From one to two; a problem of discrete resources

 There was a day, in my teenage years, when my parents bought a second car.  I don't know whether they analysed that decision.  It was probably prompted by the fact that my mother had passed her driving test (in the UK, in the fifties and sixties, not all married women could drive; driving cars was still a male preserve, and my mother, like so many others did not work after she was married).  But their decision to increase the domestic resources of cars is an example of decisions which occur in many situations.  Cars do not come in fractional parts; you either have one or you do not.  And the jump from having one resource to having two is a big one.

Why?

In many situations for management, the demand for a single resource increases steadily.  Suppose that a resource is being used 80% of the time now, and the demand increases to 90%.  Forecasting suggests that the demand will continue to  rise.  So, sooner or later, the use of the resource will exceed its capacity.  So a second resource will be needed.  However, as soon as the second resource is available, the average demand for each one will fall to (say) 50% of capacity.  And to an outside observer, that suggests inefficiency; there is unused capacity.  So, in a sense, the decision maker who obtains the second resource is "damned if (s)he does, damned if (s)he doesn't".  It is similar to Catch-22.  

(note that with more resources, the drop in average resource usage going from N resources to (N+1) is much less; naively, if N resources are all being used 100% of the time, then N+1 will be used (100N/(N+1))% of the time.)

Vehicles are an obvious example of a discrete resource.  But public facilities also fall into this problem of discreteness.  When should a new surgery be opened?  When should a new school be built?  And on a personal level, during lockdown, when should a family buy a second computer?

The vehicle problem struck me in the city earlier.  We have one council vehicle to empty the bottle-banks around us.  It is well-used; we are rather good at recycling glass.  And we don't have kerbside collection of glass in the city, yet.  The vehicle is also used to empty the recycling bins for drinks cartons (tetrapaks).  The city has chosen to stress the glass recycling over cartons, so there are many more places with glass recycling containers than carton recycling containers.  Cartons can be all dealt with in one run around the bins; glass necessitates many runs around the many sites of the glass-bins.  

So, should the city expand the number of bins for cartons?  If so, then it will need a second council vehicle to empty those bins and some of the glass-bins.  But, that's a capital cost or a vehicle, a capital cost for more carton-bins, plus wages, plus running cost, for a very small increase in the income from recycling material.  What's to be done?  At present, nothing.  The status quo remains.  But the problem will not go away.  The population of the city is growing, so demand for all kinds of recycling facilities and equipment will grow.

And an added problem about this sort of possible expansion.  There is a problem of recruitment of drivers qualified to handle heavy goods vehicles.  According to one statistic earlier this month, in the UK there are 100,000 vacancies for HGV drivers.  So even if we could justify a second vehicle, it might be hard to recruit a new driver.  

Meanwhile, I will walk to the nearest glass bin, and occasionally cycle the 4 kilometres to the most convenient carton-bin, combining such a journey with a pleasure ride.  

And I will never know how my parents justified the second car, because --- just like many two-car households --- the cars were only used for a fraction of the time.  But back then, there were no car-sharing schemes, and our home in the village didn't have a very good public transport system.