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Introduction

In a fuzzy environment, the ranking of fuzzy numbers (FNs) plays an important role in practical use and has become a prerequisite procedure for a decision-making problem. In fuzzy decision analysis, FNs are employed to describe the performance of alternatives, and the selection of alternatives will eventually lead to the ranking of corresponding FNs. However, ranking of FNs is not an easy task since FNs are represented by possibility distribution and they can overlap with each other.

Various methods for ranking fuzzy numbers (RFNs) have been developed such as distance index by [1], signed distance by [2, 3], area index by [4], and centroid index by [5]. However, no method can rank FNs satisfactorily in all cases and situations [6]. Some methods are limited to normal and trapezoidal shapes of FNs and only consider neutral decision-makers’ view. There are also methods that cannot distinguish the ranking of FNs having the same mode and symmetric spread, and some methods produce non-discriminate and nonintuitive results.

In this paper, a new method for RFNs based on Ochiai index and Hurwicz criterion is proposed. Ochiai is a similarity measure index and Hurwicz is a criterion for decision-making that compromises between the optimistic and pessimistic criteria. Thus, the proposed ranking method considers all types of decision-makers’ view such as optimistic, neutral, and pessimistic which is crucial in solving decision-making problems.

This paper is organized as follows. Section “Preliminaries” contains basic concepts and notations used in the remaining parts of the paper. In section “New Method for Ranking Fuzzy Numbers,” the new ranking method based on Ochiai index and Hurwicz criterion is proposed. Two observations on the new ranking method are presented in section “Observations on the Ochiai with Hurwicz Criterion Ranking Index.” Section “Numerical Examples” presents some numerical examples to illustrate the advantages of the proposed method. The paper ends with a conclusion in section “Conclusion.”

Preliminaries

In this section, some basic concepts and definitions on FNs are reviewed from the literature.

Definition 1

A fuzzy number is a fuzzy set in the universe of discourse X with the membership function defined as [7]

$$ {\mu}_A(x)=\Big\{\begin{array}{cc}{\mu}_A^L(x)&, \kern0.24em a\le x\le b\\ {}w&, \kern0.24em b\le x\le c\\ {}{\mu}_A^R(x)&, \kern0.24em c\le x\le d\\ {}0&, \kern0.24em \mathrm{otherwise}\end{array}. $$

where μ L A  : [a, b] → [0, w], μ R A  : [c, d] → [0, w], w ∈ (0, 1], μ L A and μ R A denote the left and the right membership functions of the fuzzy number A.

The membership function μ A of a fuzzy number A has the following properties:

  1. 1.

    μ A is a continuous mapping from the universe of discourse X to [0, w].

  2. 2.

    μ A (x) = 0 for x < a and x > d.

  3. 3.

    μ A (x) is monotonic increasing in [a, b].

  4. 4.

    μ A (x) = w for [b, c].

  5. 5.

    μ A (x) is monotonic decreasing in [c, d].

If the membership function μ A (x) is a piecewise linear, then A is called as a trapezoidal fuzzy number with membership function defined as

$$ {\mu}_A(x)=\Big\{\begin{array}{cc}w\left(\frac{x-a}{b-a}\right)&, \kern0.24em a\le x\le b\\ {}w&, \kern0.24em b\le x\le c\\ {}w\left(\frac{d-x}{d-c}\right)&, \kern0.24em c\le x\le d\\ {}0&, \kern0.32em \mathrm{otherwise}\end{array} $$

and denoted as A = (a, b, c, d; w). If b = c, then the trapezoidal becomes a triangular fuzzy number denoted as A = (a, b, d; w).

