DYNAMIC MODELING OF MACHINERY

REPLACEMENT PROBLEMS

BY

ROWLAND JERRY OKECHUKWU EKEOCHA

PG/PH.D/07/42581

DEPARTMENT OF MECHANICAL ENGINEERING

FACULTY OF ENGINEERING

UNIVERSITY OF NIGERIA

NSUKKA

OCTOBER, 2010.

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CHAPTER ONE

INTRODUCTION

Replacement of machinery is not widely discussed in Engineering Literatures. Yet it is a common occurrence in industries requiring the use of plant and equipment for the production of goods and services. Even where it is discussed, the emphasis is usually placed on minimization of total, maintenance and operating costs as well as the maximization of profit without recourse to the uncertainty resulting from the method of determining deterioration. Cognizance of the effect of deterioration on the resale value of equipment and indeed on machinery replacement date is yet to be fully appreciated. Values of deterioration are usually assumed or at best determined by methods that are highly subjective and sometimes expensive like the popular failure analysis (Sachs, 2007). Some replacement models exclude resale value in the build-up of cost. Yet it is the value that is directly affected by deterioration. This missing link will be supplied by this study.

1.1 OBJECTIVEThe following are the objectives of this study:i. To reduce the error arising from the subjective methods of

determining deterioration by generating random numbers using Monte Carlo simulation under the uniform probability distribution to produce values for deterioration.

ii. To incorporate economic parameters like the inflation rate and the rate of return on replacement investment in the model.

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iii. To develop machinery replacement model with emphasis on the effect of deterioration on the salvage (resale) value and indeed on the replacement date.

iv. To employ the dynamic programming method in the enumeration to obtain the optimal replacement date.

v. To calibrate and verify the model with field data obtained from some industries as well as compare the results of the model with those of existing models for reliability and operationability.

1.2 THE IMPORTANCE OF THE STUDYReplacement investment is one of the overheads

competing for scarce financial resources of any industry. A replacement investment should therefore be justified. An effective machinery replacement policy can be put in place to achieve this purpose. In this study, a model will be developed that will assist industry managers to make decision on replacement date when maintenance is no longer advisable. Planned replacement will reduce or perhaps eliminate unnecessary downtime arising from forced shutdowns. Total Process Reliability (TPR) is one of the methods of realizing a planned replacement of machines. Total Process Reliability (TPR) views every maintenance events as an opportunity to upgrade manufacturing processes.

Bloch et al, (2006) advise that process plants should be reliability-focused instead of repair-focused. Process plants that are repair focused have trouble surviving because they place emphasis on parts replacement and have neither time nor the inclination to make systematic improvements. They hardly

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identify the reason for parts failure and do not implement the type of remedial action that discourages the recurrence of failures. Reliability-focused plants, on the other hand, view every maintenance event as an opportunity to upgrade. Whenever cost justifies, this upgrade is achieved by adhering more closely to smarter work processes, following better procedures, selecting superior components (not parts), implementing better quality controls, using more suitable tools and choosing a suitable replacement date. These measures may reduce downtime and maximize machinery uptime.

1.3 THE METHODOLOGY:This study lays emphasis on the effect of deterioration on

salvage (resale) value and indeed on the replacement date of machinery. It also recognizes the inherent errors in the existing method of determining deterioration. These methods are highly subjective and sometimes expensive (Sachs, 2007). To reduce these errors or perhaps eliminate them the values of deterioration will be treated as stochastic variables and generated as random numbers under the uniform (rectangular) probability distribution.

The cost minimization model that will be developed will have three main cost components namely the purchasing cost, the maintenance cost and the salvage (resale) value.

Solution of the salvage value component (function) of the model is expected to produce the initial resale date that may be used to start an enumeration process. The dynamic programming technique will be adopted in the enumeration process.

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The model will incorporate economic parameters like the inflation rate and the rate of return on replacement investment.

The model will then be tested with field data to verify the reliability of its results, for possible review and adjustments.

In summary, the methodology of this study consists of two parts namely the analytical and the experimental processesa. Analytical Process: In the analytical process the following

steps are taken:i. Values of deterioration are generated as random

numbers under the uniform probability distribution.ii. The initial replacement date is obtained from the

derivative of the salvage value component of the model.

iii. The dynamic programming technique is employed as the enumeration process for obtaining the optimal replacement date starting perhaps with an initial resale date obtained from the solution of the salvage value component of the model.

b. Experimental Process: Field data obtained from different industries like the construction, pharmaceutical and plastic companies are used to test the model with the view to verifying the reliability of the results for possible review and adjustments.Finally, results from the model will be compared with those

of existing models for possible advantages in terms of operationability, reliability and perhaps savings.

1.4 THE CONTRIBUTION TO KNOWLEDGE

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This study develops a model that considers the effect of deterioration, inflation and rate of return on replacement investment on total cost and indeed on the replacement date of machinery. Thus the results of the model are intended to give answer as to whether to replace or refurbish existing machinery as well as indicate when replacement is optimal. The study therefore complements as well as improves on the existing knowledge of machinery replacement and assists industry managers in making replacement decisions especially when maintenance or refurbishment is no longer advisable. In this regard, the contribution of the study should be viewed in the following areas.i. Verification of the Salvage value to test its suitability to our

industrial environment.ii. The incorporation of economic parameters like the rate of

return on replacement investment and inflation rate in the development of the new model

iii. The reduction of error inherent in the subjective methods of determining deterioration by generating values under the uniform probability distribution to represent deterioration.

iv. The development of cost minimization model with emphasis on the effect of deterioration on the salvage value.

v. The application of dynamic programming method as the solution technique to the new model. Dynamic programming is good for problems with overlapping sub-problems and optimal substructure and therefore has an advantage over other solution techniques like the

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cumbersome forward algorithm adopted by some existing models.

vi. The calibration and verification of the model with field data and comparing its results with those of existing models for reliability and operationability.

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CHAPTER TWO

LITERATURE REVIEW

Replacement refers to a situation in which machinery is worn out and

cannot physically perform its intended function and must be

exchanged for another machinery. Replacement may also arise in

case of upgrade, the existing equipment may be functionally alright,

but output needs to be increased.

Replacement/Refurbishment has been an age long practice

especially when such machinery is not providing the required services.

Even the manual farm tools and equipment are either repaired to

improve on the services rendered or replaced completely due to

failure to provide services at all.

In modern day industries where plant and machinery are

required for the production of goods and services, the replacement of

machinery is common due to deterioration which gives rise to facility

failures.

In this chapter, brief discussion of certain terms will be

presented. It is expected that the discussion of these terms will put

this work in proper focus and simultaneously promote a better

understanding of machinery replacement practice.

The terms to be discussed are as follows:-

(i) Deterioration (Depreciation)

(ii) Probability Distribution

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(iii) Simulation

(iv) Monte Carlo Method

(v) Random Numbers

(vi) Replacement Models

2.1 DETERIORATION (DEPRECIATION)

Deterioration is defined as the loss of value (American Society of

Appraisers [ASA], 2000). Machinery deterioration is therefore the loss

of value of the machinery from all causes. Deterioration may be

curable or incurable. The types of deterioration include the following:-

(i) Physical deterioration

(ii) Functional Obsolescence

(iii) Economic Obsolescence (External Obsolescence)

2.1.1Physical Deterioration

Physical deterioration is the loss of value or usefulness of a plant

or machinery due to the using up or expiration of its useful life caused

by wear or tear, exposure to harsh environment, physical stresses and

similar factors.

Generally, physical deterioration may be caused by age, wear

and tear, fatigue, stress, exposure to harsh environment and lack of

maintenance. The inability of a plant or machinery to perform at

design capacity may be a measure of physical deterioration. Caution

should, however, be exercised in determining whether or not the

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inability to meet the design capacity is a function of the

plant/machinery or a function of the production schedule. The former

denotes physical inability while the latter denotes economic

obsolescence.

2.1.2 Functional Obsolescence

Functional Obsolescence is the loss in value or usefulness

caused by inefficiencies or inadequacies of the plant/machinery when

compared to more efficient or less costly replacement machinery that

new technology has developed.

Functionally obsolescence is the impairment of functional

capacity, inadequacies or changes in the state of the art that affect

the machinery rendering it incapable to adequately perform the

function it was initially designed to undertake. Generally, functional

obsolescence may be caused by lack of utility, change in design,

efficiency and technology change.

2.1.3 Economic Obsolescence (External Obsolescence)

Economic Obsolescence is the loss in value or usefulness caused

by factors external to the plant/machinery such as government

regulations, competition, inflation, reduced demand of product and

market accessibility/unacceptability. Generally, economic

obsolescence may be caused by management concept, government

regulation/policy, competition and similar factors.

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2.1.4DETERMINATION OF DETERIORATION

Methods of determining deterioration include the following

(i) Age/Life analysis

(ii) Use/Total Use analysis

(iii) Observation

(iv) Direct monetary measurement.

2.1.4.1 Age/Life Analysis

This is the ratio of the age to the Life of a plant or machinery.

This gives a straight-line deterioration (ASA, 2000)

Definition of some terms will be presented to facilitate the

discussion of the Age/Life analysis.

(a) Chronological age is the number of years that has elapsed

since machinery was manufactured or put in use.

(b) Effective age is the apparent age of machinery in comparison

with a new one of its kind. Effective age is indicated by the

condition of the machinery. With regular overhaul, the

effective age of machinery is normally less than its

chronological age.

(c) Normal useful life is the estimated number of years that

machinery will actually be used before retirement from service.

(Service Life).

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(d) Remaining Useful Life is the estimated period during which a

machinery of a certain effective age is expected to actually be

used before retirement from service. This is approximately

the difference between the Normal Useful Life and Effective

Age of machinery.

(e) Physical Life is the estimated number of years that new

machinery will physically endure before it deteriorates or

fatigues to an unusual condition purely from physical causes

without considering the possibility of earlier retirement due to

functional or economic obsolescence.

(f) Remaining physical life is approximately the difference

between the physical life and the effective age of machinery.

(g) Economic life is the estimated number of years that a new

machinery may be profitably used for the purpose it was

intended. In other words, economic life is the estimated

number of years new machinery can be used before it is

replaced with the most economic replacer (displacer) that

performs equivalent service. Functional and economic

obsolescence factors also affect the economic life of

machinery. Generally, the economic life of machinery is less

that its normal useful life.

(h) Remaining Economic life is approximately the difference

between the economic life and the effective age of machinery.

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In view of these definitions, the Age/Life formula for the

determination of deterioration can be simplified as follow (ASA, 2000)

EFFECTIVE AGEPHYSICALLIFE

×100=%DETERIORATION

… (2.0)

The Age/Life method is most useful for new machinery or

machinery in midlife.

2.1.4.2 Use/Total Use Analysis

This analysis is based on the use of machinery. The use of

machinery gives a good indication of physical deterioration when

requisite production statistics can be obtained. Thus the ratio

USE/TOTAL USE is an indicator of physical deterioration, where USE

and TOTAL USE are in units of time.

2.1.4.3 Observation

In this method, comparison based on experience is made by

looking at machinery and comparing it with similar new machinery.

The procedure involves actually observing those elements of wear and

tear that can be seen and converting those observations into

percentages of deterioration. This method is highly subjective.

2.1.4.4 Direct Monetary Measurement

This method is applicable to machinery deterioration that is

economically curable. Deterioration is curable when it is economically

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feasible to remedy it because the resulting increase in utility and

value is greater than the cost to cure. The concept therefore involves

the measurement of the amount expended to effect cure in

machinery. The amount must be less than the cost of a new

machinery of its kind. This monetary measurement of the curable

portion of machinery provides a clue to physical deterioration.

Generally all the methods of determining deterioration are

subjective. This gives rise for alternative method of deriving value for

deterioration to be sought. One of such alternatives is the use of

Monte Carlo approach to simulate deterioration as stochastic variable

in the machinery replacement model.

2.1.5 Machinery Failure Analysis

Sachs (2007) defined Failure Analysis as the process of

interpreting the features of a deteriorated system or component to

determine why it no longer performs the intended function. He also states

that failure analysis entails first using deductive logic to find the mechanical

and human causes of the failure and then using inductive logic to find the

latent causes. He concludes by stating that the analysis should lead to the

changes needed to prevent the recurrence of failure.

Generally, there are three main classes of failure cause or root namely

Physical, Human and Latent causes (Sachs, 2007).

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i. Physical cause is the physical mechanism that caused the

failure. It may be fatigue, overload, wear and tear, corrosion or

a combination of these. Failure analysis must start with

accurate determination of physical causes (roots) which will lead

to the detection and correction of both human and latent

causes.

ii. Human cause is the application of inappropriate human

intervention, which results in physical failure. An example of a

human root of a problem is the use of mobile phone while

driving and its effect on the rate of accident. Human causes

may be due to errors introduced by human during designing

(omission and commission), manufacturing, maintenance,

installation, operation or procrastination (situation blindness).

iii. A latent cause (system weakness) is any corporate policy or

action that allows the application of inappropriate action. A

latent error allows the human and physical causes of failure to

exist. An example is the inappropriate use of personnel to

handle jobs outside his area of specialization. For instance the

use of a Chemical Engineer to do the work of a Mechanical

Engineer.

There are multiple causes to all failures. A typical plant failure results

from culmination of a sequence of events. Timely recognition that this

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sequence has the possibility of causing a failure and making a change that

breaks the sequence may stop failure from happening. It is also significant

to note that if the chain is broken by a random incident, there is the

probability of the failure recurring.

Failure analysis looks at a problem, understands in complete detail

how and why it occurred, then tries to prevent a recurrence of the failure.

The benefit of failure analysis can be measured in terms of the increased

asset availability, increased throughput and reduced costs. The magnitude

of the benefit is proportional to the amount of financial support for the

analysis.

Sachs (2007) groups failure analysis into three categories in order of

complexity and depth of investigation. They are:

i. Component Failure Analysis (CFA) determines the specific cause of

machine part or component failure such as fatigue, overload or

corrosion. CFA uncovers the physical reasons for failures.

ii. Root Cause Investigation (RCI) goes beyond the physical root of a

problem to find the human errors involved. In fact RCI goes deeper

than a CFA and finds the major human roots but does not find all the

human roots and does not involve management system deficiencies.

iii. Root Cause Analysis (RCA) includes everything the RCI covers plus the

minor human errors and the management system deficiencies that

allow the human errors and other system weakness to exist. In fact,

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RCA finds all contributors to a failure and although the investigation is

costly, it pays for itself many times over.

Finally, the tools for finding the various roots of failure include the

logic tree and Ishikawa (Fishbone) diagram. The logic tree has superior

flexibility and the use of the tree requires the verification of each step. This

effectively counteracts the most noteworthy weakness of the other

alternatives (Sachs, 2007)

2.1.6 Failure Data Analysis

Bloch (2006) defined machinery failure as any change in a

machinery part or component which causes it to be unable to perform its

intended function or mission satisfactorily. He said that a popular yardstick

for measuring failure experience of machinery parts, assemblies,

components or system is to determine the failure rate. The failure rate is

obtained by dividing the number of failures experienced on a number of

homogeneous items, also called population within a time period, by the

population. Bloch (2006) classified failures as either chargeable or non-

chargeable for reliability assessment purposes. A chargeable failure would

be one that can be attributed to a defect in design or manufacturing while

non-chargeable failure is one caused by exposure of the part to operational,

environmental or structural stresses beyond the limits specified for the

design as well as the one attributable to operator error or improper handling

or maintenance. Bloch (2006) also defined failure mode as the appearance,

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manner or form in which a machinery component failure manifests itself and

advised that failure mode should not be confused with the failure cause, as

the former is the effect and the latter is the cause of the failure event.

