A fuzzy multiple attribute decision making method based on possibility degree

Abstract

A fuzzy multiple attribute decision making method is investigated, there the weights are given by interval numbers, the qualitative attribute values are first given by linguistic terms and then are represented as the form of triangular fuzzy numbers, and the quantitative attribute values are given by the form of triangular fuzzy numbers. A possibility degree formula for the comparison between two trapezoidal fuzzy numbers is proposed. Then, using this possibility degree formula, possibility degree matrices are built and the central dominance of one alternative outranking all other alternatives is defined under one attribute. According to the ordered weighted average (OWA) operator, an approach is presented to aggregate the possibility degree matrices based on attributes and then the most desirable alternative is selected. This fuzzy multiple attribute decision making method is used in the field of financial investment evaluation, and the set of attributes of the decision making program is built by financial analyses and accounting reports in the same industry. Finally, numerical example is provided to demonstrate the practicality and the feasibility of the proposed method.

Keywords

Possibility degree multiple attribution decision making trapezoidal fuzzy number investment options OWA operators

1 Introduction

The multiple attribute decision making (MADM) is one of the important areas of operational research. The decision process of choosing an appropriate alternative often has to take several attributes into consideration, such as goals, risks, requirement benefits and also limited resources. Several qualitative and quantitative attributes may affect mutually when alternatives are evaluated, which may make the selection process complex and challenging. MADM has become a hot topic in decision science and attracted broad studies from both theoretical and applied points of views. Recently, researchers have proposed many methods for the MADM with complete certain information. However, in fact, decision makers can not have enough knowledge about the alternative versus attributes. Moreover, crisp data is insufficient to model real-life applications under numerous conditions. Human opinions involving preferences are usually ambiguous and cannot precisely determine their preferences with numerical data. So some problems present quantitative aspects which can be assessed by means of precise numerical values with the help of fuzzy theory [1]. Fuzzy sets theory is a powerful tool to deal with fuzzy expressions and imprecise data that can protect the decision maker’s privacy or avoid influencing each other. In fact, in the real world much knowledge is fuzzy rather than precise. However, some problems present also qualitative aspects that are complex to assess by means of these numerical values. Thus, a more rational and practical approach is to utilize linguistic evaluations on the basis of linguistic terms [2, 3] instead of numerical data in order to suppose the ratings of the alternatives by decision makers versus attributes in decision making problems. Usually, the weights of attributes are not arbitrary, so using interval numbers is a more exact method. Moreover, the research on the fuzzy multiple attribute decision making with preference information on alternatives is of important theoretical significance and practical value. But given the nature of all attributes, decision makers find it extremely difficult to express the strength of their preferences on alternatives and to provide exact pair-wise comparison judgment in qualitative decision making with in the artificial intelligence and decision analysis communities.

Among several alternative approaches for modeling uncertainty, possibility theories [4 –6] have been extensively studied. Tanino T. indicated the fuzzy complementary judgment matrix for the study of fuzzy preference orderings in decision making in 1984. Facchinetti [7] proposed the method for ranking fuzzy triangular numbers, and proposed the possibility degree. Recently, the possibility degree [8 –10] are proposed to rank fuzzy numbers combining with possibility degree matrix.

Investment evaluation methods play important roles in today’s competitive environment. However, in many practical situations, the human preference model is uncertain and decision makers might be reluctant or unable to assign exact value to choose the best alternative. Given the qualitative and subjective attributes, decision makers find it extremely difficult to express the strength of their preferences or build comparison judgment matrix. Investment decision making process, which involves a number of uncertainties and is affected by subjective and objective factors, is a typical procedure of fuzzy multiple attribute decision making. In many real-life cases, such as negotiation processes, the high technology project investment of venture capital firms, decision makers cannot give exact attribute weights but can provide value ranges, which can take the form of interval numbers, and the information about attribute values usually takes the form of linguistic variables or triangular fuzzy numbers. Many studies have studied the FMADM problem with incomplete weight information [11].

When an industry meets the development needs of the community or national policy support and has received a great deal of attention, investors usually require an analysis of the investment value of the company based on financial data of the original and the various periods in the past.

Traditional fuzzy multiple attribute decision making (FMADM) usually take various normalization [12] and defuzzification [13] methods and decision makers can more thoroughly understand the risks they face under different circumstances. Imprecision may arise from a variety of reasons, such as incomplete information, unobtainable information, unquantifiable information and partial ignorance, so these evaluation methods are uncertain and imprecise. In such cases, it is useful to give an effective method which can as more as possible to protect the uncertainty of decision information.

In this paper, a fuzzy multiple attribute decision making method is developed in which the attributes are given by interval numbers and the opinions of experts are given by fuzzy numbers and linguistic settings. A possibility degree formula for the comparison between two trapezoidal fuzzy numbers is proposed. Then based on OWA aggregation operators, the most suitable alternative is selected. The aim of this paper is to establish an optimal model to solve the above problem.

