Probability approximation to multi-attribute decision making method with stochastic attribute values

Abstract

Aiming at the Multiple Attribute Decision Making (MADM) problem that each attribute value is a stochastic variable instead of a real number, this paper develops a probability approximation method to deal with the randomness of the stochastic attribute values. We consider two general cases: (1) the distributions of all stochastic attribute values are known to decision-maker, and (2) the distributions of all stochastic attribute values are unknown, but decision-maker knows the possible value set of each stochastic attribute value. As for each case, firstly, we introduce the probability approach to approximate the stochastic attribute value. Through this approximation, we make sure that the larger the transformed attribute value of one alternative is, the better it is than others. Secondly, a new method is proposed to calculate the weights of stochastic attribute values under the uncertain environment. With this method, we derive the attribute weights from two parts differences: the intra-attributes differences, which denote the weights affected by the uncertainty of the stochastic attributes themselves, and inter-attributes differences which represent the weighted affected by the different attribute differences, and then calculate the two types of attribute weights. Finally, a numerical analysis is used to illustrate the effectiveness of the method proposed.

Keywords

Multi-attribute decision-making stochastic attribute values weights determination under stochastic environment

1 Introduction

In recent several decades, Multiple Attribute Decision Making (MADM) problems have gained extensive attention of many scholars. And the MADM methods have been widely applied in healthcare [1], transportation [2] and supplier evaluation [3] fields and so on. For an extensive review of this kind of literature, we refer the reader to Tzeng et al. [4] and Kahraman et al. [5]. Specifically, considering different attributes of alternatives, decision-maker could make a more reasonable decision. Up to now, researchers have developed various MADM methods to deal with the different data forms, such as linguistic value [6], interval number [7], triangular fuzzy number [8] and trapezoidal fuzzy number [9] and so on. With the increasingly development of economy, the market environment tends to be more and more competitive. The greatest challenge a decision-maker may encounter is the uncertainty, i.e. the uncertainty of attribute values. For example, Zarghami et al. [10] point out that water resources management face various kinds of uncertainty, such as environmental uncertainty, economic or technical data uncertainty.

For traditional MADM problems, some classical methods have been proposed, such as TOPSIS method [11 –13], VIKOR method [8 , 15] and PROMETHEE method [16 –18] and so on. Unfortunately, existing MADM methods are not able to deal with the randomness of the stochastic attribute values. Almost all the traditional MADM methods aim at the deterministic attribute values. Even though the development of interval number, triangular fuzzy number and trapezoidal fuzzy number, MADM problem with stochastic attribute values can not be solved by simplified into those data forms. As a result, some stochastic MADM (SMADM) methods are proposed [19 –25]. Those SMADM methods mainly focus on the stochastic dominance (SD) rules [23 , 26] through comparison and selection, i.e., first to identify whether there exists a SD rule and second to rank alternatives based on the determined SD rule. Other SMADM methods use Probability Theory. For example, Fan et al. [27] use Probability Theory to calculate the probabilities of pairwise comparisons of alternatives, and then classify those alternatives into superior, indifferent and inferior ones, and then rank the alternatives.

Standard procedures of dealing with MDAM problems include two important steps which have a critical effect on the decision-making result. So as the SMADM problems. Firstly, normalize original attribute values. In general, There are benefit and cost attributes in the MADM problems. And several methods have been developed to normalize a decision matrix into a corresponding standardized decision matrix [11, 28]. However, those normalization methods may cause information loss during the normalized process. Because of the complexity and randomness of the stochastic attribute values, traditional normalized methods may not suit for the MADM problem with stochastic attribute values. This leads to a new challenge in the SMADM field. Secondly, determining attribute weights is also crucial to the result of decision-making. Different attribute weights may result in different decision value. Those most commonly used methods include maximizing deviation method [29], hybrid weights [30] and entropy weights [31] and so on. Aggregating the stochastic attribute value needs to consider the uncertainty of the attributes themselves, i.e. determining the weights of attributes should take the randomness into consideration. And existing methods do not consider the randomness of attribute values. This is also a great challenge in SMADM field.

Even though many studies focus on SMADM problems, those studies almost only deal with the first step, i.e., normalize original attribute values. Since the uncertainty of the attributes themselves, the traditional methods of determining attribute weights may not suit for SMADM problems, However, there are limited literature studying the attribute weights determination under stochastic attribute values [32]. Besides, there are still some limitations when the existing methods are applied in practice. For example, those methods all assume the distributions of all stochastic attribute values are known to decision-maker, but in reality, the exact distributions of attribute values are hard to obtain, which leads to a troublesome decision making.

