Application of Linear Goal Programming (LGP), the …. Appl. Environ. Biol. Sci., 4... ·...

J. Appl. Environ. Biol. Sci., 4(11)112-124, 2014

© 2014, TextRoad Publication

ISSN: 2090-4274

Journal of Applied Environmental

and Biological Sciences www.textroad.com

*Corresponding Author: Behzad Babakhani, Department of Industrial Engineering, Islamic Azad University of Arak.

E-mail: Behzad.babakhani1982 @ gmail.com

Application of Linear Goal Programming (LGP), the Fuzzy Analytic Hierarchy

Process (FAHP), the Prioritization of Factors

Case Study: Kermanshah Province Gas Company

Behzad Babakhani1*

, Emad Roghanian2

1Department of Industrial Engineering, Islamic Azad University of Arak

2Assistant Professor, Department of Industrial Engineering, Khajeh Nasirodin Toosi University

Received: May 27, 2014

Accepted: September 30, 2014

ABSTRACT

Analytical Hierarchy Process (AHP), to prioritize applications such as weighting factors have been used in various

fields. Among the methods used to cause weight gain, fuzzy pair wise comparison matrix in analytic hierarchy

process, the method of least squares is logarithmic. Approach to complex logarithmic least squares method, and other

methods, this study has tried, through a simple process, rather than the existing methods used. In this approach, a

linear goal programming model to obtain the relative importance of factors and minimize the inconsistency in the

fuzzy pair wise comparison matrix is used. Superiority of this method over other existing methods, the simplicity of

computation and minimize the amount of deviation of the matrices are incompatible. Authentication method used for

prioritization factors, the empowerment of employees, the gas company of Kermanshah Province, is investigated.

KEYWORDS: Analytical Hierarchy Process fuzzy Multi Attribute Decision Making, linear goal programming,

fuzzy normal weight, empowerment.

1 INTRODUCTION

Analytical Hierarchy Process, the first time was presented by [1]. In many areas, such as design and

development, decision-making, forecasting, etc. were used [2]. The traditional analytic hierarchy process, a certain

number of judgments expressed by the decision maker uses, but the numbers are uncertain because of the uncertainty,

a lot of stuff is not available, or if the desired results are achieved not there. To overcome this problem, the method of

AHP, fuzzy judgments, and a fuzzy analytic hierarchy process is used [3, 4, 5 and 6].

The key issue in the application of fuzzy AHP, the decision of how to calculate the relative weight vector of

paired comparisons matrix phase. Several methods of determining the relative weights for the fuzzy AHP proposed.

For instance van Laarhoven and colleagues [7], triangular fuzzy numbers are used instead of absolute numbers, the

logarithmic least squares method for fuzzy AHP proposed. Boender and colleagues [8], referring to one of the

drawbacks of Laarhoven, normal weight gain phase, a modified normalization method presented. Wang et al [9] also

showed an improved method for normalization is not true. Xu and Zhai [10], a logarithmic least squares method for

fuzzy judgment matrix based on Euclidean distance is presented. Weights obtained in this way, the characters range

from t different levels.

Buckley [11], to obtain the relative weight of the matrix of paired comparisons using the geometric mean

method. Chang [12] developed a method of analysis to obtain the final weights, the proposed fuzzy matrix of paired

comparisons. Csutora and Buckley [13], the Landa- Max (��

) to get the fuzzy weights. This method could only

non-fuzzy weight gain, some of which were very far from reality.

Among the methods mentioned above, the method of analysis developed, the computational approach is simple,

it is mostly used. While this method may provide answers wrong and unrealistic [14]. The method used in this paper,

the method proposed by Wang and Chen [15] is. The method to obtain the relative weights normalized triangular

fuzzy, fuzzy pair wise comparison matrix of a linear goal programming model is used. This method can, with relative

weights of fuzzy approach is much simpler than the above mentioned methods, such as least squares logarithmic gain.

Case Study: This article is a gas company in Kermanshah Province. Directors of the company, in line with measures

taken for the benefit of expert personnel, and high efficiency are concerned, the factors affecting the empowerment of

its staff, prioritize, and how important each of these factors, determine. Due to the aforementioned advantages of this

method over other methods, in this paper, the linear goal programming approach to priorities in the analytic hierarchy

process are used.

