A multicriteria approach to the selection of preventive maintenance intervals

-l-of

production economics Int. J. Production Economics 49 (1997) 55-64

ELSEVIER

A multicriteria approach to the selection of preventive maintenance intervals Chaichan Chareonsuk, Nagen Nagarur*, Mario T. Tabucanon Industrial Systems Engineering, Asian Institute of Technology, G.P.O. 2754, Bangkok, Thailand

Received 18 January 1996; accepted 29 October 1996

Abstract

This paper discusses the problem of determining optimal preventive-maintenance intervals for components in a production system. The maintenance planning of a paper factory is taken as a case study. The traditional approach of a single objective function for this type of a problem is described, and some of the disadvantages of taking such a single objective are discussed. The paper proposes a new model incorporating multiple criteria. For the case study, two criteria, expected costs and reliability, are taken into consideration. The MCDM method PROMETHEE is used to solve the problem. Sensitivity analysis is carried out for the variations in the subjective weights

assigned to the criteria. Keywords: Preventive maintenance; Scheduling; Multicriteria; PROMETHEE

1. Introduction In general, maintenance may be considered to be of two categories. One is failure maintenance (FM) where the maintenance is undertaken only after the equipment has failed. The other is preventive maintenance (PM) which is undertaken while the equipment is still in operating condition, so as to prevent or reduce the probability of a failure. Maintenance planning involves determining a proper level of preventive maintenance. As production systems move towards more advanced, and hence more expensive technologies,

*Corresponding author. Tel: 66-2-524-5653; Fax 66-2-524-5697; e-mail: [email protected].

proper maintenance planning becomes more important. Direct maintenance costs are increasing because of high costs of components and technical support that are needed for the advanced technologies. In addition, downtime due to system breakdown also has become expensive. Hence, a sound maintenance planning becomes imperative in a modern production system. But decision making for maintenance planning is often problematic. Maintenance is a support function of production and traditionally there exist conflicting interests between these functions. In addition, the marketing department may also like to get involved in the planning to retain their customers and markets. This necessitates maintenance planning in a multicriteria environment.

0925-5273/97/%17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PII SO925-5273(96)001 13-2

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C. Chareonsuk et aLlInt. J. Production Economics 49 (1997) 55-64

Although multiple criteria decision making (MCDM) models have been used in many applications in management science [l, 21, only a very few of such models can be found in the field of maintenance [3,4]. Traditionally the models in maintenance have dealt with a single criterion

intervals are usually dictated by the practices in the past, or taken as equal to mean time to failure. In addition, there is no monitoring of aggregate costs related to maintenance.

C&61.

3. The conventional approach to PM intervals

This paper describes the planning of preventive maintenance intervals for a production system, in a multicriteria environment. The work was inspired by an actual case study at a local paper company. This article will present a general methodology for decision making in a similar context, using the case study as an illustrative application.

2. Description of the company The company under consideration produces several kinds of paper. It has four lines of paper machines, the maintenance planning was undertaken for one of the lines. Paper making is a continuous process for a given batch, so if any of the main equipment fails, it would stop the whole production. Reliability of the equipment hence is a major concern. The production units run 24 h a day. Currently, there is no proper systematic data collection or maintenance planning at the factory. Preventive maintenance

The conventional quantitative approach to the determination of PM intervals (tJ is to look at the total expected cost for a given planning horizon and fix the interval that can minimize it. If the system exhibits aging characteristics, that is, if the failure rate is increasing with age, and if the failure maintenance costs are larger than preventive maintenance costs, then there exists a finite value of preventive maintenance interval that will yield the minimum costs. Conceptually, the behavior of the costs with respect to t, is shown in Fig. 1. If preventive maintenance level is high, i.e. if t, is small, the failure maintenance costs are low. As preventive maintenance costs are decreased by increasing t,, failure maintenance costs go up. Hence, there is a need to strike a balance between these two. Usually, failure maintenance costs are higher than preventive maintenance costs because failure maintenance is an unscheduled, and sometimes, unplanned event. Since preventive maintenance is

Cost of Maintenance

Cost of lost time due to breakdown

Maintenance Frequency Fig. 1. The cost comparison

between

failure maintenance

and breakdown

maintenance.

