A prospect theory based MADM method for solar water heater selection problems

Abstract

Recently, the solar water heaters have been widely used in China. For common consumers, the solar water heater selection problems have theoretical and practical significance. In this paper, a multiple attribute decision making method for solar water heater selection problems is proposed. Firstly, the evaluations to alternatives take the form of linguistic terms and then are transformed into triangular fuzzy numbers. Secondly, the reference points of the attributes are obtained from the decision maker. Then, based on prospect theory and the operation rules of fuzzy numbers, the linguistic rating information is converted and aggregated. Finally, the alternatives are ranked from best to worst. The first one is the optimal choice. The availability of the proposed method is illustrated through an application in solar water heater selection problems.

Keywords

Multiple attribute decision making (MADM)prospect theory linguistic evaluations solar water heater (SWH)

1 Introduction

Nowadays, with the development of the world economy, the demand for energy is higher and higher. The renewable energy technology is one of the most promising solutions for minimizing the carbon emission and environmental pollution from human activities [1]. One alternative renewable energy source is the solar energy [2]. Solar water heater is widely used in urban and rural areas [3]. Currently, the solar water heater is used to produce domestic hot water [4]. In China, the use of solar water heaters in the urban and rural areas is advocated by central and local government through a series of incentive policies [4]. Over the past two decades, the solar water heater industry in China has dramatically developed. The annual output increased from 6.1 million square meters in 2000 to 42 million square meters in 2009, with an annual growth rate of 24% [5]. Chinese government provides subsidies for rural residents to purchase home appliances. The solar water heater is included in the subsidy list in April, 2009. And the central finance provides 13% allowance of the product price to rural users for buying solar water heater products [5]. At present, there are about 3000 enterprises in the solar water heater industry, among which 1800 are solar water heater manufacturers. Especially, there are over 30 enterprises with a sales volume of more than 100 million Yuan RMB [5].

For the common consumers, the performance of solar water heaters is hard to evaluate in a simple way. Due to the diversity of the manufactures and the complexity of the products, it is difficult for the common consumers to select the proper solar water heater. In real life, it is too difficult to evaluate the performance of the products with any numerical values. So, in some multiple attribute decision making (MADM) problems, the decision makers are often reluctant to rate attributes with quantitative values. Alternatively, linguistic evaluations might be used. For example, the consumers might evaluate the performance of the solar water heaters with the linguistic terms, such as a set with 7 terms: {extremely bad, very bad, bad, common, good, very good, extremely good}. Even for the attributes able to be measured in numerical values, such as price, people would rather use natural languages in some cases. Generally, the customers, especially for those from the countryside, evaluate the solar water heaters in a rough way from several aspects. The linguistic evaluations are convenient. Consequently, the solar water heater selection problem can be considered as a MADM problem with linguistic evaluations.

Linguistic MADM is so important in both theoretic studies and practical applications. Fortunately, some researchers have made great contributions [6 –12]. Recently, MADM problems have attracted much attention [13 –16], and the MADM method has been widely applied into various areas [17 –19]. For example, fuzzy multiple criteria decision making approach is used to innovative strategies based on Miles and Snow typology [20]. The programmable logic controllers are evaluated from a fuzzy MCDM perspective [21]. In the literatures, the linguistic terms are transformed into numerical values before aggregation. Additive linguistic evaluation scales and multiplicative linguistic evaluation scales are defined respectively, through which linguistic information can be aggregated [6]. And fuzzy linguistic methodology to deal with linguistic term sets is presented [11 , 23]. In the above methods, the terms in the linguistic term sets are changed into fuzzy sets.

Another important aspect in MADM is that the risk attitude/preferences of the decision maker should be taken into account [24]. Prospect theory developed by Kahneman and Tversky [25] is a descriptive decision model in the cases under risk. Later, Tversky and Khneman [26] developed the cumulative prospect theory, which considers psychological factors in decision making. In the prospect theory, the outcomes are expressed by means of gains and/or losses from a reference point. In prospect theory, the value function is considered as a S-shape concave above the reference point, which reflects the aversion to risk in case of gains; and the convex part below the reference point reflects the risk preference in face of losses [24, 27]. In fact, prospect theory has successfully been applied as behavioral models in MADM problems[28 –33].

Therefore, based on prospect theory and the linguistic evaluation information, we propose a prospect theory based MADM method with linguistic evaluations. Firstly, the linguistic evaluation matrix is established and the reference point is provided by the decision maker. Secondly, the linguistic ratings and the reference points are transformed into fuzzy numbers. Then based on prospect theory and the operation rules of fuzzy numbers, the gains/losses of the attributes are computed, and the prospect values of the attributes are obtained. Ultimately, the weighted prospect values of the alternatives are acquired, according to which the alternatives are ranked from best to worst. The first one is the optimal choice.

