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23-Ind-A4 Production Management · May 2016

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 — May 2016 — 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)

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.

Check — two-part question
This question has two sub-parts marked 10/10: the base formulation (a) and the Edison Motors modification (b), which uses the 10,000-unit/month Edison Motors requirement.

Given.

GradeUnit profitMax monthly demandLi req. (kg/battery)
Heavy (H)\$127,000200
Medium (M)\$109,000150
Light (L)\$74,000100
PlantAssembly cap. (batteries/mo)Max Li production (kg/mo)
Quebec City (QC)5,500100,000
Toronto (TOR)7,50070,000
Seattle (SEA)2,20040,000

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 (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. Maximize total monthly profit, summing unit profit $c_p$ over every (grade, plant) combination: $$\boxed{\max Z=\sum_{p}\sum_{l}c_p\,x_{p,l}=12\!\!\sum_l x_{H,l}+10\!\!\sum_l x_{M,l}+7\!\!\sum_l x_{L,l}}.$$
  2. Demand, assembly, and Li constraints (part a). 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.
  3. Dedicated Edison Motors requirement (part b). The Edison order comes on top of the existing Heavy market, so it adds a floor without removing the 7,000-unit market ceiling. Replace the Heavy row of the demand constraint ($\sum_l x_{H,l}\le7{,}000$) with $$\boxed{10{,}000\le\sum_l x_{H,l}\le10{,}000+7{,}000=17{,}000},$$ so at least the contracted 10,000 Heavy batteries are made every month and up to 7,000 more can still be sold to the open market. (Equivalently, add Edison variables $y_l\ge0$ with $\sum_l y_l=10{,}000$, earning $\$12$ each and included in each plant's assembly and Li constraints.) All other part-(a) constraints stay unchanged; Medium and Light keep their $\le$ demand ceilings. Setting $\sum_l x_{H,l}=10{,}000$ would be wrong, because it would stop the company selling any Heavy batteries to its other customers.
ElementFormulation
Variables$x_{p,l}\ge0$, $p\in\{H,M,L\}$, $l\in\{QC,TOR,SEA\}$
Objective (a)$\max Z=\sum_{p,l}c_p x_{p,l}$
Demand (a)$\sum_l x_{p,l}\le D_p$  $\forall p$
Assembly (a)$\sum_p x_{p,l}\le A_l$  $\forall l$
Lithium (a)$\sum_p r_p x_{p,l}\le Li_l$  $\forall l$
Edison Motors (b)$10{,}000\le\sum_l x_{H,l}\le17{,}000$ (contract floor plus the 7,000 market)
Check — part (b) 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 3 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.