a transient state model for predicting maintenance requirements

Engineering Costs and fioduction Economics, 11 (1987) 87-98 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

87

A TRANSIENT STATE MODEL FOR PREDICTING MAINTENANCE REQUIREMENTS

K.W. Brammer

AT & T Technologies, Richmond, Virginia (U.S.A.)

and C.J. Malmborg

Center for Industrial and Management Engineering, Rensselaer Polytechnic Institute, Troy, NY (U.S.A.)

ABSTRACT

A microcomputer-based model is used to heterogeneity of age and deployment environ- predict the life cycle maintenance requirement of units in the population, the model ments of a population of repairable items. produces estimates of overall population By recognizing the variability of maintenance requirements. The uses of the model are requirements over the life cycle and the demonstrated in the context of a case study.

1. INTRODUCTION

The planning and management of logistics requirements for repairable item populations has become a monumental task as the size, complexity, and expense of such populations has increased. A repairable item population consists of units that, up to a point, are more economical to repair than to replace. To make effective logistics decisions in the face of com- plex logistics systems, planners need analysis tools that can help determine the maintenance requirements of recoverable item populations. The logistics planner needs a tool that provides these maintenance requirements prior to the procurement and deployment of recoverable item populations. In order to forecast the true maintenance requirements of a repairable item population, the maintenance requirements of the population during

0304-3908/87/$03.50 0 1987 Elsevier Science Publishers B.V.

its entire life cycle must be forecast. If the failure characteristics of a population vary significantly from period to period then the total population maintenance requirements can vary significantly from one time period to another.

A basic problem with logistics models intended to serve as maintenance requirement planning and analysis tools is that these models do not evaluate the complete maintenance requirements of repairable item populations over the entire life cycle of the population of items. Models have tended to focus on one particular maintenance requirement or another by making simplifying as- sumptions. One of the major simplifications made has been the assumption that systems operate in steady-state all of the time and that logistics requirements may be determined from these steady-state conditions. Under

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this assumption, an average rate of failure is applied to the total population life cycle in order to generate maintenance requirements. Some populations never attain a steady-state of operation. This could be due to characteristics inherent in the components that comprise each member of the population, or due to external stimuli such as war conditions, temperature, moisture, etc.

Logisticians such as Geisler and Murrie [ 1 ] and Kline [ 21 have called for the development of models that are useful during all phases of a population life cycle and are robust in that they address a broad range of maintenance requirements. Extensive sur- veys of maintenance models [3-61 have verified that logistics models capable of producing repairable item population maintenance requirements do not consider transient state failure characteristics.

The purpose of the present paper is to de- scribe the methodology for a logistics model that has been developed to determine and evaluate maintenance requirements of repairable item populations. The model is capable of generating maintenance requirements for the total life cycle, whether the repairable items are operating under steady-state or non-steady-state conditions. The model allows the logistics planner to develop and evaluate the desired logistics system in the face of availability constraints, budget constraints, space constraints, procurement contraints, and personnel constraints. The results produced by the model emphasize what type of data the logistician should have in order to build and operate a population at a reason- able cost and maintain the required degree of readiness.

The following sections of the paper look more closely at the maintenance requirements planning problem, described the modeling approach, and apply the model to an example problem.

2. THE MAINTENANCE REQUIREMENTS PLANNING PROBLEM

2.1. Meeting availability needs

In the model described here, availability is defined as the probability that, at any given time, there are enough end items in operation to meet a specified demand for end items. Being able to calculate the availability of an end-item population during a given time period can be of great value to the logistician in his efforts to plan logistics policy. End-item population parameters such as the number of end items in the population and the failure and repair characteristics of the end items in the population can both have an effect on the availability of the end- item population. The parameters also have an obvious effect on maintenance requirements generated by the end-item population. Thus, in order to plan maintenance requirements in the face of availability criteria, the logistics planner needs a model that will allow the evaluation of end-item availability given certain population parameters.

