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

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

2012 International Conference on Biological and Biomedical Sciences

Advances in Biomedical Engineering, Vol.9

A ong>Modelong> ong>forong> ong>Closedong>-ong>Loopong> ong>Supplyong> ong>Chainong> ong>Basedong> on Fuzzy Graph Theory ∗

Yisheng Wu

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

wyslqh@163.com

Keywords: ong>Closedong>-loop supply chain; Fuzzy graph theory; ong>Modelong>

Abstract. To describe closed-loop supply chain (CLSC) quantitatively and intuitionisticly, we used

Fuzzy graph theory to construct a model. The mode used two-level directions fuzzy graph of the

model to describe relationship among links of CLSC and used the generalized fuzzy matrix to

measure the operation time of CLSC. Thereong>forong>e this model provides a foundation ong>forong> improving

perong>forong>mance of CLSC.

1. Introduction

The closed-loop supply chain(CLSC) is a system through which material, fund and inong>forong>mation flow,

and it included ong>forong>ward supply chain(SC) and reverse SC. Furthermore, it had constituents such as

production, saling, recycling, reproduction, 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 relation,

operation mechanism and perong>forong>mance of CLSC well. Related research were as follows: Guiltinan [2]

and Pohlen [3] constructed corresponding models according to the function of various CLSC. De Brito

M P established CLSC model which contained three levels such as strategy, tactic and operation

from the view of enterprise strategy [4] . Harold Krikkle’s CLSC model was inferred from economics,

environmental protection and SC channel [5] . Fleischmann M put ong>forong>ward a continuous SC model

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

Electronic Installation Re-manufacturing Company and concluded the quantity and location of SC’s

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

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

paid little attention to operation time of CLSC. Furthermore, there are complicated relationship

among constituents of CLSC, i.e., CLSC contains not only material flow, but also fund and

inong>forong>mation flow. And it is necessary to describe relationship among constituents of CLSC based on

operation time of CLSC quantitatively. So we uses fuzzy graph theory to construct two-level

directions fuzzy graph and generalized fuzzy matrix of CLSC which will provides a method ong>forong>

improving perong>forong>mance of CLSC.

2. Process of ong>Closedong>-ong>Loopong> ong>Supplyong> ong>Chainong>

CLSC contains eleven constituents, i.e., supplying, manufacturing/remanufacturing,

distribution/redistribution, saling/resaling, consumption, callback, testing, repairing, recycling,

reusing and abandonment [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.

Redistribution means distributing the products to marks after they are remanufactured, repaired and

∗ Foundation item:The Science Foundation of Nanjing institute of technology (No. QKJB2009033)

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

290


eused. Abandonment means recycling products should be scrapped if they can not be reused

because of existing technical limitation. Figure 1 illustrates the process of CLSC.

Figure 1. Process of closed-loop supply chain

Moreover, we can understand material flowing through every constituent of CLSC clearly from

Figure 1, and we can also be easy to understand inong>forong>mation flowing through them because material

flows with inong>forong>mation at the same time. However, the relationship among constituents of CLSC are

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

testing, abandonment, repairing, reusing and recycling belong to the same entity, i.e., the reusing

enterprise which is only responsible ong>forong> fund settlement with other companies. Thereong>forong>e, no fund

flow through the constituents of abandonment, repairing, reusing and recycling. And fund flowing

through the constituents of reusing, repairing and redistribution is by the way of the reusing

enterprise paying logistics fee to the constituent of redistribution. 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 addition, the other logistics fee is from

manufacturing/remanufacturing enterprise to distribution/redistribution 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 direction

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

3.1 Definition 1.

~ ~

Supposition that V = { v1,

v2

, ,

vn}

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

V

~ , E}

represents fuzzy

graph of the set. In addition, V ~ is a fuzzy set of V and its membership function is μ v

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

,

n ,

and n represents ambiguity of node v i . Furthermore, E ~ is a fuzzy relation 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 connection where μ ij

≠ 0 and μ ij is

regarded as ambiguity between node v i and node v j . Consequently, there is no connection between

them where μ ij

= 0 and there exists definite connection where = 1 . Similarly, there is a fuzzy

connection whose size is μ ij

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

ji .

3.2 Definition 2.

Supposition that there are l connections between node v i and node v j from Definition 1. And each

connection has a direction which means v i aimming to v j or v j aimming to v i . Furthermore, E ~

from Definition 1 can be shown in another ong>forong>m:

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 , thereong>forong>e μ ij

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 directions fuzzy graph of set V .

ijk

(2)

4. ong>Modelong>

4.1 Parameter Processing

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

operation time of CLSC is usually more than 1. Thereong>forong>e we must change the time mathematically,

i.e., [0,1] processing, to construct multi-level directions fuzzy graph of CLSC and also help computer

to process the time parameter.

First we suppose T 1 is a set which includes the operation time of all CLSC constituents.

{ , }

T 1 = t 1 , t 2 , tn

(4)

Then we suppose T 2 is a set which includes the time of material, fund, and inong>forong>mation flowing

through the constituents.

{ }

T 2 = t n + 1, t n+

2 , , t n+

m

(5)

Third we suppose T is a set of CLSC operation time. Consequently, 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 ong>Modelong>

'

' ' ' '

{ 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 direction fuzzy graph, thereong>forong>e

{ , , }

= t t t

(9)

T 1 1 2 , 11

So we infers CLSC multi-level direction fuzzy graph

~ ~

G = { ,

~

T1 T , H}

(10)

In addition, T ~ 1 is a fuzzy set of set T1 . Membership function 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


Second 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 connection between node t i and node t j where μ ij

≠ 0 , and μ ijq

means there are q connections between them. Furthermore, μ ij

( i = j)

stands ong>forong> ambiguity of CLSC

nodes, and μ ij ( i ≠ j k

) connection strength

from node ti to node t j .

ij k

V 11

V 1

V 10

V 2

V 9

V 3

Material flow

Inong>forong>mation flow

V 8

V 4

Fund flow

V 7

V 6

V 5

Figure 2. Two-level deirection fuzzy graph of closed-loop supply chain

Third we use V 1 , V 2 , … , V 11 as constituents of CLSC, i.e.,supplying ,

manufacturing/remanufacturing, distribution/redistribution, saling/resaling,comsumption, callback,

testing, abandonment, repairing, reusing, recycling. Thereong>forong>e,two-level directions fuzzy graph of

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

Furthermore, we can conclude there are at most two constituents of CLSC from Figure 2.

Thereong>forong>e, the level number of Figure 2. is two. But the symbol of CLSC operation time is not

labeled on Figure 2. in order to observe the relationship among constituents 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. Conclusion

In this research, we brought ong>forong>ward a universal model of CLSC based on operation time. However,

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

adjusted to corresponding node number in its application. Furthermore, the model provides a

foundation to optimize the operation time of CLSC.

6. Acknowledgment

The author would like to express his appreciation to the referees ong>forong> their contributions in improving

the quality of the paper.

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