From scores to decisions: priority and capacity

Two different questions

A priority score answers which item is more urgent. A capacity decision answers how many of them the team can actually handle in the hours it has. Mixing the two hides the second question.

A priority score

R is normalized risk, A normalized amount, P a payment term and T normalized elapsed time; α, β, γ and δ are weights. The sign of the payment term is a policy decision: a high probability of payment can lower urgency, so in the lab we use 1 − P. Weights need calibration against outcomes; until then the score is only a ranking device.

Capacity as a knapsack

Here x_i is 1 if action i is executed, v_i is its value (for example, its priority score), a_i its effort in hours and b the hours available.

The ratio rule and its bound

Sort the actions by v_i / a_i and take them while they fit. For the fractional version, in which an action can be split, this rule is optimal, so its value is an upper bound for the 0/1 problem. For the 0/1 version the rule can be improved: dynamic programming finds the exact optimum for moderate sizes. The difference between the rule, the optimum and the bound tells you how much a better method could still gain.

Limits

• Values depend on forecasts that may be wrong.
• Interactions between actions are ignored.
• Policy limits (what must never be done) belong in the constraints, not in the weights.

Background reading

Works by other authors that treat the underlying methods. They are not Menlo Tech publications.

• Martello, S. & Toth, P. Knapsack Problems: Algorithms and Computer Implementations. Wiley, 1990.
• Bertsimas, D. & Tsitsiklis, J. Introduction to Linear Optimization. Athena Scientific, 1997.

Related

Optimization & Operations Research