chapter 3: using graphs
DESCRIPTION
Chapter 3: Using Graphs. Objectives. Create graphs of times series data Illustrate break-even analysis Show a feasible area Solve two variable linear programming problems. Time Series. Probably the most common graph Very simple to construct By hand By computer Very simple to understand - PowerPoint PPT PresentationTRANSCRIPT
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Chapter 3: Using Graphs
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Short time periods
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
What does this look like?
A
Z
20
4020
30
10
A + 2Z = 40
Feasible area
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
The Graph
A
Z
20
4020
30
10
6A + 5Z = 150
Feasible area
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Feasible Solutions
A
Z
20
4020
30
10
A + 2Z = 40
6A + 5Z = 150
Feasible Area
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
LP - Minimisation
The previous example tried to maximise profit contributionbut the technique can also be used for finding
minimum cost solutions
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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:
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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.
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
Graphical Representation
A
B
30
9020
50
Feasible Area
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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
Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE
ISBN 1-86152-991-0 © Cengage
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