The proposed method has some novelties and characteristics as follows. Firstly, the attribute values take the form of linguistic evaluation information, which is convenient for the ordinary decision maker. Secondly, the linguistic value function is expressed by means of gains and/or losses compared to the reference point, which considers the decision makers’ psychological preference in the process of decision making. Finally, an application in solar water heater selection problems is given to illustrate the feasibility of the proposed method.

Currently, the solar water heaters are widely used in China, so the research on solar water heater selection problems has theoretical and practical significance. Obviously, to evaluate a solar water heater, the consumer must take multiple aspects into consideration. Moreover, the linguistic evaluations might be used. Thus, it can be seen a linguistic MADM problem to assess the solar water heaters. In the meanwhile, when evaluating the products, everyone has his/her own psychological preferences due to the particular financial situations and individual demands. So we can apply the proposed method to the solar water heater selection problems.

The remainder of the paper is organized as follows. In Section 2, some basic concepts are reviewed, which include fuzzy sets, triangular fuzzy numbers and their operation rules, linguistic term sets and the value function based on prospect theory. In Section 3, we provide a linguistic MADM method based on prospect theory. In Section 4, an application on solar water heater selection problems is given to illustrate the practicality of the proposed method. Finally, Section 5 draws conclusions.

2 Preliminaries

In this section, some basic concepts, operation rules and theoretical methods are introduced, which conclude fuzzy sets, triangular fuzzy numbers and their operation rules, linguistic term sets and prospect theory.

2.1 Fuzzy sets and triangular fuzzy numbers

Definition 1. [34] A fuzzy set $\tilde{A}$ in a universe of discourse X is characterized by a membership function $μ_{\tilde{A}} (x)$ that assigns each element x in X a real number in the interval [0, 1]. The numeric value $μ_{\tilde{A}} (x)$ stands for the grade of membership of x in the set $\tilde{A}$ .

Definition 2. [34] A triangular fuzzy number $\tilde{a}$ is defined by a triplet $\tilde{a} = (a^{L}, a^{M}, a^{U})$ with membership given by: $μ_{\tilde{A}} (x) = {\begin{matrix} (x - a^{L}) / (a^{M} - a^{L}) a^{L} \leq x \leq a^{M} \\ (a^{U} - x) / (a^{U} - a^{M}) a^{M} \leq x \leq a^{U} \\ 0 x \in (- \infty, a^{L}) \cup (a^{U}, + \infty) \end{matrix}$ (1)

2.2 The operation rules and comparison methods of the triangular fuzzy numbers

2.2.1 The operation rules of the triangular fuzzy numbers

Suppose that $\tilde{a} = (a^{L}, a^{M}, a^{U})$ and $\tilde{b} = (b^{L}, b^{M}, b^{U})$ are two triangular fuzzy numbers. According the extension principle of fuzzy sets, the addition result of two triangular fuzzy numbers can still keep as a triangular fuzzy number. However, scalar multiplication or power result of two triangular fuzzy numbers do not produce triangular fuzzy numbers, but only the fuzzy numbers in triangular shape [35], and also require complicated computations at every α-cuts. In the present paper, for simplicity, we will adopt the following approximation formulas as [35]. The operation rules are shown as follows[34].

(1) Addition rules: $\tilde{a} + \tilde{b} = (a^{L} + b^{L}, a^{M} + b^{M}, a^{U} + b^{U}) .$

(2) Scalar multiplication rules: $k \times \tilde{a} ≅ {\begin{matrix} ({ka}^{L}, {ka}^{M}, {ka}^{U}), (k \geq 0) \\ ({ka}^{U}, {ka}^{M}, {ka}^{L}), (k < 0) \end{matrix} .$

(3) Power rules: ${(\tilde{a})}^{α} ≅ {((a^{L})}^{α}, (a^{M})^{α}, (a^{U})^{α}), (a^{L}, a^{M}, a^{U} \geq 0) .$

2.2.2 The comparison method of the triangular fuzzy numbers

Suppose that $\tilde{a} = (a^{L}, a^{M}, a^{U})$ and $\tilde{b} = (b^{L}, b^{M}, b^{U})$ are two triangular fuzzy numbers. The defuzzified value of $\tilde{a}$ is defined as: $m (\tilde{a}) = \frac{a^{L} + 2 a^{M} + a^{U}}{4}$ . Similarly, we have the defuzzified value $m (\tilde{b}) = \frac{b^{L} + 2 b^{M} + b^{U}}{4}$ . The comparison method [34] is defined as: If $m (\tilde{a}) \geq m (\tilde{b})$ , then $\tilde{a} \geq \tilde{b}$ .

