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

Abstract

The main objective of this research is to give an overview of the Analytic Hierarchy Process (AHP) in neutrosophic environment. In some realistic situations, the decision makers might be unable to assign deterministic evaluation values to the comparison judgments due to his/her limited knowledge or the differences of individual judgments in group decision making. To overcome these challenges, we have used neutrosophic set theory to handle the AHP, where each pair-wise comparison judgment is represented as a triangular neutrosophic number (TNN). In this paper, neutrosophic theory is used to form AHP decision-making model for choosing the best candidates among the applications. A real life example is developed based on expert opinions from Zagazig University, Egypt. The problem is solved to show the effectiveness of the proposed neutrosophic-AHP decision making model.

Keywords

Analytic hierarchy process multi-criteria decision making neutrosophic set theory consistency test

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.

Calculate the relative weights of the decision criteria.

Calculate the relative rankings (priorities) of 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.

Table 1
The differences between fuzzy, intuitionistic fuzzy and neutrosophic set

Set Definition

Fuzzy sets The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise.

Fuzzy sets were introduced by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set.

Fuzzy sets are sets whose elements have degrees of membership.

Intuitionistic fuzzy sets Intuitionistic fuzzy sets are sets whose elements have degrees of membership and non-membership.

Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and non-membership functions.

Intuitionistic fuzzy sets have been introduced by Krassimir Atanassov (1983) as an extension of Lotfi Zadeh’s notion of fuzzy set, which itself extends the classical notion of a set.

Triangular intuitionistic fuzzy sets A special case of the Intuitionistic fuzzy sets.

A triangular intuitionistic fuzzy number denoted as $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, β_{\tilde{a}} 〉$ is a special intuitionistic fuzzy set on a real number set R, where a₁, a₂, a₃ are the lower, median and upper of intuitionistic fuzzy number and $α_{\tilde{a}}, β_{\tilde{a}}$ are the membership and non-membership functions.

Triangular neutrosophic sets Neutrosophic sets is a generalization of classical, fuzzy and intuitionistic fuzzy sets.

Neutrosophic sets were introduced by Smarandache in1995.

Represent uncertain, inconsistent, and incomplete information about real world problems.

A single valued triangular neutrosophic number, $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ is a special neutrosophic set on the real line set R, where a₁, a₂, a₃ are the lower, median and upper of neutrosophic number and $α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}}$ are the truth- membership, indeterminacy- membership and falsity-membership functions respectively.

Represent real world problem effectively by considering all aspects of decision making situations, (i.e. truthiness, indeterminacy and falsity).

Set	Definition
Fuzzy sets	The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise.
	Fuzzy sets were introduced by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set.
	Fuzzy sets are sets whose elements have degrees of membership.
Intuitionistic fuzzy sets	Intuitionistic fuzzy sets are sets whose elements have degrees of membership and non-membership.
	Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and non-membership functions.
	Intuitionistic fuzzy sets have been introduced by Krassimir Atanassov (1983) as an extension of Lotfi Zadeh’s notion of fuzzy set, which itself extends the classical notion of a set.
Triangular intuitionistic fuzzy sets	A special case of the Intuitionistic fuzzy sets.
	A triangular intuitionistic fuzzy number denoted as $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, β_{\tilde{a}} 〉$ is a special intuitionistic fuzzy set on a real number set R, where a₁, a₂, a₃ are the lower, median and upper of intuitionistic fuzzy number and $α_{\tilde{a}}, β_{\tilde{a}}$ are the membership and non-membership functions.
Triangular neutrosophic sets	Neutrosophic sets is a generalization of classical, fuzzy and intuitionistic fuzzy sets.
	Neutrosophic sets were introduced by Smarandache in1995.
	Represent uncertain, inconsistent, and incomplete information about real world problems.
	A single valued triangular neutrosophic number, $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ is a special neutrosophic set on the real line set R, where a₁, a₂, a₃ are the lower, median and upper of neutrosophic number and $α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}}$ are the truth- membership, indeterminacy- membership and falsity-membership functions respectively.
	Represent real world problem effectively by considering all aspects of decision making situations, (i.e. truthiness, indeterminacy and falsity).