Hence in this paper, aiming at the SMADM problem that each attribute value is a stochastic variable rather than a real number, we extend the previous studies and develop a probability approximation method to deal with the randomness of the two general cases: (1) the distributions of all stochastic attribute values are known to decision-maker and each of them is mutual independent; (2) the distributions of all stochastic attribute values are unknown, but decision-maker knows the possible value set of each stochastic attribute value. As for each case, firstly, we use the probability to approximate the stochastic attribute value. Through this approximation, we make sure that the larger the transferred attribute value of one alternative is, the better it is than others. And then we discuss three common distributions in the first case. Secondly, considering not only the randomness of the stochastic attributes themselves, but the differences among attribute values. We divide the weights into two parts: the random variances part and the attribute differences part, the former can be regarded as intra-attributes differences which denotes the weights affected by the uncertainty of the stochastic attributes themselves and the latter can be regarded as inter-attributes differences which represents the weighted affected by the different attribute differences. Finally, through the result of an numerical analysis, we demonstrate the effectiveness of our method.

The remainder of this paper is organized as follows. We formulate our research problem and briefly introduce the traditional MADM method in Section 2. Our method is depicted in detail in Section 3, which includes the case that distributions of all stochastic attribute values are known and the case that distributions of all stochastic attribute values are unknown, each case includes the probability approximation method and weights determination. Then in Section 4, a numerical analysis is used to illustrate the effectiveness of proposed method. This paper ends with conclusion in Section 5.

2 Preliminaries

In this section, we will describe our research problem, and then review the traditional Multiple Attribute Decision Making(MADM) method. The symbols this paper used are described in Appendix A.

2.1 Description of research question

We first formulate our research problem. Consider a set of J alternatives to be evaluated, decision-maker needs to rank J alternatives under I attributes. As shown in Table 1, x_ij denotes the attribute value of alternative j under attribute i. However, In a specific decision-making environment, the value x_ij is not a deterministic number (includes real number, interval number and so on), but a stochastic variable. As discussed in the introduction section, there exists limited studies on this kind of decision-making problem. In this paper, we consider two general cases: (1) the distributions of all stochastic attribute values are known to decision-maker and each of them is mutual independent; (2) the distributions of all stochastic attribute values are unknown and those values are mutual independent, but decision-maker knows the possible value set of each stochastic attribute value.

Now with the I × J mutual independent stochastic variables, decision-maker needs to aggregate the attribute values of each alternative, and then makes a decision, i.e. ranks those J alternatives. So as for this kind of SMADM problem, we should deal with two challenges: the first is the randomness of the stochastic attribute values and the second is the determination of the attribute weights under the uncertain environment.

2.2 Multiple Attribute Decision Making (MADM) method

As for a traditional MADM problem, the attribute value x_ij is usually a real number, not a stochastic variable. Conventional method to rank the J alternatives contains three major procedures. Firstly, the decision-maker should normalize the primary data and obtain the normalized data which belongs to the interval between 0 and 1. In traditional MADM, we usually divide the attributes of the data into two types: cost attributes and benefit attributes, and there are some conventional methods to normalize the data. Assume the normalized data is show in Table 2. Secondly, decision-maker should determine the weights (w = w₁, w₂, ⋯ w_I) of each attribute, and there are also many methods. Finally, decision-maker can aggregate the data with the determined weights. The most common aggregation algorithm is weighted arithmetic mean method showing in Equation 1. Thus the decision-maker can rank these J alternatives based on the value of V = (V₁, V₂, ⋯ , V_J). $V_{j} = \sum_{i = 1}^{I} w_{i} \cdot r_{ij} j = 1, \dots, J .$ (1)

3 Proposed method

In this section, we consider two cases with known and unknown distributions of stochastic attribute values respectively. As discussed in the introduction, there are two challenges in dealing with the MADM problem which attribute values are stochastic variables. So in each case we first develop a probability approximation method to ‘normalize’ the stochastic attribute values. With the probability approximation, we transform the randomness of the stochastic attributes into probability comparison. This idea comes from PROMETHEEII. Specifically, the precise value (real value) is a special case of stochastic attribute values, the proposed method will simplify to PROMETHEEII when the stochastic attribute values all become real values. So our method could also handle the special case. Secondly, considering the uncertainty of stochastic attributes themselves and the difference of different attribute values, we propose a new method to determine the attribute weights. Finally, we can obtain a ranking outcome based on the probability approximation and determined weights.

3.1 The case under known distributions of stochastic attribute values

In this subsection, without loss of generality, we make an assumption to the stochastic variable of each attribute value to simplify decision-making problem: the distribution of each stochastic attribute value x_ij is known, i.e. f_ij is the density function and F_ij is the Cumulative Distribution Function of random variable x_ij, respectively. Hence we also know some parameters, including mean (μ_ij) and variance ( $σ_{ij}^{2}$ ). This assumption is reasonable because the poisson demand distribution is often assumed in some classical inventory models and the exponential distribution of service time is often assumed in the appointment scheduling problems. After making this assumption, we first develop a general probability approximation method to ‘normalize’ the stochastic attribute values. Here we discuss three common distributions. Finally, a new method is proposed to determine the attribute weights and aggregate the stochastic attribute values.

