Operation-Dependent Maintenance Scheduling in Flexible Manufacturing Systems

M. Celen1, D. Djurdjanovic

1

1 Dept. of Mechanical Engineering, University of Texas at Austin,

Austin, TX, United States

Abstract

In highly flexible and highly integrated manufacturing systems such as semiconductor manufacturing , the dynamic interactions between equipment condition, operations executed on the tools and product quality necessitate joint decision-making in maintenance scheduling and production operations. To address these problems, we devise an integrated decision-making policy optimizing a customizable objective function with respect to operation-dependent degradation models and production target. Optimization was achieved using a metaheuristic method based on the results of discrete-event simulations of a generic cluster tool. The results show that operation-dependent maintenance decision-making outperforms the case where maintenance decisions are made without considerations of operation-dependent degradation dynamics. Keywords:

Scheduling, Maintenance, Dispatching

1 INTRODUCTION

Maintenance is an essential part of manufacturing operations ensuring that adequate production resources are available to achieve desired productivity and quality in a manufacturing system. There are mainly two types of maintenance operations; reactive maintenance (RM) which occurs when a tool/machine actually fails and preventive maintenance (PM) which is performed on a tool/machine before actual failure occurs. Even though RM is unavoidable, it usually costs much more and requires more maintenance time when compared to PM [1]. For example, in an automotive assembly plant a minute of unexpected downtime can cost as much as $20000 [2]. Hence, proper scheduling of PM is always desired.

PM can be roughly characterized as reliability-based maintenance (RBM), where maintenance is performed at certain time or usage intervals, and condition-based maintenance (CBM), where maintenance is performed when the condition of machine requires a repair. RBM considers the long-run average of the system dynamics and hence the decisions made are optimal only in the steady-state. On the contrary, since CBM is more dynamic and the decisions are based on the condition of the system at that time point, the effects of those decisions can be optimized and adjusted within an arbitrarily chosen time horizon [3] . The benefits of CBM arise from the ability to preventively maintain the system only when necessary, thus, saving resources and improving system availability [4].

In Flexible Manufacturing Systems (FMS), PM decisions are considerably harder to make, as the machines have the capability of conducting different manufacturing operations and/or producing at various speeds. In such systems, degradation of a machine depends highly on the operations performed on that machine. Thus, selection of operations executed on a machine directly affects PM decisions by changing the degradation dynamics. On the other hand, PM actions interrupt production and change the system reliability and equipment availability, which in turn directly affects decisions as to which operations should be performed on which piece of equipment [5]. The necessity of joint decision making as a result of such

dynamic interaction between PM and reconfiguration actions has been addressed only recently.

Yang et al. [6] combine age-based maintenance (ABM) decisions with operations decision making in an environment where there are only two operation modes available: fast and slow. Naturally, fast operation mode leads to faster degradation but high production, and vice versa. This effect is modeled by operation specific reliability functions. In their study, possible schedules are generated via a Genetic Search Algorithm [7] and the value of each schedule is evaluated by a discrete event simulation system. It was shown in their results that jointly optimizing throughput and maintenance operations results in decreased maintenance time and increased profits.

A much more complex situation is considered in Zhou et al. [5]. They develop an integrated reconfiguration and ABM policy (IRABM) on a single-product parallel-serial system with reconfigurable capability. With reconfiguration comes the ability to improve system throughput and reduce the likelihood of a system-wide failure. Nevertheless, there is also an associated cost of operation transfer from the degraded machine to the less degraded one, necessitating a tradeoff between the benefits and drawbacks of operations reconfiguration. It has been shown in their study that IRABM outperforms ABM in terms of lower expected total cost.

The previous two publications modeled operation-dependent degradation through operation specific reliability functions, which makes them very suitable for RBM approaches. The work by Zhou et al. [8] utilizes Markov models to model the degradation of the machine conditions which makes it more suitable for CBM applications. They use simulation based optimization in a load-sharing system, where different components of the system share an overall load. In such systems, loads allocated to each machine vary, thus affecting degradation dynamics of each machine since the degradation of a machine depends on the amount of load assigned to that machine. It was shown that Integrated Load Allocation and Condition Based Maintenance Policy (ILACBM) results in increased availability of equipment when compared to the traditional CBM.