Table 2

The advantages and disadvantages of different types of AHP

AHP method	Advantages	Disadvantages
Classical AHP method	Powerful decision making methodology.	Failure to deal with imprecise and vague information which exist usually in real life situations.
	Determine priorities among different criteria.	If the pair wise comparison matrix isn’t consistent, Saaty didn’t provide any method to make it consistent.
	Comparing alternatives for each criterion.
	Determining an overall ranking of the alternatives.
	Final outcome of the AHP is the best choice among decision alternatives
Fuzzy AHP method	Has the same advantages of classical AHP beside the following advantage:	The matrices with only the membership degrees of intuitionistic preference relations are not consistent in nature.
	Deal with the imprecise or vague nature through one grade” membership degree”.
Intuitionistic AHP method	Has the same advantages of classical AHP beside the following advantage:	The matrices with only the membership degrees of intuitionistic preference relations are not consistent in nature.
	Handling vagueness and uncertainty over fuzzy AHP because it consider three different grades “membership degree, hesitancy degree and non-membership degree”.
Triangular fuzzy AHP	Has the same advantages of classical AHP beside the following advantage:	Can’t be used to express support and objection evidences simultaneously.
	Describe the preference judgment values of the decision maker efficiently.	It can’t represent real life situation efficiently, because it consider only the membership function and doesn’t take into account the indeterminacy and falsity function.
Neutrosophic AHP method	Has the same advantages of classical AHP beside the following advantages:	–
	Provides user with a richer structure framework than the classical AHP, fuzzy AHP and intuitionistic fuzzy AHP.
	Describe the preference judgment values of the decision maker efficiently.
	Handling vagueness and uncertainty over fuzzy AHP and intuitionistic fuzzy AHP because it consider three different grades “membership degree, indeterminacy degree and non-membership degree”.
	Point out how to improve inconsistent judgments.

Table 3

Saaty ranking scale for criteria and alternatives

Value of a_ij	Explanation
1	Objectives i and j are equally importance
3	Objective i is slightly more important than objective j
5	Objective i is strongly more important than objective j
7	Objective i is very strongly than objective j
9	Objective i is absolutely more important than objective j
2,4,6,8	Intermediate values

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.

2 Preliminaries

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 $α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} \in [0, 1]$ and a₁, a₂, a₃ ∈ R where a₁ ≤ a₂ ≤ a₃. Then a single valued triangular neutrosophic number, $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ is a special neutrosophic set on the real line set R whose truth-membership, indeterminacy-membership and falsity-membership functions are defined as: $T_{\tilde{a}} (x) = {\begin{matrix} α_{\tilde{a}} (\frac{x - a_{1}}{a_{2} - a_{1}}) & (a_{1} \leq x \leq a_{2}) \\ α_{\tilde{a}} & (x = a_{2}) \\ α_{\tilde{a}} (\frac{a_{3} - x}{a_{3} - a_{2}}) & (a_{2} < x \leq a_{3}) \\ 0 & otherwise, \end{matrix}$ (1) $I_{\tilde{a}} (x) = {\begin{matrix} \frac{(a_{2} - x + θ_{\tilde{a}} (x - a_{1}))}{(a_{2} - a_{1})} & (a_{1} \leq x \leq a_{2}) \\ θ_{\tilde{a}} & (x = a_{2}) \\ \frac{(x - a_{2} + θ_{\tilde{a}} (a_{3} - x))}{a_{3} - a_{2}} & (a_{2} < x \leq a_{3}) \\ 1 & otherwise, \end{matrix}$ (2) $F_{\tilde{a}} (x) = {\begin{matrix} \frac{(a_{2} - x + β_{\tilde{a}} (x - a_{1}))}{(a_{2} - a_{1})} & (a_{1} \leq x \leq a_{2}) \\ β_{\tilde{a}} & (x = a_{2}) \\ \frac{(x - a_{2} + β_{\tilde{a}} (a_{3} - x))}{(a_{3} - a_{2})} & (a_{2} < x \leq a_{3}) \\ 1 & otherwise . \end{matrix}$ (3)

Where $α_{\tilde{a}}$ , $θ_{\tilde{a}}$ and $β_{\tilde{a}}$ represent the maximum truth-membership degree, minimum indeterminacy-membership degree and minimum falsity-membership degree respectively. A single valued triangular neutrosophic number $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ may express an ill-defined quantity about a, which is approximately equal to a.

