Comput & Indu~ I::leng Vol 10 No I pp 11-20 1986 0360-8"~52,'86 $3 00 + 00 Printed m Great Britain ,~, 1986 Pergamon Press Lid

S C H E D U L I N G O F R O U T I N E M A I N T E N A N C E U S I N G

P R O D U C T I O N S C H E D U L E S A N D E Q U I P M E N T

F A I L U R E H I S T O R Y

SANJAY JOSHI Industrial Engmeenng Dept., Purdue University, West Lafayette, IN 47907. U S A.

and

RAJIV GUPTA Industrial Engineering Dept., Purdue University and GMI Engineenng & Management

Institute, 1700 West Third Ave , Flint, Michigan 48502, U.S.A

(Received for pubhcatlon 28 March 1985)

Abstract--Ideally all maintenance should be routine, and should be performed to prevent pre- mature equipment failure. From the practical viewpoint of maintenance in a manufactunng environment it is important to know when in the production cycle it is most advantageous for equipment to be removed from service for repmr

This paper describes a model that was developed for scheduling of routine maintenance at an aggregate level in a manufactunng environment. The model utilizes the production plan and the equipment failure history to develop a maintenance schedule. The objective ~s to choose a maintenance action so that the expected costs of breakdown of equipment are minimized Different maintenance schedules are generated and the schedule that gives the least expected cost is selected as the maintenance plan for the equipment The model is implemented on a North Star microcomputer.

INTRODUCTION

Fundamentally, the maintenance function should be an integral part of the production activities. Production and maintenance activities have a direct relationship. Any break- down in machine operation results in disruption of production and leads to additional cost due to downtime, loss of production, decrease m productivity and inefficient use of personnel, equipment and facilities. The cost of routine and corrective maintenance is normally lower than that of emergency maintenance and this provides the incentive for developing and Implementing a maintenance program[l].

Necessary to any planning for maintenance is the Integration of maintenance plan- ning with technological forecasting. Just as production needs to know the future plans for the firm, maintenance must know and integrate the effect of the production plans into their own planning. This paper deals primarily with the aspect of integrating the production plan into maintenance forecasting and scheduling.

Sherif and Smith[7], Pierskalla and Voelkar[6] and McCall[5] have extensively sur- veyed several maintenance policies and models. The general structure of the mainte- nance problems possesses features that are characteristic of decision theoretic models. While in operation the equipment under consideration Is assumed to occupy one of several states, the two extremes being "new" and -fai led." These two states enclose a set of intermediate states that denote different degrees of deterioration. The movement from state to state is governed by a probability mechanism. The equipment moves from one state to another until it reaches the state of failure. The behavior of the equipment can, however, be regulated by choosing a particular action at each decision point. The action taken by the decision maker reflects a maintenance policy and the difference between the regulated and uninhibited behavior of the equipment is a measure of the policy's influence.

A policy's performance can be measured by assigning an occupancy cost to each state and an intervention cost to each action. These costs are calculated to measure the downtime costs of each maintenance action as well as downtime costs associated with each operational state. The objective of the decision maker is to choose mam-

II

12 S. JOSHI and R Guvr~

tenance actions so that the expected costs of operating the equipment are minimized. Any rule that assigns a specific action in the equipment ' s life is called a maintenance policy. More precisely, a maintenance policy is a function whose range is the set of possible realizations in the equipment history, the mformation space. An optimal pohcy relative to a particular action space and a particular information space is a function whose form is best in the sense of minimizing expected cost.

This paper describes a model which integrates the production plan and maintenance pohcy, using the equipment history and cost of downtime as a hnk between the two Further, the model is ~mplemented on a North Star Horizon microcomputer w~th 64K RAM and dual 5 ] inch floppy disk drives The implementat ion of the model on a low cost mic rocompute r permits its ease of use in a dedicated stand alone mode.

MAINTENANCE PLANNING PROBLEM

The goal or obJective is to establish a maintenance schedule for a machine in the production system. Machines differ in their maintenance characteristics: some items deteriorate more rapidly than others and the cost of routine maintenance and repair varies among them. Each machine has Its own history of fadures, projected usage over the planning horizon, and hence its own frequency of repair. Ideally, eqmpment should receive routine maintenance when such maintenance causes least disruption m pro- duction. It is assumed that with the scheduling of routine maintenance of the machine before the expected time of failure, that the number of contingency failures and the associated costs of downtime can be reduced to offset costs associated with scheduled maintenance.

