novel rough delphi method for determination weights

ICMNEE 2018

The 2nd International Conference on Management, Engineering and Environment

NOVEL ROUGH DELPHI METHOD FOR DETERMINATION WEIGHTS OF CRITERIA Zeljko Stevie

Goran Petrovic 2 and Dragisa Stanujkic 3,

1 University of East Sarajevo, Faculty of Transport and Traffic Engineering Doboj, Bosnia and Herzegovina, [email protected] 2 University of Nis, Faculty of Mechanical Engineering, Serbia, pgoran1102@gm ail.com 3 University of Belgrade, Technical faculty in Bor, Serbia, [email protected] * Correspondence: [email protected] A bstract: The decision-making process in all areas requires understanding of a number of elements involved in this process. In that, very important role have the significance of the criteria on the basis of which evaluation or selection of alternatives is carried out. In this paper, the Delphi method is expanded with rough numbers for determining the relative weight of the criteria. The developed approach allows determining the significance of criteria based on an expert evaluation, which must be at least ten. The approach is demonstrated on the example of evaluating performance indicators in transport based on the assessment of 19 decision makers. A sensitivity analysis was perform ed in which the SWARA method was applied which gives very similar results with a developed approach. Taking into account both approaches, the only difference in the rankings of the criteria is at the last position. Keyw ords: Rough Delphi, weights of criteria, transport, key performance indicators.

1. Introduction Determining the significance of criteria according to Petrovic et al. [1] is one of the most important stages in the decision-making process. Practically doesn't exist the problem of multi-criteria decision-making (M CDM ) in which criteria have the equal importance. Taking into account previously said, the methods for determining the weight values are an important factor for making valid decisions. The most well-known methods belonging to the group of subjective methods for determining the weight values of the criteria are the Analytic Hierarchy Process (AHP)

method

[2],

DEM ATEL

(DEcision

MAking

Trial

and

Evaluation

Laboratory) method [3], SWARA (Step wise W eight Assessment Ratio - Analysis) method [4], BW M (Best Worst Method) [5], modified SWARA [6]. In addition to the above methods in recent years, the popularization of the rough sets theory used in MCDM for the reduction of uncertainty and im precision is obvious. For 98

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this reason, these methods have been expanded with rough numbers: Rough AHP [7], Rough SW ARA [8], Rough DEM ATEL [9, 10], Rough BW M [11]. In addition to the introductory part, the paper is structured through the following chapters. In the second chapter are shown classic Delphi method, the concept of rough sets theory and basic operations with rough numbers. In this chapter also is shown development a novel Rough Delphi approach. In the third chapter are shown results with an explanation of calculation in each step. In the fourth part of the paper, it is shown discussion and sensitivity analysis which represent checking obtained results using SWARA method. At the end of the paper, conclusions are presented with emphasis on work contributions and future research.

2. Materials and Methods 2.1. D elp h i m eth od The Delphi M ethod is studying and gives projections about uncertain or possible future situations for which we are unable to perform objective statistical legalities, to form a model, or apply a formal method. These phenomena are very difficult to quantify, because they are mainly qualitative in their nature, ie. there is not enough statistical data about them on which a study could be based. Delphi Method is one of the basic forecasting methods and is the most famous and most widely used expert judgm ent method. Methods of expert's assessments are representing significant improvement in the classical ways of obtaining the forecast by joint consultation of an expert's group for studied phenomenon. In other words, this is a methodologically organized use of expert's knowledge to predict future states and phenomena. A typical group in one Delphi session ranges from a few to thirty experts. Each interviewed expert, participant in the method, relies on knowledge, experience and his / her own opinion. The goal of Delphi Method is to exploit the collective, group thinking of experts about certain field. The goal is to reach a consensus on an event by group thinking. This is a method of indirect collective testing, but with a return link. It consists of eight steps: 1. Selection of the prognostic task, defining basic questions and fields for it; 2. Selection of experts; 3. Preparation of questionnaires; 4. Delivery of questionnaires to experts; 5. Collecting responses and their evaluating; 6 . Analysis and interpretation of responses; 7. Re-exams: 8 . Interpretation of responses and setting up final forecast. The benefits of the Delphi Method 99

It covers the large number of respondents; Expert's statements are objective because they do not know the statements of others until the end of the circle;

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-

It is possible to examine the opinion and attitude of an individual according to a task;

-

The method strengthens the sense of community and encourages thinking

about the future of the organization. Delphi M ethod disadvantages: The success of the method depends exclusively on the participants in the expert panel; -

Complicated im plem entation process; Absence of possibility to exactly identify the number of participants in the expert panel; Long duration of research.

