Developing a Decision Making Model Using Interval ...

Developing a Decision Making Model Using Interval Valued Intuitionistic Fuzzy Number Syed Abou Iltaf Hussain 1[0000-0002-5157-0607], Uttam Kumar Mandal2 and Sankar Prasad Mondal3 1, 2

3

Department of Production Engineering, National Institute of Technology, Agartala, Jirania-799046, India Department of M athematics, M idnapore College (Autonomous), West M idnapore-721101, West Bengal, India [email protected]

Abstract. A M ulti criteria decision making model is developed that considers the nondeterministic nature of decision maker along with the vagueness in the decision. Hence, the ratings of alternatives versus criteria are assessed in Parametric Interval Valued Intuitionistic Fuzzy Number (PIVIFN). The main objective of this model is to minimize the risk associated with each alternative. For this reason, the decision matrix is converted into relative benefit matrix. A defuzzification method is developed using the Riemann Integral method. In addition, different properties, theorems and operators are redefined for PIVIFN. Finally, the model is applied to solve a decision making problem

Keywords: Priority index, Degree of vagueness, Parametric Interval Value Intuitionistic Fuzzy Number, Relative Benefit matrix, Riemann Integral method

1

Introduction

Over the time Multi criteria decision making (MCDM) method has been evolved as a significant tool of modern decision making science. In MCDM method, the most appropriate alternative is chosen from a group of identical alternatives on the basis of some criteria. Some of its thriving application is pattern recognition [1], material s election [2 – 6], supplier selection [7 – 11], site selection [12 – 14] and so on. Due to the existence of vagueness or uncertainty [15] in the information as well as impreciseness in the physical nature of the problems the decision makers faces a lot of complications during the process of decision making. For obtaining a judicious result, many researchers have combined MCDM with fuzzy sets (FSs) [16 – 19], interval sets (ISs) [20 – 23], gray relation analysis (GRA) [24, 25] and others [26–29]. Out of all the methods, fuzzy integrated MCDM approach is mostly been used for tackling uncertainty based decision making problems. Atanassov [30 – 32] generalized the concept of FSs, introduced by Zadeh [33] and presented the intuitionistic fuzzy sets (IFSs). The interval-valued fuzzy sets (IVFSs) [34, 35] and vague sets (VSs) [36] were the extended and developed form IFSs. Fur-

2

ther IFSs and IVFSs are integrated which lead to the development of interval valued intuitionistic fuzzy sets (IVIFSs) [37]. With the introduction of IVIFSs, it is coupled with MCDM problems [38 – 45] for finding the most suitable alternatives in a scenario where decision variables are collected in the form of interval valued number. IVIFSs integrated MCDM techniques became a major flare in decision making b ecause the information about criteria or attribute values is usually uncertain or fuzzy due to the increasing intricacy of the socio-economic environment and the vagueness in psychological perspective of human [43]. An attempt is made in this paper to develop a model that takes into account the non-deterministic nature of decision maker along with the degree of vagueness in the decision. For this reason, in this paper, a model is developed that considers the IVIFSs in parametric form called the parametric interval valued intuitionistic fuzzy sets (PIVIFSs). Above that we have defined a defuzzification method based on Reimann integral method.

2

Preliminaries

In this section, some of our preliminary research definitions, properties and theorem are summarized. 2.1

Definition: Interval valued fuzzy number

An interval valued fuzzy number 𝐼 is represented by closed interval 〈[ 𝐼𝑙 , 𝐼𝑢 ] ; 𝜇 𝐼̃ 〉 and defined as 𝐼 = 〈[ 𝐼𝑙 , 𝐼𝑢 ]; 𝛼〉 = {𝑥; 𝜇 𝐼̃ (𝑥): 𝐼𝑙 ≤ 𝑥 ≤ 𝐼𝑢 , 𝑥 ∈ 𝑅} The membership function for this interval can be written as 𝛼, 𝐼𝑙 ≤ 𝑥 ≤ 𝐼𝑢 𝜇 𝐼𝑖 (𝑥) = { 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 where 𝑅 is the set of real numbers ; 𝐼𝑙 and 𝐼𝑢 are the left and right limit of the interval number respectively and 𝜇 𝐼̃(𝑥) as mentioned in definition 2.1. 2.2

