O~IEG.4 Int. J of Mgmt Sci. Vol. 13. No 2. pp. I25--f30. 1985 0305-0483 S5S3 00-,-000 Pnnted in Great Britain Pergamon Press Ltd

Relating Supply Performance to Aircraft Availability

W A L E W I N S

K C P A K E N H A M

Department of the Chief Scientist (RAF), Ministry of Defence, UK

(Received April 1984: in recised forrn May 1984)

This paper describes how the effectiveness of the supply of vital repairable items for an aircraft or weapon system can be measured in terms of the expected number of aircraft grounded because of supply shortages. The properties of this measure are compared with the three alternative methods of assessing supply performance, none of which are directly aircraft or weapon system related. The present applicability of this measure to various situations at RAF Squadron and station level are discussed, together with possible future areas of application.

I N T R O D U C T I O N

IN MONITORING how effective a particular set of spares for an aircraft is, or how spares should be scaled for that aircraft, it is necessary to examine how the set of spares or proposed set of spares performs against some chosen supply effectiveness measure. At present, the supply effectiveness measure employed by the RAF is 'fill-rate', defined as the proportion of demands which can be satisfied from stock on hand. A major drawback of fill-rate as an effectiveness measure is that it does not provide a value which can be related to the number of aircraft likely to be affected by a shortage of spares. The aim of this paper is to suggest an aircraft-based measure which can be used either to sup- plement decisions made on the basis of fill-rate or preferably to replace fill-rate entirely. The measure suggested provides "'the expected num- ber of aircraft not operationally ready because of supply", from now referred to as "the ex- pected number of aircraft NORS", or "expected aircraft NORS'" for short.

The term "NORS' was developed in the 1960s and appears mainly in work by the R A N D Corporat ion and the Logistics Manage- ment Institute (LMI), both American con- tractors. In particular, the expected aircraft

NORS measure was derived and used for two United States Air Force Projects [1, 2]. How- ever, the measure was derived only for a special case, as will be seen below. The method outlined in this paper provides the general form of the expected aircraft NORS measure, without the restriction of [1,2].

PRESENT RAF SCALING MODEL

Repairable items account for the vast major- ity of the RAF's aircraft spares budget. The model used to predict the number of spares required for the repairable items of a particular aircraft type is the Single Item Model (SIM), another R A N D Corporat ion product. In the SIM, demands for each item are taken to corre- spond to a Geometric Poisson distribution and, as the model 's name suggests, items are consid- ered individually and the number of spares of each item expected to be required to achieve a pre-determined fill-rate (typically around 80%) is calculated. The total number of spares re- quired is then obtained by summing the individ- ual item requirements.

This model has been employed over a number of years, despite growing dissatisfaction with the concept of fill-rate as a supply effectiveness measure. That the RAF has not discarded

126 Lea'ins. Pakenham--Relating Supply Performance to .4ircra]~ At'ailability

fill-rate in favour of expected aircraft NORS some time ago is partly because the fairly intri- cate combinatoric calculations necessary to pro- duce an analytic general expected aircraft NORS measure have not been easy to carry out, and partly because the requirement to "optimize" system performance in some sense with a stead- ily decreasing budget, has never been so strong as at present.

SUPPLY EFFECTIVENESS MEASURES

There are four main contenders from which to choose the most appropriate supply effectiveness measure. These are fill rate. the expected number of backorders (EBOs), the probability of no shortages (PNS) and expected aircraft NORS. These are defined as follows.

a.

b.

Fill-rate--the proportion of demands which can be satisfied from stock on hand.

EBOs-- the expected number of unfulfilled demands.

PNS-- the probability of all aircraft being available.

d. Expected aircraft NORS-- the expected number of aircraft unavailable for supply shortage reasons.

These terms usually refer to aircraft at a squad- ron or station level. Typically, there might be ten or twelve aircraft on a squadron and three squadrons on a station.

Intuitively, all four of the above measures would satisfy the immediately obvious criterion that would be set, in that their value would reflect greater 'effectiveness' as more spares are added. Fill-rate and PNS would increase, EBOs and expected aircraft NORS would decrease. Hence in order to differentiate between the measures, it is necessary to consider their indi- vidual attributes a little more closely.