Definition 4. [9, 10] Let $\tilde{a} = 〈 (a_{1}, a_{2}, a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ and $\tilde{b} = 〈 (b_{1}, b_{2}, b_{3}); α_{\tilde{b}}, θ_{\tilde{b}}, β_{\tilde{b}} 〉$ be two single valued triangular neutrosophic numbers and γ ≠ 0 be any real number. Then,

Addition of two triangular neutrosophic numbers $\begin{matrix} \tilde{a} + \tilde{b} = 〈 (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \\ \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \end{matrix}$

Subtraction of two triangular neutrosophic numbers $\begin{matrix} \tilde{a} - \tilde{b} = 〈 (a_{1} - b_{3}, a_{2} - b_{2}, a_{3} - b_{1}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \\ \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \end{matrix}$

Inverse of a triangular neutrosophic number $\begin{matrix} {\tilde{a}}^{- 1} = 〈 (\frac{1}{a_{3}}, \frac{1}{a_{2}}, \frac{1}{a_{1}}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉, \\ Where (\tilde{a} \neq 0) \end{matrix}$

Multiplication of triangular neutrosophic number by constant value $γ \tilde{a} = {\begin{matrix} 〈 (γ a_{1}, γ a_{2}, γ a_{3}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉 if (γ > 0) \\ 〈 (γ a_{3}, γ a_{2}, γ a_{1}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉 if (γ < 0) \end{matrix}$

Division of triangular neutrosophic number by constant value $\frac{\tilde{a}}{γ} = {\begin{matrix} 〈 (\frac{a_{1}}{γ}, \frac{a_{2}}{γ}, \frac{a_{3}}{γ}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉 if (γ > 0) \\ 〈 (\frac{a_{3}}{γ}, \frac{a_{2}}{γ}, \frac{a_{1}}{γ}); α_{\tilde{a}}, θ_{\tilde{a}}, β_{a} 〉 if (γ < 0) \end{matrix}$ Division of two triangular neutrosophic numbers $\begin{matrix} \frac{\tilde{a}}{\tilde{b}} = {\begin{matrix} 〈 (\frac{a_{1}}{b_{3}}, \frac{a_{2}}{b_{2}}, \frac{a_{3}}{b_{1}}); α_{\tilde{a}} \land α_{\tilde{b}} θ_{\tilde{a}} \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} > 0, b_{3} > 0) \\ 〈 (\frac{a_{3}}{b_{3}}, \frac{a_{2}}{b_{2}}, \frac{a_{1}}{b_{1}}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} < 0, b_{3} > 0) \\ 〈 (\frac{a_{3}}{b_{1}}, \frac{a_{2}}{b_{2}}, \frac{a_{1}}{b_{3}}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} < 0, b_{3} < 0) \end{matrix} \end{matrix}$

Multiplication of two triangular neutrosophic numbers $\begin{matrix} \tilde{a} \tilde{b} = \\ {\begin{matrix} 〈 (a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} > 0, b_{3} > 0) \\ 〈 (a_{1} b_{3}, a_{2} b_{2}, a_{3} b_{1}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}}, \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} < 0, b_{3} > 0) \\ 〈 (a_{3} b_{3}, a_{2} b_{2}, a_{1} b_{1}); α_{\tilde{a}} \land α_{\tilde{b}}, θ_{\tilde{a}} \lor θ_{\tilde{b}}, β_{\tilde{a}} \lor β_{\tilde{b}} 〉 \\ if (a_{3} < 0, b_{3} < 0) \end{matrix} \end{matrix}$

3 Methodology

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

Decomposition,

Pair-wise comparison, and

Synthesis of priorities

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.

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 $({\tilde{a}}_{ij})$ . Then construct the neutrosophic pair-wise comparison such that: $\tilde{A} = [\begin{matrix} \tilde{1} & {\tilde{a}}_{12} \dots & {\tilde{a}}_{1 n} \\ ⋮ & ⋱ & ⋮ \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} \dots & \tilde{1} \end{matrix}]$ Where ${\tilde{a}}_{ji} = {\tilde{a}}_{ij}^{- 1}$ .

