Suppose we have a number of projects on the table and a fixed maximum that we can invest. We can afford to undertake more than one project, but which ones should we choose?
Fundamental rule says select the combination that provides maximum net benefit, but even just a few options means considering a large number of possible combinations. The example in S & Z has 8 options and so that would make 2^8^ = 256 combinations.
How to proceed
1. List projects in a table with a column for (initial) cost and '''net''' benefit.
[[Image(stop-and-think-small.gif, noborder)]] Why are we using initial cost?
In the book the explanation is "because initial cost is where the shoe pinches." Another way to look at it is that the initial cost is what we are spending now and that is what our constraint reflects. To make the decision on the basis of a budget constraint now and budget constraints later would be a different problem than this one.
1. Calculate the ratio of net benefit to initial cost for each
[[Image(stop-and-think-small.gif, noborder)]] What does this ratio tell us?
This is the rate (benefit dollars per cost dollar) at which benefits are generated by each project. By sorting the projects from highest to lowest we put the project with the best "bang for buck" rate at the top.
1. Sort list according to net benefit - initial cost ratio
1. Add a cumulative cost column.
1. Select projects starting from the top of the list and continue as long as the cumulative cost is below budget constraint.
A Complication "Indivisible" Projects
Suppose Project H were twice its size (and with twice its benefit). It would still belong at position #4 in the list with a 2.0 net benefit to initial cost ratio. But, we wouldn't have enough left in our budget for its new price tag of 300. The appropriate action is to bump project E into its place. The bang-for buck is less, but the project can actually be done.
A Note on Constraints
Constraints come in many flavors and sizes. The present technique assumes constraints boil down to cost/budget and whether projects are divisible or scalable.
'''Concepts and Terminology'''
Problem Set One
Recall that each of our 8 projects has a default level of implementation (the one we assume we are stuck with prior to being told we can consider higher and lower levels). These are shown below along with the new columns we compute. Note that we have also sorted our table based on column F, the net benefit to cost ratio.