Firstly, fill-rate is a widely used measure which is both easy to calculate and readily understood. However, it does have distinct dis- advantages. Most importantly, it has no direct relation to the availability of aircraft. For ex- ample, suppose that an overall 80~/oo fill-rate is achieved. If a large proportion of the remaining 'unfilled' 20°~ is due to shortages in one or two items, then several aircraft will be missing those

items and hence will be unavailable. Ifi however. the 20°0 is spread over many items, the gaps could well be confined to relatively few aircraft. with the remainder operational. This will be emphasised if "robbing" is permitted, where rob- bing is the removal of serviceable parts from an aircraft which is unserviceable for other reasons. in order to satisfy a demand elsewhere. That is, robbing leads to the consolidation of 'gaps" or 'holes" onto fewer aircraft.

A second, though less important deficiency of fill-rate is its inability to represent satisfactorily any time factor involved. It may. perhaps, be better to have a fill-rate of, say, 70% if the remaining 30°~ can be recovered in a few days, than to have a fill-rate of 80°,0 if the remaining 20~ take several weeks to recover. However, fill-rate does not readily lend itself to this type of comparison.

The second measure, EBOs, is similar in character to fill-rate in that it considers the same aspect, but in a slightly different light. Where fill-rate measures the proportion of demands which can be satisfied t¥om stock on hand, EBOs gives the anticipated number of demands which cannot be satisfied. Thus although EBOs again provides an easily understood measure, it too has the major disadvantage of not being directly related to aircraft availability levels. EBOs does, however, have two advantages of secondary importance. Firstly, EBOs is a measure which is additive over items, that is the total number of backorders in the system is simply the sum of the backorders for each item. Secondly, it is easier to incorporate a recovery time factor to accompany the backorders mea- sure, perhaps in terms of how long a particular backorder takes to be satisfied.

PNS, thirdly, does have one distinct advan- tage over the two earlier methods in that it provides information directly related to aircraft availability, It calculates the chance of having all aircraft available for operation. However, there are difficulties with this measure. For example, the probability that there will be short- ages is (I-PNS), but this gives no value for how many aircraft there are likely to be shortages on. All that is available is the intuitive concept that the lower is the PNS, the more aircraft there are likely to be unavailable. This cannot be quantified more precisely, since the relationship is not directly one-to-one. Further, PNS is rather more nebulous than the two earlier

Omega. Vol. 13..'Vo. 2 127

measures and consequently it is a little more difficult to relate a particular value of PNS to a practical situation. For example, it is easy for an RAF officer to quantify the practical difference between three and five backorders, but it is not so easy for him to quantify the practical difference between a PNS of 0.32 and one of 0.28.

The final measure is the expected number of aircraft NORS. This is a direct availability measure, supplying the expected aircraft avail- ability level corresponding to a particular set of spares. That is. it provides a measure of oper- ational capability, rather than being purely sup- ply oriented.

An important benefit of the expected aircraft NORS measure is that, as a direct availability measure, it can examine the effect of certain realistic options which the other three measures can not. Possibly the most significant of these is robbing. Clearly, if more robbing is allowed to take place, fewer aircraft will be unavailable. Expected aircraft NORS can reflect this, whereas the other three measures remain un- changed. Fill-rate looks simply at the propor- tion of demands satisfied, which is not changed simply by moving the "holes' round, which in essence is what robbing does. Again, there is no increase in the number of uncommitted spares, hence the number of backorders will not change and, for the same reason, neither will PNS. Thus the expected aircraft NORS measure can reflect different policy decisions concerning whether all items are regarded as robbable, no items are robbable or some items robbable and others not. As will be seen, it is in this aspect that the development of the expected aircraft NORS measure derived within the Department of the Chief Scientist (RAF) discussed in this paper represents an advance over the previous work [I, 2], which assumed that all items can always be robbed.

experiencing a 'vital" (or 'mission essential') repairable item failure which can be replaced from station stores or by robbing from another aircraft (if permitted) is not classed as NORS. Hence a NORS aircraft is defined here as an aircraft which is grounded through a delay awaiting resupply of one or more vital repair- able items from further down the pipeline of repair and resupply. This definition is felt to be the most appropriate, but the measure can very easily be adapted if a different definition is felt to be more appropriate. As can be seen from this explanation, the measure concentrates on vital (mission essential) repairable items. More will be said about this later.