2.2. R o u g h set theory In rough set theory, any vague idea can be represented as a couple of exact concepts based on the lower and upper approximations. That is shown in Figure 1.

Figure 1. Elementary concept of rough set theory [12] Suppose U is the universe which contains all the objects, Y is an arbitrary object of U, R is a set of t classes { G , G

Gt} that cover all the objects in U,

R = { G , G2,...,G } . If these classes are ordered as VY e U, G e R,1 < q ^ t approximation

,

(Apr(Gq))

the

lower

G < g 2 G q}

(2)

Bnd(Gq) = U{Y e U / R(Y) * Gq} = {Y e U / R(Y) > Gq}

Then G

U {Y e U / R(Y) < Gq}

can be shown as rough number (RN (G )), which is determined by

its corresponding lower limit (Lim( G )) and upper lim it (Lim(Gq)) where: 100

(3)

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LM (G q) = = = X = ) =

e A pr(Gq)

(4 )

LUn(Gq) = = = X R(Y) = e q T ( G qq

(5)

RN(Gq ) = [Lim(Gq), Z ^ G q )]

(6)

where M L, M v are the numbers of objects that contained in Apr{Gq) and Apr(G ) , respectively. The

difference

between

them

is

expressed

as

rough

boundary

interval

(IRBnd(Gq)) : IRBnd(Gq) = ~LM(Gq) - Lim (G .) The operations for two rough numbers

(7)

R N (a) = ^Lim (a), L im (a ) ]

and

RN (P) = \_Lim(P), Lim (P)] accordingto [7] are: Addition (+) of two rough numbers RN (a ) and RN (P) RN ( a ) + RN (P ) = \^Lim ( a ) + Lim ( P), L im (a) + L im (P )]

(8)

Subtraction (-) of two rough numbers RN ( a ) and RN (P) RN ( a ) —RN (P ) = ^Lim ( a ) —Lim (P), L im (a) —Lim (P )]

(9)

Multiplication (x) of two rough numbers RN (a ) and RN (P) RN ( a ) x RN (P ) = \^Lim ( a ) x Lim ( P), L im (a) x L im (P )]

(10)

Division (/) of two rough numbers RN (a ) and RN (P) RN (a ) / RN (P ) = ^Lim ( a ) / Lim (P), L im (a) / Lim ( P )] Scalar m ultiplication of rough number RN (a ) , where i

(11) is a nonzero

constant fi x RN (a ) = [ L x Lim(a ), fi x L im (a)]

(12)

2.3. N ov el R o u g h D elp h i m eth od Novel Rough Delphi method is obtained when the integration of rough set theory and conventional Delphi method is performed. The goal is to integrate the benefits of both concepts and to easily enable the determination of the weight values of the criteria. Com bination of rough sets and Delphi method is performed in [13], but on total different way relative on this paper. The suggested Rough Delphi method consists of a total of five steps that are shown below. Step 1. Form a set of n experts to evaluate. At least 10 experts for evaluation of the criteria are recommended. Step 2. In this step, the team of n experts carries out the evaluation of criteria in the way that the assigned values make up a total of 100. 101

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Step 3. Applying equations (1)-(6) to convert all crisp matrices into single rough matrices and obtain a matrix (13). ™ ( G, ) = [ g , s j ' L

(13)

where m represents the number of criteria, while n represents the number of experts which are involved in the assessment process. Step 4. In this step it is necessary to perform summing values of rough numbers for all criteria. After that we can obtain the matrix (14). ™ ( S„) = 1 ^ : ],,m

(14)

X"'1 l : 7 ,Sij -Sij ~ X"'1 7 ,Sij:

(15)

applying equation (15): l

Sij ~

Step 5. In this step, it is necessary to normalize the value of the criteria by dividing each element of the criterion with sum. First, it is necessary to calculate the sum of all previously summing values by row using the equation (16): m m r n (t , ) = [ 'L- ‘U]

= l.> i z : (16) Jj Jj and then applying the equation (17), the normalized weights of the criteria are obtained:

rt -11 z r‘L m

j=,

(17)

‘ j

3. Results The application of the Rough Delphi method has been demonstrated in the case of evaluation and selection Key Performance Indicators (KPIs) in transport. The research was carried out in the first half of this year and the KPIs were selected using the SWARA method in [14]. In order to best demonstrate the applicability of the developed approach, the previously mentioned example was selected whose results using the SW ARA method presented in the Discussion section. A total of 62 indicators were considered, divided into groups. There are a total of 10 main indicators whose calculation using the Rough Delphi M ethod is shown below. Step 1: A team of 19 decision makers from 13 different transportation companies that operate on the territory of Bosnia and Herzegovina and Serbia is formed. It is a satisfied request for at least ten 10 experts. Step 2: In this step, the evaluation by experts was carried out so that the sum of all assessment is equal to 100, as shown in Table 1.

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T able 1. Evaluation of 10 main indicators by 19 decision makers Ei

E2

E3

E4

E5

E6

E7

E8

E9

Ei0

E 11

E 12

Ei3

E 14

E 15

E 16

E 17

E 18

E 19

C1

20

10

20

15

20

15

20

10

20

11

5

17

20

20

20

18

20

20

25

C2

10

10

10

15

5

15

10

10

20

10

15

8

20

25

20

16

10

10

10

C3

10

10

20

15

25

15

20

10

10

10

15

7

5

10

4

9

9

10

20

C4

10

10

15

15

20

15

20

10

5

11

15

7

4

10

4

5

10

10

20

C5

5

10

5

10

5

5

5

10

5

10

5

5

4

5

5

3

10

5

5

C6

5

10

5

5

5

5

5

10

5

10

5

5

4

5

5

3

10

5

5

C7

5

10

10

5

5

10

5

10

5

10

10

13

10

5

5

14

15

5

8

C8

5

10

5

5

5

5

5

10

5

9

5

13

4

5

5

13

5

5

1

C9

20

10

5

10

5

10

5

15

20

10

20

15

25

10

30

12

9

20

5

C10

10

10

5

5

5

5

5

5

5

9

5

10

4

5

2

7

2

10

1

Step 3. In this step, by applying the equations (1) - (6), converting all crisp matrices into individual rough matrices, and obtaining a matrix R N ( G -) which is shown in Table 2. Table 2. Individual rough matrices E1

E2

E3

.. .

E 17

E 18

E 19

C1

[16.72,20.45]

[8.33,17.83]

[16.72,20.45]

..

[16.72,20.45]

[16.72,20.45]

[17.16,25.00]

C2

[9.36,13.88]

[9.36,13.88]

[9.36,13.88]

..

[9.36,13.88]

[9.36,13.88]

[9.36,13.88]

C3

[8.67,14.29]

[8.67,14.29]

[11.61,21.25]

.

[6.80,13.63]

[8.67,14.29]

[11.61,21.25]

C4

[7.73,13.64]

[7.73,13.64]

[9.75,17.14]

.

[7.73,13.64]

[7.73,13.64]

[11.37,20.00]

C5

[4.79,6.47]

[6.16,10.00]

[4.79,6.47]

.

[6.16,10.00]

[4.79,6.47]

[4.79,6.47]

C6

[4.80,6.18]

[5.89,10.00]

[4.80,6.18]

..

[5.89,10.00]

[4.80,6.18]

[4.8,6.18]

C7

[5.00,8.42]

[7.38,11.20]

[7.38,11.20]

.

[8.42,15.00]

[5.00,8.42]

[5.33,10.91]

C8

[4.64,6.76]

[5.53,11.50]

[4.64,6.76]

.

[4.64,6.76]

[4.64,6.76]

[1.00,6.32]

C9

[11.82,22.5]

[7.90,16.21]

[5.00,13.47]

.

