A Model for Closed-Loop Supply Chain Based on Fuzzy Graph ...

2012 Internati**on**al C**on**ference **on** Biological and Biomedical Sciences

Advances in Biomedical Engineering, Vol.9

A **on** **Fuzzy** **Graph** Theory ∗

Yisheng Wu

Department of Ecnomics & Management, Nanjing Institute of Technology,Nanjing, China

wyslqh@163.com

Keywords: **Fuzzy** graph theory;

Abstract. To describe closed-loop supply chain (CLSC) quantitatively and intuiti**on**isticly, we used

**Fuzzy** graph theory to c**on**struct a model. The mode used two-level directi**on**s fuzzy graph of the

model to describe relati**on**ship am**on**g links of CLSC and used the generalized fuzzy matrix to

measure the operati**on** time of CLSC. There**on**

per

1. Introducti**on**

The closed-loop supply chain(CLSC) is a system through which material, fund and in**on** flow,

and it included **on**stituents such as

producti**on**, saling, recycling, reproducti**on**, resaling and so **on** [1] . And the CLSC model plays an

important role in CLSC field because the model can help us to understand structure, logic relati**on**,

operati**on** mechanism and per

and Pohlen [3] c**on**structed corresp**on**ding models according to the functi**on** of various CLSC. De Brito

M P established CLSC model which c**on**tained three levels such as strategy, tactic and operati**on**

from the view of enterprise strategy [4] . Harold Krikkle’s CLSC model was inferred from ec**on**omics,

envir**on**mental protecti**on** and SC channel [5] . Fleischmann M put **on**tinuous SC model

which described key parameter’s impact **on** SC’s cost [6] . Jayarama analyzed the SC structure of US

Electr**on**ic Installati**on** Re-manufacturing Company and c**on**cluded the quantity and locati**on** of SC’s

organizati**on**al structure [8] . node [7] . Ma Shihua’s SC model was based **on** enterprise

Thus previous research **on** CLSC model was focused **on** functi**on** and levels of CLSC, and they

paid little attenti**on** to operati**on** time of CLSC. Furthermore, there are complicated relati**on**ship

am**on**g c**on**stituents of CLSC, i.e., CLSC c**on**tains not **on**ly material flow, but also fund and

in**on** flow. And it is necessary to describe relati**on**ship am**on**g c**on**stituents of CLSC based **on**

operati**on** time of CLSC quantitatively. So we uses fuzzy graph theory to c**on**struct two-level

directi**on**s fuzzy graph and generalized fuzzy matrix of CLSC which will provides a method

improving per

2. Process of

CLSC c**on**tains eleven c**on**stituents, i.e., supplying, manufacturing/remanufacturing,

distributi**on**/redistributi**on**, saling/resaling, c**on**sumpti**on**, callback, testing, repairing, recycling,

reusing and aband**on**ment [9] . And recycling means withdrawing the useful material from castoff, e.g.,

battery's recycling use. Reusing means the products can be reused directly if they have no quality

problem after using, e.g., tray's reusing. Remanufacturing means the process of new parts replacing

breakage part of used product. Repairing means servicing the breakage part of callback goods.

Redistributi**on** means distributing the products to marks after they are remanufactured, repaired and

∗ Foundati**on** item:The Science Foundati**on** of Nanjing institute of technology (No. QKJB2009033)

978-1-61275-027-9/10/$25.00 ©2012 IERI ICBBS 2012

290

eused. Aband**on**ment means recycling products should be scrapped if they can not be reused

because of existing technical limitati**on**. Figure 1 illustrates the process of CLSC.