2- Stage fuzzy analytic hierarchy process (FAHP), based on a linear goal programming (LGP)

2.1The first step:

The fuzzy analytic hierarchy process model (FAHP), utilizes a linear goal programming (LGP), as follows:

112

Babakhani and Roghanian, 2014

Step 1: Understand the problem;

Step 2: Determine the purpose;

Step 3: Determine the criteria at this stage of the decision criteria, are identified. n standard library and researches

studies have been conducted, and the researchers are identified1 2

( , , ..., )n

C C C C= .

Step 4: Determine objects, at this stage of the decision-m options, are set. 1 2

( , , ..., )m

A A A A=

Step 5: The tree hierarchy, then the set objective standards objects decision, in the form of a hierarchy decision tree

technology. This tree includes three objectives, decision criteria and decision options professionalism.

Figure 1: Decision tree hierarchy

Step 6: formation of paired comparisons matrix phase;

At this point, the opinion of experts, the preference of each criterion relative to other criteria, according to the

purpose of the issue, as well as the preferred option compared to other options, according to the criteria in the form of

triangular fuzzy numbers, are determined. So objects these matrices, fuzzy numbers up occurs. A comparison matrix

of criteria relative to target the problem and the number of criteria, Matrixes paired comparisons between options,

compared to the standards established by the fuzzy objects screw. Paired comparisons matrix elements of triangular

fuzzy numbers, the form ��=(��,��,��)are.

If a fuzzy number to express a judgment about the priority criteria (option) i have, compared to the standard (option) j

I decide to by k ��=(��,��,��)is used, it can be a way out geometry [16] as:

M = (�∏ ��

�

�, �∏ ��

�

� ,�∏ ��

�

� )

We used to gather expert opinions.

2.2 Second, the use of linear goal programming (LGP)

Paired comparison matrices obtained from the previous steps to consider:

Table 1: Paired comparison matrix

Where we have:

)1(

The main objective

Criterion 1 Criterion2

NSelectionCri

teria

1Options Options2 mOptions

...

The first

level

The second

level

The third

level

113


Paired comparison matrix above can be converted into non-negative definite matrix with number.

Table 2: Matrix leg left, center and right

For each paired comparison matrix, a vector normal weight, is as follows:

The weight vector according to the following formula, paired comparison matrixA� is close to [9].

Fuzzy weight vector is normalif and only if:

Which is equivalent to:

Relationship between matrix elements, paired comparisons in Table(3-1), and the fuzzy weight vector can be written

as:

So it can be paired comparison matrix�� , we can write:

Table3: Paired comparison matrix

)5(

114


According to sharing rules, the triangular fuzzy number arithmetic operations, which it wrote:

According to the principle of division of the triangular fuzzy numbers, paired comparison matrix�� in Table 3, we can

write:

Table 4: Paired comparison matrix

The matrices, non-negative definite matrix into three sub-indicators are:

And can be easily fixed, the relationship between the three Matrix, and the weight vector is:

)9(

In the above equation ، و�, the three vectors are a definite weight

( ), which constitute the fuzzy weight vector.

Also non-negative definite matrix element of the three above, we can conclude the following relationships:

These conditions, compatibility conditions for fuzzy triangular matrix of paired comparisons, respectively.

However, because of the subjective judgments and uncertainties in the actual decision-makers subjective judgment,

can not always be 100% sure. In other words, these equations due to the uncertainty of subjective judgment, not

always be used, thus to solve this problem, we introduce the following error vectors.

E = (� � �) � – (n�1)

)12((�� ) – (n�1) � =�

= (� � ��)

Where I is a unit matrix is n × n and �� و �� و �� (i = 1,2, ..., n), all the variables are wrong. It is desirable that the

absolute values of the deviation variables, the size may be considered small, so we can model the nonlinear goal

programming (NGP) below, to obtain the fuzzy weight vector W build.

Minimize J = ∑ �|��| � |��| � |��|��

Subject to (� � �) � – (n�1) –� = 0.

)13. ( (�� ) – (n�1) � –� = 0

115


(�� − ��)��– ∆= 0.

