Multi-criteria group decision making based on neutrosophic analytic hierarchy process

1 Introduction

Multi-Criteria Decision Making (MCDM) is a formal and structured decision making methodology for dealing with complex problems [1]. Saaty [2] founded the Analytic Hierarchy Process in the late 1970s. Today it is the popular and the most widely used method for dealing with Multi-Criteria Decision Making. Analytic Hierarchy Process allows the use of qualitative, as well as quantitative criteria in evaluation. The basic steps of AHP algorithm is basically consist of three steps:

Decompose complex problem into a hierarchical multi-level structure of objectives, criteria and alternatives.

Vitality is measured on an integer-valued 1–9 scale, with each number having the explanation shown in Table 3. A basic, but very reasonable assumption for comparing alternatives: if attribute A is absolutely more important than attribute B and is rated at 9, then B must be absolutely less important than A and is graded as 1/9. These pair-wise comparisons are performed on all factors to be considered. In real life problems, the decision makers may be unable to allocate preference values to the objects considered. Neutrosophic set is a generalization of crisp sets, fuzzy sets and intuitionistic fuzzy sets to represent uncertain, inconsistent, and incomplete information about real world problems [3–7]. For the first time, this research attempts to introduce the mathematical representation of AHP in neutrosophic surroundings. In Table 1 a comparison between fuzzy, intuitionistic fuzzy and neutrosophic set is presented. Also the difference between classical, fuzzy and neutrosophic AHP presented in Table 2.

The organization of the research as it’s summed up:

Section 2 gives an insight into some basic definitions on neutrosophic sets. Section 3 explains the proposed methodology of neutrosophic Analytic Hierarchy Process. Section 4 introduces numerical example. Finally section 5 concludes the paper with future work.

In this section, the essential definitions involving neutrosophic set, single valued neutrosophic sets, triangular neutrosophic numbers and operations on triangular neutrosophic numbers are defined.

Definition 1. [8, 10] Let X be a space of points and x ∈ X. A neutrosophic set A in X is definite by a truth-membership function T _A  (x), an indeterminacy-membership function I _A  (x) and a falsity-membership function F _A  (x) · T _A  (x), I _A  (x) and F _A  (x) are real standard or real nonstandard subsets of]^–0, 1+[. That is T _A  (x) : X→] ^-0, 1⁺ [, I _A  (x) : X  →  ] ^-0, 1⁺ [and F _A  (x) : X → ]^ - 0,1^ + [. There is no restriction on the sum of T _A  (x) , I _A  (x) and F_A (x) , so 0 -≤  sup T _A  (x) +sup I _A  (x)  +  sup F _A  (x) ≤ 3 + .

Definition 2. [8, 11] Let X be a universe of discourse. A single valued neutrosophic set A over X is an object taking the form A ={ 〈 x, T _A  (x) , I _A  (x) , F _A  (x) 〉 : x ∈ X }, where T _A  (x) : X → [0, 1] , I _A  (x) : X → [0, 1] and F _A  (x) : X → [0, 1] with 0 ≤ T _A  (x) + I _A  (x) + F _A  (x) ≤ 3 for all x ∈ X. The intervals T _A  (x) , I _A  (x) and F _A represent the truth-membership degree, the indeterminacy-membership degree and the falsity membership degree of x to A, respectively. For convenience, a SVN number is represented by A = (a, b, c), where a, b, c ∈ [0, 1] and a + b + c ≤ 3.