The rest of the paper is organized as follows: Section 2 illustrates the solution approach; Section 3 describes the experimental results; Section 4 concludes this paper.

2 The solution approach

2.1 Preliminaries

In the following, we briefly review some basic definitions and notations which will be used throughout the paper until otherwise stated.

A fuzzy set $\tilde{a}$ in a universe of discourse X is characterized by a membership function $f_{\tilde{a}} (x)$ , which associates with each element x in X a real number in the interval [0, 1]. The function $f_{\tilde{a}} (x)$ is termed the grade of membership of x in X.

Let $\tilde{a} = [a_{1}, a_{2}] = {x | a_{1} \leq x \leq a_{2}, a_{1}, a_{2} \in R}$ , $\tilde{a}$ is an interval number. Specially, if a₁ = a₂, then $\tilde{a}$ is a real number.

Definition 1. A triangular fuzzy number $\tilde{a}$ can be defined by a triplet (a, b, c), its membership function $f_{\tilde{a}} (x)$ is given by $f_{\tilde{a}} (x) = {\begin{matrix} 0 & x < a \\ f_{\tilde{a}}^{L} (x) & x \in [a, b] \\ f_{\tilde{a}}^{R} (x) & x \in (b, c] \\ 0 & x > c \end{matrix}$ where $f_{\tilde{a}}^{L} (x) = (x - a) / - (b - a), f_{\tilde{a}}^{R} (x) = (x - c) / - (b - c) .$

Definition 2. For a trapezoidal fuzzy number $\tilde{a} = (a, b, c, d)$ , its membership function $f_{\tilde{a}} (x)$ is given by $f_{\tilde{a}} (x) = {\begin{matrix} f_{\tilde{a}}^{L} (x) & x \in [a, b) \\ 1 & x \in [b, c] \\ f_{\tilde{a}}^{R} (x) & x \in (c, d] \\ 0 & otherwise \end{matrix}$ where $f_{\tilde{a}}^{L} (x) = (x - a) / - (b - a), f_{\tilde{a}}^{R} (x) = (x - d) / - (c - d) .$

The product of two fuzzy numbers should be a more imprecise number, so we introduce a lemma as follows:

Lemma 1. [14] The product of an interval number and a triangular fuzzy number is a trapezoidalnumber.

Suppose $\tilde{a} = [a_{1}, a_{2}]$ is a random interval number and $\tilde{B} = (l, m, u)$ is a random fuzzy triangular number, then the product of $\tilde{a}$ and $\tilde{B}$ is the following trapezoidal number $a \tilde{B} = [a_{1}, a_{2}] (l, m, u) = (a_{1} l, a_{1} m, a_{2} m, a_{2} u)$ (1)

As aforementioned, when some problems present qualitative aspects that is complex to assess by means of numerical values, a linguistic approach can be first used to obtain a better solution, and then we transform nature linguistic values into triangle fuzzy numbers, as presented in Table 1.

2.2 The possibility degree of two trapezoidal fuzzy numbers

Definition 3. Suppose $\tilde{r} = (r_{1}, r_{2}, r_{3}, r_{4})$ and $\tilde{s} = (s_{1}, s_{2}, s_{3}, s_{4})$ are two trapezoidal fuzzy numbers, let $\begin{matrix} p (\tilde{r} \geq \tilde{s}) \\ = λ max {1 - max [\frac{s_{2} - r_{1}}{s_{2} - s_{1} + r_{2} - r_{1}}, 0], 0} \\ + (1 - λ) max {1 - max [\frac{s_{4} - r_{3}}{s_{4} - s_{3} + r_{4} - r_{3}}, 0], 0} \end{matrix}$ (2) and if $\frac{s_{2} - r_{1}}{s_{2} - s_{1} + r_{2} - r_{1}} = \frac{0}{0} or \frac{s_{4} - r_{3}}{s_{4} - s_{3} + r_{4} - r_{3}} = \frac{0}{0},$ then $p (\tilde{r} \geq \tilde{s}) = \frac{1}{2} .$

We call $p (\tilde{r} \geq \tilde{s})$ the possibility degree of $\tilde{r}$ prefer to $\tilde{s}$ , and call $\tilde{r} \geq \tilde{s}$ the order relationship between $\tilde{r}$ and $\tilde{s}$ , where λ ∈ [0, 1].

NOTE: λ is an index that reflects decision maker’s risk-bearing attitude. If λ < 0.5, then the decision maker is a risk averter. If λ = 0.5, then the attitude of the decision maker is neutral to the risk. If λ > 0.5, then the decision maker is a risk lover.