3.1.1 Probability approximation

In reality, the stochastic attribute x_ij can be any value of its support with a certain probability. Hence normalizing x_ij is an arduous work and in most cases, it is almost impossible. If we directly compare the mean of each stochastic attribute values, we may lose much information. Even with the same means, the stochastic attribute values may have different distributions and those values may have a great differences. In order to ‘normalize’ the stochastic attribute values, naturally, we compare these attribute values with probability, i.e. calculate the value Prob {x_ij > x_ik}. For each x_ij, the values Prob {x_ij > x_ik} , k = 1, 2 ⋯ J, k ≠ j can be calculated. Thus we can obtain I judgement matrixes of J × J dimension $\begin{matrix} P^{i} & = & (P_{jk}^{i})_{J \times J} = [\begin{matrix} P_{11}^{i} & P_{12}^{i} & \dots & P_{1 J}^{i} \\ P_{21}^{i} & P_{22}^{i} & \dots & P_{2 J}^{i} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ P_{J 1}^{i} & P_{J 2}^{i} & \dots & P_{JJ}^{i} \end{matrix}] \\ i = 1, 2, \dots, I . \end{matrix}$ where $P_{jk}^{i}$ means the probability that stochastic attribute x_ij larger than x_ik, i.e. $P_{jk}^{i} = Prob {x_{ij} > x_{ik}}$ . If we use f_ij (s) (a function of s) and f_ik (t) (a function of t) to denote the density function for stochastic attribute values x_ij and x_ik, respectively, then the probability $P_{jk}^{i}$ can be calculated by Equation 2

$\begin{matrix} P_{jk}^{i} & = & Prob {x_{ij} > x_{ik}} \\ = & \int_{- \infty}^{+ \infty} \int_{- \infty}^{s} f_{ij} (s) \cdot f_{ik} (t) dtds \\ = & \int_{- \infty}^{+ \infty} f_{ij} (s) \cdot F_{ik} (s) ds \\ or & = & \int_{- \infty}^{+ \infty} \int_{t}^{+ \infty} f_{ij} (s) \cdot f_{ik} (t) dsdt \\ = & \int_{- \infty}^{+ \infty} f_{ik} (t) \cdot \bar{F_{ij}} (t) dt . \end{matrix}$ (2) With Equation 2, we can obtain the probability value of each pair of stochastic attributes. Our purpose is to obtain the ‘normalized’ values of stochastic attributes. The basic idea is that the greater the probability of one stochastic attribute value larger than others, the greater the ‘normalized’ value it has. Conversely, the smaller the probability of one stochastic attribute value larger than others, the smaller the ‘normalized’ value it has. On this basis, we define the ‘normalized’ value of stochastic attribute as shown in Equation 3

$\begin{matrix} R_{ij} & = & \sum_{n = 1}^{J} P_{jn}^{i} - \sum_{m = 1}^{J} P_{mj}^{i} . \\ r_{ij} & = & R_{ij} - min_{i, j} R_{ij} . \end{matrix}$ (3) $\sum_{n = 1}^{J} P_{jn}^{i}$ means the total probability value of alternative j exceeding other alternatives under attribute i and $\sum_{m = 1}^{J} P_{mj}^{i}$ means the total probability value of alternative j surpassed by other alternatives under attribute i. And we define the exceeding probability subtract its surpassed probability of one alternative as the normalized stochastic attribute value. This probability approximation method makes sense intuitively, because each ‘normalized’ stochastic value is the difference of relative advantage and disadvantage under a corresponding attribute, thus represents the extent of the good or bad of one alternative, i.e. the larger the normalized value of one alternative is, the better it is than others. Finally, we transfer the R_ij into r_ij to make sure that all ‘normalized’ attribute values are non-negative. Through this transformation, the ranking results remain the same.

However, Equation 2 is a general formula to calculate the probability. In daily life, we often use some common distributions to deal with problems. Next we discuss three common distributions, including Uniform distribution, Exponential distribution and Normal distribution.

