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A Novel Method for Interval-Value Intuitionistic Fuzzy Multicriteria Decision-Making Problems with Immediate Probabilities Based on OWA Distance Operators.

1. Introduction

Atanassov and Gargov [1, 2] introduced the theory of Interval-Valued Intuitionistic Fuzzy Set (IVIFS), which is a generalization of the Intuitionistic Fuzzy Set (IFS) proposed by Atanassov [3]. The IVIFS has attracted more and more attention since its appearance. Some decision-making methods under IVIF environment have been developed by many scholars. To sum up, there are mainly four aspects on the decision-making under IVIF environment: (1) some decision-making methods are developed based on information measures (specially, distance, similarity, and entropy) because information measures for IVIFSs have great effects on the development of the IVIFS theory and its applications. For example, similarity measures [4-6], inclusion measure [7], entropy measure [8], cross-entropy measure [9], and distance measures [10] are developed and applied to corresponding MCDM and MADM problems; (2) many new aggregation operators are also investigated in the IVIFSs and applied to some decision-making problems, such as linguistic intuitionistic fuzzy power Bonferroni Mean operators [11], Hamacher aggregation operators [12], fuzzy power Heronian aggregation operators [13], fuzzy generalized aggregation operator [11, 14-18], (fuzzy Einstein) hybrid weighted aggregation operators [19, 20], fuzzy prioritized hybrid weighted aggregation operator [21], and fuzzy Hamacher ordered weighted geometric operator [22]; (3) other methods for decision-making with IVIF information are also explored, such as evidential reasoning methodology [23], particle swarm optimization techniques [4], transform technique [24], nonlinear programming methods [25], and VIKOR methods in IVIFS [26], and others methods [27-32] are also developed for decision-making problems. Distance measure has great effects on obtaining the desirable choice in some decision problems. Motivated by the OWA operator, Xu [33] introduced ordered weighted distance operator based on known Haming distance. Many extensions of distance operator have been developed; for example, Merigo et al. [34] introduced a series of aggregation operators related to distance measures [35-38] which were applied to related decision problems [37, 39, 40].

In some real decision problems, many problems are very complex. Aiming at solving some of these complicated decision problems, it is necessary to develop a new kind of decision-making method to solve this kind of problems including the following three-aspect information: (1) information represented in the form of IVIFVs not only IFVs, (2) the weighted information and the probability information, and (3) the degree of importance of each concept in the process of decision-making. Motivated by the ideas of existing operators, we propose new IVIF distance measures by using related weighted operators with probabilistic information and their applications in MCDM in the present work.

The rest of the paper is organized as follows. In Section 2, we review some related definitions on IVIFSs which are in the analysis throughout this paper. Section 3 is focused on PIVPFOWAD and IPIVIFOWAD. In Section 4, the concept of satisfaction degree is proposed and the MCDM approach based on the satisfaction degree is also constructed. Section 5, a practical example, is given to explain proposed method and compare and analyze the availability of proposed MCDM methods. This paper is concluded in Section 6.

2. IVIFSs and OWA Distance Operator

In this section, some related basic concepts of VIFSs, OWA operator, and OWAD operator are recapped.

2.1. Interval-Valued Intuitionistic Fuzzy Sets. Assume that int([0, 1]) is the collection of all closed subintervals of [0, 1], and X is a universe of discourse. An IVIFS [1, 2] on X has such a structure

[mathematical expression not reproducible]. (1)

where [[mu].sub.[??]] : X [right arrow] int([0, 1]) denotes the membership degree and [v.sub.[??]] : X [right arrow] int([0, 1]) denotes the nonmembership degree of the element x [member of] X to the set [??], respectively, with the condition that 0 [less than or equal to] sup([[mu].sub.[??]](x)) + sup([v.sub.[??]](x)) [less than or equal to] 1.

For each x [member of] X, [[mu].sub.[??]](x) and [v.sub.[??]](x) denote [mathematical expression not reproducible], respectively. Therefore, [??] may be also expressed as

[mathematical expression not reproducible]. (2)

Eq. (2) satisfies the condition [[mu].sup.+.sub.[??]](x) + [v.sup.+.sub.[??]](x) [less than or equal to] 1.

[mathematical expression not reproducible] (3)

is called the indeterminacy degree. For the convenience, [mathematical expression not reproducible] is called an interval-valued intuitionistic fuzzy value (IVIFV).

Hereafter, IVIFV denotes the collection of all IVIFVs of a IVIFS on X.

For two IVIFVs [mathematical expression not reproducible] and [mathematical expression not reproducible], a relation [less than or equal to] on the IVIFVs is defined as follows:

[mathematical expression not reproducible]. (4)

In order to compare two IVIFVs, score function and accuracy function [41] of an IVIFV are introduced.

For any IVIFV [mathematical expression not reproducible], the score function [mathematical expression not reproducible] of [??] is defined as follows:

[mathematical expression not reproducible], (5)

where [mathematical expression not reproducible].

For any IVIFV [mathematical expression not reproducible], the accuracy function [mathematical expression not reproducible] of [??] is defined as follows:

[mathematical expression not reproducible], (6)

where [mathematical expression not reproducible].

Based on the above definitions, comparison rules are defined as follows.

