Scheduling of the optimal tool replacement times in a flexible

IIE Transactions (2001) 33, 487±495

Scheduling of the optimal tool replacement times in a ¯exible manufacturing system PATRICK H. LIU, VILIAM MAKIS* and ANDREW K.S. JARDINE Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada M5S 1A4 E-mail: [email protected] Received December 1998 and accepted February 2000

In Flexible Manufacturing Systems (FMSs), a cutting tool is frequently used for di€erent operations and on di€erent part types to minimize tool change-overs and the number of tools required, and to increase part-routing ¯exibility. In such situations, the tools become shared resources and work in job-dependent, changeable and nonhomogeneous conditions. It is well known that the tool failure rate depends on both age and machining conditions and that tool reliability is a function of the duration, machining conditions, and the sequence of the operations in FMS. The objective of this paper is to obtain a schedule of the optimal preventive replacement times for the cutting tools over a ®nite time horizon in a ¯exible manufacturing system. We assume that the tool will be replaced either upon failure during an operation or preventively after the completion of each operation, incurring di€erent replacement costs. A standard stochastic dynamic programming approach is taken to obtain the optimal tool replacement times. The optimal schedule is obtained by minimizing the total expected cost over a ®nite time horizon for a given sequence of operations. A computational algorithm is developed and a numerical example is given to demonstrate the procedure.

1. Introduction The problem of the optimal tool replacement when cutting tools operate in homogeneous machining conditions has been studied extensively (see, for example, Billatos and Kendall, 1991; Gray et al., 1993; LaCommare et al., 1983; Liu et al., 1998; Pandit and Sheikh, 1980). Under this assumption, the tool replacement process is a renewal process and the optimal tool replacement time can be found by minimizing the expected average cost per unit time obtained as the expected cost of the cycle divided by the expected cycle length. In a Flexible Manufacturing System (FMS), however, the situation is di€erent. The tools become shared resources. A single tool may be required to complete several di€erent operations under di€erent machining conditions. It is a well known fact that the failure rate of a tool is not only a function of time but is also dependent on the conditions in which the tool operates. For identical tools, if their operating conditions are di€erent, they will have di€erent failure rates and their respective optimal preventive replacement times would be expected to be di€erent. In such a situation, the replacement process is no longer a renewal process and the classical approach to obtain the optimal replacement times does not apply. However, as *Corresponding author 0740-817X

Ó

2001 ``IIE''

pointed out by Gray et al. (1993), no research has been done in this area. The goal of this paper is to obtain a schedule of the optimal preventive replacement times for the tools working in changeable conditions over a ®nite time horizon in a FMS. We assume that the sequence of operations, the time to ®nish each operation, and the machining conditions for each operation are given in advance, and that a tool can be preventively replaced only after completing an operation. However, when a tool fails, it must be replaced immediately by an identical new tool at a cost which is higher than the preventive replacement cost. We consider an approach based on stochastic dynamic programming to make decisions on the optimal preventive tool replacement times. At the end of each operation, a decision is made based on the age of the tool whether to continue using the existing tool or to preventively replace the tool. The objective is to minimize the total expected cost over a ®nite time horizon for a given sequence of operations. An important part in the stochastic optimization procedure is to calculate the reliability of a tool working in changeable conditions, which is very di€erent from calculating tool reliability in constant homogeneous machining conditions where the reliability function can be represented as a function of time only. The reliability function of tools in changeable conditions in a FMS

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depends on the cumulative machining time, the sequence of operations, the length of each operation, and the machining conditions of each operation. In this paper, the computational method developed in Liu and Makis (1996) will be used to calculate the reliability of a tool operating in changeable conditions. This will be discussed in the next section. In the following sections, we will ®rst formulate the decision model and derive the formulae for the computation of the expected cost in the next period and the onestep transition probabilities. Then, we will present a computational algorithm for ®nding the optimal schedule of the tool replacement times based on stochastic dynamic programming. Finally a numerical example will be given to illustrate the computational procedure.

2. The reliability function of a tool operating in changeable conditions It is well known that the tool life is a€ected by both the cumulative operating time (or age) of the tool and the operating conditions including the cutting speed v, the feed rate f , the depth of cut d, tool gometry, contact angle, tool and work materials, etc. If we only consider the age of the tool and assume that the operating conditions do not change, the time to failure of the tool, T is a random variable. In the tool reliability literature, various forms of reliability function have been proposed, depending on the type of tool failure. In production, a tool might fail due to either over-stressing and fracture or cumulative wear. For tool failure caused by over-stressing and fracture, the tool failure time can be modeled by a two-parameter Weibull distribution function (Mazzuchi and Soyer, 1989; Pandit and Sheikh, 1980; Ramalingam, 1977, 1982; Ramalingam and Watson, 1977; Rossetto and Zompi, 1981). For continuous machining, where tool failure is due to a cumulative wear process, tool failure time is often modeled by a normal or a lognormal distribution (Ramalingam and Watson, 1977; Wager and Barash, 1971). When tool life is terminated by crater growth, a log-normal is usually used as a model. If tool failure is the result of volume detachment from the tool surface, a gamma distribution is usually a reasonable model for the data. Due to its ¯exibility and the closed-form reliability function, Weibull distribution has been used extensively in recent years to model tool failures. The Weibull reliability function for a tool in constant operating conditions can be written as R…t†  P …T > t† ˆ expfÿkta g;

