The application of linear programming to management accounting

Transcription

1 The application of linear programming to management accounting After studying this chapter, you should be able to: formulate the linear programming model and calculate marginal rates of substitution and opportunity costs using the graphical approach; construct the initial tableau using the Simplex method; explain the meaning of the entries in each column of the final tableau; describe how linear programming can be used in decision-making, planning and control; formulate the linear programming model that will maximize net present value; identify the major deficiencies of linear programming. In the previous chapters we have seen that there is an opportunity cost for scarce resources that should be included in the relevant cost calculation for decision-making and variance calculations. Our previous discussions, however, have assumed that output is limited by one scarce resource, but in practice several resources may be scarce. The opportunity costs of these scarce resources can be determined by the use of linear programming techniques. Our objective in this chapter is to examine linear programming techniques and to consider how they can be applied to some specific types of decisions that a firm may have to make. In Chapter 10 we considered how accounting information should be used to ensure that scarce resources are efficiently allocated. We established that where a scarce resource exists, that has alternative uses, the contribution per unit should be calculated for each of these uses. The available capacity for this resource is then allocated to the alternative uses on the basis of contribution per scarce resource. A typical problem is presented in Exhibit If we follow the procedure suggested in Chapter 10, we can ascertain the contribution per unit of the scarce resource. Product Y yields a contribution of 14 and uses 6 scarce labour hours. Hence the contribution is 2.33 per labour hour. Similarly, the contribution per labour hour for product Z is 2. The company should therefore allocate scarce labour hours to the manufacture of product Y. Sales of product Y, however, are limited to 420 units, which means that 2520 labour hours (420 units at 6 hours per unit) will be used. The remaining 360 hours will then be allocated to product z. As one unit of product Z requires 8 labour hours, the total output of product Z will be 45 units. Profits will be maximized when the firm manufactures 420 units of product Y and 45 units of product Z. This will give a total contribution of 6600, which is calculated as follows: Single-resource constraints SINGLE RESOURCE CONSTRAINTS C. Drury, Management and Cost Accounting J.C. Drury 1992

2 Exhibit 24.1 A single resource constraint problem LP is a manufacturing company which currently produces two products. The standards per unit of product are as follows: Product Y ( ) ( ) Product Z ( ) ( ) Standard selling price 38 Standard selling price 42 Less Standard cost: Less Standard cost: Materials {8 units at 1) 8 Materials (4 units at 1) 4 Labour (6 hours at 2) 12 Labour {8 hours at 2) 16 Variable overhead Variable overhead (4 machine hours at 1) 4 (6 machine hours at 1) Contribution 14 Contribution 16 During the next accounting period, it is expected that the availability of labour hours will be restricted to 2880 hours. The remaining production inputs are not scarce, but the marketing manager estimates that the maximum sales potential for product Y is 420 units. There is no sales limitation for product Z. Two-resource constraints Linear programming ( ) 420 units of Y at a contribution of 14 per unit units of Z at a contribution of 16 per unit Where more than one scarce resource exists, the optimum production programme cannot easily be established by the process previously outlined. Consider the situation in Exhibit 24.1, where there is an additional scarce resource besides labour. Let us assume that both products Y and Z use a common item of material and that the supply of this material in the next accounting period is restricted to 3440 units. There are now two scarce resources - labour and materials. If we apply the procedure outlined above, the contributions per unit of scarce resource would be as follows: Labour Material Product Y (14/8) Product Z (16/4) This analysis shows that product Y yields the largest contribution per labour hour, and product Z yields the largest contribution per unit of scarce materials, but there is no clear indication of how the quantity of scarce resources should be allocated to each product. In such circumstances there is a need to resort to higher-powered mathematical techniques to establish the optimal output programme. Linear programming is a powerful mathematical technique that can be applied to the problem of rationing limited facilities and resources among many alternative uses in such a way that the optimum benefit can be derived from their utilization. It seeks to find a feasible combination of output that will maximize or minimize the objective function. The objective function refers to the quantification of an objective, and usually takes the form of maximizing profits or minimizing costs. Linear programming may be used when relationships can be assumed to be linear and where an optimal solution does in fact exist. To comply with the linearity assumption, it must be assumed that the contribution per unit for each product and the utilization of resources per unit are the same whatever quantity of output is produced and sold within the output range being considered. It must also be assumed that units produced and resources allocated are infinitely divisible. This means that an optimal plan that suggests we should produce units is possible. However, it will be necessary to interpret the plan as a production of 94 units APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

3 LP is a manufacturing company that currently makes two products. The standards per unit of product are as follows: ( ) ( ) ( ) ( ) Product Y Product Z Standard selling price 38 Standard selling price 42 Less standard cost: Less standard cost: Materials (8 units at 1) 8 Materials (4 units at 1) 4 Labour (6 hours at 2) 12 Labour (8 hours at 2) 16 Variable overhead Variable overhead (4 machine hours at 1) 4 (6 machine hours at 1) Contribution 14 Contribution 16 Exhibit 24.2 Multiple resource constraint problem During the next accounting period, the availability of resources are expected to be subject to the following limitations: labour Materials Machine capacity 2800 hours 3440 units 2760 hours The marketing manager estimates that the maximum sales potential for product Y is limited to 420 units. There is no sales limitation for product Z. You are asked to advise how these limited facilities and resources can best be used so as to gain the optimum benefit from them. Let us now apply this technique to the problem outlined in Exhibit 24.1, where there is a labour restriction plus a limitation on the availability of materials and machine hours. The revised problem is presented in Exhibit The procedure that we should follow to solve this problem is, first, to formulate the problem algebraically, with Y denoting the number of units of product Y and Z the number of units of product Z that are manufactured by the company. Secondly, we must specify the objective function, which in this example is to maximize contribution (denoted by C), followed by the input constraints. We can now formulate the linear programming model as follows: Maximize C = 14Y + 16Z subject to 8Y + 4Z ~ 3440 (material constraint) 6Y + 8Z ~ 2880 (labour constraint) 4Y + 6Z ~ 2760 (machine capacity constraint) 0 ~ Y ~ 420 (maximum and minimum sales limitation) Z ~ 0 (minimum sales limitation) In this model, 'maximize C' indicates that we wish to maximize contribution with an unknown number of units of Y produced, each yielding a contribution of 14 per unit, and an unknown number of units of Z produced, each yielding a contribution of 16. The labour constraint indicates that 6 hours of labour are required for each unit of product Y that is made, and 8 hours for each unit of product Z. Thus (6 hours X Y) + (8 hours x Z) cannot exceed 2880 hours. Similar reasoning applies to the other inputs. Because linear programming is nothing more than a mathematical tool for solving constrained optimization problems, nothing in the technique itself ensures that an answer will 'make sense'. For example, in a production problem, for some very unprofitable product, the optimal output level may be a negative quantity, which is clearly an impossible solution. To prevent such nonsensical results, we must include a non-negativity requirement, which is a statement that all variables in the problem must be equal to or greater than zero. We must therefore add to the model in our example the constraint that Y and Z must be greater than or equal to zero, i.e. Z ~ 0 LINEAR PROGRAMMING