1. Probability Approximation under Uniform Distribution

In this paper, we use x ∼ D (A, B) to denote that random variable x follows a specific distribution D with mean A and variance B. In order to compare with other distributions, we only use the parameters mean (μ) and variance (σ²) to denotes a common distribution. Suppose arbitrary stochastic attribute value $x_{ij} \sim U (μ_{ij}, σ_{ij}^{2})$ , i.e., x_ij follows a Uniform Distribution with mean μ_ij and variance $σ_{ij}^{2}$ , then the density function can be denoted as Equation 4

$\begin{matrix} f_{ij} (x_{ij}) \\ = {\begin{matrix} \frac{1}{b_{ij} - a_{ij}} & a_{ij} < x_{ij} < b_{ij} \\ 0 & otherwise \end{matrix} \\ = {\begin{matrix} \frac{1}{2 \sqrt{3} σ_{ij}} & μ_{ij} - \sqrt{3} σ_{ij} < x_{ij} < μ_{ij} + \sqrt{3} σ_{ij} \\ 0 & otherwise \end{matrix} . \end{matrix}$ (4)

Arbitrary stochastic variable x_ij can be any value of the interval (a_ij, b_ij) with the same probability. Hence we should compare the intervals of each pair stochastic variable, i.e. for the stochastic variables x_ij and x_ik, we should compare the value of a_ij, a_ik, b_ik and b_ij. Then we can calculate the approximation probability $P_{jk}^{i}$ with Equation 5 $P_{jk}^{i} = {\begin{matrix} \frac{(2 b_{ij} - b_{ik} - a_{ik}) (b_{ik} - a_{ik})}{2 (b_{ij} - a_{ij}) (b_{ik} - a_{ik})} & a_{ij} < a_{ik} < b_{ik} < b_{ij} \\ 1 - \frac{(b_{ik} - a_{ij})^{2}}{2 (b_{ij} - a_{ij}) (b_{ik} - a_{ik})} & a_{ik} < a_{ij} \leq b_{ik} < b_{ij} \\ \frac{(b_{ij} - a_{ik})^{2}}{2 (b_{ij} - a_{ij}) (b_{ik} - a_{ik})} & a_{ij} < a_{ik} \leq b_{ij} < b_{ik} \\ \frac{(a_{ij} + b_{ij} - 2 a_{ik}) (b_{ij} - a_{ij})}{2 (b_{ij} - a_{ij}) (b_{ik} - a_{ik})} & a_{ik} < a_{ij} < b_{ij} < b_{ik} \\ 1 & a_{ik} < b_{ik} \leq a_{ij} < b_{ij} \\ 0 & a_{ij} < b_{ij} \leq a_{ik} < b_{ik} \end{matrix} .$ (5)

With the calculated $P_{jk}^{i}$ , we obtain the ‘normalized’ stochastic attribute value of each alternative based on the Equation 2.

2. Probability Approximation under Exponential Distribution

Now we consider the Exponential distribution. As for Exponential distribution, the mean equals to corresponding variance, i.e. μ = σ. Suppose arbitrary stochastic attribute value x_ij ∼ Exp (μ_ij), the corresponding density function can be denoted as Equation 6 $f_{ij} (x_{ij}) = {\begin{matrix} \frac{1}{μ_{ij}} e^{- \frac{x_{ij}}{μ_{ij}}} & x_{ij} > 0 \\ 0 & otherwise \end{matrix} .$ (6)

By using the result of Equation 2, we can easily derive the approximation probability $P_{jk}^{i}$ as in Equation 7

$\begin{matrix} P_{jk}^{i} & = & Prob {x_{ij} > x_{ik}} \\ = & \int_{- \infty}^{+ \infty} f_{ik} (t) \cdot \bar{F_{ij}} (t) dt \\ = & \int_{0}^{+ \infty} \frac{1}{μ_{ik}} e^{- \frac{t}{μ_{ik}}} \cdot e^{- \frac{t}{μ_{ij}}} dt \\ = & \frac{μ_{ij}}{μ_{ij} + μ_{ik}} . \end{matrix}$ (7)

Similarly, with the calculated $P_{jk}^{i}$ , we can obtain the ‘normalized’ stochastic attribute value of Exponential distribution based on the Equation 3.

3. Probability Approximation under Normal Distribution

Finally, we discuss the Normal distribution, i.e. $x_{ij} \sim N (μ_{ij}, σ_{ij}^{2})$ . It’s a hard job to directly calculate the approximation probability $P_{jk}^{i}$ by using integral method of Equation 2. With the assumption that all stochastic attribute values are mutual independent, we know that the distribution of the subtraction of two normal distributions is also a normal distribution. Let $z_{jk}^{i} = x_{ij} - x_{ik}$ , then $z_{jk}^{i} \sim N (μ_{jk}^{i}, σ_{jk}^{2 (i)})$ , where $μ_{jk}^{i} = μ_{ij} - μ_{ik}$ and $σ_{jk}^{2 (i)} = σ_{ij}^{2} + σ_{ik}^{2}$ . Thus the approximation probability $P_{jk}^{i}$ can be easily calculated with Equation 8