2.3 Linguistic term sets

Zadeh [36] presented the concept of linguistic variables, which take the form of words or sentences in a natural language instead of numbers. The fuzzy linguistic evaluations qualitatively describe the alternatives. The fuzzy linguistic approach has successfully been applied to deal with decision making problems [37]. It is very important to choose appropriate way to express linguistic term sets and their semantics.

In the literatures, linguistic variables are mapped into numerical values [6, 38]. For example, Rodriguez demonstrated a set of seven linguistic terms S by means of an ordered structure approach. It is shown in Fig. 1 as: S = { s₀, s₁, s₂, s₃, s₄, s₅, s₆ } = {nothing, verylow, low, medium, high, veryhigh, perfect} [7]. In Fig. 1, the terms in S can be transformed to fuzzy numbers. Such as, the lowest one “nothing” is denoted by a triangular fuzzy number, i.e., “s₀ = (0, 0, 0.17)”, “low” is denoted by “s₂ = (0.17, 0.33, 0.5)”, and the highest one “perfect” is denoted by “s₆ = (0.83, 1, 1)”.

2.4 Prospect theory

Prospect theory was initially proposed [25] as a descriptive theory for actual decision behavior under risk. And later they have also been applied for riskless choice [39]. Since then, a number of researches on prospect theory have been found [40, 41]. So far, prospect theory has been regarded as the most popular behavioral decision theory.

According to prospect theory, the evaluations from alternatives are regarded as gains or losses with respect to a reference point. The reference point can reflect the psychological expectations from the decision makers. When the evaluation result is better than the reference point, the decision maker can get happiness, and it can be expressed by the gain. On the contrary, when the evaluation result is less than the reference point, he/she would feel upset, and it can be expressed by the loss [30]. The prospects are evaluated by a value function and a weighting function. Then, the alternative with the highest prospect value is chosen. The value function is illustrated in Fig. 2 [29], which is an asymmetric S-shapedfunction.

In Fig. 2, x stands for the gain or loss of the outcome relative to the reference point. It denotes a gain ifx ≥ 0 and a loss if x < 0. The form of the value function is given by Kahneman and Tversky [26] as follows: $v (x) = {\begin{matrix} x^{α} & x \geq 0 \\ - λ {(- x)}^{β} & x < 0 \end{matrix} .$ (2)

Where α and β are estimable coefficients determining the concavity and convexity of the function, respectively, 0 ≤ α, β ≤ 1. λ is the parameter of loss aversion, λ > 1. In this paper, we assume α = β = 0.88, and λ = 2.25 similar to the literature [26]. In Fig. 2, for a certain gain x₀ and the corresponding loss -x₀, v (x₀) < |v (- x₀) | follows, i.e., the perceived value of a loss is larger than that of an absolutely commensurate gain.

3 The linguistic MADM method based on prospect theory

In this section, the linguistic MADM method based on prospect theory is proposed, and then the decision steps are provided.

3.1 The analysis of the linguistic MADM problems

Based on the method aforementioned, we can choose the proper term set, which encodes each word in the linguistic term set S to a triangular fuzzy number.

In the solar water heater selection problems, the decision maker is the consumer. For generality, the decision maker is used in the proposed method.The goal of the decision maker is to decide which alternative he/she should choose. In order to arrive at his/her aims, he/she must firstly rate the alternatives with respect to each attribute. His/ her ratings take the form of words, i.e., linguistic rating.

To begin with, the decision maker must choose a proper term set (S¹, S² or S³), from which he/she could select a proper term to evaluate the alternative a_i with respect to the attribute c_j. Specifically, the decision maker must fill in a table by completing the following statements: How about the attribute c_j of the alternative a_i?

