Indifferent Points in The Multicriteria Decision Making Problems (A Case Study of Suppliers’ Evaluation in Zanjan Province Gas Company)

Document Type: Original Article

Authors

1 ph.D. Student in industrial Management,Faculty of management, Science and Research

2 Department of Industrial Management, Science and Research Branch, Islamic Azad University, Tehran, Iran.

3 1. Department of Industrial Management, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

Evaluating and selecting the right contractors can increase the chances of success of a project and the organization. Considering the intense competition faced by organizations today, proper cost management to enhance profitability and customer satisfaction has attracted a lot of attention. The evaluation of contractors is usually a process thatis based on various criteria.By the end of it, theappropriate options are selected. Given the diversity in the criteria and among thedecision-making subjects, no singleway has been offered to suggest substitution between criteria.The desirability indifference on the curve ofconsumption of various goods (selection ofdecision-making options) are the same. This paper seeks to identify parallel matrices with the initial decision-making matrix of contractors that have the same results and desirability for decision-makers (indifference points). At first, the initial rating using the AHP and TOPSIS methods andthe particle swarm optimization (PSO) and genetic algorithm (GA)techniques, along withMATLAB software,was used to identify theparallel matrices. According to the obtained results, sixparallel matrixes with the initial decision-making matrix that had been prepared by experts fromthe company were produced.Out of them, the matrix related to The point of indifference is the fifth output5 AHP-PSO, based on the company experts' opinions was selected as the final version.

Keywords


 


Introduction

 

The selection of contractors playsa significant role in the success or failure of projects. Evaluating and selecting the right contractor can increase the chance of success of a project and the organization [15]. In recent times, competition has increased in all fields.The only organizations that survive are those thatuse their resources in the best and most efficient way [31]. Given the intense competition faced byorganizations today, proper cost managementto enhance profitability and customer satisfaction has attracted a lot of attention [30].

Managers and decision-makers can utilize policies to survive in this space.But the final desirable result will occur only when a detailed and comprehensive planis used [35].

Decision-making constitutes an important part of our lives.We have to makesmall and big decisions every day. The decisions we make affectour lives.We take power through selections and decisions. So far, many definitions of decision-making havebeen provided that look at it as an individual's selection.An individual chooses from among different options that may be limited or unlimited.In the most optimistic case, a few of our decisionsare fully implemented. One of the reasons for not implementing some decisions is the lack of flexibility in the organization's resources and facilities. As has been explainedin definitions related to decision-making, decision-making is aprocess through which a specific issue or solution is chosen [32].Another definition looks at it asa process in which certain practicalmethodsare chosen to resolve a particularissue [20].

In all these definitions, the selection of one option among many available options is considered. Today, there are thousands of papers and books about decision-making, especially multi-criteria decision-making (MCDM), and the number is growing every day. Only between 1987 and 1992, in the field of MCDM, about 1,216 papers, 208 books, 31 related scientific journals, and 143 conference papershave been published [3]. MCDM is categorized into two categories: multi-objective decision-making (MODM) and multi-attribute decision-making (MADM). MADM refers to certain decisions (the preferred type) such as assessments, priorities, and choosing from the available options (which sometimes may include several conflicting indices).

MADM problems in the literature on MCDM are categorized into two groups: the non-compensatory and compensatory models [4]. The compensatory model consists of methods that allow the exchange of permission among indices—that is, a change (probably small) in an index would be compensated by the opposite change in another index (or indices). Considering the available diversity in the criteria and among the decision-making subjects, a specific method for this paper has not been offered [27].Also, there is no technique that can produce scientifically points of indifference

with the modelling of the decision-making initial matrix for more action freedom of organization. Among the studies focussing on the development of MCDM models ,indifference curves, and the marginal rate of substitution ,notable are the studies of Kou, Peng, and Wang (2014); Mulliner, Malys, and Maliene (2016); Hwang, Wang,Salaty, and Makuyi (2012);Hosseini and Kazemi (2015); and Amiri and colleagues (2012) [1], [3].