In particular, when λ = 1, $p (\tilde{r} \geq \tilde{s})$ can be called pessimistic possibility degree, when λ = 0, $p (\tilde{r} \geq \tilde{s})$ can be called optimistic possibility degree at the time. In general, λ can be given by decision makers directly. Obviously, the possibility degree formula satisfies the following properties:

$0 \leq p (\tilde{r} \geq \tilde{s}) \leq 1$

If r₄ ≤ s₁, then $p (\tilde{r} \geq \tilde{s}) = 0$

If s₄ ≤ r₁, then $p (\tilde{r} \geq \tilde{s}) = 1$

$p (\tilde{r} \geq \tilde{s}) + p (\tilde{r} \leq \tilde{s}) = 1$ , especially, if $\tilde{r} = \tilde{s}$ , then $p (\tilde{r} \geq \tilde{s}) = p (\tilde{s} \geq \tilde{r}) = 0.5$

2.3 The FMADM problem

Let X ={ X₁, X₂, ⋯ , X_n } be the set of alternatives and G ={ G₁, G₂, ⋯ , G_m } be the set of attributes. There are t years’ referenced financial data and μ = (μ₁, μ₂, ⋯ , μ_t)^T be the weight vector of years, where μ_k ≥ 0 (k = 1, 2, ⋯ , t) and $\sum_{i = 1}^{t} μ_{i} = 1$ .

Let ${\hat{A}}^{(k)} = ({\hat{a}}_{ij}^{(k)})_{n \times m}$ be the decision matrix in the kth year, where $({\hat{a}}_{ij}^{(k)}) = (a_{lij}^{(k)}, a_{mij}^{(k)}, a_{uij}^{(k)})$ is an attribute value, given by the decision makers for the alternative X_i ∈ X with respect to the attribute G_j ∈ G in the kth year.

Let ${\hat{λ}}^{k} = (λ_{1}^{k}, λ_{2}^{k}, \dots, λ_{m}^{k})^{T}$ be the weight vector of attributes in the kth year, where $\sum_{j = 1}^{m} λ_{j}^{k} = 1$ . We cannot decide $λ_{j}^{k}$ directly, but we know $λ_{j}^{k} \in [c_{i}^{k}, d_{i}^{k}], 0 \leq c_{j}^{k} \leq d_{j}^{k} \leq 1 (j = 1, 2, \dots, m) .$

As $c_{j}^{k} \leq λ_{j}^{k} \leq d_{j}^{k}$ and $λ_{1}^{k} + λ_{2}^{k} + \dots + λ_{m}^{k} = 1$ , so $c_{j}^{k}$ and $d_{j}^{k}$ must satisfy the following inequalities: $(c_{1}^{k} + c_{2}^{k} + \dots + c_{m}^{k}) + max {d_{j}^{k} - c_{j}^{k} | 1 \leq j \leq m} \leq 1$ and $(d_{1}^{k} + d_{2}^{k} + \dots + d_{m}^{k}) + max {d_{j}^{k} - c_{j}^{k} | 1 \leq j \leq m} \geq 1 .$

2.4 The building of the weighted decision matrix and the possibility degree matrix

The construction of the weighted decision matrix in the kth year is as follows: $\begin{matrix} {\hat{B}}^{(k)} = ({\hat{b}}_{ij}^{(k)})_{n \times m} = (λ_{j}^{k} {\hat{a}}_{ij}^{(k)})_{n \times m} \\ = ((c_{j}^{k} {\hat{a}}_{lij}^{(k)}, c_{j}^{k} {\hat{a}}_{mij}^{(k)}, d_{j}^{k} {\hat{a}}_{mij}^{(k)}, d_{j}^{k} {\hat{a}}_{uij}^{(k)}))_{n \times m} \end{matrix}$ (3)

The construction of the weighted decision matrix of all years is as follows:

$\begin{matrix} C = (c_{ij})_{n \times m} = {(\sum_{k = 1}^{t} μ_{k} {\hat{b}}_{ij}^{(k)})}_{n \times m} \\ = ((\sum_{k = 1}^{t} μ_{k} c_{j}^{k} {\hat{a}}_{lij}^{(k)}, \sum_{k = 1}^{t} μ_{k} c_{j}^{k} {\hat{a}}_{mij}^{(k)}, \sum_{k = 1}^{t} μ_{k} d_{j}^{k} {\hat{a}}_{mij}^{(k)}, \\ {\sum_{k = 1}^{t} μ_{k} d_{j}^{k} {\hat{a}}_{uij}^{(k)}))}_{n \times m} \end{matrix}$ (4)

Many FMADM methods convert generalized fuzzy numbers into normal fuzzy through normalization process. However, Gupta and Kaufmann [15] indicated that the normalization process has a serious disadvantage, that is, the missing decision information. To overcome this shortcoming, using the possibility degree formula, we can build possibility matrices, whose every number indicates the degree of one alternative is superior to other alternatives based on each attribute.

Let R_s = {r_1s, r_2s, ⋯ r_ns} , (s = 1, 2, ⋯ m) be the collection built by the sth rank in R, build the probability matrix $P_{s} = (p_{ij}^{s})_{n \times n}, (s = 1, 2, \dots, m)$ by comparing any two elements in R_s.