The model assumes that the chance of failure of each end item is independent from the chance of failure of any other end item. The model calculates an average probability of failure for all end items operating in the same failure environment, so that all end items within the same environment can be assumed to have the same probability of failure. Further, there are only two things that can happen to an end item: either the end item experiences a failure, or the end item experiences no failure. Because of these end-item characteristics, it has been assumed that the probabilities of simultaneous failure of end items are binomially distributed.

The binomial distribution as applied to a population of end items would be defined as follows:

b b(x; n, p) = 1

x=0

where : n = total number of end items in the popula-

tion; p = probability of failure of an end item.

To calculate availability, we only need to know the probability of 0 to k end items being out of operation simultaneously. Thus the binomial calculation for availability becomes:

k

Ae = c b(x; n, pd,

x-o

where :

Ae = the availability of in environment e ;

k = Ne -De;

end items operating

De = the demand for end items operating in environment e ;

Pe = average proability of failure for end items operating in environment e;

Ne = number of end items operating in environment e during a given period.

2.2 The nature of component failures

Components are defined as the parts that make up an end item. It. is the failure of individual components that cause end items to fail. To plan maintenance requirements for populations of end items, the logistician needs to know the failure and repair characteristics of the components that comprise the end items.

This requirement results in a huge demand for failure and repair data that is not always readily available. The lack of good repair and failure data affects the way in which this maintenance-requirements planning model can be used. The model may be used as a demonstrator to show what can be done when

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good data is available, or it can be applied “as is”, if the logistician estimates failure and repair rates, uses existing data, or devises ,worst-case-best-case data that determine ranges of maintenance requirements.

In many end items, redundant and/or non-critical components exist. These are components that can fail but without causing end-item failure. The model described in this paper would require certain extensions in order to evaluate these types of conditions. At present, the model only evaluates critical component failure.

2.3 The Bill of Material concept

Since any of several individual components can be responsible for end-item failure, it is important for the logistician to know the bill of materials for the end item. This will allow him to determine which components are subject to failure and which are critical to the operation of the end item. Knowledge of the bill of materials will also help in establishing which components should be stocked as spare parts to support the end item.

Knowing the components that make up the end item is also important in determining where repairs to the end item should be made. End-item failures caused by different components may require different types of repair facilities. The repair facility at which a certain type of repair must be made should be the location at which spare parts to support that repair are kept.

3. DESCRIPTION OF THE MODELING APPROACH

3.1 Conversion of failure curves in the Markov model

In order to provide the decision maker with the output data necessary to evaluate a repairable item population operating in either transient or steady-state conditions, a framework that could handle end items

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with time-dependent and environment-dependent failure characteristics was developed. This was a system of nested Markov Chains or multiple streams, which each stream or chain representing a group of end items within the end-item population.

The multi-stream concept employed in this maintenance requirements planning model is an expansion of the multi-stream or nested Markov chain concept utilized by Frisch [7] in the development of a conceptual Markov model that evaluates the mortality characteristics of populations of spare parts. The version of the multi-stream concept employed by this model treats the failure of end items as a discrete, nonstationary, Markov process. Each step of the Markov process represents one period. The user of the model defines the length of period.

A Markov process was chosen because the life cycle of items evaluated by logistics models generally is finite. This finite period may be broken into stages or states in which the item deteriorates toward a final state where one of two things could happen, i.e. (1) it is no longer economically or oper- ationally feasible to continue to repair the end item, so it is retired from operation, or (2) the end item experiences a fatal failure. At each state leading to the final state, there is a probability of a failure and a probability that no failure will occur. The no failure/ failure random probability in each state is analogous to the birth-death process which can be suitably modeled as a Markov process. Further, the state-by-state evaluation of the unit life cycle possesses the Markov property, in that it is only necessary to know the present state of an end item in order to forecast future states. A nonstationary Markov process is utilized because the model is used to evaluate nonsteady-state conditions where there is no uniform transition probability from state to state.