[5.8,15.73]

[11.82,22.50]

[5.00,13.47]

C10

[5.79,10.00]

[5.79,10.00]

[4.15,6.73]

.

[1.67,6.06]

[5.79,10.00]

[1.00,5.79]

Step 4: Applied the equation (15) are summarized all the values of the rough numbers in rows and get the matrix R N (S ) .

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RN(c, ) = [261.51, 380.04], RN (c2) = [188.81, 315.29], RN (c3) = [165.06, 310.09], RN (c4) = [149.49, 287.20], RN (c5) = [94.78, 140.13], RN (c6) = [92.46, 132.29], RN (c7) = [121.18, 200.17], RN (c8) = [87.81, 154.05]. RN (c9) = [165.50, 359.09], RN (c10) = [76.13, 144.17]. Step 5. Using the equation (16), a matrix is obtained R N (T j) RN(Tj ) = [14002.73, 2422.52] and in order to obtain the normalized weights of the criteria, the equation is applied (17):

rwL, ] : mr‘L,‘U] 7 ft,j ]

2 61.52

380.04

24 22.52 1402.73

= [0.108,0.271]

The final normalized values of the criteria obtained in the same way are shown in Table 3. In addition, their crisp values and ranking of criteria are shown. T able 3. Weights of criteria and their ranking

104

Low

Upper

Crisp

Rank

C1

0.108

0.271

0.189

1

C2

0.078

0.225

0.151

3

C3

0.068

0.221

0.145

4

C4

0.062

0.205

0.133

5

C5

0039

0.100

0.070

8

C6

0.038

0.094

0.066

10

C7

0.050

0.143

0.096

6

C8

0.036

0.110

0.073

7

C9

0.068

0.256

0.162

2

C10

0.031

0.103

0.067

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4. Discussion Table 4 shows the calculation of the weights of the criteria using the SWARA method.

T able 4. Weights of the main indicators obtained using SWARA method Wj

Kj=Sj+1

C1 C9

Sj 1.000 0.145

C2

0.187

1.187

0.736

0.128

C3

0.132

1.132

0.650

0.113

C4

0.089

1.089

0.597

0.104

C7

0.345

1.345

0.444

0.077

C8

0.113

1.113

0.399

0.069

C5

0.036

1.036

0.385

0.067

C6

0.142

1.142

0.337

0.059

C10

0.031

1.031

0.327

0.057

1.000

qj 1.000

0.174

1.145

0.873

0.152

5.747

z Based on the table 1 it can be concluded that the most important indicator is the first one which is related to vehicle utilization. The experts estimated that the second significant criteria is related to costs in transport, following by the total number of kilometers. It is significant that the realized routes are for 0.089 more important than realized tours, which implicates that managers of transport companies, who are m onitoring performances which have influence on efficiency of management, in the m ost of cases are completely observing every trip, from start to moving till getting back to garage or to getting back on first loading place. The transport time took sixth place according to its weight in regards to rest of indicators and it is for 0.113 more significant from the value of delivery and value of damaged goods. The following are the number of loads and number of unloads. On the last place is indicator which is related to reclamations. W hen considering the difference in the applied Rough Delphi method and the results obtained using the SWARA method, it can be noticed that the results are very similar. The only difference in the rankings of the criteria is the last criterion. Using the Rough Delphi method, the sixth criterion is at the last place with the low est value, while using the SW ARA method at the last position is the tenth indicator.

5. Conclusions In this paper is performed development od a novel Rough Delphi approach for determination of criteria weights. The condition for application of this approach implies participation of at least ten experts. Verification of the proposed 105

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approach is carried out in the field of transport for evaluating key performance indicators. The validity of the developed approach has been proven by using the SWARA method in the sensitivity analysis that gives similar results. The key contribution of this paper is to enrich the field of m ulti-criteria decision-making from the aspect of developing a new approach. Rough Delphi method allows determining the weight values of the criteria, and can be used in com bination with some other approaches to determine the significance of the criteria. As a future research, Rough Delphi method can be integrated with the Rough SWARA method. In this integration, Rough Delphi can be used to determine the ranking criteria, and Rough SW ARA to determine the relative weight values of the criteria.

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