��

+ ∑ ��

��,�� ≥ 1, i= 1,…,n

��

+ ∑ ��

��,�� ≤ 1, i= 1,…,n

∑ ��

�� = 1.

�� − �� ≥ 0.

�� − �� ≥ 0. , �� ≥ 0.

The first three constraints, obtained from equation 12, and three central limits are �� fuzzy weight vector

normalization restrictions. The last three constraints, the requirement to be�� non-negative.

Considering the time we know the eigenvectors for each paired comparison ��matrix disruption, there will be a��

special vector, �� n ≥ ��

� ��Thus the error vector Δ can always be nonnegative, that is 0 ≤ Δ. However there is no

guarantee that the deviation vectors E and are also non-negative.

So you:

)14 ( i = 1, 2 ,…, n,= �� |��|

��

,= �� |��|

��

)15 (i = 1, 2 ,…, n,= �� |��|

�

,= �� |��|

��

As a result:

0 ≥(�, … , �)� =0 , ≥(��, … , ��)� =�0 , ≥(��, … , ��)� =�0 , ≥(��

�, … , ��)� =��

Based on the definitions��

and ��

, as well as �� and |��|, can be written as:

)16 (, i = 1, 2 ,…, n. ��

- ��

=��

)17. (i = 1, 2 ,…, n, ��

+��

=|��|

= 0. .The.i = 1, 2, ..., n so on, and can be expressed as:

)18 (, i = 1, 2 ,…, n. �

- ��

=�

)19 ( .i = 1, 2 ,…, n, �

+��

=|�|

That 0=��

. ��

Begin. i = 1, 2, ..., n with the results of the above equation, a non linear goal programming, � and

|�|can berewrittenas follows:

Minimize J = ∑ �� + ��

� + � + �

� + �� = ��(�� + � + � + + ∆)��

Subject to (�� − �)�� – (n-1)�� –�� +� = 0.

)20( .(�� − �)�� – (n-1)�� –� + = 0

(�� − ��)��– ∆= 0.

��

+ ∑ ��

��,�� ≥ 1, i= 1,…,n

��

+ ∑ ��

��,�� ≤ 1, i= 1,…,n

∑ ��

�� = 1.

�� − �� ≥ 0.

�� − �� ≥ 0. , ��,��,�,�,, ∆ ≥ 0.

The Nonlinear goal programming, a linear goal programming, and = (1, ..., 1) and ��And��

and ��

and can not, at

the same time as the basic variables in the simplex method selected.

It developed a model for fuzzy AHP (FAHP) Such a method employing linear programming model ideal for weight

gain, fuzzy, fuzzy pair of matrices compeer as an ideal linear programming (LGP) suggested it is. For each triangular

fuzzy comparison matrices, is quite "consistent, objective function�∗ values are always zero.

116


2-3 – third step, final weights and fuzzy, and non fuzzy

In a hierarchical structure, so that the partial fuzzy weights to the criteria in the second level and third level options,

using ideal linear programming (LGP) were optimized using fractional weights obtained, the final fuzzy weights

obtains.

Normalized triangular fuzzy weights to criteria�� (��,��

�,��

), and normalized triangular fuzzy weights for the

option�� (�� ,��

�,��

), we consider (j = 1, ..., m; i = 1, ..., n). Final fuzzy weights can be fleshed out using a

simple sum (SAW), the multi attribute decision obtained.

)21 ( �� = ∑ (��

�,��,��

�)(�� ,��

�,��)�

��

Where we have: = (�� = (��

�, ��

� , ��

�)) (equation (3-27)). So the mathematics of fuzzy numbers, the above

equation can be written as:

)22( ��

� = ∑ ��

��

, ��

� = ∑ ��

��

, ��

� = ∑ ��

��

Final fuzzy weights more precisely, you can use the following equations and linear programming models obtained.

)23 (∑ ��

��

= ��

�

)24 (��

� = ∑ ��

�� Minimize

��

≤ �� ≤ ��

Subject to

.∑ �� = 1

, j = 1,…, m

)25 ( ��

� = ∑ ��

�� Maximize

, j = 1,…, m w��

≤ w� ≤ w��

Subject to

.∑ �� = 1

To simplify the comparison of the numbers, the final decision will be final weights Non fuzzy phase, and to a certain

number of turns. The formula used in this study for Non fuzzy the fuzzy numbers is:

�� = 1

3��

� + ��

� + ��

��, � = 1, … ,�

The�� non-fuzzy weight option. Non fuzzy based weights, we can make choices to compare and prioritize up

[15].