Definition 2

Let A 1 and A 2 be two fuzzy numbers with \( {A}_{1_{\alpha }}=\left[{a}_{\alpha}^{-},{a}_{\alpha}^{+}\right] \) and A 2α  = [b − α , b + α ] be their α-cuts with α ∈ [0, 1] [8]. The fuzzy maximum of A 1 and A 2 by the α-cuts method is defined as

$$ {\left[ MAX\left({A}_1,{A}_2\right)\right]}_{\alpha }=\left[ \max \left({a}_{\alpha}^{-},{b}_{\alpha}^{-}\right), \max \left({a}_{\alpha}^{+},{b}_{\alpha}^{+}\right)\right]. $$

The fuzzy minimum of A 1 and A 2 is defined as

$$ {\left[ MIN\left({A}_1,{A}_2\right)\right]}_{\alpha }=\left[ \min \left({a}_{\alpha}^{-},{b}_{\alpha}^{-}\right), \min \left({a}_{\alpha}^{+},{b}_{\alpha}^{+}\right)\right]. $$

Definition 3

Let A 1 = (a 1, b 1, c 1, d 1; h 1) and A 2 = (a 2, b 2, c 2, d 2; h 2) be two trapezoidal fuzzy numbers [9]. The fuzzy maximum of A 1 and A 2 by the second function principle is defined as

$$ MAX\left({A}_1,{A}_2\right)=\left(a,\kern0.24em b,\kern0.24em c,\kern0.24em d;\kern0.24em h\right) $$

where

h = min{h 1, h 2}, T = {max(a 1, a 2), max(a 1, d 2), max(d 1, a 2), max(d 1, d 2)}, T 1 = {max(b 1, b 2), max(b 1, c 2), max(c 1, b 2), max(c 1, c 2)}, a = min T, b = min T 1, c = max T 1, d = max T, min T ≤ min T 1 and max T 1 ≤ max T.

The fuzzy minimum of A 1 and A 2 is defined as

$$ MIN\left({A}_1,{A}_2\right)=\left(a,\kern0.24em b,\kern0.24em c,\kern0.24em d;\kern0.24em h\right) $$

where

h = min{h 1, h 2}, T = {min(a 1, a 2), min(a 1, d 2), min(d 1, a 2), min(d 1, d 2)}, T 1 ={min(b 1, b 2), min(b 1, c 2), min(c 1, b 2), min(c 1, c 2)}, a = min T, b = min T 1, c = max T 1, d = max T, min T ≤ min T 1 and max T 1 ≤ max T.

Definition 4

The cardinality of a fuzzy number A in the universe of discourse X is defined as [10]

$$ \left|A\right|=\underset{X}{{\displaystyle \int }}{\mu}_A(x) dx. $$

New Method for Ranking Fuzzy Numbers

The new ranking method is developed based on [11] with similarity measure index defined as

$$ {S}_O\left(X,Y\right)=\frac{f\left(X{\displaystyle \cap}\;Y\right)}{\sqrt{f\left(X{\displaystyle \cap}\;Y\right)+f\left(X-Y\right)}\sqrt{f\left(X{\displaystyle \cap}\;Y\right)+f\left(Y-X\right)}} $$

and reduced to \( {S}_O\left(X,Y\right)=\frac{f\left(X{\displaystyle \cap}\;Y\right)}{\sqrt{f(X)}\sqrt{f(Y)}} \) or known as Ochiai index.

Typically, the function f is taken to be the cardinality function. The objects X and Y described by the features are replaced with FNs A and B which are described by the membership functions. The fuzzy Ochiai is defined as

$$ {S}_O\left(A,B\right)=\frac{\left|A{\displaystyle \cap}\;B\right|}{\sqrt{\left|A\right|}\sqrt{\left|B\right|}}, $$

where |A| denotes the scalar cardinality of fuzzy number A. ∩ and ∪ are the t-norm and s-norm, respectively. The fuzzy Ochiai ranking index with Hurwicz criterion is presented as follows:

  • Step 1: For each pair of the FNs A i and A j , find the fuzzy maximum and fuzzy minimum of A i and A j . The fuzzy maximum and fuzzy minimum can be obtained by the α-cuts method for normal FNs and the second function principle for non-normal FNs.

