Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers

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Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers. Advisor: Prof. Ta-Chung Chu Graduate: Elianti ( 李水妮 ) M977z240. 1. Introduction. Assume: k decision makers (i.e. D t ,t=1~ k ) m alternative (i.e. A i ,i=1~ m ) n criteria ( C j ,j=1~ n )

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### Chapter 4MODEL ESTABLISMENTThe Preference Degree of Two Fuzzy Numbers

M977z240

1. Introduction
• Assume:

k decision makers (i.e. Dt,t=1~k)

m alternative (i.e. Ai,i=1~m)

n criteria (Cj,j=1~n)

There are 2 types of criteria:

• Qualitative (all of them are benefit), Cj=1~g
• Quantitative

For benefit: Cj=g+1~h

For cost: Cj=h+1~n

2. Ratings of Each Alternative versus Criteria

Qualitative criteria Quantitative criteria

BenefitCost

Let Xijt= (aijt,bijt,cijt), i= 1,…,m, j= 1,…,g, t=1,…,k, is the rating assigned to alternative Aiby decision maker Dt under criterion Cj.

2. Ratings of Each Alternative versus Criteria

Xij= (aij,bij,cij) is the averaged rating of alternative Aiversus criterion Cjassessed by the committee of decision makers.

Then: (4.1)

Where:

j=1~g

2. Ratings of Each Alternative versus Criteria

Qualitative (subjective) criteria are measured by linguistic values represented by fuzzy numbers.

3. Normalization of the Averaged Ratings
• Values under quantitative criteria may have different units and then must be normalized into a comparable scale for calculation rationale. Herein, the normalization is completed by the approach from (Chu, 2009), which preserves by property where the ranges of normalized triangular fuzzy numbers belong to [0,1].
• Let’s suppose rij=(eij,fij,gij) is the performance value of alternative Aiversus criteria Cj, j=g+1 ~ n.
• The normalization of the rijis as follows:

(4.2)

3. Normalization of the Averaged Ratings

The fuzzy multi-criteria decision making decision can be concisely expressed in matrix format after normalization as follow:

j = 1~n

3. Averaged Importance Weights

Let j=1,…,n t=1,…,k be the weight of importance assigned by decision maker Dtto criterion Cj.

Wj = (oj,pj,qj) is the averaged weight of importance of criterion Cj assessed by the committee of k decision makers, then:

(4.3)

Where:

4. Averaged Importance Weights
• The degree of importance is quantified by linguistic terms represented by fuzzy numbers
4. Final Fuzzy evaluation Value
• The final fuzzy evaluation value of each alternative Aican be obtained by using the Simple Addictive Weighting (SAW) concept as follow:

Here, Piis the final fuzzy evaluation values of each alternative Ai.

i=1,2,…,m,

4. Final Fuzzy evaluation Value

The membership functions of the Pican be developed as follows:

and

4. Final Fuzzy evaluation Value
• By applying Eq. (4.4) and (4.5), one obtains the -cut of Pias follows:

(4.6)

There are now two quotations to solve, there are:

(4.7)

(4.8)

4. Final Fuzzy evaluation Value
• We assume:

So, Eq. (4.7) and (4.8) can be expressed as:

5. Final Fuzzy evaluation Value

The left membership function and the right membership function of the final fuzzy evaluation value Pi can be produced as follows:

(4.11)

(4.12)

Only when Gi1 =0 and Gi2=0, Pi is triangular fuzzy number, those are:

For convenience, Pi can be donated by:

(4.13)

5. An Improved Fuzzy Preference Relation
• To define a preference relation of alternative Ah over Ak, we don’t directly compare the membership function of Ph (-) Pk. We use the membership function of Ph (-) Pk. to indicate the prefer ability of alternative Ah over alternative Ak, and then compare Ph (-) Pk.with zero.
• The difference Ph (-) Pk. here is the fuzzy difference between two fuzzy numbers. Using the fuzzy number, Ph (-) Pk. , one can compare the difference between Ph and Pk. for all possibly occurring combinations of Ph and Pk.
5. An Improved Fuzzy Preference Relation
• The final fuzzy evaluation values Ph and Pkare triangular fuzzy numbers. The difference between Ph and Pkis alsoa triangular fuzzy number and can be calculated as:

Let Zhk=Ph-Pk, h,k=1,2,…m, the -cut of Zhk can be expressed as:

Where

5. An Improved Fuzzy Preference Relation
• By applying Eq. (4.6) to (4.13) to obtain results

as follows:

(4.14)

(4.15)

5. An Improved Fuzzy Preference Relation

Because the formula is too complicated, then we make some assumptions as follows:

5. An Improved Fuzzy Preference Relation
• There are two equations to solve:

(4.16)

(4.17)

Using Eq. (4.16) and (4.17), the left and right membership functions of the difference Zhk=Ph-Pkcan be produced as follows:

(4.18)

(4.19)

5. An Improved Fuzzy Preference Relation
• Obviously, Zhk=Ph-Pk may not yield a triangular shape as well. Only when Ghk1=0 and Ghk2=0, is a triangular fuzzy number, that is:
• For convenience, Zhkcan be denoted by:

(4.20)