Linear Programming and Scarce Resources
Decision Making
Linear Programming and Scarce Resources
Syllabus tag: KASNEB CPA | Advanced Level | CA34S3 Advanced Management Accounting | Topic 10 Linear Programming and Scarce Resources
Lesson objectives
By the end of this topic, you will be able to:
- Formulate a linear programming problem
- Solve a two-variable problem graphically
- Identify binding and slack constraints
- Compute and interpret shadow prices
- State the assumptions and limitations of the technique
Why this matters
CA25 ranked products by contribution per unit of one limiting factor. That method fails as soon as two or more resources are scarce, because the ranking differs depending on which resource you use. Linear programming handles them together.
Formulation
Three parts, and an examiner awards marks for each:
- Define the variables precisely, with units
- State the objective function to be maximised or minimised
- State the constraints, including non-negativity
Worked example. Chui Ltd makes two products:
| Product X | Product Y | Available | |
|---|---|---|---|
| Contribution per unit | KES 450 | KES 600 | |
| Machine hours | 4 | 5 | 2,400 |
| Labour hours | 2 | 5 | 1,800 |
Let X = units of Product X produced, Y = units of Product Y.
Maximise C = 450X + 600Y
Subject to:
- 4X + 5Y ≤ 2,400 (machine hours)
- 2X + 5Y ≤ 1,800 (labour hours)
- X, Y ≥ 0
Solving
The optimum always lies at a corner of the feasible region, so evaluate each:
| Corner | Contribution |
|---|---|
| (0, 0) | nil |
| (600, 0) | 270,000 |
| (300, 240) | 279,000 |
| (0, 360) | 216,000 |
The optimum is 300 units of X and 240 of Y, giving KES 279,000.
Checking the resources at that point:
- Machine: (4 × 300) + (5 × 240) = 2,400 hours — fully used
- Labour: (2 × 300) + (5 × 240) = 1,800 hours — fully used
Both constraints are binding. Where a constraint is not fully used, the unused amount is slack, and that resource is not restricting output.
Shadow prices
The shadow price of a resource is the increase in contribution from one more unit of it. It exists only for binding constraints — a resource with slack has a shadow price of nil, because more of it changes nothing.
Solve the dual, where m is the machine shadow price and l the labour:
- 4m + 2l = 450
- 5m + 5l = 600
m = KES 105 per machine hour, l = KES 15 per labour hour
Check: (2,400 × 105) + (1,800 × 15) = 252,000 + 27,000 = 279,000, which equals the optimum contribution. If the dual does not reconcile to the primal, the working is wrong.
Interpretation matters more than the arithmetic.
The shadow price is the maximum premium worth paying above the normal cost of the resource. A machine hour is worth up to KES 105 extra — so overtime at a premium of 80 is worthwhile and at 120 is not.
100 extra machine hours are worth 100 × 105 = KES 10,500.
Two cautions:
- The shadow price applies only within a range. Add enough of the resource and a different constraint becomes binding, at which point the shadow price changes
- It assumes the extra resource is used optimally, by reoptimising the plan
:::checkpoint A supplier offers unlimited additional labour hours at a premium of KES 20 per hour above the normal rate. Using the shadow prices above, advise whether to accept, and say what would eventually make the answer change. :::
Slack, surplus and sensitivity
Slack is unused capacity on a ≤ constraint. Surplus is the excess over a minimum on a ≥ constraint — a contractual obligation to supply at least a certain quantity, for instance.
Sensitivity analysis in linear programming asks two questions: over what range can a contribution change before the optimal mix changes, and over what range does a shadow price hold before a different constraint binds.
Assumptions and limitations
- Linearity — contribution per unit and resource usage per unit are constant. Bulk discounts and volume price reductions both break this
- Divisibility — fractional units are permitted. Where the product is indivisible, integer programming is needed and simply rounding may give an infeasible or suboptimal answer
- Certainty — all coefficients are known. In practice they are estimates
- Single objective — only contribution is maximised, so quality, service and strategic considerations sit outside the model
- Two variables only for a graphical solution. Beyond that, the simplex method or software is required
The technique's real value in an examination is less the optimum itself than the shadow prices, because they tell management where to spend money to relieve a constraint and how much it is worth spending.
:::checkpoint A company solves a linear programme and finds a labour constraint has 240 hours of slack. The production manager proposes recruiting more staff to increase output. Explain, using the shadow price concept, why this would achieve nothing. :::