Further, there is a probabili ty that the machine may fail prior to the next period during which it is scheduled for maintenance. When this happens, a cost ts incurred which includes the cost of system downtime. The cost of failure is assumed to exceed the cost associated with routine maintenance. If the cost of routine maintenance is greater than the cost associated with a failure, then there is no advantage in scheduhng maintenance. Routine maintenance is advantageous for systems and parts with m- creasing failure rates.

To achieve a balance between rehablhty and maintenance costs, several factors including the cost of downtime, mean downtime, time to repair were considered in the model. The most economical point in the planning cycle of the eqmpment was selected for performing the maintenance action Once the schedule of routine maintenance is determined the number of persons required to maintain the schedule ts determined.

The model deals with the planning problem at an aggregate level over a time span equal to the planning horizon of the master production schedule, typically one year. Since it Is desired to relate the maintenance schedule to the production schedule it ts desirable that the planning horizon should be the same for maintenance and production. The planning model specifies periods in which routine maintenance should be sched- uled, and the level of planning dictates the size of the periods used for plannmg. From the aggregate plan the details can be worked out at a lower level o~plann.ng, such as short term scheduling.

ASSUMPTIONS

The assumptions made in formulating the model are, that hfe to failure is a Welbull distribution and that reasonable est imates of the parameters , the mean downtime, mean time to repair, and mean costs of sys tem downtime and routine maintenance are avail- able. If these cannot be obtained fi'om available data then est imates based on extrinsic factors such as data from similar equipment can be used as a rough est imate, but these should be substituted with better est imates as and when they become available.

Further, routine maintenance is defined as repair, when necessary, replacement

Scheduhng of routine maintenance 13

of parts in short overhaul, so that after maintenance the equipment is b'as new" and has the same life to failure distribution as before maintenance.

The routine maintenance on the machines is assumed to be performed during the periods when the machine is idle, i.e., there is no loss of production due to the machine being down for routine maintenance. For the purpose of analysis of the failure data it is assumed that the machine is considered as a single unit, rather than as a complex system.

The material costs are not included in determining the costs for routine mainte- nance or breakdown, since these costs do not have direct bearing on the scheduling decision. Fur thermore, the planning of maintenance is performed at an aggregate level, typically on a monthly rather than on a day-to-day basis.

It is also assumed that the behavior of each equipment is economically and sto- chastically independent of every other equipment. Economic dependence means that the costs of maintenance for one equipment depends on whether it was maintained separately or jointly with other equipment, the sum of separate maintenance costs exceeding the joint cost of maintenance. Stochastic dependence can occur if the prob- ability of failure of one equipment may be affected by the failure rate of another equipment.

PROJECTION OF DOWNTIME COSTS

One of the most important factors to be considered is the effect of maintenance work on production. Any disruption of the production work due to unscheduled repair and emergency breakdown invariably results in the equipment being down when re- quired for production. The cost incurred due to this is the cost of downtime of the equipment.

In the model described later, the cost of downtime is used to link and integrate the product ion schedule with the maintenance schedule. The average cost of downtime is a function of the production schedule. The variables affecting the cost of downtime for any one machine are described below.

Let N = number of different products produced on a machine in period j. (7, = value of product i produced on the machine in period j.

Q,j -- Quanti ty of product t produced on the machine in period j. t, = time taken by product i on the machine in hours/unit.

td = mean downtime in hours.

The variables are defined with respect to a single machine and will vary from machine to machine.

The total production load on the machine in period j is given by

,%,

H , = ~ t, x Q,, t - - I

Each product contributes to the total load on the machine in each period. The contribution of the i 'j' product to production m period j on the machine is given by

t, × Q,, K I I - -

m n l , ~ \

where H . . . . is the maximum permissible load on the machine in period j. The average value of production m period j is

P, = ~ , K,, x C, I - - I

14 S. JosI-II and R Guvra

Thus the value of production is obtained as a weighted average of all the products in that period on the machine. This is used to determine the cost of downtime. It should be noted that

K,j = 1 if the machine is running to 100% capacity K , < 1 if the machine is underloaded K,, > 1 if the machine is used for more than Hm~, hours

The average cost of production computed is higher during periods of greater de- mands and lower during slack periods, thus allocating a higher cost to downtime in periods of higher demands and lower cost during slack periods.

