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 — December 2017 — 98-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); only the first five questions appearing in the answer book are marked, so candidates effectively choose 2 of 3 in Section A and 3 of 5 in Section B. 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)
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 (9,000 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 9,000 units/month splits as 6,000 Heavy and 3,000 Medium ($\tfrac23\times9{,}000=6{,}000$, $\tfrac13\times9{,}000=3{,}000$). “Must be met each month” is a hard floor. The Edison order is new business on top of the existing market, so the existing market ceilings must also be raised by the contract volume. If they stayed at 7,000 and 9,000, the Edison units would silently displace existing customers. The Heavy and Medium demand rows therefore become two-sided:
$$\boxed{6{,}000\le\sum_g x_{Hg}\le7{,}000+6{,}000=13{,}000,\qquad 3{,}000\le\sum_g x_{Mg}\le9{,}000+3{,}000=12{,}000}$$
replacing the part-(a) Heavy and Medium demand rows (objective, Light demand row, assembly and Li rows unchanged). An equivalent form uses separate Edison variables $e_{Hg},e_{Mg}\ge0$ with $\sum_g e_{Hg}=6{,}000$, $\sum_g e_{Mg}=3{,}000$, added into each plant's assembly and Li rows.
Check — part (a) solved as a bonus check, and part (b) is numerically infeasible
Solving the part-(a) LP (not required by the question, but a useful validation) shows every plant's lithium capacity binds before its assembly capacity or any grade's demand ceiling does — e.g. Quebec City's optimum is 850 Light batteries $=850\times100=85{,}000$ kg, exactly its Li cap, while assembly capacity (5,500) is barely touched. Because profit-per-kg-of-Li is $\$12/200=\$0.060$ (Heavy), $\$10/150=\$0.067$ (Medium), $\$7/100=\$0.070$ (Light), Light has the best return on the scarce resource system-wide, so the unconstrained-by-Edison optimum produces Light exclusively (3,050 units total, under its 4,000 demand cap), for $\approx\$21{,}350$/month — a useful insight for the concept box, though not what the question asks for.
The part-(b) Edison floor, however, is infeasible against the plants' own stated Li capacities: 6,000 Heavy batteries alone need $6{,}000\times200=1{,}200{,}000$ kg of Li, and the 3,000 Medium batteries need a further $3{,}000\times150=450{,}000$ kg, for $1{,}650{,}000$ kg/month — against a combined system-wide Li capacity of only $85{,}000+70{,}000+50{,}000+100{,}000=305{,}000$ kg/month, roughly $5.4\times$ too little. The formulation above is the correct graded answer to “modify your formulation”; this numeric check is a data-consistency flag, not a modelling error — as written, no feasible production plan can satisfy the Edison contract under the paper's own Li-capacity figures.
