Difficulty: Intermediate | Prerequisites: Basic microeconomics (cost curves, profit maximisation), introductory linear programming (objective functions, constraints, feasible region).
This lab sits at the intersection of cost analysis and constrained optimisation. The firm manufactures four slip ring models and must decide, for each, how many units to produce in-house versus buy from an outside supplier. The decision is not simply "which is cheaper per unit" because production capacity is limited by wiring and harnessing hours, two shared fixed resources. Linear programming (LP) formalises that trade-off: minimise total cost subject to demand requirements and capacity constraints. The lab then stress-tests the baseline solution with sensitivity scenarios (maintenance shutdowns, new contracts, workspace expansion).
A firm uses LP to decide the cheapest mix of making and buying four slip ring models, given limited wiring and harnessing hours. The optimal plan costs $593,450 and involves buying all of Model A, making all of Models B and D in-house, and splitting Model C between in-house production and outsourcing. Sensitivity analysis shows that accepting a new contract or renting extra workspace both destroy value relative to the baseline.
Linear programming (LP)
A mathematical method for finding the best outcome (e.g. lowest cost or highest profit) given a set of linear constraints. In simple terms, it is a way of asking "what mix of decisions gives me the best result without breaking any rules?"
Objective function
The formula the LP model is trying to minimise or maximise. Here the objective is to minimise total production cost across all four slip ring models.
Think of it as the scoreboard: whatever combination of make-and-buy quantities produces the lowest number on this scoreboard wins.
Decision variables
The quantities the model is free to choose. In this lab: the number of units to make in-house and the number to buy, for each slip ring model.
Constraints
Conditions the solution must satisfy. This lab has two types: demand constraints (produce at least the required number of each model) and capacity constraints (do not exceed available wiring and harnessing hours).
Think of it as the guardrails: the LP can pick any combination it likes, as long as it stays inside these limits.
Variable cost (per unit)
The cost that changes with each additional unit produced. Includes Materials 1, Materials 2, and Other Variable Costs. Each model has its own variable cost structure.
Fixed input requirements
Resources consumed per unit that are shared across all models and limited in total supply. Here: wiring hours (6,000 available) and harnessing hours (3,500 available).
Make-or-buy decision
The choice between producing a component internally ("make") and purchasing it from an outside supplier ("buy"). LP determines the cost-minimising split when capacity is limited.
Shadow price / opportunity cost of a constraint
The amount by which the objective function would improve if the constraint were relaxed by one unit. Relevant to B.3, where the firm values extra maintenance time at the cost of the outsourced units it would avoid.
Sensitivity analysis
Testing how the optimal solution changes when you alter a parameter (demand levels, available hours, additional costs). Sections B.3 through B.5 are all sensitivity exercises.
The LP model minimises total cost across four slip ring models. Each model has a per-unit variable cost to make and a per-unit cost to buy. Production is constrained by 6,000 wiring hours and 3,500 harnessing hours.
The optimal solution (total cost: $593,450.36):
Model A (500 units needed): Buy all 500. The buy price ($116.70) is lower than the in-house variable cost ($103.94 plus the fixed-input consumption). Outsourcing is simply cheaper here.
Model B (1,350 units needed): Make all 1,350. In-house cost ($101.88/unit) is roughly $27 cheaper than buying ($128.00/unit). The capacity exists, so making them all is optimal.
Model C (3,000 units needed): Make 2,386 in-house, buy 613.7 outside. In-house production is cheaper per unit ($66.39 vs $88.30), but producing all 3,000 would exceed the wiring and harnessing hour limits. The LP produces as many as the constraints allow and outsources the rest.
Model D (4,500 units needed): Make all 4,500. In-house cost ($41.10/unit) is far below the buy price ($62.00/unit), and producing all units fits within the remaining capacity.
Capacity usage at the optimum: 5,396.1 of 6,000 wiring hours used, 3,500 of 3,500 harnessing hours used. Harnessing is the binding constraint.
