Choose a low-cost package that covers every required capability

Make completeness a requirement rather than a score

Waterloo’s weighted set-cover formulation minimizes selected costs while covering every required element. Unlike additive-benefit knapsack selection, overlap does not compensate for a missing requirement.

Begin after the production owner has determined eligibility. List the capabilities that the approved brief truly requires, using descriptions precise enough that someone can substantiate each claim. Avoid combining vaguely related services under one convenient label. A package that offers consultation is not automatically evidence that it delivers the finished output the brief needs.

Use nonhuman production packages and fictional prices for the exercise. Do not turn uncertainty about a supplier into a silent negative mark, but do not count an unverified capability as present either. The calculation assumes the supplied map is correct. When that premise is unresolved, the appropriate next step is clarification rather than a more elaborate optimization.

Build the requirement matrix and inspect every selection

Invent requirements 1 through 4. A covers {1,2}, B {2,3}, C {3,4}, D {1,4}; their costs are 2, 3, 5, 3 teaching currency units. With binary choices, coverage requires A+D ≥ 1, A+B ≥ 1, B+C ≥ 1 and C+D ≥ 1.

The complete selection costs are AC: 7; BD: 6; ABC: 10; ABD: 8; ACD: 10; BCD: 11; ABCD: 13. BD is cheapest at 6.

Keep a covered-set column beside each cost. For AB, the union is {1,2,3}: requirement 2 occurs twice, while 4 is absent. Four capability mentions are not four distinct requirements covered. The local execution checked all sixteen unions against the four independent row inequalities.

Ask a reviewer to reconstruct the rejected two-package rows rather than checking only the winner. Those rows reveal why a plausible cheap shortlist fails the actual brief. Retain the empty selection as a deliberate alternative, not a missing value; it makes the complete enumeration inspectable.

Explain the recommendation without overstating the claims

The least expensive individual package need not belong to the least expensive complete recommendation. In this fixture, choosing A first is not a substitute for comparing the full alternatives. Explain that consequence to the commercial owner without implying that A is a poor product. Its suitability depends on the complete brief, not its price in isolation.

Before using an analogous real comparison, ask whether the package descriptions use the same delivery boundaries. A service that requires separately purchased finishing would need its actual cost and capability interpretation corrected. Keep the correction visible instead of quietly moving money between rows. The owner should be able to tell whether the recommendation changed because the price changed or because the original scope was misunderstood.

Confirm which practical constraints remain outside this small model. Availability, approved quality, commercial conditions and the ability to combine deliverables may rule out an otherwise attractive proposal. Do not invent numerical penalties for unresolved concerns merely to produce one tidy answer. A missing fact can justify holding the recommendation.

Keep the brief and the spending decision separate

Present the requirement definitions, supplied capability evidence and comparison together. The approving owner should see both why the proposed combination is complete and which assumptions make its price comparable. Do not send only the selected package names; that would discard the reasoning needed when a supplier updates an offer.

If the owner removes a requirement or adds a new one, preserve the earlier brief and recompute the recommendation. Treat that as a changed decision problem, not a correction to the earlier arithmetic. Similarly, a newly available package belongs in a new comparison rather than being inserted into the prior answer without checking its consequences.

This lesson does not place an order, negotiate a contract or establish supplier performance. It prepares a defensible recommendation for an authorized review. The useful result is a visible connection between the actual brief and the proposed expenditure, including the reasons inexpensive incomplete alternatives were not accepted.

Make the reasoning concrete.

A fictional studio must obtain a complete production package. One proposal emphasizes a low individual price and an impressive count of capability mentions. The owner asks whether that is enough to approve the shortlist.

The studio recommends further commercial review of the calculated alternative, without making a purchase or claiming realized savings.

Take the questions into your next brief.

Use these prompts in your own notes. Nothing is submitted from this page.

1. Brief

   Which precise capabilities are mandatory?

2. Evidence

   What establishes each supplied package claim?

3. Rows

   Can every selection be reconstructed from the supplied inputs?

4. Scope

   Which practical conditions remain outside the comparison?

5. Authority

   Who may approve the actual commercial commitment?

Watch for the wrong conclusion.

• Confusing a long feature list with an adequately specified brief.
• Hiding an unverified claim inside a numerical score.
• Changing the brief without preserving the earlier comparison.

Sources & scope

Original fictional inputs, local calculations and operational interpretation. Sources support identified mechanisms and their applications. No operational change, commercial commitment or personnel decision is made.

• Linear Programming Relaxations and Rounding →Primary instructor-authored teaching notes · Supports: Supports mandatory coverage.
• 15-451/651: Design & Analysis of Algorithms, Lecture 21 →Primary university teaching material · Supports: Supports binary requirement constraints and cost objective.