Amir Omerovic Optimization Of Transport Problems With Fuzzy Coefficients

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International University of SarajevoParomlinska 66, SARAJEVO

Optimization of Transport Problems with Fuzzy Coefficients

By: Amir Omerovic, Jasmin Jusufovic and Mehmet Can

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2 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

Optimization of Transport Problems with Fuzzy Coefficients

• In this paper, we concentrate on fuzzy linear programming problems with fuzzy right-hand sides. We consider here only the case of fuzzy numbers with linear membership functions. The symmetric method of Bellman and Zadeh [2] is used for a defuzzification of these problems. The crisp problems obtained after the defuzzification are non-linear and even non-convex in general. Finally, we give illustrative example and their numerical solutions.

3 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

Brief historical overview

• Fuzzy logic theory was initiated in 1965.• Three phases in its development:

– Academic phase ( 1965 – 1977)– Transormation phase ( 1978 – 1988)– Fuzzy boom phase ( 1989 – nowadays).

4 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

Founder of the Fuzzy theory

• Born on 4th February 1921 in Baku, Azerbaijan.

• High school graduation had in Teheran, and B.S. Degree in Electrical Engineering at the University of Teheran.

• During II World War, he moved with his family to U.S. where he persuaded S.M. Deg. and Ph.D. Deg. in electrical engineering.

• He made great breakthrough in 1965 when he published his work on Fuzzy sets.

• He lives in California, U.S.A.

5 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

njx

mibxa

xc

j

i

n

jjij

n

jjj

,...,2,10

,...,2,1,~

subject to

max

1

1

ii

iii

i

iiii

pbx

pbxb

bx

pxpbxB

if

if

if

/

0

1

)(

6 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

njx

mibxa

xc

j

i

n

jjij

n

jjj

,...,2,10

,...,2,1, subject to

max z

1

11

njx

mipbxa

xc

j

ii

n

jjij

n

jjj

,...,2,10

,...,2,1, subject to

max z

1

12

7 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

. if

if

if

/

0

1

)(

1

1

1

1

ujj

n

j

ujj

n

jl

ljj

n

j

luljj

n

j

zxc

zxcz

zxc

zzzxcG

x

n

jjijii

ii

n

jjiji

i

n

jjij

i

n

jjijiii

xapb

bpxab

bxa

pxapbD

1

1

1

1

/

0

1

)(x

8 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

GDD i

m

i

~~~1

.10,,....,2,.1,0

,....,2,.1,0

0

max

1

1

njx

mixapbp

zxczz

j

jij

n

jiii

ujj

n

jlu

9 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

Example: Fuzzy transport problem

• A company has two production centers . Markets sell the products of this company. The factory produces daily at least 200 items. This production may go to 300, depending on the raw materials and the motivation of workers. Similarly the factory produces at least 150 items a day and this production may go to 200. Market sells 250 items a day, and it may increase it to 300 items. Market sells 100 items a day, and it may increase it to 200 items.

10 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

Market PlaceM1 M2Production

Center

F1 10 8

F2 6 7

Demand

X4X3F2

X2X1F1

SupplyM2M1

Demand

X4X3F2

X2X1F1

SupplyM2M1

1

~M 2

~M

1

~F

2

~F

2121

~~~~FFCC

11 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

4321 76810 xxxxZ

,0,x,,x

,~

x

,~

x

,~

x

,~

x

such that

76810min

4321

242

131

243

121

4321

xx

Mx

Mx

Fx

Fx

xxxxZ

12 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

,300

300200

200

,

,0

100/300

,1

1

x

x

x

xxF

,200

200150

150

,

,0

50/200

,1

2

x

x

x

xxF

,300

300250

250

,

,0

50/300

,1

1

x

x

x

xxM

,200

200100

100

,

,0

100/200

,1

2

x

x

x

xxM

13 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

,0,x,,x

,100x

,250x

,150x

,200x

such that

76810min

4321

42

31

43

21

43211

xx

x

x

x

x

xxxxZ

,0,x,,x

,200x

,300x

,200x

,300x

such that

76810min

4321

42

31

43

21

43212

xx

x

x

x

x

xxxxZ

14 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

,0,150,100,100 4321 xxxx 27001 lZZ

,0,200,200,100 4321 xxxx 38002 uZZ

15 Pula, September 06 Amir Omerović, Jasmin Jusufović, Mehmet Can

.10,4,....,2,.1,0

,0200100

,030050

,020050

,0300100

038007681027003800

max

42

31

43

21

4321

jx

xx

xx

xx

xx

xxxx

j

,0,175,150,100,5.0 4321 xxxx 3250Z

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