NATIONAL BUREAU OF STANDARDS SPECIAL PUBLICATION 411, Fire Safety Research, Proceedings of a Symposium Held at NBS, Gaithersburg, Md., August 22, 1973, (Issued November 1974) SEQUENCING THE PURCHASE AND RETIREMENT OF FIRE ENGINES Patsy B. Saunders and Richard Ku National Bureau of Standards, Washington, D.C. A mathematical model and solution method are presented for the Key words: Dynamic programming; equipment replacement; fire engines. 1. INTRODUCTION This paper describes a method to determine an "optimum" manner of sequencing the purchase and retirement of fire engines (hereafter simply called "engines"), with specific application to the Washington, D.C. Fire Department. The model developed, however, has more general applicability as regards both the equipment type and the fire department. Because of the apparent similarity of the present problem to conventional equipment replacement problems, we first will review briefly some of the ideas in the equipment replacement literature. Equipment replacement problems have a long history in industrial engineering and operations research. The reader is referred to [1] for a comprehensive bibliography on this subject. One class of equipment replacement problems balances the cost of failures against the cost of planned replacements [2]. If units are to operate continuously over some time period [0, t] and are replaced upon failure, then typically the expected cost C(t) during [0, t] may be given by = per unit total cost resulting from a failure and its replacement, = per unit total cost of replacing a non-failed item (C2 N1 (t) N1 = the number of failures in [0, t], a random variable, N2 (t) (1) = the number of replacements of non-failed units, a random variable, and E denotes expected value. The problem is to minimize eq (1) over the possible replacement procedures available within a given policy of replacement. Examples of replacement policies are: strictly periodic replacement, random periodic replacement and sequentially determined replacement. Electronic components typify the equipment to which this well developed mathematical theory applies. A second class of equipment replacement problems, called "preparedness" problems, assumes that a piece of equipment is kept in a readiness state for use in case of emergency. The objective is to maintain the equipment in a state of operational readiness at minimal cost. Thus, a sequence of inspection and replacement actions that minimizes the ratio of expected cost per unit time to proportion of "good"l time, would constitute an "optimal" decision stream [1,3]. Large military hardware provides examples of the type of equipment to which this class of models may be applied. One of the basic underlying concepts of the two classes of equipment replacement models discussed so far is that of a reliability function2. This is the probability R(t) that the equipment is "good" at time t (measured from a time at which the equipment is considered to be "new") and is exemplified by the negative exponential form R(t) = exp (-λt). A closely related concept is the failure rate, defined for any reliability function R(t) as p(t) = −R' (t)/R(t), where the prime denotes the derivative. For the negative exponential, the failure rate is the constant 1. (2) A third class of equipment replacement problems deals with the replacement of items that deteriorate. Mathematical models to solve this class of problems typically trade off the increasing operational and maintenance costs (and decreasing resale value) of an aging item against the cost of a new purchase, i.e., the "optimal" replacement time is that time at which these opposing forces are equalized. Dreyfus [5] used a dynamic programming approach to solve this problem under the additional complication of technological change. The main concern of this paper is the development of a model to determine purchase and retirement decisions over a planning period, subject to certain constraints, which would minimize the cost of operation of a fleet of engines during that period. The concern of the Washington, D.C. Fire Department was not with the cost of failure or the distribution of failures of fire engines per se, primarily because of the negligible number of engine failures and the inability to measure the "cost" of a single engine failure. The model developed may be regarded as an extension of the ideas represented by the third class of equipment replacement problems discussed above. Section 2 describes a simple calculation, which serves to introduce the data at hand and compares the results of this calculation (as applied to Washington, D.C.) to those of a study [6] from which the data were obtained. A dynamic programming (DP) model is formulated and given illustrative application in section 3, and directions for further investigation are suggested in section 5. Section 4 lists formulas for computing upper and lower bounds on the variables of the dynamic programming model. For the derivation of these bounds, as well as for an integer programming analog to the DP model, the reader is referred to [7]. 2. INITIAL CONSIDERATIONS Aside from personal communications with members of the staff of the Washington, D.C. Fire Department, the main source of data was a report by Balcolm [6]. That report also proposes a model for determining the life-span of an engine, which will be described later. 1It is implicitly assumed that the equipment is either in a "good" or a "failed" state. 2 See [4] for a discussion of the statistical theory of reliability. A linear relationship between engine age and maintenance cost was used in [6], and least-squares regressions yielded three sets of coefficients, corresponding to "high usage," "medium usage," and "low usage" engines. Balcolm then obtained a "composite" equation a weighted average (by the number of engines in the three categories) which this paper also uses. This equation is of the form: u a U. = the maintenance cost of an engine entering its ath = 24.17, = U1 122.46/year. 3 year of service, Values of ua are listed in table 1. This relationship was adopted as the basis of the data for maintenance cost since it was felt that a more complex function could not be supported by the observed cost figures. A linear relationship was also used in [6] for the purchase price of a new engine, given by Values of Pt are given in table 2. The choice of the "base" year 1900 is not explained, but it accounts for the surprising (negative) value of Po. The index t refers to the year for which a value of the purchase price is desired. Using these data, simple calculation can be made to determine an "optimum" life-span for a single engine. Assuming a zero salvage value (for simplicity) 4 and a constant purchase price P, the accumulated total cost of keeping an engine for n years is Thus the average annual cost of keeping an engine for n years is AC (n) = TC (n) /n (5) Clearly, the longer an engine is kept, the longer the time to amortize the price P, so that portion of the cost per year will decrease with n. However, the maintenance costs increase year by year. Thus, with the "optimum" lifespan defined as that value of n which minimizes eq (6), the standard calculus technique of setting the derivative of (6) equal to zero and solving for n yields: Since P > 0 and n > 0, the second derivative 2P/n3 is positive so that the (8) Figure 1 indi "Constant salvage values (with respect to age) can be represented by subtracting them from Uo. For Washington, D.C., using the 1969 purchase price, eq (8) yields n = 19.6, considerably larger than the present life-span of 15 years. Balcolm [6] recommends a life-span of 10-11 years, depending on the number of years over which an engine is linearly depreciated, using as his criterion the equality of current (resale) value and accumulated repair cost, i.e., n is chosen so that where N is the number of years over which an engine is depreciated and S is the salvage value of an engine after N years. (Note that Balcolm assumes that the number of years over which an engine is depreciated (N) and the number of years it is kept (n) need not be the same.) No rationale for this criterion is offered in [6], but the large difference between the "optimum" life-span in [6] and the one derived from the present calculation [eq (8)] indicates a significant difference between the two models. some The dynamic programming (DP) model described in this section takes a what different approach to the problem of equipment replacement. Instead of determining an "optimum" life-span which would be applied to all engines, the DP model begins with the existing scenario and prescribes purchasing and retiring decisions over a T-year planning horizon. (The index t = 1,...,T is used in this model and appropriate notation changes are made in the relevant formulas presented in section 2.) In this sense, the model may be "tailored" to fit the initial state of affairs of any urban fire department. The reader interested in DP in general is referred to the text by Nemhauser [8]. For other DP formulations of equipment replacement problems, see [9] and [10]. |