The concept employed here is called multi-stream because the population of end

items being evaluated is divided into several streams depending upon such parameters as the age of the end item, the number of hours that the end item has operated, or the environment in which the end item operates. Each of these streams is a separate Markov chain consisting of end items with identical failure characteristics. In other words, items within a stream are similarly aged, are at the same stage of deterioration, and experience a similar failure environment. Grouping end items into homogeneous streams avoids the need for item-by-item tracking.

A stream begins when the first item in that grouping is procured, and it ends when the last item in that grouping is removed from operation through retirement or fatal failure. Each stream generates its own maintenance requirements. See Fig. 1 for an illustration of the stream concept.

Generally distributed probabilities of failure are applied to end item components to generate spare part and labour requirements at each step of the Markov process. The user of the model either inputs data for the model to use in computing period-by-period failure probabilities, or inputs the actual period-by- period failure probabilities for individual

t, ,12,, t. LA,

Phase I" Operating Phase out period period period

,lkXl-St~~d~-St~t~) Isteady-state -,nonsteady-state)

"o"-stead;fstate~

Fig. 1. Nested Markov chain (multi-stream) concept. Each node has a maintenance tree attached to it (see Fig. 3). Key: r, = time period over which logistician is evaluating inventory requirements (e.g. on a weekly basis, on an annual basis, etc.); G-n = age of units; the largest G-n represents the retirement age of the unit; abc = a stream or a Markov chain (all of the streams together represent the nested Markov chain concept); I = number of units procured during time period, t,,; and a,b,c > 0.

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2*MiBF t 2'Mb3F t

(a)Uniform (b) Increasing Exponential

1.0

FX(t) . . . . . . .

0.5 Lzr- 0 2 4 6 8 Z'MTBF t

(years) (c) Exponential

Z'HTBF t 2'MTBF t

(d)Approximate Normal (e) Bathtub

Fig. 2. Cumulative failure probability distributions.

components. The data that must be input by the user, if the model is to compute failure probabilities, are a failure curve choice and a mean time between failure parameter. This data must be input for each “bottom of the tree” node component (described in Sec- tion 3.2). The model user may select a cumulative distribution from the library of failure curves shown in Fig. 2. This curve should be representative of the failure characteristics of the component for which it has been chosen.

3.2 Generating maintenance requirements

In addition to the framework of the nested Markov chains, a framework is needed for evaluating the interrelationships of the failure probabilities of the various components that comprise each end item. The model operates around a maintenance tree

which is constructed using the bill of materials for the end item being evaluated. The maintenance tree presents an organized framework for determining the maintenance requirements of the components of the end item and summing these needs to generate the maintenance requirements for the total population of end items.

When an end item fails, the affected unit is removed from service and is directed to the appropriate repair facility. The maintenance tree, built from the end-item bill of materials, is used to derive the probabilities of these end item failures. The maintenance tree also contains information that governs the repair process. Figure 3 provides an illustration of maintenance-tree construction.

The maintenance tree allows the end item to be systematically divided into various major components. Below each major component will be the family of subcomponents and stockable parts that make up the major component. Each level of the maintenance tree represents a level of indenture. At some level a “bottom of the tree” node will be encountered where it is not practical to break the component down further. At this node, failure data are either available or can be estimated. Once this level of indenture is reached, a list of stockable parts must be attached to the “bottom of the tree” node. This list should include all parts that are intended to be stocked to support the spare- parts requirements of that particular “bottom of the tree” node. The sum of all of the lists of stockable parts should represent a complete list of all the parts that support the spare parts requirements of the end item.

The failures of “bottom of the tree” nodes are independent from one another, thus con- ditional probability is easily applied to evaluate the probability of failure of nodes located higher in the maintenance tree, by computing the unions of the probabilities of nodes connected directly beneath them.

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Node 6, as shown in Fig. 3, represents the probability of end-item fatal failure. Since this node is connected only to the end-item node (node l), only the probability of failure of the end item is dependent on this fatal failure probability. It is assumed that every end item has some probability of fatal failure.

The maintenance tree is also used as a frame to carry other information, such as a mean time to repair and a level of repair (LOR) for each “bottom of the tree” node. The LOR is basically the designator of the repair facility to which the end item should be sent if its failure is caused by a particular “bottom of the tree” component. The LOR concept is used by logistics models such as METRIC [ 81.