For any two IVIFVs [[??].sub.1], [[??].sub.2],

(1) if [mathematical expression not reproducible], then [[??].sub.1] < [[??].sub.2];

(2) if [mathematical expression not reproducible], then

(a) if [mathematical expression not reproducible], then [[??].sub.1] < [[??].sub.2];

(b) if [mathematical expression not reproducible], then [[??].sub.1] ~ [[??].sub.2].

2.2. OWA Distance Operator. In this section, we will review the OWAD operator and then introduce IVIFOWAD operator.

An OWA operator [42] is a function OWA : [R.sup.n] [right arrow] R that has an associated weight vector (WV) [omega] = ([[omega].sub.1], ..., [[omega].sub.n]) with [[omega].sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1][[omega].sub.j] = 1, such that

OWA([a.sub.1], [O.sub.2], ..., [a.sub.n]) = [n.summation over (j=1)][[omega].sub.j][b.sub.j] (7)

where ([a.sub.1], ..., [a.sub.n]) [member of] [R.sup.n] and [b.sub.j] is the jth largest of [a.sub.i].

Similarly, OWAD operator is introduced as a generalization of OWA operator. Let A = ([a.sub.1], ..., [a.sub.n]), B = ([b.sub.1], ..., [b.sub.n]) be two collections of arguments. An OWAD operator [36] is a function OWAD : [R.sup.n] x [R.sup.n] [right arrow] B that has an associated WV [omega] = ([[omega].sub.1], ..., [[omega].sub.n]) with [[omega].sub.j] [member of] [0, 1] and [[summation].sup.n.sub.j=1][[omega].sub.j] = 1, such that

OWAD(A, B) = [n.summation over (j=1)][[omega].sub.j][d.sub.j] (8)

where [d.sub.j] is the jth largest of [absolute value of [a.sub.i] - [b.sub.i]].

Let [mathematical expression not reproducible] and [mathematical expression not reproducible] be two collections of IVIFVs, where [mathematical expression not reproducible], and i = l, ..., n. We first recall the distance [10] between two IVIFVs [mathematical expression not reproducible] and [mathematical expression not reproducible].

[mathematical expression not reproducible]. (9)

3. IPIVIFOWA Distance Operator

In this subsection, by combining OWA operator, individual distances, and PWs, two new distances named PIVIFOWAD operator and IPIVIFOWAD operator will be introduced. PIVIFOWAD operator is defined as follows.

Definition 1. A PIVIFOWAD is a function PIVIFOWAD: IVIF[V.sup.n] x IVIF[V.sup.n] [right arrow] R, that has an associated WV [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] with [[omega].sub.i] > 0, [[summation].sup.n.sub.i=1][[omega].sub.i] = 1(i = 1, 2, ..., n), such that

[mathematical expression not reproducible], (10)

where [mathematical expression not reproducible] is the ith largest of [mathematical expression not reproducible] and each [mathematical expression not reproducible] has associated a probabilistic [p.sub.j] [member of] [0, 1], [[summation].sup.n.sub.i=j][p.sub.i] = 1.

In Definition 1, if [xi] = 1, it will be reduced to interval-valued intuitionistic fuzzy ordered weighted distant (IVIFOWAD) operator:

[mathematical expression not reproducible], (11)

where [mathematical expression not reproducible] is the jth largest of [mathematical expression not reproducible] and [mathematical expression not reproducible] is the argument variable represented in the form of individual distance between IVIFVs [mathematical expression not reproducible].

Example 2. Let

[mathematical expression not reproducible] (12)

be two collections of IVIFVs on the set X = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} and the WV is [omega] = (0.2, 0.3, 0.1, 0.4). Take [xi] = 0.3; according to Definition 1, we have

[mathematical expression not reproducible]. (13)

Now, we can also develop the IPIVIFOWAD operator by applying IVIF information, individual distance, and immediate probability (IP)[43].

Definition 3. An IPIVIFOWAD is a function LPLVTFOWAD : IVIF[V.sup.n] x IVIF[V.sup.N] [right arrow] R which has an associated WV [omega] = [([[omega].sub.1], [[omega].sub.2], ..., [[omega].sub.n]).sup.T] with [[omega].sub.i] > 0, [[summation].sup.n.sub.i=1][[omega].sub.i] = 1(i = 1, 2, ..., n), such that

[mathematical expression not reproducible], (14)

where [mathematical expression not reproducible] is the ith largest of [mathematical expression not reproducible] and [mathematical expression not reproducible] is the argument variable represented in the form of individual distance between IVIFVs [mathematical expression not reproducible] and a PW [p.sub.i] > 0, [[summation].sup.n.sub.i=1][p.sub.i] = 1. [[??].sub.i] = [[omega].sub.i][p.sub.i]/[[summation].sup.n.sub.i=1][[omega].sub.i][p.sub.i] and [p.sub.i] is the probabilistic [p.sub.j] according to [mathematical expression not reproducible], that is, according to the ith largest of the [mathematical expression not reproducible].

It is worth pointing out that IPIVIFOWAD operator is a good approach for unifying probabilities and IVIFOWAD in some particular situations. But it is not always useful. In order to show why this unification does not seem to be a final model, we could also consider other ways of representing [[??].sub.i] as in [40].