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 ${\tilde{a}}_{ij} = 〈 (a_{1}, b_{1}, c_{1}), α_{\tilde{a}}, θ_{\tilde{a}}, β_{\tilde{a}} 〉$ be a single valued triangular neutrosophic number, then, $S ({\tilde{a}}_{ij}) = \frac{1}{16} [a_{1} + b_{1} + c_{1}] \times (2 + α_{\tilde{a}} - θ_{\tilde{a}} - β_{\tilde{a}})$ (4)

And $A ({\tilde{a}}_{ij}) = \frac{1}{16} [a_{1} + b_{1} + c_{1}] \times (2 + α_{\tilde{a}} - θ_{\tilde{a}} + β_{\tilde{a}})$ (5)

Is called the score and accuracy degrees of ${\tilde{a}}_{ij,}$ respectively.

To get score and accuracy degree of ${\tilde{a}}_{ji}$ we use the following equation: $S ({\tilde{a}}_{ji}) = 1 / S ({\tilde{a}}_{ij})$ (6) $A ({\tilde{a}}_{ji}) = 1 / A ({\tilde{a}}_{ij})$ (7)

With compensation by score value of each triangular neutrosophic number in neutrosophic pair-wise comparison matrix, we have the following deterministic matrix; $A = [\begin{matrix} 1 & a_{12} \dots & a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{n 1} & a_{n 2} \dots & 1 \end{matrix}]$

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.

Take the totality row averages.

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”

Divide the elements of the weighted sum vector by the corresponding priority for each criterion.

Compute the average of the values found in step 2; this average is denoted λ_max.

Compute the consistency index (CI) as follows:

CI = \frac{λ_{\max} - n}{n - 1}

(8) where n is the number of items being compared.

Compute the consistency ratio, which is defined as:

CR = \frac{CI}{RI}

(9) where RI is the consistency index of a randomly generated pair-wise comparison matrix.

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

4 Illustrative example

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

Year of experience

Age

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).

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.

Table 4

Neutrosophic pair-wise comparison matrix of criteria

	Presentable	Years of experience	Age
Presentable	$\tilde{1}$	$\tilde{2}$	$\tilde{6}$
Years of experience	${\tilde{2}}^{- 1}$	$\tilde{1}$	$\tilde{7}$
Age	${\tilde{6}}^{- 1}$	${\tilde{7}}^{- 1}$	$\tilde{1}$

Where, $\begin{matrix} \tilde{1} & = & (o . 5, 1, 3) (0.9, 0.2, 0.3), \\ \tilde{2} & = & (0, 2, 3) (0.6, 0.5, 0.4), \\ \tilde{6} & = & (2, 6, 12) (0.8, 0.4, 0.2), \\ \tilde{7} & = & (2, 7, 15) (0.9, 0.2, 0.1) . \end{matrix}$

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

Table 5

The deterministic pair-wise comparison matrix of criteria

	Presentable	Years of experience	Age
Presentable	1	0.5	3
Years of experience	2	1	4
Age	1/3	1/4	1

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.

Take the overall row averages.

The previous pair-wise comparison matrix takes the following form: $A = [\begin{matrix} 1 & 1 / 2 & 3 \\ 2 & 1 & 4 \\ 1 / 3 & 1 / 4 & 1 \end{matrix}]$

Column sum = 10/3 7/4 8

Then, the new matrix after normalized column sums is as follows: $A = [\begin{matrix} 3 / 10 & 2 / 7 & 3 / 8 \\ 3 / 5 & 4 / 7 & 1 / 2 \\ 1 / 10 & 1 / 7 & 1 / 8 \end{matrix}]$

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: $X = [\begin{matrix} 0.37 \\ 0.55 \\ 0.125 \end{matrix}]$

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

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: $A = [\begin{matrix} 1 & 1 / 2 & 3 \\ 2 & 1 & 4 \\ 1 / 3 & 1 / 4 & 1 \end{matrix}] [\begin{matrix} 0.32 \\ 0.55 \\ 0.125 \end{matrix}] = [\begin{matrix} 0.97 \\ 1.69 \\ 0.37 \end{matrix}]$

From Equations (8) and (9), λ_max = average{0.97/0.32, 1.69/0.55, 0.37/0.125} = 3.02 and $CI = \frac{λ_{\max} - n}{n - 1}$ = (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.

