CHAPTER Dealing with curves without going round the bend 3 Chapter objectives This chapter will help you to: ■ deal with types of non-linear equations ■ interpret and analyse non-linear business models ■ apply differential calculus to non-linear business models ■ use the Economic Order Quantity (EOQ) model for stock control ■ become acquainted with business uses of EOQ model In the last chapter we looked at how linear equations, equations that represent straight lines, can be used to model business situations and solve business problems. Although important, their limitation is that the relationships they are used to model need to be linear for them to be appropriate. There are circumstances when this is not the case, for instance the sort of model economists use to represent the connection between volume of production and cost per unit. In such a model economies of scale mean that the average cost of production per unit gets lower as output increases. The equation representing the situation would be non-linear and we might be interested in analysing it to find the least cost level of output. 82 Quantitative methods for business Chapter 3 Example 3.1 A train operating company sets ticket prices using the equation: y ϭ 0.8 ϩ 0.2x where y is the ticket price in £ and x is the number of miles travelled. To use this equation to work out the cost of a ticket for a 4-mile journey we simply substitute the 4 for x: y ϭ 0.8 ϩ 0.2 * 4 ϭ 0.8 ϩ 0.8 ϭ £1.60 The cost of a 5-mile journey will be: y ϭ 0.8 ϩ 0.2 * 5 ϭ 0.8 ϩ 1.0 ϭ £1.80 The cost of a 10-mile journey will be: y ϭ 0.8 ϩ 0.2 * 10 ϭ 0.8 ϩ 2.0 ϭ £2.80 In this chapter we will look at the features of basic non-linear equa- tions and use them to analyse business operations. Following this we will consider how to find optimal points in non-linear business models using simple calculus. Later in the chapter you will meet the Economic Order Quantity model, a non-linear business model that organizations can use in determining their best stock ordering policy. 3.1 Simple forms of non-linear equations One thing you might have noticed about the linear equations that fea- tured in Chapter 2 was the absence of powers. We met terms like 60Q and 0.08x but not 3Q 2 or 3/x. The presence of powers (or for that matter other non-linear forms like sines and cosines, although we will not be concerned with them here) distinguishes a non-linear equation. In order to appreciate why, consider two possible relationships between x and y: y ϭ x y ϭ x 2 In the first case we have a linear equation: y will increase at the same pace with x however big x is. If x is 4, y will be 4. If x goes up to 5, so will y. If x is 10 and goes up to 11, so will y. Even if we had something that looks more elaborate, the effect on y that is caused by a change in x is the same, whether x is small or large. If an equation is not linear the size of the change in y that comes about when x changes does depend on how big x is. With a non-linear equation a one-unit increase in x when x is small may result in a modest change in y whereas a one-unit change in x when x is large may cause a much larger change in y. Chapter 3 Dealing with curves without going round the bend 83 The cost of an 11-mile journey will be: y ϭ 0.8 ϩ 0.2 * 11 ϭ 0.8 ϩ 2.2 ϭ £3.00 Notice that the difference an extra mile makes to the cost, £0.20, is the same whether the difference is between 4 and 5 miles or between 10 and 11 miles. This is because the equation is linear; the slope is constant so the rate at which the value of y changes when x is changed is the same however big or small the value of x. The equation is plotted in Figure 3.1. 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 x (miles travelled) y (ticket price in £) Figure 3.1 The ticket price equation in Example 3.1 Example 3.2 The inventor of the new Slugar household labour-saving gadget prepares a business plan to attract investors for the venture. She anticipates that over time sales of the prod- uct will grow according to the equation: y ϭ 2 ϩ x 2 where y is the sales in thousands of units and x is the number of years elapsed since the product launch. Show the expected sales growth over 9 years graphically. An equation that includes x to the power two is called a quadratic equation, derived from the Latin word quadrare, which means to square. Similarly an equation that includes x to the power three is known as cubic. You may also meet reciprocal or hyperbolic equations. These include x to a negative power, for instance: y ϭ x Ϫ1 ϭ 1/x 84 Quantitative methods for business Chapter 3 To plot a linear equation you only need two points since the line is straight. To plot a non-linear equation we need a series of points that track the path of the curve that rep- resents it. This entails calculating y values using the range of x values in which we are interested, in this case from 0 (product launch) to 9. These points are plotted in Figure 3.2. x (years since product launch) y (sales in 