The data required to determine failure distribution are the individual

times to failure of the equipment. The procedure is to convert the data to

become representative of the cumulative failure distribution F(t). This is

achieved by plotting times to failure against F(t) on a scale which

corresponds to the distribution to be fitted (Bloch, 2006). For the

exponential distribution Bloch (2006) gave the following equation,

F(t) = 1 – exp(-λt ) …(2.0)aConsequently: t = 1 /λln [1- F(t)] …(2.0)b

A plot of 1 / [1-F(t)] on a log scale against time on a linear scale produces a straight line. With an appropriate distribution for the failure data, the failure mode of the equipment is determined. The failure mode and the total replacement cost may lead to a replacement decision (policy) for machinery.

2.2 PROBABILITY DISTRIBUTION

Distribution function can be discrete as in counting or

continuous as in measurement. Continuous distribution can be

reduced to discrete distribution by grouping. Continuous distributions

arise in practice and therefore have advantages in real life usage over

discrete distributions. However, all the principles applicable for

discrete distributions are transferable to continuous distribution by

substituting the integral sign for the summation sign.18

Example of discrete distribution includes throwing of die (dice),

tossing of coin and drawing a card from a pack of cards and recording

the results.

For a continuous distribution, the use of probability density in

place of probability is important and necessary. If a distribution is

continuous, the probability of an event to have an exact value, say

5.00 is zero. There is infinity of possible result and the chance of

Exactly 5 is 1∞=0.

However, it is possible to consider a probability density in the

neighborhood of 5 with units of probability for occurrence within the interval (Humphreys, 1991).

If Pd (X) represents the probability density at point X, multiplied by some interval of X to convert it to probability, then Pd (X) depends on X and is a function of X.

I.e. Pd (X) = F(X) … (2.1)

Then the mean or expected value is given by

μ=E (X )=∫ X Pd(X)dx

∫Pd ( X )dx… (2.2)

And the variance is given by

σ 2=∫ ( X−μ )2 Pd (X )dx

∫Pd (X )dx

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… (2.3)

By analogy, the discrete function mean or expected value is given by

μ=E (X )=∑ XP (X )=∑1

N

X

N

… (2.4)

And the variance is given by

σ 2=∑1

N

(X−μ)2

N−1

… (2.5)The variance is a measure of the spread of a distribution from

the mean. A small variance means that the distribution is peaked while a large variance indicates a spread out from the mean.

The continuous distribution function is so expressed that the area under the probability density function is unity and the cumulative probability is unity in the limit and is given by

Pd (X )=∫−∞

+∞

Pd (X )dx=1

… (2.6)

2.2.1 Rectangular (Uniform) DistributionSuppose a distribution is uniform over a range 0 to a, then the

probability density is given by

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Pd (X )={ 1a 0≤ x≤a0elsewhere… (2.7)

Notice that the condition for Eq. (2.6) is met

∫−∞

+∞

Pd ( X )dx=∫0

a 1adx=1

… (2.8)

The cumulative probability for X or less is given by

Pc (X )=∫0

x 1adx= x

a0≤ x≤a

… (2.9)The uniform (rectangular) distribution can be used to simulate

random variable from almost any kind of probability distribution. It is also simple in application (Humphreys, 1991).

Other distributions include:(i) Exponential distribution(ii) Poisson distribution(iii) Normal distribution(iv) Binomial distribution

2.3 SIMULATIONSimulation must have taken its origin from the ancient art of

model building and has been applied to some extremely diverse forms of model building ranging from Renaissance painting and sculpture to

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scale models of supersonic jet airliners and computer models of cognitive processes. Problems with uncertainty can be quite complicated, and a solution by direct analytical methods may be very difficult if not impossible. The most general and basic method is to resort to simulation.

Walstorm (1967) described simulation as a technique that involves the setting-up of a model of a real situation and then performing experiments on the model. Simulation can also be seen as a group of numerical methods characterized by the process of making repeated solution. The input data are repeatedly selected by random numbers taken from an appropriate range of values.

Naylor et al (1969) defined simulation as a numerical technique for conducting experiments on a digital computer which involves certain types of mathematical and logical models that describe the behavior of a business or economic system (or some component thereof) over extended periods of real life.

Generally, simulation involves the construction of a model that is largely mathematical in nature. The model can then be used to describe the operation of the system of interest in terms of individual event, rather than directly describing the overall behavior of the system. The system can be subdivided into elements whose behavior can be predicted in terms of probability distribution for each of the various possible states of the system and its input data. The interrelationship of the elements can also be built into the model. Clearly, simulation offers means of dividing the model-building job into smaller component and then combining same in their natural order and allowing computer to present the effect of their interaction on each other.

After model construction, input data can then be generated to simulate the actual operation of the system over time and record its

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aggregate behavior. By repeating this operation for various alternative design configurations and comparing their performances, the most promising configuration can be identified.

Statistical error made it impossible for one to guarantee that a particular configuration with best performance is indeed the optional configuration. However, it should be near the optimal if the simulated experiment was properly designed.

If the behavior of an element cannot be predicted exactly, given the state of the system, it is better to take random observations from the probability distribution involved than to use averages to simulate this performance. This is true even when the average aggregate performance is the only issue of interest because combining average performances for the individual elements may result in some value far from the average of the overall system.

2.4. MONTE CARLO METHOD.This is a variant of simulation and is generally employed for

problems which are solved by use of a chance process.Ackoff (1963) stated that Monte Carlo is a code name given by

Van Neumann and Ulam to the mathematical technique which they applied to solving a category of nuclear-shielding problems that were too expensive for experimental solution and too complicated for analytical treatment.

Monte Carlo analysis involves the solution of a deterministic mathematical problem by simulating a stochastic process that has probability distributions satisfying the mathematical relationship of a non-probabilistic problem. Originally, the concept applied to situations in which there is a difficult deterministic problem and for which stochastic process may be invented which has probability distributions satisfying the relations of the non-probabilistic problem (National Bureau of Standards, 1961).

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Generally, the Monte Carlo method is applied to those simulations or to those portions (sub models) of large simulations in which random sampling is used.

A variable is simulated which in real life may have one or several outcomes. Repeated samples are drawn according to the probability distribution which describes the variable associated with the system or model.

2.5. RANDOM NUMBERS.Since some of the variables (inputs) in the machinery

replacement model will be generated as random numbers, it will be necessary to discuss briefly the requirement for randomness and the methods of generating Random Numbers.

2.5.1. The requirement for Random NumbersThe requirements are such that each successive number in the

sequence of random numbers must have equal probability of taking on any one of the possible values and must be statistically independent of the other number in the sequence (Willier & Lieberman, 1970).

2.5.2. Method of Generating Random Numbers.An acceptable method of generating random numbers must

yield sequences which satisfy the following conditions.(i) Uniformly distributed(ii) Statistically independent(iii) Reproducible(iv) Non-repeating for any desired length (period)(v) Capable of generating random numbers at high rates of speed

and yet require a minimum amount of computer memory capacity.

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The numbers generated by a computer are predictable and reproducible. They are therefore called Pseudo random numbers.

The following methods are used to generate random numbers.(i) The congruential method(ii) The Midsquare method(iii) The Manual method(iv) The Library (RAND) table.

2.5.2.1. The Congruential MethodThe congruential method has three variants namely the mixed,

multiplicative and the additive congruential methods. These are the most popular method of generating random numbers (Allard et al, 1963)(a) The mixed congruential method which is the most widely used congruential method, generates a sequence of random numbers by always calculating the next random number from the last one obtained, given an initial random number, called the seed.

The seed may be obtained from some published sources like the RAND table.

The recurrence relation used is given by

rn+1=(ar n+c )Modulo M

… (2.10)Where a, c and M are positive integers (a<M, c<M)

Eq. (2.10) means that rn+1 is the remainder when (arn+c) is divided by M. Thus rn+1 could take the number 0,1,2..., M-1, so that M represents the desired number of different values that could be generated for the random number (Desired length of random numbers).

There are rules for choosing values for a, c and M to ensure that rn+1 appears only once in the sequence before it starts repeating itself.

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The first rule when M = 2b (where b is the word size in the byte of a binary computer) a = 1,5,9,13... and c= 1,3,5,7,9...

The second rule when M=10d (where d is the word size of a decimal computer), a is 1,21,41,61,81... and c=1,3,7,9,11,13,17,19.. (Odd integers except those ending with 5).Random numbers with relatively small number of digits are often desired. One convention is to take the last three trailing digits and discard the rest.

(b) The multiplicative congruential method is a special case congruential

method, where c = o. Thus the recursive formula is given by rn+1=arn(Modulo M )

… (2.11)(c) The additive congruential method sets a = 1 and replaces c by some

random number preceding rn in the sequence. The method assumes K starting values where K is a positive integer and computes a sequence of number by means of the following congruency relation.

rn+1=rn+r n−k (ModuloM )

… (2.12)If K = 1, Eq. (2.12) generates the popular FIBONACCI sequence which behaves like sequence obtainable by multiplicative congruential

method with the unfavorable low multiplier,a=12 (1+√5). The statistical

properties of the sequence tend to improve as K increases. This is the only method that produces periods larger than M.

2.5.2.2. The Midsquare Method.In this method, a four digit number rn+1 is obtained from the

middle of the square of the preceding four digit random number, rn. The new random number, rn+1 which is normally extracted by a

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computer, can be used in the Monte Carlo trial to be decided at that instant and then stored for the next application.

2.5.2.3. The Manual Method.This method involves the throwing of dice or tossing of coins and

recording the outcome. 2.5.2.4. Library Table.

Library tables like those published by RAND Corporation provides random numbers which must have been generated by one of the previous methods and recorded in form of a table.

Both the manual method and the Library tables generate random numbers that are difficult to reproduce.

2.5.3. Uniformly Distributed Random Numbers.These random numbers should be random observations from a

uniform distribution and are expected to satisfy the condition of random numbers mentioned earlier.

2.6. REPLACEMENT MODELSDetermination of the service life of a machine or the machine

replacement problem has long been recognized as an important one. Analysis of this problem dates back to the works of Hotelling (1925) and Terborgh (1949). The formulation of a dynamic programming solution to the problem by Bellman (1955) led to the development of several operations research models. A survey of these models and extensive literature on the replacement problem was carried out by Rapp (1974) and Pierskalla and Voelker (1976). A generalized machine replacement model with mixed optimization techniques was developed by Sethi and Morton (1972), while the work of Sethi and Chand (1979) was on replacement planning horizon.

Generally, machine replacement models seek to achieve the following objectives.(a) Minimization of total cost (Net present value)

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(b) Minimization of operating cost(c) Maximization of profit(d) Optimal replacement time(e) Sequence of future replacements after the first replacement (Planning Horizon)

The various groups of the replacement models will be discussed under the following headings:(i) New operating machine replacement models(ii) Old operating machine replacement models.(iii) Failed machine replacement models(iv) Low cost items replacement models (Group replacement

models)(v) Planning horizon models (sequence of replacements)(vi) Simultaneous maintenance and replacement models.

2.6.1. New Operating Machine Replacement Models.This group of models makes decision on the replacement date of

an operating machine, starting with a brand new machine. Terborgh (1949) was the first to formulate a replacement theory for new operating machine. The model is based on a time dependent linear operating cost function, K (T) given by

K (T )=C (T+1 )2

+[Q−ST

−Q+S2 ]

…(2.13)

Where Q is the machine costT is the useful life of the machineC is the constant maintenance costS is the salvage value

The objective is to find T that minimizes K (T). Assuming equal depreciation payment, the optimal life of the machine is independent of the rate of interest.

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Clapham (1957) formulated a model which included capital depreciation but excluded salvage value. The model is given by

K (T )=QT

+ 1T∫0

T

f ( t )dt

… (2.14)

Where K(T) is the average sum of capital depreciation and

maintenance cost,f (t) is the maintenance effectiveness function at time t.The objective is to find T that will minimize K(T). The inflation

and interest rates are not considered because the value of money is assumed to be constant over the replacement cycles.

Fetter and Goodman (1957) formulated a model for cost minimization and a second model for profit maximization.

The model for cost minimization, which is a modification of Clapham’s model (1957), excluded the assumption of constant monetary value over the replacement time and the zero salvage value.

The cost minimization model is given by

K (T )=Q−S (T ) RT+∫

0

T

f ( t ) RT dt

1−RT

… (2.15)

Where K(T) is the present value of the total of all futurecosts when an infinite number of machine replacements are considered. S(T) is the salvage value.R is the Discount factor.

The objective is to find T that will minimize K(T).The model for profit maximization is given by

29

W (T )=∫0

T

[P ( t )−f (t ) ]RT dt+S (T )RT−Q

1−RT

… (2.16)

Where W(T) is the total discounted revenue of a machine over an infinite time spanP(t) is production rate at time t.

The objective here is to find T that will maximize W(T).Sasieni et al (1959) formulated a model which assumes that expenditure takes place at the beginning of each year. The model is given by

K (T )=Q+∑t=0

T

Rt−1 f (t )

… (2.17)

Where K(T) is the present value of the expenditure.The objective is to find T that minimizes K(T)

Eilon et al (1966) formulated two models. In both models capital allowance and taxation were included in the models by Clapham (1957) and Fetter and Goodman (1957) to arrive at two cost minimization models.

The first model which is an extension of Clapham (1957) is given by

K (T )=Q−S (T )−CrT

+ 1T ∫0

T

f ( t )dt

… (2.18)

30

Where K(T) is the average sum of capital depreciation and maintenance cost,C is the capital allowancer is the rate of taxationThe second model which is an extension of Fetter and Goodman

(1957) is given by

K (T )=Q−S (T ) RT−BY +∫

0

T

f (t )dt

1−RT

… (2.19)Where K(T) is the Present Value of the total of all future costs

when an infinite number of machine replacements are considered,B is the Present value of capital allowanceY is the taxation rate

In both models the objective is to find T that minimizes K(T).Kent (1977) developed two models. In the first model, capital

allowance and taxation were excluded to obtain the average sum of capital depreciation and maintenance cost given by

K (T )=Q−S (T )T

+ 1T ∫0

T

f (t )dt

… (2.20)In the second model, capital allowances and taxation were also

excluded to obtain the discounted average sum of capital depreciation and maintenance cost given by

K (T )=Q−S (T )RT

T+ 1T∫0

T

f (t )RT dt

… (2.21)

31

Again in both models, the objective is to find T which minimizes K(T). Kent (1977) justified the effect of discounted cash flow on optimal replacement period and explained that there is an optimal delay in replacement required by discounted cash flow with the attendant saving in real cash.

Lake and Muhlemann (1979) modified the second model of Eilon et al (1966) with the inclusion of time taken to install the machine and the exclusion of the capital allowance and taxation.

The model is given by

K (T )=Q−S (T )RT+∫

0

T

f (t )RT dt

1−RT+J

… (2.22)

Where J is the installation time of the machine. They indicated that the salvage value S(T) at any age maybe fixed irrespective of the age of the machine. They then gave the linear form of S (T) as:

S(T) = Q (1 - bt) … (2.23)Where b is set to produce some quantity of annual deterioration.