C. Chareonsuk

et al./lnt. J. Production

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49 (1997) 55-64

production and maintenance management to look at the problem at the aggregate level and come to a consensus plan. In the case study, such a criterion was developed for solving the problem. The mathematical details are presented later.

a scheduled event, there is a scope for finding an opportune time to keep the costs low. In computing the costs, the maintenance manager tends to consider the direct maintenance costs only. This practice is convenient in view of traditional compartmentalized responsibilities and accounting procedures. However, in production systems, particularly in continuous processes, the major related costs may be due to production losses during downtime. Ignoring these costs may prove to be very costly, or may jeopardize the maintenance plans. If the preventive cost per maintenance is not high, the maintenance manager schedules maintenance at very frequent intervals to keep the equipment at a high level of reliability. But if each maintenance is going to take, say about 2 h, then the production department will be reluctant to shut down the equipment for preventive maintenance. They will either force the preventive maintenance to be postponed, or to be scheduled when there is no production, for example in the night, which in turn would be inconvenient to maintenance people. So there is a need to integrate maintenance and production activities to consider maintenance scheduling in a multicriteria environment. One way to deal with the above multicriteria problem is to express the criteria in terms of costs and develop a single objective. This can be achieved by including the production losses due to maintenance downtime in the model. This will force both

3.1. A critical review

ofthe

conventional

approach

However, a drawback of an aggregate, composite objective function is that the individual criteria, and the corresponding risks lose their identities. They are not directly visible, and in the model they are dealt with in an indirect manner. The decision makers are not comfortable with such a model when the costs and risks are high. Such a scenario is illustrated in Fig. 2. The function of total costs, including the downtime costs, is convex with respect to t, . The decision would be to choose optimum t, as the interval as it minimizes the total costs. However, one can observe that the reliability of the system goes to a very low level before a preventive maintenance is undertaken. Sometimes the production managers, as it happened at the case study company, are reluctant to use such a model. Even though the reliability was built into the objective function, the manager insisted on having a direct control on it. There could be many reasons for such a demand. Managers would like to see

f

Reliability

Time Fig. 2. The relation between expected cost and reliability.

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C. Chareonsuk et al./Int. J. Production Economics

their system run as planned, and an unscheduled event such as a machine failure will disrupt the smooth running of the plant. Sometimes, the marketing department brings emergency product orders for important customers and a system failure may result in severe losses. Or, failures force the management to come up with contingency production plans and schedules, particularly in systems where the products are run in batches. It is anyway impossible to capture all these effects in a cost function. In addition, failures, even if their effects are taken into account, are not good for the morale of the workforce. Hence, the manager would like to keep reliability as a separate criterion. Similar arguments can be constructed for individual criteria for spare parts inventory, quality issues, maintenance downtime, etc. This brings us to the domain of multiple criteria decision making in maintenance planning.

49 (1997) 55-64

to minimizing expected costs per unit time (E[C(t,)]). Using renewal theory and renewal reward function. it can be shown that

mc(t,)l =

expected costs per cycle expected cycle time

= CpR(4J+ CP(4J

s

@(qJ + t.f(W 0

where Cf is the cost of unit failure maintenance, C, the cost of unit preventive maintenance, and f(t), F(t) and R(t) are the probability density distribution, cumulative distribution and reliability function of the component lifetime, respectively. The above expected costs can also be expressed in units of C,, E(C) = ---= awl

CP

4. Multicriteria for maintenance activities Depending on the type of production system, demand pattern, type of production philosophy, etc., several criteria can be identified in the domain of maintenance activities. Some of these are reliability, downtime, production loss, quality loss, expected total costs per unit time and spare parts inventory costs. For the case of the paper company, the maintenance and production managers selected two criteria: Criterion 1: Expected total costs per unit time. Criterion 2: Reliability. The conflict between these two criteria is illustrated in Fig. 2. The problem is to fix the PM interval t,. A given component will have maintenance whenever it fails or reaches the age t,, whichever comes first.

