On Improvement of Throughput for Work Centers in terms of ...



Study of Throughput Improvement for

an Unreliable Work Center in terms of

Alternating Renewal Process

Chin-Tai Chen1, John Yuan2 and Chih-Hung Tsai3

1,3 Department of Industrial Engineering and Management, Ta-Hwa Institute of Technology, Hsinchu, 307, Taiwan, R.O.C.

2 Department of Industrial Engineering and Engineering Management, National Tsing-Hwa University, Hsinchu 30047, Taiwan, ROC.

Abstract

This paper presents both numerical simulation and closed form methods to calculate performance measures in production line, such as expected the transient throughput, and the probability that measures the delivery in time, for an unreliable work center without intermediate buffers. Such approaches are based on the assumptions that (1) work center alternates between Normal and Failed; (2) up times and down times are i.i.d./independent (but with exponential distributions required in closed form method); (3) work center has the fixed production rate (. Numerical experiments that compare values of such performance measures by different methods are presented in terms of an unreliable work center cited from literature. The sensitivity of such measures with respect to either mean up time or mean down time is also presented to investigate how and how much improvement can be achieved.

Keywords: probability, transient throughput, machine failure, alternating renewal process.

1. Introduction

Many studies try to investigate the characteristic of an unreliable work center by calculating the average of the system throughput (i.e. the production rate in the long run) [1~12, 14, 16, 17], etc. However, an increasing need to calculate the variability of the output besides the average performance arises in order to measure delivery on time (due time T) and etc. Tan [14] proposed a method to calculate both average E(() and variance Var(() of the steady state throughput ( (i.e. ( = [pic]where [pic]) for an unreliable work center and a series system of unreliable machines with the same production rate ( and without buffer between machines based on that each machine has i.i.d. exponential up and down times so that the system at steady state is further modeled in an irreducible Markov chain. He also proposed to use [pic] to measure delivery on time T for the demand D in sufficiently large T case by observing that that UT(( will be very close to the normal distribution with mean E(()(T and variance Var(()(T for any large enough T by applying the Central Limit Theorem.

Without further requiring that the system is in an irreducible Markov chain as did in Tan [14], this article is: (1) to develop simulation methods to estimate Pr{UT (( (D} and [pic] for any T based on that up times and down times follow any distributions F and G respectively, (2) to develop closed form formulas to calculate them also for any T in case F and G are exponential. Then the values of each system performance measure obtained by such three methods (including Tan’s [14]) are compared numerically and the sensitivity analysis of each measure with respect to either mean up time or mean down time in the system to see how and how much improvement can be achieved is presented.

Organization of the remaining part of this paper is as follows. In Section 2, we list the model assumptions, definition of the measures, and the notations. In section 3, we develop closed forms of Pr{UT (( (D} and [pic] for any T. In section 4, basic formulas on which simulation algorithms are developed are presented. In section 5, we investigate numerically the difference of each performance measure among such three methods (including Tan’s [14]) and the sensitivity of each measure with respect to either mean up time or mean down time to see what improvement can be made. In section 6, some concluding remarks and future possible study will be presented.

2. Basic Assumptions and Notations on an Unreliable Work Center

Throughout this article, the unreliable work center without buffers must further satisfy following assumptions:

(1) The system produces only one type of products without defective.

(2) There is infinite material supply and sufficient final storage to an unreliable work center.

(3) Machine is subject to independent failures and under no maintenance.

(4) The repair is taken immediately for a work center upon failure.

(5) The operation (or normal) time and repair time of the ith machine are exponential or Gamma with rate (i and (i for each i = 1, 2,…, N respectively.

(6) Machine has the fixed production rate (.

From assumptions (3), (4) and (5), the unreliable work center can be modeled in an alternating renewal (AR) process [pic]of type (F, G) with F =[pic]and G =[pic]. That is,

(1) [pic], is i.i.d. exponential or Gamma with rate (; [pic], is i.i.d. exponential or Gamma with rate (.

2) [pic] and [pic]are independent.

Therefore, the unreliable work center can be modeled in an AR process. Let N(T) denote the total number of renewals of the renewal process [pic] during [0, T], [pic]and [pic]be the total up time of the unreliable work center during [0, T]. Then

[pic] ([pic](T-SN(T)-1)(XN(T) (1)

where

[pic]; [pic].

For convenience, we will let (T (s) =Pr{[pic](s} and [pic]. In particular, [pic]= Pr{[pic](( (D} for most continuous type F and G.

3. Closed form formulas for [pic],[pic] and [pic] (Exponential type)

First of all, we know that

[pic]= [pic]

The closed form formulas to calculate [pic] and [pic] respectively are described in Theorem C. Its proof needs to verify the following two Lemmas first.

Lemma A: Suppose that the AR process for a system is of general type (F, G). Then

(A1) [pic]= [pic] and[pic]=[pic]

(A2) Suppose that [pic]. Then

(T(s) = [pic]

for [pic]

1-(T(s) = 1-[pic]

for [pic] and

1-(T(T-) =[pic] = 1-F(T)

Proof: See Appendix A for its proof.

Lemma B: Let [pic] and [pic] be its Laplace inverse. Then

[pic]= [pic]+

[pic]+

[pic]

where

[pic][pic][pic]

[pic][pic]

for i= 0, 1,…, N-1

[pic][pic][pic][pic][pic]

for i= 0, 1,…, m-1

[pic][pic][pic][pic][pic]

for j= 1,…, N and i= 0, 1,…,[pic]

In case m = 0, [pic]0 for each i.

Proof: See Appendix B for its proof.

Theorem C: For a reliable work center,

(C1) [pic]=Pr{[pic]} = [pic]

[pic][pic]

for 0([pic] ................
................

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