They also gave the exponential form of S(T) as:S(T) = Q (1 —d)t … (2.24)Where d is the deterioration rate.The objective is to find T which minimizes K(T). It was found that

the replacement period in the model tends to increase with increasing value of d.

Schwartz and McNamara (1983) developed an optimal replacement cycle policy which assumes the existence of an active used equipment market. They also assumed that the maintenance and operating cost function is increasing and discontinuous. They

32

considered two options — when the salvage value is zero and non-zero.

The zero salvage value model is given by

U (T )=[Q+∑t=1

T

Rt f ( t )] /∑t=1T

Rt

… (2.25)The non zero salvage value model is given by:

U (T )=[Q+∑t=1

T

Rt f (t )−S (T )RT ] /∑t=1

T

Rt

… (2.26)Where in both cases U(T) is the uniform annual equivalent cost

of replacement. Again the objective is to find T that minimizes U(T).The models in this family vary in complexities because of the

relaxing of certain assumptions during formulation. They also present computational problems which vary with the complexities and may lead to poor or faulty results, if proper cost functions are not assumed. There is therefore the need to pay greater attention in the estimation of f(t) and S(T).

2.6.2. Old Operating Machine Replacement ModelsThis group of replacement models makes decision on the

replacement date of operating machine starting with an equivalent machine of any age.

Kao (1973) developed a model in this group which assumes that changes of state of a machine follow a discrete time semi — Markov process with the objective of minimizing the average cost per unit time. The model is given by

33

QRA= StKt

… (2.27)

Where QRA is long-run average cost per unit time under RA,St is the expected total cost incurred in each cycle given that RA=t

Kt is the expected number of time units spent in each cycle given that

RA = TBobos (1977) formulated a model that belongs to this group.

The model makes a number of decisions which includes replacement and non — action decisions. This decision is taken at each step of the process whether to replace the machine or not.

The model is given by

Y (i , t )=a i+M 1+R∑j=0

F

PijY (i , t−1)

… (2.28)

Where Y (i, t) is the total discounted cost for state i. ai is the occupancy cost of being in state i, M1 is the amount paid for taking decision 1, Pij is the transition probability of moving from state i to state j.The deterioration of the system, as shown by the change of the

underlying states is represented by the transition probability of a Markov chain. The objective here is to minimize Y (i,t).

Christers and Goodbody (1980) developed a model that gives the total discounted cost per unit time of an operating machine of age n for a further T1 years before replacement and then operating for a further T2 years before another replacement. This model is a

34

combination of the two models of Eilon et al (1966) and Lake and Muhlemann (1979) but excludes capital allowance, taxation and salvage value. The model takes care of a machine of age n under the period of economic uncertainty and inflation.

The model is given by

K (n ,T1 , T2 )=∫0

T

f (n+t )Rt dt+RT 1[Q+∫0

T

f ( t ) Rt dt+RT 2Q ]T 1+T2

… (2.29)Where T1 is the first replacement cycle,

T2 is the second replacement cycle.The model does not always have finite solutions for T1 and T2.

More improvements are therefore required in the solution approach.

2.6.3 Failed Machine Replacement ModelsThis group makes decision on the replacement date of failed

machine. The main focus of this family of models is to ensure that the amount spent on the refurbishment of a failed machine does not exceed the worth of an equivalent replacement machine.

Bellman (1955) developed a dynamic programming model which makes decision to retain or sell a machine after it has failed. The profit maximization model is given by

W (T )=[S (T )−Q ]RT−1+U (0 )RT−1+∑

t=1

T−1

U ( t ) Rt−1

1−RT

… (2.30)Where U (t) = P (t) - f(t) which is the difference between the revenue

and maintenance cost functions.

35

The optimal time T is obtained by successive substitution.Drinkwater and Hastings (1967) formulated a model which is another

member of this group. The model determines the upper limit for any repair cost of a machine. The repair limit equation is given by

r ( t )=θ [ g(t) ]−h(t)… (2.31)

Where r (t) is the repair limit at time t,Ө is the overall average cost per machineg (t) is the expected remaining life of the machineh (t) is the expected total cost of future repairs,r is the cost of repair due to failure

The criterion is that if a machine fails and [r + h (t)]/g(t)<O, the machine should be repaired. But if the machine fails, and [r + h (t)]/g (t)>O, then the machine should be replaced.

Glasser (1967) formulated an age replacement policy model. It involves the minimization of cost per unit utilization or equivalent maximization of utilization per unit cost over an infinite time span. The expected cost per unit utilization is given by

K (T )=C1∫0

T

g ( t )dt+C2∫0

T

g ( t )dt

… (2.32)And the utilization per unit cost is given by

U ( t )=∫0

T

tg ( t ) dt+∫T

∞

Ug (u )du

… (2.33)

Where O<t<T, T<U<∞ andt = time to failure,u = useful life,

36

C1= cost of replacement after failure per unit,C2 = cost of replacement before failure per unit,g (t) = probability density function of t, time to failure.The ratio A(T) = K(T) / U(T) is a measure of the effectiveness of a

replacement policy over an unlimited time span since it represents the long-run expected cost per unit time of utilization while the reciprocal B(T) represents the long-run expected time of utilization per unit cost. The objective of the model is to find T that minimizes

A(T) and maximizes B(T).Nakagawa and Osaki (1974) developed a minimization model

which assumes that repair work starts immediately after machine failure. The optimal replacement limit is given by

L (T )= lim C(t )t

… (2.34)Where L(t) is the expected cost per unit time for an infinite time span,

C (t) is the expected cost up to time t.The decision criterion is that if the repair is not completed

before a fixed repair limit time, the machine is replaced with a new one because the optimal repair limit replacement time minimizes the expected cost per unit time for an infinite time span. The model is appropriate for machine whose failure is not dependent on the age of the machine like the electronic equipment.

Another model on failed machine was developed by Nakagawa and Kowada (1983). The model is centered on machine with minimal failure rate and applied the replacement policy of changing the failed machine at time T or nth failure whichever comes first. The expected cost function of the model is given by:

37

K (n ,T )=C1[n−∑

j=O

n−1

(n− j) [H ( t )] j

j !e−H(t )]+C2

∑j=0

n=1

∫0

T [H (t)] j

j !e−H (t )dt

… (2.35)Where h = 1, 2, 3,

K (α ;T )=lim Kn→α (n ;T )=C1H (T )+C2

T

… (2.36)

And H (T )=∫0

t

h (u )du

Where H (T) is the cumulative hazard function assumed to be increasing and continuous. In equation (2.35), the term in bracket in the numerator is the expected number of failure until replacement while the denominator is the mean time of replacement. The objective again is to find T which minimizes K (∞; T) in equation (2.36).

This family of models focuses mainly on repair cost limit for failed machines without consideration for economic indicators like interest rate and inflation. Only the model by Bellman (1955) took economic parameters into consideration. However, the successive substitution in the model may pose some computational problem.

2.6.4. Replacement Models For Low Cost Item (Group Replacement)This family of models deals with large number of identical low

cost items with high tendency to failure with age. These items have set up cost that is independent of the number replaced. The objective is to determine the optimal time interval for group replacement if it is adjudged to be more economical than individual item replacement due to failure.

38

Ackoff and Sasiene (1967) developed a model for low cost items. They assumed that failed units are replaced as failure occurs. In addition all units are replaced at a specified interval. No record of the age of individual items is kept since the value of each item is so small. The model is given by

K (T )=Cu∑

t=1

T=1

X t

T+CgN o

T

… (2.37)Where K(T) is the total replacement cost per period through T

periodXt is the number of replacement made at the end of the tth periodN o is the initial number of unit in the groupCg is the cost of group replacement per unitCu is the cost of unit replacement.

Given the probability of failure at t, 2t, 3t they obtained a simple set of relationships from which X1 X2, X3 ... may be successively calculated. The objective is to find T which minimizes K(T).

Fabrycky et al (1987) developed a group replacement model for low cost items that modifies the model by Ackoff and Sasieni (1967). The total replacement cost per period through T periods is given by

K (T )={Cu∑t=1

T=1

N0 [P (t ) +∑

j=1

t=1

P ( j)P (t−1)]+∑b=1t=1 [∑j=1

b=1

[P ( j )P (b− j ) ]P ( t−b )+… ]+CgN O}/T… (2.38)

Where N0 is the initial number of units in the group P(t) is the probability of failure during the period t, Cu is the cost of unit replacement

39

Cg is the cost of group replacement per unitAgain, the objective is to find T which minimizes K(T).

Generally, this group of models imposes heavy penalties for not making replacements on or before failure. In practice, failure can be tolerated to a convenient time for possible replacement of failed items provided that production or services are not undermined. Again, economic parameters were not considered in the development of this family of models.

2.6.5. Planning Horizon ModelsThis group of models deals with the sequence of replacement in

a planning horizon.Sethi and Chand (1979) developed a minimization model for the

selection of optimal number of machines and their replacement times. They considered the situation of a production shop which must keep a single machine of a particular type at all times. They also considered the increasing maintenance cost of the machine as it gets older, the salvage value when sold and the cost of buying a new one for replacement.

Consequently a chain of replacement decisions has to be made for any finite or infinite planning horizon. The objective is to select the optimal number of machines and replacement times that will minimize the total cost. The model is given by

K (T )=∑i=0

n=0

Qti+1+∑i=0

n=1

∑K=ti+1

ti+1

M ti+1.k−∑i=0

n−1

S ti+1 ,t (i+1 )

… (2.39)Where ti>1, i=1,2,3 ..., (n—1),and tn=T, to =o.

Qt = Price of machine At at the beginning of period t,Mt,k = Maintenance cost of At in period kSt,k = Salvage value of At at the end of period k

40

n = Number of machine replacement during the length of the horizonAt denotes the machine of vintage t.

It is assumed that the shop goes out of business at the end of the horizon. This is not usually true in real situation. The cumbersome forward algorithm solution procedure was employed.

2.6.6. Simultaneous Maintenance and Replacement ModelsThis group of models deals with the combined problems of

maintenance and replacement by determining the optimal maintenance policy and replacement dates.

The models in this group include those of Naslund (1966), Thompson (1968), Virtanen (1982) and Mehrez et al (2000).The model by Thompson (1968) is concerned with the determination of the sequence of maintenance decisions that maximizes the present value of the machine and ensures the recovery of maintenance expenditure from the increased profitability arising from the machine maintenance before disposal. The maximization model is given by.

V (T )=S (T ) e−rt+∫0

T

Q ( t ) e−rt dt

… (2.40)Subject to: O <m (t) <M, Q (t) = P (t) S (t) — m (t); ds (t)/dt = -a (t) + f(t) m(t) and s(o) = kK = Purchase cost of the machineT = Sales date of the machiner = Constant interest rateV (t) = Present value of machine at time t.S (t) = Salvage value of machine at time t, Q (t) = Net operating receipts at time t, m (t) = Maintenance policy — hours of maintenance at time dt, f (t) = Maintenance effectiveness function at time t,

41

a(t) = Obsolescence function at time t, P(t) = Production rate at time t.

The model implies that for a machine to be held long enough after maintenance, there is the need to set aside sufficient money for the maintenance of the machine since there is the possibility of the recovery of the maintenance expenditure from the increased profitability resulting from the maintenance of the machine.

Another notable model under this group is the one developed by Tapiero and Itzhak (1979) which is an extension of the model by Thompson (1968).

They considered the simultaneous maintenance and replacement problem under uncertainty. They assumed that maintenance and deterioration have a probability effect of the Markovian type on the salvage value of the machine. They then applied the tools of optimal control theory to determine a certainty — equivalent maintenance programme and the optimal replacement for the machine.

The model by Humphreys (1991) is worth mentioning here. The model is developed from the accounting viewpoint. Decision on replacement date of an existing machine is made on a yearly basis. The present value of receipts and expenses after tax of the existing machine is compared with the cost of a new machine to show which is more economical. If the former is greater, the existing machine is replaced. It is retained if otherwise.

The present value of the cost after tax on one year basis is given by

P1=S0+r (B0−S0)−S1+r (B0−S1 )−C1(1−r)

1−R

… (2.41)Where S0 = Salvage value of the existing machine now

42

S1 = Salvage value of the existing machine one year from nowBo = Book value of the existing machine now r = Tax rateC1 = Operating cost for the coming yearR = Discount factorLet Q be the cost of the new machine (replacer), then ifP1 > Q, the existing machine is replaced.P1 <Q, the existing machine is retained for one year.

The model took care of inflation and economic parameters. It however failed to consider the possibility of machine breakdown before the end of the year, when decision is taken. It therefore means that the machine will not be in operation until the end of the year when decision to retain or replace it will be made. The model should be improved along this line of argument.

In this chapter, issues associated with machinery replacement have been discussed. In some cases, the origin of the principles and the earlier application of some of these principles were mentioned. We also discussed the objectives of various replacement models. We are now equipped with the necessary background knowledge and tools to develop a machinery replacement model. The two succeeding chapters therefore deal with the methodology and the development of the model.

43

CHAPTER THREE

METHODOLOGY

The methodology adopted in this study will be discussed first before the development of the replacement model. Issues raised in this discussion will be very significant in model development and solution technique.

It was established in the previous chapter that deterioration is a major factor in facility failure and indeed machinery replacement decisions. This is true because deterioration affects the total and maintenance costs, the salvage value and indeed the replacement date of any equipment.

Cognizance of the effect of deterioration on machinery replacement is yet to be fully appreciated. Emphasis had usually been placed on the minimization of total and maintenance costs as well as the maximization of profit to arrive at the optimal replacement date. Values for deterioration are usually assumed or at best determined by methods that are highly subjective.

Even the popular failure analysis (Sachs; 2007) like the Component Failure Analysis (CFA), Root Cause Investigation (RCI) and the Root Cause Analysis (RCA) are not only subjective, they are also expensive. Some of the replacement models even exclude the salvage value in the build-up of cost. Yet it is the value that is directly affected by deterioration.

In this study, the methodology adopted comprises:i. The analytical processii. The experimental process

3.1 THE ANALYTICAL PROCESSThe analytical process involves the following steps:i. Values for deterioration are generated as random numbers

using the Monte Carlo simulation under the uniform

44

(rectangular) probability distribution with machinery useful life as the upper limit and zero, the lower limit. Useful lives of machinery, their parts and components are published by US Treasury Department, Bureau of Internal Revenue, 1942.

Recall eq. (2.9), thus

d=Pu ( x )=∫0

x 1udx= x

u0≤ x≤u(3.1)

where d = Deterioration rateu = Useful lifex = Generated random number

ii. The salvage value function by Lake and Muhlemann (1979) is differentiated to obtain the initial replacement time for an enumeration process for the optimal replacement date of the cost minimization model.

ThusS(t) = Q (1-d)t (3.2)

Where S(t) = Salvage valueQ = Cost of Machinet = Initial replacement time

Differentiating Eq. (3.2) with respect to (w.r.t) t,

ds = Q(1-d)t ln (1-d) (3.3)dt

Therefore t = - (3.4)

Where t is the initial time which may be used to start an enumeration process for the optimal replacement date of the model.

45

logQlog(1-d)

Values of the salvage function are verified (see appendix I)

iii. The regression line for maintenance cost is obtained by least square method (see Appendix III). The maintenance cost function is given by Walker (1994).

