Limiting Factor Analysis and throughput accounting
1 Introduction
A production plan is easy when nothing is short: make as much of everything as you can sell. It stops being easy the moment one resource runs out before demand does. Then the question changes. It is no longer “is this product profitable?” but “what is the best use of the resource we have not got enough of?” — and those two questions have different answers.
A limiting factor (or scarce resource, or key factor) is any resource that stops an organisation producing enough to meet demand in full: machine hours, labour hours, a raw material, floor space, a licence. This chapter deals with the case of one limiting factor. Where two or more resources are short at the same time the technique is linear programming, which is chapter 8.
The chapter takes the same short-term decision twice. First on the conventional marginal-costing assumption, where materials, labour and other variable costs all vary with output. Then on the throughput assumption, where in the short term the only variable cost is material. The two assumptions produce different rankings, different plans and different profits from identical data, which is the point of doing both.
Syllabus
This chapter serves P1C3c — product mix decisions with constraints, and the “product mix” topic under P1C1b. Chapter 8 completes it for two or more constraints, and chapter 14 covers the other short-term decisions (make or buy, discontinuation).
2 Identifying the limiting factor
Before ranking anything, prove that a resource really is short. Compare what full demand would need with what is available:
work out the maximum demand for each product;
multiply by the resource each unit uses, and total it;
compare that total with the quantity available.
If the requirement is less than or equal to what is available the resource is not a limiting factor at all, and the plan is simply to meet demand in full. If more than one resource fails this test, there is more than one limiting factor and the technique is linear programming (chapter 8).
Demand itself is a constraint in exactly the same sense, and it is easy to forget. There is never any point in making a unit that cannot be sold, so maximum demand caps every product, however profitable it looks.
3 Ranking by contribution per unit of the limiting factor
3.1 Why contribution, and not profit
In the short term the total fixed costs do not change with the production plan. Whatever mix is chosen, the same fixed costs are incurred. So the plan that earns the most contribution is the plan that earns the most profit, and the fixed cost per unit on the cost card is simply irrelevant to the choice.
That matters here because a cost card that absorbs fixed overhead can make two products look equally attractive when they are not. In example 1 below both products show a profit of $2 a unit and they are not equally attractive at all.
3.2 Why per unit of the limiting factor
Contribution per unit answers “which product is worth more?”. That is the wrong question when a resource is short, because the products do not consume the resource at the same rate. The right question is “which product is worth more per hour (or per kilogram) of the thing we have run out of?”
Take two hours of machine time in example 1. Two hours makes one unit of A and earns $5 of contribution; the same two hours makes two units of B and earns $8. B is the better use of the machine even though A earns more per unit.
Contribution per unit of limiting factor = Contribution per unit ÷ Units of the limiting factor per unit
3.3 The method
The rule
Rank the products by contribution per unit of the limiting factor — highest first — and work down the ranking, making as much of each product as can be sold, until the limiting factor runs out.
In full, the steps are:
calculate the contribution per unit of each product (selling price less all variable costs);
identify the limiting factor and the quantity of it each unit needs;
calculate the contribution per unit of the limiting factor for each product;
rank the products on that figure, highest first;
allocate the limiting factor down the ranking, capping each product at its maximum demand, until the resource is exhausted;
if the question asks for profit, total the contribution and deduct the total fixed costs.
3.4 Fixed costs when only a rate per unit is given
A question that asks for profit, and gives fixed costs only as an absorbed amount per unit, is asking you to make an assumption — state it.
The cost card was prepared before anyone knew the resource would be short, so the absorption rate was set on budgeted output, which was full demand. The total fixed cost is therefore the rate per unit multiplied by the maximum demand, not by the units actually produced — and because it is a fixed cost it stays at that total even though the plan produces fewer units.
Do not multiply the fixed cost per unit by the units in your production plan. That treats a fixed cost as if it varied with output, which contradicts the assumption the whole technique rests on. In example 1 that error understates fixed costs by $3,000 and overstates profit by the same amount.