4 and 0 ~ Y ~ 420. The latter expression indicates that sales of Y cannot be less than zero or greater than 420 units. The model can be solved graphically, or by the Simplex method. When no more than two products are manufactured, the graphical method can be used, but this becomes impracticable where more than two products are involved, and it is then necessary to resort to the Simplex method. Graphical method Taking the first constraint for the materials input BY + 4Z ~ 3440 means that we can make a maximum of B60 units of product Z when production of product Y is zero. The B60 units is arrived at by dividing the 3440 units of materials by the 4 units of material required for each unit of product Z. Alternatively, a maximum of 430 units of product Y can be made (3440 units divided by B units of materials) if no materials are allocated to product Z. We can therefore state that when Y = 0, Z = B60 when Z = 0, Y = 430 These items are plotted in Figure 24.1, with a straight line running from Z = 0, Y = 430 to Y = 0, Z = B60. Note that the vertical axis represents the number of units of Y produced and the horizontal axis the number of units of Z produced. The area to the left of line BY + 4Z ~ 3440 contains all possible solutions for Y and Z in this particular situation, and any point along the line connecting these two outputs represents the maximum combination of Y and Z that can be produced with not more than 3440 units of materials. Every point to the right of the line violates the material constraint. The labour constraint 6Y + BZ ~ 2BBO indicates that if production of product Z is zero, then a maximum of 4BO units of product Y can be produced (2BB0 /6), and if the output of Y is zero then 360 units of Z (2BBOIB) can be produced. We can now draw a second line Y = 4BO, Z = 0 to Y = 0, Z = 360, and this is illustrated in Figure The area to the left of line 6Y + BZ ~ 2BBO in this figure represents all the possible solutions that will satisfy the labour constraint. The machine input constraint is represented by Z = 0, Y = 690 and Y = 0, Z = 460, and the line indicating this constraint is illustrated in Figure The area to the left of the line 4 Y + 6Z ~ 2760 in this figure represents all the possible solutions that will satisfy the machine capacity constraint. The final constraint is that the sales output of product Y cannot exceed 420 units. This is represented by the line Y ~ 420 in Figure 24.4, and all the items below this line represent all the possible solutions that will satisfy this sales limitation. It is clear that any solution that is to fit all the constraints must occur in the shaded Quantity of Y produced, Y Quantity of Z produced, Z Figure 24.1 Constraint imposed by limitations of materials APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

5 Quantity of Y produced, Y Quantity of Z produced, Z Figure 24.2 Constraint imposed by limitations of labour. Quantity of Y produced, Y Quantity of Z produced, Z Figure 24.3 Constraint imposed by machine capacity. Quantity of Y produced, Y y ~ Quantity of Z produced, Z Figure 24.4 Constraint imposed by sales limitation of product Y GRAPHICAL METHOD

6 Quantity of Y produced, Y D Quantity of Z produced, Z Figure 24.5 Combination of Figures area ABCDE in Figure 24.5, which represents Figures combined together. The point must now be found within the area ABCDE where the contribution Cis the greatest. The maximum will occur at one of the corner points ABCDE. The objective function is C = 14Y + 16Z, and a random contribution value is chosen that will result in a line for the objective function falling within the area ABCDE. If we choose a random total contribution value equal to 2240, this could be obtained from producing 160 units of Y at 14 contribution per unit or 140 units of Z at a contribution of 16 per unit. We can therefore draw a line Z = 0, Y = 160 to Y = 0, Z = 140. This is represented by the dashed line in Figure Each point on the dashed line represents all the output combinations of Z and Y that will yield a total contribution of The dashed line is extended to the right until it touches the last corner of the boundary ABCDE. This is the optimal solution and is at point C, which indicates an output of 400 units of Y (contribution 5600) and 60 units of Z (contribution 960), giving a total contribution of The logic in the previous paragraph is illustrated in Figure The shaded area represents the feasible production area ABCDE that is outlined in Figure 24.5, and parallel, lines represent possible contributions, which take on higher values as we move to the right. If we assume that the firm's objective is to maximize total contribution, it should operate on the highest contribution curve obtainable. At the same time, it is necessary to satisfy the production constraints, which are indicated by the shaded area in Figure You will see that point C indicates the solution to the problem, since no other point within the feasible area touches such a high contribution line. It is difficult to ascertain from Figure 24.5 the exact output of each product at point C. The optimum output can be determined exactly by solving the simultaneous equations for the constraints that intersect at point C: SY + 4Z = Y + 8Z = 2880 (24.1) (24. 2) APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

7 Quantity of Y produced, Y Quantity of Z produced, Z Figure 24.6 Contribution levels from different potential combinations of products Y and Z. We can now multiply equation (24.1) by 2 and equation (24.2) by 1, giving 16Y + B2 = 6BBO 6Y + B2 = 2BBO Subtracting equation (24.4) from equation (24.3) gives loy= 4000 and so y = 400 We can now substitute this value for Y onto equation (24.3), giving (16 X 400) + BZ = 6BB0 and so 2 = 60 (24.3) (24.4) You will see from Figure 24.5 that the constraints that are binding at point Care materials and labour. It might be possible to remove these constraints and acquire additional labour and materials resources by paying a premium over and above the existing acquisition cost. How much should the company be prepared to pay? To answer this question, it is necessary to determine the optimal use from an additional unit of a scarce resource. We shall now consider how the optimum solution would change if an additional unit of materials was obtained. You can see that if we obtain additional materials, the line BY + 42 ~ 3440 in Figure 24.5 will shift upwards and the revised optimum point will fall on line CF. If one extra unit of materials is obtained, the constraints BY + 42 ~ 3440 and 6Y + B2 ~ 2BBO will still be binding, and the new optimum plan can be determined by solving the following simultaneous equations: BY + 42 = 3441 (revised materials constraint) 6Y + B2 = 2BBO (unchanged labour constraint) The revised optimal output when the above equations are solved is units of Y and 59.B5 units of z. Therefore the planned output of product Y should be increased by 0.2 units, and planned production of Z should be reduced by 0.15 units. This optimal response from an independent marginal increase in a resource is called the marginal rate of substitution. The change in contribution arising from obtaining one additional unit of materials is as follows: GRAPHICAL METHOD

8 Increase in contribution from Y (0.2 x 14) Decrease in contribution of Z (0.15 x 16) Increase in contribution ( ) 2.80 (2.40) 0.40 Therefore the value of an additional unit of materials is The value of an independent marginal increase of a scarce resource is called the opportunity co t or shadow price. We shall be considering these terms in more detail later in the chapter. You should note at this stage that for materials purchased in excess of 3440 units the company can pay up to 0.40 over and above the present acquisition cost of materials of 1 and still obtain a contribution towards fixed costs from the additional output. From a practical point of view, it is not possible to produce units of Y and units of Z. Output must be expressed in single whole units. Nevertheless, the output from the model can be used to calculate the revised optimal output if additional units of materials are obtained. Assume that 100 additional units of materials can be purchased at 1.20 per unit from an overseas supplier. Because the opportunity cost ( 0.40) is in excess of the additional acquisition cost of 0.20 per unit ( ), the company should purchase the extra materials. The marginal rates of substitution can be used to calculate the revised optimum output. The calculation is Increase Y by 20 units (100 x 0.2 units) Decrease Z by 15 units (100 x 0.15 units) Therefore the revised optimal output is 420 outputs ( ) of Y and 45 units (60-15) of Z. You will see later in this chapter that the substitution process outlined above is applicable only within a particular range of material usage. We can apply the same approach to calculate the opportunity cost of labour. If an additional labour hour is obtained, the line 6Y + 82 ::::= 2880 in Figure 24.5 will shift to the right, and the revised optimal point will fall on line CC. The constraints BY + 42 ::::= 3440 and 6Y + 82 ::::= 2880 will still be binding, and the new optimum plan can be determined by solving the following simultaneous equations: BY + 42 = 3440 (unchanged materials constraint) 6Y + 82 = 2881 (revised labour constraint) The revised optimal output when the above equations are solved is units of Y and 60.2 units of Z. Therefore the planned output of product Y should be decreased by 0.1 units and planned production of Z should be increased by 0.2 units. The opportunity cost of a scarce labour hour is Decrease in contribution from Y (0.1 x 14) Increase in contribution from Z (0.2 x 16) Increase in contribution (opportunity cost) ( ) (1.40) Simplex method Where more than two products can be manufactured using the scarce resources available, the optimum solution cannot easily be established from the graphical method. An alternative is a non-graphical solution known as the Simplex method. This method also provides additional information on opportunity costs and marginal rates of substitution that is particularly useful for decision-making, and also for planning and control. The Simplex method involves making many tedious calculations, but there are standard computer programs that will complete the task within a few minutes. The aim of this chapter is therefore not to delve into these tedious calculations but rather to provide an understanding of their nature and their implications for management accounting. Nevertheless, to provide a basic understanding of the method, the procedure for completing the calculations must be outlined, and we shall do this by applying the procedure to the problem set out in Exhibit APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

