Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Chapter 3: Using Graphs

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © 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 © 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 Data used by kind permission of the National Gallery, New Media Department:

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

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Short time periods

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © Cengage Graphing Break-even Firstly we need to identify costs Then revenues In 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 © 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 5X If the fixed cost is 120 and unit cost is 2, then the Cost function is X Revenue = Cost 5X = X 3X = 120 X = 40 This is the break-even production figure.

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

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

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Linear Programming Linear programming is a technique which seeks the optimum allocation of scarce resources between competing products or activities It has been used in a wide range of situations in business, government and industry. Examples include: optimum product mix media selection share portfolio selection

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © 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 time Zeonites require 2 hours of labour time Total labour hours per week is 40 Ambers require 6 litres of moulding fluid Zeonites require 5 litres of moulding fluid Maximum moulding fluid per week is 150 litres Profit contribution from Ambers is £2 Profit contribution from Zeonites is £3

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © 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 available We know that Ambers take 1 hour each And Zeonites take 2 hours each

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage What does this look like? A Z A + 2Z = 40 Feasible area

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage More Limitations Moulding Fluid: We only have 150 litres per week of moulding fluid available We know that Ambers take 6 litres each And 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 © Cengage The Graph A Z A + 5Z = 150 Feasible area

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 andZ >= 0 In terms of a graph, this means that we work in the first quadrant i.e. The one where both variables are positive.

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 © Cengage Feasible Solutions A Z A + 2Z = 40 6A + 5Z = 150 Feasible Area

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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 = 240 Subtract the first equation from this one 7Z = 90 Z = Substituting gives: A =

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

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage LP - Minimisation The previous example tried to maximise profit contribution but the technique can also be used for finding minimum cost solutions

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage LP – Minimisation (2) A company has 2 machines, A & B which can each produce either the MINI or MAXI version of their product A can produce 5 MINI or 1 MAXI per session B can produce 2 MINI or 3 MAXI per session Contracts dictate that the minimum number of MINI’s must be 100 of MAXI’s must be 90 The cost of running machine A is £1000 per session The cost of running machine B is £2000 per session What is the minimum cost number of sessions for each machine? EXAMPLE:

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Formulation We can construct equations as follows: Number of MINI’s 5A + 2B >= 100 Number of MAXI’s A + 3B >=90 A, B >= 0 Costs: Minimise 1000A B Again we can use a graph.

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Graphical Representation A B Feasible Area

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © Cengage Corners & Solution The corners of the feasible area are at: A = 0, B = 50 A = 90, B = 0 AndA = 9.23, B = 26.9 (0,50)(90, 0)(9.23,26.9) The cost function is: 1000A B £100,000£90,000£63,030 MINIMUM

Jon Curwin and Roger Slater, QUANTITATIVE METHODS: A SHORT COURSE ISBN © 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, but in the long run, managers need to address the resource constraints themselves if they wish to increase production levels