Definition 3. [9, 12] suppose that α a ˜ , θ a ˜ , β a ˜ ∈ [ 0 , 1 ] and a₁, a₂, a₃ ∈ R where a₁ ≤ a₂ ≤ a₃. Then a single valued triangular neutrosophic number, a ˜ = 〈 ( a 1 , a 2 , a 3 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 is a special neutrosophic set on the real line set R whose truth-membership, indeterminacy-membership and falsity-membership functions are defined as: T a ˜ ( x ) = { α a ˜ ( x - a 1 a 2 - a 1 ) ( a 1 ≤ x ≤ a 2 ) α a ˜ ( x = a 2 ) α a ˜ ( a 3 - x a 3 - a 2 ) ( a 2 < x ≤ a 3 ) 0 otherwise ,(1)I a ˜ ( x ) = { ( a 2 - x + θ a ˜ ( x - a 1 ) ) ( a 2 - a 1 ) ( a 1 ≤ x ≤ a 2 ) θ a ˜ ( x = a 2 ) ( x - a 2 + θ a ˜ ( a 3 - x ) ) a 3 - a 2 ( a 2 < x ≤ a 3 ) 1 otherwise ,(2)F a ˜ ( x ) = { ( a 2 - x + β a ˜ ( x - a 1 ) ) ( a 2 - a 1 ) ( a 1 ≤ x ≤ a 2 ) β a ˜ ( x = a 2 ) ( x - a 2 + β a ˜ ( a 3 - x ) ) ( a 3 - a 2 ) ( a 2 < x ≤ a 3 ) 1 otherwise .(3)

Where α a ˜ , θ a ˜ and β a ˜ represent the maximum truth-membership degree, minimum indeterminacy-membership degree and minimum falsity-membership degree respectively. A single valued triangular neutrosophic number a ˜ = 〈 ( a 1 , a 2 , a 3 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 may express an ill-defined quantity about a, which is approximately equal to a.

Definition 4. [9, 10] Let a ˜ = 〈 ( a 1 , a 2 , a 3 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 and b ˜ = 〈 ( b 1 , b 2 , b 3 ) ; α b ˜ , θ b ˜ , β b ˜ 〉 be two single valued triangular neutrosophic numbers and γ ≠ 0 be any real number. Then,

Addition of two triangular neutrosophic numbers a ˜ + b ˜ = 〈 ( a 1 + b 1 , a 2 + b 2 , a 3 + b 3 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉

Subtraction of two triangular neutrosophic numbers a ˜ - b ˜ = 〈 ( a 1 - b 3 , a 2 - b 2 , a 3 - b 1 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉

Inverse of a triangular neutrosophic number a ˜ - 1 = 〈 ( 1 a 3 , 1 a 2 , 1 a 1 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 , Where ( a ˜ ≠ 0 )

Multiplication of triangular neutrosophic number by constant value γ a ˜ = { 〈 ( γ a 1 , γ a 2 , γ a 3 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 if ( γ > 0 ) 〈 ( γ a 3 , γ a 2 , γ a 1 ) ; α a ˜ , θ a ˜ , β a ˜ 〉 if ( γ < 0 )

Division of triangular neutrosophic number by constant value a ˜ γ = { 〈 ( a 1 γ , a 2 γ , a 3 γ ) ; α a ˜ , θ a ˜ , β a ˜ 〉 if ( γ > 0 ) 〈 ( a 3 γ , a 2 γ , a 1 γ ) ; α a ˜ , θ a ˜ , β a 〉 if ( γ < 0 ) Division of two triangular neutrosophic numbers a ˜ b ˜ = { 〈 ( a 1 b 3 , a 2 b 2 , a 3 b 1 ) ; α a ˜ ∧ α b ˜ θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 > 0 , b 3 > 0 ) 〈 ( a 3 b 3 , a 2 b 2 , a 1 b 1 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 < 0 , b 3 > 0 ) 〈 ( a 3 b 1 , a 2 b 2 , a 1 b 3 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 < 0 , b 3 < 0 )

Multiplication of two triangular neutrosophic numbers a ˜ b ˜ = { 〈 ( a 1 b 1 , a 2 b 2 , a 3 b 3 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 > 0 , b 3 > 0 ) 〈 ( a 1 b 3 , a 2 b 2 , a 3 b 1 ) ; α a ˜ ∧ α b ˜ , θ a ˜ , ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 < 0 , b 3 > 0 ) 〈 ( a 3 b 3 , a 2 b 2 , a 1 b 1 ) ; α a ˜ ∧ α b ˜ , θ a ˜ ∨ θ b ˜ , β a ˜ ∨ β b ˜ 〉 if ( a 3 < 0 , b 3 < 0 )

As it is shown before, the AHP include three stages:

Decomposition,

A step-by-step procedure for solving neutrosophic Analytic Hierarchy is provided in this section and the schematic diagram for solving neutrosophic AHP presented in Fig. 1.

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Fig.1

Schematic diagram for solving neutrosophic AHP.