Thus

∫0

t

B ( t )dt=b1+b2 t(3.5)

Where b1 = constant component of the maintenance cost (Preventive)

b2 = the increase per period in maintenance cost (Corrective)

The type of distribution for the maintenance cost function is presented in Appendix V.

iv. Applying the dynamic programming technique (Humphreys, 1991) and using t from eq.(3.4) as the starting time in the enumeration process, the present value of the total cost, K(T) reaches a minimum which corresponds to the optimal replacement date.

3.2 CALIBRATION OF THE MODELThe model is calibrated by the application of data from a hypothetical front-end loader. A graph of the present value of the measured and predicted total costs, K(T) of the front-end loader against time is plotted. Then the correlation coefficient, rc of the total costs, K(T) of the front-end loader is obtained to ascertain the strength and linear relationship between the two variables, as well as their degree of

46

dependence. This process is replicated with field data of the other machines.

3.3 THE EXPERIMENTAL PROCESSIn the experimental process, field data were obtained from the following companies:i. Mothercat Limited, Mando Kadunaii. Juhel Nigeria Ltd, Emene, Enuguiii. Innoson Technical and Industrial Company Nigeria Ltd, Emene

Enugu

3.4 VERIFICAITON OF MODELi. The model is tested with field data obtained from these

companies to verify the reliability of the results from the model. ii. The enumeration is presented in tabular format. The reaching of

K(T) to a minimum on the enumeration table is an indication of an optimal replacement date. Maximum K(T) suggests repairs/refurbishment.

iii. Graphs of measured and predicted K(T) against time using the field data are plotted.

iv. Correlation coefficient, rc of measured and predicted K(T) using field data from various machines are derived to ascertain the strength and linear relationship of the variables.

3.5 COMPARISON WITH EXISTING MODELSi. Results from the model are compared with the results obtained from

the model by Walker (1994) in terms of reliability, operationability and perhaps savings.

47

ii. Field data obtained from Associated Bus Company Plc in respect of MB Executive Express Bus and Toyota Hiace Shuttle Bus are used to derive values for K(T) which reach a minimum, corresponding to the replacement date.

iii. The model by Walker (1994) is applied on data obtained from an ambulance in a fleet of a local authority in Northern England to generate computer aided graphs for analysis.

iv. The model by Walker (1994) is also applied on data obtained from the Express Bus and Hiace Bus since the Ambulance data are not published.

v. The results of the model are also compared with those obtained from the Kent(1977) and Schwartz/McNamara(1983) models using the data of the Express and Shuttle buses(Appendix VI).

The development of the model and solution (Results) will be presented in the next chapter.

CHAPTER FOUR

MODEL DEVELOPMENT AND SOLUTION

The objective of machinery replacement models has always been to find a replacement date that will minimize total maintenance and operating costs and maximize profit.

Some models exclude salvage (resale) value in the cost build-up while most models trivialize the effect of deterioration on the resale value and indeed in the build-up of costs. Even in the cases where the effect is

48

considered, values of deterioration are assumed or at best, determined by methods that are highly subjective and expensive at times.

The development of the model and solution (results) follow below.

4.1 MODEL FORMULATION The model K (T) has three cost components (centre) namely

1) Purchase cost, Q2) Maintenance cost, B(t)3) Salvage (Resale) Value, S(t)

4.1.1 Assumptionsi. The purchase cost, Q is constant and may include the cost of

installation and other incidental cost before the machine is put into use.

ii. The maintenance cost, B(t) is non-linear as it increases disproportionally whenever there is refurbishment.

iii. The coefficient b1 represents the constant component of the maintenance cost (routine maintenance)

iv. The coefficient b2 represents the increase per period component of the time – dependent maintenance cost. It takes care of downtimes.(corrective maintenance).

v. The model assumes the existence of an active resale market where the resale of equipment will attract maximum salvage value.

vi. It also assumes the existence of an internal market where an in-house use of the machine for less demanding standby capacity will attract a resale value represented by the savings that ensue.

vii. The resale value is assumed to decrease with deterioration and time in an exponential form given by Lake and Muhlemann (1979)

Firstly, the maintenance costs and the salvage values at various points in time are discounted to take care of inflation and deterioration thus bringing them at par with the cost of the machine. The difference between the salvage values and the sum of the machine cost and maintenance costs gives the total cost which is discounted to arrive at its present value.

The equation for total cost is given by,

K (T )= 1R [Q+B ( t )R−S ( t )Rt ](4.1)

49

AndK (T )= 1

R [Q+(b1+b2t ) R−Q (1−d )tR t ](4.2)

Where K(T) = Present value of total cost B(t) = Time-dependent (increasing) Maintenance cost S(t) = Salvage value (Deterioration and time dependent) Q = Cost of machine (constant) R= Discount factor ( 100r+100 ) r= Rate of return on replacement investment d = Deterioration rate As usual, the objective is to find T that will minimize K (T).

4.2 THE FEATURES OF THE MODEL

The model (Eq.4.2) is a modification of two models. The first model is formulated by Lake and Muhlemann (1979) and the second by Walker (1994). The major features of the hybrid model are as follows:

i. The model is simple and incorporates basic cost components in the build-up of total cost.

ii. It lays emphasis on the effect of deterioration on resale value S ( t ) and indeed on the total cost K (T ) .

iii. Economic parameters like the rate of return on replacement investment, r and discount rate,R are considered in the model. Those costs that occurred at different past points in time need to be discounted to present values to take care of inflation. Replacement investments are forced to compete for capital with alternative investments. Rate of return on replacement investments,r takes care of the amount of investments on which the company will forgo earning elsewhere.

iv. The model is amenable to simple and straight-forward solution techniques like optimization, dynamic programming and economic lot size inventory control method.

v. In the model, deterioration is treated as a stochastic variable and generated as random number under uniform probability distribution. This reduces the error due to the usual subjective methods of determining deterioration.

50

vi. The model is amenable to review and adjustments in terms of input variables like deterioration and economic parameters. In fact the model can be restructured if the need arises.

vii. The model does not involve rigorous enumeration and computation in its solution technique.

The model has to be tested with field data to verify the reliability of its results before any possible review and adjustment.

4.3 SOLUTIONS (Results)The model is calibrated by a hypothetical machine while

the verification is done by the application of field data from three companies namely:

(i) Mothercat Limited, Kaduna (Construction)(ii)Juhel Nigeria Limited, Emene, Enugu (Pharmaceutical)(iii) Innoson Technical & Industrial, Co. Nig. Ltd, Enugu

(Plastics).Finally, the model is compared with the model by Walker

(1994) for possible advantages in terms of reliability, operationability and savings.

The solution technique adopted in this work is the use of dynamic programming method. Dynamic programming is a method of solving problems exhibiting the properties of overlapping sub-problems and optimal substructure. Optimal substructure means that optimal solutions of sub-problems can be used to find the optimal solution of the overall problem. (Wikipedia encyclopedia, 2008).

The new model exhibits the properties of overlapping sub-problems and optimal substructure.

4.4 CALIBRATION OF THE MODELThe model is calibrated by the use of a hypothetical

machine with the following technical details.

Name of machine: Front-end LoaderModel: YYear of manufacture: 1995Cost: $60,000.00

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Resale Value (Active Market): $40,000.00 (5th year)Deterioration rate: 25%Capitalization rate: 40%Minor maintenance: $2,500.00 annuallyMajor maintenance: (i) $10,000.00 (2nd year)

(ii) $3,500.00 (3rd year)(iii) $6,000.00 (4th year)(iv) $20,000.00 (5th year)

(Source: ASA, 2000)

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Q + B(t)R – S (t)Rt

R

Q + (b1 + b2t) R – Q (1-d)tRt

R

Below is the tabular presentation of the above data

Table 4.1: Data for model Y, Front-End Loader

t Q b1 b2t S(t)1 60,000 2500 - -2 60,000 2500 10,000 -3 60,000 2500 3,500 -4 60,000 2500 6,000 -5 60,000 2500 20,000 40,0006 60,000 2500 - -7 60,000 2500 - -

(Source: ASA 2000)The model is given byK(T) = (4.1)

K(T) = (4.2)

K(T) = Present Value of Total Machine CostQ = Purchase Cost

S(t) = Q(1-d)t = Resale Value

B(t) = b1 + b2t = Maintenance Cost

r = Rate of return (40%)

R=Discount rate=100140

=0.714

d = Deterioration rate = XU=56=0.83

U = Average useful life = 6 (US Treasury Dept. 1942)

53

ts = The starting time for the enumeration process

ts = − logQlog(1−d)

= −log 60,000log (1−0.83)

= −log 60,000log(0.17)

= 6.2

= 6

:. The starting time, ts for the enumeration process is 6.

4.4.1 Deterioration rate, d.

Random numbers between zero and six are generated by

the use of computer. Below is the detail of the ten random

numbers generated by computer.

TABLE 4.2: Random Numbers and Deterioration rates

Random numberX

Frequencyf

Probability of X P(X)

Deterioration d

2 1 0.1 0.333 2 0.2 0.505 3 0.3 0.836 4 0.4 1.00

Total 10 1.0 -54

55

Average X = = 4.7

= 5

d= XU

=56

= 0.83

Where U = 6 (average useful life of construction machines)

4.4.2 The enumeration process (solution)Below is a tabular presentation of the enumeration process

using the dynamic programming method with data from Table 4.1

Table 4.3: Enumeration of Data for Hypothetical front-end Loadert Q R Rt b1 b2t B(t) B(t)R d S(t) S(t)Rt K(T)1 60,000 0.71 0.71 2500 - 2500 1785 0.33 - - 865332 60,000 0.71 0.51 2500 10,000 12500 8925 0.50 - - 965333 60,000 0.71 0.36 2500 3500 6000 4284 0.50 - - 900334 60,000 0.71 0.26 2500 6000 8500 6069 0.83 - - 925335 60,000 0.71 0.19 2500 20,000 22500 16,065 0.83 40,000 7424 961366 60,000 0.71 0.13 2500 - 2500 1785 0.83 40,000 5300 79111*7 60,000 0.71 0.09 2500 - 2500 1785 1.00 40,000 3784 81234

10 60,000 0.71 0.03 2500 - 2500 1785 1.00 40,000 1376 84606

56

4710

Table 4.4: Predicted K(T) for Front-End Loader

T Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)1 60000 0.71 0.71 6653.82 4750.83 0.33 894372 60000 0.71 0.51 16653.64 1189.70 0.50 992983 60000 0.71 0.36 10153.47 7249.58 0.50 928884 60000 0.71 0.26 12653.29 9034.45 0.83 953535 60000 0.71 0.19 26653.11 19030.32 0.83 8.52 1.58 1091586 60000 0.71 0.13 6652.93 4750.19 0.83 1.45 0.19 89435*7 60000 0.71 0.09 6652.75 4750.07 1.00 0 0 89435

10 60000 0.71 0.03 6652.22 4749.69 1.00 0 0 89435

(MAINTEANCE COST REGRESSION LINE, B(t) = 6654 – 0.1778t).b1 = 6654b2 = -0.1778b2 = Cost of Refurbishment in any year of refurbishment (Overhaul)

TABLE 4.5: K(T) against Time (Measured L and Predicted M)

t 1 2 3 4 5 6 7 8K(T)L 86533 96533 90033 92533 96136 79111 81234 84606K(T)M 89437 99298 92888 95353 109158 89435 89435 89435%Dev. -3.24 -2.78 -3.07 -2.96 -11.93 -11.54 -9.17 -5.40

57

1 2 3 4 5 6 7 870000

75000

80000

85000

90000

95000

100000

105000

110000

115000

120000

Fig 4.1: Comparison of the measured and predicted total cost K(T) of the Loader

Measured DataPredicted Values

Years

Total C

ost $

4.5 VERIFICAITON OF THE MODELThe model is tested with field data of machines from three

companies to verify the reliability and operationability of the model.

The companies are as follows:(i) Mothercat Limited, Kaduna (Construction)(ii) Juhel Nigeria Limited, Enugu (Pharmaceutical)

58

(iii) Innoson Technical & Industrial Company Nigeria Limited, Enugu (Plastics)

4.6 COMPANY PROFILEName: Mothercat LimitedLocation: Mando Industrial Layout, KadunaYear of Incorporation: 1992Equity composition: 100% foreignStaff strength: 2025 (2008)Type of business: Construction

Technical Details of MachineName of Machine: CAT 140H Motor GraderSerial Number: 5HM03439Equipped with: (i) One side glass cabin

(ii) Three Shank ripper, no scarifier(iii) 14’ mouldboard

Condition: NewCost: N39, 901,680.00Resale Value of identical Grader: N27, 450,000 (3 years old)

Major Overhaul: N14, 555,656.00 (3rd year)Minor work: (i) reconditioning of injector pump

N140, 000.00 (2nd year)(ii) Electrical works N48, 000.00 (3rd year)

Running cost/preventive maintenance(i) N80,000.00 (1st year)(ii) N90,000.00 (2nd year)(iii) N105,000.00 (3rd year)

59

(iv) N120,000.00 (4th year)(v) N130,000.00 (5th year)(vi) N145,000.00 (6th year)(vii) N160,000.00 (7th year)

Deterioration rate: 12.5%Capitalization rate: 25%

(Source: Mothercat Limited, Kaduna)

The above data are presented in the table below

Table 4.6: Technical Data for CAT 140H Motor Grader

t Q b1 b2t S(t)1 39,901,680 80,000 -2 39,901,680 90,000 140,0003 39,901,680 105,000 14555656 27,450,0004 39,901,680 120,000 - -5 39,901,680 130,000 - -6 39,901,680 145,000 - -7 39,901,680 160,000 - -

(Source: Mothercat Limited, Kaduna)

See Appendix II for values for enumeration

Table 4.7: Enumeration of data for a CAT 140H Motor Grader60

t Q R Rt b1 b2t B(t)R S(t) S(t)Rt K(T)

1 39,901,680 0.80 0.80 80000 - 64000 - - 49957100

2 39,901,680 0.80 0.64 90000 140000 184000 - - 50107100

3 39,901,680 0.80 0.51 105000 14555656

11728525 27450000 14054400 46969756

4 39,901,680 0.80 0.41 120000 - 96000 27450000 11243520 35942700*

5 39,901,680 0.80 0.33 13000 - 104000 27450000 8994816 38763580

6 39,901,680 0.80 0.26 145000 - 116000 27450000 7195743 41027421

7 39,901,680 0.80 0.21 160000 - 128000 27450000 5756540 42841425

10 39,901,680 0.80 0.11 - - - 27450000 2948130 46191938

11 39,901,680 0.80 0.09 - - - 27450000 2357937 46929679

TABLE 4.8: Predicted K(T) for CAT 140H Motor Grader

t Q R Rt B(t) B(t)R S(t) S(t)Rt K(T)1 39901680 0.80 0.80 205542.12 164433.70 500826422 39901680 0.80 0.64 205542.25 164433.80 500826423 39901680 0.80 0.51 14555656 7452495.80 34210703 17515879 461768214 39901680 0.80 0.41 205542.49 164433.99 23928852 9801258 37831070*5 39901680 0.80 0.33 205542.61 164434.09 9468856 3102944 462039626 39901680 0.80 0.26 205542.74 164434.19 10459986 2741562 466556907 39901680 0.80 0.21 205542.86 164434.29 8367989 1754767 47889184

[B(t) = 205542 + 0.1227t]

