OPTIMAL PREVENTIVE AND REPLACEMENT POLICIES ... - CiteSeerX

Proceedings of the XXVI Iberian Latin-American Congress on Computational Methods in Engineering CILAMCE 2005 Brazilian Assoc. for Comp. Mechanics & Latin American Assoc. of Comp. Methods in Engineering Guarapari, Esp´ırito Santo, Brazil, 19th –21st October 2005 Paper CIL0614

OPTIMAL PREVENTIVE AND REPLACEMENT POLICIES FOR INDUSTRIAL SYSTEMS WITH BUFFERS

R. Pascual [email protected] Department of Mechanical Engineering, Universidad de Chile, Casilla 2777, Santiago, Chile P. Rey [email protected] Department of Industrial Engineering, Universidad de Chile, Casilla 2777, Santiago, Chile

Abstract. This work proposes a general framework to study and improve the effectiveness of a production system with respect to its expected life-cycle cost rate. The model that we propose considers that the system is protected against demand fluctuations and failure occurrence with elements like stock piles, line and equipment redundancy, and the use of alternative production methods. These design elements allow us to keep or minimize the effect of equipment downtime on the nominal throughput, while palliative measures are taken. The system is also subjected to an aging process which depends on the frequency and quality of preventive actions. Decisions are difficult because there exist discontinuities in direct and downtime costs and there is a limited budget. We present a non-linear mixed integer formulation that minimizes the expected global cost rate with respect to repair, overhaul and replacement times and overhaul improvement factor. The model is deterministic and considers minimal repairs and imperfect overhauls. We illustrate its application using a known example from the literature. Keywords: decision-making, downtime cost, maintenance policies, repair rate, replacement,imperfect maintenance, decision support systems, consequential cost, budget constraint 1.

INTRODUCTION

When a production equipment fails or is overhauled, the costs that the company pays may be classified in two categories. In a first category, we consider the intervention costs, which include labor and materials. The second category corresponds to the downtime costs, which are composed by the cost of lost production as well as other consequential costs like reconfiguring alternative production lines, using less efficient methods, reduced product quality, lost raw material, and so on (1). Quantifying intervention costs is quite straightforward since standard accounting procedures register it. On the other hand, downtime costs may be hard to estimate due to the dependency on several exogenous factors like production rate, stock price, and system design parameters (redundant equipment, stock piles, alternative production methods). Estimating downtime costs may be much benefitting to maintenance decision-making for several reasons: (1) it allows to measure the impact of equipment on system efficiency;

CILAMCE 2005 – ABMEC & AMC, Guarapari, Esp´ırito Santo, Brazil, 19th – 21st October 2005

(2) it may be used to assess the effectiveness of maintenance policies (key performance indicator); (3) it allows the use of a series of mathematical models applied in decision contexts like replacement policies, preventive and condition-centered maintenance strategies, spares stock levels, and so on. This work studies the effects of downtime costs and budget constraint on maintenance decision-making. For illustrative purposes, we will assume that costs curves have been estimated previously, and we focus on the effect of these discontinuities on the decisionmaking related to set overhaul frequencies and quality, service and replacement times. The article is organized as follows: first, we present a literature review. Then, we state the problem and decide services times, interval between overhauls and replacement intervals. The model is tested with an illustrative example from Zhang and Jardine (2). Finally, we comment on the results and provide possible lines of research. 2.