For example, to rate the attribute c₁ (performance), the words can be chosen from $S^{1} = {s_{0}^{1}, s_{1}^{1}, s_{2}^{1}, s_{3}^{1}, s_{4}^{1}, s_{5}^{1}, s_{6}^{1}} = {Extremly bad, Verybad, Bad, Common, Good, Very good, Extremelygood}$ . To rate attribute c₂ (price), the words are chosen from $S^{2} = {s_{0}^{2}, s_{1}^{2}, s_{2}^{2}, s_{3}^{2}, s_{4}^{2}, s_{5}^{2}, s_{6}^{2}, s_{7}^{2}, s_{8}^{2}} = {Extremely low, Very low, Low, Somewhat low, Moderate, Somewhat high, High, Very high, Extremely high}$ . And to rate attribute c₃(service), the words are chosen from $S^{3} = {s_{0}^{3}, s_{1}^{3}, s_{2}^{3}, s_{3}^{3}, s_{4}^{3}} = {Very dissatisfied, Dissatisfied, General, Satisfied, Very satisfied}$ . In brief, a MADM problem with linguistic variables is described as follows:

Suppose that A = (a₁, a₂, …, a_m) is the set of the alternatives, and C = (c₁, c₂, …, c_n) is the set of attributes. Then the decision matrix $\tilde{A}$ is constructed as follows: $\tilde{A} = ({\tilde{x}}_{ij})_{m \times n} = [\begin{matrix} {\tilde{x}}_{11} & \dots & {\tilde{x}}_{1 n} \\ ⋮ & ⋱ & ⋮ \\ {\tilde{x}}_{m 1} & \dots & {\tilde{x}}_{mn} \end{matrix}] .$ (3)

Where ${\tilde{x}}_{ij}$ is the value of attribute c_j from the alternative a_i and ${\tilde{x}}_{ij}$ takes the form of the linguistic variables. Let S be the linguistic term set and ${\tilde{x}}_{ij} \in S$ . Let w = (w₁, w₂, …, w_n) be an attribute weight vector, where w_j denotes the weight or importance of the attribute c_j, such that $\sum_{j = 1}^{n} w_{j} = 1$ and 0 ≤ w_j ≤ 1, j = 1, 2, … n. The decision making reference points of different attributes can be expressed as the linguistic variables ${\tilde{x}}_{j}^{0}$ . Based on these conditions, we can rank the possible alternatives.

3.2 The combination of linguistic MADM and prospect theory

In MADM, the risk attitude/preferences of the decision maker should be considered [24]. Fortunately, prospect theory developed by Kahneman and Tversky [25] is a descriptive model in the cases under risk, which considers psychological factors in decision making.

In the linguistic MADM problems, since the attributes are all assessed with linguistic variables, the reference points are expressed with the form of linguistic terms too. More often, before making decision, the decision maker has an ideal alternative in the heart, which is proper to his/her own psychological preference. However, the ideal alternative are difficult to be measured by numerical values precisely. For example, when selecting a solar energy heater, people might hope to find an ideal one, which is good in performance and low in price.

Similar to the attributes, the linguistic reference points can be translated into fuzzy numbers Therefore, the attributes of all alternatives can be compared with the corresponding linguistic reference point.

In the prospect theory, an essential feature is that the people would like to focus on the evaluation of changes or differences rather than to the evaluation of absolute magnitudes [25]. Hence, with the aid of prospect theory, the evaluations are expressed by means of gains and/or losses from a reference point. The value function is considered as an S-shape concave above the reference point, which reflects the aversion to risk in case of gains; and the convex part below the reference point reflects the risk preference in face of losses [24, 27]. In fact, prospect theory has successfully been applied as behavioral models in MADM problems [28 –32].

In the linguistic MADM problems, before comparison and aggregation, both the attributes and reference points should be transformed into triangular fuzzy numbers.

3.3 The steps for decision making

Based on the term sets established above, we present a novel MADM method, in which the attributes are assessed in linguistic variables and are translated into fuzzy numbers, and the decision makers’ preferences are considered.

Specifically, the decision making method consists of 8 steps as follows.

Step 1: Choosing the proper term set

Choose a proper term set S, in which each word can be mapped to a triangular fuzzy number.

Step 2: Constructing the linguistic decision matrix $\tilde{A}$

The DM’s goal is to decide which one is proper. His/her ratings take the form of linguistic variables.

The decision maker should select a proper term to evaluate the attribute c_j of the alternative a_i from one of the term sets (S¹, S² or S³).

In order to illustrate what the linguistic ratings might look like, an example is provided in Table 1. In the example, the consumer’s linguistic rating about the alternative a₁ is that the performance (c₁) is good, the price (c₂) is moderate and the service (c₃) is general. After the table is filled in, then the linguistic decision matrix $\tilde{A}$ has been constructed.