TABLE 4.9: K(T) against Time (Measured, G and Predicted, M)

t 1 2 3 4 5 6 7K(T)G 49957100 50107100 46969756 35942700 38763580 41027421 42841425K(T)M 50082642 50082642 46176821 37831070 46203962 46655690 47889184

%Dev. -0.25 0.05 1.72 -4.99 -16.10 -12.06 -10.54

61

1 2 3 4 5 6 70

10000000

20000000

30000000

40000000

50000000

60000000

Fig 4.2: Comparison of the measured and predicted total costs K(T) of the Grader

MeasuredPredicted

Years

Tota

l Cos

ts N

4.7 COMPANY PROFILEName: Juhel Nigeria LimitedLocation: 35 Nkwubor Road, Emene Enugu,Year of Incorporation: 1987Year Business Started: 1993Type of Business: Pharmaceuticals & Ozonized Table

WaterStaff Strength: 400 (2008)Equity Composition: 100% Nigerian

Technical details of MachineName: Capsule Filling MachineMake: LeoscarModel: MX-115-IISerial Number: 7298648Capacity: 1500 Capsules per minuteRating: 415V, 50HP, 1500rpm

62

Year of Manufacture: 1993Condition: NewCost: N1, 440,000.00Capitalization rate: 20%Resale Value of identical machine: N830, 000.00 (15yrs old)Maintenance History:Major refurbishment (i) N210, 000.00 (2nd year)

(ii) N105, 000.00 (6th year)(iii) N310, 000.00 (10th year)(iv) N415, 000.00 (15th year)

Routine maintenance cost

(i) N10, 000.00 (1st year)

(ii) N12, 500.00 (2nd year)

(iii) N14, 000.00 (3rd year)

(iv) N16, 500.00 (4th year)

(v) N18, 000.00 (5th year)

(vi) N19, 000.00 (6th year)

(vii) N20, 000.00 (7th year)

(viii) N21, 500.00 (8th year)

(ix) N23, 000.00 (9th year)

(x) N26, 000.00 (10th year)

(xi) N27, 000.00 (11th year)

(xii) N28, 000.00 (12th year)

(xiii) N29, 500.00 (13th year)

63

(xiv) N31, 000.00 (14th year)

(xv) N33, 000.00 (15th year)

[Source: Juhel Nigeria Limited, Emene Enugu]

64

The data above are summarized in Table 4.10 below

Table 4.10: Technical Data for Capsule Filling Machinet Q b1 b2t S(t)1 1440000 10000 - -2 1440000 12500 210000 -3 1440000 14000 - -4 1440000 16500 - -5 1440000 18000 - -6 1440000 19000 105000 -7 1440000 20500 - -8 1440000 21500 - -9 1440000 23000 - -

10 1440000 26000 310000 -11 1440000 27000 - -12 1440000 28000 - -13 1440000 29500 - -14 1440000 31000 - -15 1440000 33000 415000 830000

[Source: Juhel Nig. Limited]

Table 4.11: Enumeration of data for Capsule Filling Machine starting from the 9th year.

t Q R Rt b1 b2t B(t)R S(t) S(t)Rt K(T)9 1440000 0.83 0.19 23000 - 19090 - - 175794610 1440000 0.83 0.16 26000 31000

0278880 - - 2070947

11 1440000 0.83 0.13 27000 - 22410 - - 176194612 1440000 0.83 0.11 28000 - 23240 - - 176294613 1440000 0.83 0.09 29500 - 24485 - - 176444614 1440000 0.83 0.07 31000 - 25730 - - 176594615 1440000 0.83 0.06 33000 41500

0371840 830000 50713 2121847

16 1440000 0.83 0.05 - - - 830000 42104 1684218*17 1440000 0.83 0.04 - - - 830000 34947 1692841

65

Table 4.12: Enumeration of data for Capsule Filling MachineStarting from the first year and projecting routine maintenance costs for 16th and 17th years

t Q R Rt b1 b2t B(t)R S(t) S(t)Rt K(T)1 1440000 0.83 0.83 1000 - 8300 - - 17449462 1440000 0.83 0.69 12500 210000 184675 - - 19574473 1440000 0.83 0.57 14000 - 11620 - - 17489464 1440000 0.83 0.47 16500 - 13695 - - 17514465 1440000 0.83 0.39 18000 - 14940 - - 17529466 1440000 0.83 0.33 19000 105000 102920 - - 18589467 1440000 0.83 0.27 20500 - 17015 - - 17554468 1440000 0.83 0.23 21500 - 17845 - - 17564469 1440000 0.83 0.19 23000 - 19090 - - 175794610 1440000 0.83 0.16 26000 310000 278880 - - 207094711 1440000 0.83 0.13 27000 - 22410 - - 176194612 1440000 0.83 0.11 28000 - 23240 - - 176294613 1440000 0.83 0.09 29500 - 24485 - - 176444614 1440000 0.83 0.07 31000 - 25730 - - 176594615 1440000 0.83 0.06 33000 415000 371840 830000 50713 212184716 1440000 0.83 0.05 34000 Projected 28220 830000 42104 1718218*17 1440000 0.83 0.04 35000 Projected 29050 830000 34947 1727841

TABLE 4.13: Predicted K(T) For Capsule Filling Machine

T Q R Rt B(t) B(t)R S(t) S(t)Rt K(T)9 1440000 0.83 0.19 1199.22 995.35 1736145

10 1440000 0.83 0.16 1200.80 996.66 173614711 1440000 0.83 0.13 1262.37 997.97 173614812 1440000 0.83 011 1203.95 999.28 173615013 1440000 0.83 0.09 1205.53 1000.59 173615114 1440000 0.83 0.07 1207.11 1001.90 173615315 1440000 0.83 0.06 416208.60 345452.64 0.00005 0.00 215115516 1440000 0.83 0.05 1210.27 1004.53 0 0 173615617 1440000 0.83 0.04 1211.85 1005.84 0 0 1736158

66

[B(t) = 1185+1.5795t]

TABLE 4.14: K(T) against Time (Measured, C and Predicted, M)

t 9 10 11 12 13 14 15 16 17K(T)C 1757946 2070947 1761946 1762946 176444

61765946 2121847 1718218 1727841

K(T)M 1736145 1736147 1736148 1736150 1736151

1736153 2151155 1736156 1736158

%Dev. 1.26 19.28 1.49 1.54 1.63 1.72 -1.36 -1.03 -0.48

67

1 2 3 4 5 6 7 8 91500000

1600000

1700000

1800000

1900000

2000000

2100000

2200000

2300000

2400000

2500000

Fig 4.3: Comparison of the measured and predicted total cost K(T) of the Capsule Filling Machine

Measured DataPredicted Values

Years

Total C

ost N

4.8 COMPANY PROFILEName: Innoson Technical and Industrial Company

Nigeria LimitedLocation: KM16 Enugu/Abakiliki Expressway, Emene,

Enugu,Year of Incorporation: 2001

68

Year business started: 2004Equity Composition: 90% Nigerian

10% Foreign Staff strength: 1200 (2008)Nature of business: Plastic productsInstalled Capacity: 600Ton/monthPresent Capacity: 400Ton/monthRaw Material: (i) ABS material

(ii) Poly Propylene (PP Injection)(iii) Master batch

Technical Details of MachineName: Injection MachineMake: HAITIANModel: HTF/1000WSerial Number: 0705100019031Power rating: 166.35KWVoltage Input: 380VInjection Size: 5768 Cm3

Year: 2004Condition: NewCost: N25, 057,500Resale Value: Not Available Major Overhaul: None yetRoutine/Preventive maintenance:

(i) N50,000 (1st year)(ii) N55,000 (2nd year)(iii) N60,500 (3rd year)

69

(iv) N62,000 (4th year)Capitalization rate: 30%Deterioration rate: 15%

[Source: Innoson Technical and Industrial Company Nigeria Limited, Enugu]

Table 4.15: Technical Data for Injection Machine

t Q b1 b2t S(t)1 25057500 50000 - -2 25057500 55000 - -3 25057500 60500 - -4 25057500 62000 - -

[Source: Innoson Technical & Industrial Company Nigeria Limited, Enugu]

Table 4.16: Enumeration of Data for Injection Machine with Projected Routine Maintenance costs for the 12th, 13th & 14th years

t Q R Rt b1 b2t B(t)R d S(t) S(t)Rt K(T)1 25057500 0.77 0.77 50000 - 38500 0.06 23554050 18136619 90381582 25057500 0.77 0.59 55000 - 42350 0.12 19404528 11504945 176556443 25057500 0.77 0.46 60500 - 46585 0.18 13815904 6306960 244117444 25057500 0.77 0.35 62000 - 47740 0.24 8359727 2938444 28787922

12 25057500 0.77 0.04 90500 - 69685 0.70 13.32 0.58 3263256513 25057500 0.77 0.03 100000 - 77000 0.70 3.99 0.13 3264206614 25057500 0.77 0.03 110000 - 84700 0.82 0.0009 00000 32652066

Table 4.17: Enumeration of Data for Injection Machine starting with the 12th year and Projected Machine Overhaul in the 12th year plus Routine Maintenance in the 12th, 13th & 14th years.

70

t Q R Rt b1 b2t B(t)R d S(t) S(t)Rt K(T)12 25057500 0.77 0.04 90500 6000000 4689685 0.70 13.32 0.5779 3863253913 25057500 0.77 0.03 100000 - 77000 0.70 3.99 0.1334 32642066*14 25057500 0.77 0.03 110000 - 84700 0.82 0.01 0 32652066

TABLE 4.18:Predicted K(T) For Injection Machine (Projected Machine Overhaul in the 12th Year)

t Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)4 25057500 0.77 0.35 13901.12 10703.86 0.24 8359727 2938444 28739823

12 25057500 0.77 0.04 6013911.37 4630711.75 0.70 13.32 0.58 3855595113 25057500 0.77 0.03 13912.65 10712.74 0.70 3.99 0.13 32555978*14 25057500 0.77 0.03 13913.93 10713.72 0.82 0.01 0 32555980

[B(t) = 13896 + 1.2805t]

TABLE 4.19:K(T) against Time with Projected Machine Overhaul in Year 12TH

(Measured, I AND Predicted, M)t 4 12 13 14

K(T)I 28787922 38632539 32642066 32652066K(T)M 28739823 38555951 32555978 32555980

%Dev. 0.17 0.20 0.26 0.30

71

1 2 3 427000000

29000000

31000000

33000000

35000000

37000000

39000000

41000000

43000000

45000000

Fig 4.4: Comparison of the measured and predicted total costs K(T) of the In-jection Machine with Projected Machine Overhaul in the 12th year

Measured DataPredicted Values

Years

Total C

ost N

TABLE 4.20:K(T) against Time (Measured, N and Predicted, M)

t 4 12 13 14K(T)N 28787922 32632565 32642066 32652066K(T)M 28760534 32576681 32576681 32576682

%Dev. 0.09 0.17 0.20 0.23

72

[B(t) = 34610 + 0.4145t]

1 2 3 426000000

27000000

28000000

29000000

30000000

31000000

32000000

33000000

Fig 4.5: Comparison of the measured and predicted total costs K(T) of the In-jection Machine

Measured DataPredicted Values

Years

Total C

ost N

4.9 COMPARISON WITH OTHER MODELThe results of the model for long distance bus and a shuttle bus (Minibus) are compared to the result obtained from the model by Walker (1994). Walker (1994) conducted a replacement analysis using data from a fleet of Ford Transit ambulances of a local authority in Northern England.

The model by Walker (1994) is given by

73

R (m )=[P+∑t=1

m

A ( t )d t−1−S (m)dm]/∑t=1m

d t−10≤m≤max(4.3)

Wheremax = a management imposed upper limit on an

acceptable replacement cycle.R(m) = the equivalent average cost per period for a

replacement cycle of 0 < m < max. periods.m = economic life.r = rate of return on replacement investment, r> o.d = the discount factor = 100/(100+r), o<d<1p = the purchase and installation cost, p>o.A(t) = the age dependent running cost per period for a

machine of age t.= a1 + a2t, o< t < max.

a1 = Constant component of the age dependent running costs (intercept)

a2 = increase in age dependent running costs per period (slope)

S(t) = the resale value of a machine of age t.= PS1 (S2)t, O< S1 < 1, O< S2 < 1, 1 < t < max.

S1 = the deterioration immediately after a purchase (intercept of regression)

S2 = the periodic deterioration (slope of regression)

The associated net present value is given by NPV = R(m)/(1-d) …(4.4)

Walker (1994) used a computer program to produce graphs for the analysis. The plots (graphs) produced depended on the values of S1

and S2 with the rate of return between 10% and 20% for discounting

74

purposes (Walker, 1994, pp58-63). The local authority sells their vehicles in the seventh year usually to schools and charitable organization.This date coincides with the economic life of such vehicles. The graphical analysis indicates a replacement date between six and eight years. Walker (1994) then concludes that a replacement policy after seven years is a robust policy in the light of available data.

4.10 DATA FOR COMPARISONCompany ProfileName: Associated Bus Company PlcLocation: Km5 MCC Road, Umuoba, Uratta, Owerri

Imo StateYear of Incorporation: 5th April, 1993Staff Strength: 1210 (2008)Equity Composition: 85% Nigerian

15% ForeignNature of Business: Transport services (Passenger & Cargo)

Technical Details:a. Name of Bus: Executive Express Bus (Coach)

Make: Mercedes BenzType: 0500RSCapacity: 50- Seater (Passengers)Registration Number: XC 700 EZAChassis Number : 9BM6340116B500571Engine Number: 457932U0868802Year: 2000Condition: NewCost: N25, 833,403.96Resale value of identical bus: N2, 000,000.00 (5 yrs old)

75

Capitalization rate: 20%Major Maintenance: (i) N830, 690.00 (3rd year)

(ii) N1, 750,000.00 (5th year)Minor Maintenance: (i) N250, 000.00 (1st year)

(ii) N255, 000.00 (2nd year)(iii) N265, 000.00 (3rd year)(iv) N270, 000.00 (4th year)(v) N282, 000.00 (5th year)(vi) N291, 000.00 (6th year)(vii) N302, 000.00 (7th year)(viii) N325, 000.00 (8th year)

b. Name of Bus: Hiace Minibus (Shuttle)Make: ToyotaCapacity: 15- Seater (Passengers)Registration Number: XJ 573 EPEChassis Number: JTFJX02P605005336Engine Number: 2TR-0377458Year: 2000Condition: NewCost: N3, 950,000.00Resale value of identical bus: N850,000.00 (5yrs old)Capitalization rate: 20%Major Maintenance: (i) N362, 000.00 (3rd year)

(ii) N556, 500.00 (5th year)Minor Maintenance: (i) N120, 000.00 (1st year)

(ii) N124, 000.00 (2nd year)(iii) N131, 000.00 (3rd year)(iv) N133, 500.00 (4th year)(v) N145, 000.00 (5th year)(vi) N153, 000.00 (6th year)(vii) N159, 000.00 (7th year)(viii) N166, 500.00 (8th year)

[Source: Associated Bus Company Plc, Owerri]

76

The data above are presented in tables below

Table 4.21: Technical Data for Mercedes Benz Executive Express Bus

t Q b1 b2t S(t)1 25833404 250000 - -2 25833404 255000 - -3 25833404 265000 830690 -4 25833404 270000 - -5 25833404 282000 1750000 20000006 25833404 291000 - -7 25833404 302000 - -8 25833404 325000 - -