4. I. Criteria The first criterion which represents the total expected costs has been extensively analyzed and reported [S, 61. Minimizing this cost is equivalent

(1)

3

f

Wt,) + @(&J f Wt,)

>

(2)

+ /-VW 0

where k = Cf/C,. The lifetime probability distribution functions are obtained by fitting an appropriate distribution function for the data. For equipment which deteriorates with time, Weibull distribution usually proves to be a good fit. For the case study, a threeparameter Weibull distribution was assumed. The data and the distribution function are shown in Table 1. The parameters of the Weibull function indicate that the corresponding failure rate increases with the age of the component. The optimal solution to Eq. (2) is given by taking the derivative with respect to t,, equating it to zero and solving for t,. Unfortunately, for Weibull and other distribution, it is not possible to obtain analytical solution, and a solution is obtained by numerical procedures. However, t, is not strictly a continuous variable, in real life it is a discrete variable. And as will be explained later, we are more interested in the behavior of the cost function, in particular at the optimal point, rather than in the optimal point itself. So E[C(t,)] is plotted for various discrete

C. Chareonsuk et al. /Int.

Table 1 Failure data and estimated

Weibull

parameters

Failure

Weibull

distribution

411 32 30 70 82 70 82 70 72 71

data (days)

83 22 100 46 48 29 60 59 22 16

41 57 54 33 11 15 15 38 45 18

122 24 15 28 26 20 33

Estimated /I= 1.8 ‘I = 50.0 1’= 0

parameter

J. Production Economics 49 (1997) 55-64

values

values of t,. To make it more general, and to give more flexibility to the managers in the cost assumptions, the curve is plotted for different k. The curves are shown in Fig. 3.

59

The following observation are made about the curves. Observation 1. The curves are somewhat flat around the minimum points. This gives the maintenance manager a comfortable “window of maintenance interval”, during which the preventive maintenance can be undertaken without too much deviation from the point of minimum costs. The manager need not be concerned about staying very close to the minimum point. Observation 2. The costs per unit time are very high at small values of t,. The curve assumes flatness just before the minimum point, and maintains it for quite some time after this point. The second criterion, reliability, computed for the given Weibull distribution, is plotted against time. This curve can be seen in Fig. 3, along with the curves representing the expected costs. The two criteria are totally different in units and hence need a method that can solve such multiple criteria problems.

1.0

500 450 400 350 300

0.6

250

0.5

200

0.4

150

0.3

if 3

0.35lowerbound

100 SO 0

I” 0

0

‘I

0

10

I

I

‘II

0

bou% lower l2

(

’

I “I 30

‘I”3 40

‘II

3

SO

“If

1 60

&X~Y~)

Fig. 3. Expected

cost and reliability

for the given lifetime distribution.

I

I 70

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C. Chareonsuk et aLlInt. J. Production Economics

4.2. MCDA4 tool - PROMETHEE The method PROMETHEE is used for the MCDM. PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) is conceived by Brans [7] and is one of the outranking methods for multiple criteria problems. The method and its applications have been described in a few recent papers [8, 111. The PROMETHEE is used for problems of the type Max{fi(x),fi(x),

. . ..fk(x)Ix~A).

