The time element in Operational Research (1)

Recently, I have been reflecting on the fact that many O.R. problems are concerned with time, either in the form of "When" or "How long between" questions.  Yet it is easy to minimise this facet of O.R. when we teach it.  Obviously, forecasting models are taught, and so is the methodology of dynamic programming, but many other techniques ignore the effects of time.  (Yes, that is a gross over-simplification.) Simulation models clearly represent time and the passing of time.
I suspect that one factor which contributes to the limited discussion of problems involving time is that we know that things change over time, so we cannot assume the constancy of parameters that so many models are based on.  We teach simple inventory models in terms of "quantity ordered" rather than "Time between orders".  We build linear and integer programming models and assume that the parameters remain constant for a long time.  And if we acknowledge that things change with time, then we are indicating that the models we use are not perfect. And, imperfect models do not make for good academic papers, either.

Some years ago, we worked with an industry which used a number of heavy vehicles.  In those days, each one cost about £100,000, and their fleet size was in the low hundreds.  The normal replacement plan was for each to be replaced after four years , so they were discounted in the company accounts at this rate.  They had negligible value at the end.  The fleet size changed slowly over time, so every year, just about one quarter of the vehicles were replaced.  However, in one year, there had been government tax incentives for buying new plant.  Call that year 1.  As a result, in year 1, nearly half the fleet was replaced.  Some of the vehicles which were in good condition towards the end of year 0, the previous year, had their lives stretched, and some of the vehicles which were due for replacement in year 2 were replaced early.  So there was a blip in the number of vehicles replaced, as seen in the lines below.
Year 0 - 15% of fleet
Year 1 - 45% of fleet
Year 2 -15% of fleet
Year 3 - 25% of fleet

Now what will happen in years 4 to 7?  As the pattern has been to replace every four years, the pattern should repeat for ever.  But what is the effect on the company's profits in those years?  And would that be an acceptable solution?  We sat in with the O.R. team as they considered ways to quickly smooth the blip in the replacements "down-up-down-level" and the blip in profits "up-down a lot-up-level".  The solution was obvious, but needed to be quantified.  It was to retire some vehicles a little early, retire some a little late, but built into that recommendation was the need to inspect more carefully so as to select the vehicles whose lifetimes were not four years.  There were still blips, but they were not so substantial, and after a couple of cycles, would be negligible.

At the time, I was teaching a course which included models of replacement, which assumed that the parameters of the items being replaced, and their number, did not change with time, and so this real-life problem was useful in showing the need to question assumptions, and to think of the wider system being modelled.

More recently, we were involved with an automobile company, whose production line had over 100 workstations.  The equipment at each of these needed to be maintained in a cycle of preventive maintenance. 
We were looking at two problems.  The first, assuming that the production line would continue working at its constant rate for ages and ages, was to smooth the preventive maintenance cycle.  If machine 1 needed to be maintained every 3 weeks, machine 2 every 4 weeks, and machine 3 every 6 weeks, then every 12 weeks, machines 1, 2 and 3 would all need to be maintained at the same time.  And at other moments in time, none of these machines needed maintenance.  (But of course, there were another hundred or so workstations with independent maintenance cycles to consider.)  And sometimes there were situations where the amount of maintenance exceeded the capacity of the maintenance gangs and their equipment.  As the cycles for each machine's maintenance were not as strictly fixed as appears at first, we devised a spreadsheet model which modified some of these cycles so as to smooth the demand on the maintenance people and their equipment.  (For the record, it was a genetic algorithm based heuristic used within the spreadsheet to vary the parameters to make a smooth(ish) schedule.)  One of the complications in the model was the conflict between smoothing different resources, because there were several different types of maintenance equipment whose use had to be smoothed. 
The second problem we looked at responded to a reality that the first one ignored.  In two words, machine breakdown.  The workstations did not always perform without failure.  So if machine 1 broke down after 2 weeks, it would be repaired, and its next preventive maintenance would be due after a further 3 weeks.  The nice, smooth pattern would be disrupted.  So, either a new schedule for the whole line would be needed, or, and this was the solution we advised, when "breakdown maintenance" was needed, then other preventive maintenance should be done while the breakdown was being dealt with, according to rules which followed from the operation of the spreadsheet.  So the second problem gave rise to a model which recognised the time element in O.R. and was able to handle it. 


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