Table 6

Saaty table for calculating consistency ratio

0	2	3	4	5	6	7	8	9	10
0	0	0.58	0.90	1.12	1.24	1.32	1.4	1.45	1.49

CR = \frac{CI}{RI} = \frac{0.01}{0.58} = 0.02

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.

Table 7

Pair-wise comparison of applicants according to presentable criterion in neutrosophic environment

Presentable	A1	A2	A3	A4	A5
A1	$\tilde{1}$	$\tilde{5}$	$\tilde{3}$	$\tilde{2}$	$\tilde{4}$
A2	${\tilde{5}}^{- 1}$	$\tilde{1}$	$\tilde{6}$	$\tilde{3}$	$\tilde{7}$
A3	${\tilde{3}}^{- 1}$	${\tilde{6}}^{- 1}$	$\tilde{1}$	$\tilde{4}$	$\tilde{9}$
A4	${\tilde{2}}^{- 1}$	${\tilde{3}}^{- 1}$	${\tilde{4}}^{- 1}$	$\tilde{1}$	$\tilde{5}$
A5	${\tilde{4}}^{- 1}$	${\tilde{7}}^{- 1}$	${\tilde{9}}^{- 1}$	${\tilde{5}}^{- 1}$	$\tilde{1}$

Where, $\begin{matrix} \tilde{1} & = & (o . 5, 1, 3) (0.9, 0.2, 0.3), \\ \tilde{2} & = & (1, 2, 6) (0 . 8, 0 . 4, 0 . 2), \\ \tilde{3} & = & (0, 3, 9) (0.6, 0.3, 0.2), \\ \tilde{4} & = & (2, 4, 6) (0.7, 0.4, 0.3), \\ \tilde{5} & = & (3, 5, 15) (0.9, 0.5, 0.1), \\ \tilde{6} & = & (0, 6, 12) (0.6, 0.2, 0.1), \\ \tilde{7} & = & (3, 7, 14) (0.7, 0.4, 0.3), \\ \tilde{9} & = & (4, 9, 20) (0.8, 0.5, 0.3) . \end{matrix}$

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

Table 8

Pair-wise comparison of applicants according to presentable criterion in deterministic environment

Presentable	A1	A2	A3	A4	A5
Applicant 1	1	3.3	1.6	1.2	1.5
Applicant 2	0.3	1	2.6	1.6	3
Applicant 3	0.62	0.38	1	1.5	4
Applicant 4	0.83	0.62	0.66	1	3.3
Applicant 5	0.66	0.33	0.25	0.3	1
Column sums	3.41	5.63	6.11	5.6	12.8

Table 9

Normalized values for presentable criterion

Presentable	A1	A2	A3	A4	A5
A1	1	3.3	1.6	1.2	1.5
A2	0.3	1	2.6	1.6	3
A3	0.62	0.38	1	1.5	4
A4	0.83	0.62	0.66	1	3.3
A5	0.66	0.33	0.25	0.3	1
Column sums	3.41	5.63	6.11	5.6	12.8

Normalization of column sums is shown in Table 9.

By taking the overall row average of previous table, then priority vector as follows: $X = [\begin{matrix} 0.29 \\ 0.24 \\ 0.19 \\ 0.18 \\ 0.08 \end{matrix}]$

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

Table 10

Pair-wise comparison of applicants according to years of experience criterion in neutrosophic environment

Years of experience	A1	A2	A3	A4	A5
A1	$\tilde{1}$	$\tilde{2}$	$\tilde{4}$	$\tilde{9}$	$\tilde{5}$
A2	${\tilde{2}}^{- 1}$	$\tilde{1}$	$\tilde{7}$	$\tilde{6}$	$\tilde{2}$
A3	${\tilde{4}}^{- 1}$	${\tilde{7}}^{- 1}$	$\tilde{1}$	$\tilde{5}$	$\tilde{3}$
A4	${\tilde{9}}^{- 1}$	${\tilde{6}}^{- 1}$	${\tilde{5}}^{- 1}$	$\tilde{1}$	$\tilde{2}$
A5	${\tilde{5}}^{- 1}$	${\tilde{2}}^{- 1}$	${\tilde{2}}^{- 1}$	${\tilde{2}}^{- 1}$	$\tilde{1}$

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

Table 11

Pair-wise comparison of applicants according to years of experience criterion in deterministic environment

Years of experience	A1	A2	A3	A4	A5
A1	1	1.2	1.5	4	3.3
A2	0.83	1	3	2.6	1.2
A3	0.66	0.33	1	3.3	1.6
A4	0.25	0.38	0.3	1	1.2
A5	0.3	0.83	0.62	0.83	1
Column sums	3	3.7	6.4	11.73	8.3

Normalization of column sum is shown in Table 12.

Table 12

Normalized values for years of experience criterion

Years of experience	A1	A2	A3	A4	A5
A1	0.33	0.32	0.23	0.34	0.39
A2	0.27	0.27	0.47	0.22	0.14
A3	0.22	0.09	0.16	0.28	0.19
A4	0.08	0.10	0.05	0.08	0.14
A5	0.09	0.22	0.09	0.07	0.12

By taking the overall row average of previous table, then priority vector as follows: $X = [\begin{matrix} 0.322 \\ 0.274 \\ 0.188 \\ 0.09 \\ 0.118 \end{matrix}]$

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

Table 13

Pair-wise comparison of applicants according to agecriterion in neutrosophic environment

Age	A1	A2	A3	A 4	A5
A1	$\tilde{1}$	$\tilde{3}$	$\tilde{7}$	$\tilde{6}$	$\tilde{2}$
A2	${\tilde{3}}^{- 1}$	$\tilde{1}$	$\tilde{4}$	$\tilde{7}$	$\tilde{5}$
A3	${\tilde{7}}^{- 1}$	${\tilde{4}}^{- 1}$	$\tilde{1}$	$\tilde{3}$	$\tilde{6}$
A4	${\tilde{6}}^{- 1}$	${\tilde{7}}^{- 1}$	${\tilde{3}}^{- 1}$	$\tilde{1}$	$\tilde{9}$
A5	${\tilde{2}}^{- 1}$	${\tilde{5}}^{- 1}$	6	${\tilde{9}}^{- 1}$	$\tilde{1}$

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

Table 14

Pair-wise comparison of applicants according to agecriterion in deterministic environment

Age	A1	A2	A3	A4	A5
A1	1	1.6	3	2.6	1.2
A2	0.62	1	1.5	3	3.3
A3	0.33	0.66	1	1.6	2.6
A4	0.38	0.33	0.62	1	4
A5	0.83	0.3	0.38	0.25	1
Column sums	3.16	3.89	6.5	8.45	12.1

Normalization of column sums is shown in Table 15.

Table 15

Normalized values for age criterion

Age	A 1	A2	A3	A 4	A5
A1	0.32	0.41	0.46	0.31	0.09
A2	0.19	0.26	0.23	0.35	0.27
A3	0.10	0.17	0.15	0.18	0.21
A4	0.12	0.08	0.09	0.12	0.33
A5	0.26	0.07	0.06	0.03	0.08

By taking the overall row average of previous table, then priority vector as follows: $X = [\begin{matrix} 0.318 \\ 0.26 \\ 0.162 \\ 0.148 \\ 0.1 \end{matrix}]$

Then the relative scores for each alternative asfollows: $[\begin{matrix} 0.290.320.32 \\ 0.240.2740.26 \\ 0.190.180.16 \\ 0.180.090.15 \\ 0.080.120.1 \end{matrix}] \times [\begin{matrix} 0.33 \\ 0.50 \\ 0.125 \end{matrix}] = [\begin{matrix} 0.308 \\ 0.275 \\ 0.181 \\ 0.131 \\ 0.102 \end{matrix}]$

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

Fig.4

Ranking of alternatives with respect to overall goal.

Applicants	Priority
Applicant 1	0.31
Applicant 2	0.26
Applicant 3	0.18
Applicant 4	0.13
Applicant 5	0.10

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.

Fig.5

Dynamic sensitivity analysis.

Fig.6

Dynamic sensitivity analysis by changing priorities.

Fig.7

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

Fig.8

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

Fig.9

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

Fig.10

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

Fig.11

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

Fig.12

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

Fig.13

Gradient sensitivity for Presentable criterion with respect to overall goal.

Fig.14

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

Fig.15

Gradient sensitivity for Age criterion with respect to overall goal.

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.

5 Conclusions and future works

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.

Footnotes

Acknowledgments

We all want to thank anonymous for the constructive suggestions that improved both the quality and clarity of the research.

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