Buying Spare Parts for the Last Time
by Eric A. Snyder
The "life of type" statistical model allows the Army to determine the best way to buy parts that are going out of production.
In the world of Army wholesale supply, keeping the proper number of spare parts in stock is a battle all its own. In order to keep our equipment running, we need to make sure that parts are available when repairs are needed. Peacetime operations, though less demanding than wartime operations, still require weapon systems to be functional so that our soldiers get the training they need and are ready for potential deployment around the world.
In the past few years, because of decreases in funding and vast improvements in weapon systems, we have seen a large increase in the number of spare parts that have become "life of type" (LOT) items. LOT items are those that, for whatever reason, no longer are going to be produced; that means one final purchase of those items is needed to sustain equipment in the field. Making the most prudent decision on these types of buys clearly is essential to maintaining readiness as well as to managing available funds efficiently.
Until recently, the only mathematical model designed to make decisions on what quantities to order was the Variable Safety Level/Economic Order Quantity (VSL/EOQ) model developed by the Army Materiel Systems Analysis Activity (AMSAA) at Aberdeen Proving Ground, Maryland. This model—part of the Army's Commodity Command Standard System—assisted item managers by providing safety levels and order quantities. A safety level is the point (meaning a quantity of a particular item on hand) at which an order must be placed to ensure, to a certain specified probability, that the item will not go out of stock until that order is received in the supply system. An order quantity is the most costefficient amount to order in a single procurement.
However, the VSL/EOQ model, although quite capable of producing large savings in the supply system, was not designed to handle the unique requirements of a LOT buy. Item managers were left to make their most educated guesses at how many of an item to purchase. To fill this gap, the Army Materiel Command (AMC) tasked AMSAA to design a model specifically for the purpose of determining the most costefficient LOT purchase.
Mathematics of the Model
To discuss the LOT model in detail would require a large amount of mathematics and statistics, so I will provide a basic description of how the model works. The main concern involved in making LOT purchases can be understood best by examining the classic news vendor problem found in the study of operations research. In this problem, we have a news vendor who faces the dilemma of how many newspapers to purchase for a given day for his newsstand. If he does not purchase enough papers and thus runs out, he misses out on potential profit. If, on the other hand, he purchases too many papers, he has wasted his money on papers that will not be bought by customers. So how does the vendor decide on the correct quantity of papers to order?
In the ideal inventory world, each day would be the same: the same number of customers would come to the newsstand, and each customer would purchase the same number of papers. The vendor therefore would order this known quantity of papers, and he always would find the last customer of the day purchasing the last paper of the day. Of course, the real world has this annoying quality that we call uncertainty. To the statistician, it is known as variance.
With uncertainty introduced into his situation, the news vendor must decide which daily order quantity would yield the largest profit. To describe the variance in a system, statisticians use probability distributions. (The most famous probability distribution is the normal bellshaped curve.) These distributions are used to describe the likelihood of each possible outcome. For example, let's say that on any given day the newsstand will sell somewhere between 8 and 12 papers with equal likelihood. That is, on onefifth of the days, there will be a demand for 8 papers, on another onefifth of the days there will be a demand for 9 papers, and so on. This would thus give us a distribution of the probability of demand shown in the chart to the right.

Distribution of the probability of demand for the news vendor example. 
If we know how much the vendor pays for the papers (the wholesale price), and for how much he sells the papers to his customers (the retail price), we can determine the optimum quantity of papers to buy mathematically. For example, suppose our news vendor purchases the papers for 25 cents apiece, and, turning a nice profit, sells them for 50 cents each. We can figure his profit margin by comparing each buy possibility with each demand possibility. Multiplying the number of papers sold by the 25cent profit per paper, and then subtracting 50 cents for each unsold paper, yields the profit margin. The chart below illustrates the results.
The average profit margin for a given buy decision thus can be determined by taking the simple average of all the possible outcomes. If the outcomes had not been equally likely, as is so often the case, then we would have to weight each of the cases by the probability for that event. In the end, as the chart shows, the vendor would decide to purchase 10 papers per day since this would yield the largest expected profit margin.
Profit margin for the news vendor example. 
Applying the Theory
The LOT model, although it has its roots in the news vendor theory, has a couple of major differences. First, the danger of understocking is not the loss of potential profit but rather the loss of availability of the weapon systems that need the part. Maintaining a target availability is a chief concern of the Army supply process. Second, in the news vendor example, if the vendor understocked papers and his supply ran out, there was little he could do about it. However, if there is a shortfall in the supply of a spare part, the Army may resort to a premium purchase to correct the deficiency. Although premium buys seem to contradict the concept of a LOT buy, when the situation requires it, such purchases are made. This often will mean starting up an assembly line or purchasing a commercial product. In either case, the cost is substantially higher than the original production cost.