The points on an indifference curve indicate a combination of two goods (or alternatives) that have the same desirability in terms of consumption. One of the ways toconsider substitutation objectively among the available indices of anissue in MADM isthe marginal rate of substitution (MRS)method [6]. The substitution or exchange rate is anunderlying assumption for this procedure.It is thenecessary change amount in the present value of an index against a change unit of some other index for the existence of certain circumstances [4].

Say,two major indices—x1 and x2—havedrawnyour attention whileyou were buying a car (the effect of other indices is the same for you).You are asked, for example, if x2 as Δ increases, by how much x1 should be reduced untilyou as the decision-maker remain indifferent in your decision-making in terms of desirability? In most cases, the answer to this question will depend on the available number of x1 and x2.If,givena certainlevel of x1 and x2,you wish to reduce λΔ unit of x1 for a Δ unit increase from x2, then your MRS from x1 vis-à-visx2 is equal to λ.

In other words, λ is equal to the amount of x1 that youare wanting to lose (bypaying a fine) against gaining a unit more than x2 [4]. Usually,MADM problems, according to experts' scores ofthe identified criteria and themethods used, lead to a ranking among the options. In this paper, based on the judgment (ranking) of experts, an attempt is made to identify several conditions that brings us to this ranking. According to the conditions and resources of an organization, there may be more appropriate conditions to achieve this ranking. This paper seeks to identify the parallel matrices with the primary matrix of decision-making that have the same desirability for decision-makers(i.e. the indifference points of decision making).

 

 

Figure 1: The Marginal rate of substitution of attributes of goods / alternatives

 

 

 

 

2-Materials and Methods

2-1-Indifference Curve

The points on the indifference curve show the combination of two goods (or decision-making options) that have the same desirability in terms of consumption [20]. These curves have a negative slope and are convex. They do not cross each other and, far away from the coordination origin, show a higher level of desirability [8]. On an indifference curve, the desirability of the consumption of various commodities (or decision-making options) is the same. In other words, indifference curves are the geometric locations of different combinations of two or more products (such as MCDM) that offer the same desirability to the person. These curves are continuous. So, the position of possibility of lexicography preferences will not be obtained [9], [38], [33].

Figure 2: The graph of indifference curve

 

 

 

The utility function was proposed as a means to measure the value of the outcome of a decision, by Newman and Morgen Stern. The main idea of this approach is to maximize the utility of choosing a decision option. Despite the problems in determining the utility function, even in simple cases, it has the advantage that, if correctly identified, by solving the model, it can be assumed that the maximum satisfaction and desirability for the decision maker has been achieved. Also, the main idea Marginal Rate of Substitution( MRS) is to identify the rate of change in the consumption of a product versus the amount of loss of substitute goods in the event that the amount of final utility does not change. In the proposed method, it is possible to identify identical decision matrixes separately for each of the decision-making and decision-making indicators that so far has not existed and can improve the effectiveness of choices for supplier selection.

2-2- Meta-heuristic Algorithm

 

In the last 30years, a new kind of approximation algorithm has emerged.Its aim is to be a combination of innovative methods in bigger frameworks in order to explore efficient and effective research space. Today, these methods are named meta-heuristic methods [25]. So far, several meta-heuristic algorithms have been presented. These algorithms have proved more efficient for some problems.However, in some other problems,they face the issue of being close to the optimum answer. Most meta-heuristic methods are derived throughnatural and physical processes.

The optimization of the paper swarm optimization (PSO) algorithm, ant colony genetic algorithms (GAs), evolutionary algorithms, and simulation of refrigeration are examples of such algorithms[8]. The PSO method is a globally usedoptimization method that can deal with problems whose answer is a point or the surface in n-dimensional space. In such an environment, each paper has a position that defines the coordinates of the paper in the multi-dimensional search,which change with the motion of the paper over time and theposition of the particle. Here, xi (t) determines the position of the paper i at time t. Also,every paper needs to move in space ata certain speed. Here, vi (t) determines the velocity of the paper i at time t i. Byincreasing the speed with respect to the position of each particle, one can consider a new position for each particle. The position of updating the equation of the paper is given in the following equation.