Definition: Interval valued intuitionistic fuzzy number

An interval valued intuitionistic fuzzy number 𝐼𝑖 can be defined as 〈[ 𝐼𝑙 , 𝐼𝑢 ] ; 𝛼, 𝛽〉 . The membership function for this interval can be written as 𝐼𝑖 = 〈[ 𝐼𝑙 , 𝐼𝑢 ] ; 𝛼, 𝛽〉 = {𝑥; 𝜇 𝐼̃ (𝑥), 𝜗𝐼 𝑖 (𝑥) : 𝐼𝑙 ≤ 𝑥 ≤ 𝐼𝑢 , 𝑥 ∈ 𝑅} 𝛼, 𝐼𝑙 ≤ 𝑥 ≤ 𝐼𝑢 𝜇 𝐼𝑖 (𝑥) = { 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and the non-membership function is written as 𝛽, 𝐼𝑙 ≤ 𝑥 ≤ 𝐼𝑢 𝜗𝐼 𝑖 (𝑥) = { 1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Where 𝐼𝑙 and 𝐼𝑢 are the left and right limit of the interval number respectively and 𝜇 𝐼̃ (𝑥) and 𝜈 𝐼̃(𝑥) as mentioned in definition 2.1 & 2.2

3

2.3

Definition: Parametric interval valued intuitionistic fuzzy number

Parametric intuitionistic interval valued fuzzy number 𝐼(𝑝) is represented by 〈{𝑝 ∗ 𝐼𝑙 + (1 − 𝑝) ∗ 𝐼𝑢 }; 𝛼, 𝛽〉 = 〈{𝐼𝑢 + 𝑝 ∗ ( 𝐼𝑙 − 𝐼𝑢 )}; 𝛼, 𝛽〉 and defined as 𝐼(𝑝) = 〈{𝐼𝑢 + 𝑝 ∗ ( 𝐼𝑙 − 𝐼𝑢 )}; 𝛼, 𝛽〉 = {𝑥; 𝜇 𝐼̃ (𝑥), 𝜈𝐼̃(𝑥) : 𝑥 = {𝐼𝑢 + 𝑝 ∗ ( 𝐼𝑙 − 𝐼𝑢 )}, 𝑝 ∈ [ 0, 1]} The membership function for this interval can be written as: 𝛼, 𝑝 ∈ [0,1] 𝜇 𝐼̃ (𝑥) = { 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and the non-membership function is written as: 𝛽, 𝑝 ∈ [0,1] 𝜈𝐼̃ (𝑥) = { 1, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Where 𝐼𝑙 and 𝐼𝑢 are the left and right limit of the interval number respectively. 𝜇 𝐼̃ (𝑥) and 𝜈 𝐼̃ (𝑥) as mentioned in definition 2.1 & 2.2. 𝑝 is called the priority index of decision maker and 0 ≤ 𝑝 ≤ 1 . 2.4

Definition: Properties of PIVIFN

Let us consider the two parametric intuitionistic interval number 𝐼 = 〈{𝑝 ∗ 𝐼𝑙 + | | | | (1 − 𝑝) ∗ 𝐼𝑢 }; 𝛼1 , 𝛽1 〉 and 𝐽 = 〈{𝑝 ∗ 𝐽𝑙 + (1 − 𝑝) ∗ 𝐽𝑢 }; 𝛼2 , 𝛽2 〉. If ‖𝐼‖ = 𝐼 𝑙 + 𝐼𝑢 and ‖ 𝐽‖ =

