Question 4 of 7: Li-Ion Battery Production LP Across Three Plants
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — December 2015 — 98-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: seven questions, each worth 20 marks (sub-part weights as tabulated on the front page); only the first five questions appearing in the answer book are marked, so candidates effectively choose 5 of 7. All seven 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) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production-management systems; 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, makespan and tardiness; Hopp & Spearman, Factory Physics (3rd ed.) — variability and production-system inefficiency; Niebel & Freivalds, Methods, Standards, and Work Design — division of labour and work-design history; ISO 9001:2015 and the Toyota Production System literature — quality management, 5S/lean and TPM.
Question 4: Li-Ion Battery Production LP Across Three Plants (20 marks)
Find. (a) A profit-maximizing LP formulation across grades and plants; (b) the same formulation with an equal-capacity-utilization-fraction constraint added; (c) 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 each part's extra requirement as additional linear constraints on the same variable set — keeping every constraint linear by cross-multiplying rather than dividing by a variable.
Decision variables (part a). For grade $p\in\{H,M,L\}$ and plant $l\in\{QC,TOR,SEA\}$: $x_{p,l}\ge0$ = batteries of grade $p$ produced at plant $l$ per month.
Objective. Maximize total monthly profit, summing unit profit $c_p$ over every (grade, plant) combination:
$$\boxed{\max Z=\sum_{p}\sum_{l}c_p\,x_{p,l}=12\!\!\sum_l x_{H,l}+10\!\!\sum_l x_{M,l}+7\!\!\sum_l x_{L,l}}.$$
Demand, assembly, and Li constraints. Company-wide sales of each grade cannot exceed its market demand; each plant's total output (any grade mix) cannot exceed its assembly capacity; each plant's total lithium consumption cannot exceed its Li production limit:
$$\sum_l x_{p,l}\le D_p\ \ \forall p;\qquad \sum_p x_{p,l}\le A_l\ \ \forall l;\qquad \sum_p r_p\,x_{p,l}\le Li_l\ \ \forall l;\qquad x_{p,l}\ge0.$$
This nine-variable, nine-constraint LP is the complete part-(a) formulation.
Equal-utilization-fraction constraint (part b). Requiring the same scheduled-fraction-of-capacity $u=(\sum_p x_{p,l})/A_l$ at every plant looks nonlinear (a ratio of decision variables to a constant), but because $A_l$ is a known constant it is linearized by cross-multiplying instead of dividing — two equalities pin all three plants to one common ratio:
$$A_{TOR}\sum_p x_{p,QC}-A_{QC}\sum_p x_{p,TOR}=0,\qquad A_{SEA}\sum_p x_{p,TOR}-A_{TOR}\sum_p x_{p,SEA}=0,$$
added to the part-(a) constraint set (numerically: $7{,}500\sum_p x_{p,QC}-5{,}500\sum_p x_{p,TOR}=0$ and $2{,}200\sum_p x_{p,TOR}-7{,}500\sum_p x_{p,SEA}=0$).
Dedicated Edison Motors requirement (part c). The Edison order is a new, guaranteed customer on top of the existing Heavy market (up to 7,000/month), so Heavy output gets a hard lower bound at the contracted quantity, and its upper bound grows to the existing market plus the contract:
$$\boxed{\sum_l x_{H,l}\ge10{,}000},\qquad \sum_l x_{H,l}\le7{,}000+10{,}000=17{,}000,$$
which replaces the Heavy row of the part-(a) demand constraint. All other part-(a)/(b) constraints stay as they are (Medium and Light keep their $\le$ demand ceilings). An equivalent form uses a separate variable $e_l$ for Edison batteries built at plant $l$: add $\sum_l e_l=10{,}000$, keep $\sum_l x_{H,l}\le7{,}000$, and include $e_l$ in each plant's assembly and Li rows (profit $12e_l$). If the Edison contract were instead read as replacing the existing Heavy market, the constraint would be $\sum_l x_{H,l}=10{,}000$. Neither reading changes the feasibility result below.
Check — part (c) is numerically infeasible against the given Li limits
The 10,000-unit/month Heavy commitment requires $10{,}000\times200=2{,}000{,}000$ kg of lithium per month, but the entire company's stated maximum Li production across all three plants combined is only $100{,}000+70{,}000+40{,}000=210{,}000$ kg/month — barely a tenth of what is needed. Even devoting 100% of every plant's lithium supply to Heavy batteries alone (and producing zero Medium or Light) would yield only $210{,}000/200=1{,}050$ batteries/month, about 10.5% of the contracted 10,000. Assembly capacity is not the binding limit ($5{,}500+7{,}500+2{,}200=15{,}200$ batteries/month easily covers 10,000) — lithium supply is. This is presented as a real finding, not an authoring error: the requested formulation in Step 5 is correct and complete as written, but the exam's own numbers make the resulting model infeasible as stated. In practice this would mean the company cannot honour the Edison Motors contract without first securing substantially more lithium supply (a new supplier, a different plant, or a lower-lithium cell chemistry) — the formulation should still be submitted, with this infeasibility flagged as the operational conclusion.