9 To apply the Simplex method, we must first formulate a model that does not include any inequalities. This is done by introducing what are called to the model. Slack variables are added to a linear programming problem to account for any constraint that is unused at the point of optimality, and one slack variable is introduced for each constraint. In our example, the company is faced with constraints on materials, labour, machine capacity and maximum sales for product Y. Therefore 5 1 is introduced to represent unused material resources, 5 2 represents unused labour hours, 5 3 represents unused machine capacity and 5 4 represents unused potential sales output. We can now express the model for Exhibit 24.2 in terms of equalities rather than inequalities: Maximize C = 14Y + 16Z subject to BY + 4Z = 3440 (materials constraint) 6Y + 8Z + 52 = 2880 (labour constraint) 4Y + 6Z + 53 = 2760 (machine capacity constraint) 1 Y + 54 = 420 (sales constraint for product Y) For labour, (6 hours X Y) + (8 hours X Z) plus any unused labour hours (52) will equal 2880 hours when the optimum solution is reached. Similar reasoning applies to the other production constraints. The sales limitation indicates that the number of units of Y sold plus any shortfall on maximum demand will equal 420 units. We shall now express all the above equations in matrix form, with the slack variables on the left-hand side: First matrix Quantity y z 51= (1) (material constraint) 5z = (2) (labour constraint) 53= (3) (machine hours constraint) 54= (4) (sales constraint) c = (5) contribution Note that the quantity column in the matrix indicates the resources available or the slack that is not taken up when production is zero. For example, the 5 1 row of the matrix indicates that 3440 units of materials are available when production is zero. Column Y indicates that 8 units of materials, 6 labour hours and 4 machine hours are required to produce 1 unit of product Y, and this will reduce the potential sales of Y by 1. You will also see from column Y that the production of 1 unit of Y will yield 14 contribution. Similar reasoning applies to column Z. Note that the entry in the contribution row (i.e. the C row) for the quantity column is zero because this first matrix is based on nil production, which gives a contribution of zero. Choosing the product The next stage is to examine the matrix to determine which product we should choose. As product Z yields the highest contribution, we should choose this, but our production is limited because of the input constraints. Materials limit us to a maximum production of 860 units (3440 units/4 per unit), labour to a maximum production of 360 units (2880 hours/8 per hour) and machine capacity to a maximum production of 460 units (2760 hours/6 per hour). We are therefore restricted to a maximum production of 360 units of product Z because of a labour constraint. The procedure that we should follow is to rearrange the equation that results in the constraint (i.e. S 2) in terms of the product we have chosen to make (i.e. product Z). Therefore equation (2), which is becomes and so SIMPLEX METHOD = Y - 8Z 8Z = Y Z = iy - ~5z

10 We now substitute this value for Z into each of the other equations appearing in the first matrix. The calculations are as follows: 5 1 = Y - 4(360 - ~y - ~5 2 ) = Y Y + ~5 2 = Y + ~5 2 5l = Y- 6(360 - ~Y- ~52) = Y ~Y +~5 2 = ~ y + ~52 C = Y + 16(360 - ~y - ~5 2 ) (1) (3) (5) = Y Y = Y Note that equation (4) in the first matrix remains unchanged because Z is not included. We can now restate the revised five equations in a second matrix: Second matrix Quantity y = l 2 (1) (material constraint) 3 I z = (2) H 53= 600 +l +J (3) (machine hours constraint) = (4) (sales constraint) c = (5) The substitution process outlined for the second matrix has become more complex, but the logical basis still remains. For example, the quantity column of the second matrix indicates that 2000 units of material are unused, 360 units of Z are to be made, 600 machine hours are still unused and sales of product Y can still be increased by another 420 units before the sales limitation is reached. The contribution row indicates that a contribution of 5760 will be obtained from the production and sale of 360 units of product Z. Column Y indicates that production of 1 unit of product Y uses up 5 units of the stock of materials, but, because no labour hours are available, i units of product Z must be released. This will release three units of materials (i x 4), 6 labour hours (~ x 8 hours) and 4~ machine hours G x 6). From this substitution process we now have 8 units of materials (5 units + 3 units), 6 labour hours and 4~ machine hours. 1 From the standard cost details in Exhibit 24.2 you can see that one unit of Y requires 8 units of materials, 6 labour hours and 4 machine hours. This substitution process thus provides the necessary resources for producing 1 unit of product Y, as well as providing an additional half an hour of machine capacity. This is because production of 1 unit of product Y requires that production of item Z be reduced by ~ units, which releases 4~ machine hours. However, product Y only requires 4 machine hours, so production of 1 unit of Y will increase the available machine capacity by half an hour. This agrees with the entry in column Y of the second matrix for machine capacity. Column Y also indicates that production of 1 unit of Y reduces the potential sales of product Y (54) by 1 unit The optimum solution is achieved when the contribution row contains only negative or zero values. Because row C contains a positive item, our current solution can be improved by choosing the product with the highest positive contribution. Thus we should choose to manufacture product Y, since this is the only positive item in the contribution row. The second matrix indicates that the contribution can be increased by 2 by substituting 1 unit of Y for ~ units of Z. We therefore obtain an additional contribution of 14 from Y but lose a 12 contribution from Z G x 16) by this substitution process. The overall result is an increased contribution of 2 by adopting this substitution process. The procedure is then repeated to formulate the third matrix. Column Y of the APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

11 second matrix indicates that we should use 5 units of materials and release i units of Z to obtain an additional unit of Y, but there are limitations in adopting this plan. The unused materials are 2000 units, and each unit of Y will require 5 units, giving a maximum production of 400 units of Y. We have 360 units of Z allocated to production, and each unit of Y requires us to release i units of Z. A maximum production of 480 units of Y (360/i) can therefore be obtained from this substitution process. There is no limitation on machine hours, since the second matrix indicates that the substitution process increases machine hours by half an hour for each unit of Y produced. The sales limitation.of Y indicates that a maximum of 420 units of Y can be produced. The following is a summary of the limitations in producing product Y: 51 (materials) = 400 units (2000/5) Z (substitution of product Z) = 480 units (360/i) 54 (maximum sales of Y) = 420 units (420/1) In other words, we merely divide the negative items in column Y into the quantity column. The first limitation we reach is 400 units, and this indicates the maximum production of Y because of the impact of the materials constraint. The procedure that we applied in formulating the second matrix is then repeated; that is, we rearrange the equation that results in the constraint (S 1) in terms of the product we have chosen to make (i.e. product Y). Therefore equation (1), which is becomes and so St = Y +!Sz 5Y = !Sz Y = 400 -!St + losz Substituting for Y in each of the other equations in the second matrix, we get the following revised equations: Z = 360- i(400-!st + 1~Sz)- ksz = lost - ] 0 Sz - ksz = 60 + lost -!Sz S3 = 600 +!(400 -!St + losz) + isz = lost + ] 0 Sz + isz = lost + ~Sz 54 = 420-1(400 -!St + losz) = 20 +!S1-1~Sz C = (400-!S1 + losz) - 2Sz = ~S1 +!Sz - 2Sz = ~51-1~52 We now restate the revised five equations in a third matrix: Third matrix Quantity 51 Sz Y= (1) -5 +w Z= (2) +zo -5 53= 800 -lo +~ 5 (3) 54= (4) c = ~ (5) The contribution row (equation (5)) contains only negative items, which signifies that the optimal solution has been reached. The quantity column for any products listed on the left-hand side of the matrix indicates the number of units of the product that should be manufactured when the optimum solution is reached. 400 units of Y and 60 units of Z should therefore be produced, giving a total contribution SIMPLEX METHOD (2) (3) (4) (5)