For benefit index in the sth rank: $p_{ij}^{s} = p (r_{is} \geq r_{js}), (i, j = 1, \dots n; s = 1, \dots m)$ (5)

For cost index in the sth rank: $p_{ij}^{s} = p (r_{js} \geq r_{is}), (i, j = 1, \dots n; s = 1, \dots m)$ (6)

From the possibility degree formula, we can derive that the possibility degree matrix is also the fuzzy complementary judgment matrix.

2.5 The procedure for selecting the best alternative based on OWA operators

Yager introduced an ordered weighted averaging (OWA) [16 –19] operator, which is defined as follows:

An OWA operator of dimension n is a mapping OWA: Rⁿ→ R that has associated n vector ω = (ω₁, ω₂, ⋯ , ω_n)^T, such that $ω_{j} \in [0, 1], j = 1, 2, \dots, n, \sum_{j = 1}^{n} ω_{j} = 1$ .

Furthermore, ${OWA}_{ω} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} ω_{j} b_{j},$ where b_j is the jth largest in the collection {a₁, a₂, ⋯ , a_n}.

The OWA operator is suitable for aggregating the information taking the form of preference relation. The decision making process is described specifically as follows:

Step 1. Compute the weight vector ω = (ω₁, ω₂, ⋯ , ω_m)^T of the OWA operator aggregated by probability degree matrices of all attributes, where ω_k ≥ 0, $\sum_{k = 1}^{m} ω_{k} = 1$ $ω_{k} = Q (\frac{k}{m}) - Q (\frac{k - 1}{m}), (k = 1, \dots m)$ (7) and Q (r) is a fuzzy quantified operator, which is $Q (r) = {\begin{matrix} 0, & r < a \\ \frac{r - a}{b - a}, & a \leq r \leq b \\ 1, & r > b \end{matrix}$ (8) where a, b, r ∈ [0, 1].

In the principle of “at least half”, the respective parameters are (a, b) = (0, 0.5).

Step 2. Using an OWA operator Φ_Q, we derive a collective matrix $P^{*} = (p_{ij}^{*})_{n \times n}$ aggregated by probability degree matrices. $p_{ij}^{*} = Φ_{Q} (p_{ij}^{1}, p_{ij}^{2}, \dots, p_{ij}^{m}) = \sum_{k = 1}^{m} ω_{k} b_{ij}^{k} \forall i, j, i \neq j .$ (9) where Φ_Q is an OWA operator, $b_{ij}^{k}$ is the kth largest element in the collection ${p_{ij}^{1}, p_{ij}^{2}, \dots, p_{ij}^{m}}$ .

Step 3. Compute maximum entropy weight vector $ω^{*} = (ω_{1}^{*}, ω_{2}^{*}, \dots, ω_{n - 1}^{*})^{T}$ , which satisfies $ω_{q}^{*} \geq 0$ and $\sum_{q = 1}^{n - 1} ω_{q}^{*} = 1$ , where $ω_{q}^{*} = Q (\frac{q}{n - 1}) - Q (\frac{q - 1}{n - 1}) (q = 1, 2, \dots, n - 1)$ (10)

Fuzzy quantified operator Q (r) can be given by Equation (8).

In the principle of “as many as possible”, the respective parameters are (a, b) = (0.3, 0.8).

Step 4. Based on OWA operators and guided by “as many as possible”, compute the degree d_i of x_i prefer to other alternatives, we can derive that $d_{i} = Φ_{Q} (p_{ij}^{*}, j = 1, 2, \dots, n; j \neq i) = \sum_{q = 1}^{n - 1} ω_{q}^{*} c_{i}^{q}$ (11) where $c_{i}^{q}$ is the qth largest element in the collection ${p_{ij}^{*} | j = 1, 2, \dots n; j \neq i}$ . Choose the best alternative based on the ranking value d_i, the larger the d_i, the better the ith alternative.

3 The application in investment decision

In the process of enterprise investment decision, the choice of the decision attribute set and the solution set is to build index system of the stock investment value and give it comprehensive evaluation.

3.1 The set of decision alternatives and attributes

As different industries have different characteristics, what reflected in the financial indicators will present different criteria. So to build alternatives in the same industry is more conductive to comparison on the merits of the investment. Suppose an enterprise wants to give investment decision, and the selected scheme set within the same industry is {X₁, X₂, X₃, X₄, X₅}.

The basic situation of the companies’business activities is reflected from the companies’ profit ability, growth ability and cost. In this paper, we suppose the ability of the three aspects has equal weight. The specific indicators are as follows:

G₁: strategic fitness,

G₂: technical ability,

G₃: project market prospect,

G₄: investment profit rate,

G₅: risk present value rate,

G₆: expected net present value,

G₇: current liabilities which reflects the shorterm solvency,

G₈: investment profit tax rate,

G₉: investment payback period,

G₁₀: energy consumption of resources.

Suppose there are referenced data of three years, let μ = {0.23, 0.32, 0.45} be the weighting vector of the years.