It should be stressed that the method de- scribed here is analytic, rather than a simu- lation. At present, it can be used either on its own by using actual or possible numbers of spare items, or by linking it with an approach which provides a scale of spares (such as SIM). Further, although the term 'aircraft-based' is used throughout this paper, the method is equally applicable to overall weapon systems, provided that data on the vital repairable items of the weapons are available.

As in any mathematical representation of the real world, certain data and parameters derived from data are required to drive the model. As mentioned above, either the scale of spare items actually held, or the possible set of spares under consideration has to be known. In addition, the following information needs to be supplied in order to calculate the expected aircraft NORS measure.

a.

b.

A T T R I B U T E S OF EXPECTED A I R C R A F T c.

NORS

The method has been derived to work at RAF squadron and station level. In order to clarify d. what expected aircraft NORS measures, it is necessary to explain how a NORS aircraft is defined here. Suppose that the process is work- ing at RAF station level. Then an aircraft

The number of aircraft on the squadron or station (depending on the level under con- sideration).

The number of each item fitted on the aircraft.

The average number of flying hours per month flown or projected for each air- craft.

Parameters describing the demand distri- bution for each item. In the case of Geo- metric Poisson demand, these would be the average failure time and the variance- to-mean ratio of each item.

128 Lewins, Pakenharn--Relating Supply Perforrnance to Aircraj~ Acailability

The overall average resupply or replen- ishment time, defined as the average time taken for a replacement of a failed item to reach the squadron or station from lower lines of repair.

By mathematical modelling standards, this data requirement is relatively modest. Each of the above parameters can be obtained from existing records held within the RAF. Further- more, these parameters are already used in the SIM.

Although fairly mathematically intricate to evolve, expected aircraft NORS is an easy measure to use and to interpret. The way in which it operates is to accept as input the level of spares scaled (or suggested) for each of the aircraft's vital repairable items. From this input, the calculations outlined later are carried out, convoluting the information on each item to give the expected number of aircraft NORS, which forms the output of the process. Note that this measure is not additive. That is, whereas the total number of expected back- orders is simply the sum of the individual item expected backorders, the overall expected air- craft NORS value is not simply the sum of the component values for each item. Basically, this is because a single aircraft can be NORS due to shortages in more than one item and this poss- ible overlap must be taken into account. For example, one shortage in each of five different items could lead to anything from one to five aircraft being NORS, depending on policy and circumstances.

If desired, expected aircraft NORS can be measured against cost by reading off the cost of the particular set of spares which led to the calculated expected aircraft NORS value. Simi- larly, for any example, the expected aircraft NORS value can be measured against the calcu- lated values of fill-rate, EBOs and PNS. Al- though the expected aircraft NORS measure will of course be highly correlated with the other three measures, it is not possible in general to proceed from a particular value of expected aircraft NORS to a single value for the other three measures.

As suggested earlier, the expected aircraft NORS measure outlined here can take robbing into account. It is possible to specify whether no robbing is allowed, unlimited robbing allowed, or, the most realistic case, that some items are

robbable and others not. Hence when results are required using this last case, it is clearly neces- sary to specify which items are robbable and which not. For some aircraft types, this infor- mation can be gained directly from records and for the remainder it can be obtained from engineering experience. It is outlined later how the expected aircraft NORS measure is derived in each of the above three cases.

PRESENT AND F U T U R E SCOPE OF APPLICATION

Although this work is at an early stage of its development and it is not pretended that it can solve all of the RAF's supply scaling problems, it is felt that the expected aircraft NORS measure can already play a valuable part in assisting supply decision making. As mentioned in the introduction, the supply effectiveness measure presently employed by the RAF to assist in the scaling of vital repairable items by the SIM is fill-rate. A further drawback of fill-rate not mentioned earlier is that very many completely different sets of spares can lead to the same overall achieved fill-rate. This will not be as much of a problem with the expected aircraft NORS measure. Hence it is intended, in the first instance, that the expected aircraft NORS measure can help to select the best set of spares from sets with equal or similar fill rates. Further, it could help to examine whether the extra cost required to increase an achieved fill rate is operationally worthwhile.