[Source: Associated Bus Company Plc, Owerri]

Table 4.22: Technical Data for Toyota Hiace Bus

t Q b1 b2t S(t)1 3950000 120000 - -2 3950000 124000 - -3 3950000 131000 362000 -4 3950000 133500 - -5 3950000 145000 556500 8500006 3950000 153000 - -7 3950000 159000 - -8 3950000 166500 - -

[Source: Associated Bus Company Plc, Owerri]

77

Table 4.23 Enumeration of Data for Mercedes Benz Express Bust Q R Rt b1 b2t B(t)R S(t) S(t)Rt K(T)1 25833404 0.83 0.83 250000 - 207500 - - 313745832 25833404 0.83 0.69 255000 - 211650 - - 313796943 25833404 0.83 0.57 265000 830690 909423 - - 322203874 25833404 0.83 0.47 270000 - 224100 - - 313946945 25833404 0.83 0.39 282000 1750000 1686560 2000000 787800 322075406 25833404 0.83 0.33 291000 - 241530 2000000 653800 30627980*7 25833404 0.83 0.27 302000 - 250660 2000000 543400 307719938 25833404 0.83 0.23 325000 - 269750 2000000 450400 309070429 25833404 0.83 0.19 - - - 2000000 373800 30674330

10 25833404 0.83 0.16 - - - 2000000 310400 30750716 TABLE 4.24: Predicted K(T) for Mercedes Benz Express Bus

t Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)1 25833404 0.83 0.83 334118.18 277318.09 314588122 25833404 0.83 0.69 334118.37 277318.25 314588123 25833404 0.83 0.57 1164808.55 966791.10 322895054 25833404 0.83 0.47 334118.74 277318.55 314588135 25833404 0.83 0.39 2084118.92 1729818.70 0.71 52987.28 20871.69 331836726 25833404 0.83 0.33 334119.10 277318.86 0.71 15366.31 5023.25 31452761*7 25833404 0.83 0.27 334119.29 277319.01 0.86 27.23 7.40 314588058 25833404 0.83 0.23 334119.47 277319.16 0.86 3.81 0.86 31458813

[B(t) = 334118 + 0.1841t]

TABLE 4.25: Predicted K(T) by Walker’s Model with Data For Express Bus

t Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)1 25833404 0.83 0.83 334118.18 277318.09 0.14 314557412 25833404 0.83 0.69 334118.37 277318.25 314557413 25833404 0.83 0.57 1164808.55 966791.10 322863524 25833404 0.83 0.47 334118.74 277318.55 314557415 25833404 0.83 0.39 2084118.92 1729818.70 0.71 194.51 76.62 332054846 25833404 0.83 0.33 334119.10 277318.86 0.71 0 0 314557417 25833404 0.83 0.27 334119.29 277319.01 0.86 6.18 1.68 31455739*

78

8 25833404 0.83 0.23 334119.47 277319.16 0.86 0 0 31455741

TABLE 4.26: K(T) against Time (Measured E, Predicted M and Walkers W)t 1 2 3 4 5 6 7 8

K(T)E 31374583 31379694 32220387 31394694 32207540 30627980 30771993 30907042K(T)M 31458812 31458813 32289505 31458813 33183672 31452761 31458805 31458813K(T)w 31455741 31455741 32286352 31455741 33205484 31455741 31455739 31455741

%Deve -0.27 -0.26 -0.21 -0.20 -2.94 -2.62 -2.18 -1.75%Devw -0.26 -0.24 -0.20 -0.19 -3.00 -2.63 -2.17 -1.74

%Deve is the percentage deviation between the measured and predicted total costs while %Devw is the percentage deviation between the measured total costs and those predicted by Walker’s model for the Express bus.

1 2 3 4 5 6 7 829000000

29500000

30000000

30500000

31000000

31500000

32000000

32500000

33000000

33500000

Fig 4.6:Comparison of measured and predicted total costs K(T) of Express Bus

MeasuredPredictedWalker

Years

Tota

l Cos

ts N

Table 4.27: Enumeration of Data for Toyota Hiace Shuttle Bus

t Q R Rt b1 b2t B(t)R S(t) S(t)Rt K(T)1 3950000 0.83 0.83 120000 - 99600 - - 48790532 3950000 0.83 0.69 124000 - 102920 - - 48830533 3950000 0.83 0.57 131000 362000 409190 - - 52520554 3950000 0.83 0.47 133500 - 110805 - - 4892553

79

5 3950000 0.83 0.39 145000 556500 582245 850000 334815 50571626 3950000 0.83 0.33 153000 - 126990 850000 277865 4577275*7 3950000 0.83 0.27 159000 - 131970 850000 230945 46398058 3950000 0.83 0.23 166500 - 138195 850000 191420 46949269 3950000 0.83 0.19 - - - 850000 158865 4567649*

10 3950000 0.83 0.16 - - - 850000 131920 460011311 3950000 0.83 0.13 - - - 850000 109480 4627149

TABLE 4.28: Predicted K(T) for Toyota Hiace Shuttle Bus

t Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)1 3950000 0.83 0.83 154597.20 128315.68 49136512 3950000 0.83 0.69 154597.40 128315.85 49136513 3950000 0.83 0.57 516597.61 428776.01 52756524 3950000 0.83 0.47 154597.81 128316.18 49136515 3950000 0.83 0.39 711098.01 590211.34 0.71 8101.90 3191.34 54663816 3950000 0.83 0.33 154598.21 128316.52 0.71 2349.55 768.07 4912726*7 3950000 0.83 0.27 154598.41 128316.68 0.86 4.16 1.13 49136518 3950000 0.83 0.23 154598.62 128316.85 0.86 0.58 0.13 4913652

[B(t) = 154597 + 0.2021t]

TABLE 4.29:Predicted K(T) by Walker’s Model with Data for Hiace Shuttle Bus

t Q R Rt B(t) B(t)R d S(t) S(t)Rt K(T)1 3950000 0.83 0.83 154597.20 128315.68 0.13 49131712 3950000 0.83 0.69 154597.40 128315.85 49131713 3950000 0.83 0.57 516597.61 428776.01 52751374 3950000 0.83 0.47 154597.81 128316.18 49131715 3950000 0.83 0.39 711098.01 590211.34 0.71 19.06 7.51 54696106 3950000 0.83 0.33 154598.21 128316.52 0.71 0 0 49131727 3950000 0.83 0.27 154598.41 128316.68 0.86 0.88 0.24 4913171*8 3950000 0.83 0.23 154598.62 128316.85 0.86 0 0 4913172

TABLE 4.30:K(T) against Time (Measured S, Predicted M and Walker W)

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t 1 2 3 4 5 6 7 8K(T)S 4879053 4883053 5252055 4892553 5057162 4577275 4639805 4694926K(T)M 4913651 4913651 5275652 4913651 5466381 4912726 4913651 4913652K(T)w 4913171 4913171 5275137 4913171 5469610 4913172 4913171 4913172

%Devs -0.70 -0.62 -0.48 -0.43 -7.49 -6.83 -5.57 -4.45%Devw -0.69 -0.61 -0.44 -0.42 -7.54 -6.84 -5.56 -4.44

%Devs is the percentage deviation between the measured and predicted total costs.

%Devw is the percentage deviation between the measured total costs and those predicted by Walker’s model for the Shuttle bus.

1 2 3 4 5 6 7 84000000

4200000

4400000

4600000

4800000

5000000

5200000

5400000

5600000

Fig 4.7: Comparison of the measured and predicted total costs K(T) of Shuttle Bus

MeasuredPredictedWalkerSeries4Series5

Years

Tota

l Cos

ts N

4.11 VERIFICATION OF THE SALVAGE VALUE FUNCTION

The verification of the salvage value function to test its suitability to our industrial environment is part of the focus of this study. The salvage value component of the model is given by Lake and Muhlemann (1979)

S ( t )=Q(1−d)t (4.5)

Where Q = Cost of Machine d = Deterioration

t = Time (age)The regression line is given by

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log [S (t)Q ]=t log(1−d )(4.6)

Thus the slope of the regression line is given by

s= log (1−d )(4.7)

The Salvage value function is calibrated with S=-0.0363 and d=0.0802 for 140H Motor Grader (Appendix I)

Table 4.31:Variation of Measured and Predicted Salvage Values S(t) with Age

Q Age t

Measured S(t)

Predicted S(t)

305760 - - -305760 2 264500 258683305760 3 227750 237396305760 4 214000 218854305760 5 184750 201302305760 6 158750 185157305760 7 152000 170307305760 8 147250 156649305760 9 161000 144086305760 10 109500 132530305760 11 104500 121901305760 12 105000 112125305760 13 94250 103132

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1 2 3 4 5 6 7 8 9 10 11 120

50000

100000

150000

200000

250000

300000

Fig 4.8: Comparison of the Measured and Predicted Salvage Values S(t) of Grader

Measured S(t) -Predicted S(t) -

Years

Salvag

e Valu

es $

4.12 OBSERVATIONSThe following observations arise from the results of the model.i. Both the measured and predicted total costs, K(T) for the Front

end Loader reach a minimum at t=6 (Tables 4.3 and 4.4)ii. Fig. 4.1 shows that the measured and predicted K(T) for the

Loader are positively correlated with correlation coefficient, rc = 0.83. Table 4.5 shows the percentage deviation between the measured and predicted total costs for the Loader with absolute values ranging from 2.78 to 11.93%.

iii. The measured and predicted K(T) for the Motor Grader reach a minimum at t=4 (Tables 4.7 and 4.8).

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iv. Fig. 4.2 shows that the measured and predicted K(T) for the Motor Grader are positively correlated with correlation coefficient, rc = 0.81. Table 4.9 shows the percentage deviation between the measured and predicted total costs for the Grader with absolute values ranging from 0.25 to 16.10%.

v. From Tables 4.11 and 4.12, the measured K(T) for the Capsule filling machine reaches a minimum at t=16. From Table 4.13, the predicted K(T) apparently reaches a minimum also at t=16.

vi. Fig. 4.3 shows that the measured and predicted K(T) for the Capsule filling machine are positively correlated with rc = 0.72. Table 4.14 shows the percentage deviation between the measured and predicted total costs for the Capsule filling machine with absolute values ranging from 0.48 to 19.28%.

vii. The measured K(T) for the Injection machine does not reach a minimum in Table 4.16 but reaches a minimum at t = 13 in Table 4.17 with projected maintenance costs (preventive and corrective) in the 12th, 13th and 14th years. The predicted K(T) appears to reach a minimum at t=13 (Table 4.18)

viii. Figs. 4.4 and 4.5 show that the measured and predicted K(T) for the Injection machine are highly and positively correlated with rc = 0.99. Tables 4.19 and 4.20 show the percentage deviation between the measured and predicted total costs for the Injection machine with absolute values ranging from 0.09 to 0.30%.

ix. From Tables 4.23 and 4.24, the measured and predicted K(T) for the Express bus reach a minimum at t = 6. The K(T) of the Walker’s model (1994) with data of the Express bus reaches a minimum at t = 7 (Table 4.25)

x. Fig. 4.6 shows that the measured and predicted K(T) for the Express bus are positively correlated with rc = 0.81. The predicted K(T) by both the new model and Walker’s model (1994) are positively correlated with rc = 0.99. Similarly the

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measured K(T) and the predicted K(T) by Walker’s model for the Express bus are positively correlated with rc = 0.81. Table 4.26 shows the percentage deviation between the measured and predicted total costs for the Express bus with absolute values ranging from 1.75 to 2.94% for the new model and 1.74 to 3.00% for Walker’s model.

xi. From Table 4.27, the measured K(T) for the Shuttle bus inflects at t=6 and 9 with minimum at t=9. The predicted K(T) reaches a minimum at t=6 in Table 4.28. The K(T) of Walker model with data of the Shuttle bus reaches a minimum at t=7. (Table 4.29)

xii. Fig. 4.7 shows that the measured and predicted K(T) for the Shuttle bus are positively correlated with rc = 0.74. The predicted K(T) of both the new model and Walker’s model (1994) are positively correlated with rc = 0.99. Similarly, the measured K(T) of the Shuttle bus and that of Walker’s model (1994) are positively correlated with rc = 0.74. Table 4.30 shows the percentage deviation between the measured and predicted total costs for the Shuttle bus with absolute values ranging from 0.43 to 7.49% for the new model and 0.42 to 7.54% for Walker’s model.

xiii. Both the measured and predicted maintenance costs for the machines are generally non-linear.

xiv. Maintenance costs (preventive and corrective) increase with time due largely to inflation and deterioration rates.

xv. Salvage Value (Resale) decreases non-linearly with time due largely to deterioration rate.

xvi. Generally K(T) increases with time until convergence to a minimum due to inflation and deterioration rates.

xvii. As rate of return on investment, r increases, the discount factor R, decreases with a corresponding increase in K(T).

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xviii. The maintenance cost function appears to be normally (Lognormally) distributed. It may also fit into a gamma distribution since maintenance cost is skewed (higher) towards the end of the useful life of any machine (equipment).

xix. The measured and predicted salvage values S(t) are positively correlated with rc = 0.98. (Fig 4.8)

Detailed discussion of the results of the model is presented in the next chapter. This enables us to make inferences, draw conclusion and perhaps make recommendation for further study.

CHAPTER FIVE

DISCUSSION OF RESULTS

Details of the result of the model are discussed in this chapter. The discussion will assist in the conclusions and recommendations. The discussion is based on the observations derived from the results of the new model.

5.1 PERCENTAGE DEVIATION.

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The percentage deviation between the measured and predicted total costs for the machines in this study is generally low (0.09-19.28%). This may be an indication of the quality of the field data as well as the reliability of the new model.

5.2 SALVAGE (Resale) Value S(t)The Salvage value decreases non-linearly with time due largely to deterioration rate. The Salvage value function of the model is verified with data of the Motor Grader. The Measured and the Predicted salvage values are positively correlated with correlation coefficient, rc = 0.98. The salvage value function is therefore suitable for our industrial environment. With the existence of active resale market, maximum salvage value is realized which ensures maximum capital recovery since salvage value is related to capital depreciation.

5.3 MAINTENANCE COST, B(t)The coefficients b1 (Preventive Maintenance) and b2 (corrective Maintenance) are obtained by least square method. The coefficients b1 and b2 represent the constant and slope respectively of the regression line derived by the application of least square method on the maintenance cost data. Since refurbishment (overhaul) of machines is not a routine activity and the model is expected to predict results of real life situations, the coefficient b2 represents the cost of refurbishment in the year of occurrence and takes care of downtime.Both the measured (actual) and predicted maintenance costs for the machines are generally non linear.

The maintenance costs increase with time due largely to inflation and deterioration rates.

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The maintenance cost function appears to be normally (lognormally) distributed. Maintenance costs are higher towards the end of the useful life of any machine; its distribution may be represented by gamma distribution (see Appendix V).

5.4 TOTAL COST, K(T)The total cost, K(T) increases with time until it reaches a minimum, corresponding to the optimal replacement date. The increase of K(T) with time is due largely to inflation which affects maintenance costs and deterioration which affects Salvage value. Generally, increase in K(T) for the Injection machine does not reach a minimum(Table4.16). But with projected maintenance costs in the 12th, 13th and 14th years, the measured and predicted K(T) for the Injection machine reach a minimum at t =13 (Tables 4.17 and 4.18). This is due largely to insufficient data. The plastic plant is barely four years in operation. Sufficient cost data are yet to be generated for a complete enumeration process. Results of Tables 4.17 and 4.18 indicate that K(T) will reach a minimum whenever sufficient costs data are generated.Fig 4.2 shows some variation between the predicted and measured total costs, K(T) of the Grader for t between 4 and 6. This is due largely to the effect of the salvage values on the total costs K(T). This variation may account for the difference between the replacement dates of the Grader and Loader. This is an indication that the Motor Grader has an appreciable second hand value.