…1†

where a and k are positive constants. If we assume that the operating conditions are solely determined by the machining parameters v (speed), f (feed rate), and d (depth of cut), the expected tool life can

be expressed by the well known Taylor Life Equation (Taylor, 1906) q Expected Tool Life ˆ q q0 q ; …2† 1 v f 2d 3 where q0 , q1 , q2 and q3 are positive constants. To incorporate the e€ect of both the age of the tool and the machining conditions on the tool failure time, we consider the Accelerated Failure Time Model (AFTM) which was ®rst proposed by Pike (1966) and has been widely applied since (Kalb¯eisch and Prentice, 1982; Nelson, 1990; Newby, 1988). AFTM (also called the accelerated life model by Cox and Oakes (1984)), speci®es that the e€ect of the machining conditions is multiplicative in time. Based on (1) and (2), the reliability function of the tool can be written in the following form (Liu and Makis, 1996; Mazzuchi and Soyer, 1989)  …3† R…t† ˆ exp ÿkta vb1 f b2 d b3 ; where b1 ; b2 and b3 are positive constants. If we denote z ˆ …v; f ; d†;

…4†

w…z† ˆ …vb1 f b2 d b3 †1=a ;

…5†

and the reliability function in (1) by R0 …t†, then the reliability function in (3) can be written in the following form R…t; z† ˆ expfÿk…tw…z††a g  R0 …tw…z††;

…6†

where R0 …
† is called the base-line reliability function and w…z† is called the acceleration factor. Equation (6) has the standard form of an AFTM. The expected life of a tool based on (6) is Z‡1 E…T † ˆ 0

ÿ

 exp ÿkta vb1 f b2 d b3 dt


b1 b2 b3 ÿ1=a

ˆ kv f d

  1 C 1‡ ; a

…7†

where C…
† is the gamma function. The AFTM with Weibull base-line reliability function is also a Proportional Hazards Model (PHM) (Cox and Oakes, 1984; Crowder et al., 1991). The PHM almost always gives a reasonable measure of the importance of the covariates representing the machining conditions and is an excellent exploratory data analysis technique (Newby, 1994). Therefore the Weibull reliability function proposed in (3) or (6) combines the advantages of both AFTM and PHM and is very ¯exible in modeling failure data in real situations. In addition, the closed-form reliability function makes representation and computation very easy. In the following model development and discussions, we will assume that the tool reliability function can be represented by the AFTM in (3) or (6). Next, we study the reliability function of a tool operating in changeable machining conditions in a ¯exible

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manufacturing system. Consider a new tool which is used in operations f1; 2; . . . ; i; . . .g. Denote the operating condition of operation i by vector zi and let tiÿ1 and ti be the starting and the completion times for operation i, respectively. Given that the sequence of operations, the length and operating condition for each operation are ®xed, time t, tiÿ1 < t  ti , uniquely de®nes the operation history of the tool which can be expressed by …fz1 ; t1 ÿ t0 g; fz2 ; t2 ÿ t1 g; . . . ; fzi ; t ÿ tiÿ1 g†:

…8†

The reliability of the tool depends on both the cumulative cutting time and its operation history represented by (8). To calculate the tool reliability in any time t, we consider operation 1 and operation 2 ®rst. The calculation of the tool reliability at time 0 < t  t1 for operation 1 is straightforward R…t; z1 † ˆ R0 …tw…z1 ††:

…9†

To calculate the reliability of the tool at time t1 < t  t2 for operation 2, we de®ne ^t1 by R…t1 ; z1 † ˆ R…^t1 ; z2 †  R0 …^t1 w…z2 ††;

…10†

i.e., ^t1 is the cutting time equivalent to t1 if, instead of operation 1, an operation of length ^t1 under conditions z2 was performed starting at time zero so that the reliability value of the tool at time t1 when ®nishing operation 1 is exactly the same as the reliability value of the tool at ^t1 when starting with operation 2. ^t1 is obtained by solving (10), w…z1 † ^t1 ˆ t1 : …11† w…z2 † Let T1 be the time to failure of the cutting tool when performing operation 1, starting at time zero, and T2 be the time to failure of the cutting tool when performing operation 2 starting at time zero. At time t1 < t  t2 for operation 2, the reliability function of the tool denoted by R…t; z2 ; z1 † is by de®nition, P …T > t† ˆ R…t1 ; z1 †P …T > tjT1 > t1 † ˆ R…t1 ; z1 †P …T > tjT2 > ^t1 †;

…12†

where P …T > tjT2 > ^t1 † ˆ P …T2 > t ÿ t1 ‡ ^t1 jT2 > ^t1 † ˆ R…t ÿ t1 ‡ ^t1 ; z2 †=R…^t1 ; z2 †: …13† Hence, R…t; z2 ; z1 † ˆ R…t ÿ t1 ‡ ^t1 ; z2 † ˆ R0 ……t ÿ t1 ‡ ^t1 †w…z2 ††;

t1 < t  t2 :

…14†

Fig. 1. A sequence of n operations in a ®nite time horizon tn .