Step 3: Transforming the linguistic decision matrix to triangular fuzzy matrix

Look up each word ${\tilde{x}}_{ij}$ of the linguistic decision matrix $\tilde{A}$ in the term set, and transform it into the corresponding triangular fuzzy number, then the fuzzy decision matrix can be obtained. For convenience, the fuzzy decision matrix and its triangular fuzzy number are denoted same to the linguistic decision matrix, i.e., $\tilde{A} = ({\tilde{x}}_{ij})_{m \times n}$ and ${\tilde{x}}_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{U})$ .

Step 4: Selecting the decision making reference point

The decision making reference points are determined by the DM’s risk preference and psychology state. Since different people have different psychology aspirations for the same commodity, even the same decision maker might have different psychology aspirations as external environment has changed. Hence, the reference point is not fixed. For simplicity and rationality, we assume that the reference point is provided by the decision maker [42].

The reference points consist of all reference point values which are corresponding to each attribute respectively and expressed as linguistic terms ${\tilde{x}}_{j}^{0} (j = 1, \dots, n)$ . Each reference point value can be transformed to a triangular fuzzy number, i.e., ${\tilde{x}}_{j}^{0} = (x_{j}^{L 0}, x_{j}^{ML}, x_{j}^{U 0})$ .

Step 5: Constructing the gain or loss matrix $\tilde{R}$

Based on the fuzzy decision matrix $\tilde{A}$ and the reference point ${\tilde{x}}_{j}^{0} = (x_{j}^{L 0}, x_{j}^{M 0}, x_{j}^{U 0})$ , we can construct the gain or loss matrix $\tilde{R}$ . $\tilde{R} = ({\tilde{r}}_{ij})_{m \times n} = (r_{ij}^{L}, r_{ij}^{M}, r_{ij}^{U})_{m \times n}$ (4)

In general, there are two types for the attributes, i.e., benefit type and cost type.

For the benefit type, we assume:

$\begin{matrix} {\tilde{r}}_{ij} & = & ({\tilde{x}}_{ij} - {\tilde{x}}_{j}^{0}) \\ = & {(x_{ij}^{L} - x_{j}^{U 0}, x_{ij}^{M} - x_{j}^{M 0}, x_{ij}^{U} - x_{j}^{L 0})}_{m \times n} . \end{matrix}$ (5)

For the cost type, we assume:

$\begin{matrix} {\tilde{r}}_{ij} & = & - ({\tilde{x}}_{ij} - {\tilde{x}}_{j}^{0}) \\ = & - {(x_{ij}^{L} - x_{j}^{U 0}, x_{ij}^{M} - x_{j}^{M 0}, x_{ij}^{U} - x_{j}^{L 0})}_{m \times n} . \end{matrix}$ (6)

Step 6: Constructing the prospect value matrix $\tilde{V}$

In accordance with the formula (2), the value function of the triangular fuzzy number can be calculated as follows: ${\tilde{v}}_{ij} = (v_{ij}^{L}, v_{ij}^{M}, v_{ij}^{U}) = (v (r_{ij}^{L}), v (r_{ij}^{M}), v (r_{ij}^{U}))$ (7)

Then the prospect value matrix $\tilde{V}$ can be achieved. $\tilde{V} = ({\tilde{v}}_{ij})_{m \times n}$ (8)

Step 7: Computing the weighted prospect value of the alternatives

The weight of the attributes is provided, and then the weighted prospect value of the alternatives can be computed as follow: ${\tilde{v}}_{i} = (v_{i}^{L}, v_{i}^{M}, v_{i}^{U}) = \sum_{j = 1}^{n} (w_{j} \times {\tilde{v}}_{ij})$ (9)

Step 8: Ranking the alternatives

The weighted prospect value of the alternatives takes the form of fuzzy numbers, so the alternatives are ranked in descending order based on the defuzzified value $m ({\tilde{v}}_{i})$ to the triangular fuzzy number ${\tilde{v}}_{i}$ . $m ({\tilde{v}}_{i}) = \frac{v_{i}^{L} + 2 v_{i}^{M} + v_{i}^{U}}{4}$ (10)

Then the first one is chosen as the best decision result.

4 An application in solar water heater selection problems

In recent years, the solar water heater industry has been developed rapidly in China. Therefore, in this section, an application on solar water heater selection from real life is given to illustrate the practicality of the proposed method. Suppose a consumer is planning to buy a solar water heater. Through preliminary screening, there are four possible alternatives. But he/she is still hesitant which one is the optimal choice. The possible alternatives are denoted as a₁, a₂, a₃, and a₄. For simplicity, we assume that he/she (i.e., the DM) takes three important attributes into account, i.e., the attribute c₁ (performance), c₂ (price) and c₃ (service). According to the DM’s preference, the attribute weight vector is assumed to be w = (0.5, 0.3, 0.2). For the attributes c₁ (performance) and c₃ (service), it is suitable to be assessed with linguistic variables. For the attribute c₂ (price), though the values can be measured quantitatively, they might still be translated into qualitative evaluations for convenience.

