Modeling and Analyzing Supply Chain Reliability by Different Effects of Failure Nodes

Lun RAN*

School of Management and Economics

Beijing Institute of Technology Beijing, P.R.China ranlun@bit.edu.cn

Xujie JIA School of Science

Minzu University of China P.R.China

jiaxujie@126.com

Rongjie TIAN School of Management and

Economics Beijing Institute of Technology

Beijing, P.R.China tianmemory@163.com

Abstract—From the viewpoint of a distributed service process, an ion-channel based method was proposed to analyze the reliability of supply chains. An analytic method was presented to determine the reliability of supply chains for logistics systems. New models were built to consider the following situations: when the nodes failed, different defect states had different measures and effects on the system. A Markov model was introduced to analyze the system safety and aggregate states to get the number of repairs and length of time before the system became dangerous by using ion-channel modeling theory. Finally a numerical example was presented to illustrate the results obtained in this paper.

Keywords- Supply Chain, Failed-safe, Failed-dangerous, Markov model, Ion-channel

I. INTRODUCTION The supply chain consists of multi-level and multiple

firms, both upstream and downstream, producer, retailers, and ultimate consumers. All the firms are contact with each other and it is both meaningful and important to be able to measure the reliability of supply chains. A method for quantifying the reliability of supply chains for logistics systems is developed based on reliability interference theory.

Supply chain reliability is defined as the probability of the chain meeting mission requirements to provide the required supplies to the critical transfer points within the system [1]. To achieve supply chain reliability a number of researchers have devoted their efforts in developing models to describe the elements and activities of a supply chain. There are some models such as multi stage-supply chain model [2], economic model[3] deterministic model[4], simulation model[5] and stochastic model[6]. Varshney et al. [7] proposed a more general link-capacity model to have multiple states with different capacities in each state. Jin Liu[8] presented a relatively new graphical construct called meta-graph. Ni Wang and Jye-Chyi Lu[9] focused on developing reliability models for large-size logistics systems to approximate store location & demand data. In the paper, the supply chain encompasses several effort involved in producing and delivering the finished product from suppliers to end customers. It is a series system composed with

producer, retailers, and ultimate consumers. Figure 1 illustrates relationship in the supply chain system.

Figure 1. The Relation Block Diagram of the Supply Chain

In this paper, we study the situation that even one Node in failed-dangerous mode will lead the system risk and the situation that the system is in dangerous state when the last failed node is in dangerous. We assumed that the node states can be distinguished into three states: working, fail-safe and fail-dangerous. W represents the working state, FS for the failed-safe state, FD for the failed-dangerous state. When a node in failed-safe mode, the repair actions can be taken, but once a failed-dangerous mode of nodes occurs, then the Risk appears. Because of two failure modes for each node and the complexity of the system structure, the number of states of Markov process will be large. In the Ion-Channel modeling, Colquhoun and Hawkes [10, 11] aggregated the states of a Markov process to solve their problems. Here we reference their idea, and employ this method to aggregate states of the system to get some important distributions and metrics which are important in safety study.

Based on the above assumptions, a general model will be introduced in Section2. In Section 3, the indexes are given, and several special safety models are introduced in Section 4. Finally, a numerical example is presented to illustrate the results obtained in the paper.

II. MODEL ASSUMPTIONS AND DESCRIPTIONS The supply chain system is composed of n nodes. Both

the states of the system and nodes are classified into working, failed-safe and failed-dangerous. The nodes have the identical exponential distribution with a parameter λ . When a node fails, the probabilities to safe-failed and dangerous-failed mode are p and q , where 1p q+ = . When a node fails, different defect modes have different

1 2 n 0 …

2009 International Conference on Information Management, Innovation Management and Industrial Engineering

978-0-7695-3876-1/09 $25.00 © 2009 IEEE

DOI 10.1109/ICIII.2009.555

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2009 International Conference on Information Management, Innovation Management and Industrial Engineering

978-0-7695-3876-1/09 $26.00 © 2009 IEEE

DOI 10.1109/ICIII.2009.555

396

methods to be dealt with. For a failed-safe mode, the repair process is adopted and for a failed-dangerous mode does not. The repair time of a node also has the identical exponential distribution with a parameter μ . Based on the assumptions and the structure of the system, we can know that the process of working and repair of the system can be modeled by a continuous-time finite state Markov Chain, and then the transition rate matrix Q can be elicited. At time =0t , the n -node system is new, that is, all the nodes are operational. The failure of each node occurs independently. Once a fail-safe node has been repaired, it is as good as new. A fail-dangerous node can not be repaired, and it is in an absorbing state.

