ACCA Applied Skills · Performance Management

Limiting factors: formula sheet

Full chapter guide

Key formulas

Contribution per unit: Contribution per unit = Selling price per unit − Variable cost per unit; Use variable costs only. Fixed costs are ignored because they do not change with the production choice.
Contribution per unit of limiting factor: Contribution per limiting factor unit = Contribution per unit ÷ Units of scarce resource used per unit; For example contribution per machine hour or per kg of material. This is the ranking measure.
Resource needed for demand: Resource required = Maximum demand (units) × Resource per unit; Compare with the resource available to see whether it is truly limiting.
Limiting factor test: Resource is limiting if total required for maximum demand > resource available; Check this before ranking. If nothing is short, make everything demanded.
Ranking rule: Allocate resource in order of highest contribution per unit of scarce resource, up to maximum demand; Valid for a single limiting factor.
Contribution per unit: Contribution per unit = Selling price per unit − Variable cost per unit; Use variable costs only. Ignore fixed overheads and absorbed overheads.
Contribution per unit of limiting factor: Contribution per unit of limiting factor = Contribution per unit ÷ Units of scarce resource used per unit; This is the ranking measure. Higher is better.
Resource needed for a product: Resource required = Units to make × Resource per unit; Compare with the resource still available before allocating.
Total contribution of the plan: Total contribution = Σ (units made × contribution per unit); Deduct fixed costs only if the question asks for profit.
Extra variable cost of buying per unit: Buy price per unit − Relevant variable cost of making per unit; Use relevant costs only. Ignore absorbed fixed overheads unless they are avoidable if you buy.
Extra cost per unit of scarce resource: Extra cost of buying per unit ÷ Scarce resource used per unit made; This is the ranking figure. Use the same scarce resource for every component.
Ranking rule: Make first the component with the highest extra cost per scarce unit; buy first the lowest; Making saves the most money where this figure is highest.
Total cost of the plan: Variable cost of units made + Buy price × units bought; Check by starting from the cost of making everything and adding the extra cost of the units bought.
Objective function: Maximise C = (contribution per unit of X) × x + (contribution per unit of Y) × y; Use contribution, not profit. Fixed costs do not change the best mix.
Resource constraint: (usage per X) × x + (usage per Y) × y ≤ resource available; One inequality for each scarce resource. Define x and y clearly first.
Non-negativity: x ≥ 0, y ≥ 0; Always state it. It keeps the region in the positive quadrant.
Plotting a constraint: x-intercept = available ÷ usage per X (set y = 0); y-intercept = available ÷ usage per Y (set x = 0); Plot the two intercepts, join them, then shade the ≤ side (towards the origin).
Slack: Slack = resource available − resource used at the optimal plan; Zero for a binding constraint.
Shadow price: Shadow price = increase in optimal contribution from one extra unit of a binding resource; Valid only while the same constraints stay binding. Slack resources have a shadow price of zero.
Shadow price: Shadow price = change in total contribution ÷ change in the resource (one unit); Calculated at the optimal solution, only for a binding constraint and only within the valid range.
Slack: Slack = resource available − resource used at the optimal solution; For a maximum-type (≤) constraint. Slack on a constraint means its shadow price is zero.
Surplus: Surplus = amount used or produced − minimum required; For a minimum-type (≥) constraint. Surplus also means a zero shadow price.
Maximum premium for extra resource: Maximum extra price per unit = normal cost per unit + shadow price; Valid only for units within the shadow price range. Pay less than this and contribution rises.
Relevant cost of a scarce resource: Relevant cost = purchase cost per unit + shadow price per unit; Use when the resource is fully used up and cannot be freely bought.

Quick revision

A limiting factor is the resource that restricts activity, so contribution per unit of that resource drives the decision.
Single factor: rank by contribution per unit of scarce resource, not by contribution per product.
Allocate the resource to the top-ranked product first, up to its maximum demand, then move down.
Make-or-buy with a shortage: compare extra buying cost per unit of scarce resource saved, and buy the lowest first.
Use relevant costs only: avoidable variable costs, not allocated fixed overheads.
Linear programming needs an objective function, constraints and non-negativity conditions.
The optimal solution lies at a corner of the feasible region, or along an edge if the objective line is parallel to a constraint.
Solve intersecting constraint lines with simultaneous equations to get exact quantities.
A shadow price is the increase in contribution from one extra unit of a scarce resource.
A resource that is fully used (binding) has a positive shadow price, and a non-binding one has a shadow price of zero.
Slack is the unused amount of a resource at the optimal solution.
Always finish with a plan that states units to make and total contribution.

Common mistakes

Ranking products by contribution per unit instead of per unit of scarce resource. Fix: Always ask what is scarce first, then divide contribution by the amount of that resource each unit uses.
Including fixed costs in the contribution calculation or ranking. Fix: Use selling price less variable costs only. Subtract fixed costs once, at the end, if profit is asked for.
Ranking products by contribution per unit instead of per unit of scarce resource. Fix: Always divide by the scarce resource used per unit before you rank.
Using profit per unit or including absorbed fixed overheads in the calculation. Fix: Strip the cost back to variable costs only. Fixed costs do not change with the decision.
Ranking by the extra cost per unit instead of per scarce hour. Fix: Always divide by the scarce resource used per unit. A small extra cost on a component that uses many hours can still be the best one to buy.
Making the components with the lowest extra cost per hour first. Fix: Make first where buying hurts the most. Highest extra cost per scarce unit means make first. The lowest-ranked components are bought.
Using selling price or profit in the objective function. Fix: Use contribution per unit: selling price less variable costs. Fixed costs are irrelevant to the best mix.
Writing constraints the wrong way round, such as putting hours per unit against units available. Fix: Usage per unit goes with the variable. Total hours available go on the right-hand side.
Giving a non-zero shadow price to a constraint that has slack. Fix: Check usage against availability first. If any is left over, the shadow price is zero.
Quoting the total contribution gained instead of the gain per unit. Fix: Subtract the old contribution from the new and divide by the change in the resource. Use a one-unit change to keep it simple.

Exam tips

In objective test questions, the first line of working should be contribution per unit of the scarce resource. Wrong ranking loses all the marks because there is no partial credit.
Read carefully for more than one constraint. Demand limits are not scarce resources for ranking, but they cap how much of each product you make.
Check the resource-needed-versus-available figure early. It confirms the resource is limiting and catches arithmetic errors.
In constructed response questions, show a clear table with contribution, resource use, contribution per unit of resource and rank. Then state the plan and the total contribution.
If asked to comment, mention non-financial points: dropping a product may upset customers, shortages may be temporary, and extra resource could be bought at a premium if the extra contribution per unit exceeds the premium.
In objective test questions, the ranking is often the only thing tested. Calculate contribution per unit of scarce resource first and check the option wording against it.
Always test whether the resource is really limiting by comparing the requirement at full demand with availability. A quick check can save a lot of work.
In a constructed-response question, show a neat table of contribution, resource use, per-unit-of-resource figures and ranking. Marks are given for method even if one figure is wrong.