Example 4. In Example 2, since the following WV [omega] = (0.2, 0.3, 0.1, 0.4) and the PW (0.3, 0.2, 0.4, 0.1), now we aggregate this information according to IPIVIFOWAD, as we have calculated d([a.sub.i], [b.sub.i]) by employing (7) as follows:

d([a.sub.1], [b.sub.1]) = 0.175, d([a.sub.2], [b.sub.2]) = 0.2, d([a.sub.3], [b.sub.3]) = 0.05, d([a.sub.4], [b.sub.4]) = 0.175. (15)

According to the above distance, we reorder the PW (0.2, 0.3, 0.1, 0.4),

[5.summation over (i=1)][[omega].sub.i][p.sub.i] = (0.2, 0.3, 0.1, 0.4) [(0.2, 0.3, 0.1, 0.4).sup.T] = 0.3. (16)

Therefore, [??] = [[omega].sub.1][p.sub.1]/[[summation].sup.5.sub.i=1][[omega].sub.i][p.sub.i] = (0.2 x 0.2)/0.3 = 0.133, and similarly, we have [??] = 0.3, [??] = 0.033, [??] = 0.534. Therefore, we have

[mathematical expression not reproducible]. (17)

Monotonicity is a kind of vital property in the research of aggregation operators. The aggregation operator with monotonicity will be more reliable in decision-making process. The lack of monotonicity may depress the reliability of the final results. PIVIFOWAD and IPIVIFOWAD are new distance measure and are aggregation operators. We can prove that PIVIFOWAD and IPIVIFOWAD have the properties of boundness, monotonicity, and reflexivity. The proof of these properties is similar to Theorems 1-3 in [44].

4. Method for MCDM Based on IPIVIFOWA Operator

4.1. Formal Description of MCDM with IVIFs. The MCDM with IVIF information can be formally presented as follows.

Assume that X = {[x.sub.1], ..., [x.sub.m]} is a set of m alternatives, C = {[G.sub.1], ..., [C.sub.n]} is the collection of attributes, and [omega] = [([[omega].sub.1], ..., [[omega].sub.n]).sup.T] are the WV of all attributes, which satisfy 0 [less than or equal to] [[omega].sub.i] [less than or equal to] 1. Assume that alternative [O.sub.i](i = 1, ..., m) with respect to attribute [C.sub.j](j = 1, ..., n) is evaluated by an IVIFVs [C.sub.j]([x.sub.i]) = ([[[mu].sup.-.sub.ij], [[mu].sup.+.sub.ij]], [[v.sup.-.sub.ij], [v.sup.+.sub.ij]])(j = 1, 2, ..., n; i =, 2, ..., m) and [R.sub.mxn] = [([C.sub.j]([x.sub.i])).sub.mxn] is an IVIF decision matrix. A new kind MCDM approach will be developed based on the distance operators proposed in Section 3.

For a MCDM problem with IVIFVs, the decision matrix R = [([C.sub.j]([x.sub.i])).sub.mxn] and can be constructed or given in advance.

[mathematical expression not reproducible] (18)

We give the concepts of IVIF-PIS, IVIF-NIS, and satisfaction degree before the decision-making algorithm is given.

Considering the decision information represents the form of IVIFVs, we use (5) and (6) based on comparison approach to identify the IVIF-PIS and the IVIF-NIS. We use [O.sup.+] to represent IVIF-PIS and [O.sup.-] to represent IVIF-NIS; they are determined as follows:

[O.sup.+] = {<[C.sub.j], [max.sub.i]s([C.sub.j]([x.sub.i]))> | j = 1, 2, ..., n; i = 1, 2, ..., m}, (19)

[O.sup.-] = {<[C.sub.j], [mim.sub.i]s([C.sub.j]([x.sub.i]))> | j = 1, 2, ..., n; i = 1, 2, ..., m}. (20)

Let D be one of IPIVIFOWAD and PIVIFOWAD, and D([O.sup.+], [O.sub.i]) and D([O.sup.+], [O.sub.i]) denote the distance of [O.sup.+] and alternative [O.sub.i] and the [O.sup.-] and alternative [O.sub.i], respectively. Motivated by the well-known TOPSIS, we take both D([O.sup.+], [O.sub.i]) and D([O.sup.-], [O.sub.i]) into consideration simultaneously rather than separately. This leads naturally to the concept of satisfaction degree.

Definition 5. Let A = {[A.sub.1], ..., [A.sub.m]} be a collection of alternatives. The satisfaction degree [lambda]([O.sub.i]) of a given alternative [x.sub.i] over the criteria [C.sub.j](j = 1, 2, ..., n) is defined as

[lambda]([O.sub.i]) = (1 - [epsilon])[D([O.sup.-], [O.sub.i])]/ [epsilon][D([O.sup.+], [O.sub.i])] + (1 - [epsilon])[D([O.sup.-], [O.sub.i])], i = 1, 2, ..., m. (21)

In (21), [epsilon] denotes the risk preference of the DM and [epsilon] [member of] [0, 1]: [epsilon] > 0. means that the decision maker is pessimist, while [epsilon] < 0.5 means the opposite. [epsilon] = 0.5: satisfaction degree is of relative closeness using the classic TOPSIS method. The parameter [xi] is provided by the DM in advance. It is obvious that [lambda]([O.sub.i]) [member of] [0, 1] (i = 1, 2, ..., m). The higher the satisfaction degree, the better the alternative.

4.2. Decision Algorithm for MCDM with IVIF

Step 1. Determine the IVIF-PIS and the IVIF-NIS.

Step 2. Calculate the distance between IVIFVs in A and IVIFVs in [O.sup.+]([O.sup.-]) according to (9).

Step 3. Recalculate the probability according to distance calculated in Step 2.