Step 1. Decomposition:

The problem is formed hierarchically at various levels. The first level of hierarch represents the overall goal, the second level represents the decision criteria and sub-criteria and third level is composed of all possible alternatives.

Step 2. Comparative judgments with neutrosophic values:

After analyzing the complex multi-criteria decision making problem into three levels, the pair-wise comparisons used to generate neutrosophic judgment matrix. The vagueness of decision makers is represented by triangular neutrosophic numbers ( a ˜ ij ) . Then construct the neutrosophic pair-wise comparison such that: A ˜ = [ 1 ˜ a ˜ 12 ⋯ a ˜ 1 n ⋮ ⋱ ⋮ a ˜ n 1 a ˜ n 2 ⋯ 1 ˜ ] Where a ˜ ji = a ˜ ij - 1 .

Step 3. For calculating overall priority of each alternative and determine final ranking of all alternatives, we should first determine weights of each criterion from the corresponding pair-wise comparison matrix.

Step 4. To determine the weight of each criterion from corresponding neutrosophic pair-wise comparison matrix, we first transform neutrosophic pair-wise comparison matrix to deterministic pair-wise comparison matrix, using the following equations:

Let a ˜ ij = 〈 ( a 1 , b 1 , c 1 ) , α a ˜ , θ a ˜ , β a ˜ 〉 be a single valued triangular neutrosophic number, then, S ( a ˜ ij ) = 1 16 [ a 1 + b 1 + c 1 ] × ( 2 + α a ˜ - θ a ˜ - β a ˜ )(4)

And A ( a ˜ ij ) = 1 16 [ a 1 + b 1 + c 1 ] × ( 2 + α a ˜ - θ a ˜ + β a ˜ )(5)

Is called the score and accuracy degrees of a ˜ ij , respectively.

To get score and accuracy degree of a ˜ ji we use the following equation: S ( a ˜ ji ) = 1 / S ( a ˜ ij )(6)A ( a ˜ ji ) = 1 / A ( a ˜ ij )(7)

With compensation by score value of each triangular neutrosophic number in neutrosophic pair-wise comparison matrix, we have the following deterministic matrix; A = [ 1 a 12 ⋯ a 1 n ⋮ ⋱ ⋮ a n 1 a n 2 ⋯ 1 ]

From the previous matrix we can easily find ranking of priorities, namely the Eigen Vector X as follows:

Normalize the column entries by dividing each entry by the sum of the column.

Step 5. To measure an inconsistency within the judgments in each comparison matrix and for the entire hierarch, AHP methodology provides a consistency index (CI) [13]. To discerns if there is any inconsistency in neutrosophic judgment matrix, AHP utilizes consistency index and consistency ratio (CR). If CR is greater than 0.1, the judgments are untrustworthy because they are too near to randomness and the exercise is incorrect or must be repeated.

To calculate CI and CR do the following steps:

Multiply each value in the first column of the pair-wise comparison matrix by the priority of the first item; continue this process for all columns of the pair-wise comparison matrix. Sum the values across the rows to get a vector of values labeled “weighted sum”

Step 6. Calculate overall priority of each alternative and determine final ranking of all alternatives.

This example illustrates the evaluation process of applicants for job. Zagazig University, Egypt announced its need for a database developer to information technology center. An expert in workforce selection of the university has nominated five applicants for this job by conducting a personal interview of applicants. The expert wants to select only one applicant from the five candidates and for this reason the expert based its evaluation process on the following criteria:

Presentable

The previous criteria were considered the most important criteria for selecting the best applicant for the job according to expert opinion. The pair-wise comparison inputs were taken from workforce expert. The methodology illustrated in previous section is used to assign weights and calculate alternatives priorities.

Step 1. Decomposition:

The problem is formed hierarchically at various levels as shown in Fig. 2. The first level of hierarch represents the overall goal (evaluation of job applicants), the second level represents the decision criteria(presentable, years of experience, age) and third level is composed of all possible alternatives (applicant 1, applicant 2, applicant 3, applicant 4, applicant 5).

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Fig.2

Hierarchy tree of evaluating job applicant’s problem.

Step 2. Assign a weight for each criterion and make a rank of criteria

The neutrosophic pair-wise comparison matrix of criteria as in Table 4.