The above approach can be generalized to obtain the reliability of the tool that has been operating for t time units under conditions de®ned by (8) R…t; zi ; ziÿ1 ; . . . ; z2 ; z1 † ˆ R0 ……t ÿ tiÿ1 ‡ ^tiÿ1 †w…zi ††; where ^tiÿ1 ˆ



tiÿ1 < t  ti ;

…15†

0; i ˆ 1; …tiÿ1 ÿ tiÿ2 ‡ ^tiÿ2 †w…ziÿ1 †=w…zi †; i > 1;

…16†

and t0 ˆ 0. See Liu and Makis (1996) for a more detailed derivation and graphical illustration.

3. The stochastic dynamic decision model We consider the problem of determining the optimal tool replacement times over a ®nite time horizon tn as shown in Fig. 1. During the period ‰0; tn Š, n operations in a ®xed sequence will be completed. We assume that the sequence of the operations, and the operating condition and the processing time for each operation are determined in advance. Operation k starts at time tkÿ1 , ®nishes at time tk , and its operating condition is described by a vector zk , for k ˆ 1; 2; . . . ; n. We assume that the tool can be preventively replaced at a cost C after ®nishing each operation. When the tool fails during operation, it is replaced immediately by a new one at a cost K, where K > C. The end of each operation is a decision epoch where a decision has to be made on whether to continue using the existing tool or to preventively replace the tool. During the actual FMS production, the cumulative operating time of the tool, sk , is observed or recorded at each decision epoch tk . The optimal tool replacement decision is made by comparing the age of the tool, sk , with a control limit, tk , obtained from the stochastic dynamic decision model. If the age of the tool sk is less than or equal to the control limit tk , we will let the tool continue in the next operation; otherwise the tool has to be preventively replaced by a new one. In the stochastic dynamic decision model, the control limits on tool age are obtained by minimizing the total expected cost over the ®nite time horizon. Since the sequence of the operations, the processing times and the operating condition for each operation are ®xed and given in advance, the observed sk at decision epoch tk uniquely determines the operation history of the tool expressed by (8) and hence the reliability value in (15).

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Similarly to (8), the operation history of the tool having age sk at decision epoch tk can be expressed as

Let c…^sk ; a† be the expected cost for the next operation and P …ijqk ; a†;

…fzj ; sk ‡ tj‡1 ÿ tk g; fzj‡1 ; tj‡1 ÿ tj g; . . . ; fzkÿ1 ; tkÿ1 ÿ tkÿ2 g; fzk ; tk ÿ tkÿ1 g†;

…17†

where j is the ®rst operation for the tool. Let sh be the age of the tool at the end of operation h, j  h  k. That is, sh ˆ sk ‡ th ÿ tk ;

for j  h  k:

…18†

The reliability value of the tool at age sk can be calculated from (15) as R…sk ; zk ; zkÿ1 ; . . . ; zj‡1 ; zj † ˆ R0 ……sk ÿ skÿ1 ‡ ^skÿ1 †  w…zk †† where ^skÿ1 ˆ

for i ˆ 0; 1; . . . ; qk ‡ mk‡1 ;

…24†

be the transition probability from qk D at tk to iD at tk‡1 , where iD is the equivalent age of the tool in condition zk‡1 . Let Vtk …^sk † be the minimum expected cost in …tk ; tn Š. Then, ( qk X ‡mk‡1 P …ijqk ; 0†Vtk‡1 …^sk‡1 …i††; c…0; 1† Vtk …^sk † ˆ min c…^sk ; 0† ‡ iˆ0

‡

…19†

mk‡1 X

)

P …ij0; 1†Vtk‡1 …^sk‡1 …i†† ;

…25†

iˆ0



0;

kˆj

…skÿ1 ÿ skÿ2 ‡ ^skÿ2 †…w…zkÿ1 †=w…zk ††; k > j

where …20†

In order to compute the expected costs and the transition probability matrix in the stochastic dynamic decision model, we discretize time and use a small time interval of a constant length D as the time unit. Denote the length of operation k, k ˆ 1; 2; . . . ; n, by dk  tk ÿ tkÿ1 ˆ mk D;

…21†

where mk is a positive integer. The size of the interval D should be determined in such a way that the required computational precision be satis®ed without too lengthy computations. It depends on the shape of the tool reliability function and the cost structure and should be selected by trial and error. In practice, each operation may consist of machining many small identical parts. Suppose for operation k, mk parts are to be machined with an equal machining time Dk . If Dk is small, we can use this small interval as the discretized time unit. In order to make the one-step transition probability matrix (to be discussed in the next section) time-homogeneous for a given operation, we need to convert the age of the tool sk into its equivalent age denoted by ^sk , obtained by assuming that the tool operated the whole time in condition zk‡1 . The equivalent age ^sk determines the state of the tool and can be calculated based on the recursive formula (16). Let ^sk ˆ qk D;

…22†

where qk is positive integer. A tool having age sk at decision epoch tk is therefore in state qk . For small D, it is reasonable to assume that the tool can fail only at the end of an interval. We can take two actions, a 2 f0; 1g, at each decision epoch based on the observed age of the tool, where  0 if we use the existing tool for the next operation, aˆ 1 if we preventively replace the tool. …23†

^sk‡1 …i† ˆ iD

w…zk‡1 † : w…zk‡2 †

…26†

For the last decision epoch, Vtnÿ1 …^snÿ1 † ˆ minfc…^snÿ1 ; 0†; c…0; 1†g:

…27†

The expected costs c…^sk ; a† in (25) and (27) can be calculated from c…0; 1† ˆ C ‡ KE‰N …dk‡1 †j^sk ˆ 0Š;

