = En₂ = He₂ = 0, the cloud model calculation can be viewed as real number calculation, as d (C₁, C₂) = |Ex₁ - Ex₂|.

In normal cloud model, the contribution on the qualitative concept of each cloud drop is not the same. According to the 3En rule [25, 26], the cloud drops which have main contribution on the qualitative concept are almost in the interval [Ex - 3En, Ex + 3En]. And the cloud drops out of the interval are small probability events whose contribution can be ignored. Therefore, we give the comparison rule of normal cloud model based on the 3En rule below.

Definition 4. C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) are two one-dimensional normal clouds in domain U, now we transfer the two clouds into interval gray numbers, respectively is ⊗₁ ∈ [Ex₁ - 3En₁, Ex₁ + 3En₁] and ⊗₂ ∈ [Ex₂ - 3En₂, Ex₂ + 3En₂]. If p (⊗ ₁ ≥ ⊗ ₂) ≥0.5, C₁ ≥ C₂, on the contrary, C₁ < C₂ when ⊗₁ < ⊗ ₂. $\begin{matrix} p (\otimes_{1} \geq \otimes_{2}) \\ = \frac{min {l_{1} + l_{2}, max {({Ex}_{1} + 3 {En}_{1}) - ({Ex}_{2} - 3 {En}_{2}), 0}}}{l_{1} + l_{2}} \end{matrix}$ (2)

Formula (2) is the possible degree of ⊗₁ ≥ ⊗ ₂ [20], l₁ = (Ex₁ + 3En₁) - (Ex₁ - 3En₁), l₁ = (Ex₂ + 3En₂) - (Ex₂ - 3En₂).

From the formula (2), we can get three relationships.

0 ≤ p (⊗ ₁ ≥ ⊗ ₂) ≤1

p (C₁ ≥ C₂) + p (C₁ < C₂) =1

when p (C₁ ≥ C₂) = p (C₁ ≤ C₂) =0.5, ⊗₁ = ⊗ ₂ we call cloud C₁ and C₂ are equal.

2.1.3 cloud integration

Definition 5. [10] {C₁ (Ex₁, En₁, He₁) , C₂ (Ex₂, En₂, He₂) , ⋯ , C_n (Ex_n, En_n, He_n)} are clouds in domain U, W = {w₁, w₂, ⋯ , w_n} are the weight of each cloud. We can integrate the n clouds as a comprehensive cloud C (Ex, En, He), the three digital characteristics are as follow. ${\begin{matrix} Ex = ω_{1} {Ex}_{1} + ω_{2} {Ex}_{2} + \dots + ω_{n} {Ex}_{n} \\ En = \frac{ω_{1} {Ex}_{1} {En}_{1} + ω_{2} {Ex}_{2} {En}_{2} + \dots + ω_{n} {Ex}_{n} {En}_{n}}{ω_{1} {Ex}_{1} + ω_{2} {Ex}_{2} + \dots + ω_{n} {Ex}_{n}} \\ He = \sqrt{{He}_{1}^{2} + {He}_{2}^{2} + \dots + {He}_{n}^{2}} \end{matrix}$ (3)

2.1.4 Transforming method

According to the linguistic assessment value, firstly, we divide them into n linguistic levels, such as {⋯, Very Good, Good, ⋯, Common, ⋯, Bad, Very Bad, ⋯}, usually, n is an odd number. Secondly, we generate the effective domain U = [x_min, x_max]. At last, we use the gold segmentation method [8] to generate numerical description which can reflect the n clouds.

Take n = 5 as an example to explain how to transform the linguistic assessment value into numerical description. In this example, the linguistic levels are {Very Good(VG), Good(G), Common(C), Bad(B), Very Bad(VB)}. The calculation method is shown in Table 1.

2.2 Prospect Theory

In 1979, Kahneman and Tversky proposed Prospect Theory on the analysis of Expectation Theory and Expected Utility Theory. Prospect Theory can reflect the limited rational of decision makers when they are making decision, which is more coincidence with the reality. In the theory, the core is prospect value. It is determined by both valve function and weight function. The expression is below [14]. $V = \sum_{i = 1}^{n} π (p_{i}) υ (Δ x_{i})$ (4)

Where, V is prospect value, π (p_i) is weight function which is monotone increasing, v (Δx_i) is value function which is formed by the subjective feeling of decision makers [27].