Suppose ${\hat{λ}}^{1} = (λ_{1}^{1}, λ_{2}^{1}, \dots, λ_{10}^{1})$ is the weight vector of attributes in the first year given by decision makers, where $\begin{matrix} λ_{1}^{1} \in [0.05, 0.07], λ_{2}^{1} \in [0.07, 0.1], λ_{3}^{1} \in [0.08, 0.1], \\ λ_{4}^{1} \in [0.12, 0.17], λ_{5}^{1} \in [0.15, 0.19], λ_{6}^{1} \in [0.07, 0.08], \\ λ_{7}^{1} \in [0.13, 0.17], λ_{8}^{1} \in [0.10, 0.14], λ_{9}^{1} \in [0.06, 0.08], \\ λ_{10}^{1} \in [0.05, 0.07] . \end{matrix}$

The decision matrix ${\hat{A}}^{(1)}$ is as follows: $\begin{matrix} (\begin{matrix} H & M & VG & (11.8, 12.2, 12.6) & (42, 45, 52) & (41.5, 43, 44) \\ VH & VP & G & (11.2, 11.5, 11.8) & (37, 40, 45) & (35.2, 36.5, 37.5) \\ M & M & P & (11.3, 11.6, 11.8) & (40, 42, 50) & (41.7, 42.6, 43.7) \\ H & G & M & (11.1, 11.5, 11.9) & (38, 46, 48) & (36.6, 39, 41 \\ L & G & G & (10.8, 11.0, 11.3) & (39, 46, 49) & (34.2, 35.6, 36.4) \end{matrix} \\ \begin{matrix} (16, 16.5, 18) & (14, 14.2, 14.9) & (3.5, 4.6, 5) & L \\ (13 . 5, 14, 16) & (14, 14 . 2, 14 . 6) & (4 . 4, 5 . 4, 5 . 5) & H \\ (15, 16, 17) & (14 . 6, 16, 16 . 5) & (4, 4 . 2, 4 . 8) & M \\ (12, 15, 15 . 2) & (14 . 6, 14 . 7, 14 . 8) & (4 . 2, 4 . 6, 5 . 2) & ML \\ (14, 14 . 8, 16) & (14 . 8, 15, 15 . 2) & (5 . 6, 6, 6 . 8) & VH \end{matrix}) \end{matrix}$

Let ${\hat{λ}}^{2} = (λ_{1}^{2}, λ_{2}^{2}, \dots, λ_{10}^{2})$ be the weight vector in the second year given by decision makers, where $\begin{matrix} λ_{1}^{2} \in [0.08, 0.09], λ_{2}^{2} \in [0.08, 0.12], λ_{3}^{2} \in [0.1, 0.12], \\ λ_{4}^{2} \in [0.14, 0.18], λ_{5}^{2} \in [0.12, 0.15], λ_{6}^{2} \in [0.07, 0.1], \\ λ_{7}^{2} \in [0.1, 0.13], λ_{8}^{2} \in [0.10, 0.14], λ_{9}^{2} \in [0.06, 0.08], \\ λ_{10}^{2} \in [0.07, 0.09] . \end{matrix}$

The decision matrix ${\hat{A}}^{(2)}$ is as follows: $\begin{matrix} (\begin{matrix} VH & M & G & (11 . 8, 12 . 2, 12 . 6) & (41, 45, 52) & (41 . 5, 42, 43 . 7) \\ VH & P & M & (11 . 2, 11 . 5, 11 . 8) & (38, 40, 45) & (36, 36 . 8, 38) \\ M & G & G & (12, 12 . 6, 13) & (36, 42, 45) & (40 . 7, 42, 43 . 7) \\ VH & VG & P & (10 . 1, 11 . 5, 11 . 9) & (38, 46, 48) & (37 . 6, 39, 42) \\ VL & G & G & (10 . 8, 11 . 0, 11 . 3) & (34, 36, 39) & (34, 35 . 6, 36 . 4) \end{matrix} \\ \begin{matrix} (17, 17 . 5, 18) & (13 . 8, 14 . 2, 14 . 4) & (3 . 5, 4 . 4, 5) & L \\ (13 . 5, 14, 16 . 3) & (13, 13 . 2, 14) & (4.8, 5, 5.5) & VH \\ (15 . 8, 16, 17 . 2) & (14 . 6, 16, 16 . 5) & (5 . 5, 6 . 1, 6 . 4) & M \\ (13, 15, 15 . 2) & (14, 14 . 7, 15) & (4 . 2, 4 . 6, 5 . 2) & VL \\ (13, 13 . 8, 14) & (14, 14 . 6, 15) & (5 . 5, 6 . 1, 6 . 4) & H \end{matrix}) \end{matrix}$ Let ${\hat{λ}}^{3} = (λ_{1}^{3}, λ_{2}^{3}, \dots, λ_{10}^{3})$ be the weight vector in the third year given by decision makers, where

