Difficulty: Intermediate | Prerequisites: Basic linear programming (LP) concepts, understanding of variable vs. fixed costs, familiarity with Solver or equivalent optimisation tools.
Manufacturing firms routinely face a core question: should we produce a component in-house or buy it from an outside supplier? This assignment applies linear programming to that decision, using a firm that needs four different slip ring models in specific quantities. The LP model minimises total cost while respecting capacity constraints on wiring and harnessing labour hours. It sits squarely in the "constrained optimisation" portion of APEC 3002 and builds directly on earlier work with Solver. If you are comfortable setting up an LP objective function and constraints, this material will click quickly.
A firm uses linear programming to decide how many units of each slip ring model to make in-house versus buy from outside suppliers, minimising total cost subject to wiring and harnessing hour constraints. The optimal solution involves a mix: some models are entirely bought, some entirely made, and one is split between making and buying because labour capacity runs out. Shadow prices and reduced costs then guide follow-up managerial decisions about maintenance spending, new contracts, and facility rental.
Make-or-buy decision
The managerial choice between producing a component internally ("make") or purchasing it from an external supplier ("buy"). In simple terms, this means weighing your own production costs against a supplier's quoted price, factoring in capacity limits.
Variable cost to make (per unit)
The sum of all input costs that scale with each additional unit produced in-house: materials, labour, and other variable overheads. Think of it as what one more unit costs you on the factory floor.
Unit cost to buy
The price per unit charged by the external supplier. This is a flat rate regardless of how many you purchase.
Fixed input constraint
A capacity ceiling on a resource (such as wiring hours or harnessing hours) that limits how many total units the firm can produce in-house across all models. In simple terms, you only have so many labour hours available, and every model you make eats into that shared pool.
Shadow price (dual value)
The amount by which the objective function (total cost) would improve if the right-hand side of a binding constraint were relaxed by one unit. Think of it as the value, in cost savings, of one additional hour of wiring or harnessing capacity.
Reduced cost
The amount by which a decision variable's objective function coefficient would need to improve before that variable would enter the optimal solution at a positive level. If the reduced cost of "make Model A" is 4.20, the per-unit cost of making Model A would need to drop by at least $4.20 before in-house production becomes worthwhile.
Binding constraint
A constraint where the resource is fully used up at the optimal solution (slack = 0). When a constraint is binding, adding more of that resource would change the optimal solution.
Non-binding constraint (slack)
A constraint where the resource is not fully exhausted. There is leftover capacity, and relaxing it further would not improve the objective.
Decision variables
For each slip ring model (A, B, C, D): the number of units to make in-house and the number to buy from outside.
Objective function
Minimise total cost = sum across all models of (units made x variable cost to make) + (units bought x unit cost to buy).
The total cost at the optimal solution is $593,450.
Demand constraints
Each model has a fixed number of units needed: Model A = 500, Model B = 1,350, Model C = 3,000, Model D = 4,500.
For each model: units made + units bought = units needed. These constraints are all binding (no partial orders).
Capacity constraints (fixed inputs)
Wiring hours: each model requires a certain number of wiring hours per unit made. Total wiring used across all models cannot exceed 6,000 hours available. At the optimum, 5,396.1 hours are used.
Harnessing hours: similarly constrained at 3,500 hours available. At the optimum, all 3,500 hours are used (binding).
Variable input requirements per unit made
Input | Price | Model A | Model B | Model C | Model D |
|---|---|---|---|---|---|
Materials 1 | $6.31 | 8.49 | 9.60 | 1.56 | 1.38 |
Materials 2 | $4.25 | 8.07 | 7.73 | 10.78 | 6.53 |
Other variable costs | $1.00 | 16.07 | 8.45 | 10.73 | 4.64 |
Per unit variable cost to make | $103.94 | $101.88 | $66.39 | $41.10 | |
Unit cost to buy | $116.70 | $128.00 | $88.30 | $62.00 |
Fixed input requirements per unit made
Input | Model A | Model B | Model C | Model D | Use | Available |
|---|---|---|---|---|---|---|
Wiring (hr) | 0.97 | 0.98 | 0.50 | 0.64 | 5,396.1 | 6,000.0 |
Harnessing (hr) | 0.48 | 0.33 | 0.62 | 0.35 | 3,500.0 | 3,500.0 |
Model A (500 units needed)
Optimal decision: buy all 500 units.
Per unit variable cost to make ($103.94) is less than unit cost to buy ($116.70), but the difference is small relative to the capacity it would consume.
Reduced cost for making Model A: 4.203. This means the make cost would need to fall by $4.20/unit before making any units of A becomes worthwhile.
Reduced cost for buying: 0 (buying is already in the solution).
Shadow price of the demand constraint: $116.70.
Model B (1,350 units needed)
Optimal decision: make all 1,350 units in-house.
Per unit variable cost to make ($101.88) is substantially cheaper than buying ($128.00), a gap of over $26 per unit.
Reduced cost for making: 0 (making is in the solution).
Reduced cost for buying: 14.46. The buy price would need to drop by $14.46/unit before purchasing becomes optimal.
Shadow price of the demand constraint: $113.54.
Model C (3,000 units needed)
Optimal decision: make 2,386.3 units, buy the remaining 613.7 units.
This is the only model with a split solution. The firm makes as many as capacity allows, then buys the rest once the harnessing constraint binds.
Reduced costs for both making and buying are 0, because both activities appear in the optimal solution. This is the hallmark of a split: neither option is strictly dominated.
Shadow price of the demand constraint: $88.30.
Model D (4,500 units needed)
Optimal decision: make all 4,500 units in-house.
Per unit variable cost to make ($41.10) is far below the buy price ($62.00), and the wiring and harnessing requirements per unit are low enough that capacity is not exhausted.
Reduced cost for making: 0.
Reduced cost for buying: 8.53.
Shadow price of the demand constraint: $53.47.
These follow-up questions (B.3 through B.5) test whether you can use the LP results to make real business decisions.
B.3: Maximum maintenance cost to avoid shutdown
Question: how much should the firm pay in additional maintenance to keep production running?
Answer: up to $54,189.71.
Logic: if the production facility shuts down, the firm must buy externally every unit it currently makes in-house. The cost of switching those made units to bought units is $54,189.71. Any maintenance cost below that threshold is worth paying.
B.4: Should the firm accept a new contract for extra Model B and Model C units?
The new contract would increase demand for Models B and C, generating $416,000 in additional revenue.
Re-solving with the higher demand yields a new total cost of $1,013,820.
The cost increase: $1,013,820 minus $593,450 = $420,370.
Since $420,370 > $416,000, the incremental cost exceeds the incremental revenue.
Decision: reject the contract. The firm would lose $4,370.
B.5: Should the firm rent an additional facility for more wiring and harnessing capacity?
The extra capacity lowers the production cost to $573,623.
But the facility costs $21,500 to rent.
Total cost with rental: $573,623 + $21,500 = $595,123.
The original cost was $593,450.
Since $595,123 > $593,450, renting the facility increases total cost by $1,673.
Decision: do not rent. The capacity savings do not offset the rental expense.
Per unit variable cost to make
\text{VC}_i = \sum_j (\text{Price}_j \times \text{Requirement}_{ij})where i = model, j = input (Materials 1, Materials 2, Other variable costs).
Objective function
\min \sum_i (\text{Make}_i \times \text{VC}_i + \text{Buy}_i \times \text{BuyPrice}_i)Demand constraint (per model)
\text{Make}_i + \text{Buy}_i = \text{Demand}_iCapacity constraint (per fixed input)
\sum_i (\text{Make}_i \times \text{FixedInputReq}_{ik}) \leq \text{Available}_kwhere k = wiring or harnessing.
Incremental decision rule
For any proposed change (new contract, facility rental, maintenance spending): compare the incremental cost against the incremental benefit. Accept only if incremental benefit exceeds incremental cost.
Make-or-buy analysis with LP is standard practice in manufacturing operations. Automotive companies use exactly this framework to decide which parts to produce at their own plants versus source from tier-one suppliers, particularly when assembly line hours are constrained. The sensitivity analysis portion (what maintenance to pay, whether to accept a new contract, whether to rent extra capacity) mirrors the sort of capital allocation decisions that operations managers face quarterly.
Students often assume the cheapest-to-make model should always be made in-house. That ignores capacity constraints. Model A is cheaper to make than to buy on a per-unit basis, yet the optimal solution buys all 500 units because the labour hours are better allocated to other models.
Students confuse shadow prices with reduced costs. Shadow prices apply to constraints (resources); reduced costs apply to decision variables (activities). They answer different questions.