The average production rate in period j is given by

H/ R/ - ,v

2 Q,,

The average cost of downtime in each period is then computed as follows:

Dj = average cost of downtime m pe r iod j Dj = Pj x (ta/R,)

It is assumed that the quantity of products to be produced in one period is com- pleted in the same period. When a machine breaks down, the production that would otherwise have been available from the machine is assumed to be lost, that is, it cannot be reclaimed by subcontract ing, over t ime or by production at a later period. In actual practice the lost production can be reclaimed by a variety of actions as outlined above. These actions may vary f rom situation to situation and are dependent on extrinsic factors not directly affecting the maintenance scheduling problem. In addition, the model does not consider the value added to the part due to operat ions performed on it.

ESTIMATION OF THE PARAMETERS OF WEIBULL DISTRIBUTION

The Weibull distribution is widely used m reliability models. In the model described in this paper ~t is assumed that time between failures is a random variable having a Weibull distribution. The Weibull cumulat ive distribution is defined by

F(x) = I - e x p [ - ( x - a/b)'] f o r x > a a n d b , c > 0

= 0 otherwise

where

b > 0 is the scale parameter c > 0 is the shape parameter a is the location parameter

Since it is reasonable to assume that an item is subject to failure from the instant it is put to use, it can be assumed that a = 0. When deahng with failure rates, the shape paramete r describes the mode of failure. Thus for ¢ = I the failure rate is constant over time, a value c < 1 indicates that the failure rate is a decreasing function of time, while c > 1 means that the failure rate is increasing with time. The method proposed by Thoman, Bain and Antle[8] was used to est imate the parameters of the distribution.

Scheduhng of routine maintenance 15

MEAN DOWNTIME

The mean downtime is determined from the past history of the machine break- downs. The downtime includes the active repair time and the idle time before the machine is restored for operation.

d, = downtime for i 'h repair Nh = total number of past breakdowns Mean downtime = ta = ~,N"I (d , /Nh)

MEAN TIME TO REPAIR

When a machine fails, it is taken out of operation and repaired. More than one person may be assigned to the repair task. The elapsed time for repair is the time taken to complete the task. The actual manhours put into the repair may be different from the elapsed time, depending on the number of persons assigned to the task. The model described here uses actual man-hours to repair rather than the elapsed time.

r, = time to repair the i 'h breakdown of a machine M, = number of persons assigned to ?h repair Total manhours required to repair the machine = r, × M, Nr = total number of past repairs mean time to repair = tr = ~ , ~ (r, × M, ) /Nr

These variables are defined for each machine and will vary for different machines.

COST OF ROUTINE MAINTENANCE

The cost of routine maintenance consists of two components , the setup cost for routine maintenance activity and the labor cost. The setup costs results from the prep- aratory activities for performing the maintenance action and includes labor and ma- terials specifically required for setup. The manpower cost is the cost incurred to support the maintenance crew and is relatively fixed depending on the number of persons and their skill.

t .... = mean time for routine maintenance L = labor cost /hour C,,, = labor cost/routine maintenance C , , , = t .... × L

total cost of routine maintenance is given by C .... = C , et,o + C,,,

It is assumed that no downtime will be incurred during routine maintenance since it wdl be scheduled during the periods when the machine is idle.

THE MAINTENANCE SCHEDULING MODEL

T = time in years of the length of the planning horizon i = !. 2, 3 . . . . . N are the periods of equal intervals into which the planning

horizon is divided C,,,, = cost of routine maintenance assocmted with the equipment

D, -- average cost of contingency or emergency failure and downtime due to unscheduled repair m period i.