 

   

Figure 3: The particles move in a group

 

 

 

The best individual experience of a paper or the best met position by a paper yi (pbest) is called. Papers can meet the best Achieved place by the whole group that this position is called yi (gbest). Today,GAs are used to solvenumerousproblems in thefields of engineering andsocial science. A GA can be used for restricted and non-restricted problems [36].For standard optimization problems, this is the only way to get an answer. Also, it can be used for linear and nonlinear problems, as well as inprobable planning that involvesrandom variables and adegree of uncertainty [40]. In addition, combinatorial optimization problems that include different problems about computer sciencehavebeen used [8]. GA has provided apowerful method for the exploratory development of large-scale combinational optimization problems. The usual way of presenting chromosomes in GAs is in the form of binary strings [9].

 

2-3- Procedure of Research

Scientific research methods providethe only way to achieve acceptable and scientific achievements. The aim of the present study is to identify the parallel matrices with the primary matrices of decision-making.


 

criteria 

standards

financial stability

company records

quality

customer satisfaction and history reputation

skilled manpower

company structure

technical equipments

Weight

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

C8

alter1

44.52

48.12

61.37

40.71

42.51

67.31

50.26

41.03

alter2

62.32

35.82

50.45

52.61

55.23

52.16

63.02

44.41

alter3

67.75

61.75

62.85

59.32

61.74

60.23

46.12

49.13

alter4

45.18

48.2

37.92

60.71

52.7

71.47

59.3

39.71

alter5

47.52

49.62

44.87

64.52

46.08

64.05

51.04

40.12

alter6

65.51

55.34

64.71

57.28

52.14

56.17

48.12

43.07

alter7

56.8

47.25

58.91

49.64

55.71

52.61

51.49

48.71

alter8

42.28

52.71

54.36

60.21

61.54

60.21

53.17

38.38

alter9

43.72

55.62

48.9

52.7

51.23

64.25

48.16

46.17

alter10

62.31

51.27

53.75

48.5

60.42

58.31

57.63

40.48

Table 1: initial decision making matrix 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

To achieve the intended aim that has been accomplished as a case study regarding the evaluation of contractors of a gas company in Zanjan province, first, a primary matrix of decision-making involving10 company experts was prepared according to eight criteria (C1: standards, C2:financial stability, C3:company records, C4:quality, C5:customer satisfaction and Good record, C6:skilled manpower, C7:company structure, C8: technical equipment).This was prepared and rated separatelythrough the two methods of AHP and TOPSIS. To identify the parallel matrices, the PSO and GA techniques were used.

Consider that the number of programs running was RUN = 10, and each run was done in two ways and two different techniques were used. In general, 40 different matrices were produced.

Among them, according to the fitness function,zeroparallel matrices with the primary matrix have been identified. In the final stage, bycomparing andmatching the parallel matrices and facilities of the organization, an appropriateoption shall be selected that is more consistent with the resources and guidelines of the organization.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ranking matrix by TOPSIS