2

|𝐽𝑙 |+ |𝐽𝑢 | 2

Addition: ‖𝐼 ‖𝛼 + ‖𝐽‖ 𝛼 ‖𝐼 ‖𝛽 + ‖𝐽‖𝛽 𝐴 = 𝐼 + 𝐽 = 〈{𝑝 ∗ (𝐼𝑙 + 𝐽𝑙 ) + (1 − 𝑝) ∗ ( 𝐼𝑢 + 𝐽𝑢 )}; ‖ 1‖ ‖ ‖ 2 , ‖ 1‖ ‖ ‖ 2〉

(1)

Subtraction: ‖𝐼 ‖𝛼 + ‖𝐽‖𝛼 ‖𝐼 ‖𝛽 + ‖𝐽‖𝛽 𝑆 = 𝐼 − 𝐽 = 〈{𝑝 ∗ (𝐼𝑙 − 𝐽𝑢 ) + (1 − 𝑝) ∗ ( 𝐼𝑢 − 𝐽𝑙 )}; ‖ 1‖ ‖ ‖ 2 , ‖ 1‖ ‖ ‖ 2〉

(2)

𝐼 + 𝐽

𝐼 + 𝐽

𝐼 + 𝐽

𝐼 + 𝐽

Multiplication: 𝑀= 𝐼∗𝐽= 〈𝑝 ∗ min( 𝐼𝑙 𝐽𝑙 , 𝐼𝑢 𝐽𝑢, 𝐼𝑙 𝐽𝑢 , 𝐼𝑢 𝐽𝑙 ) + (1 − 𝑝) ∗ max ( 𝐼𝑙 𝐽𝑙 , 𝐼𝑢 𝐽𝑢 , 𝐼𝑙 𝐽𝑢 , 𝐼𝑢 𝐽𝑙 ) ; 𝛼1 𝛼2 , 𝛽1 + 𝛽2 − 𝛽1 𝛽2 〉 (3) Multiplication by constant: When 𝑘 > 0 and (𝐼𝑙 & 𝐼𝑢 ) > 0 𝑜𝑟 (𝐼𝑙 < 0 & 𝐼𝑢 > 0) 𝑜𝑟 ( 𝐼𝑙 & 𝐼𝑢) < 0 𝑀𝑐 = 𝑘𝐼 = 〈{𝑘𝑝 ∗ 𝐼𝑙 + 𝑘 (1 − 𝑝) ∗ 𝐼𝑢 }; 𝛼1 , 𝛽1 〉 (4) When 𝑘 < 0 and (𝐼𝑙 & 𝐼𝑢 ) > 0 𝑜𝑟 (𝐼𝑙 < 0 & 𝐼𝑢 > 0) 𝑜𝑟 ( 𝐼𝑙 & 𝐼𝑢) < 0 𝑀𝑐 = 𝑘𝐼 = 〈{𝑘𝑝 ∗ 𝐼𝑢 + 𝑘 (1 − 𝑝) ∗ 𝐼𝑙 }; 𝛼1 , 𝛽1 〉 (5) Inverse: When ( 𝐼𝑙 & 𝐼𝑢 ) > 0 𝑜𝑟 ( 𝐼𝑙 & 𝐼𝑢 ) < 0 1 1 ( ) 𝐻 = (𝐼) −1 = 〈{ 𝑝 ∗ ( ) + (1 − 𝑝) ∗ ( )} ; 𝛼1 , 𝛽1 〉 (6) 𝐼𝑢

𝐼𝑙

When ( 𝐼𝑙 < 0 & 𝐼𝑢 > 0) 1 1 ( ) 𝐻 = (𝐼) −1 = 〈{ 𝑝 ∗ ( ) + (1 − 𝑝) ∗ ( )} ; 𝛼1 , 𝛽1 〉 𝐼𝑢

(7)

