Optimal allocation of a finite resource

Background

This short article is based on an insight that arose when tasked with deciding how much of a fixed total amount of lift gas should be allocated to each of a number of producing wells having different response characteristics.

Apart from Solar Energy, most resources are finite. If not, there would be no issue, but our world is not like that. There are typically a number of opportunities to deploy any finite resource. In simple terms some provide more benefit than others.

Is there a simple rule or guide to decide how much i.e. what proportion of the resource to be applied to obtain each potential benefit?

This is a particular type of optimisation problem that may arise more often then we think and in diverse fields

A related but distinct question is – to obtain a certain benefit, alternative resources could be employed. Which one and how much? This question is not addressed here, nor is the ‘triage’ question of how much of a finite resource should be applied to different ‘opportunities’. I expect over-stretched A&E departments have had to face this question.

Back to the basic question, the simple case is where the benefits provided by the various opportunities are all expressed or ‘expressable’ in the same ‘currency’. If this is the case, then a simple rule seems to arise.

First, how does each benefit change as more of the resource is applied to that opportunity?

In the general case, providing more of the resource leads to more benefit, but at continuously decreasing rate.

A typical Benefit vs Applied Resource relationship.

A key variable for comparing candidates for application of a finite resource is the slope or gradient of this graph, that is (d-benefit/d-resource). It requires knowledge of this relationship.

It is found and can be quite easily proved that the optimum proportions of the finite resource that should be applied to each benefit opportunity is where the slopes on each graph are the same. This is illustrated in the next figure in a hypothetical example.

Three alternative opportunities A,B and C

For this example, A should receive 47% , B should receive 30% and C should receive 23% of the resource.

In a practical example from the oil industry, a total available quantity of high pressure lift gas can be applied to say 5 mature wells to encourage increase production. The wells each respond differently but all show a point of diminishing returns when further lift gas has little effect. The optimum allocations can be found graphically or by computation if analytical expressions are fit to the curves.