  • Step 2: Calculate the evidences of E(A i  ≻ A j ), E(A j  ≺ A i ), E(A j  ≻ A i ) and E(A i  ≺ A j ) which are defined based on fuzzy Ochiai index as

    $$ E\left({A}_i\succ {A}_j\right)={S}_O\left( MAX\left({A}_i,{A}_j\right),{A}_i\right), $$
    $$ E\left({A}_j\prec {A}_i\right)={S}_O\left( MIN\left({A}_i,{A}_j\right),{A}_j\right), $$
    $$ E\left({A}_j\succ {A}_i\right)={S}_O\left( MAX\left({A}_i,{A}_j\right),{A}_j\right), $$
    $$ E\left({A}_i\prec {A}_j\right)={S}_O\left( MIN\left({A}_i,{A}_j\right),{A}_i\right), $$

    where \( {S}_O\left({A}_i,{A}_j\right)=\frac{\left|{A}_i{\displaystyle \cap }{A}_j\right|}{\sqrt{\left|{A}_i\right|}\sqrt{\left|{A}_j\right|}} \) is the fuzzy Ochiai index and |A i | denotes the scalar cardinality of fuzzy number A i .

    To simplify, C ij and c ji are used to represent E(A i  ≻ A j ) and E(A j  ≺ A i ), respectively. Likewise, C ji and c ij are used to denote E(A j  ≻ A i ) and E(A i  ≺ A j ) respectively.

  • Step 3: Calculate the total evidences E total (A i  ≻ A j ) and E total (A j  ≻ A i ) which are defined based on the Hurwicz criterion concept as

    $$ {E}_{total}\left({A}_i\succ {A}_j\right)=\beta {C}_{ij}+\left(1-\beta \right){c}_{ji} $$
    (83.1)
    $$ {E}_{total}\left({A}_j\succ {A}_i\right)=\beta {C}_{ji}+\left(1-\beta \right){c}_{ij} $$
    (83.2)

    β ∈ [0, 0.5), β = 0.5 and β ∈ (0.5, 1] represent pessimistic, neutral, and optimistic criteria, respectively.

    To simplify, E O (A i , A j ) and E O (A j , A i ) are used to represent E total (A i  ≻ A j ) and E total (A j  ≻ A i ), respectively.

  • Step 4: For each pair of the FNs, compare the total evidences in Step 3 which will result the ranking of two FNs A i and A j as follows:

    • $$ 1.\kern0.46em {A}_i\succ {A}_j\;\mathrm{if}\;\mathrm{and}\;\mathrm{only}\;\mathrm{if}\;{E}_O\left({A}_i,{A}_j\right)>{E}_O\left({A}_j,{A}_i\right) $$
      (83.3)
    • $$ 2.\kern0.46em {A}_i\prec {A}_j\;\mathrm{if}\;\mathrm{and}\;\mathrm{only}\;\mathrm{if}\;{E}_O\left({A}_i,{A}_j\right)<{E}_O\left({A}_j,{A}_i\right) $$
      (83.4)
    • $$ 3.\kern0.46em {A}_i\approx {A}_j\;\mathrm{if}\;\mathrm{and}\;\mathrm{only}\;\mathrm{if}\;{E}_O\left({A}_i,{A}_j\right)={E}_O\left({A}_j,{A}_i\right) $$
      (83.5)

Observations on the Ochiai with Hurwicz Criterion Ranking Index

The ranking results of the proposed method were observed based on the values of d ij , β ij , n ij and Equations (3.1), (3.2), (3.3), (3.4), and (3.5), with d ij  = C ij  − c ji  − C ji  + c ij , n ij  = c ij  − c ji and \( {\beta}_{ij}=\frac{c_{ij}-{c}_{ji}}{d_{ij}} \). The observation can be divided into six cases as follows.

Case 1:

Let d ij  ≠ 0, β = β ij and β ij  ∈ [0, 1].