Step 4. Compute the distance D([O.sup.+], [O.sup.i]) of the positive ideal IVIFS [O.sup.+] and alternative [O.sub.i] and the distance D([O.sup.-], [O.sub.i]) of the negative ideal IVIFS [O.sup.-] and alternative [O.sub.i], respectively.

Step 5. Calculate the satisfaction degree [lambda]([O.sub.i]) according to Definition 5. And get the priority of the alternative [O.sub.i](i = 1, ..., m) by ranking [lambda]([O.sub.i])(i = 1, ..., m); the bigger the satisfaction degree [lambda]([O.sub.i]), the better the alternation [O.sub.i].

Step 6. End.

5. Case Study

In this section, we will give a practical example about the optimal invest strategy to show the application of the proposed IPIVIFOWAD and PIVIFOWAD.

Technological innovation not only is directly related to the survival and development of an enterprise, but also affects the economic development of a region or even a country. As we all know, the management of an enterprise's technological innovation activities is an important manifestation of its technological innovation capability. In evaluating the technological innovation capability of enterprises, the following evaluation index system should be considered:

(1) [G.sub.1]: innovation system construction, attitude to innovation failure, and incentives for innovation by the enterprise distribution system;

(2) [G.sub.2]: establishment and implementation of technological innovation strategy, the formation, and maintenance of enterprise innovation culture;

(3) [G.sub.3]: the feasibility of research and development project feasibility report;

(4) [G.sub.4]: the completeness of the monitoring;

(5) [G.sub.5]: evaluation system and innovation awareness of leaders and staff.

DM assesses the technical innovation management of 5 large enterprises [O.sub.i](i = 1, ..., 5) by questionnaires survey and discussion and uses the IVIFVs for evaluation and constructs the IVIF decision matrix as shown in Table 1.

To find the desirable alternative, the experts give the probabilistic weight information as follows: p = (0.3, 0.3, 0.2, 0.1, 0.1). They assume that the importance degree of each characteristic is w = (0.2, 0.3, 0.1, 0.3, 0.1).

5.1. Decision-Making Using IPIVIFOWAD Operator

Step 1. Determine the IVIF-PIS [O.sup.+] and the IVIF-NIS [O.sup.-] by (5) and (6) which are shown in Table 2.

We can see from Table 2 that [s.sub.j]([G.sub.1])(j = 1, 2, 3, 4, 5) all are different, so do [s.sub.j]([G.sub.2]), [s.sub.j]([G.sub.3]), [s.sub.j]([G.sub.4]), [s.sub.j]([G.sub.5])(j = 1, 2, 3, 4, 5). Therefore, we do not need to compute the accuracy function. And so, IVIF-PIS [O.sup.+] and IVIF-NIS [O.sup.-] are obtained, respectively, and shown as follows:

[mathematical expression not reproducible]. (22)

Step 2. Denote [mathematical expression not reproducible]. Now we calculate the distance [mathematical expression not reproducible] between IVIFVs [mathematical expression not reproducible] between the IVIFVs [mathematical expression not reproducible], respectively. The results can be found in Tables 3 and 4.

Step 3. Calculate IP by using the above probabilities and weights according to Tables 3 and 4. The results are shown in Tables 5 and 6.

Step 4. Calculate the IPIVIFOWAD([O.sub.i], [O.sup.+]) and IPIVIFOWAD([O.sub.i], [O.sup.-]) according to Step 2 and Step 3. For convenience, we denote IPIVIFOWAD([O.sub.i], [O.sup.+]) and IPIVIFOWAD([O.sub.i], [O.sup.-]) as D([O.sub.i], [O.sup.+]) and D([O.sub.i], [O.sup.-])(i = 1, 2, ..., 5), respectively. The results are as follows.

[mathematical expression not reproducible]. (23)

Step 5. Calculate the satisfaction degree according to the distance in Step 4. The results can be found in Table 7 under different risk preference [epsilon].

It follows from Table 7 that the order of alternatives is consistent with results by using IPIVIFOWAD when parameter changes. We can obtain the ranking [O.sub.1] > [O.sub.5] > [O.sub.3] > [O.sub.4] > [O.sub.2]. All of the results show that [O.sub.1] is the desirable alternative. Such a conclusion can be drawn directly from Figure 1.

5.2. Decision-Making Using PIVIFOWAD. Now we use the PIVIFOWAD to this decision-making problem. According to the distance matrices in Tables 3 and 4, note that the WA has an importance of 40 percent and the probabilistic information has an importance of 60 percent. We can rearrange the probabilistic information according the distance. The results can be found in Tables 8 and 9.

Therefore, we can calculate PIVIFOWAD distances D([O.sub.i], [O.sup.+]), D([O.sub.i], [O.sup.-]) (i = 1, ..., 5) as follows:

[mathematical expression not reproducible]. (24)

Therefore, we can obtain the satisfaction degree under difference risk preference parameters e; please refer to Table 10.

It follows from Table 10 that the ranking is consistent with the results by using the PIVIFOWAD operators when parameter changes. We can obtain the order of alternatives: [O.sub.1] > [O.sub.3] > [O.sub.5] > [O.sub.4] > [O.sub.2]. All of the results show that [O.sub.1] is the desirable one. Such a conclusion can be also drawn directly from Figure 2.

