PROMETHEE V (Brans & Mareschal, 1992)

An organisation's annual investment budget is limited. A decision is to be made on which of three projects will be funded together. There are three criteria: expected-return score, strategic-fit score and feasibility score; all three are benefit criteria. The criteria are weighted 0.4, 0.3 and 0.3 respectively. Every project has a cost, and the total budget is 17 units.

The method first computes base PROMETHEE II's net flow. The incoming and outgoing flow of every project relative to the other two is found, and the difference is taken.

These net flows then become the input to a 0-1 program that finds the subset maximising the total under the budget constraint. The program tries the six combinations that fit the budget, and all of them, except taking all three together, fit the budget.

The combination giving the highest total is P2 alone.

The result reads as follows. P2 has the highest net flow on its own (0.30) and comfortably fits the budget; its cost is 6, below the budget. Taking P2 and P3 together fits the budget too (17) but lowers the total net flow (0.20). P3's negative net flow reduces P2's contribution. This is why P3, alone or together with P2, is not part of the selection.

The organisation's hesitation: if P1 and P2 were taken together, the cost would be 14, still under the budget, but the total net flow would be only 0.10. This is lower than the 0.30 given by P2 alone. Leaving part of the budget unspent gives a better result than fitting a low-net-flow project into the budget. Spending the entire budget is not a goal in itself.

In the report: "Under the budget constraint, the set that maximises the total net flow is P2 alone. P1 and P3 each fit the budget individually but were not selected because they lower the total return."

Source: DecisionMind's validation example prepared for the PROMETHEE V engine. It is based on Brans and Mareschal's (1992) budget-constrained selection problem and is not the paper's own figures. The net flow and the selection were computed by independently running the engine.

A freight company is to decide, with a limited annual investment budget, in which of four candidate regions to open a new distribution centre. The criteria are estimated daily parcel-volume score, proximity to the existing network score, and labour-availability score; all three are benefit criteria. Every centre has a setup cost.

The method computes the net flow for every region, then finds, among the combinations that fit the budget, the set that maximises the total net flow. Suppose the region with the highest individual net flow has the highest setup cost and, on its own, consumes most of the budget. The set the program proposes instead contains two regions with medium net flow, taken together.

The company's hesitation is this: should the investment go to the single highest-scoring region, or to the two medium regions that together provide a higher return? Without the budget constraint, the answer would be the first region; under the constraint, the answer changes. This shows where PROMETHEE V's real difference from base PROMETHEE lies: the best individual alternative is not always the best selection.

In the report: "Under the budget constraint, the set that maximises the total net flow is the two medium-sized regions. The single highest-scoring region is not part of this set, owing to its high setup cost."

Ranking the net flows in the investment table (P2, P3, P1) and taking them in sequence until the budget runs out: first P2 (cost 8, remaining budget 9), then P3 (cost 9, remaining budget 0). This greedy selection gives P2 and P3 together, for a total net flow of 0.20. Yet the program selects P2 alone and gives a net flow of 0.30. The second error is fabricating a result when the constraint system turns out infeasible, for instance if the budget is lower than even the cheapest project's cost. In that case no selection can be made and the budget must be reviewed. The third error is reporting the rejected P1 as a "weak project." P1's net flow is negative, but this only means it would rank low even without a budget; it is not a weakness independent of the budget constraint.