2.4 Case Study prioritization factors, the gas company employee empowerment

Step 1: Understand the problem;

Empowering employees of the gas company preferences of Kermanshah province, in the context of human

resource management. Empowerment and the factors affecting it are the issues of interest to corporate executives, and

the influencing factors on the evaluation and Rankings empowering employees and as a result is known in the

organization.

Step 2: Determine the purpose;

The aim of the present study, prioritize the most important factors, the empowerment of employees, the gas

company of Kermanshah Province.

Step 3: Determine the criteria;

According to studies conducted library, empowerment of employees, many felt the five criteria of merit (��), the

feeling of having a choice (� ), perceived effectiveness (��), a sense of significance (��), and a sense of confidence

(��) as standards, empowering employees, have been considered. In this research, these criteria as the main criteria

for the analysis of hierarchical structure, are considered.

Step 4: Determine objects;

9 important factors have been identified, using Pareto charts, as the options in the hierarchical analysis, are

used. The nine factors are:

Use of meritocracy (��), performance-based bonuses (� ), education (��), Delegate (��), personal motivation

(��), personal skills (��), sense of responsibility, (��), education level (��), and experiments (��).

Step 5: The tree hierarchy;

117


Step 6: formation of paired comparisons matrix phase;

At this stage, to obtain the relationship between options and decision criteria, using a questionnaire that included

questions on paired comparisons between alternatives with respect to criteria for decision-making, as well as paired

comparisons between the decision criteria, the goal is to decide, human resources experts from the four gas companies

of Kermanshah province, located. In the questionnaire, as in the following examples, the respondents are asked about

the options listed towards the decision criteria as the quality of speech.

Example) The following options can be paired to the competency criteria to compare.

Absolutely "More important Very important More Relatively "More important Equal

importance Relatively "More important More Very important Absolutely "More important

Use of meritocracy Bonuses based on performance

Figure2: Comparison of paired two options

In step 6, the respondents 9 option decision, compared to the five decision criteria, the couple, and then the decision

criteria as well, towards the objective of the decision, compare, and questionnaires were completed, which contains

information quality, the researchers are doing. Since Thursday, paired comparison matrix, decision criteria, and a

paired comparison matrix, based on objective decision comes as a result of responses per respondent, for up to 6

paired comparison matrix is used. For example, first responders, 9 options, make decisions, to measure competency,

using a paired comparison questionnaire, and the results will be available to researchers, with putting Quality answers

in paired comparison matrix, the matrix the following is obtained:

Prioritize the factors

affecting employee

empowerment

Merit)��( Choice)��

(

Effective)

��(

Significan

t)��(

Having

confidence

Meritoc

racy

)��(

Rew

ard

)��(

Educ

ation

)��(

Deleg

ation

)��(

Motiv

ation

)��(

Skills

)�(

Respo

nsibilit

y

)�(

Educat

ion)

��(

Experi

ences

)��(

Figure 3: Decision tree showing the three levels

118


Table 6: Paired comparison matrix with linguistic values

As you can see, the information obtained from pair wise comparison matrix, as their language. To convert qualitative

data into numerical values phase, we use the following table [15]

Table 7: Variable fuzzy language and equivalents Variableloss Fuzzyvaluesof

Equalimportance 1�= (1,1,2)

Relatively"more 3�= (2,3,4)

More important 5�= (4,5,6)

Veryimportant 7�= (6,7,8)

Quiteimportant 9�= (8,9,9)

The median valuesare2�= (1,2,3),4�= (3,4,5),6�= (5,6,7),8�= (7,8,9)

Using the above table, the values of qualitative comparison matrix pair are converted into numerical values phase.

