Question 8 of 8: Integer Programming — Mutual Fund Allocation with Disjunctive Rules

Question 8: Integer Programming — Mutual Fund Allocation with Disjunctive Rules (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Total to invest $T$ = $100,000; returns 5%/7%/10% on bonds/mainstream/speculative; each category capped at 40% of $T$ ($40,000); at least 2 of 3 named rules must hold.

Find. An IP model (continuous allocation variables plus binary rule-indicators with big-M logic) that maximizes expected return.

Approach. Model the allocation as three nonnegative continuous variables summing to $T$ and capped at 40% each; attach one binary indicator per rule with a big-M constraint that forces the indicator to 0 whenever its rule is violated, and require at least two of the three indicators to equal 1.

1. Decision variables. $B,S,P\ge0$ = dollars allocated to bonds, mainstream stocks, speculative stocks; binary $y_a,y_b,y_c\in\{0,1\}$ = 1 if rule (a)/(b)/(c) is satisfied.
2. Budget and category caps: $$B+S+P=100{,}000,\qquad B,S,P\le40{,}000$$
3. Big-M linking of each binary to its rule (M = 100,000, the largest any allocation can be): $$B\ge25{,}000-M(1-y_a)\qquad\text{[rule a: }B>25\%\text{]}$$ $$S\le50{,}000+M(1-y_b)\qquad\text{[rule b: }S<50\%\text{]}$$ $$P\le12{,}000+M(1-y_c)\qquad\text{[rule c: }P<12\%\text{]}$$ Each inequality is vacuous (non-binding) when its $y=0$ and enforces the literal rule when $y=1$.
4. "At least 2 of 3" logic and objective: $$y_a+y_b+y_c\ge2$$ $$\boxed{\max Z=0.05B+0.07S+0.10P}$$ subject to Steps 2–4, $B,S,P\ge0$, $y_a,y_b,y_c\in\{0,1\}$.
5. Validating solve. Because the 40% cap already forces $S\le40{,}000<50{,}000$, rule (b) is always satisfied regardless of allocation — so "at least 2 of 3" reduces to needing rule (a) or rule (c) as well. Evaluating both remaining combinations directly: {a,b} ($B\ge25{,}000$, others free up to cap) is feasible and, to maximize return, pushes the highest-return category (speculative) to its 40% cap and the lowest-return category (bonds) down to its 25% floor: $B=25{,}000,\ P=40{,}000,\ S=35{,}000$, return = $7,700. {a,c} and {b,c} both force $B+P\le52{,}000$ or less while needing $S\le40{,}000$ too, which cannot reach the required $100,000 total — both are infeasible. So the optimum is unique.