Rank = 1 Alter = 3 Score = 0.13885

Rank = 2 Alter = 6 Score = 0.12422

Rank = 3 Alter = 10 Score = 0.10622

Rank = 4 Alter = 8 Score = 0.10137

Rank = 5 Alter = 7 Score = 0.094438

Rank = 6 Alter = 5 Score = 0.092619

Rank = 7 Alter = 9 Score = 0.09074

Rank = 8 Alter = 4 Score = 0.089866

Rank = 9 Alter = 2 Score = 0.084263

Rank = 10 Alter = 1 Score = 0.07741

Ranking matrix by AHP

Rank = 1 Alter = 3 Score = 0.11111

Rank = 2 Alter = 6 Score = 0.10501

Rank = 3 Alter = 10 Score = 0.10173

Rank = 4 Alter = 8 Score = 0.099919

Rank = 5 Alter = 7 Score = 0.099179

Rank = 6 Alter = 4 Score = 0.098012

Rank = 7 Alter = 2 Score = 0.097471

Rank = 8 Alter = 9 Score = 0.097327

Rank = 9 Alter = 5 Score = 0.097023

Rank = 10 Alter = 1 Score = 0.093211

2-4- Data

The primary matrix of the company contractors’ evaluation decision-making based on the company’s criteria and in thepresence of experts has been presented below and ranked,using AHP and TOPSIS in MATLABsoftware.

3-Discussion& Results

In this line of research regarding the ability of meta-heuristic algorithms,PSO and GAshave been used through the AHP and TOPSIS methodsin MATLAB software to generate 10 parallel matrices(RUN =10).These have been implemented and their results are given in Table 2 and Graph 2.According to the above table,the best fitness = 0;that is as its parallel matrix with initial decision making matrix. These matrices have been presented in Table 3.

Considering that one of the objectives of this research is the development of scientific issues in the field of multi-criteria decision making, the results of this research are in line with Salati&Makoui research. With the difference that they sought to provide a value function (utility) using the UTA method, the present research seeks to identify the points of indifference in decision-making issues. Since, in the indifference curves, all points on each curve provide the same utility with different constituents. Each Parallel Matrixes represents a point on the indifference curve. Given that most of the decisions and strategies adopted by organizations are in the process of implementation, the identification of different options for decision making increases the flexibility of the organization in the implementation phase and increases the chance of realizing decisions and strategies. The results of this study are closely related to Xiaohan et al. The distinction of the present study is to find the final rate of succession between the criteria and to find indifferent points and to use particle swarm algorithms and genetic algorithms and information on hospital purchases have been the main constraints of this study. Therefore, it is suggested that the calculation of the final rate of succession between several commodities / methods be simultaneously studied.

 

 

Table 3: - output status matrixes produced with GA and PSO techniques&TOPSIS

 

 

TOPSIS

RUN

PSO

GA

Iter = 100 BEST = 4 MEAN = 4.66

Best Fitness = 4

Iter = 100 BEST = 6

Best Fitness = 6

1

Iter = 100 BEST = 2 MEAN = 3.06

Best Fitness = 2

Iter = 100 BEST = 4

Best Fitness = 4

2

Iter = 7 BEST = 0 MEAN = 22.24

Best Fitness = 0

Iter = 100 BEST = 16

Best Fitness = 16

3

Iter = 100 BEST = 4 MEAN = 4.36

Best Fitness = 4

Iter = 100 BEST = 24

Best Fitness = 24

4

Iter = 10 BEST = 0 MEAN = 27.86

Best Fitness = 0

Iter = 100 BEST = 22

Best Fitness = 22

5

Iter = 100 BEST = 6 MEAN = 8.2

Best Fitness = 6

Iter = 100 BEST = 20

Best Fitness = 20

6

Iter = 100 BEST = 6 MEAN = 8.34

Best Fitness = 6

Iter = 100 BEST = 18

Best Fitness = 18

7

Iter = 3 BEST = 0 MEAN = 40.78

Best Fitness = 0

Iter = 100 BEST = 4

Best Fitness = 4

8

Iter = 100 BEST = 8 MEAN = 8.28

Best Fitness = 8

Iter = 100 BEST = 20

Best Fitness = 20

9

Iter = 100 BEST = 2 MEAN = 2.34

Best Fitness = 2

Iter = 100 BEST = 8

Best Fitness = 8

10

 Table 3: - output status matrixes produced with GA and PSO techniques&TOPSIS

 

Table 4: - output status matrixes produced with GA and PSO techniques&AHP

 

 

 

 

 