Figure 1. Process of closed-loop supply chain

Moreover, we can understand material flowing through every c**on**stituent of CLSC clearly from

Figure 1, and we can also be easy to understand in**on** flowing through them because material

flows with in**on** at the same time. However, the relati**on**ship am**on**g c**on**stituents of CLSC are

very complicated, so the focus of analysis is **on** fund flow. We suppose the c**on**stituents of callback,

testing, aband**on**ment, repairing, reusing and recycling bel**on**g to the same entity, i.e., the reusing

enterprise which is **on**ly resp**on**sible

flow through the c**on**stituents of aband**on**ment, repairing, reusing and recycling. And fund flowing

through the c**on**stituents of reusing, repairing and redistributi**on** is by the way of the reusing

enterprise paying logistics fee to the c**on**stituent of redistributi**on**. Similarly, the fee of resaling

products is from resaling enterprise to the reusing enterprise, and the fee of callback parts is from

remanufacturing enterprise to the reusing enterprise. In additi**on**, the other logistics fee is from

manufacturing/remanufacturing enterprise to distributi**on**/redistributi**on** enterprise, and the other fee

of resaling products is from resaling enterprise to the manufacturing/remanufacturing enterprise.

3. **Fuzzy** **Graph** Theory

The primary coverage of fuzzy graph theory are generalized fuzzy matrix and multi-level directi**on**

fuzzy graph, and their principles are as follows [10] :

3.1 Definiti**on** 1.

~ ~

Suppositi**on** that V = { v1,

v2

, ,

vn}

is a set including n nodes, and tuple G = { V,

V

~ , E}

represents fuzzy

graph of the set. In additi**on**, V ~ is a fuzzy set of V and its membership functi**on** is μ v

~ ( v i ) , i = 1,2,

,

n ,

and n represents ambiguity of node v i . Furthermore, E ~ is a fuzzy relati**on** of V × V and it can be

expressed as follows:

⎡μ11

⎢

~ ⎢

μ21

E =

⎢

⎢

⎢⎣

μn

1

μ

μ

μ

12

22

n2

μ1

n⎤

⎥

μ2n⎥

⎥

⎥

μnn⎥⎦

Moreover, 0 ≤ μ ij

≤ 1 , i = 1,2,

,

n , j = 1,2,

,

n . There is a c**on**necti**on** where μ ij

≠ 0 and μ ij is

regarded as ambiguity between node v i and node v j . C**on**sequently, there is no c**on**necti**on** between

them where μ ij

= 0 and there exists definite c**on**necti**on** where = 1 . Similarly, there is a fuzzy

c**on**necti**on** whose size is μ ij

. And it is obvious that E ~ is symmetrical, i.e., μij = μ

ji .

3.2 Definiti**on** 2.

Suppositi**on** that there are l c**on**necti**on**s between node v i and node v j from Definiti**on** 1. And each

c**on**necti**on** has a directi**on** which means v i aimming to v j or v j aimming to v i . Furthermore, E ~

from Definiti**on** 1 can be shown in another

μ ij

(1)

291

⎡μ11

⎢

~ ⎢

μ 21

H =

⎢

⎢

⎢⎣

μ n1

μ

μ

μ

12

22

n2

μ1n

⎤

⎥

μ 2n⎥

⎥

⎥

μ nn⎥⎦

And μ ( i = 1,2,

, n;

j = 1,2, n)

is a set whose number is less than l , there

is given by

ij

,

{ μ , μ 2 , μ }

μ = , 0 ≤ μ ijk

≤ 1 , ( k = 1,2, ,

l)

(3)

ij ij1 ij

,

~ ~

So G = { V , V

~ , H}

is defined as muti-level directi**on**s fuzzy graph of set V .

ijk

(2)

4.

4.1 Parameter Processing

We are known that ambiguity of fuzzy graph is more than 0, and less than 1 from secti**on** III. But

operati**on** time of CLSC is usually more than 1. There

i.e., [0,1] processing, to c**on**struct multi-level directi**on**s fuzzy graph of CLSC and also help computer

to process the time parameter.

First we suppose T 1 is a set which includes the operati**on** time of all CLSC c**on**stituents.

{ , }

T 1 = t 1 , t 2 , tn

(4)

Then we suppose T 2 is a set which includes the time of material, fund, and in**on** flowing

through the c**on**stituents.

{ }

T 2 = t n + 1, t n+

2 , , t n+

m

(5)

Third we suppose T is a set of CLSC operati**on** time. C**on**sequently, T has

Finally, we suppose t = max t , i ∈ [ 1 , n + m]

, and

i

{ , , , , , }

n + m time parts.