R, = rehabflity of the equipment inperiod i R, = I - exp( Y,/b)' assuming a Weibull distribution

16 S JosHiand R GuerA

Y, = total running time of the machine in hours, up to period i. since the last repair

p, = probabili ty of failure of the eqmpment in period i p , = I - R , x, = O, I variable .~, = 0 if routine maintenance not scheduled in period t

= I ff routine maintenance scheduled in period i

The expected cost of downtime in period i is

C, = D, × p,

The expected cost m period i ~s

E C , = C , + x , x C ....

The total expected cost of a maintenance policy ~s

,q

T E C = ~'~ (D, × p, + x, x C,,,,) I - - I

The decision variable in the above equation is the period or periods m which main- tenance should be scheduled. This decision will subsequently affect the probabdlty of failure which is a function of the period i in which the last routine maintenance was performed. The object ive is to determine the periods in which routine maintenance can be performed so that the total expected cost is minimized.

Let M be the minimum mterval between two successive routine maintenances. Once maintenance is performed on a machine, the machine is expected to be running for a certain amount of ume before it deteriorates to the point where repair could be necessary. The possibility of scheduhng maintenance within this interval is not included while evaluating the possible alternatives. It is reasonable to assume that if a machine needs maintenance more frequently, then other factors such as replacement of the machine, etc. . may have to be looked into.

Initially the first routine maintenance activity is scheduled in period A, then the subsequent routine maintenance activities are performed in period A + M , A + 2M.

. . . . A + nM such that A + n M is less than or equal to the length of the planmng horizon. The probabilities of failure and costs associated with the policy are calculated. The interval between the (n + 1)" and the n 'h routine maintenance is increased by one and the costs recalculated. The process is repeated until the last routine maintenance is performed m the N th period Next, the interval between the (n - 2) ''d and (n - 1)" maintenance is increased and costs computed. The process is repeated until all possible solutions are evaluated. The maintenance pohcy assocmted with the minimum cost is selected and maintenance is scheduled in those periods which gwe the minimum cost solution. The process Is repeated for all machines m the system until the main- tenance schedule ts completed

The next step is to determine the manpower reqmrements to maintain the schedule. The maintenance schedule can be broken down into the number of man hours required m each period.

Let W be the max imum working hours of a maintenance person m a period

to,,,, = mean time for routine maintenance of machine t

n, = total number of machines to be maintained in period j total repairing time reqmred m period j

= ~ ' ? , t, . . . . .

number of persons required in period j = M, = ~ " - - , It,,,,,,/W)

Scheduling of routine maintenance 17

the minimum number of men required to maintain the schedule is MAX{M, + } where M, ÷ is M, rounded up to nearest integer.

To determine the best maintenance schedule it is necessary to not only minimize the expected cost of breakdown, but also to minimize the total cost associated with maintaining the schedule obtained by the model. A certain number of persons would be reqmred to maintain the schedule. This manpower need is based on the optimal schedule, but it may not be the optimum schedule for the manpower planning since the work loads may not be level in each period resulting in idle time in some periods. To solve this problem the total costs associated with the maintenance policy are reev- aluated. The total relevant system costs over the planning horizon for a maintenance policy ~s the sum of the expected costs of maintaining the machines according to a particular schedule and the cost of idle manpower. An interactive program was written in BASIC which allows the user to vary the maintenance work load in each period, and simulate the effect on the total cost in order to obtain the best possible combination of schedule and manpower requirements.

COMPUTER MODEL

Maintenance management and control involves an appreciable number of data rec- ords. The need for such voluminous, current information makes maintenance man- agement suitable for computer implementation.

The model discussed here has been implemented on a North Star Horizon micro- computer , and the programs written in BASIC[3]. The information needed by the model was accessed from the database created for the model. The database was designed

PRODUCTION I ROUTING I PART I MACHINE SCHEDULE I I INFORMATION NFORMATIO~ IINFORMAT'ONI

ESTIMATE COST

MAINTENANCE I SCHEDULING ]

MODEL I

INFORMATIONI I HISTORY ][BREAKDOWNI

ESTIMATE DISTRIBUTION PARAMETERS, MEAN TIME TO REPAIR, MEAN DOWN TIME

J MANPOWER PLANNING J

FINAL ] SCHEDULE I

I AND I

Fig I Flow chart fol computer model

18 S JosHl and R. GUFF~

using a relational approach and consisted of a set of files containing the production schedule, routing information, part information, machine mnformauon and equipment breakdown history. These files form the tnput to the programs to estimate cost of downtime and projected hours of use and estimate distribution parameters. The output of these programs is input to the maintenance schedule generation program. The main- tenance schedule generated is input to the manpower planning program to determine the final schedule and total costs (Fig. i).