For instance, the prices from 3500 to 3600 Yuan are all very moderate, and almost indifferent in preference. Therefore, in the example, the attribute value of each alternative is expressed as the linguistic variables. And the linguistic terms are chosen from the term set S = S¹ ∪ S² ∪ S³. Where $S^{1} = {s_{0}^{1}, s_{1}^{1}, s_{2}^{1}, s_{3}^{1}, s_{4}^{1}, s_{5}^{1}, s_{6}^{1}} = {Extre - melybad, Very bad, Bad, Common, Good, Verygood, Extremely good}$ , $S^{2} = {s_{0}^{2}, s_{1}^{2}, s_{2}^{2}, s_{3}^{2}, s_{4}^{2}, s_{5}^{2}, s_{6}^{2}, s_{7}^{2}, s_{8}^{2}} = {Extremely low, Very low, low, Somewhat low, Moderate, Somewhat high, High, Very high, Extremely high}$ and $S^{3} = {s_{0}^{3}, s_{1}^{3}, s_{2}^{3}, s_{3}^{3}, s_{4}^{3}} = {Very dissatisfied, Dissatisfied, General, Satisfied, Very satisfied}$ . Please help the consumer choose the most proper alternative according to the above decision information.

Remark. For some attributes, even though they can be measured by numerical values, in some cases, people often prefer to evaluate them in the form of linguistic variables. Specifically, when assessing the attribute “price”, obviously, it could be measured precisely by numerical values. However, we often consider the objects with the similar prices as almost undifferentiated. For instance, there are two alternatives with the prices 3500 and 3600 Yuan, respectively, the decision maker might feel moderate in price to them indiscriminatingly, rather than compare the price in detail. Under such cases, it is proper to assess the attribute “price” in linguisticvariables.

4.1 Analysis to the problem

In this problem, the attributes c₁(performance) and c₃(service) are difficult to rate in numerical values, even for the attribute c₂(price), in real rating, most people are prone to evaluate them qualitatively. Therefore, all the attributes could be rated in natural language. Concretely, the first attribute (c₁ : performance) is rated with the terms from S¹, the second attribute (c₂ : price) is rated with the terms from S² and the last attribute (c₃ : service) is rated with the terms from S³. It is a typical MADM problem with the linguistic variables. So the proposed method above can be used to make decisions. Similar to the method in the literature [7], the linguistic evaluation information can be transformed into numerical values. Therefore, the term sets S = S¹ ∪ S² ∪ S³ are constructed, which are shown in Tables 2–4.

4.2 The decision steps

The decision steps are shown as follows:

Step 1: Choosing the proper term set

All the terms possible to be used can be transformed into corresponding triangular fuzzy numbers. The term sets S = S¹ ∪ S² ∪ S³are constructed firstly, which are shown in Tables 2–4.

Step 2: Constructing the linguistic decision matrix $\tilde{A}$

Let the decision maker fill in the Table 5 by completing the following statements: How about the attribute c_jof the alternative a_i? (i = 1, …, 4 ; j = 1, 2, 3).

After that, the linguistic decision making matrix $\tilde{A}$ can be obtained, which is shown in Table 1.

Step 3: Transforming the linguistic decision matrix to triangular fuzzy matrix

Look up each element ${\tilde{x}}_{ij}$ of the linguistic decision matrix $\tilde{A}$ in the term sets S established in Step 1, and transform it into the corresponding triangular fuzzy number. Then, the fuzzy decision matrix $\tilde{A}$ can be obtained as shown in Table 6.

Step 4: Selecting the decision making reference points

For simplicity, we assume that the reference points are provided by the decision maker. they are expressed as linguistic variables: ${\tilde{x}}_{1}^{0} = {s_{3}^{1}} = {Good}$ , ${\tilde{x}}_{2}^{0} = {s_{6}^{2}} = {High}$ and ${\tilde{x}}_{3}^{0} = {s_{2}^{3}} = {General}$ . Therefore, with respect to the term set S, they are transformed into fuzzy numbers, i.e., ${\tilde{x}}_{1}^{0} = (0.5, 0.67, 0.83)$ , ${\tilde{x}}_{2}^{0} = (0.625, 0.75, 0.875)$ , and ${\tilde{x}}_{3}^{0} = (0.25, 0.5, 0.75)$ .

