Limiting Factors: Allocating a Single Scarce Resource to Maximise Contribution

A budget normally assumes that the company can make as much as it can sell. Sometimes it cannot. A brewery in Ibadan finds that malt barley is rationed, a clothing factory in Aba has only so many skilled machinists. A printing press can run only so many hours a week.

When one resource limits output, the question is no longer how much the customers want, but which products deserve the scarce resource. The wrong answer is to favour the product with the highest contribution per unit, because it may eat most of the scarce resource. The right answer is to favour the product that earns the most contribution from each unit of the scarce resource.

This topic explains the idea of a limiting factor, how to identify it, how to rank products and build a production plan that maximises contribution, how to deal with a resource given as a naira value, and what to do when more than one resource is limited. It builds on the multi-product CVP topic. The next topic covers linear programming: formulation and graphical solution.

The limiting factor

Budgeting often assumes that a company can produce as many units as it needs to meet demand, so sales demand sets the limit. Sometimes a key resource, such as an item of direct materials, skilled labour or machine capacity, is in short supply. Then the limit on sales and profit is the amount of the resource, because sales cannot exceed what can be made.

A limiting factor is the item that restricts or constrains the production or sale of a product or service.

If the company makes one product, profit is maximised by making as many units as the scarce resource allows. If it makes several products that use the same scarce resource, the problem is to decide how many of each to make so as to maximise profit.

Identifying the limiting factor

A question may say that a resource is in short supply without saying which one. Find it by comparing, for each resource, the amount needed to meet the full sales demand with the amount available. A resource for which the need is greater than the supply is limiting. If two are short, there is more than one limiting factor and the problem needs linear programming.

The method for one limiting factor

1. Contribution per unit of each product (selling price less variable costs).
2. Identify the scarce resource.
3. Resource used per unit of each product.
4. Contribution per unit of scarce resource = contribution per unit ÷ resource per unit.
5. Rank the products, highest first.
6. Build the production plan. Work down the ranking, making each product up to its sales demand, until the scarce resource is used up. The last product made may be made in a smaller quantity than its demand.
7. Total contribution = Σ units × contribution per unit. Subtract fixed costs for profit.

Fixed costs are assumed to be the same whatever the plan, so only contribution matters. Ranking by contribution per unit of product, or by profit per unit, is wrong, because it ignores how much of the scarce resource each unit uses.

Materials by value, and more than one limit

A resource given as a naira value

A question may give the available quantity of a material as a naira amount, not as kilograms. The analysis is the same, using the cost of the material in each unit as the amount of the resource used: contribution per ₦1 of material = contribution per unit ÷ material cost per unit.

Other points

• Labour. If labour is the scarce resource, find the hours per unit as labour cost ÷ pay rate, and treat labour cost as a variable cost in the contribution as usual.
• Final answer. The plan should use exactly the available resource, unless demand is met before the resource runs out.
• Value of extra resource. The contribution per unit of the scarce resource of the last product made is the most the company should pay, over the normal price, for one more unit of the resource.
• Beyond the numbers. Dropping a product altogether may lose customers, and some customers may need a minimum supply. Management may keep a small output of low-ranked products for strategic reasons.
• More than one limit. If two or more resources are scarce, the ranking can differ for each, and no single ranking gives the best plan. Linear programming is then used, as the next topic explains.

Layouts

1. Resource needed for full demand = Σ (demand × resource per unit), compared with the amount available.
2. Contribution per unit of resource = contribution per unit ÷ resource per unit (for minutes, convert to hours).
3. Plan table: product, rank, units, resource used, contribution per unit, total contribution; running total of resource used = limit.
4. Last product = remaining resource ÷ resource per unit.

Money is in naira and answers are exact.

Worked examples

1. Machine 1 for full demand.
   X 18,000 × 8 ÷ 60 = 2,400 hours
   Y 12,000 × 5 ÷ 60 = 1,000 hours.
   Total 3,400 hours against 2,400 available: short.
2. Machine 2. X 1,200 hours; Y 2,000 hours; total 3,200 against 3,400: enough. Machine 1 is the limiting factor.
3. Per machine 1 hour.
   X 12 ÷ (8 ÷ 60) = ₦90
   Y 9 ÷ (5 ÷ 60) = ₦108.
   Rank Y first, X second.
4. Plan. Y: 12,000 units use 1,000 hours.
   Remaining 1,400 hours for X: 1,400 ÷ (8 ÷ 60) = 10,500 units, less than demand.
5. Contribution.
   Y 12,000 × 9 = ₦108,000
   X 10,500 × 12 = ₦126,000.
   Total ₦234,000.
6. Comment. X has the higher contribution per unit but Y earns more per scarce hour, so Y is made in full and X fills the rest.