TOPSIS

RUN

PSO

GA

Iter = 100 BEST = 4 MEAN = 4.66

Best Fitness = 4

Iter = 100 BEST = 6

Best Fitness = 6

1

Iter = 100 BEST = 2 MEAN = 3.06

Best Fitness = 2

Iter = 100 BEST = 4

Best Fitness = 4

2

Iter = 7 BEST = 0 MEAN = 22.24

Best Fitness = 0

Iter = 100 BEST = 16

Best Fitness = 16

3

Iter = 100 BEST = 4 MEAN = 4.36

Best Fitness = 4

Iter = 100 BEST = 24

Best Fitness = 24

4

Iter = 10 BEST = 0 MEAN = 27.86

Best Fitness = 0

Iter = 100 BEST = 22

Best Fitness = 22

5

Iter = 100 BEST = 6 MEAN = 8.2

Best Fitness = 6

Iter = 100 BEST = 20

Best Fitness = 20

6

Iter = 100 BEST = 6 MEAN = 8.34

Best Fitness = 6

Iter = 100 BEST = 18

Best Fitness = 18

7

Iter = 3 BEST = 0 MEAN = 40.78

Best Fitness = 0

Iter = 100 BEST = 4

Best Fitness = 4

8

Iter = 100 BEST = 8 MEAN = 8.28

Best Fitness = 8

Iter = 100 BEST = 20

Best Fitness = 20

9

Iter = 100 BEST = 2 MEAN = 2.34

Best Fitness = 2

Iter = 100 BEST = 8

Best Fitness = 8

10

RUN

AHP

PSO

GA

1

Iter = 100 BEST = 2 MEAN = 2.92

 Best Fitness = 2

Iter = 100 BEST = 6

 Best Fitness = 6

2

Iter = 100 BEST = 6 MEAN = 10

 Best Fitness = 6

Iter = 100 BEST = 6

 Best Fitness = 6

3

Iter = 100 BEST = 2 MEAN = 2.18

 Best Fitness = 2

Iter = 100 BEST = 22

 Best Fitness = 22

4

Iter = 100 BEST = 2 MEAN = 11.94

 Best Fitness = 2

Iter = 100 BEST = 26

 Best Fitness = 26

5

Iter = 4 BEST = 0 MEAN = 50.02

Best Fitness = 0

Iter = 100 BEST = 14

 Best Fitness = 14

6

Iter = 11 BEST = 0 MEAN = 44.7

Best Fitness = 0

Iter = 100 BEST = 12

 Best Fitness = 12

7

Iter = 100 BEST = 6 MEAN = 9.66

 Best Fitness = 6

Iter = 100 BEST = 30

 Best Fitness = 30

8

Iter = 100 BEST = 4 MEAN = 4

 Best Fitness = 4

Iter = 100 BEST = 10

 Best Fitness = 10

9

Iter = 45 BEST = 0 MEAN = 10.5

Best Fitness = 0

Iter = 100 BEST = 24

 Best Fitness = 24

10

Iter = 100 BEST = 4 MEAN = 5.6

 Best Fitness = 4

Iter = 100 BEST = 18

 Best Fitness = 18

Table 5: Indifference point initial decision making matrix

TOPSIS- PSO OUT PUT 3 - :Indifference point 1

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

Ater1

4

0

0

82

57

50

0

0

Ater2

34

0

0

88

0

46

78

18

Ater3

118

124

30

119

0

118

93

0

Ater4

77

6

0

6

41

100

0

3

Ater5

48

0

43

130

0

13

103

81

Ater6

19

74

126

80

79

8

14

41

Ater7

86

58

25

100

10

0

94

30

Ater8

83

63

0

70

108

31

26

77

Ater9

40

0

35

106

0

85

64

93

Ater10

81

2

108

62

24

79

89

81

 