Why Model C splits: this is the key insight. When a resource constraint binds, the LP cannot push all production in-house even though the per-unit cost favours it. The model optimally allocates the scarce resource (harnessing hours) across models and outsources only the marginal units it cannot afford to produce internally.
Scenario: the firm considers shutting down production for maintenance, which would reduce available capacity and force more outsourcing.
The question is: how much should the firm be willing to spend on maintenance to avoid the shutdown?
Answer: $54,189.71. This figure comes from calculating the additional outsourcing cost the firm would incur if capacity were reduced. Multiply the number of extra units that would need to be bought externally by their per-unit buy price.
The logic: if avoiding maintenance costs less than $54,189.71, the firm should pay for maintenance and keep producing at the current optimal plan. If avoiding maintenance costs more than that, the firm should accept the shutdown and absorb the higher outsourcing expense.
This is an application of opportunity cost reasoning. The value of the capacity that would be lost equals the cost of replacing it through outside purchasing.
Scenario: a customer offers $416,000 in additional revenue if the firm increases production of Slip Rings B and C beyond the current demand.
With the increased demand constraints, re-running the LP produces a new total cost of $1,013,820, up from the baseline $593,450. That is an incremental cost of $420,370.
The incremental revenue is $416,000. The incremental cost ($420,370) exceeds the incremental revenue ($416,000) by $4,370.
Decision: reject the contract. The firm would lose money on the deal.
This is a textbook marginal analysis application. The relevant comparison is always incremental revenue versus incremental cost, not total revenue versus total cost. The cost increase is large because the extra units push production further past the binding constraints, forcing expensive outsourcing or requiring resources that are already fully committed.
Scenario: the firm considers renting additional workspace to gain more wiring and harnessing hours, which would allow it to produce everything in-house (eliminating all outsourcing).
With the extra capacity, the in-house production cost falls to $573,623. Add the rent of $21,500, and the total becomes $595,123.
The baseline cost (with partial outsourcing of Model C) is $593,450.
$595,123 > $593,450, so the expansion costs $1,673 more than the current plan.
Decision: do not rent the additional workspace. The savings from eliminating outsourcing do not cover the rent.
The lesson here is that eliminating a constraint is only worthwhile if the cost of doing so is less than the savings it unlocks. The outsourcing "penalty" on Model C's 613.7 units is real, but it is smaller than $21,500 in rent.
Per-unit variable cost to make
(Materials 1 quantity x $6.31) + (Materials 2 quantity x $4.25) + (Other Variable Costs quantity x $1.00)
Each model has different input requirements per unit, so the per-unit variable cost differs across models.
Objective function (minimise total cost)
Sum over all models of: (Units Made x Variable Cost per Unit) + (Units Bought x Buy Price per Unit)
Demand constraint (per model)
Units Made + Units Bought >= Units Needed
Capacity constraints
Sum over all models of (Units Made x Wiring Hours per Unit) <= 6,000
Sum over all models of (Units Made x Harnessing Hours per Unit) <= 3,500
Incremental analysis (B.4 contract decision)
Accept if: Incremental Revenue > Incremental Cost
Here: $416,000 < $420,370, so reject.
Expansion analysis (B.5)
Expand if: (New In-House Cost + Rent) < Baseline Cost
Here: $595,123 > $593,450, so do not expand.
Students often think the cheapest-per-unit option always wins. It does not. Model A is cheaper to make on a pure variable-cost basis, but when you account for the fixed-input hours it consumes and the opportunity cost of those hours across all models, buying is optimal. The LP weighs system-wide trade-offs, not model-by-model comparisons.
Students assume a binding constraint means the firm should always invest to relax it. B.5 shows this is wrong: the cost of relaxing the constraint (rent) can exceed the savings it produces.