Step 5: Constructing the gain and/or loss matrix $\tilde{R}$

Based on the fuzzy decision matrix $\tilde{A}$ shown in Table 5 and the decision making reference points ${\tilde{x}}_{j}^{0} = (x_{j}^{L 0}, x_{j}^{M 0}, x_{j}^{U 0}) (j = 1, 2, 3)$ achieved in Step 4, we can construct the gain and/or loss matrix $\tilde{R}$ shown in Table 7. The details are illustrated as follows.

The attributes c₁(appearance) and c₃ (service) are benefit type variables, so according to the formula (8), we can obtain ${\tilde{r}}_{ij} (i = 1, \dots, 4; j = 1, 3)$ respectively. For example, ${\tilde{r}}_{12}$ can be computed. ${\tilde{r}}_{12} = - ({\tilde{x}}_{12} - {\tilde{x}}_{2}^{0}) = (0.375, 0.5, 0.625) - (0.625, 0.75, 0.875) = (- 0.5, - 0.25, 0)$ . Similarly, other elements for the attribute c₁ (appearance) and c₃ (service) can be achieved too.

The attribute c₂ (price) is cost type variable, according to the formula (9), we can obtain ${\tilde{r}}_{ij} (i = 1, \dots, 4; j = 2)$ respectively.

It is important to note that some elements ${\tilde{x}}_{ij}$ might be same as the reference value ${\tilde{x}}_{j}^{0}$ , but the corresponding gain or loss value ${\tilde{r}}_{ij}$ is not always (0, 0, 0). For instance, both the term ${\tilde{x}}_{22}$ and the reference point ${\tilde{x}}_{2}^{0}$ are $s_{6}^{2} = {High}$ , however, ${\tilde{r}}_{22} = (- 0.25, 0, 0.25)$ .

Step 6: Constructing the prospect value matrix $\tilde{V}$

In accordance with the formula (2) and (10), the prospect value matrix $\tilde{V}$ can be achieved as shown in Table 8. Here, we assume α = β = 0.88, and λ = 2.25 [26]. The details are illustrated through an example. For instance, the element ${\tilde{r}}_{23} = (- 0.25, 0.25, 0.75)$ from the gain/loss matrix $\tilde{R}$ is transformed into an element ${\tilde{v}}_{23}$ of the prospect value matrix $\tilde{V}$ as follows.

Obviously, $r_{23}^{L} = - 0.25 < 0$ , and $r_{23}^{M} = 0.25 > 0$ , $r_{23}^{U} = 0.75 > 0$ , we have $v_{23}^{L} = - 2.25 \times 0 . 25^{0.88} = - 0.66$ , $v_{23}^{M} = 0 . 25^{0.88} = 0.3$ and $v_{23}^{U} = 3 . 6^{0.88} = 0.78$ . So, the triangular fuzzy number ${\tilde{v}}_{23} = (- 0.66, 0.3, 0.78)$ . Similarly, other elements in the prospect value matrix $\tilde{V}$ can be computed.

Step 7: Computing the weighted prospect value of the alternatives

The weight of the attributes is assumed to be (0.5, 0.3, 0.2). Then, based on the formula (12), the weighted prospect value of each alternative can be computed as shown in Table 9:

For the alternative a₃, we have $\begin{matrix} {\tilde{v}}_{3} & = & (v_{3}^{L}, v_{3}^{M}, v_{3}^{U}) = \sum_{j = 1}^{3} (w_{j} \times {\tilde{v}}_{3 j}) \\ = & 0.5 \times (- 1.56, - 0.87, 0) \\ + 0.3 \times (- 0.95, - 0.66, 0) \\ + 0.2 \times (- 1.75, - 0.66, 0.3) \\ = & (- 1.42, - 0.77, 0.06) . \end{matrix}$

Similarly, the weighted prospect value of other alternatives can also achieved. $\begin{matrix} {\tilde{v}}_{1} & = & (v_{1}^{L}, v_{1}^{M}, v_{1}^{U}) = (- 0.67, 0.09, 0.46), \\ {\tilde{v}}_{2} & = & (v_{2}^{L}, v_{2}^{M}, v_{2}^{U}) = (- 0.94, - 0.18, 0.35), \\ {\tilde{v}}_{4} & = & (v_{4}^{L}, v_{4}^{M}, v_{4}^{U}) = (- 0.27, 0.32, 0.66) . \end{matrix}$

Step 8: Ranking the alternatives

According to the formula (14), the defuzzified value $m ({\tilde{v}}_{i})$ of the triangular fuzzy number ${\tilde{v}}_{i}$ can be obtained.

