Question 4 of 8: LP Formulation — Corn Buy/Sell/Storage Plan
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016 — 98-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 160 marks across 8 questions (each worth 20) and only 100 marks are required, so a candidate would normally answer 5 — all eight are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming and the simplex method & sensitivity analysis (ch. 3–4/6), network optimization & PERT/CPM (ch. 9–10), integer programming (ch. 12), Markov chains (ch. 16), decision analysis (ch. 15). Nahmias, Production and Operations Analysis — the single-period (newsvendor) inventory model.
Question 4: LP Formulation — Corn Buy/Sell/Storage Plan (20 marks)
Given. Starting position: 50 tons of corn and $1,000 cash on Jan 1. Warehouse capacity 100 tons. Cash pays for purchases at the time of purchase (no credit).
Given data — monthly buy/sell prices ($/ton)
Month
Buy price
Sell price
January
300
250
February
350
400
March
400
350
April
500
550
Find. The linear program (decision variables, objective, constraints) that maximizes Alexis's cash on hand at the end of April — formulate only, do not solve.
Approach. Model each month $t=1,\dots,4$ (Jan–Apr) with a buy quantity, a sell quantity, an end-of-month corn inventory, and an end-of-month cash balance; chain the inventory and cash balances month to month, cap inventory at the warehouse limit, and require cash to stay non-negative at the moment of each purchase.
Define decision variables. For $t=1,\dots,4$: $B_t\ge0$ = tons bought on the 1st of month $t$; $S_t\ge0$ = tons sold on the last day of month $t$; $I_t\ge0$ = tons of corn in the warehouse at the end of month $t$ (after that month's buy and sell); $C_t\ge0$ = cash on hand at the end of month $t$. Given: $I_0=50$ tons, $$C_0=1{,}000$ (starting position, before January's transactions).
Inventory balance (corn on hand carries from month to month, buy adds, sell removes), each month $t$:
$$I_t = I_{t-1} + B_t - S_t \qquad (t=1,\dots,4)$$
with the warehouse cap applied to the stock actually held after buying (before that month's sale), i.e. $I_{t-1}+B_t\le 100$, and feasibility of the sale itself, $S_t\le I_{t-1}+B_t$.
Cash balance (buying happens on the 1st and must be paid for immediately out of cash carried from the prior month; selling happens on the last day and adds to that month's cash):
$$\text{buy price }p^B_t\cdot B_t \le C_{t-1}\qquad(\text{pay-cash-at-purchase constraint})$$
$$C_t = C_{t-1} - p^B_t B_t + p^S_t S_t \qquad (t=1,\dots,4)$$
using $(p^B_1,\dots,p^B_4)=(300,350,400,500)$ and $(p^S_1,\dots,p^S_4)=(250,400,350,550)$.
Assemble the complete LP.
$$\text{Maximize } Z=C_4$$
$$\text{s.t.}\quad I_t=I_{t-1}+B_t-S_t,\quad I_{t-1}+B_t\le100,\quad S_t\le I_{t-1}+B_t,$$
$$p^B_t B_t\le C_{t-1},\quad C_t=C_{t-1}-p^B_tB_t+p^S_tS_t \quad (t=1,\dots,4),$$
$$I_0=50,\ C_0=1000,\quad B_t,S_t,I_t,C_t\ge0.$$
This is 4 copies of the buy/sell/inventory/cash relations chained by $I_{t-1}, C_{t-1}$, exactly what "formulate the LP" asks for — it is not solved here, per the instruction.
Final results — Question 4 (model summary, not solved)