$\begin{matrix} P_{jk}^{i} & = & Prob {x_{ij} > x_{ik}} \\ = & Prob {z_{jk}^{i} > 0} \\ = & 1 - Φ_{jk}^{i} (0) . \end{matrix}$ (8)

Where Φ represents a cumulative distribution function (CDF) of a Normal distribution and $Φ_{jk}^{i} (0) = \frac{1}{\sqrt{2 π} σ_{jk}^{(i)}} \int_{- \infty}^{0} e^{- \frac{(t - μ_{jk}^{i})^{2}}{2 σ_{jk}^{2 (i)}}} dt .$

Similarly, with the calculated $P_{jk}^{i}$ , we can obtain the ‘normalized’ stochastic attribute value of Normal distribution based on the Equation 3.

3.1.2 Determining the attribute weights

As discussed in the introduction section, the weights of stochastic attributes have a critical impact on the decision-making result. However, existing SMADM studies just use traditional methods, such as maximizing deviation method and entropy weights,to determine the attribute weights. Since the uncertainty of stochastic attribute values, the traditional methods may not suit to determine the attribute weights. For stochastic attribute values, there exist two kind of differences among attributes: intra-attributes differences and inter-attributes differences. However, those existing methods do not consider the first kind of differences, i.e., the randomness of stochastic attributes themselves,or the variance of each stochastic attribute value. In order to obtain a set of reasonable weights, we should consider not only the randomness of the stochastic attributes themselves, but the differences among attribute values. Unfortunately, existing studies only consider the second part. Here we should stress that the randomness of attribute values is the special characteristic in this type of decision-making areas. Hence we divide the attribute weights into two parts: the random variances part w^d and attribute differences part w^s. The former denotes the weights affected by the uncertainty of the stochastic attributes themselves and the latter represents the weighted affected by the different attribute differences.

Firstly, we discuss the first kind of attributes differences, the weights of the random variances part. Here we assume that this kind of attribute weights are not the same for different alternatives, because some alternatives have little variances under all attributes while others may have a large variances under some attributes. Our basic idea is that one attribute should be given a larger weight if it has a little variance than others under a certain alternative. In other words, if the variance of one stochastic attribute value of a certain alternative is very small, we can regard this stochastic attribute value as a relative solid value under this alternative. Thus the smaller the variance, the more you can rely on the stochastic attribute value and the larger weight it should be allocated. Hence we use the parameter standard deviation (σ) to define the weights of the random variances part and the weights are negative correlate with the variances. So the weighs of the random variances part w^d can be defined as follows $w_{ij}^{d} = \frac{1 / σ_{ij}}{\sum_{k = 1}^{I} 1 / σ_{kj}} i = 1, \dots, I; j = 1, \dots, J .$ (9)

In Equation 9, we use the reciprocal of standard deviation to denote the weight of the corresponding attribute under a certain alternative. This kind of idea makes sense and indicates that the more stable attribute should be given a larger weight.

Next we discuss the second kind of attributes differences, weights of the attribute differences part. In general, if the values of one attribute have little difference (the extremely case is that all the attribute values under one attribute are the same), this attribute should be given a relative small weight, since it has a little contribution to obtain the ranking result. Conversely, the attribute which has a bigger differences among whose attribute values should be given a relative larger weight. In order to depict such relationship, we use the mean of stochastic attribute values to deal with the average differences among different attributes. Hence the weighs of attribute differences part w^s can be defined as follows $\begin{matrix} w_{i}^{s} = \frac{Δ_{i}}{\sum_{k = 1}^{I} Δ_{k}} i = 1, \dots, I . \end{matrix}$ (10)

Where $Δ_{i} = \frac{1}{J * (J - 1)} Σ_{j = 1}^{J} Σ_{k = 1}^{J} | μ_{ij} - μ_{ik} | i = 1, \dots, I .$

In Equation 10, Δ_i means the average differences among all stochastic attribute values of the ith attribute. And we use the ratio of one attribute’s differences to the whole attributes’ differences as the the weight. This method makes sure that the bigger the difference of one attribute values, the larger the weight it has.

Finally, we incorporate the two parts weights together. We introduce the coefficient λ to denote the ratio of the two parts. Hence the weights of attributes can be defined as

$\begin{matrix} w_{ij} & = & λ w_{ij}^{d} + (1 - λ) w_{i}^{s} \\ i = 1, \dots, I; j = 1, \dots, J . \end{matrix}$ (11)

The coefficient λ can be regarded as risk coefficient. If the decision maker is risk-averse, then he can choose a small λ, i.e., λ < 0.5; On the contrary, if the decision maker is risk preference, then he can choose a large λ, i.e., λ > 0.5. Through this combination, we can capture the attribute differences from both inter-attributes and intra-attributes. Most importantly, this kind of differences of intra-attributes is the first time to be considered in this paper. Since the differences of intra-attributes is the inherent character of stochastic attribute values, we should not ignore this kind of differences.