𝐼𝑙

Division: 𝐼 𝐼 𝐼 𝐼 𝐼 𝐼 𝐼 𝐼 𝐼 𝐷 = = 〈{ 𝑝 ∗ min ( 𝑙 , 𝑢 , 𝑙 , 𝑢) + (1 − 𝑝) ∗ max ( 𝑙 , 𝑢 , 𝑙 , 𝑢)} ; 𝛼1 𝛼2 , 𝛽1 + 𝛽2 − 𝐽

𝛽1 𝛽2 〉

𝐽𝑙 𝐽𝑢 𝐽𝑢

𝐽𝑙

𝐽𝑙

𝐽𝑢 𝐽𝑢

𝐽𝑙

(8)

4

2.5

Definition: Defuzzification of PIVIFN

A parametric interval value intuitionistic fuzzy number 𝐼 = 〈{𝑝 ∗ 𝐼𝑙 + (1 − 𝑝) ∗ 𝐼𝑢 }; 𝛼, 𝛽〉 is converted into crisp number ∁ using Riemann integral which is as follows: (1+𝛼−𝛽) 𝑝=1 ∁= × ∫𝑝=0 {𝑝 ∗ 𝐼𝑙 + (1 − 𝑝) ∗ 𝐼𝑢 }𝑑𝑝 ∁= ∁= ∁= ∁= ∁=

2 (1+𝛼−𝛽) 2 (1+𝛼−𝛽) 2 (1+𝛼−𝛽) 2 (1+𝛼−𝛽)

×[

𝑝2 2 1

∗ ( 𝐼𝑙 − 𝐼 𝑢 ) + 𝑝 ∗ 𝐼 𝑢 ]

𝑝=1 𝑝=0

× [{ ∗ ( 𝐼𝑙 − 𝐼𝑢 ) + 1 ∗ 𝐼𝑢 } − 0] 1

2

× [ ∗ (𝐼𝑙 + 𝐼𝑢 )]

2 2 (1+𝛼−𝛽) ×(𝐼 𝑙+𝐼𝑢 )

(9)

4

Theorem: If there are two PIVIFNs 𝐼 1 and 𝐼 2, then the relation between them is totally defined by their crisp number 𝐶 1 and 𝐶 2 as follows:

2.6 1. 2. 3.

3.

𝑝=1

× ∫𝑝=0 {𝑝 ∗ ( 𝐼𝑙 − 𝐼𝑢 ) + 𝐼𝑢 }𝑑𝑝

If 𝐶 1 ≻ 𝐶 2 then 𝐼 1 ≻ 𝐼 2 . If 𝐶 1 = 𝐶 2 then 𝐼 1 = 𝐼 2. If 𝐶 1 ≺ 𝐶 2 then 𝐼 1 ≺ 𝐼 2 .

Model Establishment

Step 1: Creation of the decision matrix(𝐷) . Decision from each of the decision makers is taken into consideration and the decision matrix (D)is created using the IVIWAA operator having η number of alternatives and γ number of criteria. 𝐷 = [ 𝑑 𝑗𝑖 ] η×γ Step 2: Converting the decision matrix into relative benefit matrix (𝑅) 𝑑 𝑗𝑖 − min (𝑑 𝑗𝑖 ) , ∀ 𝑏𝑒𝑛𝑒𝑓𝑖𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 , 𝑖 ∈ [ 1, 𝛾] 𝑖 𝑅 = [ 𝑟 𝑗𝑖 ] η×γ = { max ( 𝑑 𝑗𝑖 ) − 𝑑 𝑗𝑖 , ∀ 𝑐𝑜𝑠𝑡 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 , 𝑖 ∈ [ 1, 𝛾] 𝑖

Step 3: Calculation of the score of each alternative (σ) 𝛾 𝑗

𝜎 =∑ 𝑖=1

𝜔 𝑖 . 𝑟 𝑗𝑖 ∑𝜂 ( 𝑟𝑗𝑖 ) 𝑗=1

Step 4: Defuzzification and Ranking of alternatives

5

The 𝜎 𝑗 is converted into crisp value (𝐹𝑗 ) according to the eq 9. The ranking of alternatives are done in ascending order of the 𝐹𝑗 values.