…28†

c…^sk ; 0† ˆ KE‰N …dk‡1 †j^sk Š;

…29†

and where E‰N …dk‡1 †j^sk Š is the expected number of failures in …tk ; tk‡1 Š given that, at decision epoch tk , the equivalent age of the tool in condition zk‡1 is ^sk . If the lengths of the operations are relatively short, we can assume that the tool can fail at most once in each operation and the expected costs in (28) and (29) can be calculated by using the following approximations: c…0; 1† ˆ C ‡ K  ‰1 ÿ R0 …dk‡1  w…zk‡1 ††Š;

…30†

  R0 ……^sk ‡ dk‡1 †  w…zk‡1 †† : c…^sk ; 0† ˆ K  1 ÿ R0 …^sk  w…zk‡1 ††

…31†

and

4. Transition probability matrix Assume that the current operation is k ‡ 1 and the equivalent age of the tool in condition zk‡1 is iD. Then, using discretization described in the previous section, after one step, the equivalent age of the tool is …i ‡ 1†D with probability Pi;i‡1 and is equal to zero with probability Pi;0 , where 8 if i ˆ 0; > < R0 …Dw…zk‡1 ††; …32† Pi;i‡1 ˆ R0 ……i ‡ 1†Dw…zk‡1 †† > ; if i > 0; : R0 …iDw…zk‡1 ††

Optimal tool replacement times

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and Pi;0 ˆ 1 ÿ Pi;i‡1 8 > < 1 ÿ R0 …Dw…zk‡1 ††; ˆ R0 ……i ‡ 1†Dw…zk‡1 †† > ; :1 ÿ R0 …iDw…zk‡1 ††

Step 1.

if i ˆ 0, if i > 0;

…33†

where R0 …tw…z†† is de®ned by (6). The other transition probabilities are equal to zero. The mk‡1 -step transition probability matrix P…mk‡1 † ˆ …m † fPij k‡1 g is then obtained by multiplying the one-step transition probability matrix P ˆ fPij g by itself mk‡1 -1 times, P…mk‡1 † ˆ Pmk‡1

Step 2.

…34†

From (34), we can obtain the mk‡1 -step transition probabilities P …ijqk ; a† in Equation (25). …m † Let Nqk ;0k‡1 be the number of times the process visits state 0 (failure state) in mk‡1 steps, given that initially the process was in state qk . Then we have (see for example, Bhat (1984)). …m

†

E‰N …dk‡1 †j^sk Š  E‰Nqk ;0k‡1 Š ˆ

mk‡1 X lˆ1

…l†

Pqk ;0 ;

…35†

which can be substituted into (28) and (29) to calculate the expected costs c…^sk ; a† for long operations where (30) and (31) cannot give accurate approximations. We have now completely speci®ed all the terms in Equations (25) and (27) for the stochastic dynamic decision model. We can proceed to describe in detail the algorithm to obtain the control limits for making optimal tool replacement decisions at the decision epochs.

5. The computational algorithm and implementation procedure The computational results of the above stochastic dynamic decision model are the total expected costs associated with the replacement decision based on the age of the tool and the control limit at each decision epoch. To compute the control limit at decision epoch tk , k ˆ 1; 2; . . . n ÿ 1, we can use the following algorithm: Step 0. Preliminaries ± Select a time unit D. A general guidline for choosing a proper D is that D should be small enough to satisfy the required computational precision and at the same time the computation can be completed in a timely fashion. This can be done through experience and by trial and error. After D has been selected, calculate w…zk † using Equation (5) for each operation k, k ˆ 1; 2; . . . ; n. For each decision epoch tk , k ˆ 1; 2; . . . ; n ÿ 1, calculate the equivalent cutting time ^tk by using the recursive formula (16).

Step 3. Step 4.

Step 5.

Set the maximum equivalent cutting time max…^sk † ˆ ^tk for k ˆ 1; 2; . . . ; n ÿ 1. For the last decision epoch tnÿ1 : 1. Calculate c…0; 1† in Equation (27) using (28), or using (30) if operation n is short. 2. Calculate c…^snÿ1 ; 0† in Equation (27) using (29) with ^snÿj ˆ iD, starting from i ˆ 0, or using (31) if operation n is short. 3. If c…0; 1†  c…^snÿ1 ; 0† or iD  max…^snÿj †, go to Step 3; otherwise set i ˆ i ‡ 1 and go to 2 of Step 1. For decision epoch tnÿj (starting from j ˆ 2): 1. Calculate the second term in Equation (25) using (28) and (32)±(35). If operation n ÿ j is short, (30) can be used instead of (28). 2. Calculate the ®rst term in Equation (25) using (29) and (32)±(35) with ^snÿj ˆ iD, starting from i ˆ 0. If operation n ÿ j is short, (31) can be used instead of (29). 3. If the second term  the ®rst term or iD  max…^snÿj †, go to Step 3; otherwise set i ˆ i ‡ 1 and go to 2 of Step 2. If n ÿ j  1 go to Step 4; otherwise set j ˆ j ‡ 1 and go to 1 of Step 2. Convert ^sk back into sk . The results produced by the above algorithm are the costs Vtk …^sk † associated with di€erent values of the equivalent cutting times ^sk at each decision epoch and the optimal decisions. We need to convert the equivalent cutting times ^sk into the actual cumulative cutting time sk using the formula in (16). Find the control limits tk for each decision epoch tk , k ˆ 1; 2; . . . ; n ÿ 1. tk is obtained by comparing the two terms in (25) or (27). tk is the value of sk converted from ^sk in (25) or (27) when term 1 equals term 2 or exceeds term 2 the ®rst time. Linear interpolation may be used to improve the accuracy of the control limits for D not small enough as will be illustrated in the following numerical example.