3.2 The case under unknown distributions of stochastic attribute values

In this subsection, we relax the assumption that the distribution of each stochastic attribute value x_ij is known which is a valid assumption in many situations. Fitting distributions for a stochastic variable requires a large amount of data and sometimes it is hard to estimate because of lack of data. This motivates us to study the MADM problem that the distributions of stochastic attribute values are unknown. In order to simplify decision-making problem, we assume that the possible value sets of stochastic attribute values is known to the decision-maker and he makes a decision based on some historical data, i.e. the possible value sets of stochastic attribute values. Similarly, with this assumption, we first develop a specific probability method to approximate the ‘normalized’ stochastic attribute values. And then we propose a new method to determine the attribute weights. On this basis, we aggregate the stochastic attribute values with the ‘normalized’ value and determined weights.

3.2.1 Probability approximation

As for the case that the distributions of stochastic attribute values are unknown, we do not know the exactly distribution of each stochastic attribute value, but know the possible value set of each stochastic attribute value, i.e. $x_{i, j} \in S_{ij} = {d_{ij}^{1}, \dots, d_{ij}^{h}, \dots}$ and $x_{i, k} \in S_{ik} = {d_{ik}^{1}, \dots, d_{ik}^{l}, \dots}$ . This means the stochastic attribute value x_ij can be any one in the set S_ij. In order to ‘normalize’ the stochastic attribute value, we follow the idea used in the previous case. So we also calculate the probability that stochastic attribute value x_ij better than x_ik, i.e. $P_{jk}^{i} = Prob {x_{ij} > x_{ik}}$ . Firstly, as for an arbitrary element $d_{ij}^{h} \in S_{ij}$ and the set S_ik, we define a set $S_{jk}^{ih}$ as in Equation 12

$\begin{matrix} S_{jk}^{ih} & = & {d_{ik}^{l} : d_{ij}^{h} > d_{ik}^{l}, d_{ik}^{l} \in S_{ik}, l = 1, \dots, | S_{ik} |} \\ d_{ij}^{h} \in S_{ij}, h = 1, \dots, | S_{ij} | . \end{matrix}$ (12)

Where the symbol |S| means the number of the elements in set S.

The set $S_{jk}^{ih}$ includes these elements in set S_ik which smaller than $d_{ij}^{h}$ in set S_ij. In other words, for two stochastic attribute value x_ij and x_ik, x_ij takes a value in set S_ij and x_ik takes a value in set S_ik. Now for an arbitrary element $d_{ij}^{h} \in S_{ij}$ , we can find a set $S_{jk}^{ih}$ in which all elements are smaller than $d_{ij}^{h}$ . And we have the relationship $S_{jk}^{ih} \subset S_{ik}$ . So we can define the set $S_{jk}^{ih}$ and set $S_{kj}^{il}$ . With a pair set of stochastic attribute value x_ij and x_ik, we can define the probability $P_{jk}^{i}$ and $P_{kj}^{i}$ as in Equation 13

$\begin{matrix} P_{jk}^{i} & = & \frac{\sum_{h = 1}^{| S_{ij} |} | S_{jk}^{ih} |}{| S_{ij} | \cdot | S_{ik} |} \\ and P_{kj}^{i} & = & \frac{\sum_{l = 1}^{| S_{ik} |} | S_{kj}^{il} |}{| S_{ij} | \cdot | S_{ik} |} \\ i = 1, \dots, I; j = 1, \dots, J . \end{matrix}$ (13)

As for the first equation of 13, the denominator denotes all the possible combinations of the pair stochastic attribute value x_ij and x_ik, the numerator denotes the cumulative number that values in set S_ij larger than values in S_ik. Hence we can use this ratio to approximate the probability. Here we should emphasize that $P_{jk}^{i} + P_{kj}^{i} = 1$ not always hold, because there may exist same value in both sets. However, this equation holds in the previous case.

Secondly, following the idea used in previous case, we can also use Equation 3 to obtain the ‘normalized’ stochastic attribute values.

