chapter 3: using graphs

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE

ISBN 1-86152-991-0 © Cengage

Chapter 3: Using Graphs


ISBN 1-86152-991-0 © Cengage

Objectives

• Create graphs of times series data

• Illustrate break-even analysis

• Show a feasible area

• Solve two variable linear programming problems


ISBN 1-86152-991-0 © Cengage

Time Series

• Probably the most common graph• Very simple to construct

– By hand– By computer

• Very simple to understand• Works for annual, quarterly, monthly weekly,

daily or even hourly data


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Annual Data

• Graph has years on X-axis• Data values on the y-axis• Annual data smooths out

short-term effects• Often used to consider

long-term trends in the

data• If they exist

National Gallery Visitor Numbers

0

1

2

3

4

5

6

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

Data used by kind permission of the National Gallery, New Media Department: http://www.nationalgallery.org.uk/default.htm


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Quarterly Data

• Quarterly data will show seasonal patterns

• Many of these are obvious, eg. coat sales

• Knowledge of patterns helps in planning for the business

0

50

100

150

200

250

300

1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,4 4,1 4,2 4,3 4,4


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Short time periods


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Break-even Analysis

• Tries to answers the fundamental question• “How many do we have to make/serve to cover our costs?”• Any business which cannot do this will, in the long run,

fail• Not to mention the cash flow problems in the short term.• For a single product company, the calculation is simple• Much more difficult of a large, multi-product company

since you then need to address the accounting question of

allocation of overheads.


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Graphing Break-even

Firstly we need to identify costsThen revenuesIn the simplest case, both of these will be linear functions

Output

£Revenues

Costs

Break-even

Break-even is where the two functions cross


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Calculating Break-even

It will usually be easier, and more convenient, to calculate the break-even figure

To do this we use X for the output and set up cost and revenue functions

Now put them equal to each other to find the X-value

If we sell the product at £5, then the Revenue function is 5XIf the fixed cost is 120 and unit cost is 2, then the Cost function is 120 + 2X

Revenue = Cost5X = 120 + 2X

3X = 120X = 40

This is the break-even production figure.


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Calculating Break-even (2)

An alternative, and quicker, way to calculate break-even is to use

the accounting concept of contribution

First step is to find the difference between the price per unit and the

cost per unit

eg. If P = 40 and C = 25

then the contribution (from each unit sold) is 15

Then divide the Fixed Cost by the contribution

eg. If Fixed Cost is 3000

then the break-even is 3000/15 = 200


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Break-even with Non-linear functions

• If the cost function is non-

linear, then we can still

graph the cost and revenue

functions• Break-even will still be

where R = C• For a quadratic cost

function, there may be two

break-even points0

500

1000

1500

2000

2500

3000

3500

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97


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Break-even with Non-linear functions (2)

• An alternative is to define a Profit Function as

• Profit = Total Revenue – Total Cost

• Then graph this function

• Break-even is where it crosses the X-axis

• (if it does)

Profit

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

1 7

13 19 25 31 37 43 49 55 61 67 73 79 85 91 97


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Linear Programming

Linear programming is a technique which seeks the optimum allocation of scarce resourcesbetween competing products or activities

It has been used in a wide range of situations in business, government and industry.

Examples include:optimum product mixmedia selectionshare portfolio selection


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Feasible Area

We are trying to create a graph which shows all

feasible mixes of the products, media types or

shares.

We will limit our analysis to only two items, but

you should note that the techniques will work in

much more complex situations

The first step is always to formulate the problem

i.e. to write out equations


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An Example

A small company (Singletons & Co.) make two products. They are asking for your advice on what mix of products to make, and have been able to provide the following information:

Ambers require 1 hour of labour timeZeonites require 2 hours of labour timeTotal labour hours per week is 40Ambers require 6 litres of moulding fluidZeonites require 5 litres of moulding fluidMaximum moulding fluid per week is 150 litres

Profit contribution from Ambers is £2Profit contribution from Zeonites is £3


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Formulating the Problem

What are we trying to achieve?

Probably maximum profit

Where does this profit come from?

The two products we produce

How much profit do we make?

Profit = £2A + £3Z


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Limitations

If there were a plentiful supply of everything we needed, then there would be no problem!

This is never the case!

Labour

Total hours used will be:A + 2Z But this total must be less

than or equal to 40

So: A + 2Z <= 40

We only have 40 hours per week of labour availableWe know that Ambers take 1 hour eachAnd Zeonites take 2 hours each


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What does this look like?

A

Z

20

4020

30

10

A + 2Z = 40

Feasible area


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More Limitations

Moulding Fluid:

We only have 150 litres per week of moulding fluid availableWe know that Ambers take 6 litres eachAnd Zeonites take 5 litres each

Total Fluid used will be:6A + 5Z

But this must be less than or equal to 150 litres

So: 6A + 5Z <= 150


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The Graph

A

Z

20

4020

30

10

6A + 5Z = 150

Feasible area


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An Assumption

We assume that it is only possible to get answers which are either zero or are positive

This means that:A >= 0

and Z >= 0

In terms of a graph, this means that we work in the first quadranti.e. The one where both variables are positive.


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Output Combinations

We need to find combinations of outputs which are feasible under all constraints

i.e. those which use no more than the labour available and no more than the moulding fluid available

Since we have graphs of each constraint, we can bring these together to find

The feasible area for the whole problem


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How many do we produce?On the graph, the feasible area has several “corners”

One where we produce only Zeonites (0,20)One where we produce only Ambers (25, 0)And one where we produce a combination of the two

where the two constraints cross:

6A + 5Z =150 A + 2Z = 40

Multiply by 66A + 12Z = 240Subtract the first equation from this one

7Z = 90Z = 12.857

Substituting gives: A = 14.2857


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Profit Levels

We can evaluate the profit contribution at each “corner” of the feasible area.

(0,20) (25, 0) (14.29,12.86)

Profit Contribution = 2A + 3Z

Highest

60 50 67.14


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LP - Minimisation

The previous example tried to maximise profit contributionbut the technique can also be used for finding

minimum cost solutions


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LP – Minimisation (2)

A company has 2 machines, A & B which can each produce either the MINI or MAXI version of their productA can produce 5 MINI or 1 MAXI per sessionB can produce 2 MINI or 3 MAXI per sessionContracts dictate that the minimum number

of MINI’s must be 100of MAXI’s must be 90

The cost of running machine A is £1000 per sessionThe cost of running machine B is £2000 per sessionWhat is the minimum cost number of sessions for each machine?

EXAMPLE:


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Formulation

We can construct equations as follows:

Number of MINI’s5A + 2B >= 100

Number of MAXI’sA + 3B >=90A, B >= 0

Costs:Minimise 1000A + 2000B

Again we can use a graph.


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Corners & Solution

The corners of the feasible area are at:A = 0, B = 50A = 90, B = 0

And A = 9.23, B = 26.9

(0,50) (90, 0) (9.23,26.9)

The cost function is: 1000A + 2000B

£100,000 £90,000 £63,030

MINIMUM


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Conclusions• Linear programming provides a method of solution for

a wide range of problems• It is not limited to two items and a few constraints, as

in our example• Computer based solutions are easily available

- for small problems you can use an add-in to Excel - for large problems there is specialist software

• It provides a short to medium term solution , butin the long run, managers need to address the resource constraints themselves if they wish to increase production levels

chapter 3: using graphs

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