Limiting factors

1 Introduction

We have already looked at how to deal with one limited resource – key factor analysis and throughput accounting.

In this chapter we will look at the situation where there is more than one limited resource, and a technique known as linear programming.

2 Linear Programming

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If there are two or more scarce resources then we are unable to use the Key Factor approach. Instead, we must use Linear Programming.

2.1 The steps are as follows:

Define the unknowns in terms of symbols
Formulate equations for the constraints
Formulate an equation for the objective
Graph the constraints and the objective
Find the optimum solution

Peter makes two types of chair – the ‘Executive’ and the ‘Standard’.

The data relating to each as follows:

	Standard	Executive
Materials	2 kg	4 kg
Labour	5 hours	6 hours
Contribution	$6	$9

There is a maximum of 80 kg of material available each week and 180 labour hours per week. Demand for ‘Standard’ chairs is unlimited, but maximum weekly demand for ‘Executive’ chairs is 10.

Find the optimal production plan and the maximum contribution that this will generate.

Answer to example 1

Let S = number of standard chairs produced per week

E = number of executive chairs produced per week

Constraints:

Materials:	2S + 4E ≤ 80
Labour:	5S + 6E ≤ 180
Demand:	E ≤ 10
Non-negativity:	S ≥ 0; E ≥ 0

Objective:

Maximise C = 6S + 9E

Feasible area: A, B, C, D, 0

Maximum contribution occurs at point B (using the objective function).

At B,	2S + 4E = 80	(1)
	5S + 6E = 180	(2)
(1) × 2.5:	5S + 10E = 200	(3)
(3) – (2):	4E = 20
	E = 5
In (1):	2S + 20 = 80
	2S = 60
	S = 30
	C = 6S + 9E
	= 180 + 45
	= $225

Produce 5 Executive chairs and 30 standard chairs per week.

Maximum contribution is $225 per week.

3 Spare capacity

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In the previous example, there were limits on the resources available. However, there was no requirement to use all of the resources – only that we could not use more than the maximum available.

If the optimum solution results in using less that the maximum available of a particular resource, then we have spare capacity of that resource or slack.

Using the information from example 1, calculate the slack for each of the constraints i.e. for materials, for labour, and for demand for ‘Executive’ chairs.

4 Shadow prices

In real life there are unlikely to be any truly limited resources – it will almost always be possible to get more, but we are likely to have to pay a premium for it. For example, the supply of labour may be limited by the length of the normal working week, but we can get more hours if we are prepared to pay overtime.

The shadow price (also known as the dual price) of a limited resource is the most extra that we would be prepared to pay for one extra unit of the limited resource. We calculate it by calculating the extra profit that would result if we have one extra unit of the limited resource.

Answer to example 3

(a) If there was 1 more kg of material available, then the material constraint becomes:

2S + 4E ≤ 81

Point B will still be the optimum solution, and therefore this will be when:

2S + 4E = 81 (1)

5S + 6E = 180 (2)

(1) × 2.5 5S + 10E = 202.5 (3)

(3) – (2) 4E = 22.5

E = 5.625

in (1) 2S + 22.5 = 81

2S = 58.5

C = 6S +9E

= 175.5 + 50.625

= 226.125

Shadow price of material = extra contribution

= 226.125 – 225

= $1.125 per kg

(b) If there was 1 more hour of labour available, then the labour constraint becomes:

5S + 6E ≤ 181

Point B will still be the optimum solution, and therefore this will be when:

2S + 4E = 80 (1)

5S + 6E = 181 (2)

(1) × 2.5 5S + 10E = 200 (3)

(3) – (2) 4E = 19

E = 4.75

in (1) 2S + 19 = 80

2S = 61

S = 30.5

C = 6S +9E

= 183 + 42.75

= 225.75

Shadow price of labour = 225.75 – 225

= $0.75 per hour

The shadow price of demand for executive chairs is $0, because there is already spare demand.

Answer the questions one at a time. Your progress is saved so you can leave and come back.

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