THE LEARNING CURVE IN MAINTENANCE ANALYSIS

The paper applies the theory of learning curves to explain the tendency toward decreasing downtime observed in the maintenance of many systems. Maintainability demonstrations performed under current military specifications assume stationary distributions of maintenance times; that is, the demonstrated times are considered as samples drawn from populations having time-invariant parameters. Statistical evidence, based on observed maintenance actions on military electronic equipments, is presented to contest the validity of these assumptions. The author shows that maintenance times decrease for iterative tasks in accordance with the negative logarithmic relationship, which is the accepted form of the human learning curve, historically applied in manufacturing operations. A s the number of performances of a task increases, the time required for each iteration is progressively reduced in accordance with a constant experience factor. Although the experience factor is, in a strict sense, a unique characteristic of a specific worker at a specific task, it is shown that a satisfactory composite factor may be computed for a maintenance crew or department, and may be employed for prediction of maintainability improvemept . Possible objections to the accepted form of the learning curve stem from the fact that it is empirically based, and from intuitive realization that it can not hold true in the limit, since the function approaches zero asymptotically. Studies are described showing that the accepted form of the function gives excellent correlation coefficients out to the region of x = lo4, and that none of several alternative functions investigated gave better fits. The use of learning curve analysis in maintainability demonstration testing is recommended, and it is argued that present procedures reduce the probability of acceptance of newly developed equipments. Introduction Tests to demonstrate the maintainability @) of newly developed systems are required under current military specifications (MIL-M-26512, MIL-M-23313, MIL-STD-471). The sampling procedures set forth in these documents are predicated on the implicit assumption that maintenance performance time distributions are stationary; that is, the demonstraed times are considered as samples drawn from population distributions having time-invariant parameters. Statistical evidence is presented here to contest the validity of the foregoing assumption, and to support a thesis that maintenance times decrease for iterative tasks in accordance with the accepted form of the human learning curve. By analysis of task times observed during g demonstration testing, the applicable learning curve can be plotted, the rate of learning estimated, and the g to be expected of a system during its service life can be predicted. Confidence statements can also be associated with such predictions. Nature of Learning Curves The validity of the learning curve, particularly in manufacturing operations, has been verified b more than a quarter century of experience. ( 1 , 2 9 3 3 Thus it is one of the few empirical relationships so firmly established by repeated observations that it can be safely extrapolated to predict future behavior. The learning curve describes the human capacity for progressive learning of repetitive tasks. It indicates (Figs. 1 and 2) that as the number of performancesof a task increases, the time required for each performance is progressively reduced in accordance with the relationship log t(x) = log t(1) + m log x (la) log 2 where : x 2 number of performances t(1) = time required for' first performance t(x) = time required for xth performance b = experience factor or, in antilog form The average time also decreases. t(1,x) = Cto X Each time the number of performances is doubled, the time required is reduced to a constant fraction of,the previous time. This fraction by expressed as b = * (3) is called the experience factor and describes the rate of learning. Values of b observed in industrial operations usually lie between .70 and .90. In the strictest sense an experience factor describes only the. behavior of a particular worker learning a specific task. However, a quite satisfactory composite experience factor can often be computed for a department, for workers of a iven grade or skill level, or for an entire plantf3). In many industries such composite factors serve as fundamental indicators for predicting the effects of production run lengths on production costs and, consequently, for establishing price policies. The accepted form of the learning curve, as embodigd in equation (l), could be questioned on the grounds that is is empirically based, and from the realization that it cannot continue to apply