Each period, the model uses the MTBF and failure-curve data to determine the probability of failure of each “bottom of the tree” node. From these probabilities of failure, the spare-part requirements generated by each “bottom of the tree” node component are calculated as follows:

k

j-l

where :

N, =

P(fl,i =

Qij =

k =

number of end items in stream s; probability of failure of component i; number of stockable part i required to support component i; number of stockable parts required to support component i.

By utilizing the MTTR information input into the maintenance tree for each “bottom of the tree” node component, the expected manpower demand at the individual repair facilities is generated in a manner similar

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to the expected spare-parts demand. The following equation illustrates how manpower demand for the repair phase may be generated:

C Ns (C (p(flsi l MTTRsiJs),

S i

where MTTRsi = mean time to repair an end item in stream s if the failure is caused by component i.

The model automatically divides this sum among the repair facilities according to which repair facility performs the repair. The spare part demands and resulting warehouse space requirements are also allocated, by the model, to the appropriate repair facility. The model then utilizes the calculated probabilities of failure and MTTR data to compute the cost of logistics policy per environment per period and the availability of end items per environment per period.

3.3 Uses of the model

The model should be able to satisfy many of the logistician’s planning needs. In general, the model will generate the maintenance requirement demand of a finite population of repairable items. This will include spare-parts requirements in the form of EOQ inventory level policies, manpower hours required, the staff maintenance facilities, warehouse space required for spare parts inventories, the costs associated with the operation and maintenance of the designed logistics system, and the availability of units under the proposed logistics policies. Model outputs such as end-item population availability and cost ‘of logistics policy are produced on an environment-by-environment basis each period. These outputs serve as performance meas- ures to be used by the logistician in judging the effectiveness of proposed logistics policy. They allow the logistician to evaluate the

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effects of cost and availability constraints, as well as warehousing, personnel, and procurement policy constraints.

The model in its present form uses Eco- nomic Order Quantity Theory to calculate inventory lot sizing for the spare parts needed to meet the maintenance requirements of the end-item population. This gives the logistician an estimate of the costs involved with carry- ing an inventory of spare parts in support of the end-item population. Various other. methods of spare-parts inventory lot sizing could easily be incorporated into the model.

The model is structured to handle more than one level of repair simply by specifying, in the maintenance tree, what type of facility is required for a specific repair. The model produces spare-parts requirements based on the repair facility at which the repair must be made, so that spare-parts inventory requirements for separate levels of repair are totaled separately. By comparing the spare-parts requirements at the various levels of repair, common spare-parts requirements can be identified.

The model considers the MTTR associated with the repair being made at the specified repair facility. From this information the expected demand on various repair facilities is calculated. This can assist in the design and construction, or alteration, of repair facilities. Knowing the requirements ahead of time would also help with staffing and training. Since non-steady-state conditions are being evaluated, the characteristics of the repair channels can be adjusted to keep pace with the needs of the population. The same ad- justments may be made to the spare-parts inventory.

The logistics decision maker can use the model as an aid in making traditional repair or replace decisions. By varying the retirement age of end items, the user of the model can view the cause-and-effect relationships between retirement age and cost, availability, spare-parts inventory requirements and main-

tenance facility space, and manpower requirements. Decision variables besides the retirementsage, such as the total number of units in the population, may also be varied.

4. DESCRIPTION OF AN EXAMPLE PROBLEM

4.1 The SLWT system

The model was applied to a case population consisting of 220 Side Loadable Warping Tugs (SLWTs). The SLWT is a component of the Container Offloading and Transfer System (COTS) utilized by the U.S. Navy. The COTS is a deployable system designed to provide seabom military forces of the U.S. Navy with logistics support during and sub- sequent to entry on land masses, where suitable port facilities are not available. The logistics support provided is basically the unloading (from ships) and transfer to shore of the required quantities of bulk dry cargo and vehicles. The COTS is designed to operate beginning on the day after land entry by military forces (D-day + 1) to D-day + 180 or longer.

