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; (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.
3. One other important cost factor (part a). The objective uses one unit profit per grade at every plant, so it ignores costs that differ by plant. The most important is the cost of producing Li at each site: Li is the binding resource, and a kg of Li made in Quebec City need not cost the same as one made in Vancouver. Replacing the fixed profit with (price − plant-specific assembly and Li cost) makes the model send production to the cheapest sites. Other valid answers are the transportation cost from each plant to the vehicle makers, and the changeover cost each time a plant switches grade.
4. Edison Motors order (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. The order is new business on top of the existing market, and it must be met every month. So each of these grades gets a floor equal to the Edison quantity, and its ceiling rises to market demand plus the Edison quantity (replacing the part-(a) ceilings for H and M): $$\boxed{9{,}000\le\sum_g x_{Hg}\le17{,}000+9{,}000,\qquad 6{,}000\le\sum_g x_{Mg}\le19{,}000+6{,}000}$$ All other constraints and the objective are unchanged. An equivalent form uses separate Edison variables $e_{Hg},e_{Mg}$ with $\sum_g e_{Hg}=9{,}000$ and $\sum_g e_{Mg}=6{,}000$, counted in each plant's assembly and Li rows.