Question 5 of 9: LP Formulation — Assembly of Two Products with a Make-or-Buy Raw Material
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2019 — 17-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 135 marks across 9 questions (all worth 15 marks) and only 100 marks are required, so a candidate would normally answer a subset — all nine are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming & the simplex method (ch. 3–4), duality & sensitivity analysis (ch. 6), integer programming (ch. 12), network optimization & CPM/PERT (ch. 9–10), queueing theory (ch. 17), decision analysis (ch. 15–16), Markov chains (ch. 16), equipment replacement (ch. 11/19). Nahmias, Production and Operations Analysis — single-period (newsvendor) inventory models.
Question 5: LP Formulation — Assembly of Two Products with a Make-or-Buy Raw Material (15 marks)
Given. Selling prices $\$11$/unit A, $\$23$/unit B; A needs 2 hr on line 1 + 1 unit raw material; B needs 2 hr on line 2 + 2 units raw material and 1 already-produced unit of A as a component; line 1 capacity 1,300 hr, line 2 capacity 500 hr; raw material either purchased at $\$5$/unit or self-produced using 2 hr of line-1 time (no cash cost, but it competes with product-A production for line-1 hours).
Find. An LP that maximizes profit — formulation only, not solved.
Approach. Because a unit of B consumes a unit of A as an input (not just raw material), track total A production separately from A sold directly, and give raw material its own supply-vs-demand balance since it has two competing sources (purchase or in-house production, the latter costing line-1 hours instead of dollars).
Define the decision variables. Let $S_A\ge0$ = units of Product A sold directly; $x_B\ge0$ = units of Product B produced & sold; $P_A\ge0$ = total units of Product A produced (sold directly plus consumed as a component of B); $r_b\ge0$ = units of raw material purchased; $r_p\ge0$ = units of raw material produced in-house.
Objective — maximize profit. Revenue comes only from units actually sold (A consumed inside a B is not sold separately — its value is embedded in B's $23 price); the only cash cost is purchased raw material (in-house raw material has no dollar cost, only an hours cost captured by the line-1 constraint):
$$\text{Maximize } Z = 11\,S_A + 23\,x_B - 5\,r_b$$
Product-A allocation constraint. Every unit of A produced is either sold directly or consumed by a B:
$$P_A = S_A + x_B$$
Raw-material balance. Supply (bought + self-made) must cover demand (1 unit per A produced, 2 units per B produced):
$$r_b + r_p \ge P_A + 2\,x_B$$
Line-1 hours (shared by A-assembly and raw-material production).
$$2\,P_A + 2\,r_p \le 1300$$
Line-2 hours (used only by B-assembly).
$$2\,x_B \le 500$$
Assemble the complete model (all variables $\ge0$):
$$\boxed{\begin{aligned}
\text{Maximize}\quad & Z = 11S_A + 23x_B - 5r_b\\
\text{s.t.}\quad & P_A = S_A + x_B\\
& r_b + r_p \ge P_A + 2x_B\\
& 2P_A + 2r_p \le 1300\\
& 2x_B \le 500\\
& S_A,\,x_B,\,P_A,\,r_b,\,r_p \ge 0
\end{aligned}}$$
($P_A$ can be eliminated by substituting $P_A=S_A+x_B$ throughout, giving an equivalent 4-variable model; it is kept explicit above because it mirrors the problem statement directly and makes the raw-material and line-1 balances easier to read.)
Check: the model above is a genuine, bounded, non-degenerate LP — solving it (not required by the question, done here only to confirm the formulation is sound) gives $S_A=400,\ x_B=250,\ r_b=1{,}150,\ r_p=0$, profit $=\$4{,}400$, with all three constraints (raw material, line 1, line 2) binding simultaneously. This cross-check is reported; it is not part of the requested deliverable.
Final results — Question 5 (model summary, not solved)