12 of This agrees with the results we obtained using the graphical method. When an equation appears for a slack variable, this indicates that unused resources exist. The third matrix therefore indicates that the optimal plan will result in 800 unused machine hours (5 3) and an unused sales potential of 20 units for product Y (54). The fact that there is no equation for 5 1 and 5 2 means that these are the inputs that are fully utilized and that limit further increases in output and profit. Interpreting the final matrix Exhibit 24.3 The effect of removing 1 unit of material from the optimum production programme The 5 1 column (materials) of the third matrix indicates that the materials are fully utilized. (Whenever resources appear as column headings in the final matrix, this indicates that they are fully utilized.) So, to obtain a unit of materials, the column for 5 1 indicates that we must increase production of product Z by ~~ of a unit and decrease production of product Y by ~ of a unit. The effect of removing one scarce unit of material from the production process is summarized in Exhibit Let us focus on the machine capacity column of Exhibit If we increase production of product Z by ]c 1 of a unit then more machine hours will be required, leading to the available capacity being reduced by 10 of an hour. Each unit of product Z requires 6 machine hours, so :itj of a unit will require 10 of an hour (]0 x 6). Decreasing production of product Y by ~ unit will release ~ of a machine hour, given that 1 unit of product Y requires 4 machine hours. The overall effect of this process is to reduce the available machine capacity by frj of a machine hour. Similar principles apply to the other calculations presented in Exhibit Increase product Z 5:3 $4 $1 ~ Machine Sales capacity of Y Materials Labour Contribution ( ) by i> of a unit - {o!~o x 6) -it&, X 4) - l~{~0 x 8) +2~(i> X 16) Decrease product Y byi of a unit +~(l X 4) +1 ~ =ro ~ + 1!(~ X 8) + 1!(~ X 6) - 2!(~ X 14) Net effect 1 t l +1 Nil _l_ Let us now reconcile the information set out in Exhibit 24.3 with the materials column (5 1) of the third matrix. The 5 1 column indicates that to release 1 unit of materials from the optimum production programme we should increase the output of product Z by ] 0, and decrease product Y by ~ of a unit. This substitution process will lead to the unused machine capacity being reduced by frj of a machine hour, an increase in the unfulfilled sales demand of product Y (5 4) by k of a unit and a reduction in contribution of ~. All this information is obtained frl;m column 5 1 of the third matrix, and Exhibit 24.3 provides the proof. Note that Exhibit 24.3 also proves that the substitution process that is required to obtain an additional unit of materials releases exactly 1 unit. In addition, Exhibit 24.3 indicates that the substitution process for labour gives a net effect of zero, and so no entries appear in the 5 1 column of the third matrix in respect of the labour row (i.e. 5 2). Opportunity cost The contribution row of the final matrix contains some vital information for the accountant. The figures in this row represent opportunity costs (also known as shadow prices) for the scarce factors of materials and labour. For example, the reduction in contribution from the loss of 1 unit of materials is ~ (40p) and from the loss of one labour hour is 1g ( 1.80). Our earlier studies h a~e indicated that this information is vital for decision-making, and we shall use this information again shortly to establish the relevant costs of the resources. The proof of the opportunity costs ca n be found in Exhibit From the contribution column we can see that the loss of one unit of materials leads to a loss of contribution of 40p APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

13 Management may be able to act to remove a constraint which is imposed by the shortage of a scarce resource. For example, the company might obtain substitute materials or they may purchase the materials from an overseas supplier. A situation may therefore occur where resources additional to those included in the model used to derive the optimum solution are available. In such circumstances the marginal rates of substitution specified in the final matrix can indicate the optimum use of the additional resources. However, when additional resources are available it is necessary to reverse the signs in the final matrix. The reason is that the removal of one unit of materials from the optimum production programme requires that product Z be increased by { 0 of a unit and product Y decreased by ~ of a unit. If we then decide to return released materials to the optimum production programme, we must reverse this process- that is, increase product Y by ~ of a unit and reduce product Z by { 0 of a unit. The important point to remember is that when considering the response to obtaining additional resources over and above those specified in the initial model, the signs of all the items in the final matrix must be reversed. We can now establish how we should best use an additional unit of scarce materials. Inspection of the third matrix indicates that product Y should be increased by ~ of a unit and product Z reduced by { 0, giving an additional contribution of Note that this is identical with the solution we obtained using the graphical method. Note that this process will lead to an increase in machine hours of lo hour (5 3) and a decrease in potential sales of product Y by ~ (5 4). Similarly, if we were to obtain an additional labour hour, we should increase production of Z by ~ of a unit and decrease production of product Y by 1~ of a unit, which would yield an additional contribution of These are the most efficient uses that can be obtained from additional labour and material resources. From a practical point of view, decisions will not involve the use of fractions; for example, the company LP considered here might be able to obtain 200 additional labour hours; the final matrix indicates that optimal production should be altered by increasing production of product Z by 40 units (200 x ~ of a unit) and decreasing production of product Y by 20 units. This proc-ess will lead to machine capacity being reduced by 160 hours and potential sales of product Y being increased by 20 units. Calculation of relevant costs The calculation of relevant costs is essential for decision-making. When a resource is scarce, alternative uses exist that provide a contribution. An opportunity cost is therefore incurred whenever the resource is used. The relevant cost for a scarce resource is calculated as acquisition cost of resource + opportunity cost When more than one scarce resource exists, the opportunity cost should be established using linear programming techniques. Let us now calculate the relevant costs for the resources used by the company LP. The costs are as follows: materials = 1.40 ( 1 acquisition cost plus 40p opportunity cost) labour = 3.80 ( 2 acquisition cost plus 1.80 opportunity cost) variable overheads = 1.00 ( 1 acquisition cost plus zero opportunity cost) fixed overheads = nil Because variable overheads are assumed to vary in proportion to machine hours, and because machine hours are not scarce, no opportunity costs arise for variable overheads. Fixed overheads have not been included in the model, since they do not vary in the short term with changes in activity. The relevant cost for fixed overheads is therefore zero. Substitution process when additional resources are obtained Uses of linear programming Selling different products Let us now assume that the company is contemplating selling a modified version of product Y (called product L) in a new market. The market price is 50 and the product requires 10 units input of each resource. Should this product L be manufac-

14 tured? Conventional accounting information does not provide us with the information necessary to make this decision. Product L can be made only by restricting output of Y and Z, because of the input constraints, and we need to know the opportunity costs of releasing the scarce resources to this new product. Opportunity costs were incorporated in our calculation of the relevant costs for each of the resources, and so the relevant information for the decision is as follows: Selling price of product L Less relevant costs: Materials (10 X 1.40) Labour (10 x 3.80) Variable overhead (10 X 1.00) Contribution ( ) (f) Total planned contribution will be reduced by 12 for each unit produced of product L. Maximum payment for additional scarce resources Opportunity costs provide important information in situations where a company can obtain additional scarce resources, but only at a premium. How much should the company be prepared to pay? For example, the company may be able to remove the labour constraint by paying overtime. The matrix indicates that the company can pay up to an additional 1.80 over and above the standard wage rate for each hour worked in excess of 2880 hours and still obtain a contribution from the use of this labour hour. The total contribution will therefore be improved by any additional payment below 1.80 per hour. Similarly, LP will improve the total contribution by paying up to 40p in excess of the standard material cost for units obtained in excess of 3440 units. Hence the company will increase short-term profits by paying up to 3.80 for each additional labour hour in excess of 2880 hours and up to 1.40 for units of material that are acquired in excess of 3440 units. Control Opportunity costs are also important for cost control. In Chapter 19 we noted that standard costing could be improved by incorporating opportunity costs into the variance calculations. For example, material wastage is reflected in an adverse material usage variance. The responsibility centre should therefore be identified not only with the acquisition cost of 1 per unit but also with the opportunity cost of 40p from the loss of one scarce unit of materials. This process highlights the true cost of the inefficient usage of scarce resources and encourages responsibility heads to pay special attention to the control of scarce factors of production. This approach is particularly appropriate where a firm has adopted an optimized production technology (OPT) strategy because variances arising from bottleneck operations will be reported in terms of opportunity cost rather than acquisition cost. Transfer pricing Linear programming can be used to determine the correct transfer price when goods are transferred within the divisions of a company. However, we shall not discuss the application of linear programming to transfer pricing until we have considered the principles necessary for establishing a sound transfer pricing system as this would be inappropriate. Our discussion of how linear programming can be used in setting transfer prices will be deferred to Chapter 26. Sensitivity analysis Opportunity costs are of vital importance in making management decisions, but production constraints do not exist permanently, and therefore opportunity costs cannot be regarded as permanent. There is a need to ascertain the range over which 708 APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