4.

Illustrative Example

The best alternative is to be computed amongst a set of 4 alternatives on the basis of 4 criteria. The 3rd and 4th criterion is a cost criterion and the remaining are benefit criteria. The weightage of the criteria are 𝜔 𝑖 = [ 0.3 0.25 0.35 0.1]. The decision matrix is shown in table 1. Table 1: Decision M atrix 1

A2

Criteria

A

C1

〈{ 𝑝 ∗ 4.2 + (1 − 𝑝) ∗ 6} ; 0.6, 0.4〉

〈{𝑝 ∗ 5 + (1 − 𝑝) ∗ 6.7} ; 0.5,0.5〉

2

〈{ 𝑝 ∗ 8 + (1 − 𝑝) ∗ 9.2} ; 0.7, 0.3〉

〈{𝑝 ∗ 7.3 + (1 − 𝑝) ∗ 8.2} ;0.7, 0.3〉

3

〈{ 𝑝 ∗ 7.2 + (1 − 𝑝) ∗ 8} ; 0.5, 0.5〉

〈{𝑝 ∗ 6.2 + (1 − 𝑝) ∗ 7.8} ;0.6, 0.4〉

4

C

〈{𝑝 ∗ 8.2 + (1 − 𝑝) ∗ 8.5 };0.6, 0.4〉

〈{ 𝑝 ∗ 7.2 + (1 − 𝑝) ∗ 8 }; 0.7, 0.3〉

Criteria

A3

A4

C1

〈{𝑝 ∗ 7.3 + (1 − 𝑝) ∗ 8.5 };0.7,0.3〉

〈{𝑝 ∗ 7.1 + (1 − 𝑝) ∗ 8.5} ;0.7, 0.3〉

2

〈{𝑝 ∗ 7.7 + (1 − 𝑝) ∗ 8.5 };0.6, 0.4〉

〈{ 𝑝 ∗ 6.6 + (1 − 𝑝) ∗ 7 }; 0.5, 0.5〉

3

〈{𝑝 ∗ 6.5 + (1 − 𝑝) ∗ 7.5 };0.5, 0.5〉

〈{𝑝 ∗ 8.7 + (1 − 𝑝) ∗ 9.5} ;0.8, 0.2〉

4

〈{ 𝑝 ∗ 7 + (1 − 𝑝) ∗ 8.2} ; 0.5, 0.5〉

〈{ 𝑝 ∗ 7.6 + (1 − 𝑝) ∗ 8 }; 0.6, 0.4〉

C C

C C C

Table 2: Relative benefit matrix Criteria

A1

C1

〈{p*(-2.5)+(1-p)*1.0};0.55,0.45〉

〈{p*(-2.5)+(1-p)*1.0};0.55,0.45〉

C2

〈{p*(-0.5)+(1-p)*1.5};0.65,0.35〉

〈{p*(-0.5)+(1-p)*1.5};0.65,0.35〉

C3

〈{p*(-1.7)+(1-p)*1.7};0.50,0.50〉

〈{p*(-1.7)+(1-p)*1.7};0.50,0.50〉

C4

〈{p*(-1.2)+(1-p)*0.5};0.65,0.35〉

〈{p*(-1.2)+(1-p)*0.5};0.65,0.35〉

Criteria

A

3

A2

A4

C1

〈{p*(-1.8)+(1-p)*0.6};0.55,0.45〉

〈{p*(-1.3)+(1-p)*(-0.2)};0.65,0.35〉

C2

〈{p*(-1.6)+(1-p)*1.6};0.60,0.40〉

〈{p*(-0.8)+(1-p)*(0.8)};0.70,0.30〉

3

〈{p*(-1.3)+(1-p)*1.3};0.55,0.45〉

〈{p*(-1.0)+(1-p)*(1.0)};0.60,0.40〉

4

〈{p*(-2.0)+(1-p)*0.8};0.55,0.45〉

〈{p*(-2.0)+(1-p)*0.8};0.55,0.45〉

C C

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Table 3: Score and rank of alternatives Alternatives