In actual practice, the relevant data can be obtained from the company's corn-

Table I Production plan

part number

Perlod 00001 00002 00003

1 1500 1000 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 1500 --- 1500 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . . . . . . . . .

. . . . . . . . . . . | . . . . . . . . .

4 2 0 0 0 2 0 0 0 . . . . . . . . . . . i . . . . . . . . .

5 I 1500 1000 1000

. . . . . . . . . I . . . . . . . . .

00004 I 00005

--- I 1000

1500 I ---

1000 I 2000

- - - I lO00

lOOO 5 0 0

6 I 1000 500 2000 2000

. . . . . . . . . . . I . . . . . . . . . ' . . . . . . . . .

7 I 2000 2000 2000

8 I 1500 500 500

9 I 2 0 0 0 1 0 0 0 - - - 5 0 0 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 I 1000 - - - 500

11 I 1000 ---

12 I 1000 ---

. . . . . . . . . . . I . . . . . . . . .

500

500

1000 ---

1000 1000 1000

Table 2 Results for machine # 1000

P e r z o d H o u r s o f

u s e

I 2 2 5

2 150

Cost of

Downtime

Prob. of

failure M=3

522.6 .121

735 .192 . . . . . . . . , . . . . . . . . . . i . . . . . . . . . .

3 2 5 0 2 6 0 . 4 . 2 6 6 . . . . . . . . , . . . . . . . . . .

4 3 0 0 7 9 3 . 3 •

5 2 2 5 7 3 4 . 2 . 0 8 3 . . . . . . . . , . . . . . . . . . . i . . . . . . . . . .

6 175 7 7 0 . 1 5 8

7 100 2 8 0 . 6 • 2 1 0

8 150 484.2

9 225 7 6 5 . 3

10 50 2 8 0

11 50 2 8 0 . . . . . . . . , . . . . . . . . . . i . . . . . . . . . .

12 200 4 5 1 . 5

0

• 0 5 6

• 1 0 6

. 1 2 5

• 1 7 2

. . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . .

This table suggests maintenance of the machine should be scheduled at the start of period 4 & 8 Total expected cost for the maintenance pohcy = 1253 3

Scheduling of routine maintenance 19

prehensive production database. The programs are written in a modular manner and can be easily incorporated into a larger maintenance management and control system.

E X A M P L E PROBLEM

To dlustrate the model a simple hypothetical example ~s presented. The hypo- thetical system is assumed to constst of five machines, and five different parts are produced on these machines in varying quantities in each period. The planning horizon is assumed to be one year and Is divided into twelve periods of one month each. Each

Table 3 New produchon plan

p a r t nuraber l . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I P e r x o d 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5

I 1 5 0 0 1 0 0 0 1 0 0 0 1 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I 2 1500 --- 1500 1500 --- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 . . . . . . . . . 1000 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 --- 2000 . . . . . . 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 1500 1000 1000 1000 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 1000 500 2000 2000 --- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 2000 2000 - - - . . . . . . . . . . . i . . . . . . . . . i . . . . . . . . . i . . . . . . . . . ~ . . . . . . . . . i . . . . . . . . .

8 500 --- 500 500 500

9 2000 1000 --- 500 500

10 1000 . . . . . . . . .

11 1000 --- 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . . ;

12 1000 --- 1000 1000 1000

Table 4 Results for machine # 1000.

Perxod Hours o f Cost o f Prob. o f use Downtxme f a x l u r e

1 175

2 150

3 2 5 0

4 2 0 0

2 8 0 . 1 1 2

7 3 5 . 1 7 3

2 6 0 . 4 . 2 4 8

2 9 4 . 3 2 8

5 2 2 5 7 3 4 . 2 •

6 175 7 7 0 . 0 6 1

7 1~0 2 8 0 I

8 100 2 2 5 . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

9 2 2 5 7 6 5 . 3

10 5 0 2 8 0

• 1 1 1

• 149

0

• 0 3 9

11 5 0 2 8 0 . 0 5 6 . . . . . . . . . . ! . . . . . . . . . . . . . . . . . . . . .

