Question 4 of 7: Li-Ion Battery Production LP Across Three Plants

Question 4: 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) A profit-maximizing LP formulation across grades and plants; (b) the same formulation with a guaranteed 10,000-unit/month Heavy order added, plus a feasibility check against the given resource limits.

Approach. Define one continuous decision variable per (grade, plant) pair, build the objective and the three natural resource limits (demand, assembly capacity, lithium supply), then add part (b)'s guaranteed-order requirement as an additional constraint on the same variable set.

1. Decision variables and objective. Let $x_{gp}\ge0$ be the number of batteries of grade $g\in\{H,M,L\}$ produced at plant $p\in\{QC,TOR,SEA\}$ per month. The objective is $$\max Z=\sum_{g,p}\pi_g x_{gp}=12\sum_p x_{Hp}+10\sum_p x_{Mp}+7\sum_p x_{Lp}.$$
2. Demand constraints (one per grade, summed across all plants). $$\sum_p x_{Hp}\le7{,}000,\qquad\sum_p x_{Mp}\le9{,}000,\qquad\sum_p x_{Lp}\le4{,}000.$$
3. Assembly-capacity constraints (one per plant, summed across all grades). $$\sum_g x_{gQC}\le5{,}500,\qquad\sum_g x_{gTOR}\le7{,}500,\qquad\sum_g x_{gSEA}\le2{,}200.$$
4. Lithium-supply constraints (one per plant, weighted by each grade's Li requirement). $$200x_{HQC}+150x_{MQC}+100x_{LQC}\le100{,}000,$$ $$200x_{HTOR}+150x_{MTOR}+100x_{LTOR}\le70{,}000,$$ $$200x_{HSEA}+150x_{MSEA}+100x_{LSEA}\le40{,}000,$$ $$x_{gp}\ge0\ \ \forall g,p.$$ This is the complete part-(a) formulation: 9 decision variables, 3 demand constraints, 3 assembly constraints, 3 lithium constraints, and non-negativity.
5. Edison Motors modification (part b). The Edison order is a new customer on top of the existing Heavy market, so the Heavy demand constraint becomes a two-sided bound: a floor that guarantees the contract is met every month, and a ceiling equal to the existing market plus the contract: $$10{,}000\le\sum_p x_{Hp}\le7{,}000+10{,}000=17{,}000.$$ All other constraints (Medium and Light demand, assembly capacity, lithium supply, non-negativity) are unchanged. An equivalent form adds a separate variable $e_p$ for Edison batteries made at plant $p$, with $\sum_p e_p=10{,}000$, and adds $e_p$ to each plant's assembly constraint and $200e_p$ to its lithium constraint. Writing $\sum_p x_{Hp}=10{,}000$ would be wrong, because it drops the existing 7,000-unit Heavy market.
6. Feasibility check on the Edison Motors requirement. Total lithium capacity across all three plants is $100{,}000+70{,}000+40{,}000=210{,}000$ kg/month. Producing 10,000 Heavy batteries alone requires $10{,}000\times200=\boxed{2{,}000{,}000\ \text{kg Li/month}}$ — roughly ten times the plant network's entire lithium supply. Equivalently, even devoting all available lithium to Heavy batteries and nothing else, the network could produce at most $210{,}000/200=\boxed{1{,}050\ \text{Heavy batteries/month}}$, barely a tenth of the 10,000 required. $$\boxed{\text{The Edison Motors requirement is infeasible against the given resource limits.}}$$