Where, 1 ˜ = ( o . 5 , 1 , 3 ) ( 0.9 , 0.2 , 0.3 ) , 2 ˜ = ( 0 , 2 , 3 ) ( 0.6 , 0.5 , 0.4 ) , 6 ˜ = ( 2 , 6 , 12 ) ( 0.8 , 0.4 , 0.2 ) , 7 ˜ = ( 2 , 7 , 15 ) ( 0.9 , 0.2 , 0.1 ) .

By using Equations (4) and (6), the previous neutrosophic pair-wise comparison matrix transformed to deterministic pair-wise comparison matrix as in Table 5.

To calculate the weight of each criterion, we apply the following steps:

Normalize the column entries by dividing each entry by the sum of the column.

The previous pair-wise comparison matrix takes the following form: A = [ 1 1 / 2 3 2 1 4 1 / 3 1 / 4 1 ]

Column sum = 10/3  7/4  8

Then, the new matrix after normalized column sums is as follows: A = [ 3 / 10 2 / 7 3 / 8 3 / 5 4 / 7 1 / 2 1 / 10 1 / 7 1 / 8 ]

Column sum = 1 1 1

From the previous matrix we can easily find ranking of priorities, namely the Eigen Vector X by taking overall row averages as follows: X = [ 0.37 0.55 0.125 ]

The ranking of priorities according to the Eigen Vector X are shown in Fig. 3.

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Fig.3

Priorities of criteria with respect to goal.

It can be seen from Fig. 3 that years of experience is the most important selection criteria with a weight of 0.500 followed by presentable. Age is the least important criterion.

To ensure that all inputs of expert are consistent we should make a consistency test as follows:

Calculate consistency index and consistency ratio as follows: A = [ 1 1 / 2 3 2 1 4 1 / 3 1 / 4 1 ] [ 0.32 0.55 0.125 ] = [ 0.97 1.69 0.37 ]

From Equations (8) and (9), λ _max  = average{0.97/0.32, 1.69/0.55, 0.37/0.125} = 3.02 and CI = λ max - n n - 1 = (3.02–3)/(3–1) = 0.01.

To calculate the consistency ratio (CR), we used Table 6 which derived from Saaty book. The upper row is the order of the random matrix, and the lower row is the corresponding index of consistency for random judgments.

Because 0.02<0.1, so the evaluations are consistent.

Step 4. Ranking alternatives

Neutrosophic pair-wise comparisons of applicants according to presentable criterion are shown in Table 7.

Where, 1 ˜ = ( o . 5 , 1 , 3 ) ( 0.9 , 0.2 , 0.3 ) , 2 ˜ = ( 1 , 2 , 6 ) ( 0 . 8 , 0 . 4 , 0 . 2 ) , 3 ˜ = ( 0 , 3 , 9 ) ( 0.6 , 0.3 , 0.2 ) , 4 ˜ = ( 2 , 4 , 6 ) ( 0.7 , 0.4 , 0.3 ) , 5 ˜ = ( 3 , 5 , 15 ) ( 0.9 , 0.5 , 0.1 ) , 6 ˜ = ( 0 , 6 , 12 ) ( 0.6 , 0.2 , 0.1 ) , 7 ˜ = ( 3 , 7 , 14 ) ( 0.7 , 0.4 , 0.3 ) , 9 ˜ = ( 4 , 9 , 20 ) ( 0.8 , 0.5 , 0.3 ) .

By using Equations (4) and (6), the previous neutrosophic pair-wise comparison table transformed to deterministic pair-wise comparison as shown in Table 8.

Normalization of column sums is shown in Table 9.

By taking the overall row average of previous table, then priority vector as follows: X = [ 0.29 0.24 0.19 0.18 0.08 ]

By comparing applicants according to years of experience criterion in neutrosophic environment, the results are shown in Table 10.

By using Equations (4) and (6), the previous neutrosophic pair-wise comparison table transformed to deterministic as shown in Table 11.

Normalization of column sum is shown in Table 12.

By taking the overall row average of previous table, then priority vector as follows: X = [ 0.322 0.274 0.188 0.09 0.118 ]

By comparing applicants according to age criterion in neutrosophic environment, the results are shown in Table 13.