T = t t t t t , t

(6)

1, 2 n n+

1 n+

2 n+

m

t ' = ti / t

(7)

So the time parameter processing has been completed after above steps, and the results is

expressed as follows:

4.2

'

' ' ' '

{ t t , , t , t , t , t }

'

1, 2 n n+

1 n+

2 n+

m

T

' = ,

(8)

First we define T 1 as ambiguity of CLSC multi-level directi**on** fuzzy graph, there

{ , , }

= t t t

(9)

T 1 1 2 , 11

So we infers CLSC multi-level directi**on** fuzzy graph

~ ~

G = { ,

~

T1 T , H}

(10)

In additi**on**, T ~ 1 is a fuzzy set of set T1 . Membership functi**on** of T ~ 1 is μ ~ ( ti)

T 1

ambiguity of node ti , i = 1,2,

,

11. And H ~ is generalized fuzzy matrix of set T1× T1.

1

that represents

292

Sec**on**d we define μ ij

as a set of membership

μ

⎡ μ11

μ12

μ1,11

⎤

⎢

⎥

~ ⎢

μ 21 μ 22

μ 2,11

H =

⎥

⎢ ⎥

(11)

⎢

⎥

⎢⎣

μ11,1

μ11,2

μ11,11⎥

⎦

{ μ μ }

= , 0 < μ ≤ 1(

k = 1,2 )

(12)

ij ij , 1

ij2

Moreover, there is certain c**on**necti**on** between node t i and node t j where μ ij

≠ 0 , and μ ijq

means there are q c**on**necti**on**s between them. Furthermore, μ ij

( i = j)

stands

nodes, and μ ij ( i ≠ j k

) c**on**necti**on** strength

from node ti to node t j .

ij k

V 11

V 1

V 10

V 2

V 9

V 3

Material flow

In**on** flow

V 8

V 4

Fund flow

V 7

V 6

V 5

Figure 2. Two-level deirecti**on** fuzzy graph of closed-loop supply chain

Third we use V 1 , V 2 , … , V 11 as c**on**stituents of CLSC, i.e.,supplying ,

manufacturing/remanufacturing, distributi**on**/redistributi**on**, saling/resaling,comsumpti**on**, callback,

testing, aband**on**ment, repairing, reusing, recycling. There**on**s fuzzy graph of

CLSC, i.e., G ~ , is shown as Figure 2.

Furthermore, we can c**on**clude there are at most two c**on**stituents of CLSC from Figure 2.

There**on** time is not

labeled **on** Figure 2. in order to observe the relati**on**ship am**on**g c**on**stituents of CLSC clearly.

Moreover we infers H ~ which represents generalized fuzzy matrix of CLSC from (11) and Figure

2. H ~ is shown as (13).

⎡ μ11

μ12

0 0 0 0 0 0 0 0 μ1,11

⎤

⎢

⎥

⎢

μ21

μ22

μ23

0 0 μ26

μ27

0 0 0 0

⎥

⎢ 0 μ32

μ33

μ34

0 0 0 0 μ39

μ3,10

0 ⎥

⎢

⎥

⎢ 0 μ42

μ43

μ44

μ45

μ46

0 0 0 0 0 ⎥

⎢ 0 0 0 μ

⎥

54 μ55

μ56

0 0 0 0 0

~ ⎢

⎥

H = ⎢ 0 0 μ63

0 μ65

μ66

μ67

0 0 0 0 ⎥ (13)