For the alternative a₁, we have $\begin{matrix} m ({\tilde{v}}_{1}) & = & \frac{v_{1}^{L} + 2 v_{1}^{M} + v_{1}^{U}}{4} \\ = & \frac{(- 0.67) + 2 \times 0.09 + 0.46}{4} = - 0.01 \end{matrix}$

Similarly, the defuzzified values of other alternatives can also be achieved. $\begin{matrix} m ({\tilde{v}}_{2}) & = & \frac{v_{2}^{L} + 2 v_{2}^{M} + v_{2}^{U}}{4} = - 0.23 \\ m ({\tilde{v}}_{3}) & = & \frac{v_{3}^{L} + 2 v_{3}^{M} + v_{3}^{U}}{4} = - 0.72 \\ m ({\tilde{v}}_{4}) & = & \frac{v_{4}^{L} + 2 v_{4}^{M} + v_{4}^{U}}{4} = 0.26 \end{matrix}$

Then rank the alternatives in descending order based on the defuzzified value $m ({\tilde{v}}_{i})$ : a₄ ≻ a₁ ≻ a₂ ≻ a₃, so the alternative a₄ is chosen as the best decision result.

In this problem, all the attributes (c₁ (performance), c₂ (price), c₃ (service)) are rated in linguistic terms, which are transformed into fuzzy numbers based on the term set. Then, based on prospect theory, alternatives are ranked according to the overall prospect value.

The proposed method transforms the linguistic terms into fuzzy numbers, and considers the DM’s psychological preference. So the process of decision making seems to be more consistent with people’s real decision behavior.

Intuitively, for the alternative a₄, the attribute c₁(performance) is “very good”, c₂(price) is “low” and c₃(service) is “satisfied”, respectively. Thus, the alternative a₄ is superior to others. For the alternative a₁, the attribute c₁(performance) is “good”, c₂(price) is “moderate” and c₃(service) is “general”, respectively. Similarly, for the alternative a₂, the attribute c₁(performance) is “common”, c₂(price) is “high” and c₃(service) is “satisfied”, respectively. So the alternative a₁ and a₂ are moderate in overall evaluation. Moreover, the alternative a₁ is better than a₂ due to the larger weight of the attribute c₁(performance). Yet, for the alternative a₃, the attribute c₁(performance) is “bad”, c₂(price) is “extremely high”, and c₃(service) is “dissatisfied”, respectively. Thus, the alternative a₃ is inferior to others. Obviously, it is much consistent with the real world.

5 Conclusions

We have presented a novel MADM method based on prospect theory to solve solar water heater selection problems. In the solar water heater selection problems, the attributes are evaluated in linguistic terms. Thus, the linguistic rating information must be transformed into numerical values before aggregation. In this paper, the linguistic evaluation information is transformed into triangular fuzzy numbers. Meanwhile, the decision makers’ psychological preferences are taken into account through combining prospect theory into the MADM method. With the aid of prospect theory, decision maker’s aspiration-levels are considered as reference points. And the gains/losses of alternatives with respect to each attribute are assessed by measuring the perceived difference between the attribute value and the reference point. Then the overall prospect values of all alternatives are computed. Ultimately, the alternatives are ranked in descending order according to overall prospect values. The first one is the optimal choice.

Compared with the existing methods [28 –33], the proposed method has some novelties and characteristics as follows.

Firstly, the attribute values take the form of linguistic evaluation information. As shown in the example, for some attributes such as performance and service, it is so difficult to assess in numerical values. For some attribute such as price, even though they could be measured in numbers, people often take the form of words instead of numerical values for convenience. So it is valuable to study MADM problems with linguistic evaluations.

Secondly, the linguistic terms are transformed into triangular fuzzy numbers. And the linguistic evaluations are transformed into linguistic value function based on prospect theory. Then, the linguistic value function is expressed by means of gains and/or losses compared to the reference point.

Finally, the proposed method considers the decision makers’ psychological behavior in the process of decision making. According to prospect theory, the consumer’s aspiration-levels are considered as the reference points. By measuring the perceived difference between the attribute value and the reference point, the gains/losses and the prospect value of each alternative is obtained. Based on the prospect values, the consumer can rank alternatives and select the optimal choice.

An application in solar water heater selection problems is given to illustrate the feasibility of the proposed method.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (71371049), Jiangsu Provincial Graduate Research Innovation Plan (KYLX15_0190) and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1567).

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