LITERATURE REVIEW

An important amount of research in the area of reliability engineering considers the study of maintenance policies to prevent system failures and improve system availability. Maintenance modelling has been recognized as a growing subject and also as a late developer (3). The situation is partially explained because maintenance take actions on plant equipment and not directly on product. As a consequence, it is perceived as a marginal activity of the company. Scarf (3) and Dekker and Scarf (4) present very good reviews in the area of maintenance models, assessment of their impact and applications. Maintenance decision-making concentrates in general on costs, thus, it is necessary to analyze them. Komonen(5; 6) proposes a classification of costs related to maintenance considering two groups: (1) direct costs; that appear due to maintenance operations (administrative costs, labor, material, subcontracting), (2) lost production costs due to equipment failures and lost quality production due to equipment malfunctioning. This classification emphasizes two confronting maintenance objectives: maximize equipment availability while reducing maintenance related costs. Traditional cost models assume that lost production dominates the downtime cost and neglect discontinuities due to stock-piles exhaustion, lost raw material, etc. As they assume constant production rates and unit prices, downtime costs rise linearly with maintenance service time (see Jardine (7)). Vorster and De la Garza (1) segregate different sources for the consequential costs and propose a non-linear time-dependency of the downtime costs. They classify the consequential costs when a machine breaks down in four sub-component costs associated with lack of availability and downtime. Associated resource impact costs concern the effects of the failure on other components of the team. Lack of readiness costs accrue as a result of resources that could be used not being in a state of repair such that they can be used. Service level impact costs measure the decreased productivity of a fleet of equipment when a portion of that fleet has failed. Alternative method impact costs occur when a different method of production must be used due to the failure of a given component of the original production team. Mitchell (8) uses regression analysis to represent costs of equipment in terms of age. After identifying the model parameters, Vorster’s cumulative cost model is used to identify optimum economic decisions for repair policies, and overhaul and replacement intervals. Analytical results to quantify the downtime costs are difficult to obtain in general. For


example, Roman and Daneshmend (9) use simulation to study the effect of contractors on service level in open-pit mines. Cor (10) uses a similar strategy to quantify the time and economic impacts of work operational changes. Changes may include alterations to the sequence of work, to the design, or changed conditions. As a consequence, it may affect the quantity and type of the resources required to perform the works, and logically, on equipment idle time and usage. A complementary strategy considers the use of past information: Edwards et al. (11) use regression analysis to predict the expected downtime cost rate for tracked hydraulic excavators in opencast mining industry. Their metamodel shows that in that case, machine weight is an excellent predictor of downtime cost. Vorster and Sears (12) propose what they call failure cost profiles which measure the expected cost per unit time in terms of the duration of the interval out of service. For a fixed repair time, they introduce a cost related criterion that also takes into account the relative productivity of an equipment that may be assigned to different kind of works. By doing so, they are able to decide replacement and task assignments for a fleet of similar equipment. Operator-induced consequential costs are studied by Edwards et al. (13). The work includes skill level of operators, fatigue, morale, and motivation. Edwards accuses operator’s skill has the most important factor concerning the performance of the equipment. Considering operational poor practices, Pathmanathan (14) reports the increased frequency and cost of equipment downtime induced by the negligence of the operator and lack of proper training and knowhow on the part of equipment supervisor. Arditi et al. (15) also consider operation uncertainty (i.e. different environmental conditions) as well as design complexity as causes of greater risk for equipment downtime. Nepal and Park (16) explore the impact of downtime on construction projects duration and related costs. Their analysis highlights how various factors and processes interact with each other to create downtime, and mitigate or exacerbate its impact on project performance. Concerning preventive maintenance policies, Kobbacy et al. (17; 18) consider decisions for multicomponent systems. The problem is to determine the optimal maintenance preventive times which minimize the sum of down time due to both kind of interventions for a large set of components. A decision support system is used to carry out an automated analysis of the maintenance history data. Maintenance data are presented to the system and the most suitable mathematical model from a model-base is identified. Other approach considers the use of expert systems for maintenance project planning. Moynihan et al. (19) applies the concept to an automotive components manufacturer. In addition to the cost categories listed previously we may add other two, which we will consider constant in our model: spare holding costs and costs associated to investment on assets that are decided on maintenance criteria (see Jourden et al. (20)). Pham and Wang (21) review the literature on imperfect maintenance. They classified maintenance works according to the degree to which the operating conditions of an item is restored by maintenance as: a) perfect maintenance: the system is restored to an operating condition as good as new and has the same lifetime distribution and failure rate function as a brand new one; b) minimal maintenance: the system is restored but the failure rate is the same it had when it just prior to the failure; c) imperfect maintenance: the system is not like as good as new, but younger; d) worse maintenance: the system failure rate or actual age increases but the system does not break down. As a consequence, the operating


Downtime costs Cd (mu/intervention)

Cld,4

Cld,3 Crd,2 Cld,2 Cld,1

Trl,1

Crd,1 Trl,2

Trl,3 Ti (ut)

Trl,4

Figure 1: Downtime cost vs maintenance service time condition becomes worse than that just prior to thins kind of work; e) worst maintenance: a maintenance action which undeliberately makes the system fail. The failure/repair process for repairable items is also well explained by Calabria et al. (22). Martorell et al. (23) considered an equipment age-dependent reliability model for nuclear power plant operation. In particular, they perform a sensitivity study and conclude that maintenance effectiveness has an importante impact on the selection of an appropriate maintenance strategy. 3.