15 the opportunity cost applies for each input. This information can be obtained from the final matrix. For materials we merely examine the negative items for column 5 1 in the final matrix and divide each item into the quantity column as follows: y = 400/( -~) = = 800/( -/ 0 ) = The number closest to zero in this calculation (namely ) indicates by how much the availability of materials used in the model can be reduced. Given that the model was established using 3440 units of materials, the lower limit of the range is 1440 units ( ). The upper limit is determined in a similar way. We divide the positive items in column 5 1 into the quantity column as follows: Z = 601] 0 = = 20/~ = 100 The lower number in the calculation (namely 100) indicates by how much the materials can be increased. Adding this to the 3440 units of materials indicates that the upper limit of the range is 3540 units. The opportunity cost and marginal rates of substitution for materials therefore apply over the range of 1440 to 3540 units. Let us now consider the logic on which these calculations are based. The lower limit is determined by removing materials from the optimum production programme. We have previously established from the final matrix and Exhibit 24.3 that removing one unit of material from the optimum production programme means that product Y will be reduced by ~ and machine capacity will be reduced by lo of an hour. Since the final matrix indicates an output of 400 units of product Y, this reduction can only be carried out 2000 times (400/~) before the process must stop. Similarly, 800 hours of machine capacity are still unused, and the reduction process can only be carried out 8000 times (800/l 0 ) before the process must stop. Given the two constraints on reducing materials, the first constraint that is reached is the reduction of product Y. The planned usage of materials can therefore be reduced by 2000 units before the substitution process must stop. The same reasoning applies (with the signs reversed) in understanding the principles for establishing the upper limit of the range. Similar reasoning can be applied to establish that the opportunity cost and marginal rates of substitution apply for labour hours over a range of 2680 to 3880 hours. For any decisions based on scarce inputs outside the ranges specified a revised model must be formulated and a revised final matrix produced. From this matrix revised opportunity costs and marginal rates of substitution can be established. In Chapter 15 we discussed capital rationing and identified this as being a situation where there is a budget ceiling or constraint on the amount of funds that can be invested during a specific period of time. In such a situation we should select the combination of investment proposals that provide the highest net present value, rt'ji subject to the budget constraint for the period. In Chapter 15 we assumed that investment funds were restricted to one period only, but it was suggested that in... practice more than a one period constraint must be considered. Where there is multi-period capital rationing, we should use linear programming techniques to maximize the net present value. Let us consider the example set out in Exhibit 24.4 to illustrate the application of linear programming to capital rationing where there are budget constraints for three periods. 2 We can formulate the linear programming model by representing each of the projects numbered 1,..., 6 by x 1 (where j = 1,..., 6); X1 represents investment number 1, X 2.represents investment number 2, and so on. Our objective is to maximize the net present value subject to the budget constraints for each of the three periods. The model is as follows: Maximize 14X1 + 30X2 + 17X3 + 15X4 + 40Xs + 6X6 subject to:.-. '_....-~~_.,.: ~< ::-... _--.~~_:_.:;. _~::~., The use of linear,:'. ~... _. - f - - :. :...;..'"' - -:-- -. _'prqgrar:nmhlg ii't..:... -,capital: -: ~.... :bu~geting ;.. _:. -~ ~~-'-- : : :., ~. - - USE OF LINEAR PROGRAMMING IN CAPITAL BUDGETING

16 Exhibit 24.4 Multi-period capital rationing Present value Present value Present value Investment of outlay in of outlay in of outlay in Net present value project period 1 period 2 period 3 of investment ( 000) ( 000) ( 000) ( 000) The present value of the outlays for the budget constraints for each of periods 1-3 are as follows: ( 000) Period 1 35 Period 2 20 Period 3 20 You are required to formulate the linear programming model that will maximize net present value. Exhibit 24.5 Optimum values for multiperiod capital rationing problem 12X X 2 + 6X 3 + 6X X 5 + 6X = 35 (period 1 constraint) 3X Xz + 6X 3 + 2X4 + 35X5 + 10Xh + 52 = 20 (period 2 constraint) 5X 1 + 4X 2 + 6X 3 + 5X X 5 + 4Xh + 51 = 20 (period 3 constraint) o ~ xi ~ 1 (j = 1,..., 6) The final term in the model indicates that X, may take any value from 0 to 1. This ensures that a project cannot be undertaken more than once, but allows for a project to be partially accepted. The terms 5 1, 5 2 and 5 3 represent the slack variables (i.e. unused funds) for each of the three periods. It is assumed that the budgeted capital constraints are absolute and cannot be removed by project generated cash inflows. The solution to the problem is presented in Exhibit X,= 1.0 s, = X2 = o.o7 ~ = x3 = 1.0 X4 = 1.0 X5 = Xs = o.o 5:J = 0.0 Objective function = You can see from these figures that we should fully accept projects 1, 3 and 4, 7'X, of project 2, 23.5% of project 5 and zero of project 6. Substituting these values into the equations for the objective function gives a net present value of (in OOOs). The slack variables indicate the opportunity costs of the budget constraints for the various future periods. These variables indicate the estimated present value that can be gained if a budget constraint is relaxed by 1. For example, the slack variable of for period 1 indicates that the present value is expected to increase by if the budget of funds available for investment in period 1 is increased by 1, while the slack variable of indicates that present value can be expected to increase by if the budget is increased by 1 in period 2. If the budget is increased by 1000 in period 1, the present value is expected to increase by 408. The slack variables can also indicate how much it is worth paying over and above the market price of funds that are used in the net present value calculation for additional funds in each period. A further use of the opportunity costs is to help in appraising any investment projects that might be suggested as substitutes for projects 1-6. For example, assume that a new project whose cash inflows are all received in year 3 is expected to yield a net present value of 4 for an investment of 5 for each of years 1 and 2; this project should be rejected because the opportunity cost of the scarce funds will be 6 (5 x x ), and this is in excess of the net present value. So far we have assumed that capital constraints are absolute and cannot be re APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

17 moved by project-generated cash inflows. Let us now assume that project-generated cash inflows are available for investment and that the cash inflows for period 2 are 5000, 6000, 7000, 8000, 9000 and respectively for projects 1-6. The revised constraint for period 2 is 3X1 + 10X2 + 6X3 + 2X4 + 35X5 + 10X = Xt + 6Xz + 7X3 + 8X4 + 9Xs + 10X6 You can see that the cash inflows are entered on the right-hand side of the equation, and thus increase the amount of funds available for investment. The same approach should be adopted for cash infjows arising in periods 1 and 3. Note that it is assumed that any unused funds cannot be carried forward and used in future periods. For an illustration of how unused funds can be carried forward to future periods and also how project-generated cash flows are incorporated into the LP model see answers to Questions and in the Students' Manual. Note that in formulating the model we have assumed that the investment projects were divisible in the sense that a partial acceptance of an investment proposal was possible. In the optimal solution both projects 2 and 5 were fractional. However, in practice, investment projects are unlikely to be divisible - acceptance will involve acceptance of the full amount of the investment and rejection will involve zero investment. To overcome this problem, it is possible to use an integer programming model by requiring that Xj be an integer - either 0 or 1. This process excludes fractional investments. For an illustration of integer programming see Carsberg (1969). The model can also be modified to take account of mutually exclusive projects. For example, if projects 1, 3 and 6 are mutually exclusive, we can simply add the constraint X1 + X3 + X 6 :s 1. When this constraint is used with integer programming, we are assured that only one of these projects will appear in the final solution. Also, if project 2 is contingent upon the acceptance of project 1, the constraint X2 :!S X1 ensures that the contingency is recognized in the final solution of an integer programming model. The major problem with the application of linear programming to the capital budgeting process is that it is based on the assumption that future investment opportunities are known. However, management may be aware of future investment opportunities for the earliest years only. Budget constraints for later years are likely to be utilized only as new investment proposals are generated, and they are unlikely to be binding. To overcome this problem, the selection process must be revised continually. For a more detailed discussion of the application of linear programming to capital investment decisions see Weingartner (1974). In practice, linear programming models are likely to be far more complex than the models that have been presented in this chapter. Practical linear programming... models can be very large, having at least a few hundred constraints and variables. However, these models can be quickly solved using a computer. For an example of a linear programming model developed for BP Ltd see Deam et al. (1975). A further problem with the application of linear programming is that the objective function assumes a constant contribution per unit of output, regardless of the level of activity. The implications of this are that the selling price and the variable cost per unit remain constant. It is therefore assumed that a constant quantity of resources is used per unit for all levels of output and that there will be no savings in the usage of inputs per unit of output when producing large output levels or large batches instead of small ones. In practice, labour may be more efficient when working on large production runs, or savings may be made with long production runs because there is less starting and stopping of machines. Another assumption is that all costs either vary with a single volume-related cost driver output measure or they are fixed for the period under consideration. For example, in Exhibit 24.2 we assumed that costs either varied with the number of units produced or were fixed. The singlecost-driver assumption may be valid for short-run resource allocation decisions. For decisions over longer-term horizons, however, the single-cost-driver assumption is unlikely to be appropriate. PRACTICAL PROBLEMS IN APPLYING LINEAR PROGRAMMING Practical.... problems Jn applying linear _ progra~ming 711 I

18 Application to other business problems In the linear programming models presented in this chapter it has been assumed that output and use of resources are perfectly divisible. For example, the marginal rates of substitution for input resources and the output of products included fractions. We can avoid this problem by using integer programming. Here suitable constraints are added to the linear programming model to permit acquisition of whole numbers of a resource or the production of a complete product. In addition, integer programming can also handle step fixed costs. Most of the problems with applying linear programming stem from the linearity assumption, but note that a number of statistical studies suggest that, within a fairly wide relevant range, cost functions can be reasonably approximated by linear expressions (for examples see Johnston, 1972). For a discussion of mathematical programming where there are non-linear relationships see Carsberg (1979) and Baumol (1977). In this chapter we have illustrated the use of linear programming as a method of establishing the optimum output level that maximizes profit. However, for other business problems such as transport routing production scheduling and personal - resource planning, the objective function of the linear programming model will be to minimize costs. Alternatively, management may have an objective to achieve a number of goals concurrently. In this type of situation there is no single function to be optimized - instead, we must attempt to achieve a series of goals simultaneously within specified constraints. The problem can be resolved by using a special kind of linear programming procedure known as goal programming. (For a discussion of goal programming see Salkin and Kornbluth (1973).) With goal programming, the decision-makers must either rank or weight the quantitatively expressed goals in terms of their subjecti ve opinion of their relative importance. An objective function is then formulated that incorporates this ranking or weighting. For example, we could incorporate in the model that goal 1 is of overriding importance and must be satisfied as far as possible, irrespective of other goals included in the model. Alternatively, it is possible to specify in the model that the profit goal is twice as important as a second goal. Linear programming is not just a theoretical tool; a number of firms have found it to be a useful technique for planning their activities. See Grinyer and Wooller (1975) and Moores and Hodges (1970) for evidence on the use of linear programming models. The survey by Drury eta/. (1992) of UK manufacturing companies reported that linear programming was widely used by S'X, of the respondents and occasionally used by a further 13'/'o. A number of linear programming software packages can now be purchased, and the complexity of developing in-house programmes is no longer a barrier to using linear programming to solve business problem s. SELF-ASSESSMENT QUESTIONS You should atlemptto answer these questions yourself before looking up the suggested answers, which appear on pages If any part of your answer is incorrect, check back carefully to make sure you understand where you went wrong. 1 LP Ltd is a manufacturing company that currently produces three products. The standards per unit of product are as follows: Product X Product Y Product Z ( ) ( ) ( ) ( ) ( ) ( ) Standard selling price Less standard costs: Materials 2 units at 1 = 2 3 units at 1 = 3 4 units at 1 = 4 Labour 4 hours at 1 = 4 1 hour at 1 = 1 1 hour at 1 = 1 Machine time 1 hour at 1 = hours at 1 = hour at 1 = Standard profit per unit 2 1=.!1 1 = = 712 APPUCAT10N OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

19 Note: The charge for materials and labour is based on their cost of acquisition. The charge for machine time is based on the allocation of 8000 overheads (all of which are fixed costs) over an estimated capacity of 8000 hours. During the next cost period, the availability of resources is expected to be subject to the following limitations: Materials 9000 units Labour 9200 hours Machinery 8000 hours Furthermore, the output of the products is limited because of a marketing constraint. The marketing manager estimates the maximum sales of the 3 products to be as follows: Product X 2100 units Product Y 1400 units Product Z 380 units During the next period, the standard costs and selling prices are expected to remain unchanged. The managing director is aware that a rival company uses mathematical programming to plan its production, and has asked you to apply the same technique to the above data. You have used a computer package, and the following final linear programming model has just been received from the computer: M (Materials) SY (Sales of Y) 8 X (Product X) SX (Sales of X)' Y (Product Y) Z (Product Z) L (labour) M/c (Machinery) SZ (Sales of Z) 8 7/19 10/19 3.2/19-1/19 4/19-3/19-5/ /19 5/ / ,'19 3/ /19-13/19-14/19 a SX, SY and SZ relate to slack variables for the sales demand for each product Required: (a) Prepare the first tableau of the linear programming model to which the solution is given. (5 marks) (b) The managing director has requested an explanation of the meaning of the final tableau You are required to provide an explanation of the meaning of each of the terms in the final tableau, and also to provide any other information that would be of use that can be derived from the final tableau. (14 marks) (c) Discuss briefly the possible uses of linear programming for short-term planning, and indicate whether there are any limitations in using a model of this kind. (6 marks) 2 Brass Ltd produces two products, the Masso and the Russo. Budgeted data relating to these products on a unit basis for August 1980 are as follows: Masso Russo ( ) ( ) Selling price Materials Salesmen's commission Each unit of product mcurs costs of machining and assembly. The total capacity available in August 1980 is budgeted to be 700 hours of machining and 1000 hours of assembly, the cost of this capacity being fixed at 7000 and respectively for the month, whatever the level of usage made of it. The number of hours required in each of these departments to complete one unit of outpults as follows: Machining Assembly Masso Russo Under the terms of special controls recently introduced by the Government in accordance with EEC requirements, selling prices are fixed and the maximum permitted output of either product in August is 400 units (i.e. Brass Ltd may produce a maximum of 800 units of product). At the present controlled selling prices the demand for the products exceeds this considerably. APPLICATION TO OTHER BUSINESS PROBLEMS