Each pair comparison matrix element that has the quality to become fuzzy number, and the element, it is symmetric

about the main diagonal inverse of the fuzzy number, is placed. Due to the mentioned issues, the expert opinion is:

119


Competency Use

ofmerit

ocracy

Bonusesba

sed on

performan

ce

Educatio

n

Delegation Theremoti

vatestaff

Interperson

alskills

Sense

ofresponsibil

ity

Level of

Educatio

n

Experiences

1A

2A

3A

4A

5A

6A

7A

8A

9A

Use ofmeritocracy

1A

l 1 4.1195 3.7224 0.6389 1.0746 1 1

1.5197

2.8284

m 1 5.2068 4.7867 0.6687 1.3151 1.2359 1.0878 2.0062 3.3437

u 1 6.2603 5.8259 1.1892 1.8612 1.5651 1.6818 2.4495 4.5590

Bonusesbased on

performance 2A

l 0.1597 1 3.7224 0.4855 0.9036 0.4518 0.5373 2 1.0746

m 0.1921 1 4.7867 0.5886 1.1583 0.5774 0.6687 2.5900 1.3161

u 0.2427 1 5.8259 0.8409 1.6818 0.7825 1 3.7224 1.8612

Education

3A

l 0.1716 0.1716 1 0.1716 0.3195 1.1892 0.3799 0.5000 1.1892

m 0.2089 0.2089 1 0.2089 0.3861 1.4316 0.5081 0.5774 1.3161

u 0.2686 0.2686 1 0.2686 0.5946 2 0.7071 1 2.3784

Delegation

4A

l 0.8409 1.1892 3.7224 1 2 2.0598 0.9036 2.6321 2

m 1.4953 1.6990 4.7867 1 2.2361 2.6457 1.0878 3.2011 2.2361

u 1.5651 2.0598 5.8259 1 3.4641 3.3636 1.3161 4.4267 3.4641

Theremotivatestaff

5A

l 0.5373 0.5946 1.6818 0.2887 1 1.1892 0.6389 0.7598 2.6321

m 0.7598 0.8633 2.5900 0.4472 1 1.4316 0.6687 0.9573 3.2011

u 0.9306 1.1067 3.1302 0.5000 1 2 1.1892 1.1892 4.4267

Interpersonalskills

6A

l 0.6389 1.2779 0.5000 0.2973 0.5000 1 0.3433 1.1892 1.8612

m 0.8091 1.7321 0.6985 0.3780 0.6985 1 0.4111 1.3161 2.1407

u 1 2.2134 0.8409 0.4855 0.8409 1 0.5000 2.3784 3.3636

Sense ofresponsibility

7A

l 0.5946 1 1.4142 0.7598 0.8409 2 1 2.7108 2.6321

m 0.9193 1.4953 1.9680 0.9193 1.4935 2.4323 1 3.2700 3.2011

u 1 1.8612 2.6321 1.1067 1.5651 2.9130 1 4 4.4267

Level of Education

8A

l 0.4082 0.2686 1 0.2260 0.8409 0.4204 0.1768 1 1

m 0.4984 0.3860 1.7321 0.3124 1.0450 0.7598 0.3058 1 1.1362

u 0.6580 0.5000 2 0.3799 1.3161 0.8409 0.3689 1 1.8612

Experiences

9A

l 0.2193 0.5373 0.4204 0.2887 0.2259 0.2973 0.2259 0.5373 1

m 0.2991 0.7598 0.7598 0.4472 0.3124 0.4671 0.3124 0.8801 1

u 0.3536 0.9306 0.8409 0.5000 0.3799 0.5373 0.3799 1 1

Competency

Use of

meritocr

acy

Bonuses

based on

perform

ance

Educatio

n

Delegati

on

There

motivate

staff

Interper

sonal

skills

Sense of

responsi

bility

Level of

Educatio

n

Experien

ces

1A

2A

3A

4A

5A

6A

7A

8A

9A

Use of

meritocracy 1A

(1,1,1)

(6,7,8)

(4,5,6)

(1,1,2)

(�

�,�

�,�

�)

(�

�,�

�,�

�)

(�

�,�

�,�

�)

(8,9,9)

(4,5,6)

Bonuses based

on

performance

2A

(�

�,�

�,�

�) (1,1,1) (4,5,6) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (4,5,6) (

,

�,

�)

Education 3

A

(�

�,�

�,�

�)