The implementation procedure of the decision model is very simple. At each decision epoch tk , k ˆ 1; 2; . . . ; n ÿ 1, observe the age of the tool sk and compare it with tk . If sk  tk , continue using the existing tool, otherwise replace the tool. The decision on the length of the ®nite time horizon tn should be based on both the real production situation and the computational complexities associated with dynamic programming. The length tn can be chosen for a shift (8 hours), a day (24 hours), or a week to facilitate production planning. However, since the computational demands of dynamic programming grow at an exponential rate 2n with increasing number of operations n (or decision epochs) over time tn , n may not exceed, say, 40 operations.

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6. A numerical example In the following example, we consider making optimal tool replacement decisions over a 2-hour production period during which six operations with di€erent processing times and machining conditions are to be completed using identical tools. This is illustrated in Fig. 2. The sequence of the six operations, the processing times, and the machining condition for each operation are given in Table 1. The preventive tool replacement cost C ˆ $10, and the tool failure replacement cost K ˆ $100. We choose the sequence, lengths and operating conditions for these six operations and the costs associated with tool replacements in this example solely for the purpose of demonstrating the computational procedures. We assume that the tool reliability function can be represented by the AFTM in (3) or (6) and we consider the tool failure data in Taraman (1974). The unknown parameters have been estimated by using the method of maximum likelihood (see Liu and Makis (1996) for detailed estimation procedures). The estimated reliability function is given by n R…t† ˆ exp ÿ1:2757  10ÿ2 …t=10†8:7325 …v=1000†11:2760 o …100f †2:0843 …100d†1:7942 : …36† A Kolmogorov±Smirnov goodness-of-®t test of the ®tted distribution has been performed and the detailed test is given in Appendix 1. Standard deviations of the estimated parameters have also been calculated and are given in Appendix 2. Column 6 in Table 1 gives the calculated expected tool life based on (36) for each of the six operating conditions.

There are ®ve decision epochs in this example. We consider an interval of a constant length D ˆ 5 minutes as the time unit. The acceleration factor for each operation is calculated and listed in column 7 of Table 1. For each decision epoch tk , k ˆ 1; 2; . . . ; 5, we have calculated the equivalent cutting time ^tk using (16): ^t1 ˆ 21:87, ^t2 ˆ 40:06, ^t3 ˆ 73:75, ^t4 ˆ 59:94, and ^t5 ˆ 95:55. These are the maximum equivalent cutting times max…^sk † for k ˆ 1; 2; . . . ; 5. Following Steps 1±3 in the computational algorithm, we have calculated the two cost terms of the cost function Vtk …^sk † in (25) and (27) for di€erent values of ^sk as listed in column 2 of Table 2 for k ˆ 1; 2; . . . ; 5. The computational results are shown in column 4 of Table 2. The equivalent cutting time (or tool age), ^sk , is then converted back to the actual age of the tool, sk , using (16) and is shown in column 3 of Table 2. The control limit tk on the age of tool at each decision epoch is obtained by ®rst comparing the values of the two cost terms in (25) or (27) as shown in column 4 of Table 2. tk is the value of sk when the value of the ®rst term equals or exceeds the value of the second term for the ®rst time, shown as bold in column 3 of Table 2. The resulting values for the control limits are shown in column 5 of Table 2. Based on the results in Table 2, we have used linear interpolation to improve the precision of the control limits. The improved control limits and the replacement decisions are shown in Table 3. We have used the time unit D ˆ 5 minutes in this example only to illustrate the computational procedure. For a smaller value of D, say D ˆ 1 minute in this example, the results would have been more precise and the linear interpolation would not have been necessary.

Fig. 2. A sequence of six operations and the processing times. Table 1. Operation times and machining parameters for a sequence of six operations Operation sequence 1 2 3 4 5 6

Operation time (minutes)

Speed v (fpm)

30 25 15 20 10 20

305 440 440 340 440 440

Feed f (ipr) 0.009 0.009 0.009 0.014 0.017 0.009

05 05 05 16 32 05

Depth d (in)

Calculated expected life (minutes)

Acceleration factor w(á)

0.0290 0.0135 0.0290 0.0210 0.0290 0.0135

59.18 43.14 36.87 49.39 31.58 43.14

253.658 347.960 407.152 303.928 475.381 347.960

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Table 2. Replacement decisions and the corresponding costs at the decision epochs Decision epoch t1 ˆ 30 minutes

t2 ˆ 55 minutes

t3 ˆ 70 minutes

t4 ˆ 90 minutes

t5 ˆ 100 minutes

Equivalent tool age

Age of tool

Costs

^s1 ^s1 ^s1 ^s1

ˆ5 ˆ 10 ˆ 15 ˆ 20

s1 s1 s1 s1

ˆ 6:89 ˆ 13:72 ˆ 20:58 ˆ 27:44

vt1 …^s1 † ˆ minf38:86; 47:10g vt1 …^s1 † ˆ minf44:87; 47:10g vt1 …^s1 † ˆ minf60:49; 47:10g vt1 …^s1 † ˆ minf88:78; 47:10g