Score in PIVIFN

Score in crisp value

Rank

A1

〈{𝑝 ∗ (−0.404 ) + (1 − 𝑝) ∗ 0.48 };0.35, 0.65 〉

0.153

3

A2

〈{𝑝 ∗ (−0.43 ) + (1 − 𝑝) ∗ 0.409 };0.34, 0.66 〉

0.143

1

A

3

〈{𝑝 ∗ (−0.55 ) + (1 − 𝑝) ∗ 0.39} ; 0.35,0.65 〉

0.152

2

A

4

〈{𝑝 ∗ (−0.64 ) + (1 − 𝑝) ∗ 0.32} ; 0.34,0.66 〉

0.161

4

The ordering of the alternatives is done in ascending order of their 𝐶 𝑗 value. 𝐴2 ≻ 𝐴4 ≻ 𝐴1 ≻ 𝐴3

5.

Results and Discussions

5.1 Validation of proposed models The result from the proposed algorithm is compared with the result from the esta blished models like MOORA, COPRAS, VIKOR and TOPSIS. Table 4: Comparison table Rank Alternatives

Proposed model 3

M OORA

COPRAS

VIKOR

TOPSIS

3

3

2

3

A

1

A

2

1

1

4

1

4

A3

2

2

1

4

1

4

4

4

2

3

2

A

Some of the points observed during the study are as follows: i.

The value of 𝐼𝑙 & 𝐼𝑢 are not constrained except 𝐼𝑙 < 𝐼𝑢.

ii.

Non-deterministic nature of decision maker is measured by priority index (𝑝) and degree of vagueness in decision is the degree of confidence and the degree of scepticism of decision makers’ decision.

iii.

The aggregated decision matrix is converted into relative benefit matrix which implies that the relative benefit for not selecting the alternative with least benefit and highest cost for a benefit and cost criterion respectively.

iv.

Risk minimization is the basic idea for the conversion of the aggregated d ecision matrix into relative benefit matrix.

7

v.

Defuzzification of PIVIFN is done using the Reimann integral approach for 𝑝 varying from 0 to 1.

vi.

Two interval value intuitionistic numbers are said to be equal, lesser or greater than each other if their crisp number are equal, lesser or greater respectively.

vii.

From table 5.1 it is validated that the proposed algorithm could be used for the decision making.

6.

Conclusions

In the present study, a robust MCDM model is developed that can be used in decision making considering the non-deterministic nature of decision maker along with the vagueness in decision. The priority index is the measure of non -deterministic nature of decision maker whereas degree of confidence and scepticism is the measure of degree of vagueness in decision. For this reason the ratings of alternatives versus criteria are assessed in PIVIFN. The PIVIFN defined in the paper is the generalised form of IVIFSs. In the proposed model the aggregated decision matrix is converted into relative benefit matrix which implies the relative benefit for not selecting the alternative with least benefit or highest cost. Risk minimization is the basic idea for the conversion of the aggregated decision matrix. The relative weight of each altern ative is calculated which is in the form of PIVIFN. As a result of which the relative weights are defuzzified using the Reimann integral approach for the value of 𝑝 varying from 0 to 1. The final ranking and comparisons of different alternatives is done in descending order. Finally, the proposed model is used for solving a numerical exa mple. In order to study the stability of the model result obtained from the proposed algorithm is compared with other decision making models and it is noticed that the model 60% of the time returns the same best answer. Hence we can accept the model for ranking of alternatives when the decision matrix is created in PIVIFN form.

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