(�

�,�

�,�

�) (1,1,1) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (6,7,8) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (1,1,2)

Delegation 4

A

(�

�,1,1)

(4,5,6) (4,5,6) (1,1,1) (4,5,6) (6,7,8)

(�

�,�

�,�

�) (6,7,8) (1,1,2)

There

motivate staff 5A

(4,5,6) (4,5,6) (4,5,6) (�

�,�

�,�

�) (1,1,1) (6,7,8) (1,1,2) (6,7,8) (6,7,8)

Interpersonal

skills 6A

(4,5,6)

(2,3,4) (�

�,�

�,�

�) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (1,1,1) (

�

�,�

�,�

�) (1,1,2) (6,7,8)

Sense of

responsibility 7A

(4,5,6) (4,5,6) (4,5,6) (4,5,6) (�

�,1,1) (4,5,6) (1,1,1) (6,7,8) (6,7,8)

Level of

Education 8A

(�

,�

,�

�)

(�

�,�

�,�

�) (2,3,4) (

�

�,�

�,�

�) (

�

�,�

�,�

�) (

�

�,1,1) (

�

�,�

�,�

�) (1,1,1) (1,1,2)

Experiences

9A

(

,

�,

�)

(4,5,6) (

�,1,1) (

�,1,1) (

�,

�,

) (

�,

�,

) (

�,

�,

) (

�,1,1) (1,1,1)

120


The matrix, the matrix of paired comparisons between factors affecting employee empowerment, to measure

competency, which, according to information obtained from questionnaires completed by the first respondent, and

transforming them into fuzzy numbers, is obtained. Similarly, can be paired comparison matrices options than the

standard competency, to acquire other experts, and then using the geometric mean method, integration experts, and the

final phase paired comparison matrix for each criterion obtained.

Similarly, what criteria do we feel merit, can be paired comparison matrix between factors affecting

empowerment, empowerment than other criteria are also obtained. The analytic hierarchy process, then the paired

comparisons between options, decisions, decisions were made about the criteria should then paired comparisons

among the criteria for the decision, rather than deciding the main purpose of this research. "prioritization factors,

based empowerment ", is also carried out. At the end of step 6, all paired comparisons matrices for use in later stages

of the fuzzy analytic hierarchy process, the obtained. The six paired comparison matrices, as an input to the next phase

of the research will be used.

Priority ideal method of linear programming (LGP), a fuzzy AHP (FAHP)

After forming the matrix of paired comparisons, the next phase of this study is to calculate the normalized weight

vector elements in the matrix of paired comparisons, using linear programming priorities Armani (LGP), a fuzzy AHP

(FAHP), we . To calculate the weight vector (relative weight), the matrix of paired comparisons in this method, the

original phase matrix into three matrix with non-negative definite numbers, analyze them. For example, the paired

comparisons matrix, options, based on the competency criteria into three definite matrix with non-negative integers,

we decompose. The three matrices, respectively, are the matrix elements of the left leg fuzzy numbers, matrix

elements of the middle leg and the right leg of fuzzy numbers. After analyzing the fuzzy matrix of paired

comparisons, the three matrices left foot, mid-foot and right foot, and put the matrices in equation (20), and using the

software LINGO 11, the weights of the paired comparisons matrix, obtained come.

Relative weightsobtainedcompetency-based criteria )�∗�( Factors

affectingempowerment

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

�

�

�

�

�

�

�

�

Relative weightsobtained, according to the feeling ofhaving a

choice) ∗�(

Factors


(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

��

��

��

��

��

��

��

��

�

121


Relative weights obtained, according to the sense of

significance) ∗�(

Factors affecting

empowerment

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

(�.�� ,�.�� ,�.��)

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Relative weights obtained, according to a sense of confidence) ∗�( Factors affecting

empowerment

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Relative weights obtained, according to feel the effect of) ∗�( Factors


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122


And so on, using a linear goal programming (LGP), which measures the relative weight of the paired comparisons

matrix, we get:

Calculation of the fuzzy weights and the weights Non fuzzy

After calculating the relative weight options, and criteria for paired comparisons matrix, linear goal programming

method, the final weight factors, the empowerment of employees (making choices), using the existing relationship

obtains. After calculating the weights of the items, in order to facilitate the comparison of fuzzy weights, converted to

absolute numbers, and then we prioritize options.