$υ (Δ x) = {\begin{matrix} (Δ x)^{α}, Δ x \geq 0 \\ - λ (- Δ x)^{β}, Δ x < 0 \end{matrix}$ (5)

$π (p) = {\begin{matrix} \frac{p^{γ}}{(p^{γ} + (1 - P)^{γ})^{1 / γ}}, Δ x \geq 0 \\ \frac{p^{δ}}{(p^{δ} + (1 - P)^{δ})^{1 / δ}}, Δ x < 0 \end{matrix}$ (6)

Δx is the differential value of the decision criterion value related to the reference point. When Δx ≥ 0, it represents benefit and on the contrary, it is loss. α and β are the risk attitude coefficient with the value 0 ≤ α, β ≤ 1. The bigger the value of α and β, it shows that the decision maker more prefer adventuring. When α = β = 1, the decision maker is risk neutral. λ is loss aversion coefficient. p is the probability of solution under a certain status. γ and δ are used to control the curvature of the weight function [28]. They represent the given value of benefit and loss by decision maker respectively.

Definition 6. C₁ (Ex₁, En₁, He₁) and C₂ (Ex₂, En₂, He₂) are two one-dimensional normal clouds in domain U, viewing C₂ as the reference point, and based on the calculation and comparison rules of cloud model, the prospect value function of C₁ is below.

$υ (C_{1}) = {\begin{matrix} (C_{1} - C_{2})^{α}, C_{1} \geq C_{2} \\ - λ (C_{1} - C_{2})^{β}, C_{1} < C_{2} \end{matrix}$ (7)

3 Decision-making steps about wind power investment

3.1 Problem description

There is a linguistic assessment information risky decision-making about wind power investment. And assume that the alternative solution set is A = {a₁, ⋯ , a_i, ⋯ a_m}, the decision criteria areZ = {z₁, ⋯ , z_j, ⋯ , z_n}, and the weight vector is W = {ω₁, ⋯ , ω_j, ⋯ , ω_n} which satisfies the condition $\sum_{j = 1}^{n} w_{j} = 1, w_{j} \in G$ . G presents part of weight information given by decision maker. The risk status are θ = {θ₁, ⋯ , θ_k, ⋯ , θ_s}. In the risk status, θ_k is the kth status with the probability p_k. The linguistic assessment value of solution a_i in criterion z_j and the status θ_k is x_ijk, and 1 ≤ i ≤ m, 1 ≤ j ≤ n, 1 ≤ k ≤ s. Thus, we can get a linguistic assessment decision-making matrix X = (x_ijk) _m×n×s. At last, we should order thesolutions.

3.2 Decision-making steps

Step 1 transfer the assessment information

Based on the Table 1 and the linguistic levels, we transform the linguistic assessment decision-making matrix X = (x_ijk) _m×n×s into cloud matrix C = (c_ijk) _m×n×s.

Step 2 calculate cloud prospect decision-making matrix

In Prospect Theory, benefit and loss are both relative value relative to reference point. In this paper, we use the dynamic reference point as in literature [19], taking the rest alternative solutions as references. According to formula (4), we can calculate the cloud prospect value of each solution in each criterion andstatus.

$\begin{matrix} V_{ij} (C) & = & \sum_{k} \sum_{l} v (C_{ijl}^{k}) π_{ijl} (p_{k}), \\ l & = & 1, 2, \dots, m, l \neq i \end{matrix}$ (8)

The cloud prospect value function $v (c_{i j l}^{k})$ and the probability weight function are,

$υ (C_{lji}^{k}) = {\begin{matrix} (C_{lj}^{k} - C_{ij}^{k})^{α}, C_{ij}^{k} \geq C_{lj}^{k} \\ - λ (- (C_{lj}^{k} - C_{ij}^{k}))^{β}, C_{ij}^{k} \leq C_{lj}^{k} \end{matrix},$ (9)

$π_{lji} (p_{k}) = {\begin{matrix} \frac{p_{k}^{γ}}{(p_{k}^{γ} + (1 - p_{k})^{γ})^{1 / γ}}, C_{ij}^{k} \geq C_{lj}^{k} \\ \frac{p_{k}^{δ}}{(p_{k}^{δ} + (1 - p_{k})^{δ})^{1 / δ}}, C_{ij}^{k} \leq C_{lj}^{k} \end{matrix} .$ (10)

And then, we can get the cloud prospect decision-making matrix V = (V_ij (C)) _m×n.

Step 3 calculate the criteria weights

In order to distinguish the advantage and disadvantage of each solution, we use an optimization programming model which satisfies the algorithm of maximizing deviation and the decision maker’s subjective information to calculate the criteria weights. Shown as follow.

$\begin{matrix} max Z = \sum_{i = 1}^{m - 1} \sum_{k = i + 1}^{m} \sum_{j = 1}^{n} d (V_{ij} (C), V_{kj} (C)) ω_{j} \\ s . t . \sum_{j = 1}^{n} ω_{j} = 1, ω_{j} \geq 0, ω_{j} \in G \end{matrix}$ (11)

We can use matlab or lingo software to program and get the optimal criteria weights $W^{*} = (ω_{1}^{*} ω_{2}^{*}$ $\dots ω_{n}^{*})$ .