000s) 02 13 26 311 418 527 638 751 866 983 0 10 20 30 40 50 60 70 80 90 0610 x (years since product launch) y (sales in 000) 482 Figure 3.2 The sales growth equation in Example 3.2 Chapter 3 Dealing with curves without going round the bend 85 Example 3.3 An economist studying the market for a certain type of digital camera concludes that the relationship between demand for the camera and its price can be represented by the equation: y ϭ 800/x where y is the demand in thousands of units and x is the price in £. To plot this equation we need a series of points such as the following: The equation is plotted in Figure 3.3. x (price in £) y (demand in 000s) 100 8.000 200 4.000 300 2.667 400 2.000 500 1.600 600 1.333 700 1.143 800 1.000 0 1 2 3 4 5 6 7 8 9 0 200 400 600 800 1000 x (price in £) y (demand in 000) Figure 3.3 The demand equation in Example 3.3 Some curves feature peaks and troughs, known as maximum and minimum points respectively. These sorts of points are often of particu- lar interest as they may represent a maximum revenue or a minimum cost, indeed later in the chapter we will be looking at how such points can be identified exactly using calculus. Figure 3.4 show the sort of curve that economists might use to repre- sent economies of scale. The minimum point represents the minimum cost per unit and the point below it on the horizontal axis the level of output that should be produced if the firm wants to produce at that cost. Other models economists use include maximum points. 86 Quantitative methods for business Chapter 3 Example 3.4 The project manager of the new Machinar car plant suggests to the board of directors that the production costs per car will depend on the number of cars produced accord- ing to the equation: y ϭ x 2 Ϫ 6x ϩ 11 where y is the cost per car in thousands of pounds and x is the number of cars produced in millions. The equation is plotted in Figure 3.4. 0 2 4 6 8 10 12 01357246 x (millions of cars) y (cost per car in £000) Figure 3.4 The curve representing the equation in Example 3.4 Example 3.5 Pustinia plc sell adventure holidays. The company accountant believes that the rela- tionship between the prices at which they could sell their holidays and the total revenue that the firm could earn is defined by the equation: y ϭϪx 2 ϩ 4x where y is the total revenue in millions of pounds and x is the price per holiday in thousands of pounds. Chapter 3 Dealing with curves without going round the bend 87 The equation is plotted in Figure 3.5. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 01 324 x (price in £000) y (revenue in £m) Figure 3.5 The curve representing the equation in Example 3.5 The maximum point in Figure 3.5 represents the maximum revenue the firm could earn and the point below it on the horizontal axis the price the firm should set in order to maximize its revenue. 3.2 Basic differential calculus We can find the approximate location of the maximum and minimum points in Examples 3.4 and 3.5 by studying the graphs carefully, marking the maximum or minimum point then identifying the values of x and y at which the point is located using the scales along the axes. At best this would give us an idea of where the point lies in relation to x and y, but it is almost impossible to pinpoint it accurately by inspecting the graph. To find the precise location of maximum and minimum points we can use techniques from the branch of mathematics known as calculus. The word calculus conveys an impression of mystery to some people. In fact it, like the word calculate, is derived from the Latin word calculare which means to reckon with little stones, a reflection of the method of counting in ancient times. Calculare is related to the Latin word calx which means a stone, the source of the word calcium. Calculus has two branches, differential calculus and integral calculus. The former, which involves the process of differentiation, is about finding how curves change, whereas the latter is about finding areas underneath curves. Our concern in this section is with differentiation. If you would like to find out about integration you may find Croft and Davison (2003) helpful. Differentiation is concerned with slopes, and slopes in equations reflect the way that the x variable changes the y variable. In a simple equation such as: y ϭ 5 ϩ 3x the slope, ϩ3, tells us that an increase of one in the value of x will result in the value of y increasing by three. The slope is the rate of change in y that is brought about by a unit change in x. The other number in the equation, 5, is a constant or fixed component; whatever the value of x, the amount added to three lots of x to get y is always 5. The slopes of more elaborate equations are not so straightforward. If you look carefully at Figure 3.5 you will see that the slope changes as the line moves from left to right across the graph. It begins by climbing upward then reaches a maximum before altering course to descend. To start with it has a positive