As machine degrades, the outgoing product quality (yield) usually decreases. The earliest work considering the effects of equipment condition on product quality and incorporating it into maintenance scheduling is reported in [9]. A decision making policy that simultaneously determines maintenance and production schedules for a multiple-product single machine system was developed by considering the fact that machine condition can affect the yield of different product types. In a later study, Sloan [12] extends the work in [9] by considering varying production targets and multiple maintenance actions. Sloan and Shanthikumar [10] consider multiple machines and add job dispatching decisions to the decision-making process developed in [9]. They compare the performances of some predetermined dispatching rules (such as FCFS, SPT, selecting the lot with highest yield, etc.) and some special maintenance policies (such as fixed state, fixed time, etc.). Lee et al. [3] present a more elaborate work on CBM and dispatching with yield considerations in a semiconductor manufacturing environment. In this work, as an improvement to [9], there are no priori predetermined CBM policies. Instead, CBM policies for different wafer types are determined via discrete event simulation and genetic algorithm based optimization of product-specific maintenance triggering states. Dispatching of jobs to the machines is also obtained using a genetic algorithm. The authors report that using wafer type-dependent CBM policies results in increased yields. By not restricting dispatching and maintenance policies as in [9], Lee et al. work on a system which is a close representation of an actual cluster tool in a semiconductor fab. However, they overlook the fact that degradation is an operation-dependent process and assume that each operation affects the degradation of machines in the same way.

In our study, we consider a multiple-product/multiple-machine system where each product requires several operations for completion. These operations are executed on non-identical machines, each of which can execute a certain subset of operations. Degradation processes of the machines are modeled as operation dependent Markov models, with the output quality of the products decreasing as the condition of the machine degrades according to a known product-specific stochastic model. Our aim is to facilitate maintenance decision-making in highly flexible manufacturing systems (where one has the ability to do multiple operations in multiple stations), based on the aforementioned operation-dependent degradation models. The decision-making will be done by maximizing a customizable reward function, taking into account rewards of production and the costs of maintenance operations.

2 PROBLEM STATEMENT

In this study, we focus on a flexible manufacturing system with m manufacturing stations labeled

and we

assume that in those stations we are producing a set of product types . Let be the set of all operations that can be executed by the stations of that manufacturing system. Each product type , is associated with a sequence of operations

, where is the number of

operations needed to manufacture that product type. Since any two product types and may have some

common operations, the intersections

,

are not necessarily empty sets. It is assumed that each

station can execute operations

,

where is the number of operations that can be

executed by that station. Any two stations and may

be able to execute some common operations, which means that

, , are not

necessarily empty sets. That means some operations can be executed by more than one station, making it necessary to choose which station to use for a given operation. The goal will be to produce of each product

type , , within a certain mission time .

In this paper, we acknowledge that different operations have different degradation effects on the stations and model the degradation process via concatenated operation-dependent Markov models. The degradation process of each station is characterized by a set of operation-dependent Markov Chains defined over a

common state space , in which each of the states denotes a degradation state of the station (1 denoting the "good as new" state and denoting the

"failed" state). For any station, the probability of transitioning from state to state if operation is

executed in it is defined as

, together these

probabilities form the operation-dependent state transition

matrices . All Markov chains are assumed to be unidirectional, modeling the well-accepted intuition that the condition of a station can only worsen over time, unless a maintenance operation is done.

Let denote the state of station that triggers

preventive maintenance when operation is executed in

it. It means that if the current station’s degradation state

is greater than or equal to , then operation

could not be performed in that station, unless a maintenance operation is done on it beforehand. Thus, as time passes and stations degrade, each station "looses" more and more operations that can be done in it, unless a maintenance action is invoked on it to restore its condition.

In addition, let us take into account the fact that as the degradation level of a station increases, the probability of success of any operation executed in that station decreases. We assume that for any station the probability of success of a given operation is a known function of the operation and the station state , denoted as

.

Our objective is to find a combined policy of maintenance triggering and dispatching of operations across a system that maximizes a reward function that considers the benefits of production, costs of maintenance and penalties for unmet production goals. It is assumed that the sequence in which products are manufactured is fixed. Let

denote this fixed product

sequence meaning that first of products will be

made, followed by of products , etc. Hence, our

problem can be expressed as follows:

where

Reward for each completed wafer of type ,

Cost of reactive maintenance

Cost of reactive maintenance

Unmet production goal penalty unit penalty for wafer

type ,

Number of good wafers produced of type during

mission time T

][ to subject

(P1)

)( maximize

21

21

222

21

112

11

w

Nom

NoNo

om

oo

om

oo

N

Wjjjjjpprr

Wjjj

SSS

SSS

SSS

,w,,ww

pnNacnmcnmnRE

Total number of reactive maintenance

Total number of preventive maintenance

{

The set of operation-dependent preventive maintenance

triggering states obtained by solving problem (P1) will

be referred to as the Operation-dependent CBM Policy.