In order to study the effects by node failure modes, we divide the space of supply chain system states into three subsets： W , S and U . Subset W consists of the states that the system is working. Subset S consists of the states that the system is failed-safe, and U , failed-dangerous, respectively.

The generator matrix Q can now be partitioned as follow according to the subsets of states defined above.

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

WW WS WU

SW SS SU

UW US UU

Q Q QQ Q Q Q

Q Q Q.

When the system is in a failed-dangerous state, the repair actions do not be adopted. Therefore, the states in subset U are absorbing states. So we get = = =UW US UUQ Q Q 0 , and the indexes are as follows:

A. Probabilities that the system remains within set W throughout time t We can get the probabilities that the system remains

within the set W throughout time t by referencing [10] which is the following equation,

( ) ( ) exp( )R t t t= =WW WWP Q . (1)

Similarly, the probabilities that the system remains within the other two subsets can be gotten.

B. Distribution of the time leaving the set W for a state in set S The density that describes the probability of staying

within the subset of states W for a time t and then leaving W for a state in subset S can be elicited as follow,

( ) ( )t t=WS WW WSG P Q . (2)

Similarly, we can get ( ), ( ), ( )t t tWU SW SUG G G . The Laplace transform of (2) is

* * 1 ( ) ( ) ( )s s s −= = −WS WW WS WW WSG P Q I Q Q . (3)

* 1(0) −≡ = −WS WS WW WSG G Q Q . (4)

WSG has the elements that give the probabilities of leaving set W for a state in set S .

C. The number of repairs before the system first gets to a dangerous state The probability of r repairs before the system first gets

into an unsafe state is

1

1

( ) [( )

( ) ]

( ) ( ) ,

r

r

r

P R r−

−

= =

+

= +

WS SW WU

WS SW WS SU U

WS SW WS SW WU WS SU U

G G GG G G G eG G G G G G G e (5)

where Ue is a unit vector with the number of set U . The expectation of the number of repairs,

1 02

( ) ( ) ( )

( ) ( ) .r r

E R rP r P R r∞ ∞

= =

−

= = >

= − +

∑ ∑

WS SW WS SW WU WS SU UI G G G G G G G e (6)

D. The length of time that the system is in safe states before the system becomes dangerous

1* * *

* * * * *

*

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

f s s s

s s s s s

s

−⎡ ⎤= −⎣ ⎦

⎡ ⎤+⎣ ⎦+

WS SW

WS SW WU WS SU U

WU U

I G G

G G G G G e

G e

.

(7)

The inversion of (7) gives the p.d.f of the length of time that the system is in safe states before the system becomes unsafe. Further, the expected length of time can be calculated.

III. EASE OF USE

A. The supply chain system I The supply chain system is composed of n nodes and

only one repairman whose repair time has an exponential

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distribution with parameter μ . In the system, the system is in working state when all the nodes are working. And once a node is failed-safe, the repair actions can be taken and the system is in the failed-safe state. But once a failed-dangerous mode of nodes occurs, it can not be repaired. Then the system is dangerous Situation. Let ( , )i j denotes the

state that in the system there are i failed-safe nodes and j failed-dangerous nodes. Based on these assumptions, the relationships of the states and the transition rates are shown in Figure 2.

Figure 2. The Transition Diagram of the Parallel Strict Safety System

Then the generator matrix Q is, (0,0) (1,0) ( ,0) (0,1) (1,1) ( ,1) ( -1,1)

(0,0) 0 0(1,0) ( 1) 0 ( 1) 0

( ,0) (0,1) 0 0 0 0 0

(1,1) 0 0 0 0 0( ,1) 0 0 0( ,1) 0 0 0

nn np nq

n n q

n

λ λ λμ λ μ λ

−⎡ ⎤⎢ ⎥− − − −⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q

(8) Dividing the space of system states like in Section 2, we

can get the partition matrix of Q , then the reliability indexes can be given by using the corresponding equations provided in Section 2.