Since \( {\beta}_{ij}=\frac{c_{ij}-{c}_{ji}}{d_{ij}} \) we have

$$ \beta =\frac{c_{ij}-{c}_{ji}}{d_{ij}}=\frac{c_{ij}-{c}_{ji}}{C_{ij}-{c}_{ji}-{C}_{ji}+{c}_{ij}}, $$

β(C ij  − c ji  − C ji  + c ij ) = c ij  − c ji , and rearranging the equation will give

β C ij  + (1 − β) c ji  = βC ji  + (1 − β)c ij which implies E O (A i  ≻ A j ) = E O (A j  ≻ A i )

and, therefore, A i  ≈ A j .

Thus, if d ij  ≠ 0, β = β ij and β ij  ∈ [0, 1], then A i  ≈ A j .

Case 2:

Let d ij  ≠ 0, β > β ij and β ij  ∈ (−∞, 1).

Since \( {\beta}_{ij}=\frac{c_{ij}-{c}_{ji}}{d_{ij}} \) we have \( \beta >\frac{c_{ij}-{c}_{ji}}{d_{ij}}=\frac{c_{ij}-{c}_{ji}}{C_{ij}-{c}_{ji}-{C}_{ji}+{c}_{ij}} \).

For d ij  < 0,

β(C ij  − c ji  − C ji  + c ij ) < c ij  − c ji , and rearranging the inequality will give

β C ij  + (1 − β)c ji  < β C ji  + (1 − β)c ij which implies E O (A i  ≻ A j ) < E O (A j  ≻ A i )

and, therefore, A i  ≺ A j .

For d ij  < 0,

β(C ij  − c ji  − C ji  + c ij ) > c ij  − c ji , and rearranging the inequality will give

β C ij  + (1 − β) c ji  > β C ji  + (1 − β) c ij which implies E O (A i  ≻ A j ) > E O (A j  ≻ A i )

and, therefore, A i  ≻ A j .

Thus,

  1. 1.

    If d ij  > 0, β > β ij and β ij  ∈ (−∞, 1), then A i  ≻ A j .

  2. 2.

    If d ij  < 0, β > β ij , and β ij  ∈ (−∞, 1), then A i  ≺ A j .

Case 3:

Let d ij  ≠ 0, β < β ij and β ij  ∈ (0, + ∞).

Since \( {\beta}_{ij}=\frac{c_{ij}-{c}_{ji}}{d_{ij}} \) we have \( \beta <\frac{c_{ij}-{c}_{ji}}{d_{ij}}=\frac{c_{ij}-{c}_{ji}}{C_{ij}-{c}_{ji}-{C}_{ji}+{c}_{ij}} \).

For d ij  > 0,

β(C ij  − c ji  − C ji  + c ij ) < c ij  − c ji , and rearranging the inequality will give

β C ij  + (1 − β) c ji  < β C ji  + (1 − β) c ij which implies E O (A i  ≻ A j ) < E O (A j  ≻ A i )

and, therefore, A i  ≺ A j .

For d ij  < 0,

β(C ij  − c ji  − C ji  + c ij ) > c ij  − c ji , and rearranging the inequality will give

β C ij  + (1 − β) c ji  > β C ji  + (1 − β) c ij which implies E O (A i  ≻ A j ) > E O (A j  ≻ A i )

and, therefore, A i  ≻ A j .

Thus,

  1. 1.

    If d ij  > 0, β < β ij , and β ij  ∈ (0, + ∞), then A i  ≺ A j .

  2. 2.

    If d ij  < 0, β < β ij , and β ij  ∈ (0, + ∞), then A i  ≻ A j .

Case 4:

Let d ij  = 0 and n ij  > 0.

Then, for all β ∈ [0, 1],

$$ \beta {d}_{ij}<{n}_{ij}. $$

Since d ij  = C ij  − c ji  − C ji  + c ij and n ij  = c ij  − c ji , then

β(C ij  − c ji  − C ji  + c ij ) < c ij  − c ji , and rearranging the inequality will give

βC ij  + (1 − β)c ji  < βC ji  + (1 − β)c ij which implies E O (A i  ≻ A j ) < E O (A j  ≻ A i )

and, therefore, A i  ≺ A j .

