Question 5 of 8: Li-Ion Battery Production LP Across Three Plants

Question 5: Li-Ion Battery Production LP Across Three Plants (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. (a) An LP formulation allocating the three grades among the three plants to maximize total monthly profit, plus one other important cost factor the formulation omits; (b) the formulation modified so the Edison Motors order (15,000 units/month, Heavy:Medium $=3$:2) 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. Although the question asks only for the formulation, solving both LPs (unconstrained, and with the Edison floor added) is a cheap and decisive check on whether the new contract can actually be satisfied.

1. Decision variables and objective (part a). Let $x_{ig}\ge0$ be the number of grade-$i$ batteries produced at plant $g$, for $i\in\{H,M,L\}$, $g\in\{QC,TOR,VAN\}$. Maximize total monthly profit: $$\boxed{\max Z=12\sum_g x_{Hg}+10\sum_g x_{Mg}+7\sum_g x_{Lg}}.$$
2. 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 three 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 $3\times3=9$ variables, 3 assembly constraints, 3 Li constraints, and 3 demand constraints. One other important cost factor omitted here: inter-plant/inter-grade changeover or setup cost — the formulation implicitly assumes a plant can freely mix all three grades within a month at no switching penalty, but a real assembly line likely incurs a setup cost or lost capacity each time it changes grade, which would need either a fixed-charge term per (grade, plant) pair used or a minimum-run-length constraint to model realistically; transportation cost of finished batteries from each plant to the customer/distribution network is a second reasonable answer.
3. Edison Motors floor (part b). A 3:2 Heavy:Medium mix totalling 15,000 units/month splits as $\tfrac35\times15{,}000=9{,}000$ Heavy and $\tfrac25\times15{,}000=6{,}000$ Medium. “Must be met each month” is a hard floor. The Edison order is new business on top of the existing market, so the market ceilings must also rise by the contract volume. If they stayed at 17,000 and 19,000, the Edison units would silently displace existing customers. The Heavy and Medium demand rows therefore become two-sided: $$\boxed{9{,}000\le\sum_g x_{Hg}\le17{,}000+9{,}000=26{,}000,\qquad 6{,}000\le\sum_g x_{Mg}\le19{,}000+6{,}000=25{,}000}$$ 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 $\sum_g e_{Hg}=9{,}000$ and $\sum_g e_{Mg}=6{,}000$, added into each plant's assembly and Li rows.