slope then at the maximum it has a zero slope, it ‘stands still’, and finally it has a negative slope. The nature of the slope therefore depends on where we are along the horizontal scale, in other words, the value of x. For the lower values of x the slope is positive, for the higher ones it is negative. Figure 3.4 shows a similar pattern. To begin with the slope is down- wards, or negative, then it ‘bottoms out’ at a minimum and finally becomes upwards, or positive. In this case the minimum is the point where the slope is momentarily zero, it is a point of transition between the negative-sloping and positive-sloping parts of the curve. The max- imum point in Figure 3.5 is a similar point of transition, or turning point in the line. So how does differential calculus help us find slopes? It consists of a fairly mechanical procedure that is applied to all the components in an equation that contains x in one form or another; either simply with a coefficient, like 3x, or perhaps raised to a power, like x 2 . The procedure involves taking the power to which x is raised and making it the new coefficient on x then reducing the original power by one. When this is applied to x 2 the result is 2x. The power, 2, is placed in front of the x, forming the coefficient, and the power reduced from 2 to 1: x 2 becomes 2x 2Ϫ1 or simply 2x This means that for the equation: y ϭ x 2 88 Quantitative methods for business Chapter 3 the slope, or the rate at which y changes in response to a unit change in x, is 2x. This is the difference to y that a change in x makes, the differen- tial of the equation, which is represented as dy/d x: You may find it helpful to associate the process of differentiation as finding such a difference. A more exact definition is that it is the marginal change in y resulting from an infinitely small marginal change in x. Because the differential is derived from the original equation it is also known as the derivative. If the expression already includes a coefficient for x, like 2x 2 , we multi- ply the power by the existing coefficient to give us the new coefficient: In this case the differential includes x, so the slope of the equation depends on the size of x; whatever x is, the rate of change in y arising from a change in x is four times the value of x. If we apply the same pro- cedure to 3x, the result is a constant, 3. In carrying out the differentia- tion of 3x remember that 3x is 3 times x to the power one so the power reduces to zero, and that any quantity raised to the power zero is one: In this case the slope is simply 3, and does not depend on the value of x: whether x is small or large the slope will always be 3. The other type of differentiation result that you should know con- cerns cases where x is raised to a negative power, such as the reciprocal of x, 1/x, which can be written as x Ϫ1 . The process is the same, take the original power and multiply it by the existing coefficient then reduce the power by one: When you differentiate expressions like this the key is to remember that a constant divided by x raised to a positive power is simply the con- stant times x raised to the negative power. When x is taken above the line the power becomes negative. Some of the equations you may have to differentiate will consist of several different parts, not necessarily all involving x. For the types of equation we shall look at you need only deal with the parts one at a time to reach the differential. y x x y x xx x 2 2 d d 2*2 4 2 2213 3 ϭϭ ϭϪ ϭϪ ϭ Ϫ ϪϪϪϪ ( ) 4 yx y x xx 3 d d 1* 3 3 3 1110 ϭϭϭϭ Ϫ yx y x xx 2 d d 2*2 4 2 ϭϭϭ yx y x x d d 2 2 ϭϭ Chapter 3 Dealing with curves without going round the bend 89 Note that in Example 3.6 the constant of 11 is not represented in the derivative. The derivative tells us how y changes with respect to x. When x changes so will x 2 and 6x but the constant remains 11. At this point you may find it useful to try Review Question 3.1 at the end of the chapter. When an equation has a turning point, a maximum or a minimum, we can use the differential to find the exact position of the turning point. At the turning point the slope is zero, in other words the differ- ential, the rate of change in y with respect to x, is zero. Once we know the differential we can find the value of x at which the slope is zero sim- ply by equating it to zero and solving the resulting equation. 90 Quantitative methods for business Chapter 3 Example 3.6 The production cost equation for the car plant in Example 3.4 was: y ϭ x 2 Ϫ 6x ϩ 11 where y is the cost per car in thousands of pounds and x is the number of cars produced in millions. Differentiate this equation. d d 2 6 2 6 21 11 y x xxxϭϪϭϪ ϪϪ Example 3.7 Find the location of the turning point of the production cost equation for the car plant in Example 3.4. From Example 3.6 we know that the derivative is: The value of x at which this is equal to zero is the position of the turning point along the horizontal axis: 2x Ϫ 6 ϭ 0so2x ϭ 6 and x ϭ 3 The turning point is located above 3 on the horizontal axis. If you look back to Figure 3.4 you can see that the plotted curve reaches its minimum at that point. We can conclude that the minimum production cost per car will be achieved when 3 million cars are produced. d d 2 6 y x xϭϪ [...]