STATEMENT OF THE PROBLEM

Let us consider a situation where an equipment may suffer failures in time according to a distribution with probability density function f (t). The equipment receives corrective actions and also may be subjected to periodic overhauls. After some time Tl (to be decided), the equipment is replaced by a new one. Repairs and overhauls take Tr and To time units to be performed respectively, and cost Cr and Co money units (mu). Maintenance decision-makers decide on these values in search of maximum profitability for the company. Downtime costs also depend on the maintenance service times. Some discontinuities appear (see figure 1) on the curve due to the exhaustion of stock piles, use of alternative, less efficient methods, and so on. In general this function is non decreasing with respect to Tr and To . Intervention costs may also be estimated from the service times. We will consider, that for the same set of maintenance actions, the intervention cost is a non increasing function of Tr (figure 2) and may also be discontinuous due to scale economies, use of extra resources and more expensive repair methods. There could be segments where maintenance actions are not possible (i.e., it takes at least Tr min time units to repair). We also consider the following conditions: 1. corrective actions are minimal.


Intervention costs Cr (mu/intervention)

Cri,1

Cli,2

Cri,2 Cli,3

Cli,4 Clr,1

Tr,1l

Tr,2l

Tr,3l Tr (ut)

Tr,4l

Figure 2: Intervention cost vs maintenance service time 2. opportunistic maintenance practices may be disregarded due to the lack of scale economies with grouped maintenance actions (24; 25). 3. preventive actions are applied at constant time intervals Ts . 4. life cycle duration corresponds to nTs time units. Consequently, there are n − 1 overhauls during it. 5. The objective is to minimize the expected global cost rate in order to decide overhaul frequency and maintenance service time. Other possible objectives are: • Maximizing availability: Applies to critical equipment. This criterion is appropriate when downtime costs are much larger than intervention costs. • Minimizing intervention costs: Once availability is modelled, it is straightforward to estimate intervention costs and fix a budget. This criterion does not take into account downtime costs explicitly. For that, it is necessary to impose a minimum availability constraint. The difficulty of using the global cost rate as objective comes from the identification of downtime costs, which may be hard to estimate. The amount of information needed to use this objective is much larger that using others. Obtaining the required data from the information system could be difficult or costly. • Intervention costs depend on the time interval required to perform the activities. It is defined by a curve which is possibly discontinuous. Each segment is determined l r l y Ci,j on the left and right side of segment j. , and costs Ci,j by Tr,j


• Global replacement cost corresponds to CR,g : CR,g = CR,i + CR,d • No salvage value is considered. 3.1 Variables • Ts , time interval between two preventive interventions, • To , time interval required for a preventive action, To,j =

l l To if To,j ≤ To ≤ To,j+1 0 -

(1)

• Tr , time interval required for a repair, Tr,j =

l l Tr if Tr,j ≤ Tr ≤ Tr,j+1 0 -

(2)

• Co,i , preventive intervention cost; • Co,d , downtime cost for overhauls; • Cr,i , repair intervention cost; • Cr,d , downtime cost for repairs; • Co,g , global cost of an overhaul (intervention+downtime); Co,g = Co,i + Co,d • Cr,g , global cost of a repair (intervention+downtime); Cr,g = Cr,i + Cr,d • δjo and δjr are auxiliary binary variables, such that:

δjo

δjr

=

=

l l 1 if To,j ≤ To ≤ To,j+1 0 -

(3)

l l 1 if Tr,j ≤ Tr ≤ Tr,j+1 0 -

(4)

• nr , number of failures during the life cycle.


