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The mathematics of maintenance  
This
explanation of the mathematics applicable to equipment maintenance
will interest anyone employed in the maintenance of large electromechanical
installations. The principles here discussed (with the sample study
of a nuclear generating station), can with advantage be applied
to any large plant and equipmen. That includes oil refineries,
offshore drilling rigs, petrochemical installations, hospitals,
airports, manufacturing plants and oceangoiong vessels. The work of I. Walker, former superintendent of quality assurance of Ontario Hydro, in developing the probability analysis of equipment and coauthor of the original paper read at the 16th InterRAM conference is fully acknowledged. All the costs quoted in this article are expressed in Canadian dollars. In fact, as will be recognized, the actual figures quoted have little bearing on the methods and principles, for those values are relative values only. 

This article discusses RAM analysis (RAM meaning reliability, availability and maintainability), Poisson distribution, and offers a solution to the balance of maintenance load and capacity. As an introduction to aspects of equipment maintenance mathematics, the loadcapacity equation for computerized management is discussed first. Organizing and maintaining large electromechanical equipment is a major undertaking. It requires managers to control the work, trades people to do it, subcontractors for specialized services not available inhouse, purchasing agents for a spare parts and material programme, storage and distribution management. This all adds up to many hours of work, which is an element of the mathematical equation. All in all, the total hours required to maintain a given installation is represented by the letter L while the total capacity of the organization in work hours to meet that load is expressed as C. That is, ideally, the maintenance load in L hours must equal the maintenance work capacity in C hours for the highest efficiency to be achieved; i.e. L = C. The equation is true only when the total maintenance load in work hours is balanced by the available capacity in work hours. Both L and C, however, are variables and cannot be otherwise. That is, L is variable because equipment breakdown cannot with accuracy be predicted; C is variable because those who perform the work are available or not available when needed for a variety of reasons. Nevertheless, the L = C statement provides a platform on which to develop a programmable computerized system. Maintenance Load L: The total load in any installation is the sum of three components, which are:
That is, L = R_{m} + P_{m} + B_{m} Maintenance capacity C: The capacity (C) of the enterprise to meet the total maintenance load (L) is, as earlier stated, variable, but can be calculated using several elements. That is:
Therefore, the total available capacity C = S_{h} + O_{h} + C_{h} or
In each of the elements R_{m}, P_{m} and B_{m} a further breakdown of hours required is possible and defined. For example, in known periodic or regular maintenance tasks (R_{m}) the required mix of hours by trades is available from existing hard copy or electronic records. For given periodic maintenance on a valve, say, a crew of two fitters and one electrician (for control and signaling connections) might require an elapsed time of eight hours. That is, the work comprises two electrical hours (one to disconnect, another to connect and test) and two fitters for six hours. In other words, within the limits defined, the elapsed time in hours required by the R_{m} load is fixed. In the case of the example R_{m} = 8 hours. The available standard hours (S_{h}) in this instance are E electrical hours 2, and F fitter hours or 2 x 6 = 12 hours. Hence R_{m} = S_{h} = 2E + 12F = 14 work hours to be satisfied. To measure the capacity of the system, each element of the available capacity in work hours by trades must be plotted into the programme: welders, fitters, electricians, control and computer system technologists, machinists, carpenters and the entire range of specialist trades that make up a given maintenance work force. The same applies to the available overtime hours for each trade and the subcontract and specialist hours that are available. By developing a programme of maintenance work by trades and elapsed hours required, and measuring or estimating the available hours to meet the work load, using the principles here outlined, a sensible and workable maintenance equation can be programmed. Such programs can be, and have been, implemented. Balancing load and capacity is but one technique used to minimize the total maintenance load. 

This in turn reduces the maintenance cost. Figure 1 shows the expected trend in direct maintenance cost that can result from an effective program. By changing from unplanned maintenance (related to equipment failure  see below) to a 70% preventive maintenance element, the cost saving in a Canadian nuclear generating station was 30% of the original estimated cost. By increasing the preventive maintenance component from the 30% level in 1986 to 70% by 1990, the estimated reduction in cost was 15% or $2.3m a year. This was a net saving. Furthermore, there was no evidence of a reduction in production reliability. 
Figure
1, Expected trend 

This in turn reduces the maintenance cost. Figure 1 shows the expected trend in direct maintenance cost that can result from an effective program. By changing from unplanned maintenance (related to equipment failure  see below) to a 70% preventive maintenance element, the cost saving in a Canadian nuclear generating station was 30% of the original estimated cost. By increasing the preventive maintenance component from the 30% level in 1986 to 70% by 1990, the estimated reduction in cost was 15% or $2.3m a year. This was a net saving. Furthermore, there was no evidence of a reduction in production reliability. A computerized maintenance system of the type described is one way to raise efficiency of the overall maintenance program. A parallel program that works well is an analytical system that is known as the RAM technique, where RAM is an abbreviation of reliability, availability and maintainability. The RAM method of analysis was used at a Canadian nuclear generating station to improve maintenance practices on specific types of equipment. They were applied to:
RAM analysis was part of a reliabilitycentred approach to improve the overall maintenance program. If it worked successfully on selected systems and equipment it could be applied to any highcost installation. The effectiveness of the method was reflected in generating unit availability, maintenance cost, and unit energy costs. RAM analysis is also a form of probability examination as used in the example that follows. To conduct this analysis and apply it to a moderator cooling system, specific steps were described, steps that might also be regarded as objectives on which to base the analysis. These were to:







