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11 Maintenance Optimization Models

Andrew K. S. Jardine

Maintenance optimization is all about getting the best result possible, given one or more assumptions. In this chapter, we introduce the concept of optimization through a well-understood traveling problem: identifying the best mode of travel, depending on different requirements. We also examine the importance of building mathemati-cal models of maintenance decision problems to help arrive at the best decision.

We look at key maintenance decision areas: component replacement, capital equip-ment replacement, inspection procedures, and resource requirements. We use opti-mization models to find the best possible solution for several problem situations.

11.1 what Is oPtIMIzatIon all about?

Optimal means the most desirable outcome possible under restricted circumstances. For example, following a reliability-centered maintenance (RCM) study, you could conduct condition monitoring maintenance tactics, time-based maintenance, or time-based discard for specific parts of a machine or system. In this chapter, we introduce maintenance decision optimization. In the next chapter, we discuss detailed models for asset maintenance and replacement decision making.

To understand the concept of optimization, consider this travel routing problem: you have to take an airplane trip, with three stops, before returning home to Chicago. The first destination is London, followed by Moscow and then Hawaii. Before pur-chasing a ticket, you would weigh a number of options, including airlines, fares,

contents

11.1 What Is Optimization All About? ................................................................ 25111.2 Thinking Optimization ................................................................................. 25211.3 What to Optimize ......................................................................................... 25311.4 How to Optimize .......................................................................................... 25311.5 Key Maintenance Management Decision Areas ..........................................256References .............................................................................................................. 257

252 Asset Management Excellence: Optimizing Equipment Life-Cycle Decisions

and schedules. You’d make your decision based on factors such as economy, speed, safety, and extras:

If • economy is most important, you’d choose the airline with the cheapest ticket. That would be the optimal solution.If • speed was it, you’d consider only the schedules and disregard the other criteria.If you wanted to optimize • safety you’d avoid airlines with a dubious safety record and pick only a well-regarded carrier.If you wanted a free hotel room (an • extra) for three nights in Hawaii, you’d opt for the airline that would provide that benefit.

This list illustrates the concept of optimization. When you optimize in one area—economy, for example—then almost always you get a less desirable (suboptimal) result in one or more of the other criteria.

Sometimes, you have to do a trade-off between two criteria. For example, though speed may be most important to you, the cost of traveling on the fastest schedule could be unacceptable. The solution is somewhere in the middle—providing an acceptable cost (but not the very lowest) and speed (but not the very fastest).

In any optimization situation, including maintenance-decision optimization, you should do the following:

Think• about optimization when making maintenance decisions.Consider • what maintenance decision you want to optimize.Explore • how you can do this.

11.2 thInkIng oPtIMIzatIon

Thinking about optimization means considering trade-offs: the pros and cons. Optimization always has to do with getting the best result where it counts most while consciously accepting less than that elsewhere.

A customer service manager was asked by the vice president of marketing what he thought his main mission was. His answer: To get every order for every cus-tomer delivered without fail on the day the customer specified, 100% of the time. To achieve this goal, the inventory of ready-to-ship goods would have to include every color, size, and style in sufficient quantities to ensure that no matter what was called for, it could be shipped. In spite of unusually big orders, a large number of customers randomly wanting the same thing at the same time, or machinery failure, the manager would have to deliver. His inventory would have had an unacceptably high cost.

The manager failed to realize that a delivery performance just slightly less, say 95%, would be better. In fact, it would be a profit-optimization strategy, the best trade-off between the cost of inventory and an acceptable and competitive customer satisfaction level.

Maintenance Optimization Models 253

11.3 what to oPtIMIze

Just as in other areas, you can optimize in maintenance for different criteria—including cost, availability, safety, and profit.

Lowest-cost optimization is often the maintenance goal. The cost of the component or asset, labor, lost production, and perhaps even customer dissatisfaction from delayed deliveries are all considered. Where equipment or component wear-out is a factor, the lowest possible cost is usually achieved by replacing machine parts late enough to get good service out of them but early enough for an acceptable rate of on-the-job failures (to attain a “zero” rate, you’d probably have to replace parts every day).

Availability can be another optimization goal: getting the right balance between taking equipment out of service for preventive maintenance and suffering outages due to breakdowns. If safety is most important, you might optimize for the safest possible solution but with an acceptable impact on cost. If you optimize for profit, you would take into account not only cost but also the effect on revenues through greater customer satisfaction (better profits) or delayed deliveries (lower profits).

11.4 how to oPtIMIze

One of the main tools in the scientific approach to management decision making is building an evaluative model, usually mathematical, to assess a variety of alternative decisions. Any model is simply a representation of the system under study. When applying quantitative techniques to management problems, we frequently use a sym-bolic model. The system’s relationships are represented by symbols and properties described by mathematical equations.

To understand this model-building approach, examine the following maintenance stores problem. Although simplified, it illustrates two important aspects of model use: constructing the problem being studied and its solution.

A Stores Problem

A stores controller wants to know how much to order each time the stock level of an item reaches zero. The system is illustrated in Figure 11.1.

The conflict here is that the more items ordered at any time, the more order-ing costs will decrease, but holding costs increase, since more stock is kept

Stoc

k Le

vel Q

Time

Order

FIgure 11.1 An inventory problem.

254 Asset Management Excellence: Optimizing Equipment Life-Cycle Decisions

on hand. These conflicting features are illustrated in Figure 11.2. The stores controller wants to determine which order quantity will minimize the total cost. This total cost can be plotted, as shown in Figure 11.2, and used to solve the problem.

A much more rapid solution, however, is to construct a mathematical model of the decision situation. The following parameters can be defined:

D = total annual demandQ = order quantityCo = ordering cost per orderCh = stockholding cost per item per year

Total cost per year of ordering and holding stock = Ordering cost per year + Stockholding cost per year

Now,

Ordering cost/year = Number of orders placed per year × Ordering cost per order =

DQ

Co

Stockholding cost/year = Average number of items in stock per year (assuming linear decrease of stock) × Stockholding cost per item

Q

Ch2

Therefore, the total cost per year, which is a function of the order quantity and is denoted C(Q), is

Cost

s

Ordering cost

Holding cost

Total cost

Optimalorder

Order Quantity

FIgure 11.2 Economic order quantity.

Maintenance Optimization Models 255

C QDQ

CQ

Co h( ) = +2

(11.1)

Equation 11.1 is a mathematical model of the problem relating order quan-tity, Q, to total cost, C(Q). The stores controller wants to know the number of items to order to minimize the total cost, that is, the right-hand side of Equation 11.1. The answer is obtained by differentiating the equation with respect to Q, the order quantity, and equating the answer to zero as follows:

dC Q

dQD

QC

Co

h( ) = − + =2 2

0

Therefore,

D

QC

Co

h2 2

=

QDCC

o

h

= 2 (11.2)

Since the values of D, Co, and Ch are known, substituting them into Equation 11.2 gives the value of the order quantity Q. You can check that the value of Q obtained from Equation 11.2 is a minimum and not a maximum by taking the derivative of C(Q) and noting the positive result. This confirms that Q is optimal.

example

Let D = 1,000 items, Co = $5, Ch = $0.25

Q = 2 1000 50.25

= 200 items× ×

Each time the stock level reaches zero, the stores controller should order 200 items to minimize the total cost per year of holding and ordering stock.

Note that the various assumptions that have been made in the inventory model may not be realistic. For example, no consideration has been given to quantity discounts, the possible lead time between placing an order and its receipt, and the fact that demand may not be linear or known for certain. The purpose of the previous model is simply to illustrate constructing a model and attaining a solution for a particular problem. There is abundant literature about stock control problems without many of these limitations. If you are interested in stock control aspects of maintenance stores, see Nahmias.1

It’s clear from the previous inventory control example that we need the right kind of data, properly organized. Most organizations have a computerized maintenance management system (CMMS) or an enterprise asset management

256 Asset Management Excellence: Optimizing Equipment Life-Cycle Decisions

(EAM) system. The vast amount of data they store makes optimization analyses possible.

Instead of building mathematical models of maintenance decision problems, software is available to help you make optimal maintenance decisions. This is covered in Chapter 12.

11.5 key MaIntenance ManageMent decIsIon areas

There are four key decision areas that maintenance managers must address to opti-mize their organization’s human and physical resources. These areas are depicted in Figure 11.3. The first column deals with component replacement, the second with inspection decisions, including condition-based maintenance and the third with establishing the economic life of capital equipment. The final column addresses decisions concerning resources required for maintenance and their location.

To build strong maintenance optimization, you need an appropriate source, or sources, of data. The foundation for this, as shown in Figure 11.3, is the CMMS/EAM system/enterprise resource planning (ERP) system.

Optimizing Equipment Maintenance & Replacement Decisions

ComponentReplacement

1. Best preventive replacement time

(a) Deterministic performance deterioration(b) Replace only on failure(c) Constant interval(d) Age-based

2. Spare parts provisioning3. Repairable systems4. Glasser’s graphs5. Software SMS and OREST

Probability and statistics(Weibull analysis)including software

Weibullsoft

Stochastic processes(for CBM optimization)

Time value of money(discounted cash flow)

Database (CMM/EAM/ERP System)

Queuing theorysimulation

1. Inspection frequency for a system

1. Economic life 1. Workshop machines/crew sizes

(a) Profit maximiza- tion

(a) Constant annual use

(a) Own equipment(b) Contracting out peaks in demand

(b) Varying annual use(c) Technological improvement

(b) Availability maximization

2. A, B, C, D class inspection intervals 2. Repair vs. replace

2. Right sizing equipment

3. Software PERDEC and AGE/CON4. Software workshop simulator and crew size optimizer

3. Lease/buy3. FFI’s for protective devices4. Condition-based maintenance5. Blended health monitoring and age replacement6. Software EXAKT

InspectionProcedures

Capital EquipmentReplacement

ResourceRequirements

FIgure 11.3 Key areas of maintenance and replacement decisions.

Maintenance Optimization Models 257

In Chapter 12, we discuss optimization of key maintenance decisions of com-ponent replacement, inspection procedures, and capital equipment replacement (Columns 1, 2, and 3 of Figure 11.3). The framework, foundation, or database is discussed in detail in Chapter 5.

Extensive development and discussion of models, including case studies, is pro-vided in Jardine and Tsang.2

reFerences

1. Nahmias, S., Production and Operations Analysis, 3d ed., Irwin/McGraw-Hill, 1997. 2. Jardine, A.K.S., and A.H.C. Tsang, Maintenance, Replacement and Reliability: Theory

and Applications, CRC Press, 2006.