A common error is treating a split solution (Model C) as a mistake. It is not. When a constraint binds partway through producing a model, the optimal response is to make what you can and buy the rest.
Students sometimes forget to include all costs in a sensitivity analysis. In B.5, the production cost drops, but the rental cost must be added back. Comparing only the production cost would lead to the wrong decision.
Expect to be asked to set up and interpret an LP make-or-buy model from scratch, not just read Solver output.
Shadow prices and reduced costs are high-frequency exam topics. Be ready to explain what each one means in words for a specific scenario.
Sensitivity analysis questions ("should the firm accept this contract?") require incremental reasoning: compare the change in cost to the change in revenue, not total cost to total revenue.
The concept of a binding constraint driving a split solution (some made, some bought) is a key analytical insight. If you can explain why Model C is split and Model B is not, you understand the core logic.
True or false: if a model's per-unit variable cost to make is lower than its buy price, the firm should always make it in-house.
False. Capacity constraints may force the firm to buy even when making is cheaper per unit.
Fill in the blank: the amount by which total cost would decrease if one more hour of harnessing were available is called the ______.
Shadow price (of the harnessing constraint).
True or false: when both making and buying a model appear in the optimal solution, both activities have a reduced cost of zero.
True.
Fill in the blank: in B.4, the firm should reject the contract because the incremental cost of $______ exceeds the incremental revenue of $416,000.
$420,370.
True or false: renting the additional facility in B.5 is profitable because it lowers production cost.
False. The production cost drops, but the rent ($21,500) pushes total cost above the original.
Q: Why does the LP solution buy all 500 units of Model A rather than making them, even though the variable cost to make ($103.94) is lower than the buy price ($116.70)?
A: Because the labour hours Model A would consume (wiring and harnessing) generate more cost savings when allocated to other models. The reduced cost of 4.203 on the "make" variable tells us the make cost would need to drop by at least $4.20/unit before making Model A becomes optimal. The capacity is better spent on models with a larger make-vs-buy cost gap.
Q: Explain what it means for Model C to have a reduced cost of zero on both the make and buy variables.
A: When a decision variable is in the optimal solution at a positive level, its reduced cost is zero. Model C is the only model where both making and buying appear in the solution (2,386.3 made, 613.7 bought). Neither activity is strictly dominated, so neither has a positive reduced cost. This happens because the binding harnessing constraint forces the split at exactly the point where the marginal cost of making one more unit (including the opportunity cost of the capacity it uses) equals the buy price.
Q: A new contract offers $416,000 in additional revenue for extra units of Models B and C. The re-solved LP shows total cost rising from $593,450 to $1,013,820. Should the firm accept?
A: No. The incremental cost is $1,013,820 minus $593,450 = $420,370. The incremental revenue is $416,000. Since $420,370 > $416,000, the firm loses $4,370 by accepting.
Q: The firm can rent extra wiring and harnessing capacity for $21,500, which would reduce production cost to $573,623. Is this worthwhile?
A: No. Total cost with rental = $573,623 + $21,500 = $595,123. Original cost = $593,450. The rental increases total cost by $1,673.
Q: What is the maximum the firm should pay in maintenance costs to keep its production facility running?
A: $54,189.71. This is the additional cost the firm would incur if it had to buy externally every unit it currently makes in-house. Any maintenance expense below that threshold avoids a larger outsourcing bill.
This material connects directly to the LP and Solver modules earlier in APEC 3002. The constraint structure (limited labour hours shared across products) is the same framework used in product-mix problems, just reframed as a cost-minimisation rather than profit-maximisation. Shadow prices here link back to the concept of opportunity cost from introductory microeconomics: the shadow price of harnessing hours is the opportunity cost of one more hour of that resource. The sensitivity analysis questions (B.3 through B.5) are applications of marginal/incremental analysis, a thread running through the entire course.
Make or buy, make-or-buy decision, outsourcing decision, linear programming, LP cost minimisation, Solver, slip ring, shadow price, dual value, reduced cost, binding constraint, non-binding constraint, slack, sensitivity analysis, incremental analysis, marginal analysis, capacity constraint, wiring hours, harnessing hours, fixed input, variable cost, unit cost to buy, APEC 3002, managerial economics, applied microeconomics, Ragsdale spreadsheet modeling, constrained optimisation, product mix, opportunity cost of capacity