The SLWT is a powered causeway section that can serve several functions as a part of COTS. When equipped with a winch and A- frame, one of the major functions of the SLWT is to assist in the construction of the causeway on piers that connects ship to shore. The example problem is fabricated and does not necessarily represent real events, nor does the data used necessarily represent the present SLWT used by the Navy.

The SLWT is virtually immune to fatal failure when maintained properly. An ex- ception to this is when the SLWT is operating in a militarily hostile environment where an SLWT may be damaged irreparably by enemy attack. In the example problem, a population of 220 SLWTs is evaluated with some of the units operating in a support area environment that is removed from hostile military action

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and the rest of the units operating in a combat zone.

The combat zone differs from the support area in two ways. The major difference between the two environments is in fatal failure probabilities. The SLWTs operating in the support area environment are not likely to suffer fatal failure, which is represented by a mean time between fatal failures of 100,000 operating hours per unit. The cumulative rate of fatal failures follows an increasing exponential function. This means that, at the start, the probability of fatal failure will be almost nonexistent. On the other hand, SLWTs operating in the combat zone environment have a mean time between fatal failuress of 7200 operating hours per unit. The cumulative rate of fatal failures in this case follows a uniform distribution because fatal failure occurs due to enemy attack.

The second major difference is that SLWTs work a different number of hours per day in each environment. An SLWT works an average of 8 hours per day in the support area environment, and an average of 12 hours per day in the combat zone environment. This gives the various components (“botton of the tree” nodes) of the SLWT a different probability of failure in each environment.

The maintenance requirements of the SLWT population are evaluated over twelve 30day periods using an EOQ inventory lot sizing policy. The population of SLWTs will be evaluated as if the 220 end items were procured in a block procurement at time 0 and then operated for the next 12 consecutive periods.

There are three levels of repair that apply to SLWT. Organizational level repairs will be performed on board the SLWT with the necessary spare parts carried on board. Inter- mediate repairs will be performed by a repair facility set up on shore, with that facility stocking the necessary spare parts. The third level of repair, depot level repair, will be necessary only in the event of serious failures.

The depot is a centrally located facility and stocks its own parts. It is assumed that, regardless of the environment in which the units operate, they require the same levels of repair dependent on the type of component failure that occurs.

Since the SLWTs are virtually indestruc- tible by natural causes, the retirement age of the units can be taken as 500 periods, or roughly 40 years. To ensure success of the support area operation, a total of 140 SLWTs must be operating, and to ensure success of the combat zone operation, a total of 60 units must be operating. The set of information contained in Table 1 represents general population data that was input into the model for the case study.

The maintenance tree in Fig. 3 represents the hierarchical structure of the SLWT. Since the SLWTs operate in two separate environments, “bottom of the tree” node data for each environment had to be provided. In order to determine stockable-part lot sizing,

TABLE 1

General population data

Period parameter Period length Length of model run

of failure environments

First cost per SLWT Salvage value per SLWT

(upon fatal failure) Procurement costs Repair costs per SLWT

per day Shortage cost per day Time value of money

Days 30 12

2

$350,000

$ 8000 $ 10000

5 500 % 30000 10%

Model mode Proposed population Spare parts and warehouse requirements EOQ Procurement policy one initial block procurement Procurement amount 220 Divide procurement Support: 144 Combat: 76

Failure environments: Support area Combat zone Retirement age per

SLWT 500 periods 500 periods Retirement book value $lSOOO/SLWT $15OOO/SLWT Demand for SLWT’s 140 60 I

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inventory data had to be input for each “bottom of the tree” node. Inventory is input only once, regardless of the number of environments in which end items operate.