⎢ 0 μ

⎥

⎢

72 0 0 0 μ76

μ77

μ78

μ79

μ7,10

μ7,11

⎥

⎢ 0 0 0 0 0 0 μ87

μ88

0 0 0 ⎥

⎢

⎥

⎢

0 0 μ93

0 0 0 μ97

0 μ99

0 0

⎥

⎢ 0 0 μ

⎥

10,3 0 0 0 μ10,7

0 0 μ10,10

0

⎢

⎥

⎢⎣

μ11,1

0 0 0 0 0 μ11,7

0 0 0 μ11,11⎥⎦

And μ = { t , t , t } where i = j , and the values of μ ij

were as follows where i ≠ j

ij

1 2 , 11

μ = { μ } μ = { μ }

12 1 12 2

12

, μ

21 1 21 2

21

, μ

293

μ 32

= { μ32 1

, μ32 2

} μ 42

= { μ 421 }

μ 26

= { μ 261 }

μ 27

= { μ 271 }

μ 72

= { μ72 1

, μ72 2

} μ 34

= { μ34 1

, μ34 2

}

μ 43

= { μ 431 }

μ 63

= { μ 631 }

μ 39

= { μ 391 }

μ 93

= { μ93 1

, μ93 2

}

μ 3,10

= { μ3,10 1

}

μ 10 ,3

= { μ10,3

1

, μ10,

3 2

}

μ 45

= { μ 45 1

, μ 45 2

} μ 54

= { μ54 1

, μ54 2

}

μ 46

= { μ 461 }

μ 56

= { μ56 1

, μ56 2

}

μ 65

= { μ65 1

, μ65 2

} μ 67

= { μ 67 1

, μ 67 2

}

μ 76

= { μ 761 }

μ 78

= { μ78 1

, μ78 2

}

μ 87

= { μ 871 }

μ 79

= { μ79 1

, μ79 2

}

μ 97

= { μ 971 }

μ 7 ,10

= { μ7,10

1

, μ7,

10 2

}

μ 10,7

= { μ10,7 1

}

μ 7 ,11

= { μ7,11

1

, μ 7, 11 2

}

μ 11,7

= { μ11,7 1

}

μ 1 ,11

= { μ1,11

1

, μ1,

11 2

}

μ = { μ }

11 ,1 11,1 1

, μ11,

1 2

5. C**on**clusi**on**

In this research, we brought **on** operati**on** time. However,

there are many kinds of CLSC according to various enterprise characteristic. So the model should be

adjusted to corresp**on**ding node number in its applicati**on**. Furthermore, the model provides a

foundati**on** to optimize the operati**on** time of CLSC.

6. Acknowledgment

The author would like to express his appreciati**on** to the referees **on**tributi**on**s in improving

the quality of the paper.

References

[1] MARGARETE A S , KEN P. “Meeting the closed-loop challenge: the case of

remanufacturing”, Cali

[2] Guiltinaa J P, Nwokoye N G. “Developing distributi**on** channels and systems in the

emerging recycling industries”, Internati**on**al Journal of Physical Distributi**on**, vol. 6(1), pp.28-

38, 1975.

[3] Pohlen T L, Farris II M. “Reverse logistics in plastic recycling”, Internati**on**al Journal of

Physical Distributi**on** & Logistics Management, vol. 22(7), pp.35-47, 1992.

[4] De Brito M P, Dekker R. “Reverse Logistics-a framework”, Ec**on**ometric Institute Report EI

2002-38, Erasmus University Rotterdam, the Netherlands, pp.98, 2002.

[5] Krikke H., Pappis C.P., Tsoulfas G.T., “Design Principles

optimizing ec**on**omic logistics and envir**on**mental per

Research in Management, ERs-2001-62-LIs, the Netherlands, pp34, 2001.

294

[6] Fleischmann M, Jo Van N., Ben G. “Integrating

Management at IBM”, ERIM Report Series Research in Management, 2002, ERS-2002-107-L

IS, Erasmus University Rotterdam, the Nethlands, pp.76, 2002.

[7] Jayaraman V, Guide J r VDR, Srivastava R. “A closed-loop logistics model

remanufacturing”, Journal of the Operati**on**al Research Society, vol. 50(5), pp.497-508, 1999.

[8] Ma Shihua. “

[9] Vlachos D, Georgiadis P, Iakovou E. “A system dynamics model

planning of remanufacturing in closed-loop supply chain”, Computers & Operati**on**s Research,

vol.21(34), pp.368-370, 2007.

[10] Wu Yi-sheng. “**on** Logistics System **on** **Fuzzy** **Graph**

Theory”,Journal of Applied Sciences,vol. 23(4), pp.417-418, 2005.

295