3.2 Constraints • In order to force To,j to be active only in the corresponding interval, l ≤ To,j for j = 1..J δjo Ti,j

(5)

l To,j ≤ δjo To,j+1 for j = 1..J

and for repairs, l ≤ Tr,j for j = 1..J δjr Tr,j

(6)

l for j = 1..J Tr,j ≤ δjr Tr,j+1

• Computing downtime costs, " Co,d =

X

l Cd,j

# r l Cd,j − Cd,j l + l To,j − To,j δjo l To,j+1 − To,j

(7)

l Cd,j

# r l Cd,j − Cd,j l + l Tr,j − Tr,j δjc l Tr,j+1 − Tr,j

(8)

j

" Cr,d =

X j

• Computing intervention costs, " Co,i =

X

l Co,j

j

# r l Co,j − Co,j l + l To,j − To,j δjo l To,j+1 − To,j

" Cr,i = κ

X

l Cr,j

j

# r l Cr,j − Cr,j l + l Tr,j − Tr,j δjc l Tr,j+1 − Tr,j

(9)

(10)

• Only one active interval, X

δjo = 1

(11)

δjc = 1

(12)

j

X j

• All variables are non negative.


-1

10

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λ (1/days)

10

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0

200

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Time (days)

Figure 3: Model from Zhang and Jardine vs proposed (from the example, p = 0.7) 4.

FAILURE RATE MODEL

Zhang and Jardine (2) propose a failure rate model to describe the situation where the equipment receives imperfect preventive maintenance actions at discrete instants in time, with a fixed frequency. Corrective actions are minimal in their model. In the model considered here, after an overhaul, the equipment is between as good as before and as good as after previous overhaul. Let λk (t) be the failure rate after the k − th overhaul. We will express the failure rate as: ¯ λ0 (t) = λ(t) λk (t) = pλk−1 (t − Ts ) + (1 − p)λk−1 (t),

k = 1, . . . , n − 1,

(13)

where p ∈ (0, 1) corresponds to an improvement factor that models the degree of im¯ perfection of the overhaul; and λ(t) is a baseline failure rate function that models the situation before the first overhaul (and determines the behavior of λk (t), k = 1, ..., n − 1). The failure rate function is discontinuous and presents a behavior of the kind shown in figure 3. The curve with no discontinuities represents an equivalent for λ(t) in terms of the expected number of failures and its parameters are estimated in Pascual and Ortega (26). We may express the expected global cost per unit time for this model: cg (Ts , To , Tr , n) =

nr (nTs ) (Cc,d + Cc,i ) + (n − 1) (Co,d + Co,i ) + CR,g nTs

(14)

4.1 Budget constraint Figure (4) represents the long term trend of intervention costs. We observe how the increasing failure rate forces more repairs as the equipment gets older. Let us consider that a constant budget B is allocated at the beginning of a given strategic time unit (i.e. each year).


Intervention costs (mu)

Overhauls

1 year

Repairs

Time (ut)

Figure 4: Intervention cost vs time 40 Overhaul Repair Downtime

35

Cost (mu/action)

30

25

20

15

10

5

0

0

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40

Downtime (days)

Figure 5: Expected costs vs downtime (mu/action) The worst situation appears last part of the life cycle and specifically during the last year. As the budget is constant, the available money limits the permissible number of failures during the last year (we recall that the life cycle corresponds to the multiplication of two decision variables, n and Ts ). The condition can be written as: Z

nTs

λdt + Cio nol ≤ B

Cir

(15)

nTs −k

where nol corresponds to the number of overhauls during the last year. As the budget constraint holds for the last year, previous budgets are not fully used unless they are applied for preventive actions which improve the failure rate, and specifically during the last year.


120 Overhaul Repair

Global cost (mu/action)

100

80

60

40

20

0

0

10

20

30

40

50

60

Downtime (days)

70

80

90

100

Figure 6: Expected global costs (mu/action) vs downtime Trl 0.1 2 14 90 10000

l Cfl Cfr Cio 0 0 35 0.2 2 12 3 100 6 100 1000 1

r Cio 12 6 2 1

Cirl 35 1.8 1 0.5

Cirr 2 1 0.5 0.5

Table 1: Cost parameters 5.