Table 16: Final fuzzy weights for the nine factors affecting employee empowerment, and rank them

��

��

��

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�

The

relative

weights

of the

criteria

Criteria

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�0.05790.0668

0.0703

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�0.14490.1642

0.1642

�

�0.08680.0936

0.1053

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�0.10020.1060

0.1120

�

�0.18300.1993

0.2194

�

�0.02350.0345

0.0668

�

�0.10390.1039

0.1180

�

�0.19240.1924

0.2170

�

�0.20050.2005

0.2413

�

��.��.��.��

�0.03420.0524

0.0531

�

�0.09150.1055

0.1070

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�0.06970.0776

0.0827

�

�0.07240.0743

0.0825

�

�0.32970.3489

0.3489

�

�0.00000.0109

0.0507

�

�0.11180.1126

0.1126

�

�0.13910.1555

0.1642

�

�0.12490.1694

0.2146

� �

��. 0��.��.��

�0.03980.0464

0.0520

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�0.06280.0735

0.0780

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�0.05170.0574

0.0644

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�0.08510.0932

0.1008

�

�0.22750.2578

0.2578

�

�0.00060.0227

0.0512

�

�0.17330.1851

0.2036

�

�0.19860.2148

0.2629

�

�0.09050.1190

0.1682

�

�

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�0.01310.0298

0.0423

�

�0.10120.1256

0.1256

�

�0.04210.0474

0.0539

�

�0.14580.1698

0.1815

�

�0.17570.2012

0.2143

�

�0.03740.0393

0.0477

�

�0.13100.1517

0.1643

�

�0.18950.1936

0.2379

�

�0.22010.2750

0.2750

�

��.��.��.��

�0.01160.0256

0.0320

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�0.06200.0799

0.0825

�

�0.03700.0416

0.0460

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�0.09410.1100

0.1128

�

�0.20940.2453

0.2453

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�0.03470.0371

0.0453

�

�0.19000.2195

0.2347

�

�0.18530.2035

0.2298

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�0.20780.2361

0.2361

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�

��.��.��.�� 0.02830.0420

0.0484

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�0.09250.1129

0.1119

�

�0.05490.0616

0.0677

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�0.10060.1176

0.1194

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�0.21100.2430

0.2450

�

�0.02080.0310

0.0519

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�0.13640.1555

0.1602

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�0.17910.1918

0.2174

�

Finalfuzzyweights

0.0404 0.0396 0.1058 0.0614 0.1125 0.2330 0.0346 0.1507 0.1961

Non-fuzzy

weights.

Relative weights obtained, based on objective criteria enabling

prioritization factors) ∗�(

Empowerment

criteria

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(�.�� ,�.�� ,�.��)

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123


Final ranking factors, the empowerment approach using FAHP-LGP

Table17: Final ranking factors on employee empowerment Rating Enablers Weight

1 entrusting 0.2330

2 Use meritocracy 0.1961

3 bonuses based on performance 0.1507

4 motivate staff 0.1125

5 a sense of responsibility 0.1058

6 Individual Skills 0.0614

7 experiences 0.0404

8 levels of education 0.0396

9 education 0.0346

3. Conclusions and Recommendations

Fuzzy analytic hierarchy process, a technique widely used in various sciences. The huge volume of articles, in

which Fuzzy Analytic Hierarchy Process is used, the logarithmic methods such as least squares, total single weighty,

development, analysis, and other methods to obtain the relative weights of fuzzy, fuzzy pair wise comparison matrix

used, while some of the methods, procedures are complex, and some results are far from reality, and often provide

inaccurate. To use properly, and the Fuzzy Analytic Hierarchy successful, a linear goal programming approach, we use.

This study was conducted in Kermanshah Province Gas Company. And using the above mentioned factors on

the empowerment of employees in the company were prioritized. The results show that the method can be linear goal

programming, Fuzzy relative weights precise, perfectly adapted for paired comparisons matrix, and the optimal

weights normalized fuzzy, with the lowest variance for paired comparisons matrix inconsistent obtained. This case

study shows that, this method factors prioritize the empowerment of employees, as well as answers.

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