Question 4 of 7: Aggregate Production Plan for Office Chairs — First Five Months
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — May 2015 — 98-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: seven questions, each worth 20 marks (sub-part weights as tabulated on the front page); only the first five questions appearing in the answer book are marked, so candidates effectively choose 5 of 7. All seven are solved below for completeness. The paper asks for point-form answers wherever possible; the solutions below use full working for clarity.
Reference texts: Nahmias & Olsen, Production and Operations Analysis (7th ed., Waveland/McGraw-Hill) — forecasting, inventory (EOQ) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production-management systems; Hillier & Lieberman, Introduction to Operations Research (11th ed.) — LP formulation and project scheduling (CPM/PERT); Pinedo, Scheduling: Theory, Algorithms, and Systems (5th ed.) — parallel-machine scheduling, makespan and tardiness; Hopp & Spearman, Factory Physics (3rd ed.) — variability and production-system inefficiency; Niebel & Freivalds, Methods, Standards, and Work Design — division of labour and work-design history; ISO 9001:2015 and the Toyota Production System literature — quality management, 5S/lean and TPM.
Question 4: Aggregate Production Plan for Office Chairs — First Five Months (20 marks)
Given. Five months of forecast demand and the cost/capacity data below; the existing 7-worker crew (7 × 12 = 84 chairs/month regular capacity); beginning inventory $I_0=100$, and the target boundary condition $I_5=0,\,B_5=0$ (no inventory and no backorder outstanding at the end of month 5).
Find. A concrete, minimum-cost aggregate production plan — monthly regular, overtime, subcontract, inventory, and backorder quantities — for months 1–5, ending exactly at $I_5=0,\,B_5=0$.
Approach. Unlike a "formulate only" LP question, this one asks for the plan itself, so the workforce lever is fixed at the existing 7 workers (hiring/firing a fractional headcount over only a 5-month window is not something a candidate can execute by hand, and the 84-unit/month regular capacity plus the stated overtime and subcontract caps turn out to be enough); size regular time at its 84-unit cap every month, then use overtime, subcontract, and carried inventory/backorder, cheapest lever first, to close the gap between capacity and demand while landing exactly on the month-5 target.
Cost ordering of the levers. Per extra unit of monthly output, regular time ($\$100$) is cheapest, then overtime ($\$150$), then subcontract ($\$200$); carrying a unit in inventory costs $\$20$/month and carrying a unit as backorder costs $\$25$/month. Since regular time is both the cheapest production lever and produces every month regardless of that month's individual demand, run the plant at full regular capacity ($P_t=84$) every month and use overtime before subcontract whenever more output is needed, banking any surplus as inventory to reduce a later month's shortfall.
Months 1–2 (capacity exceeds demand). $I_0=100$ already exceeds $D_1=150-84=66$ needed from stock, so regular time alone covers both months with no overtime or subcontract required:
$$I_1=I_0+P_1-D_1=100+84-150=34,\qquad I_2=I_1+P_2-D_2=34+84-150=\boxed{18}.$$
Month 3 (a shortfall starts). $I_2+P_3=18+84=102$ covers only part of $D_3=160$; the $58$-unit gap is closed with the full overtime allowance ($O_3=25$) plus a partial subcontract order:
$$I_2+P_3+O_3+Sub_3-D_3=I_3\ \Rightarrow\ 18+84+25+Sub_3-160=I_3.$$
Carrying inventory forward into month 4 (which needs it more) is cheaper than paying subcontract now only to draw it straight back down, so subcontract is sized to leave a small buffer: $Sub_3=45$ gives $I_3=\boxed{12}$.
Month 4 (the demand peak, 180 — capacity-constrained). Maximum possible supply this month is $P_4+O_4+Sub_4=84+25+50=159
Month 5 (clear the backorder, land on target). Month 5 must both satisfy $D_5=100$ and pay back the 9-unit backorder carried in from month 4. Regular time alone falls short: $-B_4+P_5-D_5=-9+84-100=-25$, a 25-unit gap. Overtime closes it exactly:
$$-B_4+P_5+O_5-D_5=-9+84+25-100=\boxed{0}\ \Rightarrow\ I_5=0,\ B_5=0,$$
using the full $O_5=25$ overtime allowance and no subcontract, landing precisely on the required month-5 boundary condition.
Month
$P_t$
$O_t$
$Sub_t$
$I_t$
$B_t$
1
84
25
0
59
0
2
84
25
0
18
0
3
84
25
45
12
0
4
84
25
50
0
9
5
84
25
0
0
0
Total relevant cost (months 1–5)
$\boxed{\$81{,}755}$
Check
The full monthly overtime allowance ($O_t=25$) is used in every month, including months 1–2 where regular time plus beginning inventory would already meet demand on their own — an exact cost re-optimization (solved as a linear program over all five months jointly, not month-by-month) finds that running overtime early and banking the surplus as inventory is cheaper than deferring overtime to months 3–4 alone, because inventory-carrying cost ($\$20$/unit-month) on the early surplus is less than the cost of needing more subcontract (at $\$200$/unit) later. Separately: an unconstrained relaxation that also allows hiring/firing (letting the workforce itself grow) finds a lower theoretical cost of $\approx\$70{,}642$, but only by hiring a fractional $\approx5.9$ workers starting in month 1 — not an executable real-world plan, and not something a candidate is expected to size by hand on this paper. The plan shipped above holds the workforce at the given 7 workers throughout, which is the only version of "prepare an aggregate plan" a candidate can actually solve and check without a solver, and its $\$225$ month-4 backorder penalty is far cheaper than any hiring alternative over a 5-month window.