Step 4 order solutions

On the basis of cloud prospect decision-making matrix and criteria weights, we use Definition 5 to integrate the solutions as a comprehensive cloud. And then, order the solutions according to Definition 4.

4 Numerical example

In this paper, we intend to invest a wind power plant among three solutions, named {A₁, A₂, A₃}. According to the opinion of experts, the wind resource (Z₁), the plant condition (Z₂) and the reliability of wind turbine (Z₃) are viewed as three mainly criteria when assess the solutions. The linguistic assessment levels are {Very Good(VG), Good(G), Common(C), Bad(B), Very Bad(VB)}.The risk status are Great, Normal and Poor. The linguistic assessment to the solution of each expert is in Table 2. Besides, we give part of weight information, G = {0.2 ≤ ω₁ ≤ 0.6, ω₃ ≤ ω₂ ≤ ω₁, ω₃ + ω₂ + ω₁ = 1}. On this condition, we will give the optimal solution via the method given in this paper.

Firstly, pursuant to the linguistic level, we use Table 1 to generate five relative clouds {C₊₂, C₊₁, C₀, C_-1, C_-2} in domain U = [X_min, X_max] = [0, 100] shown in Table 3. And we can get the normal cloud model of each solution in each criterion and status in Table 4 to 6.

Secondly, based on the formula (8) to formula (10) in step 2, we calculate the cloud prospect decision-making matrix V = (V_ij (C)) _3×3. The matrix is about Table 7. As for the parameters, α = β = 0.88, λ = 2.25, γ = 0.61, δ = 0.69 [15].

And then, according to algorithm of maximizing deviation and the decision makers’ subjective information, the following model can be obtained by formula (11).

$\begin{matrix} max Z = 88.42 ω_{1} + 224.33 ω_{2} + 104.16 ω_{3} \\ s . t . {\begin{matrix} 0.2 \leq ω_{1} \leq 0.6 \\ ω_{2} \leq ω_{1} \\ ω_{3} \leq ω_{2} \\ ω_{1} + ω_{2} + ω_{3} = 1 \end{matrix} \end{matrix}$ (12)

In this paper, we use matlab software to program and get the result of W^* and Z^*. W^* = (0.549, 0.312, 0.139), Z^* = 136.75.

At last, we integrate the cloud of each solution in each criterion and status in Table 7 as a comprehensive cloud on the base of formula (3) in Definiton 5. , , . Till now, we can use the comparison rule of cloud to compare the three clouds. , , , and we can get . We can get the order of the three solutions A₂ ≻ A₃ ≻ A₁, and the optimal one is A₂.

Besides, to verify the feasibility and reliability of the method, we take the literature [9] as a comparison. We use the same criteria weights W = (0.549, 0.312, 0.139), and the risk status are Great, Normal and Poor. The order is below.

By comparing the results, we get three aspects to prove that the method in this paper is more valuable.

In this paper, we take the three risk status in consideration and get a comprehensively optimal solution A₂. However, in literature [9], it can just order the solutions in each risk status, and obviously, the order is not the same which lead the decision maker hard to choose.

We consider the objective risk attitude of the decision maker and give the base that the decision maker is limited rational which is more coincidence with the reality. Yet, literature [9]’s research is in the view that people are totally rational. In another word, in literature [9], the decision maker’s risk attitude is the same when facing benefit and loss which deviates from the reality.

As for the weight, we build an optimization programming model which considers the subjective and objective information. But in literature [9], the weight is given by decision maker and ignore the objective messages from historical datum. On this condition, we cannot figure out the source and it has poor repeatability.

5 Conclusion

In reality, the linguistic assessment and having a preference for risk are more coincidence with the decision-making process. Thus, in this paper, we discuss the linguistic assessment risky multi-criteria decision-making about wind power investment and introduce the Cloud Model Theory and Prospect Theory to transfer the linguistic assessment information and solve the risk preference problem respectively. In this case, the decision-making result is more consistent with the actual situation. Besides, we build an optimization programming model which satisfies the algorithm of maximizing deviation and the decision maker’s subjective information to calculate the criteria weights.It increases the reliability and feasibility of the method in choosing the optimal wind power plant. At last, we give a result comparison between this paper and literature [9], the conclusion of the comparison shows that the method in this paper can be more valuable in in the wind power investment decision-making.

Footnotes

Acknowledgments

The authors would like to acknowledge the supports by National Natural Science Foundation of China (No. 71271084).

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