Thus, if d ij  = 0 and n ij  > 0, then for all β ∈ [0, 1], A i  ≺ A j .

Case 5:

Let d ij  = 0 and n ij  < 0. Then, for all β ∈ [0, 1],

$$ \beta {d}_{ij}>{n}_{ij}. $$

Thus, β(C ij  − c ji  − C ji  + c ij ) > c ij  − c ji , and rearranging the inequality will give

βC ij  + (1 − β)c ji  > βC ji  + (1 − β)c ij which implies E O (A i  ≻ A j ) > E O (A j  ≻ A i )

and, therefore, A i  ≻ A j .

Thus, if d ij  = 0 and n ij  < 0, then for all β ∈ [0, 1], A i  ≻ A j .

Case 6:

Let d ij  = 0 and n ij  = 0.

Then, for all β ∈ [0, 1],

$$ \beta {d}_{ij}={n}_{ij}. $$

Thus, β(C ij  − c ji  − C ji  + c ij ) = c ij  − c ji , and rearranging the equation will give

βC ij  + (1 − β)c ji  = βC ji  + (1 − β)c ij which implies E O (A i  ≻ A j ) = E O (A j  ≻ A i )

and, therefore, A i  ≈ A j .

Thus, if d ij  = 0 and n ij  = 0, then for all β ∈ [0, 1], A i  ≈ A j .

The ranking result of the proposed method can be classified as having two main observations which are Observations 4.1 (covers cases 1–3) and 4.2 (covers cases 4–6). The two main observations are presented as follows.

Observation 4.1

For two FNs A i and A j with d ij  ≠ 0, the ranking results for Ochiai index are as follows:

  1. 1.

    If d ij  ≠ 0 and β = β ij , then A i  ≈ A j .

  2. 2.

    If d ij  > 0 and

    1. (a)

      β > β ij , then A i  ≻ A j .

    2. (b)

      β < β ij , then A i  ≺ A j .

  1. 3.

    If d ij  < 0 and

    1. (a)

      β > β ij , then A i  ≺ A j .

    2. (b)

      β < β ij , then A i  ≻ A j .

Observation 4.2

For two FNs A i and A j with d ij  = 0, the ranking results for Ochiai index are as follows:

  1. 1.

    If n ij  > 0, then for all β ∈ [0, 1], A i  ≺ A j .

  2. 2.

    If n ij  < 0, then for all β ∈ [0, 1], A i  ≻ A j .

  3. 3.

    If n ij  = 0, then for all β ∈ [0, 1], A i  ≈ A j .

Numerical Examples

In this section, four sets of numerical examples are presented to illustrate the validity and advantages of fuzzy Ochiai ranking index. For two FNsA 1 and A 2, d 12 = C 12 − c 21 − C 21 + c 12, \( {\beta}_{12}=\frac{c_{12}-{c}_{21}}{d_{12}} \), and n 12 = c 12 − c 21 and C 12, c 21, C 21 and c 12 denoted the evidences E(A 1 ≻ A 2), E(A 2 ≺ A 1), E(A 2 ≻ A 1) and E(A 1 ≺ A 2) respectively.

Example 1

Consider the FNs in [12], i.e., A 1 = (0.1, 0.3, 0.5) and A 2 = (0.2, 0.3, 0.4).

Since A 1 and A 2 have the same mode and symmetric spread, a number of the existing ranking methods cannot discriminate them, such as [1–5, 13–16]. However, [12, 17–21, 22] produce A 1 ≺ A 2. By the proposed method, we obtain d 12 = 0.098 > 0 and β 12 = 0.5.