If we change the weight important degree [xi], we can obtain other satisfaction degrees listed in Table 11. From Table 11, we can see that the desirable alternative is consistent with the ranking obtained by PIVIFOWAD when parameter [xi] changes although the ranking of alternatives is not the same. All of the results show that [O.sub.1] is the desirable one. Such a conclusion can be drawn directly from Figure 3.

5.3. Effectiveness Test of the Proposed Method. For MCDM problems, Wang and Triantaphyllou [45] established assessing criteria (please refer to [45]) to assess the effectiveness of MCDM methods. In what follows, we will use the above MCDM criteria to test our proposed methods in Section 4. As far as the proposed method based on IPIVIFOWAD is concerned, we choose the satisfactory degree [epsilon] = 0.5 to analyze the above criteria.

Validity Test for Criterion 1. In Section 5.2, we obtained that [O.sub.1] is the desirable one and the order of alternatives is [O.sub.1] > [O.sub.5] > [O.sub.3] > [O.sub.4] > [O.sub.2]. In order to test the effectiveness of the developed IPIVIFOWAD method under criterion 1, we construct the modified IVIF decision matrix (Table 12) by interchanging the membership and nonmembership of alternatives [O.sub.3] and [O.sub.4] in Table 1.

Repeating the same steps 1-2 in Section 5.1, we can obtain the modified IVIF-PIS [O.sup.+] and the IVIF-NIS [O.sup.-] which are listed as follows:

[mathematical expression not reproducible]. (25)

Using Step 3-Step 5 of IPIVIFOWAD method, the IPIVIFOWAD distances D([O.sub.i], [O.sup.+]) between alternatives [O.sub.i] and [O.sup.+] and the IPIVIFOWAD distances D([O.sub.i], [O.sup.-]) between alternatives [O.sub.i] and [O.sup.-] are calculated, respectively, where i = 1, 2, ..., 5.

[mathematical expression not reproducible]. (26)

According to the satisfaction degree formula when [epsilon] = 0.5, we have

[mathematical expression not reproducible]. (27)

We can see from the above satisfaction degrees that the rank is [O.sub.1] > [O.sub.5] > [O.sub.2] > [O.sub.3] > [O.sub.4]; that is, [O.sub.1] is the best one. Therefore, the best alternative coincides with the best alternative obtained in Section 5.1 by the same method, and the relative orders of the rest of the unchanged alternatives keep constant. That is, Criterion 1 is suitable for the proposed method.

Validity Test for Criterion 2 and Criterion 3. According to the requirements of criterion 2 and test criterion 3 introduced in [45], the original problem should be decomposed into two smaller MCDM problems, such as {[O.sub.1], [O.sub.2], [O.sub.3], [O.sub.4]} and {[O.sub.1], [O.sub.3], [O.sub.4], [O.sub.5]}. For the subproblem {[O.sub.1], [O.sub.2], [O.sub.3], [O.sub.4]}, we can obtain the satisfaction degree by repeating Step 1 to Step 6 as follows:

[mathematical expression not reproducible]. (28)

Therefore, the ranking of the subproblem is [O.sub.1] > [O.sub.3] > [O.sub.4] > [O.sub.2]. For the subproblem {[O.sub.1], [O.sub.3], [O.sub.4], [O.sub.5]}, we can obtain the satisfaction degree by repeating Step 1 to Step 6 as follows:

[mathematical expression not reproducible]. (29)

The ranking of the subproblem {[O.sub.1], [O.sub.3], [O.sub.4], [O.sub.5]} is [O.sub.1] > [O.sub.5] > [O.sub.3] > [O.sub.4].

We obtain the final ranking [O.sub.1] > [O.sub.5] > [O.sub.3] > [O.sub.4] > [O.sub.2] by combining the order of alternatives of subproblems {[O.sub.1], [O.sub.2], [O.sub.3], [O.sub.4]} and {[O.sub.1], [O.sub.3], [O.sub.4], [O.sub.5]}, and the final order is the same as the order of original decision problem and it also exhibits transitive property. Criterion 2 and criterion 3 proposed in [45] are also suitable for the proposed method.

5.4. Comparison with Existing Work. For the comparison with Hadi-Vencheh and Mirjaberi's method [46], in the classical TOPSIS method, we often need to compute the relative closeness of the alternative [O.sub.i] with respect to the PIS [O.sup.+] as below:

RC([O.sub.i]) = D([O.sub.i], [O.sup.-])/[D([O.sub.i], [O.sup.-]) + D([O.sub.i], [O.sup.+])] (30)

D(.) is a distance measure. The ranking of all alternatives can be determined according to the closeness index RC([O.sub.i]). If [epsilon] = 0.5 in our proposed equation (21), then (21) will be (30).

However, Hadi-Vencheh and Mirjaberi [46] suggested that one may use the following formula instead of the relative closeness index:

[zeta]([O.sub.i]) = [D([O.sub.i], [O.sup.-])/[D.sub.max]([O.sub.i], [O.sup.-])] - [D([O.sub.i], [O.sup.+])/[D.sub.min]([O.sub.i], [O.sup.+])] (31)

[D.sub.max]([O.sub.i], [O.sup.-]) = [max.sub.1[less than or equal to]i[less than or equal to]m]{D([O.sub.i], [O.sup.-])} and [D.sub.min]([O.sub.i], [O.sup.+]) = [min.sub.1[less than or equal to]i[less than or equal to]m]{D([O.sub.i], [O.sup.+])}. Equation (31) is called the revised closeness used to measure the extent to which the alternative [O.sub.i] is close to the PIS 0+ and is far away from the NIS [O.sup.+], simultaneously. By (31),

[mathematical expression not reproducible] (32)

Therefore, the ranking of ([O.sub.1], [O.sub.2], [O.sub.3], [O.sub.4], [O.sub.5]} is arranged [O.sub.1] > [O.sub.5] > [O.sub.3] > [O.sub.4] > [O.sub.2] which coincides with our proposed method.

