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Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers

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Chapter 4MODEL ESTABLISMENTThe Preference Degree of Two Fuzzy Numbers

Advisor: Prof. Ta-Chung Chu

Graduate: Elianti (李水妮)

M977z240

- 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

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.

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

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

- 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)

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

j = 1~n

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:

- The degree of importance is quantified by linguistic terms represented by fuzzy numbers

- 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,

The membership functions of the Pican be developed as follows:

and

- 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)

- We assume:

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

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)

- 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.

- 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

- By applying Eq. (4.6) to (4.13) to obtain results
as follows:

(4.14)

(4.15)

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

- 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)

- 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)