where A is a set of decision alternatives, andJ(x), i=l , . . ., k is the set of criteria for which each alternative is to be evaluated. The ranking of alternatives is carried out by pairwise comparison of the alternatives for each criterion. The comparison is measured using a predefined preference function. For a preference function P, alternatives a and b, and criterionj, Pj(a, b) = Pj[dj(Q,

bN>

(3)

where dj(a, b) =fj(a) -f,(b) gives the difference in measurement for a criterion j. The PROMETHEE gives a choice of six generalized criteria to define the preference function. These generalized criteria are shown in Fig. 4. The aggregate ranking or preference of the two alternatives is determined by summing up the weighted values of the preference functions of the complete set of criteria. That is the overall measure is given by II(u, b) = i wj Pj(a, b),

(4)

j=l

where Wj is the weight given to criterion j. weights are obtained from the decision maker they are normalized to sum up to unity. If the number of alternatives is more than the overall ranking is done by aggregating measures of pairwise comparisons. For each alternative a E A, the following two ranking dominance flows can be obtained respect to all the other alternatives x E A: $‘(a) = &

XTAZl (a, x)

(leaving flow).

The and two, the outwith

(5)

49 (1997) 55-64

The leaving flow is the sum of the values of the arcs leaving node a and therefore provide a measure of the outranking character of a. The higher, the 4+(u), the better the alternative a.

C(a) = -&

1 n(x, a)

(entering flow).

(6)

XEA

The entering flow measures the outranked character. The smaller $-(a), the better the alternative a. The complete ranking of the set of alternatives is obtained by computing, for each alternative, the next outranking flow given by 4(a) = 4+(a) - 4-(a)

(7)

The higher the net flow, the better the alternative. This is the PROMETHEE II version of the method.

4.3. Alternatives The alternative decisions are the set of the preventive maintenance intervals. The candidate list is prepared on the basis of current maintenance practices, certain system constraints, and the behavior of the cost structure and reliability functions. The list went through several iterations, the process of which is briefed below. A current practice of the company is to shut down the plant every ten days for cleaning and maintenance. This is a technical requirement, to keep the process free of fungi and other contaminants. It is not unusual to also undertake some maintenance activities during this shutdown. It is desired that the planned PM activities also avail of this opportunity. So the decision alternatives are kept in multiples of 10 days. For the reliability, it is felt that its value should not go below 0.35. This is taken as the lower bound for the reliability. In the region where the system exhibits a high degree of reliability, that is at low values of t,, the total cost tends to be very high (Observation 2). Hence, this region is taken out from the solution space. The lower bound on t, is fixed at 12 days. Following the above procedures, as can be seen from Fig. 3, the solution space is reduced to four

C. Chareonsuk

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49 (1997) 55-64

Fammeten

Types of generalized criteria

I. Usual criterion H(d)=

,;,=>OO

\ 0 1

--j-=

d

-

II. Quasi-criterion H(d) =

0

Idl<=q

ld1>q

1

fid

’ ._.-

III. Criterion with linear preference P H(d) =

I d I/ P

d

Idi<=p Idl ’ P

1

0

P

IV. Level criterion H(d) =

I 0 / l/2 I I

ldl<=q I <= p Idl ’ P

9. P

q < (d

d 0

V. Criterion with preference and indifference

P

linear area

Idl<=q

I 0 H(d)=

q

IdI-q

q<=idi<

9. P

;

P-9

ji

-

ldl ’ P

I

oq

P

I

VI. Gaussian criterion

H(d) =

1-

s

-d’ls=

63

Fig. 4. The six types of generalized

alternatives Al : 20 A2: 30 A3 : 40 A4: 50

of t, values: days days days days

The pruning of the solution set was a result of many discussions carried out with the decision makers. Some of the constraints could have been directly included in the model as the PROMETHEE

criteria

[7]

has the capability to take care of certain types of constraints (PROMETHEE V; [12]); however, it was felt that the involvement of decision makers in the determination of the set of alternatives will facilitate the implementation of the result of the model.