6. Conclusion

IVIFSs, which are a generalization of the IFSs, have been used widely in decision problems. IVIFS permits the membership degrees and nonmembership degrees to a given set to have an interval value in [0, 1] and can be considered as a powerful tool to express complex information in the human decision-making process. In this paper, we introduced some new distance measures, namely, PIVIFOWAD operator and IPIVIFOWAD operator, while, with respect to probabilistic decision-making problems with IVIF information, some new probabilistic decision-making analysis methods are developed. The new distance operators such as IVIFOWAD operator, PIVIFOWAD operator, and IPIVIFOWAD operator have been developed in this paper, while the concept of satisfaction degree of alternatives has been introduced based on some distance measures and applied to MCDM problem with IVIF information.

https://doi.org/10.1155/2018/1359610

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 61673320); The Scientific Research Project of Department of Education of Sichuan Province (15TD0027, 18ZA0273); Natural Science Foundation of Guangdong Province (2016A030310003).

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Ya Qin [ID], (1,2) Yi Liu [ID], (1) and Jun Liu (3)

(1) Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang, Sichuan 641000, China

(2) School of Mathematics and Information, Neijiang Normal University, Neijiang, Sichuan 641000, China

(3) School of Computing, Ulster University, Jordanstown Campus, UK

Correspondence should be addressed to Yi Liu; liuyiyl@126.com

Received 17 December 2017; Revised 1 June 2018; Accepted 19 June 2018; Published 5 July 2018

Academic Editor: Peide Liu

Caption: Figure 1: Satisfaction degree obtained by IPIVIFOWAD under different [epsilon].

Caption: Figure 2: Satisfaction degree obtained by PIVIFOWAD under different [epsilon] and [xi] = 0.4.

Caption: Figure 3: Satisfaction degree obtained by PIVIFOWAD under different [xi] and [epsilon] = 0.5.
Table 1: IVIF decision matrix.

                    [G.sub.1]                  [G.sub.2]

[O.sub.1]     ([0.6,0.7], [0.2,0.3])     ([0.6,0.7], [0.1,0.2])
[O.sub.2]     ([0.3,0.4], [0.5,0.6])     ([0.2,0.3], [0.6,0.7])
[O.sub.3]     ([0.5,0.6], [0.2,0.4])     ([0.5,0.7], [0.2,0.3])
[O.sub.4]     ([0.3,0.4], [0.4,0.5])     ([0.5,0.6], [0.3,0.4])
[O.sub.5]     ([0.4,0.6], [0.2,0.4])     ([0.5,0.6], [0.2,0.3])

                    [G.sub.3]                  [G.sub.4]

[O.sub.1]     ([0.5,0.6], [0.3.0.4])     ([0.4,0.5], [0.2,0.3])
[O.sub.2]     ([0.4,0.6], [0.2,0.3])     ([0.7,0.7], [0.1,0.2])
[O.sub.3]     ([0.4,0.5], [0.3,0.4])     ([0.2,0.3], [0.5,0.6])
[O.sub.4]     ([0.5,0.6], [0.2,0.3])     ([0.5,0.6], [0.2,0.3])
[O.sub.5]     ([0.6,0.7], [0.2,0.3])     ([0.3,0.4], [0.5,0.6])

                    [G.sub.5]

[O.sub.1]     ([0.5,0.6], [0.3,0.4])
[O.sub.2]     ([0.4,0.6], [0.1,0.2])
[O.sub.3]     ([0.3,0.4], [0.4,0.5])
[O.sub.4]     ([0.5,0.6], [0.2,0.3])
[O.sub.5]     ([0.6,0.7], [0.2,0.3])

Table 2: The results by using score function.

             [G.sub.1]    [G.sub.2]    [G.sub.3]    [G.sub.4]

[s.sub.1]       0.4          0.5          0.2          0.2
[s.sub.2]      -0.2         -0.4          0.25         0.6
[s.sub.3]       0.25         0.35         0.1         -0.3
[s.sub.4]      -0.1          0.2          0.3          0.3
[s.sub.5]       0.2          0.3          0.4         -0.2

             [G.sub.5]

[s.sub.1]       0.2
[s.sub.2]       0.35
[s.sub.3]      -0.1
[s.sub.4]       0.3
[s.sub.5]       0.4

Table 3: The distance between [[??].sub.ij] and [[??].sub.j].

             [[??].sub.1]    [[??].sub.2]    [[??].sub.3]

[O.sub.1]          0               0              0.1
[O.sub.2]         0.3             0.5            0.15
[O.sub.3]         0.1             0.1             0.2
[O.sub.4]         0.3             0.2             0.1
[O.sub.5]        0.15             0.1              0

             [[??].sub.4]    [[??].sub.5]

[O.sub.1]         0.3             0.1
[O.sub.2]          0             0.25
[O.sub.3]         0.5             0.3
[O.sub.4]         0.2             0.1
[O.sub.5]         0.4              0

Table 4: The distance between [[??].sub.ij] and [[??].sub.j].

