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Ming-Hung Shu

1

and Peng-Jen Chen

2,3

Performance Evaluation and Loss Measures for

the Deterioration Process

ABSTRACT: This paper considers a problem of loss measures and setup determination for a process under performance deterioration. Traditionally, to evaluate reliability and performance of the process, a binary-state model, a working (success) state or a failure state, is used to classify its conditions. However, in many cases, the process is deterioration over time, providing that a multiple-state model could be a more realistic model to capture the process deterioration conditions. Hence, the process under performance deterioration is considered as a general Markovian model, which means that its length of time staying in some state depends not only on its present state, but also on how long it has been in the present state. We present an integration method to find the probability function sojourning in each state at some point in time for this deterio-ration process. Based on probability functions of states, we first present a number of performance-evaluation methods. Then, we integrate a general class of loss functions to construct loss measures for assessing how severe the cost is that this process of deterioration causes at some point in time, as well as the cost over the entire deterioration process. With this measure, the setup time can be determined by the total expected cost over the operation period exceeding a preset threshold value. A tool-wear problem of the friction-drilling process is illustrated throughout the paper. KEYWORDS:deterioration process, performance evaluation, general loss function, setup time determination

Introduction

Many manufacturing processes, production systems, and struc-tural systems bearing usage and age suffer from increasing deteri-oration, an inevitable phenomenon wherein the key parameter of the process/system gradually worsens over time, and, if left unat-tended, leads to complete process/system failure, sometimes bringing serious losses to companies, facilities, and human lives. [5,17,27]. For example, a sharp cutting edge of a cutting tool, a tool-wear process of a manufacturing system, is gradually worn down and finally breaks [24]. A corrosion process of pipelines is a chemically induced damage, resulting in deteriorating failure of their materials and properties [7]. Because of the corrosive envi-ronment, cyclic loading, and concrete cracking, structural deterio-ration, affecting long-term infrastructures performance, becomes a major issue for structural safety [6,18]. Apparently, the above processes/systems possess similar deterioration properties. For brevity, we focus on the manufacturing process under perform-ance deterioration in this paper. For other applications, the methods provided here can be similarly applied.

To accurately evaluate reliability and performance of the manufacturing process under deterioration, an effective model to capture its deterioration properties is required. Traditionally, either a binary-state model, a working (success) state or a failure state,

or a homogeneous multi-state model, with known transition prob-abilities of the states that are independent of time, is used to cap-ture this deterioration conditions [2,13,21]. However, the deterioration process is performed with dynamic behaviors whose important quality characteristic deteriorates over time. For exam-ple, a sharp edge of a machining tool in a cutting manufacturing process is quickly worn down at the beginning, a finite wear land is established at the flank face, then the wear progresses with a slightly increasing rate, lasting for a certain period, and finally the wear rate increases and breakdown occurs [2,13,23]. Obviously, in this case, neither a binary state nor the homogeneous multi-state model is sufficient to capture the process deterioration condi-tions, referring to Fish et al. [4], Li and Yuan [15], Moriwaki and Tobito [20], and Sun et al. [23].

Therefore, to consider a problem of cost measures and setup time determination for a manufacturing process under perform-ance deterioration, a multi-state model following a general Marko-vian model (GMM) could be a more realistic model to evaluate the process deterioration, whose length of time staying in some state depends not only on its current state, but also on how long it has been in the current state [22]. Based on GMM, we present an integration method to obtain the probability function of remaining in each state at time t for this deterioration model. Based on proba-bility functions of these states, we first present a number of performance-assessment methods. Then, we incorporate a general class of loss functions to build cost measures with respect to the assessment of how severe the loss is that this process deterioration causes at some point in time, as well as over the entire deteriora-tion process. Finally, to prevent losses from an unnecessary process setup or replacement of work pieces, the setup time is determined. Last, a real application is illustrated for the applicabil-ity of our proposed methods.

Manuscript received August 24, 2011; accepted for publication January 10, 2012; published online March 2012.

