LP Model for Maximizing NPV of Two Investments: 23-Ind-A1 Operations Research, May 2016, Q1

Question 1: LP Model for Maximizing NPV of Two Investments (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. Two investment plans with cash flows at Oct. 1 of years 0, +1, +2 and +3; discount rate $i=0.04$ (4% per year); cash available for investment is $10,000 at year 0 and $7,000 at year +1; either investment can be purchased in any fraction between 0 and 1 (a "fraction of an investment" cannot exceed the deal offered).

Given data — cash flows (CAD)
Investment	Yr 0	Yr +1	Yr +2	Yr +3
A	(6,000)	(5,000)	7,000	9,000
B	(8,000)	(3,000)	9,000	7,000

Find. The LP model (decision variables, objective, constraints) that decides what fraction of each investment to buy so as to maximize total NPV — formulate only, do not solve.

Fig. 1a — cash-flow diagram, Investment A (outflows down, inflows up).

Fig. 1b — cash-flow diagram, Investment B.

Approach. Discount each investment's own 4-year cash-flow stream to an NPV-per-unit-purchased coefficient, then let a continuous fraction variable for each investment carry that coefficient in the objective, tied down by the two cash-availability constraints.

Define decision variables. Let $x_A$ = fraction of Investment A purchased and $x_B$ = fraction of Investment B purchased, $0\le x_A\le 1$, $0\le x_B\le 1$.
Discount each investment's cash-flow stream to its own NPV coefficient at $i=0.04$ (4%): $$NPV_A = -6000 - \dfrac{5000}{1.04} + \dfrac{7000}{1.04^2} + \dfrac{9000}{1.04^3} = \boxed{3{,}665.17\ \text{CAD}},$$ $$NPV_B = -8000 - \dfrac{3000}{1.04} + \dfrac{9000}{1.04^2} + \dfrac{7000}{1.04^3} = \boxed{3{,}659.37\ \text{CAD}}.$$ These are the objective-function coefficients: buying a fraction $x_A$ of A contributes $NPV_A\cdot x_A$ to total NPV, and likewise for B.
Write the year-0 and year-+1 cash-availability constraints. The up-front outlay for both investments together cannot exceed the $10,000 on hand at year 0, and the second instalment cannot exceed the further $7,000 on hand a year later: $$6000\,x_A + 8000\,x_B \le 10{,}000 \qquad(\text{year 0 budget}),$$ $$5000\,x_A + 3000\,x_B \le 7{,}000 \qquad(\text{year }+1\text{ budget}).$$
Assemble the complete LP model. $$\text{Maximize } Z = 3665.17\,x_A + 3659.37\,x_B$$ $$\text{s.t.}\quad 6000\,x_A+8000\,x_B\le 10{,}000,\quad 5000\,x_A+3000\,x_B\le 7{,}000,\quad 0\le x_A\le 1,\ 0\le x_B\le 1.$$ The question asks only for this formulation — it is not solved here, per the instruction.

Check: the year-+1 budget is modelled as an independent $7,000 allowance rather than $7,000 plus any cash left unspent from year 0 rolled forward at 4% — the question states the two amounts as separate resources ("$10,000 available ... a year later you will have $7,000 available") rather than describing a rollover/reinvestment mechanism, so the simpler two-constraint reading is used.

Final results — Question 1 (model summary, not solved)
Item	Value
Decision variables	$x_A,x_B\in[0,1]$ — fraction of each investment purchased
Objective coefficient, A	$NPV_A=3{,}665.17$ CAD
Objective coefficient, B	$NPV_B=3{,}659.37$ CAD
Constraints	2 budget constraints (yr 0, yr +1) + 2 upper bounds
Solved numerically?	No — formulation only, per instructions

Topic: Capital-budgeting LP built from discounted cash flows.

Key relations: Present worth of a cash flow $F_t$ at year $t$: $PW=F_t/(1+i)^t$. NPV of a project is the sum of its own discounted cash flows; an LP that allocates a limited budget across several such projects maximizes $\sum_j NPV_j\,x_j$ subject to a cash constraint at every period money changes hands.

Why this works: Because NPV is additive across projects once each project's own stream has been discounted to a single number, the only coupling between investments is the shared, period-by-period cash budget — which is exactly what the two constraints encode. Allowing fractional purchase makes the feasible region a convex polytope, so the (unsolved) problem is a genuine LP rather than an integer program.

Common pitfall: Discounting the coefficients correctly but then writing the budget constraints in discounted dollars too — the budget constraints must stay in actual (undiscounted) dollars, because that is the cash physically on hand at each date, while only the objective is discounted.

Source: Hillier & Lieberman, Introduction to Operations Research (11th ed.), ch. 3 (LP formulation) and the capital-budgeting/PW material shared with the economics stream.

Question 1 of 8: LP Model for Maximizing NPV of Two Investments

Question 1: LP Model for Maximizing NPV of Two Investments (20 marks)