Question 5 of 8: Li-Ion Battery Production LP Across Four Plants
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — May 2018 — 17-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: eight questions, each worth 20 marks (sub-part weights 10/10 as tabulated on the front-page marking scheme); candidates do two questions from Section A and three from Section B, and only the first five questions appearing in the answer book are marked. All eight 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/EPQ) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production scheduling, JIT/kanban and shop-floor implementation gaps; 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 and days-off workforce scheduling; Hopp & Spearman, Factory Physics (3rd ed.) — variability, buffering, and production scheduling; Liker, The Toyota Way, and Shingo, A Revolution in Manufacturing: The SMED System — 5S, Five Whys, SMED and lean root-cause analysis.
Question 5: Li-Ion Battery Production LP Across Four Plants (20 marks)
The model has four plants and three battery grades, with a per-grade profit and Li requirement, two constraints per plant, and the Edison Motors 2:1 Heavy:Medium contract. This sitting’s Edison Motors floor (10,000 Heavy $+$ 5,000 Medium $=15{,}000$ units/month) is Li-feasible against the system’s own capacity — confirmed by the direct LP solve below.
Given.
Grade
Unit profit
Max demand (units/mo)
Li requirement (kg/battery)
Heavy (H)
$\$12$
17,000
200
Medium (M)
$\$10$
19,000
150
Light (L)
$\$7$
14,000
100
Plant
Assembly cap. (batteries/mo)
Max Li production (kg/mo)
Quebec City (QC)
15,500
850,000
Toronto (TOR)
17,500
700,000
Vancouver (VAN)
12,200
500,000
Seattle (SEA)
16,000
1,000,000
Find. (a) An LP formulation allocating the three grades among the four plants to maximize total monthly profit; (b) the formulation modified so the Edison Motors order (15,000 units/month, Heavy:Medium $=2$:1) is guaranteed met every month.
Approach. Define a decision variable for each (grade, plant) pair; each plant has two independent capacity limits (assembly units, and kg of Li) since Li requirement differs by grade, and each grade has an independent market-demand ceiling summed across all plants.
Decision variables and objective (part a). Let $x_{ig}\ge0$ be the number of grade-$i$ batteries produced at plant $g$, for $i\in\{H,M,L\}$, $g\in\{QC,TOR,VAN,SEA\}$. Maximize total monthly profit:
$$\boxed{\max Z=12\sum_g x_{Hg}+10\sum_g x_{Mg}+7\sum_g x_{Lg}}.$$
Constraints (part a). Each plant's total assembled units cannot exceed its assembly capacity; each plant's total Li consumed (grade-specific requirement $\times$ quantity) cannot exceed its Li capacity; and each grade's total production across all four plants cannot exceed its monthly demand ceiling:
$$\sum_i x_{ig}\le \text{cap}_g\ \ \forall g,\qquad \sum_i \ell_i\,x_{ig}\le \text{LiMax}_g\ \ \forall g,\qquad \sum_g x_{ig}\le \text{Dem}_i\ \ \forall i,\qquad x_{ig}\ge0,$$
where $\ell_H=200,\ell_M=150,\ell_L=100$ kg/battery. This gives $3\times4=12$ variables, 4 assembly constraints, 4 Li constraints, and 3 demand constraints.
Edison Motors floor (part b). A 2:1 Heavy:Medium mix totalling 15,000 units/month splits as 10,000 Heavy and 5,000 Medium ($\tfrac23\times15{,}000=10{,}000$, $\tfrac13\times15{,}000=5{,}000$). Edison's order is new volume on top of the existing market, and it must be met in full every month. So each of these two grades gets a floor equal to the contract, and its ceiling rises by the contract so existing customers can still be served up to their old limit:
$$\boxed{10{,}000\le\sum_g x_{Hg}\le17{,}000+10{,}000=27{,}000}$$
$$\boxed{5{,}000\le\sum_g x_{Mg}\le19{,}000+5{,}000=24{,}000}$$
These replace the part-(a) demand rows for Heavy and Medium; the objective, Light's ceiling and all plant constraints are unchanged. Keeping the old $\le17{,}000$ and $\le19{,}000$ ceilings would wrongly make Edison's batteries displace existing customers.
$\approx\$208{,}000$/mo (Medium+Light only — see callout)
Bonus: part-(b) LP solved with Edison floor, optimal profit
$\approx\$191{,}000$/mo (feasible — see callout)
Check — both LPs solved as a bonus check
Solving the part-(a) LP (not required by the question, but a useful validation) shows Li capacity, not assembly floor space, is the binding resource system-wide: since profit-per-kg-of-Li is $\$12/200=\$0.060$ (Heavy), $\$10/150=\$0.067$ (Medium), $\$7/100=\$0.070$ (Light), the unconstrained optimum produces zero Heavy, all of Medium's and Light's demand it can fit (Medium 11,000, Light 14,000 units), for $\approx\$208{,}000$/month.
Adding the Edison contract ($10{,}000\le H\le27{,}000$, $5{,}000\le M\le24{,}000$) and re-solving gives a feasible optimum of $H=10{,}000$, $M=5{,}000$, $L=3{,}000$ (profit $\approx\$191{,}000$/month, lower than the unconstrained optimum since the floor forces low-profit-per-kg Heavy into the mix) — system-wide Li usage in this solution is $3{,}050{,}000$ kg, all of the capacity (Edison needs $2{,}750{,}000$ kg; the remaining $300{,}000$ kg makes 3,000 Light), i.e. the contract is achievable with essentially zero Li to spare.