             [[??].sub.1]    [[??].sub.2]    [[??].sub.3]

[O.sub.1]         0.3             0.5             0.1
[O.sub.2]          0               0              0.1
[O.sub.3]        0.25             0.4              0
[O.sub.4]         0.1             0.3             0.1
[O.sub.5]        0.25             0.4             0.2

             [[??].sub.4]    [[??].sub.5]

[O.sub.1]         0.3             0.2
[O.sub.2]         0.5             0.3
[O.sub.3]          0               0
[O.sub.4]         0.3             0.2
[O.sub.5]         0.1             0.3

Table 5: The IP according to Table 3.

              [G.sub.1]    [G.sub.2]    [G.sub.3]    [G.sub.4]

[IP.sub.1]      0.1053       0.1579       0.1053       0.4737
[IP.sub.2]      0.2609       0.3913       0.0435       0.2609
[IP.sub.3]      0.1053       0.1579       0.1053       0.4737
[IP.sub.4]      0.2607       0.3913       0.0435       0.2607
[IP.sub.5]      0.0952       0.4286       0.1429       0.2857

              [G.sub.5]

[IP.sub.1]      0.1579
[IP.sub.2]      0.0435
[IP.sub.3]      0.1579
[IP.sub.4]      0.0435
[IP.sub.5]      0.0476

Table 6: The IP according to Table 4.

              [G.sub.1]    [G.sub.2]    [G.sub.3]    [G.sub.4]

[IP.sub.1]      0.3158       0.1579       0.0526       0.3158
[IP.sub.2]      0.1053       0.1579       0.1053       0.4737
[IP.sub.3]      0.2857       0.4286       0.0952       0.1429
[IP.sub.4]      0.2857       0.1429       0.0476       0.4286
[IP.sub.5]      0.3158       0.1579       0.1579       0.3158

              [G.sub.5]

[IP.sub.1]      0.1579
[IP.sub.2]      0.1579
[IP.sub.3]      0.0476
[IP.sub.4]      0.0952
[IP.sub.5]      0.0526

Table 7: Satisfaction degree obtained by IPIVIFOWAD under different
risk preference parameter [epsilon].

                     [lambda]([O.sub.1])     [lambda]([O.sub.2])

[epsilon] = 0.1            0.9747                  0.7696

[epsilon] = 0.3            0.9088                  0.4641

[epsilon] = 0.5            0.8103                  0.2707

[epsilon] = 0.7            0.6468                  0.1372

[epsilon] = 0.9            0.3219                  0.0396

                     [lambda]([O.sub.3])     [lambda]([O.sub.4])

[epsilon] = 0.1            0.9154                  0.8976

[epsilon] = 0.3            0.7372                  0.6943

[epsilon] = 0.5            0.5459                  0.4933

[epsilon] = 0.7            0.3400                  0.2944

[epsilon] = 0.9            0.1178                  0.0976

                     [lambda]([O.sub.5])

[epsilon] = 0.1            0.9560

[epsilon] = 0.3            0.8492

[epsilon] = 0.5            0.7070

[epsilon] = 0.7            0.5084

[epsilon] = 0.9            0.2115

                                   Ranking

[epsilon] = 0.1       [O.sub.1] > [O.sub.5] > [O.sub.3]
                           > [O.sub.4] > [O.sub.2]
[epsilon] = 0.3       [O.sub.1] > [O.sub.5] > [O.sub.3]
                           > [O.sub.4] > [O.sub.2]
[epsilon] = 0.5       [O.sub.1] > [O.sub.5] > [O.sub.3]
                           > [O.sub.4] > [O.sub.2]
[epsilon] = 0.7       [O.sub.1] > [O.sub.5] > [O.sub.3]
                           > [O.sub.4] > [O.sub.2]
[epsilon] = 0.9       [O.sub.1] > [O.sub.5] > [O.sub.3]
                           > [O.sub.4] > [O.sub.2]

Table 8: The probabilities according to Table 3.

                [G.sub.1]    [G.sub.2]    [G.sub.3]    [G.sub.4]

[[??].sub.1]       0.14         0.18         0.16         0.3
[[??].sub.2]       0.26         0.3          0.1          0.24
[[??].sub.3]       0.14         0.18         0.16         0.3
[[??].sub.4]       0.26         0.3          0.1          0.24
[[??].sub.5]       0.14         0.3          0.22         0.24

                [G.sub.5]

[[??].sub.1]       0.22
[[??].sub.2]       0.1
[[??].sub.3]       0.22
[[??].sub.4]       0.1
[[??].sub.5]       0.1

Table 9: The probabilities according to Table 4.

                [G.sub.1]    [G.sub.2]    [G.sub.3]    [G.sub.4]

[[??].sub.1]       0.26         0.18         0.1          0.24
[[??].sub.2]       0.14         0.18         0.16         0.3
[[??].sub.3]       0.26         0.3          0.16         0.18
[[??].sub.4]       0.26         0.18         0.1          0.3
[[??].sub.5]       0.26         0.18         0.22         0.24

                [G.sub.5]

[[??].sub.1]       0.22
[[??].sub.2]       0.22
[[??].sub.3]       0.1
[[??].sub.4]       0.16
[[??].sub.5]       0.1

Table 10: Modified IVIF decision matrix.