1

Dept. of Industrial Engineering and Management, National Kaohsiung

Univ. of Applied Sciences, Taiwan (Corresponding author), e-mail:

workman@cc.kuas.edu.tw

2

Dept. of Industrial Engineering and Management, National Kaohsiung Univ. of Applied Sciences, Taiwan

3

Metal Industries Research and Development Center, Taiwan

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Process Deterioration

To evaluate reliability and performance of a manufacturing pro-cess, traditionally, its condition is classified by using a binary-state model, a functioning binary-state at time t denoted by ~/ðtÞ ¼ 1, or a failure state at time t denoted by ~/ðtÞ ¼ 0. However, after being used, most manufacturing processes deteriorate so gradually that their working efficiency performs at intermediate states between working perfectly and complete failure; consequently, a multiple-state model, /(t) [f0, 1, 2, …, mg, where m  2, could be a more realistic model to capture the process deterioration conditions [16]. GMM is a stochastic process model with discrete states and continuous time to capture the process with an age effect on the state-to-state change, which has been widely employed as a gener-alization of the traditional Markov processes [22,26]. Therefore, we utilize GMM to model a gradual deterioration process whose performance gradually deteriorates from the perfect state / (0)¼ m to lower states /(t) [ fm  1, m  2, …, 0g, as shown in Fig. 1. Then, an integration method is presented to obtain the probability function of staying in each state at some point in time t.

General Markovian Model

GMM is a discrete-state and continuous-time stochastic process model, /(t) [f0, 1, 2, …, mg whose conditional probability mass function satisfies the following relation:

Prf/ðtnÞ ¼ inj/ðtn1Þ ¼ in1;…; /ðt2Þ ¼ i2;/ðt1Þ ¼ i1g ¼ Prf/ðtnÞ ¼ inj/tn1Þ ¼ in1g (1) where t1<t2<… < tn-1<tn and in, in-1, …, i1 [ f0, 1, …, mg [1,16,22]. Let tn-1¼ t and tn¼ t þ Dt; then, the expression (1) is simplified to

Prf/ðt þ DtÞ ¼ jj/ðtÞ ¼ ig ¼ pi; jðt; DtÞ (2)

for i, j [f0, 1, …, mg. Thus, pi,j(t, Dt) is a transition probability that a process moves from state i to state j, satisfying the following conditions

pi; jðt; DtÞ  0 for t; D t > 0 (3)

Xm j¼0

pi; jðt; DtÞ ¼ 1 for t; D t > 0 (4)

By incorporating the Markov property and the total probability rule, we have the Chapman- Kolmogorov equation [8,22]

pi; jðt; DtÞ ¼X m k¼0 pi; kðtÞ  pk; jðt; Þ for; t; D t > 0 (5) By expanding Eq5, we have pi; jðt; DtÞ  pi; jðtÞ ¼  pi; jðtÞ X m k¼0; k6¼j pj; kðt; DtÞ þ X m k¼0; k6¼j pi; kðtÞ  pk; jðt; DtÞ (6)

Let Eq 6divide by Dt and then limit Dt! 0. We have the state equation for this deterioration process

p0i; jðtÞ ¼ pi; jðtÞ X m k¼0; k6¼j kj; kðtÞ þ X m k¼0; k6¼j pi; kðtÞ  kk; jðtÞ (7)

where ki,j(t) and kj,j(t) are the transient deterioration rates from state i to state j and from state j to state j, respectively, to capture the deterioration conditions,

ki; jðtÞ ¼ lim Dt!0 Prð/ðt þ DtÞ ¼ jj/ðtÞ ¼ iÞ Dt ¼ lim Dt!0 pi; jðt; DtÞ Dt (8) kj; jðtÞ ¼  X m k¼0; k6¼j kj; kðtÞ (9)

Let us now assume that / (0)¼ i ¼ m, namely that the deteriora-tion process is in state i¼ m at time t ¼ 0. Then we can simplify the notation of the state Eq.7by omitting the index i, that is,

p0jðtÞ ¼ pjðtÞ X m k¼0; k6¼j kj; kðtÞ þ X m k¼0; k6¼j pkðtÞ  kk; jðtÞ (10)

where pm(0)¼ 1 and pk(0)¼ 0 for k = m and pj(t) is the probabil-ity that the process is in state j at time t.

Gradual Deterioration Process

An m 2 multi-state manufacturing process with gradual deterio-rations followed by GMM is shown in Fig.1. One can see that in state m there is only one transition from this state to the state m 1 with the transient deterioration rate of km,m-1(t) and there are no transitions to the state m. In each state i within 0 < i < m, there is one transition to this state from the previous state iþ 1, with the transient deterioration rate kiþ 1,i, and there is one transition from this state to state i – 1, with the transient deterioration rate ki,i1. Observe that there are no transitions from state 0, called the absorbing state. This means that if the process enters state 0, it never leaves.

From the state Eq10, we can rewrite it explicitly as a matrix equation:

FIG. 1—A multi-state manufacturing process with gradual performance deteriorations.