4.4. MCDM problem

solution

The PROMETHEE method ponding software PROMCALC

and its the corresgive the ranking of

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C. Chareonsuk et al./Int. J. Production Economics 49 (1997) L-64

the alternatives for the model, once all the parameters and the values are presented. For the case study, the criteria and the alternatives have already been described. The generalized criteria for the pairwise comparison need to be defined. The generalized criterion of type 3 was selected for the first criterion, the expected costs. The selection was made because the difference in the total costs moves in a narrow range for the different alternatives. It was felt that a preference function that is proportional up to a relative cost difference of 10 units (p-value), and is unity thereafter is appropriate. In the case of the second criterion, reliability, the generalized criterion of type 5 was chosen, with its parameters 4 set at 0.01 and p at 0.15. The decision makers thought that very small differences in the reliabilities, up to 0.01, can be ignored. After that the preference function can be assumed to be proportional for the range of low values of not more than 0.15. Any difference greater

The

PFIWR~E

t&:Cktimr/Critoria

II

N-t

Walking

than 0.15 would be significant and the preference function would be set equal to 1.0. The problem is now solved using the software PROMCALC. Fig. 5 shows the output screen for a k value of 4. The alternative 30 days is listed as the first preference. Since a solution of wider applicability was desired, the two criteria were evaluated for various values of k, and for different sets of intervals of preference weights for the two selected generalized criteria. The results are shown in Table 2. The decision makers can obtain a solution for chosen sets of cost values and preference weights. These results also provide an opportunity to analyze the sensitivity of the decisions or rankings to these parameters. From the above results it can be seen that, for a given cost ratio, as the weight of the first criterion, the expected total costs per unit time, increases, the top preference tends to change from lower values of

Yciets

Flow

<*.-),(*,/):Scnsitiuity

Fig. 5. PROMCALC

output

screen with rankings

F1:Helo

of the alternatives.

tEsc:nalu

C. Chareonsuk et aLlInt. .I Production Economics

Table 2 The preference

Ranking

of Alternatives

Preference weight for total cost

Preference weight for reliability

Preference sequence

k=4

co,0.41 (0.4, 0.63 (0.6, 0.83 (0.8,I.01

[l.O,0.61 (0.6, 0.41 (0.4, 0.21 (0.2, 0.01

Al>A2>A3>A4 A2>Al>A3>A4 A2>A3>Al>A4 A2>A3>A4>Al

k=6

ro, 0.41 (0.4, 0.61 (0.6, 0.81 (0.8,I.01

[l.O,0.61 (0.6, 0.41 (0.4, 0.23 (0.2. 0.01

Al >A2>A3>A4 Al>A2>A3>A4 Al>A2>A3>A4 Al >A2>A3>A4

k=8

co, 0.41 (0.4, 0.61 (0.6, O.S] (0.8,I.01

[l.O,0.61 (0.6, 0.41 (0.4, 0.23 (0.2, 0.01

Al >A2>A3>A4 Al>A2>A3>A4 Al>A2>A3>A4 Al>A2>A3>A4

IO

co,0.41 (0.4, 0.61 (0.6, 0.83 (0.8,1.01

[l.O,0.61 (0.6, 0.41 (0.4, 0.23 (0.2, 0.01

Al >A2>A3>A4 Al>A2>A3>A4 At >A2>A3>A4 Al>AZ>A3>A4

k = 20

co, 0.41 (0.4, 0.61 (0.6, 0.83 (0.8,1.01

[l.O,0.63 (0.6, 0.41 (0.4, 0.21 (0.2, 0.01

Al>A2>A3>A4 Al>AZ>A3>A4 At>A2>A3>A4 Al>A2>A3>A4

k = 30

co, 0.41 (0.4, 0.61 (0.6, 0.81 (0.8,1.01

[l.O,0.61 (0.6, 0.41 (0.4, 0.21 (0.2, 0.01

Al >A2>A3>A4 Al >A2>A3>A4 Al >A2>A3>A4 Al>A2>A3>A4

days; A3,40

days; A4, 50 days.

cost ratio

k=

Note: Al, 20 days; A2,30

for various

weights

49 (1997) 55-64

63

interval, for chosen values of k and preference weights.