Thus, by Observation 4.1 the ranking order is produced as \( \begin{array}{cc}{A}_1\prec {A}_2&, \beta \in \left[0,0.5\right)\\ {}{A}_1\approx {A}_2&, \beta =0.5\\ {}{A}_1\succ {A}_2&, \beta \in \left(0.5,1\right]\end{array} \), where A 1 ≺ A 2 for pessimistic decision-makers, A 1 ≈ A 2 for neutral decision-makers, and A 1 ≻ A 2 for optimistic decision-makers. The ranking result is affected by decision-makers’ perspective, and this shows that the proposed method has strong discrimination ability.

Example 2

Consider the FNs in [15], i.e., A 1 = (0.3, 0.5, 0.9) and A 2 = (0.155, 0.645, 0.8). References [1, 4] rank them as A 1 ≺ A 2, while [12, 15] produce A 1 ≻ A 2. By the proposed method, we obtain d 12 = − 0.006 < 0 and β 12 = 0.833, and by Observation 4.1 the ranking order is produced as \( \begin{array}{cc}{A}_1\succ {A}_2& \kern0.84em ,\beta \in \left[0,0.833\right)\\ {}{A}_1\approx {A}_2&, \beta =0.833\\ {}{A}_1\prec {A}_2& \kern0.72em ,\beta \in \left(0.833,1\right]\end{array} \). Both neutral and pessimistic decision-makers rank A 1 ≻ A 2 while optimistic decision-makers rank them in three different results. The result shows that the equal ranking does not necessarily occur for neutral decision-makers.

Example 3

Consider the FNs in [5], i.e., A 1 = (6, 7, 9, 10; 0.6) and A 2 = (5, 7, 9, 10; 1) as shown in Fig. 83.1.

Fig. 83.1
figure 1

Fuzzy numbers in Example 3

Full size image

Some of the existing ranking methods such as [2, 3, 15, 16, 23–25] can only rank normal FNs and, thus, fail to rank the FNs A 1 and A 2. Moreover, [1, 5] rank them as A 2 ≺ A 1, while [4] ranks them as A 1 ≺ A 2. By the proposed method, d 12 = 0.434 > 0 and β 12 = 0.350, thus, obtain the ranking result as A 1 ≺ A 2 for β ∈ [0, 0.350), A 1 ≈ A 2 for β = 0.350 and A 1 ≻ A 2 for β ∈ (0.350, 1]. Similarly, the ranking result is affected by decision-makers’ perspective.

Example 4

Consider the FNs in [24], i.e., A 1 = (1, 2, 5) and A 2 = (1, 2, 2, 4) as shown in Fig. 83.2, with the membership function of A 2 defined as \( {\mu}_{A_2}(x)=\Big\{\begin{array}{cc}\sqrt{1-{\left(x-2\right)}^2}&, \left[1,2\right]\\ {}\sqrt{1-\frac{1}{4}{\left(x-2\right)}^2}&, \left[2,4\right]\\ {}0&, \mathrm{else}\end{array} \).

Fig. 83.2
figure 2

Fuzzy numbers in Example 4

Full size image

Some of the existing ranking methods such as [12, 17, 18] can only rank trapezoidal FNs and, thus, fail to rank the FNs A 1 and A 2. By using the proposed method, we have d 12 = − 0.015 < 0 and β 12 = 3.88. Therefore, the ranking order is A 1 ≻ A 2 regardless of the decision-makers’ perspective, as shown in Table 83.1. The ranking result of the proposed method is consistent with human intuition and other ranking methods in Table 83.1.

Table 83.1 Ranking results of Example 4
Full size table

Conclusion

This paper presents a new method for RFNs using Ochiai index and Hurwicz criterion. Two observations that can simplify the ranking procedure are produced. The observations have rendered the proposed ranking index as an advantageous method since the ranking results can be obtained for all continuous values of β ∈ [0, 1]. The proposed method can overcome certain shortcomings that exist in the previous ranking methods such as can rank both non-normal and general shapes of FNs and can discriminate the ranking of FNs having the same mode and symmetric spreads which fail to be ranked by the previous ones. The proposed method can be highly applied in solving decision-making as it has strong discrimination ability which is a crucial criterion in solving decision-making problems.