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Study on the Reliability of Hot Standby

Repairable Supply System Based on Markov ModelShuhang Ren, Cunlu Zhang

AbstractStarting from the analysis of cooperation relationsof members in the supply system, the paper studies the reliability

of repairable supply system. Based on Markov model, a reliability

evaluation method of repairable supply system is given. The

transient and steady reliabilities of hot standby repairable supply

system with a main supplier and a spare supplier are found. The

conclusion is tested by a case study. The contribution is that it

provides a quantitative method to increase the understanding of

dynamic reliability of enterprise supply system.

Index TermsHot standby supply system, Markov model,Repairable, Reliability

I. INTRODUCTIONn 2000, Ericsson withdrew from mobile phone market

because its single supplier of one of the core components

suffered from fire disaster, which in the meantime aroused

public concern about the reliability of supply system.

Nowadays, as the commercial competition is becoming fiercer

and fiercer, the supply system management which has been

playing a more and more important role in business activities

has become an indispensible part in competitive advantages.

Therefore, its getting increasingly urgent to evaluate and

improve the reliability of enterprise supply system.

To study the reliability of supply system is to evaluate thegeneral effect after taking measures. Thus, the standard of

reliability evaluation, i.e. the goal of reliability control, which

is based on the planned target of supply system, should be set

first. Mr. Zhang Cunlu has offered the principles to set the

standard of the reliability of supply system, including the

available operation target, planned target, the expected

customer service level, the present management level, etc [1].

However, how to set the standard of reliability evaluation is

not the key point in this paper.

A number of domestic and overseas scholars have studied on

the supply chain and the reliability of its operation activities

from many perspectives and have obtained many

achievements. Document 2 has offered an evaluation matrix ofthe reliability of supply chain, regarding various factors

affecting reliability,

Shuhang Ren, is with management science department, school of

management,XiamenUniversity,CO361005 China(mobile

phone:13950159344,e-mail:renshuhang2006@yahoo.com.cn).

Cunlu Zhang, is with management science department, school of

management, Xiamen University,CO361005 china. He is the vice-

professor.(mobile phone:13850002630,e-mail:zclu@sina.com).

supply chain. Then, the reliability level of the whole supply

chain can be worked out according to general series principle.

Document 3 simply describes the single-level supply chain and

evaluates the reliability of multi-level supply chain based on

the Markov process. Furthermore, some suggestions on how to

avoid risk in supply chain and improve reliability are also

provided. Document 4 puts forward the general structure of

suppliers network risk management as well as the solutions

under complex network environment and demonstrates the

challenge of suppliers network cooperation to the supply chain

risk management. Document 5 starts from the workflow of

distribution service and applies relevant theories of Markov

process to put forward a forecast method of reliability of

distribution service based on Vector Markov Chain. Thus,

distribution service under steady state and the reliability of its

business processes are obtained. Document 6 first derives the

expression of reliability level of single-part repairable system

based on Markov process, and deduces a formula of

availability and reliability of the standby redundancy repairable

system.

According to the available document research, studies on the

reliability of hot standby repairable supply chain are quite few

at present. Besides, on analyzing and evaluating the reliability

of supply chain, the present documents mainly concern two

aspects: one is to make comprehensive evaluations in terms ofexternal factors index, including the concrete method of

analytical hierarchy process, fuzzy comprehensive assessment

and so on, or to make analyses in terms of internal factors, such

as fault tree analytical method; the other is to analyze the

structure of supply chain according to the theory of system

reliability and divide it into serial system, parallel system,

compound system, redundant system and network system to

study system reliabilities on the basis of their internal logical

relationships. However, the present researches show that the

first method is more often used, although its subjectivity of

weight setting to some extent makes inaccurate results; while

the second one seldom differentiates the repairability of supply

chain system, which means it doesnt consider the reoperationafter the systems repair. In fact, supply chain is not able to

rebuild its system immediately encountering malfunction or

invalidation, so its more practical to resort to scientific

repairing, which has just offered a cut-in point for my paper to

evaluate the reliability of supply system under repairable

circumstances.

Therefore, starting from the corporative relations of

members of the supply system, this paper examines the

I

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reliability of hot standby repairable supply system which

consists of a main supplier and a spare supplier. Besides, based

on the Markov model which emphasizes state conversion

instead of internal mechanism, the paper also brings forward a

calculation method of evaluating reliability of hot standby

repairable supply system. This method not only provides some

evidence for evaluating the present reliability level, but also

compares the result with the previous evaluation standard or

much higher target to make sure whether the deviation has

been controlled in a certain scope.

II. THE DEFINITION OF RELIABILITY OF SUPPLY SYSTEMFrom the perspective of system engineering, reliability

refers to the probability that parts, components, products or

systems complete their prescribed functions smoothly under

the set condition within the fixed time [7].