20 You are required: (a) to calculate Brass Ltd's optimal production plan for August 1980, and the profit earned, (10 marks) (b) to calculate the value to Brass Ltd of an independent marginal increase in the available capacity for each of machining and assembly, assuming that the capacity of the other department is not altered and the output maxima continue to apply, (10 marks) (c) to state the principal assumptions underlying your calculations In (a) and (b) above, and to assess their general significance. (5 marks) ICAEW Management Accounting July 1980 SUMMARY When there is more than one scarce input factor, linear programming can be used to determine the production programme that maximizes total contribution. This information can be obtained by using either a graphical approach or the Simplex method. The graphical approach, however, is inappropriate where more than two products can be produced from the scarce inputs, and in such a situation the Simplex method should be used. This method has the added advantage that the output from the model provides details of the opportunity costs and the marginal rates of substitutions for the scarce resources. Linear programming can be applied to a variety of management accounting problems. In particular, the technique enables the relevant costs of production inputs to be computed. This information can be used for decision-making, standard costing variance analysis and the setting of transfer prices in divisionalized companies. It can also be applied to capital budgeting in multi-period capital rationing situations. Linear programming has a number of limitations when applied to real world situations, but some of these problems can be overcomes by establishing more complex models and using integer programming techniques. KEY TERMS AND CONCEPTS capital rationing (p. 709) goal programming (p. 712) integer programming (p. 711) linear programming (p. 696) marginal rate of substitution (p. 701) objective function (p. 696) opportunity cost (p. 702) shadow price (p. 702) simplex method (p. 702) slack variables (p. 703). RECOMMENDED READING For an illustration of how linear programming can be solved using a computer application see Clarke (1984). You should refer to Carsberg (1979) for a discussion of mathematical programming when there are non-linear relationships. lt is also recommended that you read Dev (1980). This article develops some aspects of linear programming further using the graphical approach. REFERENCES AND FURTHER READING Barron, M.J. (1972) Linear programming dual prices in management accounting, Journal of Business Finance, Spring, Baumol, W.J. (1977) Economic Tlleory and Operatron Analysis, Prentice-Hall. Bernhard, R.N. (1968) Some problems in applying mathematical programming to opportunity costing, Journal of Accounting Researcll, 6(1), Spring, Bhaskar, K.N. (1974) Borrowing and lending in a mathematical programming model of capital budgeting, Journal of Business Finauce and Accounting, 1(2), Summer, Bhaskar, K.N. (1976) Linear programming and capital budgeting: A reappraisal, journal of Business Fiunnce and Accountiug, 3(3), Autumn, Bhaskar, K.N. (1978) Linear programming and capital budgeting: the financing problem, journal of Business Finm1ce nnd Accouutmg, 5(2), Summer, Carsberg, B. (1969a) Att Introduction to Matllematical Programming for Accountants, Allen & Unwin, pp Carsberg, B. (1969b) On the linear programming approach to asset valuation, ]oumal of Accounting Research, 7(2), Autumn, Carsberg, B. (1979) Economics of Business Decisio11s, Penguin, Chs 11 and 12. Clarke, P. (1984) Optimal solution? Try the linear programming way, Accollntallctj, December, Deam, R.J., Bennett, J. W. and Leather, J. (1975) Firm: A Computer Model for Fimmcial Plmming, lnstitute of Chartered Accountants in England and Wales Research Committee Occasional Paper No APPLICATlON OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

21 Dev, S. (1978) Linear programming dual prices in management accounting and their interpretation, Accounting and Business Research, 9(33), Winter, Dev, S. (1980) Linear programaung and production planning, in Topics in Mnnageme11t Accounting (eds J. Arnold, B. Carsberg and R. Scapens), Philip Allan. Drury, C., Braund, S., Osborne, P. and Tayles, M. (1992) A Survey of Management Accounting Practices i11 UK Manufacturing Companies, ACCA Research Occasional Paper, Chartered Association of Certified Accountants. Grinyer, P.H. and Wooller, J. (1975) Corporate Models Today, Institute of Chartered Accountants in England and Wales. Hughes, j.s. and Llewellen, W.G. (1974) Programming solutions to capital rationing problems, foumal of Business Finance a11d Accounti11g, 1(1), Spring, ljiri, Y., Levy, F.K. and Lyon, R.C. (1963) A linear programming model for budgeting and financial planning, journal of Accounting Research, 1(2), Autumn, Jensen, R.E. (1968) Sensitivity analysis and integer linear programming, The Accounting Ret ieu, July, Johnston, J. (1972) Statistical Cost Analysis, McGraw-Hill. Kaplan, R.S. (1982) Advn11ced Management Accounting, Prentice-Hall, Chs 5, 8, Lin, W.T. (1979) Applications of goal programming to accounting, foumal of Business Filmnce and Accozmting, 6(4), Winter, Moore, P.G. and Hodges, S.D. (1970) Programming for Optimal Decisions, Penguin. Salkin, G. and Kornbluth, J. (1973) Linear Programmi11g in Financial Plmming, Prentice-Hall, Ch. 7. Samuels, J.M. (1965) Opportunity costing: an application of mathematical programming, jouma/ of Accozmting Research, 3(2), Autumn, ; also in Studies in Cost Analysis (ed. D. Solomons), Sweet & Maxwell. Weingartner, H.M. (1974) Mathematical Programming and the Analysis of Capital Budgeting Problems, Kershaw Publishing. KEY EXAMINATION POINTS A common error is to state the objective function in terms of profit per unit. This is incorrect, because the fixed cost per unit is not constant. The objective function should be expressed in terms of contribution per unit. You should note that there are several ways of formulating the tableaux lor a linear programming model. The approach adopted in this chapter was to formulate the first tableau with positive contribution signs and negative signs lor the slack variable equations. The optimal solution occurs when the signs in the contribution row are all negative. Sometimes examination questions are set that adopt the opposite procedure. That is, the signs are the reverse of the approach presented in this chapter. For an illustration of how to cope with this situation you should refer to the answers to Questions 24.9 and in the Students Manual. Most examination questions include the final tableau and require you to interpret the figures. You may also be requ1red to formulate the initial model. It is most unlikely that you will be required to complete the calculations and prepare the final tableau. However, you may be asked to construct a graph and calculate the marginal rates of substitution and opportunity costs. QUESTIONS *Indicates that a suggested solution is to be found in the Students Manual Intermediate: Optimal output decision using contribution per scarce factor and graphical approach (a) Discuss the importance of key factors or limiting factors in an organization. (b) E Limited manufactures two products, K and L, the selling prices of which are 48 and 72 respectively. Standard cost data are as follows: Per article Direct material Direct wages, 4 per hour; Department: 1 2 Product K ( ) Product L ( ) Variable overhead Per annum Fixed overhead E Limited operates a 40-hour week for 50 weeks each year. Currently the employees in each department are: You are required to state: 12 2 Department: (1) if one product only were to be made: (i) which product would give maximum profit and the problems that you envisage could arise; 16 6 QUESTIONS

22 (ii) which product should be made and the amount of profit per annum resulting, assuming that products K and L use the same direct material and that there is a shortage of the material with supply limited at current price to a max1mum of per annum; (iii) which product should be made and the amount of profit per annum resulting, assuming that there is a shortage of persons possessing the skills required in department 2, with the result that the number of employees there cannot be increased; (2) how many of each product should be made and the amount of profit resulting, assuming that the number of employees cannot be increased or transferred from one department to another (30 marks) CIMA Cost Accounting 2 November Advanced: Optimal output using the graphical approach and the impact of an Increase In capacity A company makes two products, X andy. Product X has a contribution of 124 per unit and product Y 80 per unit. Both products pass through two departments for processing and the times in minutes per unit are: Department1 Department 2 Product X Product y Currently there is a maximum of 225 hours per week available 1n department 1 and 200 hours in department 2. The company can sell all it can produce of X but EEC quotas restrict the sale of Y to a maximum of 75 units per week. The company, which wishes to maximize contribution, currently makes and sells 30 units of X and 75 units of Y per week. The company is considering several possibilities including (i) altering the production plan if it could be proved that there is a better plan than the current one; (ii) increasing the availability of either department 1 or department 2 hours. The extra costs involved in increasing capacity are 0.5 per hour for each department; (iii) transferring some of their allowed sales quota for Product Y to another company Because of commijments the company would always retain a mimmum sales level of 30 units. You are required to (a) calculate the optimum production plan using the existing capacities and state the extra contribution that would be achieved compared with the existing plan; (8 marks) (b) advise management whether they should increase the capacity of either department 1 or department 2 and, if so, by how many hours and what the resulling increase in contribution would be over that calculated in the improved production plan (7 marks) \t) ca\c\1\a\e \Tie minimum pnce pe1 un l't \or w'n'tch \'ney could sell the rights to their quota, down to the mimmum level, given the plan in (a) as a starting point. (5 marks) (Total 20 marks) CIMA Stage 3 Management Accounting Techniques May Advanced: Maximizing profit and sales revenue using the graphical approach Goode, Billings and Prosper pic manufactures two products, Razzle and Dazzle. Unit selling prices and variable costs, and daily fixed costs are: Razzle Dazzle ( ) ( ) Selling price per unit Variable costs per unit 8 20 Contribution margin per unit Joint fixed costs per day 60 Production of the two products is restricted by limited supplies of three essential inputs: Aaz, Ma, and T az. All other inputs are available at prevailing prices without any restriction. The quantities of Aaz, Ma, and T az necessary to produce single units of Razzle and Dazzle, together w1th the total supplies available each day, are Raz Ma Taz lb per unit required Razzle Dazzle Total available (lb per day) William Billings, the sales director, advises that any combination of Razzle and/or Dazzle can be sold wijhout affecting their market prices. He also argues very strongly that the company should seek to maxim1ze its sales revenues subject to a minimum acceptable profit of 44 per day in total from these two products. In contrast, the financial director. Silas Prosper, has told the managing director, Henry Goode, that he believes in a policy of profit maximization at all times. You are required to: (a) calculate: (i) the profit and total sales revenue per day, assuming a policy of profit maximization, (10 marks) (ii) the total sales revenue per day, assuming a policy of sales revenue maximization subject to a minimum acceptable profit of 44 per day, (10 marks) (b) suggest why businessmen might choose to follow an objective of maximizing sales revenue subject to a minimum profit constraint (5 marks) 15 (Total 25 marks) /CAEW Management Accounting July Advanced: Optimal output, shadow prices and decision making using the graphical approach The 1nstruments department of Max ltd makes two products: the Xl and \he 'fm Standard revenues and costs per unit lor these products are shown below: APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