This lecture teaches the conventional method through example 1 and is a good foundation. Two points before you watch. At around the point where the plan is set out the tutor ranks B first and then says “produce as many A’s as we can, which is 10,000” — he means B; the arithmetic he then does is B’s, and the notes and the answer below have it right (10,000 B, then 19,000 A). The assumption behind the $80,000 of fixed costs is explained in the recording and is written out in section 3.4 and in Answer 1 below.
Conventional limiting factor analysis
Pi plc manufactures 2 products, A and B.
The cost cards are as follows:
$ per unit | A | B |
Selling price | 25 | 28 |
Materials | 8 | 20 |
Labour | 5 | 2 |
Other variable costs | 7 | 2 |
Fixed costs absorbed | 3 | 2 |
Total cost | 23 | 26 |
Profit per unit | 2 | 2 |
Machine hours per unit | 2 | 1 |
Maximum demand (units) | 20,000 | 10,000 |
The total machine hours available are 48,000.
Calculate the optimum production plan and the maximum profit using conventional limiting factor analysis.
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4 Theory of constraints and throughput accounting
4.1 The theory of constraints
Goldratt’s theory of constraints starts from the observation that a system’s output is set by its slowest step. That step is the bottleneck. Improving anything else does not increase output; it only builds inventory in front of the bottleneck. So management should organise everything else around the bottleneck to keep it working, and then work to remove it — at which point some other step becomes the bottleneck and the exercise begins again.
Goldratt measures performance by throughput — the rate at which the system turns purchases into sales revenue — rather than by output or by unit cost, because producing something that has not been sold generates no money at all.
4.2 The throughput assumption
Over the next week or month, most organisations cannot change their labour force: the same people are paid whether the week is busy or slack. The same is true of most of what is normally called variable overhead. Materials are different — make fewer units and you buy less material.
Throughput accounting therefore assumes that in the short term the only variable cost is direct material, and that every other production cost — labour included — is fixed. Those other costs, added together, are the total factory cost.
The measure that replaces contribution is throughput:
Throughput = Sales revenue − Direct material cost
and the ranking rule is unchanged in form — rank by throughput per unit of the bottleneck resource, which is called the return per factory hour:
Return per factory hour = Throughput per unit ÷ Bottleneck hours per unit
4.3 Cost per factory hour and the throughput accounting ratio
The cost of running the factory for one bottleneck hour is the total factory cost spread over the bottleneck hours available:
Cost per factory hour = Total factory cost ÷ Total bottleneck hours available
Comparing the two gives the throughput accounting ratio (TPAR):
TPAR = Return per factory hour ÷ Cost per factory hour
Interpreting the ratio
A TPAR greater than 1 means an hour of the bottleneck spent on that product returns more than it costs to run the factory for that hour. The product is viable.
A TPAR less than 1 means the reverse: that hour loses money.
The higher the ratio the better the product, so the TPAR ranking is the same ranking as return per factory hour — it carries no new information about the ordering, only about viability.
A ratio below 1 can be improved in four ways, and each is worth naming because they are the four terms in the ratio:
raise the selling price (raises throughput per unit);
reduce the material cost per unit (raises throughput per unit);
reduce the time the product spends on the bottleneck (raises return per factory hour);
reduce the total factory cost (lowers cost per factory hour).
A TPAR below 1 is not an instruction to withdraw the product. Products are often sold as a range, or together, and a customer who cannot buy one may buy neither. A low price may be deliberate — a loss leader, penetration pricing, or an introductory price. And withdrawing a product only helps if the bottleneck hours it frees can be used for something better; if there is nothing else to make, the hours are simply idle and the factory cost is incurred anyway. Chapter 14 takes discontinuation decisions properly.
This lecture takes Example 1’s figures again on the throughput assumption and works Example 2 below: throughput of $17 and $8 a unit, a return per factory hour of $8.50 and $8.00, a plan of 20,000 A and 8,000 B, a total factory cost of $360,000 and a profit of $44,000, then the reasons for keeping a product whose ratio is below 1. Two points before you play it.