NUMERICAL EXAMPLE

Let us consider the example of Zhang & Jardine. We modify the conditions of the case to consider the effect of a annual budget B and the times to repair and overhaul the equipment. Once decided, Tr and To remain constant along the life-cycle. In order to reduce the search space and take into account that strategic decisions are made on a annual basis we consider: 365 days υ Tl = 365κ days

Ts =

where υ and κ are positive integers. Thus, decision variables are: Tr , To , υ and κ. Intervention and downtime costs are shown in figure (5). These costs vary linearly on each segment according to data given in table (1). The improvement factor is set at p = 0.7, as in Zhang & Jardine. We observe that during the first time units no downtime costs exists due to the existence of contingency measures like stock piles and alternative equipment. Combining both direct and indirect cost we obtain the profiles of global costs for both corrective and preventive interventions as it is shown in figure (6). We notice that minimal global costs


Figure 7: Expected global cost (mu/year) for κ = 6 and υ = 2 are Cg,r = 2 mu/repair for the case of corrective actions (at Tr = 2 days) and Cg,o = 8 mu/overhaul (service time To = 14 days). If budget is enough large, the maintenance department would repair and overhaul the equipment according to these times to maximize company’s profitability. If we fix a life cycle of 6 years and 2 overhauls/year (κ = 6 and υ = 2) and no budget restriction applies, we obtain the cost surface shown in figure 7. We corroborate that the optimal decision for the company is to repair and overhaul at Tr = 2 days and To = 14 days at an expected global cost of 50.44 mu/year. In order to study the effect of budge, figure (8) superimposes the budget constraint (labelled hyperbola-like curves) over the isocost lines of the expected global cost per unit time. If B is below 25 mu/year the global minimum is no longer feasible and service times are affected by the available money to perform them. A sensitivity analysis is shown in figure (9) where optimal values for the decision variables and the objective are plotted against B. The optimal life-cycle ranges between 3 and 6 years. The budget constraint acts when B < 25 mu/year. Below 9 mu/year, To has to be incremented. Tr is more sensitive and shows discontinuities for different life cycle durations (Tl = 3, 4, 5, 6 years). 6.

FINAL COMMENTS AND FURTHER WORK

Real industrial systems almost always have design alternatives to handle load fluctuations and failure occurrence. This situation implies discontinuities in the cost functions that measure the impact of maintenance and replacement policies. This work develops a framework to handle the problem through the use of a non-linear mixed integer programming formulation. Feasibility has been shown through a numerical example. The model let us improve maintenance decision support-systems to take into account the effect of palliative measures in real industrial processes. We have studied a situation where Tr and To are fixed in order to minimize the expected global cost rate. Further developments would let these decisiones variables variate according to observed failure rates, consider time-dependent budgeting, set variable overhaul frequency, and use discounted values.


100 10

1

25

90 30

10

80 70

15

60

To

25 50 30 40

15

30 20 20

25

10

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25 30

10

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30

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Tr Figure 8: Expected global cost (mu/year) for various annual budgets levels


140 c

g

υ* κ* Tr* To*

120

Corresponding units

100

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20

0

0

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30

Budget (mu)

Figure 9: Effects of budget on costs and times REFERENCES [1] Vorster,M.C., De la Garza, J.M., 1990. Consequential equipment costs associated with lack of availaibility and downtime, Journal of Construction Engineering and Management-ASCE, vol. 116, n. 4, pp. 656-669. [2] Jardine A.K.S. Zhang, F., 1998. Optimal maintenance models with minimal repair, periodic overhaul and complete renewal. IIE Transactions, vol. 30, pp. 1109-1119. [3] Scarf, P.A., 1997. On the application of mathematical models in maintenance, European Journal of Operational Research, vol. 99, pp. 493-506. [4] Dekker, R., Scarf, P.A., 1998. On the impact of optimisation models in maintenance decision making: the state of the art, Reliability Engineering and System Safety, vol. 60, pp. 111-119. [5] Komonen,K., 1998. The structure and effectiveness of industrial maintenance, Acta Polytechnica Scandinavica, Mathematics, Computing and Management in Engineering Series, vol. 93. [6] Komonen, K., 2002, A cost model of industrial maintenance for profitability analysis and benchmarking, International Journal of Production Economics, vol. 79, n.1, pp. 15-31. [7] Jardine, A.K.S., 1973. Maintenance, Replacement and Reliability, Pitman Publishing, Marshfield, MA.


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