In order to evaluate the benefits of operation-dependent maintenance decision-making, we will compare solutions of (P1) with the traditional CBM policy. In traditional Operation-independent CBM Policy, it is assumed that the maintenance triggering states for each station are

independent of the operation executed in it (i.e.

for each station i). This problem can

be expressed as follows:

In both problems (P1) and (P2), for the operations that can be executed by more than one station, operations dispatching is based on the intuitive paradigm of always dispatching it to the least degraded station. This dispatching policy is supported by results from [3].

Objective function values obtained from (P1) and (P2) will be used to evaluate relative efficacy of the two policies and explore conditions when one outperforms the other.

3 SOLUTION PROCEDURE

3.1 Operation-independent CBM policy determination

For a manufacturing system with stations, where each station has M degradation states, the solution space for

problem (P2) consists of candidate solutions

(since degradation state 1 would never be observed in a CBM policy as it denotes a trivial policy of maintaining the station after each operation). The solution space for a simple 5-station manufacturing system with 5 degradation

states has candidate solutions. Since our focus in this study was on a small manufacturing system (see Results section), problem (P2) can be solved using complete enumeration.

The expected profit of each candidate solution is determined by evaluating it via discrete-event simulation of the target system over multiple replications.

3.2 Operation-dependent CBM policy determination

Solution Representation

For an m-station manufacturing system with N operations that can be executed in these stations, a solution for operation-dependent CBM policies can be represented with a matrix illustrated in Figure 1. In this matrix,

th column represents the maintenance triggering states

of station for each operation and similarly th row

represents the triggering states of each station when operation is produced in them.

There are obviously up to candidate solutions in the solution space, though it can be decreased by acknowledging that not all operations can be executed in all stations. Nevertheless, even the aforementioned reduction usually results in a solution space that is too large for complete enumeration, especially when candidate solutions are evaluated via replicated discrete-event simulations.

Candidate solution matrix Maintenance policy for representation operation

Figure 1: Solution representation for operation-dependent CBM policy.

Hence, in order to find practical, applicable and near optimal solution, we used a Tabu Search algorithm [11] based on the results of discrete-event simulations. As illustrated in Figure 2, a set of feasible solutions is generated by the tabu search algorithm and fed into the discrete-event simulator. The objective function value relevant to each candidate solution is obtained from multiple replications of discrete-event simulations of the target manufacturing system. This expected value is fed back into the tabu search algorithm as the objective function of the optimization process.

Figure 2: Decision-making by tabu search algorithm based on simulations

Tabu Search Algorithm

Tabu search (TS) is a local search technique that enhances the exploration performance by using advanced memory structures of a computer. Once a candidate solution has been determined, it is marked as 'tabu' so that the same solution is not visited by the algorithm over a certain number of iterations. The search starts from an initial solution (randomly seeded or chosen based on some problem specific information), moves iteratively from a solution to a “non-tabu” solution in

the local neighborhood of and terminates when some stopping criterion is satisfied. The flowchart of the tabu search algorithm implemented in this study is given in Figure 4. The solution representation, initial solution determination, neighborhood generation, tabu and aspiration conditions, and stopping criteria used in this study are explained below.

Solution Representation: The representation of a solution in matrix form was explained thoroughly in Solution Representation section.

Initial Solution Determination: The initial solution for the TS is based on the solution of the problem (P2). The result of (P2), denoted as , is converted into the operation-dependent CBM solution representation as shown in Figure 3.

Figure 3: Conversion of operation-independent CBM policy solution to initial solution of TS

Neighborhood Generation: A cell in the matrix representing the current solution is selected randomly and its value is perturbed. An example of neighborhood generation for a system with 3 stations, 4 operations and 5 degradation states is illustrated in Figure 5.

][ to subject

(P2)

)( maximize

21

21

w

m

N

Wj

jjjjpprr

Wj

jjSSS

,w,,ww

pnNacnmcnmnRE

Maintenance policy for chamber 𝑐𝑖

At each iteration, random cell selection is repeated for times and thus candidate

solutions are generated at each iteration.

Figure 4: Flowchart of tabu search algorithm for operation-dependent CBM policy optimization

Figure 5: Illustration of neighbourhood generation in TS algorithm

Tabu Condition: Returns to previous candidate solutions are prohibited for the next iterations.

Aspiration Condition: If the best solution among candidate solutions is obtained by a tabu move, but yields an expected profit higher than the best profit obtained so far, then that solution is selected as the next incumbent solution regardless of its tabu status.

Stopping Criterion: The algorithm terminates whenever maximum number of iterations is reached or a non-improving move is made for a certain number of consecutive iterations.

