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An indivisible-budget ratio counterexample

With budget 100, projects B and C produce total entered NPV 64, while a highest-ratio-first choice of A produces only 40.

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Calculate the net NPV-to-cost ratios

Synthetic independent indivisible proposals
ProjectCostNPVNet ratio
A60400.666667
B50320.64
C50320.64

Follow the ratio-ranked choice

A has ratio 40/60 ≈ 0.666667, slightly above 32/50 = 0.64 for B and C. Selecting A first uses 60 and leaves 40, which cannot fund either remaining whole project. Its total NPV is 40.

List all eight candidate sets

Exact subset enumeration, budget 100
SetInitial costTotal entered NPVFeasible
Empty00Yes
A6040Yes
B5032Yes
A + B11072No
C5032Yes
A + C11072No
B + C10064Yes
A + B + C160104No

Identify the objective within this model

B + C costs 100 and produces total NPV 64. It is the highest value among the feasible sets in this example. The result assumes independent additive NPVs, no prerequisites and no later-period funding constraints.

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