$\begin{matrix} λ_{1}^{3} \in [0.06, 0.08], λ_{2}^{3} \in [0.08, 0.12], λ_{3}^{3} \in [0.09, 0.1], \\ λ_{4}^{3} \in [0.13, 0.15], λ_{5}^{3} \in [0.18, 0.19], λ_{6}^{3} \in [0.08, 0.09], \\ λ_{7}^{3} \in [0.12, 0.14], λ_{8}^{3} \in [0.10, 0.14], λ_{9}^{3} \in [0.05, 0.07], \\ λ_{10}^{3} \in [0.05, 0.07] . \end{matrix}$ The decision matrix ${\hat{A}}^{(3)}$ is as follows: $\begin{matrix} (\begin{matrix} VH & G & G & (11 . 8, 12, 12 . 6) & (41, 45, 53) & (41 . 9, 43, 44) \\ H & P & M & (11 . 2, 11 . 5, 12) & (40, 42, 46) & (36, 36 . 8, 37 . 4) \\ M & VG & G & (12, 12 . 6, 13 . 4) & (36, 42, 43) & (41, 42, 44) \\ H & G & VP & (10, 11 . 2, 11 . 7) & (38, 46, 48) & (37 . 6, 39, 41) \\ L & VG & VG & (10 . 5, 11 . 0, 11 . 6) & (35, 36, 44) & (34, 35 . 6, 36 . 4) \end{matrix} \\ \begin{matrix} (17 . 4, 17 . 8, 18) & (14, 14 . 2, 14 . 6) & (3 . 5, 4 . 4, 5) & H \\ (13 . 8, 14 . 8, 16) & (13 . 3, 13 . 8, 14) & (4 . 8, 5, 5 . 3) & VH \\ (15 . 8, 16, 17 . 2) & (14 . 6, 15, 16) & (4, 4 . 2, 5) & M \\ \begin{matrix} (13, 14 . 6, 15 . 2) \\ (13, 13 . 5, 14 . 5) \end{matrix} & \begin{matrix} (14, 14 . 5, 14 . 6) \\ (14, 14 . 6, 15) \end{matrix} & \begin{matrix} (4 . 2, 4 . 6, 5) \\ (5, 5 . 9, 6) \end{matrix} & \begin{matrix} VL \\ H \end{matrix} \end{matrix}) \end{matrix}$

3.2 Decision procedure

Step 1. Transform linguistic values into triangle fuzzy numbers using Table 1, then build the weighted decision matrix C using Equations (1), (3) and (4).