                    [G.sub.1]                  [G.sub.2]

[O.sub.1]     ([0.6,0.7], [0.2,0.3])     ([0.6,0.7], [0.1,0.2])
[O.sub.2]     ([0.3,0.4], [0.5,0.6])     ([0.2,0.3], [0.6,0.7])
[O.sub.3]     ([0.2,0.4], [0.5,0.6])     ([0.2,0.3], [0.5,0.7])
[O.sub.4]     ([0.4,0.5], [0.3,0.4])     ([0.3,0.4], [0.5,0.6])
[O.sub.5]     ([0.4,0.6], [0.2,0.4])     ([0.5,0.6], [0.2,0.3])

                    [G.sub.3]                  [G.sub.4]

[O.sub.1]     ([0.5,0.6], [0.3.0.4])     ([0.4,0.5], [0.2,0.3])
[O.sub.2]     ([0.4,0.6], [0.2,0.3])     ([0.7,0.7], [0.1,0.2])
[O.sub.3]     ([0.3,0.4], [0.4,0.5])     ([0.5,0.6], [0.2,0.3])
[O.sub.4]     ([0.2,0.3], [0.5,0.6])     ([0.2,0.3], [0.5,0.6])
[O.sub.5]     ([0.6,0.7], [0.2,0.3])     ([0.3,0.4], [0.5,0.6])

                    [G.sub.5]

[O.sub.1]     ([0.5,0.6], [0.3,0.4])
[O.sub.2]     ([0.4,0.6], [0.1,0.2])
[O.sub.3]     ([0.4,0.5], [0.3,0.4])
[O.sub.4]     ([0.2,0.3], [0.5,0.6])
[O.sub.5]     ([0.6,0.7], [0.2,0.3])

Table 11: Satisfaction degree obtained by PIVIFOWAD under
different [xi].

                [lambda]([O.sub.1])     [lambda]([O.sub.2])

[xi] = 0.1            0.7762                  0.2740

[xi] = 0.3            0.7584                  0.3136

[xi] = 0.5            0.7419                  0.3509

[xi] = 0.7            0.7267                  0.3859

[xi] = 0.9            0.7126                  0.4190

                [lambda]([O.sub.3])     [lambda]([O.sub.4])

[xi] = 0.1            0.5053                  0.4923

[xi] = 0.3            0.4766                  0.5027

[xi] = 0.5            0.4487                  0.5128

[xi] = 0.7            0.4217                  0.5231

[xi] = 0.9            0.3955                  0.5333

                [lambda]([O.sub.5])

[xi] = 0.1            0.7008

[xi] = 0.3            0.6921

[xi] = 0.5            0.6835

[xi] = 0.7            0.6751

[xi] = 0.9            0.6667

                               Ranking

[xi] = 0.1       [O.sub.1] > [O.sub.5] > [O.sub.3] >
                        [O.sub.4] > [O.sub.2]
[xi] = 0.3       [O.sub.1] > [O.sub.5] > [O.sub.4] >
                        [O.sub.3] > [O.sub.2]
[xi] = 0.5       [O.sub.1] > [O.sub.5] > [O.sub.4] >
                        [O.sub.2] > [O.sub.2]
[xi] = 0.7       [O.sub.1] > [O.sub.5] > [O.sub.4] >
                        [O.sub.3] > [O.sub.2]
[xi] = 0.9       [O.sub.1] > [O.sub.5] > [O.sub.4] >
                        [O.sub.2] > [O.sub.3]

Table 12: Modified IVIF decision matrix.

                    [G.sub.1]                  [G.sub.2]

[O.sub.1]    ([0.6,0.7], [0.2,0.3])     ([0.6,0.7], [0.1,0.2])
[O.sub.2]    ([0.3,0.4], [0.5,0.6])     ([0.2,0.3], [0.6,0.7])
[O.sub.3]    ([0.2,0.4], [0.5,0.6])     ([0.2,0.3], [0.5,0.7])
[O.sub.4]    ([0.4,0.5], [0.3,0.4])     ([0.3,0.4], [0.5,0.6])
[O.sub.5]    ([0.4,0.6], [0.2,0.4])     ([0.5,0.6], [0.2,0.3])

                    [G.sub.3]                  [G.sub.4]

[O.sub.1]    ([0.5,0.6], [0.3,0.4])     ([0.4,0.5], [0.2,0.3])
[O.sub.2]    ([0.4,0.6], [0.2,0.3])     ([0.7,0.7], [0.1,0.2])
[O.sub.3]    ([0.3,0.4], [0.4,0.5])     ([0.5,0.6], [0.2,0.3])
[O.sub.4]    ([0.2,0.3], [0.5,0.6])     ([0.2,0.3], [0.5,0.6])
[O.sub.5]    ([0.6,0.7], [0.2,0.3])     ([0.3,0.4], [0.5,0.6])

                    [G.sub.5]

[O.sub.1]    ([0.5,0.6], [0.3,0.4])
[O.sub.2]    ([0.4,0.6], [0.1,0.2])
[O.sub.3]    ([0.4,0.5], [0.3,0.4])
[O.sub.4]    ([0.2,0.3], [0.5,0.6])
[O.sub.5]    ([0.6,0.7], [0.2,0.3])
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Title Annotation:Research Article
Author:Qin, Ya; Liu, Yi; Liu, Jun
Publication:Mathematical Problems in Engineering
Date:Jan 1, 2018
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