12 2 0 0 4 5 1 . 5 . 1 0 2 . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This table suggests maintenance of the machine should be scheduled at the ~tart of period 5 & 9. Total expected cost of the pohcy = 1055 36

20 S JosH~and R Gt~r'TA

machine is assumed to have a ddTerent failure distribution. The shape and scale pa- rameters of the We~bull dtstrtbutton ~s computed from the t~me between fatlure data obtamed from the machine breakdoss n h~story.

The production plan regarding the quantity of each part to be produced m each period ts g~ven m Table 1. The data regarding the routing mformat~on, cost of part, etc., Is accessed from the production database created for the example problem. It is assumed that before the start of the planning, the machines under consideration have been running for some t~me and tile mformatton on how many hours a machine has run prior to the start of the planning period ~s also obtained from the database.

The results for a machine m the sys tem are shown m Table 2. The proce,,s ~s repeated tbr all machmes m the ~ystem to obtain the maintenance schedule for all machines m the producuon system.

Smce this model integrates the production plan and the maintenance plannmg function, it is expected that a change m production schedule should affect the main- tenance decision. Using a different productton schedule as shown m Table 3 the new production mamtenance pohcy for the machine # 1000 ~s shown m Table 4.

DISCUSSIONS AND C O N C L U S I O N

From the results obtamed ~t can be seen that the model ts sensitive to changes in the production schedule. Any change m the production schedule results in a change in the hours of use, and hence in the probability of failure. It also results in changes in the cost of downtime, thus resulting in a different expected cost and possibly in a new schedule. Smce the production schedule is dependent on other plannmg activities such as capaci ty planning and master production scheduhng, the maintenance planning model may have to be run several times during the planning horizon to incorporate the changes made to the input paramete r of the model.

The choice of m i m m um interval between successwe routine maintenance is a man- agement decision. It is generally desired that once routine maintenance is performed on a machine the machine would be running for a certain amount of time before it deteriorates to the point of repair. The time intervals between two successive mainte- nance actions also reflects the extent to whtch a machine is over or under maintained. Too short an interval would tend to indicate that the machine needs repair frequently and would necessi tate looking into other factors such as the replacement of the machine, etc. In the model the minimum mterval ~s input as data and any possibility of scheduling the maintenance w~thin this interval ts not included whde evaluating the possible alternatives.

The maintenance scheduhng model has been designed to be independent of the methods of estimating the cost of downtime, probabd~ty of failure and average ttme to repair. The relevant data needed for the computat ions can be accessed from a production database or a small data base can be created as was done to test the model.

R E F E R E N C E S

I R L Bovlard, Characteristic, , of optimal mamtenence pohcle,, ~,lanagement 5(t 7, 238-253 (1961) 2 A S Corder. Maintenance Managemen t Techmque~, McGraw Hdl, New York(1976) 3 S Joshl. "'A Microcomputer Based Sys tem for Scheduling of Preventwe Maintenance '" Unpubl ished

mas te r ' s thesis. S UNY at Buffalo. Buffalo. New York (1983) 4 L Mann, Jr . M a m t e n a m e Manal,,ement. Lexington Books. D C Health and Co Lexington. MA (1976) 5 J J McCall. Maintenance pohcles fol ,;tochastically fading eqmpment a surve} Management .$~ 11,

493-524 ( 19651 6 W P Plerskalla & J V Voelker. ,k survey of maintenance models the control and survedlance of

deteriorating sys tems Naval Re~eal~ h Logt~tt~ ~ Qmute~lv 22, 353-388 I1976) 7 Y S S h e n f & M L Smith. Optimal maintenance models for sys tems subject to f a d u r e - - A review Naval

Re~eat~ h Loet~tt( ~ QttalteJIv 28, 47-74(1981) 8 D R Thoman . L J Barn & C E Antle. Inferences on the parameters of the W~ebull dtstrtbutlon

Te( hnomeo tc s l 1 ,445-460 (1969) 9 E Turban. The complete computer ized maintenance s} s tem Indust tml Engmeettnt~ March. 20-27 (1969)