Stock control --- rubbish!

 There have been numerous books and research papers on aspects of stock control, starting with the classical EOQ model (economic order quantity) and developing modifications to that and other models.  Stock control models have led to the concept of Just In Time production --- something which backfired during the COVID pandemic when supply chains were disrupted and demand for many items was more irregular than in less troubled times.  

Recently I went on a site visit which revealed an interesting variant on many of the usual introductory models.  Let me explain.  Here in Exeter, our domestic waste is collected by the local authority.  We sort it into four categories: 

(1) recyclables (paper, card, plastics), 

(2) garden waste 

(3) waste food

(4) non-recyclable stuff

Category (1) is sorted into its various categories, (2) is composted (in such quantities that harmful bacteria are killed by the heat), (3) goes to a digester system, (4) to the "energy from waste plant"

The site visit was to the energy from waste plant (EFWP), where the waste material is burnt in a very hot furnace to produce electricity.  Now the output from the EFWP needs to be relatively constant, but the input is variable.  So the stock control problem is concerned with the "stock" of stuff that is going to be burnt.  It isn't collected on Sundays (and not always on Saturdays)  so each week, the amount that is used has to be about 70% of what has come in, so that there is a supply at the end of deliveries on Friday evening to last to the following Monday.  Except that the daily supply is variable, and there are seasonal variations.  But on top of this, there are general holiday dates with no collections on one or more holiday dates.  Fortunately, these are known in advance.  So the basic amount of rubbish being held will follow a time line like this:

I didn't put a great deal of daily variation in that one.  With a more variable daily variation, we get:

and it is apparent that the amount available at the start of the weekend is also variable.  

But both clearly show that the storage space for "rubbish" has to be considerable.

The public holidays make things worse.  That is left to the reader as an exercise.  

How is the stock control managed?  By rules of thumb; the controller knows about the holidays, and the daily "demand" so works to ensure that there is stock to meet the demand.  And the controller does not have a degree in O.R.; I wonder what would ensue if an O.R. specialist tried to convince that team that a textbook formula would help!  

footnote: but of course!  This is a variant of the reservoir control problem!  We looked at that ages ago in http://orindevon.blogspot.com/2011/09/water-supply-and-environment.html.  There, each month had a target level for the water in the reservoir.  If the water was below that, then the reservoir was allowed to fill up, if above, more water was released.  Here, the problem is essentially the same, except that the target levels of waste in the incinerator are for each day, and those targets follow some kind of trend in the run-up to holidays.