Current syllabus: Limiting-factor analysis and linear programming remain explicitly examinable in ACCA PM for September 2026 to June 2027. The syllabus includes one scarce resource, make-or-buy decisions, graphical linear programming, simultaneous equations, shadow prices and slack. The simplex method and sensitivity to changes in objective-function coefficients are excluded.
A limiting factor is a resource or demand constraint that prevents an organisation from doing everything it would otherwise choose to do. The exam task is not just to identify the shortage: it is to use the scarce resource where it earns the greatest contribution and to explain what the resulting plan means.
1. One limiting factor: rank the scarce resource
When there is only one scarce resource, rank products by contribution per unit of the limiting factor, not contribution per unit of product.
Product A contributes $30 and needs three skilled-labour hours, so it earns $10 per scarce hour. Product B contributes $24 and needs two hours, so it earns $12 per scarce hour. If labour is the only limiting factor, B ranks ahead of A despite its lower contribution per unit.
Calculate contribution per unit for each product.
Divide by units of the scarce resource used per product.
Rank from highest to lowest contribution per scarce-resource unit.
Allocate the scarce resource in that order, respecting maximum demand and other stated constraints.
For a make-or-buy question, compare the saving from making rather than buying per unit of the scarce resource. The objective is to minimise total relevant cost, which is equivalent to using internal capacity where it saves the most purchasing cost.
2. Why linear programming is needed
When two or more resources are simultaneously scarce, a single ranking is not sufficient. Linear programming identifies the best feasible combination subject to all constraints.
Use this five-stage method:
Define the variables. State exactly what each symbol represents.
Formulate the constraints. Translate resource limits, demand limits and non-negativity into inequalities.
Formulate the objective. Usually maximise total contribution.
Graph the constraints. Draw boundary lines and identify the feasible region.
Test the corner points. A linear objective reaches its optimum at a corner of the feasible region.
3. Complete graphical example
Peter makes Standard chairs (S) and Executive chairs (E):
Per chair | Standard (S) | Executive (E) |
|---|---|---|
Material | 2 kg | 4 kg |
Labour | 5 hours | 6 hours |
Contribution | $6 | $9 |
Weekly availability is 80 kg of material and 180 labour hours. Demand for Standard chairs is unlimited, but Executive demand is no more than 10 chairs.
The model is:
Maximise contribution: C = 6S + 9E
Material: 2S + 4E ≤ 80
Labour: 5S + 6E ≤ 180
Executive demand: E ≤ 10
Non-negativity: S ≥ 0 and E ≥ 0
To draw a boundary line, temporarily replace ≤ by = and find two points. For example, the material line passes through (S = 40, E = 0) and (S = 0, E = 20). Because resource use cannot exceed availability, the feasible side is towards the origin.
Interactive optimisation lab
Change the three limits and the two contribution figures. The shaded feasible region, iso-contribution line and optimum marker recalculate together. Try removing the third constraint, then add it back and reduce its limit.
Three quick experiments: cut material availability, increase Standard-chair contribution, then switch off the Executive-demand constraint. Predict the binding constraints and optimum before reading the new marker.
The material and labour lines intersect where both resources are fully used:
2S + 4E = 80
5S + 6E = 180
Solving simultaneously gives E = 5 and S = 30. Contribution is (30 × $6) + (5 × $9) = $225.
4. Test every feasible corner
Corner (S, E) | Why it is a corner | Contribution |
|---|---|---|
(0, 0) | Origin | $0 |
(0, 10) | E-axis and demand limit | $90 |
(20, 10) | Demand and material limits | $210 |
(30, 5) | Material and labour limits | $225 |
(36, 0) | Labour line and S-axis | $216 |
The optimum plan is therefore 30 Standard and 5 Executive chairs. Do not select a point merely because it looks furthest from the origin. Test the objective function at all feasible corners.
Practise testing the feasible corner points
Enter the objective-function formula in each yellow cell. The largest feasible contribution identifies the optimum.
| Standard chairs (S) | Executive chairs (E) | Your total contribution | Feasible? | Corner |
| 0 | 0 | Yes | Origin | |
| 0 | 10 | Yes | Demand/axis | |
| 20 | 10 | Yes | Demand/material | |
| 30 | 5 | Yes | Material/labour | |
| 36 | 0 | Yes | Labour/axis |
5. Slack: unused capacity
Slack is the unused amount of a ≤ constraint at the optimum.
Material used = (30 × 2) + (5 × 4) = 80 kg, so slack is zero.
Labour used = (30 × 5) + (5 × 6) = 180 hours, so slack is zero.
Executive demand used = 5 out of 10, so demand slack is 5 chairs.
A binding constraint has zero slack. A non-binding constraint has slack. Be careful: “surplus” is normally used for a ≥ constraint, whereas PM questions on resource maxima usually involve slack.
6. Shadow prices
A shadow price is the increase in the optimal objective value from one extra unit of a binding resource, while the current basis remains valid. It is therefore the maximum extra amount worth paying for one additional unit of that resource, over and above its normal cost already included in the contribution calculation.
For the chair example, increasing material from 80 kg to 81 kg and resolving the two binding equations gives S = 29.25 and E = 5.625. Contribution becomes $226.125, so the material shadow price is $1.125 per kg.
Increasing labour from 180 to 181 hours gives S = 30.5 and E = 4.75. Contribution becomes $225.75, so the labour shadow price is $0.75 per hour. The Executive-demand constraint is non-binding, so its shadow price is zero at the current optimum.
Do not use a shadow price indefinitely. It is valid only over the range in which the same constraints remain binding and the same corner structure applies. Also distinguish the shadow price from the normal purchase price: it is the maximum premium over the normal cost if that normal cost is already included in contribution.
7. Common exam errors
Ranking products by contribution per unit instead of contribution per scarce-resource unit.
Writing a constraint with the coefficients reversed.
Forgetting a demand constraint or non-negativity.
Shading the wrong side of a boundary line.
Testing an infeasible intersection.
Calling every unused amount a shadow price.
Applying a shadow price beyond its valid range.
Attempting simplex calculations, which are outside the stated PM syllabus scope.