The first of these points can be described by considering a trivial example. Suppose that there are three aircraft, each with three vital repairable components "X', 'Y' and 'Z' say, and suppose robbing is allowed.

If we suppose that there are 2 demands for 'X' l demand for 'Y' and 1 demand for 'Z' then:

a. if the spares pack comprised only 3 'X', a fill-rate of 50~/o would be achieved and there would be I aircraft NORS; and

b. if the spares pack comprised '2 'Y' and 1 'Z', a fill-rate of 50~ would again be achieved, but this time there would be 2 aircraft NORS, since 2 aircraft would be lacking an 'X'.

Obviously, it is not claimed that the above example is realistic, but it serves to illustrate an

Omega. Vol. 13. No. 2 129

immediate application to which the expected aircraft NORS measure can be put. The set of spares taken away' by a group of aircraft on detachment from their normal station is known as a "fly-away pack" (FAP). Scaling calculations for FAPs are presently obtained from the SIM, based on fill-rate and cost targets. Hence any FAPs produced by this method, with similar expected achieved fill-rates or which meet cost criteria or both. could use expected aircraft NORS as an aircraft-based means of differentiation.

Two points should be re-emphasised when considering the measure's future applicability. Firstly, having given the above example, recall that the expected aircraft NORS measure is not restricted to use with the SIM, it can be used either with any scale producing model or with a reading from a scale. Also, the measure is in an early stage of its development. Whilst it may already be regarded as a useful measure, the scope for its application in the future is far wider. Possibly its main limitation at present is the requirement for a scale of spares. The measure is now being extended to give the option of providing a scale of spares which can give the smallest expected NORS value for a given budget, or of minimising the cost of providing spares for a given aircraft NORS level.

Before its full potential can be reached, there are a number of other limitations which need to be overcome. At present, the process is limited to squadron or station level. Ultimately, it would be hoped to be able to examine the entire fleet of any weapon system and allocate spares optimally on a global basis from the main repair and maintenance depots right down to individ- ual squadrons. The major difficulty which needs to be overcome before this is possible is that of dealing with multi-applicable items, that is, those items which can be fitted on more than one aircraft type. At this stage, we are restricted to regarding each item as earmarked for a particular aircraft type. Since work is now mainly centred on squadron level this does not present a problem, since once on a squadron, it can reasonably be assumed that only one air- craft type is available. Such an assumption must be avoided if it is wished to take into account the situation at central depots supplying squad- rons of several different aircraft types.

Finally, as suggested earlier, the process con-

centrates on vital (or mission essential) repair- able items. It would be useful to extend the work to cover both mission essential and mission desirable items, since, as yet, the only way of considering mission desirables is to draw no distinction between desirables and essentials. As a corollary to this, it is theoretically feasible to take into account a complete hierarchy of essen- tiality. However. it is anticipated that this would run into problems of a practical rather than a mathematical nature, since all interested parties from different specialisations would need to agree on the level of essentiality of each item.

L O G I C OF M A T H E M A T I C A L D E R I V A T I O N

The mathematical derivation of the expected aircraft NORS measure depends largely on the convolution of a number of combinatoric argu- ments. Whilst it is not felt that it would be suitable to reproduce the mathematics here, an indication of the logic and procedures followed may be of interest. Basically, the contribution made by individual items is calculated first of all, both for robbable and for unrobbable items. This information concerning the individual items is then convoluted to form the expected aircraft NORS measure for the aircraft as a whole.

Firstly, then, we calculate the cumulative probability of M aircraft NORS due to a rob- bable item i taken in isolation. This is a simple procedure for a robbable item, since "holes" (unsatisfied demands) due to the item are taken to be concentrated to ensure that the minimum number of aircraft possible are grounded. The cumulative probability of M aircraft NORS due to a robbable item i, p~rl (M) can be written as a simple function of M, the number of spares of item i and the fit of item i (that is, the number of item i on each aircraft).