Check one figure against Answer 2: the throughput accounting ratios are 1.13 for A and 1.07 for B ($8.00 ÷ $7.50). Both are above 1, which is the point the recording goes on to make.
Coverage: it does not teach the theory of constraints in section 4.1 — the bottleneck and Goldratt’s measure of throughput. Read that section on its own.
Beyond P1. P1C3c asks for product mix decisions with constraints, and the examinable core of this chapter is the ranking rule — contribution, or throughput, per unit of the scarce resource. The throughput accounting ratio and the wider theory of constraints are developed at P2. They are kept here because they are the same argument under a different cost assumption, and because the lecture teaches them.
Throughput accounting
Pi plc manufactures 2 products, A and B. The cost cards are the same as in example 1:
$ per unit | A | B |
Selling price | 25 | 28 |
Materials | 8 | 20 |
Labour | 5 | 2 |
Other variable costs | 7 | 2 |
Fixed costs absorbed | 3 | 2 |
Total cost | 23 | 26 |
Profit per unit | 2 | 2 |
Machine hours per unit | 2 | 1 |
Maximum demand (units) | 20,000 | 10,000 |
The total machine hours available are 48,000. Machine time is the bottleneck resource.
(a) Calculate the optimum production plan and the maximum profit, on the assumption that in the short term only material costs are variable, i.e. using a throughput accounting approach.
(b) Calculate and interpret the throughput accounting ratios (TPAR) for A and B.
(c) Suggest some business reasons why management might decide NOT to withdraw an unprofitable product from sale.
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Three products, one scarce material
Kappa Co makes three products, X, Y and Z. All three use material M, which costs $2 per kg. Only 6,000 kg of M can be obtained next period. There are no other constraints.
$ per unit | X | Y | Z |
Selling price | 40 | 30 | 55 |
Material M | 8 | 6 | 20 |
Other variable costs | 17 | 14 | 15 |
Fixed overhead absorbed | 6 | 5 | 8 |
Total cost | 31 | 25 | 43 |
Profit per unit | 9 | 5 | 12 |
Maximum demand (units) | 500 | 800 | 300 |
(a) Prepare the optimum production plan and calculate the profit it earns.
(b) A second supplier will sell Kappa Co additional material M at $3.20 per kg. State, with a calculation, how many extra kilograms Kappa Co should buy.
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5 The two approaches compared
Examples 1 and 2 use identical data and reach opposite rankings. That is not a contradiction: they answer the same question under different assumptions about which costs vary in the short term. Neither profit figure is “the” profit; each is the profit consistent with its own assumption.
Conventional (marginal) | Throughput | |
What varies in the short term | Materials, labour and other variable costs | Materials only |
Measure per unit | Contribution — price less all variable costs | Throughput — price less material cost |
Ranked on | Contribution per unit of the limiting factor | Return per factory hour |
Ranking here | 1st B, 2nd A | 1st A, 2nd B |
Plan | 10,000 B and 19,000 A | 20,000 A and 8,000 B |
Profit on its own assumption | $55,000 | $44,000 |
The test that each ranking really is best under its own assumption is to cost the other plan the same way:
Conventional plan | Throughput plan | |
Profit on the marginal assumption | $55,000 | $52,000 |
Profit on the throughput assumption | $43,000 | $44,000 |
(Throughput plan on the marginal assumption: contribution (20,000 × $5) + (8,000 × $4) = $132,000, less $80,000 fixed = $52,000. Conventional plan on the throughput assumption: throughput (19,000 × $17) + (10,000 × $8) = $403,000, less $360,000 total factory cost = $43,000.) Each plan wins under its own assumption — so in an exam question the assumption stated in the requirement decides the method, and the two are never mixed.
6 Test your knowledge
Two quick checks before you move on: work through the flashcards to fix this chapter’s key terms and definitions, then sit the objective questions for exam-style practice. Both mark themselves and explain the answers as you go.
Limiting Factor Analysis and throughput accounting
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