UAL2-U3-L08 · University Accounting Level 2
Product mix under a constrained resource
Learning goals
- Rank products by contribution per scarce-resource unit.
- Build an optimal simple product mix subject to demand and capacity.
- Calculate the value of one additional constraint unit and test assumptions.
Prerequisite check
The product with highest contribution per unit is not always first. When machine time is scarce, contribution per machine hour—not per product—determines how each bottleneck hour contributes to profit.
Vocabulary
- Constraint/bottleneck: resource limiting total output.
- Contribution per constraint unit: unit contribution divided by scarce-resource use.
- Shadow value: incremental benefit of one more constraint unit within a relevant range.
- Demand limit: maximum units that can be sold at assumed economics.
- Throughput: sales less truly variable costs in a specified constraint approach.
Core idea
With one binding constraint, satisfy demand in descending contribution per constrained resource. Fixed costs unchanged by the mix do not enter the ranking. Real decisions also require minimum commitments, multiple constraints, setup effects, and customer strategy.
Why this treatment makes sense
A bottleneck hour spent on one product cannot be spent on another. Ranking by total unit contribution ignores this opportunity cost; ranking per bottleneck hour reveals which use creates the greatest incremental benefit.
A repeatable method
- Confirm demand, selling price, truly variable cost, and constraint use per product.
- Compute unit contribution.
- Divide by constraint units and rank.
- Allocate scarce capacity to demand in rank order.
- Calculate total contribution and subtract unchanged fixed costs.
- Value extra capacity using the marginal product displaced or added.
- Test minimum orders, setups, quality, multiple bottlenecks, and demand uncertainty.
Worked example
Thunder Bay Woodcraft has 4,000 finishing hours.
| Product | Price | Variable cost | Contribution | Hours/unit | Contribution/hour | Max demand |
|---|---|---|---|---|---|---|
| Desk | $500 | $320 | $180 | 3 | $60 | 700 |
| Shelf | $260 | $140 | $120 | 1.5 | $80 | 1,600 |
Make shelves first: 1,600 × 1.5 = 2,400 hours, contribution $192,000. Remaining 1,600 hours make 533 whole desks (1,599 hours), contribution $95,940. Total contribution $287,940 with one unused hour. Although desks have higher unit contribution, shelves create more per scarce hour.
The $60 contribution-per-hour figure is a divisible-output shadow value. With whole units and one idle hour, one extra hour has no immediate value because a desk needs three hours. Buying two usable extra hours would combine with the idle hour to complete one desk, adding $180 contribution before overtime, quality, or fixed effects. Across usable three-hour blocks, the upper average value is $60 per hour while unmet desk demand and all other assumptions hold.
Journal, ledger, and statement connection
The mix model uses controlled sales, routing, and cost data but creates no entry. Actual production and contribution later flow through normal ledgers. Track bottleneck hours, rejected units, and demand lost to validate assumptions.
Common mistakes
- Ranking by unit contribution instead of contribution per constraint.
- Producing above demand limits.
- Treating allocated fixed cost as variable in the ranking.
- Claiming a shadow value remains constant beyond the range where the constraint binds.
Guided practice
Product A contributes $50 and uses 2 hours = $25/hour. B contributes $42 and uses 1 hour = $42/hour. With one constraint, B ranks first despite lower unit contribution.
Independent practice
Level 1 — rank: X contributes $90 using 3 hours; Y contributes $56 using 1.4 hours. Rank.
Level 2 — mix: Capacity 1,000 hours. Y demand 400 units at 1.5 hours and $60 contribution; X uses 2 hours, contributes $70, demand 500. Find mix and contribution.
Level 3 — decision: Overtime must be purchased in usable two-hour blocks because X requires two hours per whole unit. Using your Level 2 result, state the maximum average overtime cost per hour while unmet X demand remains and name two cautions.
Self-check and solutions
Level 1: X $30/hour; Y $40/hour; rank Y first.
Level 2: Y uses 600 hours and contributes $24,000. Remaining 400 hours make 200 X, contributing $14,000. Total $38,000.
Level 3: Each two-hour block completes one X and adds $70, so the maximum average incremental capacity cost is $35/hour. Paying $45/hour would destroy $20 per completed unit before other effects. Check quality, worker fatigue, setup, and whether demand and prices hold.
Retrieval practice
- What denominator determines rank?
- What limits production after ranking?
- When is shadow value valid?
Answers: scarce resource per unit; demand and available capacity; only while assumptions and the same binding constraint hold.
Exam-style application
Rank three products, allocate one bottleneck, compute total contribution, value a capacity proposal, and explain how a minimum customer commitment changes the solution.
Lesson summary
Product mix maximizes contribution from the true bottleneck, subject to demand and operational realities, and values capacity only over its valid range.