3.2.2 Determining the attribute weights

Similarly, we divide the attribute weights into two parts, the random variances part w^d affected by the uncertainty of the stochastic attributes themselves and the attribute differences part w^s affected by the different attribute differences. As for the former part, we also assume that they are not the same for different alternatives. In order to depict the relationship that one attribute should be given a larger weight if it has less variance than others under a certain alternative, we use the number of the elements in set S_ij, rather than the variance of x_ij, to illustrate the random variance of stochastic attribute values themselves. So the weighs of the random variances part w^d can be defined in Equation 14 $w_{ij}^{d} = \frac{1 / | S_{ij} |}{\sum_{k = 1}^{I} 1 / | S_{kj} |} i = 1, \dots, I; j = 1, \dots, J .$ (14)

In Equation 14, we use the reciprocal of the number of elements in set S_ij to denote the weight of the corresponding attribute under a certain alternative. This means that the more elements one set has, the more ‘unstable’ the corresponding stochastic attribute value is, the smaller weight it should has. And this idea makes sense.

As for the the weights of the attribute differences parts, we can also use the method used in previous case. Based on the value in the corresponding possible value set, we can calculate the mean of each stochastic attribute value. And then we can obtain the weights of the attribute differences parts w^s based on Equation 10.

After calculating the weights of random variance part and attribute differences part, we can incorporate the two parts weights by using Equation 11.

3.3 Probability approximation to multi-attribute decision making method

Based on the above analysis, the steps of the multi-attributes decision-making method of stochastic attribute values are given as follows:

Step 1: Identify the type of the MADM problem, if it belongs to case with known distributions, then goes to Step 2; if it belongs to the case with unknown distributions, then goes to Step 4.

Step 2: Obtain the distributions of each attribute values, goes to Step 3.

Step 3: Transfer the primary stochastic attribute values into probability with Equation 2, if there exists some common distributions,such as the Uniform distribution, the Exponential distribution or the Normal distribution, we can use Equations 5, 7 or 8 instead, goes to Step 5, and then goes to Step 6.

Step 4: Calculate the approximation probability by defining the corresponding sets and using Equation 13, goes to Step 5, and then goes to Step 7.

Step 5: ‘Normalize’ the stochastic attribute values with probability approximation based on Equation 3.

Step 6: Calculate the weights of the random variances part based on Equation 9, and then goes to Step 8.

Step 7: Calculate the weights of the random variances part based on Equation 14, and then goes to Step 8.

Step 8: Calculate the weights of attribute differences part based on Equation 10, and then incorporate the two parts together with Equation 11, and then goes to Step 9.

Step 9: Rank alternatives based on the Equation 1.

4 Numerical analysis

In order to illustrate the effectiveness of the method proposed, we present a numerical analysis to analyse the case that the distributions of stochastic attributes are known. In this section, suppose a decision-making problem has three alternatives (denote as A₁, A₂ and A₃, J = 3) and four attributes (I = 4). As for this problem, each attribute value is a stochastic variable instead of a real number. And we know the distribution of each stochastic attribute value. For comparison, we discuss three cases which include Uniform distribution, Exponential distribution and Normal distribution, respectively. In addition, we assume that the standard deviations are the same with the corresponding means in terms of comparing with Exponential distribution, i.e. μ = σ. The data of the mean and variance under each attribute is shown in Table 3:

Firstly, we calculate the weighs of attribute under each alternative. And we obtain the weights of attribute differences part based on Equation 10: w^s = (0.1020, 0.3878, 0.1837, 0.3265). Through Equation 9, the weights of the random variances part w^d are shown in Table 4.

Now we incorporate the two parts weights together with Equation 11. Assume the coefficient is: λ = 0.5, the finally weight of each attribute under each alternative is shown in Table 5.

After calculating attributes weights of alternatives, we should ‘normalize’ the stochastic attribute values. We consider three common cases: Uniform distribution, Exponential distribution and Normal distribution, respectively.

1. Uniform Distribution

By using Equation 5, we can calculate the approximated probabilities of the stochastic attribute values under Uniform distribution as shown in Table 6. Then the ‘normalized’ attribute values of each alternative can be obtained in Table 7.

With the normalized attribute values and incorporated weights, we can obtain the ranking values of these alternatives based on Equation 1: V^U = (0.6396, 0.6193, 0.7451). Hence A₃ ≻ A₁ ≻ A₂.

2. Exponential Distribution

Similarly, by using Equation 5, the approximated probabilities of the stochastic attribute values under Exponential distribution are calculated in Table 8. And the ‘normalized’ attribute values of each alternative are shown in Table 9.

With the normalized attribute values in Table 9 and the incorporated weights in Table 5, we can use the Equation 1 to obtain the ranking result: V^E = (0.7836, 0.8181, 0.9538). Hence A₃ ≻ A₂ ≻ A₁.

3. Normal Distribution

By using Equation 5, the approximated probabilities of the stochastic attribute values and the ‘normalized’ attribute values of each alternative are shown in Tables 10, 11, respectively.

Thus we could obtain the ranking result by those corresponding information: V^N = (0.6065, 0.6264, 0.7330). Hence A₃ ≻ A₂ ≻ A₁.