TABLE 2

4.2 Description of example problem results

Table 2 summarizes the spare parts and warehouse requirements for the total popula-

Period 5: Spare parts and warehouse requirement summary

Node Part Demand W’house EOQ Total no. (tuft) cost

Repair level

c

Eng. oil cooler 2101 3 5.00 10 114.49 1 Freshwa. pump 21211 9 12.00 12 2310.00 2 Seawater pump 21281 4 12.00 6 1256.51 2 Seawater pump 21282 4 3.00 15 35.49 2 Seawater pump 21283 1 4.40 22 134.14 2 Seawater pump 21284 1 4.40 22 120.14 2 Hyd. pump 21291 2 6.00 6 1228.28 1 Eng. coolant HE 28321 2 9.00 3 2048.99 2 Freshwa. HE 28331 2 8.00 4 140.00 1 Hyd. fluid HE 28341 2 8.00 4 940.00 1 Pres. red. valve 332521 1 2.60 26 25.65 1 Pres. red. valve 312522 1 0.11 11 1.22 1 Dir. con. valve 312531 41 16.90 169 1246.94 1 Dir. con. valve 312532 82 1.01 101 43.91 1 Steering motor 312541 3 6.00 6 902.40 2 Filter 313581 66 30.40 152 156.40 1 Starter 313591 5 19.00 19 968.11 1 Auto lub. 314621 13 21.50 43 1061.33 1 Auto lub. 314622 13 8.80 88 61.33 1 Timer 314631 38 21.50 110 2421.51 .l Manifold 314641 0 0.00 0 0.00 2 Manifold 314642 0 0.00 0 0.00 2 Solenoid 314651 38 1.30 13 196.41 1 Waterjet pump 420921 0 0.00 0 0.00 3 Disch. elbow 420931 1 4.00 4 68.11 2 Disch. elbow 420932 1 0.92 92 5.42 2 Disch. elbow 420933 1 3.50 118 3.91 2 Steer. nozzle 420941 0 0.00 0 0.00 2 Booster pump 5231011 3 4.50 9 262.91 2 Fuel/wa. Sep. 5231081 13 21.50 43 801.33 1 Inverter 5241121 8 3.30 33 160.13 1 Battery 5241131 41 50.00 50 2509.60 1 Battery 5241132 82 31.40 151 523.31 1 Alternator 5241141 8 4.90 98 44.90 1 Alternator 5241142 15 6.10 134 36.11 1 Alternator 5241143 8 6.90 69 114.93 1 Hyd. motor 5251181 31 22.00 22 24911.36 2 Hyd. pump 5251191 41 26.00 26 24128.06 2 Cable 5251201 0 0.00 0 0.00 1 Engine on/off 5261231 1 2.60 26 23.65 1 Throttle 5261241 0 0.00 0 0.00 1 Steer. lever 5261251 0 0.00 0 0.00 1 Clutch lever 5261261 0 0.00 0 0.00 1

Total warehouse requirement repair level 1 = Total warehouse requirement repair level 2 = Total warehouse requirement repair level 3 =

296.08 111.12

0.00

tion of SLWTs for time period 5. This type of summary is produced for each period. These period summary charts give the logistician a tool to use in planning his stockable- part ordering schedule to ensure that the proper repair facilities are stocked at a level that should satisfy the maintenance requirement for spare parts generated by the population of SLWTs.

In addition to period spare parts and warehouse summaries, end of the model run summaries are produced. Included among these are the manpower requirements per repair facility per period. Information supplied in the manpower requirement summary includes the number of man-days of repair time per period required at each level of repair by the total population of SLWTs. The summary produced by the case study showed significant period-to-period fluctuations in the expected man-days required. If the logistics decision-maker has some idea of the expected repair-time requirements, it may be possible to prevent repair delays due to staffing short- ages, or prevent overstaffing due to sudden drops in repair demand.

An additional model summary shows the period-by-period probabilities of failure of the SLWT. If the model user chooses the print option, period-by-period probabilities of failure for each node are tracked and printed. Plots of the point probabilities and cumulative probabilities of failure for selected “bottom of the tree” nodes mirror the failure curves originally chosen by the model user. Plots of the point probabilities of nodes higher in the maintenance tree yield useful information on general failure patterns of major components of the SLWT.

