Strategic Cost Management · Decisions involving Alternative Choices

Product Mix with Limiting Factors: Method and Solved Problems

Updated 11 October 2026 · Fact-checked

A limiting factor is a scarce resource that caps output. With one such factor, rank products by contribution per unit of that factor and produce in that order until the resource or demand runs out. With two or more constraints, formulate a linear programme and solve it, usually graphically.

Understand Product Mix with Limiting Factors

Every business has resources it cannot expand in the short run: machine hours, skilled labour, a scarce raw material, or demand itself. When one of these is short, you cannot make everything the market wants. You must choose which products get the scarce resource.

The wrong instinct is to favour the product with the highest contribution per unit. That product may eat up a lot of the scarce resource. The right question is: how much contribution does each unit of the scarce resource earn? A product that earns ₹40 per machine hour beats one that earns ₹15 per machine hour, even if the second has a higher contribution per unit.

This is the key factor or limiting factor rule. It works cleanly when there is only one binding constraint. You rank products, allot the resource in rank order, and respect each product's maximum demand. Fixed costs stay unchanged across the choice, so they do not affect the ranking.

When two or more resources are scarce at once, ranking can fail, because a product that is best on one resource may be poor on another. Then you use linear programming: maximise total contribution subject to every constraint. With two products, the graphical method finds the best corner of the feasible region. The result also shows which constraints bind and what an extra unit of a scarce resource is worth.

Key rules to remember

Contribution per unit: Contribution = Selling price − Variable cost per unit; Use only variable costs. Allocated fixed costs are irrelevant to the ranking.
Contribution per unit of limiting factor: Contribution per key factor unit = Contribution per unit ÷ Units of limiting factor used per unit of product; Rank products on this figure, highest first. Valid for a single binding constraint.
Production plan under one constraint: Units of a product = Lower of (maximum demand, remaining resource ÷ resource per unit); Allot in rank order until the resource is exhausted.
Profit: Profit = Total contribution − Fixed costs; Fixed costs are deducted once, after the mix is decided.
LP objective function: Maximise Z = c1x1 + c2x2 + …, where c = contribution per unit; Use contribution, not selling price, as the coefficient.
LP constraints: a1x1 + a2x2 + … ≤ b (for each resource); x1, x2 ≥ 0; Add demand limits as extra constraints such as x1 ≤ maximum demand.
Shadow price of a resource: Shadow price = Increase in total contribution from one extra unit of a binding resource; It is the most you would pay above the normal price for extra units of that resource. It is zero for a non-binding resource, and it holds only within a limited range.

How to solve Product Mix with Limiting Factors questions

Use this sequence for any product mix question. First decide whether there is one binding constraint or several.

1Read the question and list every scarce resource and every demand or contract limit. Note which costs are variable and which are fixed.
2Compute contribution per unit for each product as selling price less variable cost. Ignore apportioned fixed overheads.
3Find the resource needed per unit of each product. Check that the total needed to meet full demand exceeds the resource available. If it does not, there is no limiting factor and you make the full demand.
4If only one resource is binding, compute contribution per unit of that resource and rank the products.
5Allot the resource in rank order, giving each product the lower of its maximum demand and what the remaining resource allows. Keep a running total of resource used.
6Compute total contribution, deduct fixed costs, and state the profit.
7If two or more resources bind, define variables, write the objective function and constraints, plot the constraints, and evaluate Z at each corner point of the feasible region.
8Pick the corner with the highest Z, state the units of each product, the contribution, and which resources are fully used. Close with a clear recommendation.

Quickest way: Rank-and-fill table for a single constraint

When to use it: Use when the question has one scarce resource and a demand limit for each product. It takes about five minutes.

Draw one table with columns: product, contribution per unit, resource per unit, contribution per resource unit, rank.
Write the resource available at the top and tick off the usage as you fill in rank order.
For each product in rank order, compute the units as the lower of maximum demand and remaining resource ÷ resource per unit.
Multiply units by contribution per unit, add up, then subtract fixed costs.
If a second constraint appears, switch to corner point evaluation. Test only the corners where the constraint lines meet each other or the axes.

Common mistakes in Product Mix with Limiting Factors

Ranking products by contribution per unit instead of per unit of the limiting factor.
Contribution per unit is the first figure you compute and it looks like the answer.
Fix: Always divide by the units of scarce resource used and rank on that figure.
Ignoring the maximum demand for the top-ranked product.
You pour all the resource into the best product without checking the market limit.
Fix: Cap each product at its demand, then pass the leftover resource to the next-ranked product.
Deducting allocated fixed overheads before ranking or treating them as product cost.
Absorption costing habits carry over to a decision problem.
Fix: Rank on contribution only. Deduct total fixed costs once at the end, if profit is asked.
Applying the single key factor ranking when two resources are binding.
The ranking method is quick, so students use it without checking all constraints.
Fix: Test the plan against every resource. If a second resource is exceeded or also fully used, formulate an LP.
Using selling price or profit instead of contribution in the LP objective function.
The problem lists selling prices and costs together, and students mix them up.
Fix: Compute contribution per unit first and use it as the coefficient of each variable.
Missing a contractual minimum or a compulsory order that must be met first.
The condition is buried in a note below the table.
Fix: Underline such conditions while reading. Allot the resource to compulsory output first, then rank the rest.