^s2 ^s2 ^s2 ^s2 ^s2

ˆ5 ˆ 10 ˆ 15 ˆ 20 ˆ. 25 ..

s2 s2 s2 s2 s2

ˆ 5:85 ˆ 11:70 ˆ 17:55 ˆ 23:40 ˆ .30:83 ..

vt2 …^s2 † ˆ minf26:88; 36:65g vt2 …^s2 † ˆ minf28:40; 36:65g vt2 …^s2 † ˆ minf35:02; 36:65g vt2 …^s2 † ˆ minf54:78; 36:65g vt2 …^s2 † ˆ minf89:60; 36:65g .. .

^s3 ^s3 ^s3 ^s3 ^s3

ˆ5 ˆ 10 ˆ 15 ˆ 20 ˆ. 25 ..

s3 s3 s3 s3 s3

ˆ 3:73 ˆ 7:46 ˆ 11:20 ˆ 14:93 ˆ .19:27 ..

vt3 …^s3 † ˆ minf16:75; 26:63g vt3 …^s3 † ˆ minf19:73; 26:63g vt3 …^s3 † ˆ minf21:56; 26:63g vt3 …^s3 † ˆ minf26:70; 26:63g vt3 …^s3 † ˆ minf38:94; 26:63g .. .

^s4 ^s4 ^s4 ^s4 ^s4

ˆ5 ˆ 10 ˆ 15 ˆ 20 ˆ. 25 ..

s4 s4 s4 s4 s4

ˆ 7:82 ˆ 15:64 ˆ 22:58 ˆ 28:42 ˆ .34:26 ..

vt4 …^s4 † ˆ minf10:09; 19:10g vt4 …^s4 † ˆ minf10:98; 19:10g vt4 …^s4 † ˆ minf16:61; 19:10g vt4 …^s4 † ˆ minf37:84; 19:10g vt4 …^s4 † ˆ minf74:96; 19:10g .. .

^s5 ^s5 ^s5 ^s5 ^s5

ˆ5 ˆ 10 ˆ 15 ˆ 20 ˆ. 25 ..

s5 s5 s5 s5 s5

ˆ 3:66 ˆ 7:32 ˆ 11:53 ˆ 17:26 ˆ .22:98 ..

vt5 …^s5 † ˆ minf 0:50; 10:01g vt5 …^s5 † ˆ minf 2:46; 10:01g vt5 …^s5 † ˆ minf 9:10; 10:01g vt5 …^s5 † ˆ minf26:34; 10:01g vt5 …^s5 † ˆ minf57:35; 10:01g .. .

Table 3. Improved control limits and replacement decisions at the decision epochs Decision epoch

Control limit t1 (minutes)

Replacement decision

t1 ˆ 30 minutes

14.70

t2 ˆ 55 minutes

18.22

t3 ˆ 70 minutes

14.90

t4 ˆ 90 minutes

23.26

t5 ˆ 100 minutes

11.83

if age £ 14.70 minutes continue; o/w replace if age £ 18.22 minutes continue; o/w replace if age £ 14.90 minutes continue; o/w replace if age £ 23.26 minutes continue; o/w replace if age £ 11.83 minutes continue; o/w replace

7. Summary and conclusions In this paper, we have developed a method based on stochastic dynamic programming for obtaining the optimal preventive tool replacement times over a ®nite time horizon in a ¯exible manufacturing system. An algorithm has been presented to ®nd the minimum expected costs,

Control limit

t1 ˆ 20:58

t2 ˆ 23:40

t3 ˆ 14:93

t4 ˆ 28:42

t5 ˆ 17:26

Decision Continue Continue Replace Replace Continue Continue Continue Replace Replace .. . Continue Continue Continue Replace Replace .. . Continue Continue Continue Replace Replace .. . Continue Continue Continue Replace Replace .. .

the control limit, and the optimal tool replacement decision at each decision epoch. The ®nal computational results of the stochastic dynamic decision model can be put into a simple table containing only the decision epochs and the corresponding control limits, as illustrated in Table 2 in the numerical example. The table of the control limits can be calculated before the production period. Once this table is obtained, the optimal tool replacement decisions can be implemented easily by the shop ¯oor production personnel or can be carried out automatically.

References Bhat, U.N. (1984) Elements of Applied Stochastic Processes, John Wiley & Sons, Toronto, p. 105. Billatos, S.B. and Kendall, L.A. (1991) A general optimization for multi-tool manufacturing systems. Journal of Engineering for Industry, 113, 10±16. Cox, D.R. and Oakes, D. (1984) Analysis of Survival Data, Chapman and Hall, New York. Crowder, M.J., Kimber, A.C., Smith, R.L. and Sweeting, T.J. (1991) Statistical Analysis of Reliability Data, Chapman and Hall.