1. Contribution per ₦1 of material.
   W 3,200 ÷ 400 = 8.0
   Z 2,100 ÷ 350 = 6.0
   X 1,500 ÷ 300 = 5.0
   Y 2,800 ÷ 700 = 4.0.
   Rank W, Z, X, Y.
2. W.
   Demand 400 units × ₦400 = ₦160,000 of material.
3. Z.
   500 × 350 = ₦175,000.
   Cumulative ₦335,000.
4. X.
   300 × 300 = ₦90,000.
   Cumulative ₦425,000.
5. Y.
   Remaining 519,500 − 425,000 = ₦94,500
   ÷ 700 = 135 units, less than the demand of 200.
6. Contribution.
   W 400 × 3,200 = ₦1,280,000
   Z 500 × 2,100 = ₦1,050,000
   X 300 × 1,500 = ₦450,000
   Y 135 × 2,800 = ₦378,000.
   Total ₦3,158,000.
7. Comment. The product with the lowest ratio, Y, is cut back; the highest, W, is made in full. The material used adds up to the amount available.

Expect variants: labour hours as the scarce resource, minutes to convert to hours, a minimum output for one product, and a question that asks for the most that should be paid for an extra unit of the resource.

Choosing one product under several limits, and overcoming constraints

Sometimes only one of two products can be made, and there are several limits (two machine lines, a scarce material, market demand). This is not a linear programming problem: work out the maximum output of each product under each limit separately, take the lowest figure, find the contribution at that output, and compare the products. Example. Machine A has 10,000 hours, machine B 12,000 hours, material 90,000 kg.

Product P: 0.20 and 0.30 hours, 2 kg, demand 50,000, contribution ₦260. Product Q: 0.25 and 0.25 hours, 2.5 kg, demand 40,000, contribution ₦300. P: A 50,000; B 40,000; material 45,000; demand 50,000, so 40,000 units, contribution ₦10.4m. Q: A 40,000; B 48,000; material 36,000; demand 40,000, so 36,000 units, contribution ₦10.8m. Make Q.

Surplus hours. If spare machine hours can be sold at ₦400 an hour (A) and ₦350 an hour (B): Q leaves 1,000 hours of A and 3,000 of B, worth 400,000 + 1,050,000 = ₦1.45m, giving ₦12.25m; P leaves 2,000 hours of A and none of B, worth ₦0.8m, giving ₦11.2m. Q is still better.

Ways to overcome a capacity constraint, with the costs. Hire or buy more machines (hire or financing cost); raise output per hour by faster running (breakdowns, lost output); add shifts or overtime (premium pay, wear and breakdowns); sub-contract or outsource (higher unit cost, risk to quality and delivery); find another source of the scarce material (higher price, delivery cost); and for a demand limit, spend on marketing or cut price (advertising cost, lower margin).

Each is worth doing only if the extra contribution exceeds the extra cost.

The minimum penalty that makes meeting a contract worthwhile

A major customer’s order is at a discount, and the scarce material means that it may not be possible to supply everything. Treat the discount as a variable cost of the special order only.

Compare the best plan without a penalty (which may supply only part of the order) with the best plan that meets the contract in full. The lowest penalty that would make the customer’s order be met in full is the difference. Example. Material 9,000 kg. Product F: contribution ₦700, 5 kg; product G: ₦540, 4 kg. Normal demand F 1,000 and G 500 (7,000 kg). Special order: 400 units of each at a discount of ₦50 a unit (3,600 kg).

Without a penalty: make the normal demand (₦970,000) and use the 2,000 kg left on the special order, ranked per kg after discount (F 650 ÷ 5 = 130; G 490 ÷ 4 = 122.5): 400 units of F, ₦260,000; total ₦1,230,000. Meeting the order in full: special order 400 × 650 + 400 × 490 = ₦456,000 uses 3,600 kg; the remaining 5,400 kg make 1,000 F (5,000 kg) and 100 G: ₦754,000; total ₦1,210,000. The penalty must be at least 1,230,000 − 1,210,000 = ₦20,000.

Constraints given as maximum revenue, and several machine types

If demand is given as maximum revenue, divide by the selling price to get maximum units. Calculate the hours needed for maximum demand on each machine type and compare with the hours available: only a type that is short is a limiting factor (a type with spare hours is not).

Rank products by contribution per hour of the short machine, make the best-ranked up to demand and the last with the hours that remain. Fixed costs of each machine block are deducted at the end; a fixed cost attributable to a product is relevant only if the product could be dropped.

Common mistakes

• Ranking by contribution per unit of product. Rank by contribution per unit of the scarce resource.
• Ranking by profit per unit. Fixed costs are the same in every plan.
• Not identifying the limiting factor first. Compare need with supply for each resource.
• Not converting minutes to hours.
• Exceeding demand. Stop each product at its demand.
• A plan that does not use exactly the resource available. Check the total.

Quick revision

• Limiting factor: resource that restricts output. Compare need for full demand with supply.
• One limit: contribution per unit of scarce resource; rank; plan to demand until resource is gone.
• Materials as a value: contribution per ₦1 of material.
• Two or more limits: linear programming.

How the marks are earned

• Identify: a mark for each resource compared and the limiting one named.
• Rank: contribution per unit of resource for each product.
• Plan: units, resource used, contribution, and a total that ties to the limit.

Practice questions

These are original questions, written in the style of the Skills examination. Work each one before you open the answer.

Recall check: multiple choice

Section B style: written questions

Model answers

Spotted a mistake? Tell us and we will check it.