TOPSIS- PSO  OUT PUT 5 -:Indifference point 2

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

Ater1

42

0

0

40

0

0

7

12

Ater2

73

2

0

13

103

18

127

51

Ater3

26

122

123

91

99

105

66

87

Ater4

75

0

52

122

5

12

91

12

Ater5

20

66

8

74

22

61

59

77

Ater6

132

55

130

0

0

113

97

52

Ater7

39

49

29

100

112

0

103

98

Ater8

85

47

35

97

0

120

92

16

Ater9

20

69

18

83

103

52

0

31

Ater10

99

12

99

97

101

51

116

61

 

TOPSIS- PSO OUT PUT 8 -:Indifference point 3

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

alter1

90

0

50

8

28

0

40

4

alter2

0

72

61

0

32

32

0

63

alter3

109

124

33

119

9

105

19

96

alter4

91

97

66

0

0

143

0

0

alter5

46

90

90

21

50

32

103

0

alter6

132

27

9

74

105

0

97

82

alter7

85

95

84

0

112

19

83

0

alter8

85

0

87

121

0

115

69

0

alter9

83

70

67

0

0

129

97

0

alter10

0

75

8

80

121

117

116

0

 

Indifference point 4: OUT PUT 5- AHP-PSO

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

Ater1

50

9

0

0

86

135

0

27

Ater2

70

10

95

2

53

36

59

43

Ater3

136

124

13

110

124

106

38

78

Ater4

48

39

76

37

15

30

43

47

Ater5

87

20

28

0

34

124

24

0

Ater6

132

0

126

115

87

99

83

87

Ater7

40

95

118

19

0

66

0

62

Ater8

0

60

109

55

36

121

54

0

Ater9

50

17

0

38

0

0

97

71

Ater10

125

58

0

40

53

65

116

73

 

AHP-PSO OUT PUT 6 -: Indifference point 5

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

alter1

56

9

0

82

0

0

101

83

alter2

23

72

0

0

111

0

95

89

alter3

136

105

7

119

95

121

93

7

alter4

82

26

62

0

0

143

119

80

alter5

0

28

32

85

0

99

0

81

alter6

132

20

130

0

0

113

97

67

alter7

0

95

0

100

112

13

103

0

alter8

0

0

109

121

0

73

107

77

alter9

45

0

0

32

103

82

97

93

alter10

125

0

50

86

75

117

116

13

 

AHP-PSO   Indifference point 6:OUT PUT 9 -

وزن

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

 

c1

c2

c3

c4

c5

c6

c7

c8

alter1

0

34

0

0

86

93

1

30

alter2

1

20

101

76

3

3

0

17

alter3

136

110

92

20

95

99

1

99

alter4

37

35

41

16

74

127

0

5

alter5

0

32

18

59

9

59

0

6

alter6

59

11

114

52

89

87

79

63

alter7

97

7

8

78

28

43

103

20

alter8

33

98

109

0

79

13

94

31

alter9

5

1

94

9

37

44

0

93

alter10

45

39

61

56

121

49

113

0

 

Table 6: alter 3 position in output results of Indifference pointmatrixes

 

 

Table 5: alter 3 position in output results of Indifference pointmatrixes

 

 

Weight

0.125

0.15

0.125

0.15

0.1

0.125

0.125

0.1

criteria

c1

c2

c3

c4

c5

c6

c7

C8

TOPSIS- PSO   OUT PUT 3-

118

124

30

119

0

118

93

0

  OUT PUT 5-TOPSIS- PSO

26

122

123

91

99

105

66

87

TOPSIS- PSO  OUT PUT 8-

109

124

33

119

9

105

19

96

AHP-PSO  OUT PUT 5-

136

124

13

110

124

106

38

78

AHP-PSO   OUT PUT 6-

136

105

7

119

95

121

93

7

AHP-PSO  OUT PUT 9-

136

110

92

20

95

99

1

99

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  1. Xiaohan Yu, SuojuanZh, XianglinL,Xiuli Q. ELECTRE Methods in Prioritized MCDM Environment. Information Sciences Journal. January 2018;424: 301-316https://doi.org/10.1016/j.ins.2017.09.061