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Gray, A.E., Seidmann, A. and Stecke, K.E. (1993) A synthesis of decision models for tool management in automated manufacturing. Management Science, 32(5), 549±567. Gertsbakh, I.B. (1989) Statistical Reliability Theory, Marcel Dekker, New York. Kalb¯eisch, J.D. and Prentice, R.L. (1982) The Statistical Analysis of Failure Time Data, John Wiley, New York. LaCommare, U., LaDiega, S., Noto, S. and Passannanti, A. (1983) Optimal tool replacement policies with penalty cost for unforeseen tool failure. International Journal of Machine Tool Design Research, 23, 237±243. Law, A.M. and Kelton, W.D. (1991) Simulation Modeling and Analysis, 2nd edn, McGraw-Hill, New York. Liu, H. and Makis, V. (1996) Cutting-tool reliability assessment in variable machining conditions. IEEE Transactions on Reliability, 45(4), 573±581. Liu, P.H., Makis, V. and Jardine, A.K.S. (1998) Optimal tool replacement based on surface roughness in ®nish machining. IMA Journal of Mathematics Applied in Business and Industry, 9(3), 223±239. Mazzuchi, T.A. and Soyer, R. (1989) Assessment of machine tool reliability using a proportional hazards model. Naval Research Logistics, 36, 765±777. Nelson, W. (1990) Accelerated Testing: Statistical Models, Test Plans and Data Analysis, John Wiley & Sons, New York. Newby, M. (1994) Perspective on the Weibull proportional-hazards modes. IEEE Transactions on Reliability, 43(3), 217±223. Newby, M. (1988) Accelerated failure time models for reliability data analysis. Reliability Engineering and System Safety, 20, 187±197. Pandit, S.M. and Sheikh, A.K. (1980) Probability and optimal replacement via coecient of variation. Journal of Mechanical Design, 102, 761±768. Pike, M.C. (1966) A method of analysis of a certain class of experiments in carcinogenesis. Biometrics, 22, 142±161. Ramalingam, S. (1977) Tool-life distributions, part 2: multiple-injury tool-life model. Journal of Engineering for Industry, 99B(3), 523± 528. Ramalingam, S. (1982) Tool wear, tool life distribution, and consequencies, in On the Art of Cutting Metals-75 Years Later, Kops, L. and Ramalingam, S. (eds.), ASME, New York. pp. 149±188. Ramalingam, S. and Watson, J.D. (1977) Tool life distributions. part 1: single injury tool-life model. Journal of Engineering for Industry, 99B(3), 519±522. Rossetto, S. and Zompi, A. (1981) A stochastic tool-life model. Journal of Engineering for Industry, Series B, 103, 126±130. Taraman, K. (1974) Multi machining output ± multi independent variable turning research by response surface methodology. International Journal of Production Research, 12(2), 233±245. Taylor, F.W. (1906) On the art of cutting metals, Transactions of the ASME, 28, 31±35. Wager, J.G. and Barash, M.M. (1971) Study of the distribution of the life of the HSS tools. Journal of Engineering for Industry, 73, 1044±1049.

Appendices Appendix 1: Kolmogorov±Smirnov goodness-of-®t test of the ®tted distribution The reliability function (36) is estimated based on the tool failure data in Taraman (1974) as shown in Table A1. It can be written as R…t; z† ˆ exp…ÿH …t; z††;

Table A1. Tool failure data from Taraman (1974) Test number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Speed v (fpm) 340 570 340 570 340 570 340 570 440 440 440 440 305 635 440 440 440 440 305 635 440 440 440 440

Feed f (ipr) 0.006 0.006 0.014 0.014 0.006 0.006 0.014 0.014 0.009 0.009 0.009 0.009 0.009 0.009 0.004 0.017 0.009 0.009 0.009 0.009 0.004 0.017 0.009 0.009

30 30 16 16 30 30 16 16 05 05 05 05 05 05 72 32 05 05 05 05 72 32 05 05

Depth d (in)

Failure time (minutes)

0.0210 0.0210 0.0210 0.0210 0.0400 0.0400 0.0400 0.0400 0.0290 0.0290 0.0290 0.0290 0.0290 0.0290 0.0290 0.0290 0.0135 0.0455 0.0290 0.0290 0.0290 0.0290 0.0135 0.0455

70 29 60 28 64 32 44 24 35 31 38 35 52 23 40 28 46 33 46 27 37 34 41 28

where H …t; z† is the cumulative hazard function H …t; z† ˆ 0:012757…t=10†8:7325 …v=1000†11:2760  …100f †2:0843 …100d†1:7942 :

…A1†

If the reliability function (36) is adequate, the cumulative hazard H …T ; z† at failure time T is exponential with parameter 1. In the following, we perform a Kolmogorov± Smirnov goodness-of-®t test to check the null hypothesis that the H …T ; z† is distributed exponentially with parameter 1. First, we calculate H …T ; z† for each tool failure time and rank the results in ascending order, i ˆ 1; 2; . . . ; 24. Let Xi represent the cumulative hazard corresponding to order i. We have X1 < X2 <


< X24 . The empirical distribution function F …Xi † is calculated as F …Xi † ˆ i=24 and the ®tted distribution function F^…x† is calculated as F^…x† ˆ 1 ÿ exp…ÿx†, for x ˆ Xi . The K±S test statistic is calculated as D ˆ maxfD‡ ; Dÿ g, where   i ‡ ^ ÿ F …X…i† † ; D ˆ max 1i24 24   iÿ1 ÿ ^ : D ˆ max F …X…i† † ÿ 1i24 24 The calculations are shown in Table A2. The computed value of the K±S test statistic is D ˆ 0:1421. The value of the adjusted test statistic is