$\begin{matrix} C = (\begin{matrix} \begin{matrix} (0.049, 0.055, \\ 0.059, 0.075) \end{matrix} & \begin{matrix} (0.03828, 0.04605, \\ 0.0685, 0.08) \end{matrix} & \begin{matrix} (0.0582, 0.0673, \\ 0.0791, 0.0897) \end{matrix} & \begin{matrix} (1.5446, 1.5853, \\ 1.9892.2.0689) \end{matrix} & \begin{matrix} (6.344, 6.926, \\ 7.782, 9.309) \end{matrix} \\ \begin{matrix} (0.046, 0.0522, \\ 0.0656, 0.0737) \end{matrix} & \begin{matrix} (0.01232, 0.0209, \\ 0.0300, 0.0415) \end{matrix} & \begin{matrix} (0.04, 0.0491, \\ 0.0578, 0.0684) \end{matrix} & \begin{matrix} (1.4661, 1.4878, \\ 1.8883, 1.9512) \end{matrix} & \begin{matrix} (5.9757, 6.318, \\ 7.259, 8.06) \end{matrix} \\ \begin{matrix} (0.0256, 0.0321, \\ 0.04, 0.049) \end{matrix} & \begin{matrix} (0.0506, 0.0584, \\ 0.087, 0.0985) \end{matrix} & \begin{matrix} (0.0472, 0.0563, \\ 0.0653, 0.0759) \end{matrix} & \begin{matrix} (1.5514, 1.6217, \\ 2.0298, 2.1147) \end{matrix} & \begin{matrix} (5.6784, 6.4635, \\ 7.4424, 8.022) \end{matrix} \\ \begin{matrix} (0.0436, 0.05, \\ 0.0624, 0.0705) \end{matrix} & \begin{matrix} (0.0517, 0.0595, \\ 0.088, 0.1) \end{matrix} & \begin{matrix} (0.0138, 0.0229, \\ 0.0275, 0.0382) \end{matrix} & \begin{matrix} (1.3438, 1.4878, \\ 1.8681, 1.9405) \end{matrix} & \begin{matrix} (5.8482, 7.094, \\ 8.183, 8.506) \end{matrix} \\ \begin{matrix} (0.0077, 0.01411, \\ 0.0185, 0.0266) \end{matrix} & \begin{matrix} (0.0322, 0.04895, \\ 0.05918, 0.0707) \end{matrix} & \begin{matrix} (0.0626, 0.0717, \\ 0.0835, 0.0941) \end{matrix} & \begin{matrix} (1.3962, 1.4399, \\ 1.8062, 1.8757) \end{matrix} & \begin{matrix} (5.486, 5.885, \\ 6.816, 7.775) \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} (3.106, 3.181, \\ 3.877, 3.99) \end{matrix} & \begin{matrix} (1.962, 2.0146, \\ 2.4946, 2.5866) \end{matrix} & \begin{matrix} (1.3936, 1.42, \\ 1.988, 2.0447) \end{matrix} & \begin{matrix} (0.1943, 0.239, \\ 0.286, 0.365) \end{matrix} & \begin{matrix} (0.0248, 0.0305, \\ 0.0409, 0.0477) \end{matrix} \\ \begin{matrix} (2.669, 2.737, \\ 3.174, 3.421) \end{matrix} & \begin{matrix} (1.5812, 1.6692, \\ 2.0661, 2.3117) \end{matrix} & \begin{matrix} (1.3365, 1.37, \\ 1.9177, 1.9793) \end{matrix} & \begin{matrix} (0.261, 0.283, \\ 0.385, 0.409) \end{matrix} & \begin{matrix} (0.0023, 0.0079, \\ 0.0109, 0.0185) \end{matrix} \\ \begin{matrix} (3.059, 3.148, \\ 3.842, 3.984) \end{matrix} & \begin{matrix} (1.8073, 1.8544, \\ 2.2992, 2.4638) \end{matrix} & \begin{matrix} (1.46, 1.5573, \\ 2.1802, 2.2785) \end{matrix} & \begin{matrix} (0.2508, 0.2726, \\ 0.3727, 0.4188) \end{matrix} & \begin{matrix} (0.0226, 0.0282, \\ 0.0382, 0.0458) \end{matrix} \\ \begin{matrix} (2.785, 2.906, \\ 3.545, 3.759) \end{matrix} & \begin{matrix} (1.4768, 1.7169, \\ 2.1303, 2.1842) \end{matrix} & \begin{matrix} (1.4138, 1.461, \\ 2.0454, 2.0684) \end{matrix} & \begin{matrix} (0.2331, 0.2553, \\ 0.3473, 0.3863) \end{matrix} & \begin{matrix} (0.0451, 0.0508, \\ 0.0688, 0.0764 \end{matrix} \\ \begin{matrix} (2.537, 2.652, \\ 3.236, 3.309 \end{matrix} & \begin{matrix} (1.5366, 1.614, \\ 2.0035, 2.1215) \end{matrix} & \begin{matrix} (1.4184, 1.4692, \\ 2.057, 2.106) \end{matrix} & \begin{matrix} (0.2871, 0.3327, \\ 0.4525, 0.4779) \end{matrix} & \begin{matrix} (0.018, 0.0236, \\ 0.026, 0.0399) \end{matrix} \end{matrix}) \end{matrix}$