4 RESULTS

Operation-dependent maintenance decision-making was tested on an example of a cluster tool, a highly sophisticated and integrated machine routinely used in semiconductor manufacturing. A cluster tool is a “mini” manufacturing system of interacting subsystems (chambers and material handling system) and can be seen as a quintessential FMS since each chamber can be used to execute various operations with various operating parameters. Each chamber of this cluster tool can be seen as a manufacturing station and different wafer layers produced in these chambers can be seen as different operations performed in these chambers, while different wafer types pushed through this system can be

seen as different product types produced in this system. In order to assess the performance of the newly proposed optimization methodology, we used the AutoMod software package [13] to simulate a 5-chamber cluster tool to produce 3 types of wafers. The parameters used are given in Tables 1, 2 and Figure 6.

Table 1: Summary of parameters

1

0.98 0.02 0 0 0

0 0.98 0.02 0 0

0 0 0.98 0.02 0

0 0 0 0.98 0.02

0 0 0 0 1

OpP

2

0.90 0.08 0.02 0 0

0 0.90 0.08 0.02 0

0 0 0.90 0.08 0.02

0 0 0 0.93 0.07

0 0 0 0 1

OpP

3

0.93 0.05 0.02 0 0

0 0.93 0.05 0.02 0

0 0 0.93 0.05 0.02

0 0 0 0.97 0.03

0 0 0 0 1

OpP

4

0.94 0.04 0.015 0.005 0

0 0.94 0.04 0.015 0.005

0 0 0.94 0.04 0.02

0 0 0 0.94 0.06

0 0 0 0 1

OpP

5

0.98 0.015 0.005 0 0

0 0.98 0.015 0.005 0

0 0 0.98 0.015 0.005

0 0 0 0.98 0.02

0 0 0 0 1

OpP

6

0.95 0.04 0.01 0 0

0 0.95 0.04 0.01 0

0 0 0.95 0.04 0.01

0 0 0 0.95 0.05

0 0 0 0 1

OpP

7

0.93 0.04 0.02 0.01 0

0 0.93 0.04 0.02 0.01

0 0 0.93 0.06 0.01

0 0 0 0.97 0.03

0 0 0 0 1

OpP

8

0.97 0.03 0 0 0

0 0.97 0.03 0 0

0 0 0.97 0.03 0

0 0 0 0.97 0.03

0 0 0 0 1

OpP

9

0.90 0.05 0.03 0.02 0

0 0.90 0.05 0.03 0.02

0 0 0.90 0.07 0.03

0 0 0 0.90 0.10

0 0 0 0 1

OpP

10

0.99 0.01 0 0 0

0 0.99 0.01 0 0

0 0 0.99 0.01 0

0 0 0 0.99 0.01

0 0 0 0 1

OpP

11

0.89 0.07 0.03 0.01 0

0 0.89 0.07 0.03 0.01

0 0 0.91 0.07 0.02

0 0 0 0.91 0.09

0 0 0 0 1

OpP

12

0.92 0.06 0.02 0 0

0 0.92 0.06 0.02 0

0 0 0.92 0.06 0.02

0 0 0 0.95 0.05

0 0 0 0 1

OpP

13

0.95 0.03 0.02 0 0

0 0.95 0.03 0.02 0

0 0 0.95 0.03 0.02

0 0 0 0.96 0.04

0 0 0 0 1

OpP

Figure 6: Operation-dependent transition probability matrices

Table 2: Probability of success information for each operation

As mentioned before, the results for operation-independent CBM policy were obtained by complete enumeration. 40 replications of each candidate solution were simulated to obtain the expected profit of that solution and was determined to be the

best operation-independent CBM policy.

In order to optimize the operation-dependent CBM policy, tabu search algorithm was run for 20 iterations (tests with longer iterations showed that 20 iterations were enough to obtain the best results). At each iteration 10 random cells were selected for neighborhood generation. Hence, 30 candidate solutions were generated at each iteration. 40 replications of each solution were simulated to obtain the expected profit of that solution. Admissible moves were marked as tabu for = 5 iterations. As

illustrated in the first two rows of Table 3, operation-dependent CBM policy results in higher expected profit when compared to the operation-independent CBM policy.

