chapter 4 model establisment the preference degree of two fuzzy numbers
Download
Skip this Video
Download Presentation
Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers

Loading in 2 Seconds...

play fullscreen
1 / 20

Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers - PowerPoint PPT Presentation


  • 105 Views
  • Uploaded on

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 )

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Chapter 4 MODEL ESTABLISMENT The Preference Degree of Two Fuzzy Numbers' - lilly


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
chapter 4 model establisment the preference degree of two fuzzy numbers

Chapter 4MODEL ESTABLISMENTThe Preference Degree of Two Fuzzy Numbers

Advisor: Prof. Ta-Chung Chu

Graduate: Elianti (李水妮)

M977z240

1 introduction
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
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 criteria1
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 criteria2
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
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 ratings1
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
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
4. Averaged Importance Weights
  • The degree of importance is quantified by linguistic terms represented by fuzzy numbers
4 final fuzzy evaluation value
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 value1
4. Final Fuzzy evaluation Value

The membership functions of the Pican be developed as follows:

and

4 final fuzzy evaluation value2
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 value3
4. Final Fuzzy evaluation Value
  • We assume:

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

5 final fuzzy evaluation value
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
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 relation1
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 relation2
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 relation3
5. An Improved Fuzzy Preference Relation

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

5 an improved fuzzy preference relation4
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 relation5
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)

ad