Optimal tool replacement times

495

Table A2. K±S test calculations t

Xi

F …Xi †

F^ …Xi †

D

46 28 31 37 52 28 40 29 35 35 41 33 23 44 38 46 34 24 28 70 27 32 60 64

0.0657 0.1203 0.1304 0.1575 0.1916 0.2075 0.3111 0.3554 0.3764 0.3764 0.3801 0.5052 0.6024 0.6866 0.7719 1.0383 1.1306 1.1702 1.4151 2.3036 2.4434 2.6681 3.2426 3.3470

0.0417 0.0833 0.1250 0.1667 0.2083 0.2500 0.2917 0.3333 0.3750 0.4167 0.4583 0.5000 0.5417 0.5833 0.6250 0.6667 0.7083 0.7500 0.7917 0.8333 0.8750 0.9167 0.9583 1.0000

0.0636 0.1134 0.1223 0.1457 0.1744 0.1874 0.2673 0.2991 0.3137 0.3137 0.3162 0.3966 0.4525 0.4967 0.5379 0.6460 0.6772 0.6897 0.7571 0.9001 0.9131 0.9306 0.9609 0.9648

)0.0219 )0.0300 0.0027 0.0210 0.0339 0.0626 0.0243 0.0342 0.0613 0.1030 0.1421 0.1034 0.0892 0.0866 0.0871 0.0207 0.0312 0.0603 0.0346 )0.0668 )0.0381 )0.0139 )0.0026 0.0352

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

‡

D

ÿ

0.0636 0.0717 0.0390 0.0207 0.0077 )0.0210 0.0173 0.0075 )0.0196 )0.0613 )0.1004 )0.0617 )0.0475 )0.0449 )0.0455 0.0210 0.0105 )0.0186 0.0071 0.1084 0.0798 0.0556 0.0443 0.0065

p p … 24 ‡ 0:12 ‡ 0:11= 24†  D ˆ 0:7164; which is less than the critical value c0:90 ˆ 1:224 at level a ˆ 0:10 (see Table 6.14 in Law and Kelton (1991). Therefore we cannot reject the null hypothesis that the Xi 's are exponentially distributed with parameter 1. Hence, the distribution determined by (A1) is a good description of the tool failure data in Table 4. For a reference, see Gertsbakh (1989). Appendix 2: standard deviations of the estimators of unknown parameters The reliability function (36) can be written as n o ^ ^ ^ R…t† ˆ exp ÿ^ k…t=10†^a …v=1000†b1 …100f †b2 …100d†b3 ; …A2† where ^ ˆ 11:2760; ^a ˆ 8:7325; ^ k ˆ 1:2758  10ÿ2 ; b 1 ^ ^ b ˆ 2:0843; b ˆ 1:7942: 2

3

The standard deviations of the estimators were obtained by ®nding the inverse of the observed informa-

tion matrix and taking the square roots of the diagonal elements: r^a ˆ 1:427; r^k ˆ 3:812  10ÿ3 ; rb^1 ˆ 2:135; rb^2 ˆ 0:692; rb^3 ˆ 0:713: (See Liu and Makis (1996) for a detailed description of the estimation procedure.) The 95% con®dence intervals for the estimated parameters are a : …4:935; 10:530†; k : …5:287  10ÿ3 ; 2:023  10ÿ2 †; b1 : …7:092; 15:460†; b2 : …0:729; 3:440†; b3 : …0:396; 3:192†:

Biographies Patrick H. Liu is currently an operations research scientist at OPUS2 Revenue Technologies Inc., Portsmouth, New Hampshire, USA. He is interested in reliability, maintenance, production scheduling, optimization and revenue management. He received his Ph.D. in 1997 from the University of Toronto, and a Master's degree in Engineering Production from the University of Birmingham, UK. His papers have appeared in IEEE Transactions on Reliability and in the IMA Journal of Mathematics Applied in Business and Industry. Viliam Makis is an Associate Professor in the Department of Mechanical and Industrial Engineering, University of Toronto. His teaching and research areas include quality and reliability engineering with special interest in modeling and optimization of stochastic systems. His articles have appeared (or will appear) in Mathematics of Operations Research, Technometrics, IIE Transactions, IEEE Transactions on Reliability, Journal of Applied Probability, Naval Research Logistics, EJOR, INFOR, Journal of the OR Society, International Journal of Production Economics, IMA Journal, Kybernetika, etc. He is a Senior Member of the Institute of Industrial Engineers and of the American Society for Quality. Dr. Andrew Jardine is a Professor in the Department of Mechanical and Industrial Engineering at the University of Toronto and principal investigator in the Condition-Based Maintenance Laboratory where the EXAKT software has been developed. He also serves as a Senior Associate Consultant to the Global Leader of the Physical Asset Management Practice of Pricewaterhouse Coopers. Dr. Jardine is the author of the AGE/CON and PERDEC life-cycle costing software licensed to British Airways, Canada Post, the Hong Kong Mass Transit Authority and other organizations in North America and globally. Dr. Jardine was the 1993 Eminent Speaker to the Maintenance Engineering Society of Australia and was the ®rst recipient of the Sergio Guy Memorial Award from the Plant Engineering and Maintenance association of Canada in recognition of his outstanding contribution to the Maintenance profession. Contributed by the Reliability Modeling and Optimization Department.