Students confuse total cost with incremental cost in the contract decision (B.4). The relevant question is whether the additional revenue covers the additional cost, not whether total revenue covers total cost.
Students sometimes forget that the LP can produce fractional solutions (e.g. 2,386.3 units made, 613.7 bought for Model C). In a continuous LP, fractional units are valid. Integer constraints would be needed if the firm cannot produce partial units, but that is a separate modelling choice.
Make-or-buy with capacity constraints is a classic LP application in managerial economics exams. Be ready to set up the objective function, identify constraints, and interpret the solution.
Know how to identify which constraint is binding (here, harnessing hours are fully used at 3,500; wiring hours have slack). A binding constraint drives the need to outsource.
Sensitivity analysis questions often ask: "Should the firm accept a new contract?" or "Should the firm invest to relax a constraint?" The method is always the same: compare incremental cost to incremental benefit.
The B.3 maintenance question tests whether you understand the value of capacity in dollar terms. The willingness to pay for maintenance equals the cost of the outsourcing you would otherwise be forced into.
Expect to be asked to explain why a model splits between make and buy. The answer always involves a binding constraint preventing full in-house production.
True or False: If a product is cheaper to make per unit than to buy, the firm should always make all units in-house. (False: capacity constraints may prevent it.)
Fill in the blank: The binding constraint in the baseline scenario is ______ hours. (Harnessing)
True or False: The firm should accept the B.4 contract because total revenue exceeds total cost. (False: the incremental cost of 420,370 exceeds the incremental revenue of 416,000.)
Fill in the blank: The firm would be willing to pay up to ______ for maintenance to avoid a capacity shutdown. (54,189.71)
True or False: Renting extra workspace in B.5 saves the firm money overall. (False: the total cost with rent is 595,123, which is 1,673 more than the baseline.)
Q: Why does the LP buy all 500 units of Model A rather than making any in-house?
A: The per-unit cost to buy (116.70) is lower than the per-unit variable cost to make (103.94) once the consumption of scarce wiring and harnessing hours is factored in. Those hours are better allocated to models where in-house production saves more money.
Q: Explain why Model C's production is split between make and buy.
A: Model C is cheaper to produce in-house (66.39 vs 88.30 per unit), but producing all 3,000 units would exceed the available harnessing hours. The LP makes 2,386 units in-house (using all remaining harnessing capacity) and outsources the remaining 614 units.
Q: A customer offers 416,000 for additional units of Slip Rings B and C. Should the firm accept? Show your reasoning.
A: Re-optimising the LP with the higher demand yields a total cost of 1,013,820, up from 593,450, an incremental cost of 420,370. Since 420,370 > 416,000, the incremental cost exceeds the incremental revenue. The firm should reject the contract.
Q: The firm can rent extra workspace for 21,500 to gain enough wiring and harnessing hours to produce everything in-house. Is this worthwhile?
A: With extra capacity, in-house production cost is 573,623. Adding 21,500 rent gives a total of 595,123. The baseline cost is 593,450. The expansion costs 1,673 more than the current plan, so it is not worthwhile.
Q: What does the maintenance willingness-to-pay figure of 54,189.71 represent economically?
A: It represents the opportunity cost of lost capacity. If the firm shuts down for maintenance, it must outsource additional units at a higher per-unit cost. The 54,189.71 is the total additional outsourcing expense, so any maintenance expenditure below this amount is justified.
This connects to cost curves and profit maximisation because the make-or-buy decision is fundamentally about comparing marginal costs across production methods. The LP formalises what you do intuitively when you compare MC to price in a perfectly competitive market.
This connects to constrained optimisation (Lagrangian methods, shadow prices) from earlier in the course. The shadow price of the harnessing constraint is the implicit cost per additional hour, which drives the B.3 maintenance valuation.
This connects to capacity planning and operations management in the broader field. Real firms routinely use LP models to decide which components to manufacture internally versus outsource, especially in aerospace, automotive, and electronics manufacturing.
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