5. Conclusions

PM interval to higher values. For example, at k = 4, for the preference weight of [0,0.4] range for the cost criterion, the first preference is to keep the PM interval at 20 days. At the same level of k, when the weight is increased to (0.4,0.6] range, the first preference is to keep the maintenance interval at 30 days. When the value of k increases, the preference order changes to favor lower values of PM interval, so as to avoid the higher costs of failure maintenance. At k = 4, for a preference weight of (0.4,0.6] for the cost criterion, the first choice is a maintenance interval of 30 days, whereas for the same range of weights, but at a k value of 6, the first preference is an interval of 20 days. Using Table 2, the decision maker can make an appropriate choice of PM

Maintenance, like any other operations management entity, requires its decisions to be made in a multicriteria environment. It needs coordination between various functional groups like production and maintenance, since it is a support function of production activity. In the present paper, which follows a case study at a paper factory, such an illustrative scenario is presented, clearly showing the need for cooperation between production and maintenance. For the optimal preventive maintenance interval, the shortcoming of a single objective function aggregating all the different criteria is discussed. The managers would like to keep reliability as a separate criterion, for better control. The problem is reformulated as a multicriteria decision-making problem with two criteria: expected total costs per unit time, and reliability of production system. The MCDM problem is solved by the PROMETHEE method. The method, with its pairwise comparisons and its choices of generalized criteria for the decision-making criteria proved to be easy for understanding and applications among the decision makers. The advantages of MCDM methodology and the PROMETHEE method were that the decision makers could input criteria of their interest into the model as separate entities. The impact of the decisions on these criteria could be perceived directly. They could also examine the sensitivity of the decisions to the changes in the subjective weights given to the criteria.

References [l]

Saaty, T.L., 1980. Analytical Hierarchy Process, McGrawHill, New York. [2] Tabucanon, M.T., 1988. Multi Criteria Decision Making in Industry. Elsevier, Amsterdam. [3] Golabi, K., 1983. A Markov decision modeling approach to a multi-objective maintenance problem. Essays and Surveys on Multiple Criteria Decision Making, SpringerVerlag, Berlin, 115-125.

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[4] Jedrzejowicz, P., 1988. An overview on available models

[S] [6] [7]

[S]

and techniques for the multicriteria reliability problem with emphasis on a potential use in a DSS for this problem type. Ann. Oper. Res., 16: 413-424. Barlow, R. and Hunter, L., 1960. Optimum preventive maintenance. Oper. Res., 8: 90-l 10. Jardine, A.K.S., 1973. Maintenance Replacement and Reliability. Pitman, USA Brans, J.P., Vincke, P.H. and Mareschal, B., 1988. How to select and how to rank project: The PROMETHEE method. European J. Oper. Res., 24: 228-238. Briggs, T., Kunsch, P.L., and Mareschal, B., 1990. Nuclear waste management: An application of the multicriteria PROMETHEE method. European J. Oper. Res., 44: I-10.

Economics 49 (1997) 55-64

[9] Du Bois, D., Brans, J.P., Cantraine, F. and Mareschal, B.,

1989. Medicis: an expert system for computer-aides diagnosis using the PROMETHEE multicriteria method. European J. Oper. Res., 39: 284-292. [lo] Brans, J.P. and Mareschal, B., 1992. PROMETHEE V: MCDM problems with segmentation constraints. INFOR, 30(2): 8596. [ll] Pandey, P.C. and Athakorn, K., 1995. Selection of an automated inspection system using multiattribute decision analysis. Int. J. Prod. Econom., 39: 289-298. [12] Brans, J.P. and Mareschal, B., 1994. The PROMCALC & GAIA decision support system for multicriteria decision aid. DSS. 12: 297-310.