Generally, total cost K(T) increases as rate of return on investment, r increases and discount factor, R, decreases.

5.5 CORRELATION COEFFICIENT, rc

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The correlation coefficient, rc is an indication of the linear relationship between two variables as well as the strength and direction of their dependence. Thus the sign (positive or negative) is an indication of the direction whether increasing (positive) or decreasing (negative) linear relationship. The value of rc gives the strength or magnitude of linear dependence of the variables.

The correlation coefficients, rc of the predicated and measured total costs are generally positive and high in value between 0.72 (fig. 4.3) and 0.99 (figs 4.4 and 4.5). Thus the measured and predicated total costs K(T) in this study are largely dependent and have increasing linear relationship. This agreement may be due largely to the quality of cost data obtained from the field.

5.6 OPTIMAL REPLACEMENT DATE, TThe optimal replacement date, T is indicated whenever the total cost K(T) reaches a minimum on the enumeration table.

The measured and predicted replacement date for the loader is at t=6 which corresponds to the useful life of construction machines. (US Treasury Department Bureau of Internal Revenue, 1942). The result shows that the derived values for the coefficients b1 and b2 of the maintenance cost function as well as the values of the Salvage Value, S(t) are satisfactory. Hence the new model can predict replacement date for construction machines.The measured and predicted replacement date for the Grader is at t=4. The useful life for construction machines is 6 years. The difference may be attributed to the unavailability of all maintenance costs associated with the Grader. There exists the problem of allocating costs to a single machine when other machines are involved. Thus all such costs are not considered in the cost build-up

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for the Grader. Secondly, the Grader has an appreciable second hand value, this accounted for the drop of the total cost at t=3 and subsequently reaching a minimum at t=4. The difference, notwithstanding, the results are within reasonable and acceptable limit. The replacement date for construction machines is corroborated by a survey on Monier Construction Company (MCC) Ltd in 2001 which indicates an average useful life of 5-7 years for various construction machines (Asomugha, 2002).

The measured and predicted replacement date for the Capsule Filling machine falls at t=16 while the useful life for pharmaceutical machines is 20 years (US Treasury Department, 1942). The difference may be due to the unavailability of maintenance costs data after the 15th year. Some maintenance costs for the machine are not recorded or covered by invoices especially when items are obtained from the company’s material store to effect minor repairs. However the indicated results are within reasonable and acceptable limit.

The measured and predicted replacement date for the Injection machine falls at t=13 when provisions are made for the maintenance costs in the 12th, 13th and 14th years. Otherwise no replacement date is indicated on the enumeration table. The plastic plant is in its 4th

year of operation and sufficient cost data are yet to be generated for complete enumeration process. The indicated result shows that a replacement date will be achieved whenever there are sufficient cost data.

The measured and predicted replacement date for the Express bus falls at t=6 while the useful life for long distance buses is 7 years (US Treasury Department, 1942). The difference may be attributed to the unavailability of some cost data. This is due to the problem of

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allocating costs to a single bus when other buses are involved. Such costs are not considered in the build-up of cost for the Express bus.

The measured and predicted replacement date for the Shuttle bus falls at t=6. Useful life for minibuses is 8 years (US Treasury Department, 1942). Cost data are yet to be generated for t>8 because the Shuttle bus is in its 8th year of service. If cost data are generated in the 9th year, the replacement date for the Shuttle bus may fall on the 9th year as indicated on the enumeration table. The indicated results are satisfactory, reasonable and fall within acceptable limit.

5.7 THE MODEL BY WALKER (1994)i. Walker (1994) conducted a replacement analysis using data

from a fleet of Ford Transit ambulances of a local authority in Northern England.

ii. Walker (1994) used a computer program to produce graphs for the analysis. The plots (graphs) depend on the values of S1 and S2 with rates of return between 10% and 20% for discounting purposes (Walker, 1994, pp58-63)

iii. The data used for the computer aided graphical analysis are not published. The total cost, K(T) of the model is not available for comparison with the predicted K(T) for both the Express and Shuttle buses.

iv. The local authority sells their vehicles in the seventh year usually to Schools and Charitable organization. This coincides with the economic life of such vehicles.

v. The graphical analysis indicates a replacement date of between 6 and 8 years. Walker (1994) then concludes that a replacement policy after 7 years is a robust policy in the light of available data. (Comparison with other models, Appendix VI)

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5.8 INFERENCESThe following inferences are drawn from the discussion of the results.i. The optimal replacement date for construction machines is

between 4 and 6 years assuming regular maintenance and proper operation of the machines. The Motor Grader has an appreciable second hand value.

ii. The optimal replacement date for Pharmaceutical Machines is between 16 and 20 years assuming regular maintenance and proper operation of the machines.

iii. The optimal replacement date for plastic and rubber machines is between 13 and 17 years assuming regular maintenance and proper operation of the machines.

iv. The optimal replacement date for long distance buses is between 6 and 7 years assuming regular maintenance and proper use of the buses.

v. The optimal replacement date for Shuttle buses (Minibuses) is between 6 and 9 years assuming regular maintenance and proper use of the minibuses.

vi. The optimal replacement date for ambulances is between 6 and 8 years assuming regular maintenance and proper use of the ambulances.

vii. The new model appears to be more suitable for machines with smaller economic life like the construction machines than those with longer economic life like the pharmaceutical equipment, non-available of sufficient cost data notwithstanding.

viii. The results of the model compare favourably with those obtained from the model by Walker (1994). Thus the new model is reliable and operational.

5.9 TRIVIAL SOLUTION (Trivial Replacement Policy)

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There are trivial solutions to machinery replacement problems. The followings are some of the trivial solutions

i. Replace with higher capacity machine whenever there is an increase demand of products or services provided the fund is available. Place existing machine on standby duty.

ii. Replace with higher performance machine with new technology.iii. Replace with lower capacity machine when demand of products

or services drops.Trivial replacement policy may not require the application of the replacement model.

CHAPTER SIX

CONCLUSIONS AND RECOMMENDATIONS

6.1 CONCLUSIONSThe study is introduced in chapter one which includes the objectives, relevance and contribution of the study to knowledge. The methodology adopted for the study is briefly mentioned.

The literature review of chapter two addresses issues related to machinery replacement. The origin of the principles and earlier application of some of these principles are mentioned. An overview of the relevance of these principles or methods to this study is highlighted. In addition, various existing replacement models, their limitations and objectives are discussed.

Details of the methodology adopted in the study are presented in chapter three. The Analytical and Experimental processes are part of the methodology of the study. The Dynamic programming method is the solution technique adopted for the study.

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The model is developed in chapter four. The assumptions and features of the model are also highlighted. The model is calibrated and verified with field data from three Industries namely Construction, Pharmaceuticals and Plastics. Finally the model is compared with other models and found to predict results that are satisfactory. The results are presented in tables and graphs.

The results of the model are discussed in details in chapter five. The effects of inflation and deterioration rates on relevant variables are highlighted. The variation of these variables with time is also mentioned.

The summary of the findings of the study follows.

6.2 SUMMARY OF FINDNGSi. The new model is simple, reliable and operational.ii. It is amenable to review and adjustment.iii. It is amenable to solution techniques like optimization, dynamic

programming and economic lot size inventory control method.iv. The results of the new model are comparable to those of other

models like the Walker’s (1994) model.v. The new model appears to be suitable for machines with small

economic life. However, with sufficient cost data the new model will also be applicable to the replacement problems of machines with long economic life.

vi. The new model can address the replacement problems of some industries like construction, pharmaceutical, plastic and transport services. The application of the model can be extended to other industries provided that sufficient cost data are available.

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vii. The results of the model depend on regular preventive maintenance and timely corrective maintenance of the machines. Proper maintenance record with costs should be kept.

viii. It is also assumed that the machines will be operated by trained operators with proper work ethics for handling the machines while in service to avoid unnecessary downtime arising from forced shutdowns. The latter impacts negatively on the maintenance costs and indeed the total cost.

ix. The simplicity in the application of the model may lead to savings in terms of cost and time. In fact, the use of Monte Carlo simulation to generate values for deterioration rates eliminates the cost incurred through failure analysis.

x. It is expected that the model will assist industry managers to make effective machinery replacement decisions, since the model predicts machinery replacement date that reflects real life situation.

xi. Maximum total cost K(T) may suggest refurbishment.

6.3 RECOMMENDATION FOR FURTHER STUDYTo improve on the model, further study in the following area is encouraged and recommended.i. Generate random numbers under other probability distributions

like the normal distribution and poisson distribution.ii. The application of the economic lot size inventory control

method as the solution technique.iii. Introduction of parameters like capital allowance, taxation rate

and book value of the existing machine in the development of the model.

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iv. Obtain failure data and analyze same to determine the failure mode which gives the shape parameter that may lead to a replacement policy.

v. Other statistical tests in addition to the regression analysis should be used to test the model’s performance.

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APPENDIX I

VERIFICAITON OF THE SALVAGE VALUE FUNCTION

The Salvage Value is given by Lake and Muhlemann (1979)

S(t) = Q(1-d)t

Where Q = Cost of machined = Deterioration ratet = Time

Applying the least square method, the following values are generated for the regression line.

Table A: Regression Value for the Salvage Value of Grader

S(t) S(t)/QX Y

Log S(t)/QXY X2

248500 0.813 2 -0.0899 -0.1798 4230000 0.752 3 -0.1238 -0.3714 9210000 0.687 4 -0.1630 -0.6520 16194000 0.634 5 -0.1979 -0.9895 25163500 0.535 6 -0.2716 -1.6296 36161000 0.527 7 -0.2782 -1.9474 49155250 0.508 8 -0.2947 -2.3576 64161000 0.527 9 -0.2782 -2.5038 81137500 0.450 10 -0.3468 -3.4680 100104500 0.342 11 -0.4659 -5.1249 121105000 0.343 12 -0.4647 -5.5764 14498500 0.322 13 -0.4921 -6.3973 169

TOTAL 90 -3.4668 -31.1977 818

The slope of the regression line is given by,

106

S=N∑ XY−∑ X∑ Y

N∑ X2−¿¿¿

S=12 (−31.1977 )−90(−3.4668)

12 (818 )−(90)2

= -0.0363 Log(1-d) = -0.0363

1-d = 10-0.0363=0.9198 d = 1-0.9198

= 0.0802

107

APPENDIX IIVALUES FOR ENUMERATION PROCESS

1. CAT 140H MOTOR GRADER

(a) Deterioration rate, d= XU

=56=0.83

Where X = 5 (Average of 10 random numbers between zero & six)

u = 6 (Average useful life of construction machines)

(b) Starting time t s= −logQlog (1−d )

¿− log39901680log(1−0.83)

= 9.9

≅ 10(c) Discount factor, R = 100/100 + r

¿ 100125

=0.8

Where r = 25% - (Rate of return on investment)

2. CAPSULE FILLING MACHINE

(a) d= XU

=1620

= 0.8

X = 16 (Average of 50 random numbers between zero & twenty)

U = 20 (Average useful life for pharmaceutical machines)

(b) t s=−log1440000log(1−0.8)

108

= 8.8

= 9

(c) R=100120

= 0.83

r = 20% (Capitalization rate)

3. INJECTION MACHINE

(a) d= XU

=1317

=0.76

X = 13 (Average of 50 random numbers between zero and seventeen)

u = 17 (Average useful life of rubber/plastic machines)

(b) t s=−log25057500log(1−0.76)

= 8.8 = 9

(c) R = 100130

= 0.77

r = 30%

4. EXPRESS BUS

(a) d= XU

=67=0.86

X = 6 (Average of 10 random numbers between 109

zero and seven)U = 7 (Average useful life of long distance

buses)

(b) t s=−log25833404log(1−0.86)

= 8.7= 9

(c) R = 100120 = 0.83

r = 20%

5. SHUTTLE BUS

(a) d= XU

=68=0.75

X = 6 (Average of 10 random numbers between zero and eight)

U = 8 (Average useful life of Shuttle Buses)

(b) t s=−log3950000log(1−0.75)

= 10.9= 11

(c) R = 100120

= 0.83r = 20%

APPENDIX III

LINEAR REGRESSION

In statistics, Linear regression is a form of regression analysis in which the relationship between one or more independent variables and

110

another variable (dependent) is modeled by a least square function called Linear regression equation.

This function is a linear combination of one or more model parameters called regression coefficients. A Linear regression equation with one independent variable represents a straight line. The results are subject to statistical analysis.

Applying the least square method the slope of the linear regression equation is given by

S=N∑ XY−∑ X∑ Y

N∑ X2−(∑ X )2

The constant K=∑ X2∑Y−∑ X∑ XY

N∑ X2−(∑ X)2

and log K=∑YN

−S∑ XN

where x = Independent variable

y = Dependent variable.

The linear regression equations for the maintenance cost function from data of various machines are presented as follows:

1. Front – end Loader

Table B: Regression Values for Loader

t B(t)L Xlog t

ylog B(t)

xy X2

1 2500 0 3.3980 0 0

111

2 12500 0.3010 4.0969 1.2332 0.09063 6000 0.4771 3.7781 1.8025 0.22764 8500 0.6020 3.9294 2.3655 0.36245 22500 0.6990 4.3522 3.0422 0.48866 2500 0.7781 3.3980 2.6440 0.60547 2500 0.8451 3.3980 2.8716 0.714210 2500 1.0000 3.3980 3.3980 1.0000

Total 4.7023 29.7486 17.3570 3.4888

S=8 (17.357 )−(4.7023 )(29.7486)

8 (3.4888 )−(4.7023)2

¿ 138.856−139.886827.9104−22.1116

¿−1.03085.7988

= - 0.17776

log k=29.74868

+0.1778 (4.7023)8

= 3.7186 + 0.1045= 3.8231

:. K = 103.8231

= 6654The regression equation for the maintenance cost function is given by

B(t) = 6654 – 0.1778t.

