The effects of different production rate measures and cost structures on rate adjustment models

The effect of production rate on the cost of weapon systems has attracted much attention in the cost estimating community in recent years. A variety of adjustments to weapon systems cost models have been proposed to reflect the impact of different production rates. The most popular solution is to add a rate term to the traditional learning curve model. This paper examines the effects of different rate measures and cost structures on rate adjustment models. Numerical examples illustrate that the production rate term should be measured as a ratio and not as an absolute quantity of a production lot or a period. The paper also points out that a rate adjustment model is appropriate only with data collected from plants which have not undergone changes in cost structure. THE EFFECTS OF DIFFERENT PRODUCTION RATE MEASURES AND COST STRUCTURES ON RATE ADJUSTMENT MODELS The effect of production rate on the cost of weapon systems has attracted much attention in the cost estimating community in recent years. A variety of adjustments to weapon systems cost models have been proposed to reflect the impact of different production rates. The most popular solution is to add a rate term to the traditional learning curve model. The resulting learning curve model augmented with the production rate variable is usually referred to as a rate adjustment model. The purpose of this paper is to examine the theoretical underpinning of the production rate effect on weapon system cost and illustrate that the popular solution to the rate problem may result in erroneous conclusions. Numerical examples will be used to illustrate the potential problems of the popular approach to production rate adjustment. The paper concludes with a discussion of the scenarios in which the rate adjustment models may be utilized. CONCEPTUAL FOUNDATION OF PRODUCTION RATE EFFECT The conceptual foundation of the production rate impact on cost is related to economies of scale. In many industries that effect is well understood. High production rates allow greater use of facilities and greater specialization of labor. The increased volume of materials purchased reduces their unit cost. The increased volume of production activities spreads fixed overhead costs over a larger quantity of products produced. Taken together. all these effects work to increase efficiency and lower production costs (Bemis, 1981; Large, et al., 1974; Under and Wilbourn, 1973). It should be noted, however, that a plant with a higher production rate does not necessarily produce at a lower unit cost when compared to another plant. This point is illustrated in Figure 1 . Assume there are three plants capable of producing the same item, such as a missile. The Average Unit Cost curve for each plant is shown as AUC1, AUC2, and AUC3, respectively. If the output quantity were fixed at 25 units, then Plant 1 is the most efficient of the three plants. However, if the output level were fixed at the rate of 40 units per period, Plant 1 's unit cost would be higher than that of Plant 2, which is the most efficient of the three at that production quantity. This is consistent with economic theory, which says that, in general, there are both economies and diseconomies of scale. This phenomenon is recognized by the above analysts and is reflected in their use of this familiar U-shaped average cost curve to incorporate the effect of production rate into weapon systems cost models. Figure 1 Average Unit Cost and Production Capacity Unit Cost 10 30 40 50 60 70 Output Quantity 80 90 The same theory of economies and diseconomies of scale is applicable to a single plant's expansion when it is operating beyond its efficient capacity level. This scenario has significant implications in weapon systems cost estimation. Recent experience has shown that production rates of major weapon systems are subject to continual adjustment, sometimes significantly. At the low end of the spectrum is the initial production rate. This is usually a function of early procurement funding constraints and the technical risk of building substantial numbers of newly developed Items before the design has fully matured. Thus low rate initial production avoids the risk of incurring costly retrofits to early production units. During this early stage of production, the amount of fixed costs may vary from period to period because of the changing production setup. At the upper boundary is the limitation of available plant capacity and the requirement for additional investments in tooling and facilities for capacity expansion. Additional investments in tooling and facilities alter the cost structure of the plant. The unit cost curve of a plant expanding its investment in tooling and facilities is equivalent to changing from AUC1 to AUC2 as shown in Figure 1. REVIEW OF RATE ADJUSTMENT MODELS Although studies of the effect of production rate change on weapon systems cost began as early as the 1950s (Hirsch, 1952; Alchian, 1963), and various models had been proposed, the most widely used rate adjustment model in use today was developed by augmenting the traditional learning curve model with a production rate term: Z = aXR = YR (1) where, Z = unit cost of the Item with production rate as well as learning considered, X = cumulative quantity produced, R = production rate measure, Y = unit cost of the item with only learning considered, a = a constant, usually called the theoretical first unit cost, b = a parameter, usually called the slope of the learning curve, c = a parameter, usually called the slope of the production rate curve. Empirical work on this production rate/learning model was first conducted at RAND, but the model was later popularized by Bemis (1981). Large, et al. (1974) attempted to develop this model for various production cost elements. They were forced to conclude, however, that the production-rate/cost relationship could not be predicted with any reasonable degree of confidence. For production planning purposes, they recommended that production rate effects in aircraft production programs be ignored because they were dominated by other effects. They also suggested that production rate is subject to change and, hence, is difficult to predict. Further work on the production rate/learning model was carried out by Smith (1976). He analyzed three aircraft programs for which a large number of data values were available due to long production periods. Where the data permitted, Smith applied his model separately to fabrication and assembly labor hours. He then compared his production rate/learning model to a reduced, learning-only model. Smith found that the rate term was an important contributor to the explanatory power of the model. However, he obtained a surprisingly large variation in parameter values for cases with similar production quantities and rates. Additional efforts using this approach were carried out by Bemis (1981), Cox and Gansler (1981), and others. If one recognizes the inherent rate instability scenario of major weapon systems production and the resultant changing cost structure discussed in the preceding section, then none of the inconclusive findings discussed above would be surprising. In the following sections, we will examine the issues of alternative production rate measures and changing cost structures, and we will discuss other major considerations that must be addressed before one can use the rate adjustment model in weapon systems cost estimation. ALTERNATIVE PRODUCTION RATE MEASURES Although the concept of production rate is clear, its measurement is by no means unambiguous. Several alternatives have been used as surrogate measures of production rate. The two primary measures are lot size and annual/monthly production quantity. We will first discuss these two and related measures, along with the difficulties of their use. We then discuss a third alternative, a ratio measure which we believe will avoid some of the difficulties of the measures used to date. Using Lot Size or Annual/Monthly Quantity As the Rate Measure Hirsch (1952), Cox and Gansler (1981), and Bohn and Kratz (1984) all used lot size as their measure of production rate. Hirsch was careful to note that his lot intervals were fairly stable; however, this has not been the case with almost all more-recent aircraft programs. Since the time (and, hence, cost) required to produce sequential, similarly-sized lots often changes over the life of the program, it is unclear what is being measured by the lot size proxy. Perhaps the most common measure of rate is that of production quantity in some time interval. The time period involved is usually selected as a function of data availability. Most studies use annual quantities as a measure of production rates. An inverse of the quantityper-unit-time measure has also been used; Large, et ai. (1974) used the number of months required to reach a certain cumulative production quantity as their inverse measure. Some studies, such as Womer (1984), use monthly data. Womer notes that if there is substantial work-in-progress and the production period is long compared to the period of observation, then units produced in the following time period actually reflect work performed in the preceding time period, and this can result in substantial bias in estimation. Since this problem is especially critical for monthly data, Womer used a lagged model of production to obtain his estimates. When analyzing a cross-section of programs, it is possible to use an average rate for each program. Because the production rate may change in a typical production run, an average rate for an individual program is usually used in these cross-section analyses. Use of an average may understate the effects of these disruptive rate changes, but we do not expect it to mask the effect of production rate itself. Large, et al. (1974) used this approach in their examination of several programs. Gulledge and Womer (1986) noted that cumulative quantity is hi