Step 2. Compute the probability degree matrices P₁, P₂, ⋯ , P₁₀ by Equations (2), (5) and (6), where λ = 0.4. There P_i is the possibility degree matrix built by ith (i = 1, 2, ⋯ , 10) rank in matrix C. $\begin{matrix} P_{1} = (\begin{matrix} 0.5 & 0.529 & 1 & 0.628 & 1 \\ 0.471 & 0.5 & 1 & 0.73 & 1 \\ 0 & 0 & 0.5 & 0 & 1 \\ 0.318 & 0.27 & 1 & 0.5 & 1 \\ 0 & 0 & 0 & 0 & 0.5 \end{matrix}) \\ ∥ P_{2} = (\begin{matrix} 0.5 & 1 & 0 & 0 & 0.768 \\ 0 & 0.5 & 0 & 0 & 0 \\ 1 & 1 & 0.5 & 0.4327 & 1 \\ 1 & 1 & 0.5673 & 0.5 & 1 \\ 0.232 & 1 & 0 & 0 & 0.5 \end{matrix}) \\ ∥ P_{3} = (\begin{matrix} 0.5 & 1 & 1 & 1 & 0.279 \\ 0 & 0.5 & 0.459 & 1 & 0 \\ 0 & 0.541 & 0.5 & 1 & 0 \\ 0 & 0 & 0 & 0.5 & 0 \\ 0.721 & 1 & 1 & 1 & 0.5 \end{matrix}) \\ ∥ P_{4} = (\begin{matrix} 0.5 & 1 & 0.265 & 1 & 1 \\ 0 & 0.5 & 0 & 0.716 & 1 \\ 0.735 & 1 & 0.5 & 1 & 1 \\ 0 & 0.284 & 0 & 0.5 & 0.918 \\ 0 & 0 & 0 & 0.082 & 0.5 \end{matrix}) \\ ∥ P_{5} = (\begin{matrix} 0.5 & 0.928 & 0.897 & 0 & 0 \\ 0.072 & 0.5 & 0 & 0.719 & 0.824 \\ 0.103 & 1 & 0.5 & 0.305 & 0.8 \\ 1 & 0.281 & 0.695 & 0.5 & 0.99 \\ 1 & 0.176 & 0.2 & 0.01 & 0.5 \end{matrix}) \\ ∥ P_{6} = (\begin{matrix} 0.5 & 0 & 0.646 & 1 & 1 \\ 1 & 0.5 & 0 & 0 & 0.747 \\ 0.354 & 1 & 0.5 & 1 & 1 \\ 0 & 1 & 0 & 0.5 & 1 \\ 0 & 0.253 & 0 & 0 & 0.5 \end{matrix}) \\ ∥ P_{7} = (\begin{matrix} 0.5 & 1 & 1 & 1 & 1 \\ 0 & 0.5 & 0.418 & 0.671 & 0.744 \\ 0 & 0.582 & 0.5 & 1 & 1 \\ 0 & 0.329 & 0 & 0.5 & 0.827 \\ 0 & 0.256 & 0 & 0.173 & 0.5 \end{matrix}) \end{matrix}$ $\begin{matrix} P_{8} = (\begin{matrix} 0.5 & 1 & 0 & 0.0337 & 0.008 \\ 0 & 0.5 & 0 & 0 & 0 \\ 1 & 1 & 0.5 & 0.997 & 0.997 \\ 0.9663 & 1 & 0.003 & 0.5 & 0.095 \\ 0.992 & 1 & 0.003 & 0.905 & 0.5 \end{matrix}) \\ P_{9} = (\begin{matrix} 0.5 & 0 & 0 & 0.124 & 0 \\ 1 & 0.5 & 0.602 & 0.982 & 0 \\ 1 & 0.398 & 0.5 & 0.863 & 0 \\ 0.876 & 0.008 & 0.137 & 0.5 & 0 \\ 1 & 1 & 1 & 1 & 0.5 \end{matrix}) \\ P_{10} = (\begin{matrix} 0.5 & 1 & 0.88 & 0 & 1 \\ 0 & 0.5 & 0 & 0 & 0 \\ 0.12 & 1 & 0.5 & 0 & 0.916 \\ 1 & 1 & 1 & 0.5 & 1 \\ 0 & 1 & 0.084 & 0 & 0.5 \end{matrix}) \end{matrix}$

Step 3. In the principle of “at least half”, compting the respective weight vector ω = (ω₁, ω₂, ⋯ , ω₁₀) of the OWA operator based on the probability matrices P₁, P₂, ⋯ , P₁₀ by using Equations (7) and (8), we get $\begin{matrix} ω_{1} = 0.2, ω_{2} = 0.2, ω_{3} = 0.2, ω_{4} = 0.2, \\ ω_{5} = 0.2, ω_{6} = 0, ω_{7} = 0, ω_{8} = 0, ω_{9} = 0, ω_{10} = 0 . \end{matrix}$

Step 4. Computing the aggregated matrix $P^{*} = (p_{ij}^{*})_{5 \times 5}$ by Equation (9), we get $P^{*} = (\begin{matrix} 0.5 & 1 & 0.955 & 0.926 & 0.954 \\ 0.509 & 0.5 & 0.496 & 0.829 & 0.863 \\ 0.818 & 1 & 0.5 & 0.999 & 1 \\ 0.968 & 0.866 & 0.652 & 0.5 & 0.998 \\ 0.789 & 1 & 0.463 & 0.632 & 0.5 \end{matrix})$

Step 5. Computing the corresponding weights of the OWA operator based on the principle of “as many as possible” by using Equations (8) and (10), we get that $ω_{1}^{*} = 0, ω_{2}^{*} = 0.4, ω_{3}^{*} = 0.5, ω_{4}^{*} = 0.1 .$

According to Equation (11), we can obtain that, $\begin{matrix} d_{1} = 0.9516, d_{2} = 0.6357, d_{3} = 0.9813, \\ d_{4} = 0.8854, d_{5} = 0.6779 . \end{matrix}$

Thus, the best alternative is X₃.

4 Conclusion remarks

Based on time weight vector and the conclusion that the product of an interval number and a triangle number is a trapezoidal number, a novel ranking method from a set of mutually exclusive alternatives is presented by combining with OWA operators and using possibility degree formula of comparing two trapezoidal given in this paper. The superiority of the proposed approach is that it does not require the normalization and defuzzification process and thus avoids the loss of information results from transforming fuzzy numbers to normal form or real-value. Moreover, possibility degree ranking method has advantage of less calculation and simplicity. Finally, a practical example is given to illustrate the effectiveness of the approach.

The National Natural Science Foundation Project of China (No. 07XMZ027).

National Natural Science Foundation Project of China (No. 11271041).

Open issue of meteorological disaster forecasting and emergency management (ZHYJ15-YB12).

The Project of Chengdu University of Information Technology (N0. CRF201508).

Footnotes

Acknowledgments

The work was supported by the foundations as follows:

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