Comparing with the results from three different distributions, we find that the ranking results are different among three distributions. The alternative 3 is the best one among three distributions, alternative 1 is the worst one in the Exponential distribution and the Normal distribution, while it is not the worst one in the Uniform distribution. This outcome indicats that different distributions may result in different ranking results, even the different distributions have the same parameter μ and σ². Hence it’s not a wise method to compare alternatives by simply using means directly.

5 Conclusion

With the increasingly development of social economy, the market environment becomes more and more competitive. As for MADM area, the decision-making problems are transforming from deterministic attribute values into stochastic attribute values. This leads to a great challenge for decision-maker with uncertain attribute values. Aiming at the SMADM problem that each attribute value is a stochastic variable rather than a real number, this paper develops a probability approximation method to deal with the randomness of the stochastic attribute values under two cases: (1) the distributions of all stochastic attribute values are known to decision-maker; (2) the distributions of all stochastic attribute values are unknown, but decision-maker knows the possible value set of each stochastic attribute value. To the best of our knowledge, the method is the first of its kind. The contributions of this paper are as follows:

Firstly, as for each case, we use the probability to approximate the attribute value by comparing with the other stochastic attribute values. By this kind of transformation, the stochastic attribute values can be transferred into a deterministic attribute values which could represent the primary alternative preference. So we can make sure that the larger the transferred attribute value of one alternative is, the better it is than others. As for three common distributions, Uniform distribution, Exponential distribution and Normal distribution, we also present the specific transformation method instead of the general one.

Secondly, we develop a new method to deal with the attributes weights under the uncertain environment of stochastic attribute values. Because of the uncertainty of stochastic attribute values themselves, we should consider not only the randomness of the stochastic attributes themselves, but also the differences among attribute values. Hence we divide the weights into two parts. The random variances part denotes the weights affected by the uncertainty of the stochastic attributes themselves and the attribute differences part represents the weighted affected by the different attribute differences. With this new method, we can obtain a relative reasonable decision-making weights.

Finally, through the numerical analysis, we find that different distributions even with same parameters may result in different ranking results. So we can conclude that it’s not reasonable to make a decision based on the means of stochastic attribute values directly. And this outcome demonstrates the effectiveness of our method from another aspect.

However, there are also some limitations in this paper. Since the decision-making problems become more and more complicated, we need to update the history data, in order to obtain more accurate distributions of stochastic attribute values. On this basis, we should make a dynamic decision instead of a static decision. How to deal with such stochastic MADM problem in dynamic environment is also a great challenge. Besides, when making a decision, different decision-maker has different risk preference. This paper does not consider the risk preference problem which is an interesting perspective and worth studying. we will deal with those limitations in the future work.

Footnotes

Appendix A

Symbol descriptions

Symbol	Description
A _j	The jth alternative
I	The number of attributes
J	The number of alternatives
x _ij	Stochastic variable, the attribute value of alternative j under attribute i
S _ij	The Possible value set of Stochastic variable x_ij
$d_{ij}^{h}$	The value of the hth element in set S_ij
$S_{jk}^{ih}$	A subset of set S_ik, all elements in set $S_{jk}^{ih}$ are smaller than $d_{ij}^{h}$
\|S_ij\|	The number of elements in set S_ij
r _ij	Real number, the normalized attribute value of alternative j under attribute i
f _ij	The probability distribution function(P.D.F) of stochastic attribute value x_ij
F _ij	The cumulative distribution function(C.D.F) of stochastic attribute value x_ij
μ_ij	The mean of stochastic attribute value x_ij
$σ_{ij}^{2}$	The variance of stochastic attribute value x_ij
P ⁱ	The probability approximation judgement matrix of the i^th attribute
$P_{jk}^{i}$	The probability that x_ij larger than x_ik
Φ	The cumulative distribution function(C.D.F) of a Normal distribution
w	The attribute weights
w ^d	The attribute weights of random variances part
w ^s	The attribute weights of attribute differences part
$w_{ij}^{d}$	The attribute weights of attribute i under alternative j to the random variances part
$w_{i}^{s}$	The attribute weights of attribute i to the attribute differences part
λ	The coefficient of the attribute differences part weights
V	The vector of ranking values
V ^U	The vector of ranking values under Uniform distribution
V ^E	The vector of ranking values under Exponential distribution
V ^N	The vector of ranking values under Normal distribution

Acknowledgments

The authors are grateful to the anonymous referees and associate editor for their careful and obviously time-consuming and constructive reading of this paper. They also thank Jingcheng Yang and Panpan Wang for their discussion and valuable suggestions on improving this paper. This paper is supported by the National Natural Science Foundation of China (71671189, 71361021, 41661116) and Social Science Research Plan Project of Jiangxi Province China (15ZQZD01).

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