In Fig. 4, it is interesting to note that the plot of point probabilities of failure for node 1 (the end item or SLWT) shows that the failure rate tends to approach a steady-state in each environment after about 3 months of operation. This is not really very soon con-

0.24

0.20 Support area

0.16

0.12

0.08

0.04

0 2 4 6 8 10 12 (periods)

(Z.1

NODE2, Point probabilities of failure

1.0 support area

0.8

0.6

0.4

0.2 b- 1.0 Combat zone

0.8

0.6

0.4

0.2 i’- I

0 2 4 6 8 10 12 0 2 4 6 8 10 12

Iperiods) (periods)

(C) Id)

NODEl Point probabilities NODE1 Point probabilities of failure of failure

F@. 4. Probability of failure plots.

0 2 4 6 (periods)

(b) NODEZ3 Point probabll1ties

of failure

sidering that the SLWT is designed to operate for only l-6 months. The plots of point probabilities of failure for node 23 illustrates that some of the component probabilities of failure underlying the end-item probability of failure may never reach a steady-state failure rate during the evaluation period.

Additional end-of-the-model run information output by the model includes .the availability of SLWTs in their respective environments. The results of the case study showed that, by the 10th month, the SLWT population in the combat zone was operable less that 14% of the time. Because of fatal failures, the number of SLWTs operating in the combat zone dropped below the demand figure of 60 by the end of period 10. Fluc- tuations in end-item availability for the support area were due totally to minimal failure of end items, as no end items in this environment failed fatally during the model run.

Finally, the model produced a summary of the period-by-period costs to operate the

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population of SLWT’s in each environment. A period equivalent cost per environment was also given. The summary illustrated that shortage penalties played a significant role in the total costs and the operating cost per unit for the SLWT’s operating in both environments. The costs of operation per environment serves as an index that the logistics decision maker can now use to evaluate the effects of altering various decision variables.

5. CONCLUSIONS

The failure behaviour of repairable items in real life very often has a transient, rather than steady-state, character. The example problem illustrated how maintenance requirements can vary from period to period as a result of transient-state failure characteristics. If a logistician is to be effective in providing support to handle maintenance requirements generated by repairable item populations, some means of evaluating maintenance requirements produced during transient periods, as well as steady-state periods of operation, must be made available. The model methodology described in this paper illustrates some basic concepts that may be applied in the evaluation of maintenance requirements produced under transient or steady-state failure conditions.

Finally, the use of concepts illustrated by the model discussed in this paper, can help

the logistics planner to meet availability requirements imposed on the populations of end items that must be managed. More effective satisfaction of requirements means being able to predict the availability of end- item populations more accurately, thus allowing more cost-effective design of populations and logistics support systems.

References

Geisler, M.A. and Murrie, B.L., 1981. Assessment of Aircraft Logistics Planning Models, Omega, 9: 59- 69. Kline, M.B., 1982. A survey of logistics analysis models. In: Proc. 17th Annual Int. Logistics Symp., August, 1982. Sot. Logistics Engineers, Boston, Mass. Pierskalla, W.P. and Voelker, J.A., 1976. A survey of maintenance models: The control and surveiIlance of deterioration systems. Nav. Res. Logistics Q., 23: 353-388. Barlow, R.E. and Proschan, F., 1967. Mathematical Theory of Reliability. Wiley, New York. McCall, J.J., 1965. Maintenance policies for stochast- ically failing equipment: A survey. Manage. %I., 11: 493-524. Sherif, Y.S. and Smith, M.L. 1981. Optimal maintenance models for systems subject to failure - A review. Nav. Res. Logistics Q., (March): 47-74. Frisch, F.A.P., 1983. Mortality and spareparts: A conceptual analysis. In: Proc. 1983 Federal Acquisition Research Symp. Sherbrooke, C.C., 1968. METRIC: A multi-echelon technique for recoverable item control. Oper. Res., 16: 122-141. Giesler, M.A. (Ed.), 1975. Preface. In: Logistics North- Holland/TIMS Studies in the Management Science Series, Vol. I. North-Holland, Amsterdam.

a transient state model for predicting maintenance requirements

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