... It costs the council £50 per litre to store the paint for a year Each time a purchase order for paint is processed the cost to the council is £20 What is the optimal order quantity of paint that will allow the council to minimize its total stock cost? 106 Quantitative methods for business 3. 13 3.14 3. 15 3. 16 3. 17 3. 18 Chapter 3 The annual demand for dog biscuits at Dogwatch Security amounts to 1690... 633 – 636 3. 1* Differentiate each of the equations listed below on the left and match your answers to the derivatives listed on the right dy Ϫ1 ϭ 2 dx x (i) y ϭ 3x (a) (ii) y ϭ 4x2 (b) dy ϭ 2x 3 dx (iii) y ϭ 2x3 (c) dy ϭ 8x dx (iv) y ϭ 1 x (d) dy ϭ 2x ϩ 4 dx (v) y ϭ x4 2 (e) dy Ϫ10 ϭ 3 Ϫ1 dx x (vi) y ϭ 3x3 ϩ 2 (f) dy 3 dx (vii) y ϭ x2 ϩ 4x (g) dy ϭ 6x 2 dx 5 Ϫx x2 (h) dy Ϫ2 ϭ 2 ϩ 10x dx 3x 2 ϩ 5x 2 3x... stock-holding cost for a given order quantity, for instance if the order quantity is 5 the total annual stock-holding cost will be £125 The expression is plotted in Figure 3. 7 Total annual stock-holding cost in £ 400 35 0 30 0 250 200 150 100 50 0 0 5 10 Q (Order quantity) 15 Figure 3. 7 Total annual stock-holding cost and order quantity in Example 3. 10 96 Quantitative methods for business Chapter 3 The cost... y ϭ2 dx 2 The inclusion of the 2s on the left hand side signifies that the process of differentiation has been applied twice in reaching the result 92 Quantitative methods for business Chapter 3 Example 3. 9 The total revenue for the firm in Example 3. 5 was: y ϭ Ϫx2 ϩ 4x where y is the total revenue in millions of pounds and x is the price per holiday in thousands of pounds Find the first order derivative... dx 3x 2 ϩ 5x 2 3x (i) dy ϭ 16x 3 ϩ 9x 2 dx (viii) y ϭ (ix) y ϭ dy ϭ 9x 2 dx Ackno Fenestrations manufacture garden conservatories to order Their production costs per unit vary with the number of conservatories they produce per month according to the equation: (x) y ϭ 4x4 ϩ 3x3 3. 2* ( j) y ϭ 2x2 Ϫ 280x ϩ 16000 Chapter 3 Dealing with curves without going round the bend 3. 3 where y is the cost per conservatory... was set up to manufacture motorcycles, jet skis and snowmobiles for the American market and had a troubled early history The way Butt used EOQ as a reference point is described in Schonberger (1982) Hall (1987) provides a fuller account of the improvements introduced at the Kawasaki factory 102 Quantitative methods for business Chapter 3 Review questions Answers to these questions, including fully... want to set a price for a season ticket that will maximize their revenue from season tickets 104 Quantitative methods for business Chapter 3 Past experience suggests that total season ticket revenue in £, y, is related to the price in £, x, in keeping with the equation: y ϭ Ϫ3x2 ϩ 30 00x 3. 7 (a) What season ticket price will maximize revenue? (b) Show that the revenue equation will be maximized at this... stock costs for a specific order quantity, for instance 5 gallons: Total annual stock cost ϭ 30 00 ϩ 25 * 5 ϭ 600 ϩ 125 ϭ £725 5 Figure 3. 9 shows the relationship between total annual stock cost and order quantity The uppermost line in Figure 3. 9 represents the total annual stock cost The lower straight line represents the total annual stock holding cost, the lower curve represents Chapter 3 Dealing with.. .Chapter 3 Dealing with curves without going round the bend 91 The cost per car at that level of production is something we can establish by inserting the value of x at the turning point into the original production cost equation: y ϭ x2 Ϫ 6x ϩ 11 Minimum cost ϭ 32 Ϫ 6 * 3 ϩ 11 ϭ 9 Ϫ 18 ϩ 11 ϭ 2 If 3 million cars are produced the cost per car will be £2000 In Example 3. 7 we were able... Figure 3. 6 The repeating saw-tooth pattern that you can see in Figure 3. 6 consists of a series of vertical lines each of which represents a delivery of the order quantity, Q The stock level peaks at Q at the point of each delivery and then declines at a constant rate as the material is taken out of the store and used until the stock level is zero and another delivery comes in Quantitative methods for business . and solving the resulting equation. 90 Quantitative methods for business Chapter 3 Example 3. 6 The production cost equation for the car plant in Example 3. 4 was: y ϭ x 2 Ϫ 6x ϩ 11 where y is the. twice in reaching the result. d d 2 2 2 y x ϭ 92 Quantitative methods for business Chapter 3 Example 3. 9 The total revenue for the firm in Example 3. 5 was: y ϭϪx 2 ϩ 4x where y is the total revenue. following: The equation is plotted in Figure 3. 3. x (price in £) y (demand in 000s) 100 8.000 200 4.000 30 0 2.667 400 2.000 500 1.600 600 1 .33 3 700 1.1 43 800 1.000 0 1 2 3 4 5 6 7 8 9 0 200 400 600 800 1000 x