The next stage is to calculate the probability of M aircraft NORS due to an unrobbable item j taken in isolation and then to form the cumu- lative probability P}"~ (M). This is a little more complicated than the first stage, since 'holes' cannot be consolidated: whether failures of item j occur on the same aircraft or on different aircraft needs to be considered. However, the probability of M aircraft NORS due to the unrobbable item j can be calculated by the

130 Lewins. Pakenham--Relating Supply Performance to Aircraft drailability

iteration of a conditioning argument and then P '~ ( M) obtained by summation.

From the above procedures, the expected number of aircraft NORS due to a robbable item i taken in isolation, and the expected number of aircraft NORS due to an unrobbable item j taken in isolation are

E i ( M ) = ~ (1-Pl '~(- 'v/)) .I,I ~ 0

and

Ej(/W)= ~ (I - P!") (M)), - - g

M = 0

where n is the total number of aircraft under consideration.

The next stage is to calculate the cumulative probability of M aircraft NORS due to the first i robbable items, QI '~ (M) say. This is done by building up iteratively and using Q~'~(M)= P]'~ (M). With the added assumption, which is used throughout, that different item failures are independent, and extending over all N, robbable items, it can be shown that

V r

Q(n ( Iv[ )= I-I p ! r ) (M) . ,V r - - t

i = l

The corresponding result for unrobbable items Q~'~ (M) is obtained. This is more complex to )"u

evolve than the cumulative probability for the robbable items because of the non- consolidation of holes. It is calculated by build- ing up iteratively over the individual items and obtaining recurrence relations, and by the application of combinatoric and conditioning arguments.

Finally, the last two expressions are con- voluted to give the cumulative probability of M aircraft NORS due to the whole set of N items on the aircraft, where N = 3;, + N,. This is a simple procedure. Consider the two sub-packs of items (that is, the set of robbable and the set of unrobbable items) as two "composite' items, one robbable and the other not. In effect, this means that they can be treated as if they were two robbable items, since only one of the com- posite items needs to be moved to consolidate the 'holes' on the fewest aircraft possible. Thus F(M), the probability of at most M aircraft NORS is just

F ( M ) = t3~,~ (M) t3~,~ ~,v, ~,% (M).

Finally, if E(NORS) represents the expected number of aircraft NORS taken over all items (that is, over the whole aircraft or weapon system), then

E(NORS) = ~ ( 1 - F(M)). ),1 = 0

It should also be mentioned that because of the way in which the result has been derived, it is a simple matter to extract the probability distribu- tion of the number of aircraft that are NORS. Hence the mean, variance and other moments can be calculated and their meaningfulness examined.

CONCLUSION

This paper suggests an aircraft-based supply performance measure, with the logic of its deri- vation outlined above. In its present form, it is suggested that it can be of use in aiding decision making in relation to supply scaling by using it as a measure which differentiates between scales derived from other arguments. However, it has potential to be developed in several ways, for example to calculate scales which optimize the set of spares provisioned within certain con- straints, such as for a budget or for a given acceptable aircraft-on-ground rate. Various de- partments within the Royal Air Force are now considering the range of applications to which the measure can best be put.

A C K N O W L E D G E M E N T

The authors wish to express their thanks to the Chief Scientist (RAF) for permitting the release of this paper for publication. The opinions expressed are their own and should not be attributed to the Department of the Chief Scientist (RAF).

REFERENCES

l. Brooks RS, Gillen CA and Lu JY (1969) Alternatire Measures of Supply Performance." Fills, Backorders, Operational Rate and NORS. RAND Memorandum RM-6094-PR, Rand Corporation, Santa Monica.

2. Sherbrooke CC (1968) MINE: Multi-lndenture NORS Eealuator. RAND Memorandum RM-5826-PR, Rand Corporation, Santa Monica.

A D D R ~ FOR CORRESPONDENCE: Dr WA Lewins, Science 1 (RAF), Department of the Chief Scientist (RAF), Room 12,-°3. MOD Main Building, Whitehall SWIA 2HB. UK.