B. The supply chain system II A parallel supply chain system with the repair process is

more complex than a series system. Here we only consider the situation that all nodes have the same exponential distribution with parameter λ , and there is only one repairman whose repair time has exponential distribution

with parameter μ . The same coverage of each node to two modes are p and q , ( 1p q+ = ), respectively. Let state 0 be the state in which all nodes are operational. Let the i th state denote the state in which there are i nodes in the failed-safe modes and the n i+ th state denote one of

1n i− + operational nodes will be in failed-unsafe mode. Based assumptions, the relationships of the states and the transition rates are shown in Figure 3.

Figure 3. State Transition Diagram for the Parallel System.

Then the generator matrix Q is,

μ

( 1)n pλ− 0 1 2

npλ

μ i

1n + 2n + 3n + 1n i+ +

nqλ ( 1)n qλ− ( 2)n qλ− ( )n i qλ−

1n − n

2n

μ

pλ

qλ

(0,0) (1,0) ( 1,0)n − ( ,0)n( ,0)inpλ

λpλ

μ μ

(0,1) (1,1) ( ,1)i ( 1,1)n −

nqλλ

( 1)n qλ−λ

( )n i qλ−λ

qλλ

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0 0 0( 1) ( 1) 0 ( 1) 0

( 2) 0 0 0

0 0 0 00 0 0 0 00 0 00 0 0

n np nqn n p n q

n p

λ λ λμ λ μ λ λ

μ λ μ

μ μ

−⎡ ⎤⎢ ⎥− − − − − −⎢ ⎥⎢ ⎥− − −⎢ ⎥⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q

(9)

Dividing the space of system states like in Section 2, we can get the partition matrix of Q , then the reliability indexes can be given by using the corresponding equations provided in Section 2.

IV. A NUMERICAL EXAMPLE

A. A Series System There are two nodes in a series system and the

parameters are

1 2λ =， 2 3λ =

， 1 0.9p =，

2 0.8p = , 1 1μ = , 2 0.1μ = , with equation (8), the generator matrix Q of the system

is 5 1.8 2.4 0.2 0.6

1 1 0 0 02 0 2 0 00 0 0 0 00 0 0 0 0

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q

. The expectation of the number of repairs before the

system becomes unsafe is

( ) 5.25E R = .

The p.d.f of the length of time that the system is in safe states before the system becomes unsafe,

5.967 1.619

0.414 5

( ) 0.653 0.03600.111 0.8

t t

t t

f t e ee e

− −

− −

= ++ − .

The expectation of the time that the system is in safe states before the system becomes unsafe is 0.648 .

B. A parallel System There are two identical nodes and a repairman in a

parallel system and the parameters are

2,λ = 0.8p = , 3μ = ,

with equation (9), the generator matrix Q of this system is

4 3.2 0 0.8 03 5 1.6 0 0.40 3 3 0 00 0 0 0 00 0 0 0 0

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q

. The expectation of the number of repairs before the

system becomes unsafe,

( ) 1.92E R = .

The p.d.f of the length of time that the system is in safe states before the system becomes unsafe,

7.052 1.966 5.019( ) 0.038 0.00021 0.8384t t tf t e e e− − −= − + + The expectation of the time that the system is in safe

states before the system becomes unsafe is 0.033 .

V. CONCLUSIONS A model is built in the paper for supply chain in terms

of some practical backgrounds and the model described in this paper provides a situation for assessing the effectiveness of a supply chain. By introducing the model of ion-channel into the studies of supply chain, the Markov chain based method is suggested. And the important reliability indexes such as the probability of the supply chain system failure, mean time to failure, the reliability function, and steady probability of the state and so on are obtained when the distribution center operates steadily. The results from this study are of great value and have important meaning for developing plans in a supply chain. They can be used in evaluating and measuring the performance of the supply chain and can provide significant help and guidance.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (No.60776817) and the

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Excellent Young Scholars Research Fund of Beijing Institute of Technology (No. 2007YS0807).

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[10] Colquhoun, D. & Hawkes, A.G. (1977). Relaxation and Fluctuations of Membrane Currents that Flow Through Drug-operated Ion Channels, Proc. R. Soc. Lond. B 199, pp. 233-262.

[11] Colquhoun, D. & Hawkes, A.G.(1982). On The Stochastics Properties of Bursts of Single Ion Channel Openings and of Clusters of Bursts, Phil. Trans. R. Soc. B300, pp.1-59.

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