LCL PROFIT UCL STDDEV

Original

Parameters

Operation-

independent 3532.5 4337.4 5142.2 402.4

Operation-

dependent 3732.7 4605.6 5478.6 436.5

Identical

Wafer

Reward

Operation-

independent 9175.5 10282.5 11389.5 553.5

Operation-

dependent 9077.6 10443.8 11809.9 683.1

Identical

PM and

RM Cost

Operation-

independent 4111.2 4756 5400.8 322.4

Operation-

dependent 4252.6 5006 5759.4 376.7

Very

Expensive

RM Cost

Operation-

independent 3148.8 4049.6 4950.5 450.4

Operation-

dependent 3146.6 4173.1 5199.7 513.3

Table 3: Comparison of the results of operation-dependent and operation-independent CBM policies

For the best operation-dependent CBM policy, it was

observed that the maintenance triggering states and

have been changed from 3 (the state implied by the

operation-independent CBM policy) to 4 and 5, respectively. Both operations degrade the chambers slowly and this result proves the intuition that less frequent maintenance corresponding to later triggering of maintenance is optimal for slower degrading operations. In addition to that, both operations are executed in the manufacturing of cheaper wafers. The optimized maintenance triggering states also confirm our intuition that as the wafer reward decreases, the effect of completing that wafer successfully on the profit decreases, thus favoring later triggering of maintenance.

In order to see the performance of our proposed method in the presence of different parameters, we have created three case studies by changing the wafer rewards and reactive maintenance costs.

4.1 Case 1: Identical Wafer Rewards for All Wafers

In this case we wanted to see the effects of changes in wafer rewards on operation-dependent CBM policies. In the previous section, it was seen that a decrease in wafer rewards would lead to later triggering of maintenance. To further prove this intuition, we set and

tested the proposed method with this setting. For this parameter setting, was determined as the

best operation-independent CBM policy via complete enumeration.

(a)

(b)

(c)

(d)

Table 4: Comparison of maintenance triggering states for different parameter sets

As shown in the 3rd and 4th rows of Table 3, with identical wafer reward setting, our proposed method still yields a higher expected profit than the operation-independent CBM policy. Comparing the maintenance triggering states in Tables 4.a and 4.b, it can be observed that increasing wafer rewards results in earlier triggering of maintenance, which further confirms our intuition.

4.2 Case 2: Identical PM and RM Cost

When the costs of PM and RM are the same and if the probability of success (operation yield) was not affected by chamber degradation, we would let all the chambers run until failure. However, since we assume that success probabilities are affected by chambers' conditions, we cannot expect the chambers to run until failure. Nevertheless, intuition suggests that later triggering of maintenance would occur if the RM cost is decreased to be the same as the PM cost, since the difference between scheduled and unscheduled maintenance becomes smaller. To test this intuition, we decreased the cost of RM and set . For this parameter

setting, was chosen as the best operation-

independent CBM policy via complete enumeration. The effect of the RM cost decrease can immediately be seen when the operation-independent CBM policy of this case is compared to the policy obtained with the original parameters ( was increased by 2). Comparison of the

profits in the 5th and 6th rows of Table 3 shows that our method yields higher expected profit. In addition, it can be observed by comparing Tables 4.a and 4.c that most of the maintenance triggering states call for later triggering of maintenance, agreeing with our intuition that a decrease in RM cost would result in later maintenance triggering.

4.3 Case 3: Very Expensive RM

When the cost of reactive maintenance is extremely high, it would be intuitive to avoid reactive maintenance as much as possible, even if it means sacrificing the production, at the expense of frequent PM actions that avert costly RM. After increasing the RM cost from 250 to 1000, we obtained the best operation-independent CBM policy as . In compliance with our intuition,

comparison of Tables 4.a and 4.d shows that a high RM cost results in earlier triggering of maintenance. Even so, comparison of the expected profits in rows 7 and 8 of Table 3 shows that operation-dependent CBM policy yields higher expected profits when compared to operation-independent CBM policy.

5 SUMMARY

This paper presents a solution methodology to determine an intelligent CBM policy for a manufacturing system composed of multiple manufacturing stations that can execute multiple operations. To the best of our knowledge, this is the first maintenance scheduling study in a manufacturing system consisting of multiple manufacturing stations that considers both operation-dependent degradation models and a model of the probability of manufacturing success yield, which is both operation and degradation dependent. A simulation model incorporating the degradation and success probability information is developed to obtain the expected profits for candidate CBM policies. As a benchmark, operation-independent CBM policies are determined via complete enumeration. These policies are compared with the newly proposed methodology that optimizes the operation-dependent maintenance policy via a tabu search algorithm. For all the scenarios considered, the operation-dependent CBM policies obtained by the newly proposed solution methodology are

shown to outperform the benchmark operation-independent CBM policies.

One should note that in this paper, we assume that the sequence in which different product types are produced is a priori given. However, there exists a potential benefit in simultaneously optimizing the decisions of maintenance and product type sequence. Therefore, development of an integrated decision making policy for maintenance scheduling and product dispatching, and evaluating the effects of different parameters on this policy will be the topics of the future research.

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