Choosing the statistical distribution method for the LOT model was an important consideration. In the news vendor example, the distribution was very simple and very limited, with only five, equally likely demand possibilities. In the real world, the number of possibilities for demand for a given part is vast and not as easily predicted. Experience gained in developing the VSL/EOQ model helped to determine the distribution of demand in the LOT model. A cataloging approach based on the negative binomial distribution was selected for the LOT model. [A negative binominal distribution represents how many failures must occur until a predetermined number of successes have occurred.]
The model contains many other features that extend beyond the original theory. The easiest way to understand all the flexibility of the model is to look at the inputs and parameters that the user must supply to the model. In most cases, the user is the item manager directly responsible for the LOT purchases.
Inputs
There are seven inputs to the model for a given part: national stock number (NSN), expected life, unit price, premium price ratio, average yearly demand, average requisition size, and assets on hand. The NSN is a unique 13digit identifier used in the model to track a particular part. The expected life is an estimate of the number of years the weapon system that contains the part will remain in operation; this number is only an estimate, and it creates an uncertainty that will be discussed below under the "robustness" parameter. The unit price is the current cost to produce the item.
The premium price ratio is the anticipated ratio of the premium price to the unit price. For example, if a part currently can be produced for $10 and later would cost $50 as a premium buy, this ratio would be set at 5. This again is only an estimate. Most of AMC's major subordinate commands—such as the Tankautomotive and Armaments Command, Aviation and Missile Command, and CommunicationsElectronics Command—have historical estimates on premium buys available, which can be used to approximate this number.
Average yearly demand is the expected quantity of a part requisitioned from the wholesale level of supply each year based on historical demand data. The average requisition size is the expected number of items requested in a single requisition. This number is important only in that the distribution of demand for a given item is correlated with the average requisition size. Finally, the assets on hand represent the number of items currently in the supply system. These include items that are serviceable (ready for issue), unserviceable (in need of repair), and due in (en route to the inventory control point).
Parameters
While the inputs represent information for a given part, the parameters represent matters of policy that affect how the model runs. For some parameters, policy can be set on an Armywide basis; for others, it can vary by major subordinate command; and for still others, policy can be the responsibility of the user. There are seven parameters that control the processing of the model: interest rate, storage cost rate, obsolescence risk rate, low procurement cost, high procurement cost, procurement cost breakpoint, and robustness.
The interest rate is important since it further refines cost optimization. If we are comparing the cost of making a larger buy now with making a premium buy later, we need to convert all calculations to presentyear dollars. Storage cost rate is expressed as a percentage of the unit price and represents the yearly cost of storing excess stock. Obsolescence risk rate is the chance, expressed as a percentage, that any single item will become obsolete in a given year. An example of obsolescence would be a part no longer being used because of improvements in technology.
The procurement cost is the amount of money involved in each individual procurement decision. This is a fixed cost associated with the process and not the part itself. The Army has two procurement costs associated with each major subordinate command. The low procurement cost represents buys that fall below a certain amount, and the high procurement cost represents buys that exceed that amount. The "amount" in question is the procurement cost breakpoint and is an Army supply policy standard.
Finally, robustness is a special parameter that allows the user to decide on a measure of uncertainty, expressed as a percentage, in the expected life of the item. When set at zero, it assumes that the estimate of expected life is accurate. A robustness factor of 10 represents an uncertainty of 10 percent in either direction from the expected life estimate. For example, if we select an expected life of 10 years and a robustness factor of 30, we are saying that the true life of the item may fall anywhere from 7 to 13 years. This gives the model an added degree of power by compensating for the erratic nature of the expected life input.
Use of the Model
In December 1996, the first LOT model was released to AMC's major subordinate commands. After getting feedback from those commands, some improvements were made, and a second version of the model was released in January 1998. Only the second version contained all of the inputs and parameters described above.
More recently, the model has undergone improvement in the area of user interface. Although the mathematical portion of the model is written in FORTRAN, a Clipper user interface originally was included to give users a menudriven interface. A more advanced Visual Basic interface, which provides a Windows environment for the user, was added in a third version released in March.
Initial results from the field show cost savings for LOT items. Savings are based partly on the ability to predict correctly the optimum quantity of parts to buy and partly on providing discipline to a component of the Army supply process that previously had none. More efficient use of the dollars dedicated to the spare parts wholesale supply system inevitably will free up funding in these times of decreasing defense budgets. Greater availability of the materiel our soldiers depend on, through a more accurate LOT supply system, will increase our Nation's readiness. ALOG
Eric A. Snyder is an operations research analyst at the Army Materiel Systems Analysis Activity at Aberdeen Proving Ground, Maryland. He has a bachelor's degree in mathematics from Pennsylvania State University and a master's degree in statistics from the University of Delaware. He is a graduate of the Army Management Staff College.