At its most basic level, prioritizing projects is often accomplished by ranking them by decreasing benefit-to-cost (B/C) ratio. The projects featuring the biggest “bang for the buck” are then at the top of the list, and all that is left to do is determine how many top projects can be funded given the available budget.

The following table illustrates where to draw the funding line, assuming a $10 million budget.

Notice in passing that we would not want to go below a B/C ratio of 1, since funding such a project would yield less benefit than it would cost.

What are the problems with ranking?

Ranking furnishes a fine first cut at an optimal portfolio, but fails to fully address the following situations — all encountered by our clients.

• Ranking cannot handle interdependencies. If project X is required before project Y can truly yield benefits, or even be undertaken at all, clearly looking at the B/C ratio of X alone is inadequate. A mesh of interdependencies is common with complex infrastructure or IT projects, for examples. In that case, a comprehensive optimization of this network of projects is indispensable.
• More commonly, ranking fails to consider alternative funding levels. Imagine that each of the projects on the list, instead of an all-or-nothing choice, could be funded in various cheaper alternatives than the all-out version, featuring a spectrum of lower costs and lower benefits. Conventional ranking would simply pick, for each project, the one alternative with the highest B/C ratio. However, it can sometimes be optimal to try to cut costs on one project to enable another project to be funded in a more expensive alternative.
• Finally, ranking ignores fine-tuning, i.e., the so-called knapsack problem. The knapsack problem is often perceived to be an academic and unnecessary complication for real-life portfolio optimization. In the remainder of this article, we go through a real-life case study to evaluate the validity of this perception.

What is the knapsack problem?

Bear with us as we indulge in one short theoretical paragraph. The knapsack problem goes as follows: imagine you have a bunch of objects of various values and weights, from which you have to select any number to fit into a knapsack. Your goal is to create the most valuable knapsack possible, without of course exceeding the allowable weight capacity.

Source: Dake under Creative CommonsAttribution-Share Alike 2.5 Generic license.

The parallel with portfolio optimization is obvious: value is the benefit, weight is the cost, and the weight capacity of the knapsack is your budget constraint.

A ranking approach to this problem, therefore, would order the objects by decreasing density: the yellow object right underneath the pack has the highest “value density” — i.e., B/C ratio — (2.5$/kg) and will go in first: it adds a lot of value relative to a small weight consumed. The green box at the top left will go last, if at all: its value density is the lowest (.33$/kg).

Now imagine we start fitting these objects into the knapsack by picking the highest-density objects first, very much like we ranked projects by B/C ratio in the table above and started funding them, until we hit our limit.

As many of us know from experience, we might then decide to make some changes “on the margin”: even though object 26 was the last one to go in, it leaves a lot of unused room in the knapsack. It turns out if I removed object 23 and placed object 27 instead, I would fill the entire sack, even though I have sacrificed a higher B/C object for a lower one. The degree of “shuffling on the margin” is difficult to build a good intuition for. Many practitioners dismiss this phenomenon as, precisely, marginal.

We wanted to put this question to the test: is ranking really insufficient? Do we really need to optimize and deal with the nagging knapsack problem, or can we be content with the simpler ranking method?

Does optimization really matter in real life?

Optimization really does better than ranking in “real life.” The following example is taken from a portfolio of 901 real client projects (the data has been disguised by linear rescaling, but the shape of the curves and the conclusions are strictly identical).

The red curve represents the efficient frontier of the simple ranking method. The blue curve represents the efficient frontier of a full-blown optimization.

Notice how an optimization is able to extract more value for certain budget levels than a straightforward ranking. Why is that? There are two main reasons why optimization fares better:

• Optimization switches alternative funding levels in and out as appropriate. Even though a given project may have one clear winner funding level, featuring a higher B/C ratio than other funding level alternatives for this project, it is sometimes appropriate to be less “greedy” and fund a less costly version so as to enable another high-yielding project to be funded. This is the dominant effect in area C of the curve, where optimization systematically beats ranking by about $8 million.There are actually very few projects featuring multiple, alternative funding levels in this database.
• Optimization can handle big discrepancies in project spending. Of course, the most dramatic benefit of optimization over ranking can be seen at A. A single, very expensive project “holds up” other projects from being funded in the ranking method because the expensive project still comes first due to its high B/C ratio. In the optimization solution, rather than holding up the budget gap, other, lower B/C ratios are “filled into the knapsack” until such time as the big project can be funded. At that point, the red curve abruptly catches up with the blue one. The same phenomenon happens at B with another project, and, to a smaller extent, in other places along the curve where kinks are visible. The more heterogeneous the project sizes, the more value a straight ranking method is likely to leave on the table. At its worst discrepancy ca. a $46 million budget, 19% of the value is missed by the ranking method.

This case study should not be interpreted as an indictment of B/C-ranking. But it clearly illustrates one of the advantages of optimization in the presence of multiple funding levels and heterogeneously sized projects.