2. Motor Grader

112

Table C: Regression Values for Grader

t B(t)G Xlog t

ylog B(t)

xy X2

1 80000 0 4.9031 0 02 104000 0.3010 5.0170 1.5101 0.09063 1466065

60.4771 7.1662 3.4190 0.2276

4 120000 0.6020 5.0792 3.0577 0.36245 130000 0.6990 5.1139 3.5746 0.48866 145000 0.7781 5.1614 4.0161 0.60547 160000 0.8451 5.2041 4.3980 0.7142

Total 3.7023 37.6449 19.9755 2.4888

S=7 (19.9755 )− (3.7023 )(37.6449)

7 (2.4888 )−(3.7023)2

¿ 0.45583.7146

= 0.1227

log K=37.64497

−0.1227 (3.7023)7

= 5.3129:. K = 105.3129

= 205542Thus B(t) = 205542 + 0.1227t

3. Capsule Filling Machine

Table D: Regression Values for Capsule Filling Machine

t B(t)c Xlog t

ylog B(t)

xy X2

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9 23000 0.9542 4.3617 4.1619 0.910510 336000 1.0000 5.5263 5.5263 1.000011 27000 1.0414 4.4314 4.6149 1.084512 28000 1.0792 4.4472 4.7994 1.164713 29500 1.1139 4. 4698 4.9789 1.240814 31000 1.1461 4.4914 5.1476 1.313515 448000 1.1761 5.6513 6.6465 1.3832

Total 7.5109 33.3791 35.8755 8.0972

S=7 (35.8755 )−(7.5109 )(33.3791)

7 (8.0972 )−(7.5109)2

¿ 0.42140.2668

= 1.5795

log K=33.37917

−1.5795 7.51097

= 3.0736

:. K = 103.0736

= 1185

:. B(t) = 1185 + 1.5795t

114

4. Injection Machine

(a)Table E: Regression Values for Injection Machine

(Projected)

t B(t)I Xlog t

YLog B(t)

xy X2

4 62000 0.6020 4.7924 2.8850 0.362412 6090500 1.0792 6.7847 7.3220 1.164713 100000 1.1139 5.0000 5.5695 1.240814 110000 1.1461 5.0414 5.7779 1.3135

Total 3.9412 21.6185 21.5544 4.0814

S=4 (21.5544 )−(3.9412 )(21.6185)

4 (4.0814 )−(3.9412)2

=1.2805

log K=21.61854

−1.2805(3.9412)4

= 4.1429

K = 104.1429

= 13896

B(t) = 13896 + 1.2805t

115

(b). Injection Machine

Table F: Regression Values for Injection Machine

t B(t)I Xlog t

ylog B(t)

Xy X2

4 62000 0.6020 4.7924 2.8850 0.362412 90500 1.0792 4.9566 5.3492 1.164713 100000 1.1139 5.0000 5.5695 1.240814 110000 1.1461 5.0414 5.7779 1.3135

Total 3.9412 19.7904 19.5816 4.0814

S=4 (19.5816 )− (3.9412 )(19.7904 )

4 (4.0814 )−(3.9412)2

= 0.4145

log K=19.79044

−0.4145 3.94124

= 4.5392

K = 104.5392

B(t) = 34610 + 0.4145t

116

5. Express Bus

Table G: Regression Values for Express Bus

t B(t)E Xlog t

ylog B(t)

xy X2

1 250000 0 5.3979 0 02 255000 0.3010 5.4065 1.6274 0.09063 1095690 0.4771 6.0397 2.8815 0.22764 270000 0.6020 5.4314 3.2697 0.36245 2032000 0.6990 6.3079 4.4092 0.48866 291000 0.7781 5.4639 4.2515 0.60547 302000 0.8451 5.4800 4.6311 0.71428 325000 0.9031 5.5119 4.9778 0.8156

Total 4.6054 45.0392 26.0482 3.3044

S=8 (26.0482 )−4.6054(45.0392)

8 (3.3044 )−(4.6054)2

¿ 0.96215.2255

=0.1841

log K=45.03928

−0.1841 (4.6054)8

= 5.5239

K = 105.5239

= 334118

B(t) = 334118 + 0.1841t

117

6. Shuttle Bus

Table H: Regression Values for Shuttle Bus

t B(t)S Xlog t

ylog B(t)

xy X2

1 120000 0 5.0792 0 02 124000 0.3010 5.0934 1.5331 0.09063 493000 0.4771 5.6928 2.7160 0.22764 133500 0.6020 5.1255 3.0855 0.36245 701500 0.6990 5.8460 4.0863 0.48866 153000 0.7781 5.1847 4.0342 0.60547 159000 0.8451 5.2014 4.3937 0.71428 166500 0.9031 5.2214 4.7154 0.8156

Total 4.6054 42.4444 24.5662 3.3044

S=8 (24.5662 )−4.6054(42.4444)

8 (3.3044 )−(4.6054)2

¿ 1.05625.2255

=0.2021log K=42.4444

8−0.2021 (4.6054)

8

= 5.1892

K = 105.1892

= 154597

B(t) = 154597 + 0.2021t

118

APPENDIX IV

CORRELATION

Correlation, in general statistical usage, refers to the departure of two random variables from independence.

Correlation coefficient indicates the strength and direction of a linear relationship between two random variables.

The correlation coefficient between two random variables x and y with expected values µx and µy and standard deviations

x and y is given by

rc=cov (X ,Y )σ xσ y

=E ( X−μX )(Y−μY)

σ x σ y

¿∑ (X−X )(Y−Y )

√∑ (X−X )2∑ (Y−Y )2

where Cov (x,y) =Covariance of x and y variables

x = Mean of x variable

y = Mean of y variable

The correlation is 1 in the case of increasing linear relationship, -1 in the case of a decreasing linear relationship and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either -1 or 1, the stronger the correlation between the variables.

If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables.

119

The correlation between two variables is diluted in the presence of measurement error around estimates of one or both variable in which case disattenuation provides a more accurate coefficient.

120

APPENDIX VDISTRIBUTION FUNCTIONS

(1) NORMAL DISTRIBUTION

Normal distribution has a very wide range of applications in statistics, including the testing of hypothesis.

The probability density function is given by

Pd (X )= 1σ √2 π

e−[ X−μ

2σ2 ]

Where µ = Mean

2 = Variance

The measured and predicted maintenance costs for the Motor Grader are as follows:

Table I: Maintenance costs for Motor Grader

t Measured Predicted1 80000 205542.122 104000 205542.253 14660656 205542.374 120000 205542.495 130000 205542.616 145000 205542.747 160000 205542.86

Mean 2199951 205542.50S. Dev. 5494717 0.26542

The Normal cumulative distribution function at 145000 with mean of 2199951 and standard deviation of 5494717 is 0.354 for the measured maintenance cost of the Grader

121

The Normal cumulative distribution function at 205542.74 with mean of 205542.50 and standard deviation of 0.26542 is 0.774 for the predicted maintenance cost of the Grader.

2. Lognormal Distribution The lognormal distribution is used to analyze data that has been logarithmically transformed.

Table J: Logarithm of Maintenance cost for MotorGrader

t B(t)Grader Log B(t)1 80000 4.90312 104000 5.01703 14660656 7.16624 120000 5.07925 130000 5.11396 145000 5.16147 160000 5.2041

Mean 2199951 5.3778S. Dev. 5494717 0.7947

The cumulative lognormal distribution function at 145000 with mean of 5.3778 and standard deviation of 0.7947 is 1.

Since the value of the cumulative distribution function is not zero, the maintenance cost data follow a normal distribution. The data also fit into lognormal distribution.

122

3. Gamma DistributionGamma distribution can be used to study variables that may have a skewed distribution. It is commonly used in queuing analysis.

The standard gamma probability density function is given by

Pd (X )= Xa−1 e−N

Γ (α )

When alpha α = 1, gamma distribution becomes exponential with λ=1βWhere = beta parameter.

Gamma distribution is also called Erland distribution,when alpha is a positive integer.

4. Weibull Distribution

Weibull distribution is used in reliability analysis such as the calculation of the mean time to failure of a device.

The Weibull probability density function is given by

Pd (X )=α Xa−1 e−( X

β)α

βa

When alpha = 1, Weibull becomes exponential distribution with λ=1β

The Weibull cumulative distribution function is given by

Pc (x) = 1 – e- (x/)

5. Poisson Distribution

123

Poisson distribution is applicable only when the events occur completely at random and the number that occurs is small compared to the potential number that could occur.

The Poisson distribution is given by

P (X )= e−μμX

X !

Where P(x) = probability of exactly x occurrencese = Naperian Constant (2.71828)µ = Expected or average number of

occurrences.

The mean and variance of Poisson function are both µ.

6. Binomial DistributionBinomial distribution applies to events that can take on only two values such as the head or tail for a tossed coin or accept or reject for an object.The binomial distribution is given by

P (X )= N !X ! (N−X ) !

PX (1−P)N−X

Where P(x) = Probability of exactly x occurrence in N trials.

P = Probability of success in one trial

In the case of an unbiased coin P and (1-p) are both 0.5, but in most problems p will not be 0.5. The binomial distribution is symmetrical if and only if p = 0.5The mean and variance of a binomial distribution respectively are

124

µ = Np2 = Np(1-p)

The binomial distribution assumes that the trials are independent.

APPENDIX VI

COMPARISON WITH OTHER MODELS

The model was compared with the Kent (1977) model and Schwartz/McNamara (1983) model in addition to the Walker (1994) model.

125

Maintenance costs and salvage values of the Express and Shuttle buses were used in the comparison. The Kent (1977) model is given by

K (T )=Q−S (T )RT

T+ 1T∫0

T

f (t )RT dt

Where K(T) is the Discounted average sum of capital depreciation and maintenance cost and f(t) is the maintenance effectiveness function.

The Schwartz and McNamara (1983) model is given by

U (T )=[Q+∑t=1

T

Rt f (t )−S (T )RT ] /∑t=1T

Rt

Where U(T) is the Uniform annual equivalent cost of replacement. In both cases the objective is to find T that minimizes K(T) and U(T) respectively. Kent (1977) verified the effect of discounted cash flow on optimal replacement period and explained that there is an optimal delay in replacement required by discounted cash flow with the attendant saving in real cash. Schwartz and McNamara (1983), on the other hand, developed an optimal replacement cycle policy which assumes the existence of an active used equipment market. They also assumed that the maintenance and operating cost function is increasing and discontinuous. The models vary in complexities because of the relaxing of certain assumptions during formulation. These complexities present computational problems which may lead to poor or faulty results, if proper cost functions are not assumed. There is therefore the need to pay greater attention in the estimation of the maintenance cost f(t) and salvage value S(T)

126

Comparison using Express Bus Data

Table K: Predicted K(T) by Kent(1977) model with Express Bus Datat Q R Rt f(t) f(t) Rt d S(t) S(t) Rt K(T)1 25833404 0.83 0.83 334118.18 277318.08 314587012 25833404 0.83 0.6889 668236.55 460348.15 316792193 25833404 0.83 0.5718 1833045.10 1048135.1 323873964 25833404 0.83 0.4746 2167163.84 1028535.9 323637835 25833404 0.83 0.3939 4251282.76 1674580.2 .71 52987.27 20871.69 331170026 25833404 0.83 0.3269 4585401.86 1498967.8 .71 15366.31 5023.25 329245167 25833404 0.83 0.2727 4919521.15 1336633.9 .86 27.23 7.40 327349778 25833404 0.83 0.2252 5253640.62 1183119.7 .86 3.81 0.86 32550027

Table L:Predicted U(T) by Schwartz/McNamara(1983)with Express Bus Data t Q R Rt f(t) f(t) Rt d S(t) S(t) Rt U(T)1 25833404 0.83 0.83 334118.18 277318.09 314587012 25833404 0.83 0.6889 334118.37 507492.23 382361673 25833404 0.83 0.5718 1164808.55 1173529.7 472314344 25833404 0.83 0.4746 334118.74 1332102.4 572387395 25833404 0.83 0.3939 2084118.92 2153036.8 .71 52987.27 20871.69 709966206 25833404 0.83 0.3269 334119.10 2262260.3 .71 15366.31 5023.25 859303797 25833404 0.83 0.2727 334119.29 2353040.5 .86 27.23 7.40 10374103

08 25833404 0.83 0.2252 334119.47 2428284.2 .86 3.81 0.86 12549594

0

Table M: K(T) against Time (Measured, Predicted, Walker, Kent, Schwartz)t 1 2 3 4 5 6 7 8

Measured 31374583 31379694 32220387 31394694 32207540 30627980 30771993 30907042Predicted 31458812 31458813 32289505 31458813 33183672 31452761 31458805 31458813Walker 31455741 31455741 32286352 31455741 33205484 31455741 31455739 31455742Kent 31458701 31679219 32387396 32363783 33117002 32924516 32734977 32550027

Schwartz 31458701 38236167 47231434 57238739 70996620 85930379 103741030 125495940

127

1 2 3 4 5 6 7 80

20000000

40000000

60000000

80000000

100000000

120000000

140000000

Fig 1: Comparison of the measured and predicted total costs K(T) of the Express Bus

MeasuredPredictedWalkerKentSchwartz

Years

Tota

l Cos

t N

Comparison using Shuttle Bus Data

Table N: Predicted K(T) by Kent (1977) model with Shuttle Bus Datat Q R Rt f(t) f(t) Rt d S(t) S(t) Rt K(T)1 3950000 0.83 0.83 154597.20 128315.67 49136342 3950000 0.83 0.6889 309194.60 231004.15 50156673 3950000 0.83 0.5718 825792.21 472187.98 53279374 3950000 0.83 0.4746 980390.02 465293.10 53196305 3950000 0.83 0.3939 1691488.0 666277.12 .71 8101.90 3191.34 55579356 3950000 0.83 0.3269 1846086.2 603485.57 .71 2349.55 786.07 54852027 3950000 0.83 0.2727 2000684.6 543586.00 .86 4.16 1.13 54139588 3950000 0.83 0.2252 2155283.2 485369.77 .86 0.58 0.13 5343819

Table P:Predicted U(T) by Schwartz/McNamara(1983) with Shuttle Bus Datat Q R Rt f(t) f(t) Rt d S(t) S(t) Rt U(T)1 3950000 0.83 0.83 154597.20 128315.67 49136342 3950000 0.83 0.6889 154597.40 234817.81 60746383 3950000 0.83 0.5718 516597.61 530208.32 78352714 3950000 0.83 0.4746 154597.81 603580.44 95945645 3950000 0.83 0.3939 711098.01 883681.94 .71 8101.90 3191.34 122632416 3950000 0.83 0.3269 154598.21 934220.09 .71 2349.55 786.07 149386727 3950000 0.83 0.2727 154598.41 976224.47 .86 4.16 1.13 181311118 3950000 0.83 0.2252 154598.62 1011040.00 .86 0.58 0.13 22029484 Table Q: K(T) against Time (Measured, Predicted, Walker, Kent, Schwartz)

T 1 2 3 4 5 6 7 8

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Measured 4879053 4883053 5252055 4892553 5057162 4577275 4639805 4694926Predicted 4913651 4913651 5275652 4913651 5466381 4912726 4913651 4913652Walker 4913171 4913171 5275137 4913171 5469610 4913172 4913171 4913172Kent 4913634 5015667 5327937 5319630 5557935 5485202 5413958 5343819

Schwartz 4913634 6074638 7835271 9594564 12263241 14938672 18131111 22029484

1 2 3 4 5 6 7 80

5000000

10000000

15000000

20000000

25000000

Fig 2: Comparison of the measured and predicted total costs K(T) of the Shuttle Bus

MeasuredPredictedWalkerKentSchwartz

Years

Tota

l Cos

ts N

The tables and figures show that K(T) for Kent(1977) model appears to reach a minimum at t = 8 for both the Express and Shuttle buses while U(T) for Schwartz/McNamara(1983) model is ever increasing with no minimum in sight. Generally, both models predicted higher values of total costs than those predicted by the new model with correlation coefficients of 0.47 and -0.12 for Kent (1977) and Schwartz/McNamara (1983) models respectively. The new model compares fairly with the Kent (1977) model as indicated by the positive value of the correlation coefficient but has negative linear relationship with the Schwartz/McNamara (1983) model. As earlier mentioned, there is the need to pay greater attention to the estimation of the maintenance costs and salvage values in the optimization of the models to improve on the results obtained from them.

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