Worked examples

Example 1

Navkar Industries makes three products. Machine hours are limited to 3,000 per month. Fixed costs are ₹30,000 per month.

Product A: selling price ₹100, variable cost ₹60, machine hours 2 per unit, maximum demand 800 units.
Product B: selling price ₹150, variable cost ₹90, machine hours 4 per unit, maximum demand 500 units.
Product C: selling price ₹120, variable cost ₹80, machine hours 1 per unit, maximum demand 1,000 units.

Find the profit-maximising mix and the monthly profit.

Show the solution

Contribution per unit: A = 100 − 60 = ₹40. B = 150 − 90 = ₹60. C = 120 − 80 = ₹40.
Hours needed for full demand: A 1,600 + B 2,000 + C 1,000 = 4,600. This exceeds 3,000, so machine hours are the limiting factor.
Contribution per machine hour: A = 40 ÷ 2 = ₹20. B = 60 ÷ 4 = ₹15. C = 40 ÷ 1 = ₹40.
Ranking: C first, A second, B third.
Product C: 1,000 units × 1 hour = 1,000 hours. Remaining hours = 2,000.
Product A: demand 800 units needs 1,600 hours, which is within 2,000. Make 800 units. Remaining hours = 400.
Product B: 400 ÷ 4 = 100 units, which is below its demand of 500.
Contribution: C 1,000 × 40 = ₹40,000. A 800 × 40 = ₹32,000. B 100 × 60 = ₹6,000. Total = ₹78,000.
Profit = 78,000 − 30,000 = ₹48,000.

Answer: Make 1,000 units of C, 800 units of A and 100 units of B. Total contribution is ₹78,000 and monthly profit is ₹48,000.

Example 2

Kaveri Tools makes products X and Y. Contribution is ₹30 per unit of X and ₹40 per unit of Y. Each unit of X needs 1 machine hour and 3 labour hours. Each unit of Y needs 2 machine hours and 2 labour hours. Available per month: 80 machine hours and 120 labour hours. Find the mix that maximises contribution.

Show the solution

Let x = units of X and y = units of Y. Maximise Z = 30x + 40y.
Constraints: machine x + 2y ≤ 80. Labour 3x + 2y ≤ 120. x, y ≥ 0.
Axis points for machine: (80, 0) and (0, 40). Axis points for labour: (40, 0) and (0, 60).
Feasible axis corners: (0, 0). On the x-axis, labour limits x to 40, so (40, 0). On the y-axis, machine limits y to 40, so (0, 40).
Intersection of the two constraints: subtract machine from labour: (3x + 2y) − (x + 2y) = 120 − 80, so 2x = 40 and x = 20. Then 20 + 2y = 80, so y = 30. Corner (20, 30).
Check feasibility of (40, 0): machine use 40 ≤ 80. OK. Check (0, 40): labour use 80 ≤ 120. OK.
Evaluate Z: (0, 0) = 0. (40, 0) = 1,200. (0, 40) = 1,600. (20, 30) = 600 + 1,200 = 1,800.
At (20, 30): machine use = 20 + 60 = 80 and labour use = 60 + 60 = 120. Both resources are fully used.

Answer: Make 20 units of X and 30 units of Y. Maximum contribution is ₹1,800 per month, with both machine and labour hours fully used.

Exam tips

Check first whether full demand actually exceeds the resource. If it does not, say there is no limiting factor and produce to demand. Examiners test this.
Show the ranking table and the running resource balance. Marks go to the method even if one figure slips.
When two constraints appear, state the LP formulation clearly before solving. Formulation usually carries its own marks.
Always finish with a recommendation sentence: which products, how many units, total contribution or profit. Add a comment on any idle resource or unmet demand.
In MCQs, watch for the trap options: the product with the highest contribution per unit, and the mix that ignores the demand cap.

Practice questions from Decisions involving Alternative Choices

Product Mix with Limiting Factors in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Product Mix with Limiting Factors: frequently asked questions

What is a limiting factor in cost accounting?

It is a scarce resource that restricts output or sales, such as machine hours, labour hours, material or demand. It stops you from making all the products the market wants. Decisions then focus on using it where it earns the most contribution.

Why do we rank products by contribution per limiting factor and not per unit?

Because the scarce resource, not the number of units, is what runs out. A product with high contribution per unit may use up much of the resource. Contribution per unit of the resource shows which product gives the best return on what is short.

What if there are two limiting factors?

Ranking by one resource may give a plan that breaks the other constraint. Formulate a linear programme and solve it, graphically for two products. Evaluate the objective at each corner of the feasible region and choose the highest.

Do fixed costs affect the optimal product mix?

Not in the short run, if fixed costs do not change with the mix. The ranking and the choice depend only on contribution. Fixed costs matter only when you compute the final profit.