Although many experts and scholars have studied on the

reliability of supply chain and provided its definitions from

various points of view, there hasnt been an authoritative

definition so far. Thomas (2002) claims that supply chain

reliability refers to the probability of delivering supply to thekey points of supply chain system to satisfy demands [8]. Mu

Dong and Du Zhiping (2004) defines the term as the ability of

supply chain to complete the demand of orders under the set

time and condition, which is generally presented as the

accessibility and the minimum acceptable service level of

supply chain [9]. Based on the theory of system reliability

engineering, Mr. Liu Yuanhong (2005) defines it as the

measurement of supply chain systems fault-free working

ability, which refers to the normal operation ability within

fixed time on the basis of completely competitive market [10].

According to the trading theory and system reliability theory,

Tian Guiliang (2007) defines supply chain coordination

reliability as the ability of each enterprise in the supply chainto fulfill their order demands to ensure the smooth operation of

the whole supply chain, on the condition that the supply chain

system is not disturbed by external factors [11].

On the basis of the above definitions of system engineering

and supply chain reliability, this paper adopts the following

definition of supply system reliability: it refers to the

probability of suppliers to deliver correct materials to

customers under the right time and place, with right product

package as well as right document files.

III. STUDY ON THE RELIABILITY OF HOT STANDBYREPAIRABLE SUPPLY SYSTEM BASED ON MARKOV MODEL

A. Markov Model and the BasicHypothesesMarkov model, exploring the interconversion among various

states, was put forward by a Russian named Markov in 1907. If

at a certain moment, the conversion probability from one state

to another is only related to the present state instead of the

previous state, that is it has nothing to do with the finite

previous state, then this process is called Markov process [7].

For the requirement of model and calculation, the paper

provides the following hypotheses:

Hypothesis one: This paper assumes that supply system

consists of a main supplier and a spare supplier.

As manufacturers core components will involve various

factors such as suppliers patent, property right and high-level

technology, single supplier strategy should be adopted to

maintain the partnership or strategic alliance with suppliers. To

manufacturers, the products and services that strategic

suppliers provide are of great importance and value and will

probably have impact on their own products and operation

processes or even influence the ultimate capacity to satisfy the

demands o f cus tomer s . B ecause o f t he i r s t r ong

competitiveness, these products and services which are

generally aimed at manufacturers specific requirement have

achieved high personalization and differentiation. However,

there are comparatively less strategic suppliers who can satisfy

their own demands, and their conversion cost is also pretty

high. Therefore, in consideration of the risk, spare suppliers

should be set to deal with the problems incase main suppliers

malfunction. According to the above considerations, it isnecessary to evaluate the reliability of supply system which

cons is t s of a main suppl ier and a spare suppl ier .

Hypothesis two: This paper assumes the supply system is

repairable.

Nowadays, as competition has become globalized,

manufacturers and suppliers gradually resort to cooperation

strategy. Thus the factors influencing supply system reliability

also include the long-term cooperation ability of the chosen

supplier. As for those who have encountered problems,

manufacturers tend to choose joint effort to improve suppliers

operation level and enhance their partnership instead of

changing suppliers like before. It is the strategic partnership of

upstream and downstream members that makes their hand-in-hand improvement, with the goal of long-term cooperation and

mutual win. From this point, supply system can be regarded as

a repairable system, which is alternated by normal operation

and breakdown maintenance.

Hypothesis three: As an enterprise, suppliers will take some

risk to cooperate with manufacturers in the supply chain.

Suppliers may be at standby state in one supply chain, but at

working state in another, so this paper assumes that suppliersare possible to malfunction no matter what kind of state they

are at in supply chain. That is to say, what we are discussing

here is hot standby repairable system.

We define the main supplier in supply system as unit 1 and

spare supplier as unit 2, with a constant failure andimprovement rate. At working state, their failure rates are

respectively 1 and 2 , and improvement rates 1 and 2 . As

the system is hot standby system, failure rate is 3 at standby

state( 1 , , 01 2 3 > > 1 , 01 2 > > .

Hypothesis four: If unit 1 (the main supplier) fails and unit 2

(the spare supplier) alternates to function, manufacturers and

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suppliers cooperate to solve problems and satisfy demands,

then this supply system is standby repairable system.

Hypothesis five: As long as one unit is working, the supply

system is at normal state. However, if both unit 1 and unit 2 are

ineffective at the same time, then the supply system

malfunctions.

Hypothesis six: If unit 2 fails and unit 1 is still at working

state, manufacturers should contact unit2 immediately and

solve problems together.

Hypothesis seven: The successful maintenance of unit 1 and

failure of unit 2 will not happen simultaneously and the two

will not fail at the same time as well.

B. Evaluation on the Reliability of Hot Standby Repairable SupplySystem

From the above hypotheses, we can figure out four possible

states of supply system:

0: unit 1 is at working state while unit 2 at standby state.

Now the system is normal;

1: unit 2 is at working state while unit 1 at the state of

being improved. Now the system is normal;

2: unit 1 is at working state while unit 2 at the state ofbeing improved. Now the system is normal;

3: unit 1 is at the state of being improved and unit 2 is the

same. Now the system is ineffective.