SHU AND CHEN ON PERFORMANCE EVALUATION AND LOSS 455

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p0 mðtÞ p0 m1ðtÞ .. . p0 1ðtÞ p00ðtÞ 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ¼ km; mðtÞ 0 0    0 0 km; m1ðtÞ km1; m1ðtÞ 0    0 0 0 km1; m2ðtÞ km2; m2ðtÞ .. . ... ... 0 0 km2; m3ðtÞ .. . 0 0 .. . .. . .. . . . . k1; 1ðtÞ 0 0 0 0    k1; 0ðtÞ 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5  pmðtÞ pm1ðtÞ .. . p1ðtÞ p0ðtÞ 2 6 6 6 6 6 4 3 7 7 7 7 7 5

Because state 0 is an absorbing state, all the transition rates from this state are equal to zero. Thus the elements of the column corre-sponding to the absorbing state are all equal to zero. Without los-ing any information about the probability that the deterioration process is in each state at time t, pm(t), pm1(t), …, p0(t), we can reduce the matrix equation to

p0ðtÞ ¼ RðtÞ  pðtÞ; (11) where p0(t)¼ [p0m(t), pm-1(t), …, p0 02(t), p01(t)]T, pðtÞ ¼ ½pmðtÞ; pm1ðtÞ; …; p2ðtÞ; p1ðtÞT; and RðtÞ ¼ km; mðtÞ 0 0    0 km; m1ðtÞ km1; m1ðtÞ 0    0 0 km1; m2ðtÞ km2; m2ðtÞ .. . ... .. . .. . .. . . . . 0 0 0 0 k2; 1ðtÞ k1; 1ðtÞ 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 (12)

calls the transient deterioration rate matrix. Suppose now R(t) has full rank, then the p0(t) can be found from p0(t)¼ 1 - p1(t) … -pm(t).

Probability Functions of States

In practice, finding the probability of the deterioration process sojourning in each state j at time t, pj(t) for j [fm, m  1, …, 0g is somewhat difficult by simultaneously solving several differen-tial equations described by Eq11with regard to R(t) being a func-tion of t. In this secfunc-tion, a basic integrafunc-tion technique is presented to develop a straightforward and general calculation formula, as shown in Theorem 2.1.

From a state-by-state procedure, we can have the following: • The probability that the deterioration process will be in state

m at time t is given by pmðtÞ ¼ e Ðt 0km; mðlÞdl (13) where km; mðtÞ ¼ km; m1ðtÞ

The probability that the deterioration process is state m 1 at time t is pm1ðtÞ ¼ ðt 0 e Ðs1 0 km; mðlÞdl km; m1ðs1Þ  e Ðt s1km1; m1ðlÞdlds1 (14)

The probability of the deterioration process staying state m 2 at time t, in this case it deteriorates from m to m 1 and then from m  1 to m  2, i.e., m ! m  1 ! m 2. p1m2ðtÞ ¼ ðt 0 ðs2 0 e Ðs1 0 km; mðlÞdl km; m1ðs1Þ  e Ðs2 s1km1; m1ðlÞdl km1; m2ðs2Þ  e Ðt s2km2; m2ðlÞdlds1ds2 (15)

where p1m-2(t) is the probability that the manufacturing process goes from state m to state m 2 with one transition state m  1, l¼ 1, where l stands for the number of intermediate states.

To find a general formula for pj(t), the probability that the dete-rioration process remaining in state j [ fm  3, …, 1g at time t with intermediate states l [f2, …, m  j  1g, the following defi-nition is very helpful.

Definition 1—A deterioration process moves from state m

to lower state j.

Let S¼ fm  1, m  2, …, j þ 1g be a set of intermediate states containing m  j  1 elements, i.e., s(1) ¼ m  1, s(2)¼ m  2, …, s(m  j  1) ¼ j þ 1.

With l [f2, 3, …, m  j  1g intermediate states, then the probability that the deterioration process transit from state m to state j is given by pljðtÞ ¼ ðt 0    ðs2 0 e Ðs1 0 km; mðlÞdl km; sð1Þðs1Þ  Y l q¼1 e Ðsqþ1 sq ksðqÞ; sðqÞðlÞdl ! Yl1 q¼1 ksðqÞ; sðqþ1Þðsqþ1Þ !  ksðqþ1Þ; jðslþ1Þ  e Ðt slþ1kj; jðlÞdlds1…dslþ1 (16) where j [fm  3, m  4, …, 1g

Based on above the generalization term pjl(t), we can state the following theorem.

Theorem 1—For a (m

þ 1)  2 gradual deterioration process followed by GMM with transient deterioration rates from state j to state k denoted kj,k(t) for j, k [f0, 1, …, mg, the probability that the system working in state j at time t, pj(t), is given by

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production performance, evaluate production cost because of the deterioration, and determine when is the time to replace the dril-ling tool.

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