By using Equations (4) and (6), the previous neutrosophic pair-wise comparison table transformed to deterministic as shown in Table 14.

Normalization of column sums is shown in Table 15.

By taking the overall row average of previous table, then priority vector as follows: X = [ 0.318 0.26 0.162 0.148 0.1 ]

Then the relative scores for each alternative asfollows: [ 0.290.320.32 0.240.2740.26 0.190.180.16 0.180.090.15 0.080.120.1 ] × [ 0.33 0.50 0.125 ] = [ 0.308 0.275 0.181 0.131 0.102 ]

Finally, the AHP ranking of decision alternatives are shown in Fig. 4.

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Fig.4

Ranking of alternatives with respect to overall goal.

From Fig. 4 the AHP results indicate that Applicant 1 is the best applicant for job and according to these results, Zagazig University should select Applicant 1 as a database developer of information technology center.

In order to test persistence of the priority ranking, the sensitivity analysis is performed to the AHP results. A weight of 90% is assigned to one criterion and the remaining ratio distributed among the other criteria in the ratio of the oldest weights; this done for all criteria and the final result of AHP is calculated in each case. Different types of sensitivity graphs of AHP results are shown form Figs. 5 to 16.

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Fig.5

Dynamic sensitivity analysis.

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Fig.6

Dynamic sensitivity analysis by changing priorities.

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Fig.7

Two dimensional sensitivity analysis of alternatives according to Presentable and Years of experience criteria.

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Fig.8

Two dimensional sensitivity analysis of alternatives according to Age and Years of experience criteria.

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Fig.9

Head-to-Head sensitivity between Applicant 1 and Applicant 2.

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Fig.10

Head-to-Head sensitivity between Applicant 1 and Applicant 3.

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Fig.11

Head-to-Head sensitivity between Applicant 1 and Applicant 4.

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Fig.12

Head-to-Head sensitivity between Applicant 1 and Applicant 5.

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Fig.13

Gradient sensitivity for Presentable criterion with respect to overall goal.

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Fig.14

Gradient sensitivity for Years of experience criterion with respect to overall goal.

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Fig.15

Gradient sensitivity for Age criterion with respect to overall goal.

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Fig.16

Sensitivity analysis graph of the AHP results.

From Fig. 5 the priorities of the alternatives will change in the right column (alternatives column) by dragging the objectives priorities forward and backward in the left column (criteria column).

The decision maker can drag objective’s bar forward or backward to increase or decrease the priority of objectives and see the impact on alternatives as in (Fig. 6).

The second type of sensitivity analysis is the two dimensional sensitivity. The two dimensional sensitivity compare each alternative according to criteria in a two dimensional manner. From Figs. 7 and 8 it’s obvious that the first alternative (Applicant 1) is the best alternative according to years of experience, presentable and Age criteria.

Another type of sensitivity analysis is the Head to Head sensitivity analysis. The Head to Head sensitivity analysis of the AHP results is shown from Figs. 9 to 12. In Head to Head sensitivity each alternative compare with another alternative according to all criteria.

From Figs. 9 to 12 we can conclude that according to Head-to-Head sensitivity of AHP results, Applicant 1 is the best alternative compared to other alternatives.

Figures 13 to 15 illustrates the gradient of each alternative according to each criterion. The previous figures of gradient sensitivity show that Applicant 1 is the best alternative according to allcriteria.

We can gather the previous results of sensitivity in the following graph:

The overall sensitivity graph of the AHP results is shown in Fig. 16. The scores of the alternatives are on the y-axis and the criteria are on the x-axis. From Fig. 16 it’s obvious that Applicant 1 is the best alternative among the five alternatives.

Neutrosophic set includes classical set, fuzzy set and intuitionistic fuzzy set as it doesn’t mean only truth-membership and falsity- membership but also considers indeterminacy function which is very obvious in real life situations. In this research, we have considered parameters of AHP comparison matrices as triangular neutrosophic numbers and we used score function to transform neutrosophic AHP parameters to deterministic values. In the future this research will be extended to trade with different MCDM techniques such as alpha-discounting, nonlinear alpha-discounting and interval alpha-discounting methods.