The result of supply system must be one of the four kinds if

states upwards at a certain point. However, the result of supply

system may transfer from one state to another during a period

of time. Besides the result of every time the state of supply

system is only related with the process of current operation of

supply system, and will not influence the other operation of

supply system. Therefore the state transition of supply system

can be regarded as Markov Chain.

Figure 1 shows the model of state transition within the time

of t :

Figure1The Model of State Transition

If the malfunction is repaired immediately, then a micro-

modulus transfer matrix within the time of ~t t t+ isderived from figure1:

01 ( ) 1 31 3

1 ( ) 02 1 21

0 1 ( )1 2 12

1 ( )0 2 1 2 1

t tt

t ttt

t tt

t t t

+

+ =

+

+

P( )(1)

( ( )P ti j , 0,1,2,3i j= indicates the transition probability

from state i to j )

On the basis of micro-modulus transfer matrix, we get the

following state transition equation from 0t :

0( ) 1 31 3

( ) 02 1 21( ), ( ), ( ), ( )0 1 2 3

0 ( )1 2 12

( )0 2 1 2 1

' ' ' '( ), ( ), ( ), ( ) ( ), ( ), ( ), ( )0 1 2 3 0 1 2 3

P t P t P t P t

P t P t P t P t P t P t P t P t

+

+ +

+

= = A

(2)

( )P tj , j=0,1,2,3 shows the supply system transient

probability at any moment t,

i.e. '( ) ( )t t=P A P (3)

The reliability of supply network can be analyzed from thefollowing two aspects. If the systems initial state is given, the

transient probability of any normal working state can be

figured out from the following procedure [12]:

1) After making Laplace transform to equation (3), we get

( )( ) (0)s s =P I A P ( (0)P means the initial state of system), that

is 1( ) (0)( )s s = P P I A (4)

2) Work out ( )sI-A and its inverse matrix 1( )s I-A

3) Expand 1( )s I-A through partial fraction method, and

insert it in formula (4). After making Laplace transform, well

finally get the result of [ ]( ) ( ), ( ), ( ), ( )0 1 2 3t P t P t P t P t =P . Thus, thetransient reliability should be

( ) ( ) ( ) ( )0 1 2R t P t P t P t= + + . (5)

Although there is a solution to work out transient reliability,this method involves matrix inversion which is generally

complicated, so its not easy to derive the theoretical model

solution of transient reliability. However, as for some special

data, matrix inversion can be easily solved through EXCELand MATLAB, and transient reliability is then figured out

based on the above calculation processes. Below is the solving

procedure of steady reliability when the system is at a steady

state.The reliability of supply system under steady state refers to

the probability of supplied state when the operation of supplysystem reaches a steady state, which means no matter what

state the initial supply system is, its probability of beingvarious states tends to be steady after a long-time operation. As

a constant, this probability has nothing to do with the initialstate.

Let lim ( ) , 0,1,2,3P t jj j

t= =

0 1

32

2 t

1 t

1 t

1 t

2t

1 t

1 31 ( )t t +

1 21 ( )t t + 1 21 ( )t t +

2 11 ( )t t +

2 t 3 t

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as/

lim ( ) 0, 0,1, 2, 3P t jj

t= =

,

10 1 2 3 + + + = (6)

Let t , then the system tends to be steady, and the above

equation can be transformed into the following linear equation:

[ ] [ ]

0( ) 1 31 3

( ) 02 1 21, , , 0,0,0,00 1 2 30 ( )1 2 12

( )0 2 1 2 1

+

+ = +

+

( 7 )

Obviously, the four equations in the system are linearly

dependent, so we have to add 10 1 2 3 + + + = and solve

the linear system as follows:( )1 2 1 2 1 2

0 ( ) [ ( ) ( ) ( ) ]1 1 2 2 1 2 1 23 3

( )1 2 1 1 231( ) [ ( ) ( ) ( ) ]1 1 2 2 1 2 1 23 3

( )1 1 2 2 1 23 3 32

( ) [ ( ) ( ) ( ) ]1 1 2 2 1 2 1 23 3( ) ( )1 2 1 1 2 1 23

3

+ + +=

+ + + + + +

+ + +=

+ + + + + +

+ + +=

+ + + + + +

+ + +=

( ) [ ( ) ( ) ( ) ]1 1 2 2 1 2 1 23 3 + + + + + +

(8)

Thus, the steady reliability of supply system should be:

( ) ( ) ( ) ( )0 1 2 0 1 2R P P P = + + = + + (9)

Compare the calculated steady reliability with the previously

set reliability evaluation standard to check the reliability level

of repairable supply system at a certain time. (see table 1)

TABLE1. the Standard of Reliability Evaluation

Reliability

(R)

R

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[8] M. U. Thomas. Supply chain reliability for contingency operations.Proceedings Annual Reliability and Maintainability Symposium,no.1,pp.61-67,2002.

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