23 XL YM ( ) ( ) ( ) ( ) Selling price Variable costs: Material A ( 10 per kg) (40) (40) Direct labour ( 8 per hour) (32) (16) Plating (12 per hour) (12) (24) Other variable costs (76) (70) (160) (150) Fixed overheads (allocated at 7 per direct labour hour) _@ ~ Standard profit per unit Plating is a separate automated operation and the costs of 12 per hour are for plating materials and electricity. In any week the maximum availability of inputs is limited to the following: Material A Direct labour Plating time 120kg 100hours 50 hours A management meeting recently considered ways of increasing the profit of the instrument department. II was decided that each of the following possible changes to the existing situation should be examined independently of each other (1) The selling price of product YM could be increased. (2) Plating time could be sold as a separate service at 16 per hour. (3) A new product, ZN, could be sold at 240 per unit. Each unit would require the following: Material A Direct labour Plating time Other variable costs 90 Skg Shours 1 hour (4) Overtime could be introduced and would be paid at a premium of 50% above normal rates. Requirements: (a) Fonnulate a linear programme to determine the production policy which maximizes the profits of Max Ltd in the present situation (i.e. ignoring the alternative assumptions in 1 to 4 above), solve, and specify the optimal product mix and weekly profit. (6 marks) (b) Detennine the maximum selling price of YM at which the product mix calculated for requirement (a) would still remain optimal. (3 marks) (c) Show how the linear programme might be modified to accommodate the sale of plating time at 16 per hour (i.e. formulate but do not solve). (3 marks) (d) Using shadow prices (dual values), calculate whether product ZN would be a profitable addition to the product range. (4 marks) (e) Ignoring the possibility of extending the product range, detennine whether overtime working would be worthwhile, and if so state how many overtime hours should be worked. (3 marks) (f) Discuss the limitations of the linear programming approach to the problems of Max Ltd. (6 marks) (Total 25 marks) ICAEW P2 Management Accounting November * Advanced: Relevant material costs, optimal output and shadow prices using the graphical approach The Milton Carpet Company has been manufacturing two ranges of carpet for many years, one range for commercial use, the other for private use. The main difference between the two ranges is in the mix of wool and nylon; with the commercial range having 80% wool and 20% nylon and the private range 20% wool and 80% nylon. The designs of each range are the same and each range can be made in 5 different colours. There are variations in the cost of the dyes used for the different colours in the range, but they are all within 5% of each other so the accountant takes an average cost of dyeing in her costing. The Board has just decided to use up its remaining stocks of wool and nylon and transfer its production over to making acrylic carpets m three months' time. The company's objective is to maximize the contribution from the running down of the stocks of wool and nylon over that period subject to any operating constraints. Data concerning its present carpet range is given below. It is assumed that sufficient demand exists to ensure all production can be sold at the stated price. Selling price Manufacturing costs: Material- Wool -Nylon Direct labour Variable production costs Fixed production overheads based on200% direct labour Standard full cost Production requirements: Wool(lb) Nylon (lb) Direct machine time (hours) Private use ( ) Per roll Commercial use ( ) There are lb of wool and lb of nylon in the stores to be used up. At the end of the quarter, when it changes production to the new carpet, any wool or nylon left can be sold for 1 per lb. The production manager forecasts that the machines can operate for a total of 6600 hours during the next quarter. You are required to: (a) Formulate the above problem in a linear programing fonnat. Solve the problem and provide the production manager with the required output mix of rolls of private and commercial 36 QUESTIONS

24 carpets. State whether any of the raw material has to be sold off as scrap at the end and thus what the total contribution for the quarter's production should be. (10 marks) (b) Show whether or not it will be necessary to recompute the optimum solution if due to economic difficulties, the cost of the dyes for the carpets are to be increased by 11 0 and 40 per roll of private and commercial carpet respectively. (4 marks) (c) Describe what would happen in physical and financial terms if the availability of one of the fully utilized resources were to be increased by a small amount. (6 marks} (d) Comment on the advisability of introducing the concept of opportunity costs into the budgetary control framework, by using the output from the linear programming solution to the optimum production mix. (5 marks) (Total25 marks) ICAEW P2 Management Accounting July * Advanced: Optimal production and shadow prices using the graphical approach plus contribution range using probability theory (a) Blanton Ltd manufactures two enriched powder products known as Yin and Zee. After charging all direct costs, including the variable element of labour time and machine time, the products make a contribution per kg of output of 22 for Yin and 15 for Zee. The total fixed manufacturing capacity available in the first week of 1982 is 1200 hours of labour and 1000 hours of machine time. The number of hours of each of these resources required to produce a kilogram of output is as follows: Labour Machine time Yin 4 2 Zee 2 5 Any fraction of a kilogram can be produced for either product You are required: (I) to calculate Blanton Ltd's optimal production plan for the first week of 1982, and the contribution earned from this plan, (5 marks) (ii) to calculate the related shadow price per hour of labour time and of machine time respectively, (5 marks) (iii) to state clearly and concisely the significance of the shadow price of resources. (5 marks) (b) Hinton Ltd also has the capacity to manufacture Yin and Zee. Until recently Hinton earned the same contribution per kg of output as Blanton, and required the same amount of resources as Blanton to produce a kilogram of output. However, its weekly capacity was 800 hours of labour and 2450 hours of machine time. Its optimal production plan was to produce 400 kg of Zee and no Yin, giving a contribution of Hinton's engineering superintendent has recently made some irreversible adjustments to the machines to try to reduce the amount of machine time needed to make Zee. In addition to the fixed costs of the machines, the variable cost of machine time actual used was 4 per machine hour. As this variable cost was charged before arriving at the contribution per kg of output, any reduction in the machine time required would directly increase the contribution per unit of output of Zee. These engineering adjustments would have no effect on the production of Yin, if that were to be undertaken. The adjustments have had only limited success. A small sample of observations taken at different times has led to the estimate that the average machine time per kg of output of Zee is still expected to be 5 hours, but that due to the wide variation of machining times in the sample the standard error of this estimate is 1 hour. Your are required: (i) to calculate the range within which there is a 95.4% probability that Hinton ltd's weekly contribution will lie under the above conditions, and also the expected value of the contribution, if it continues to produce as much Zee as possible per week and no Yin, (5 marks) (ii) to calculate Hinton Ltd's optimal production plan for both Yin and Zee if the machine time required per kg of Zee produced were to be assumed to be 7 hours for certain, all other matters remaining unchanged. (5 marks) (Total 25 marks) ICAEW Management Accounting December Advanced: Optimal output, shadow prices and decision making using the graphical approach Riverside Processors pic recycles by-products from nearby industrial sites. It produces two products, Ackney and Boylle. The current level of production is 1 00 tonnes of each product per week, but there remains some unused production capacity. Due to restrictions on the availability of an essential ingredient, Zalium, the maximum weekly output of Ackney is 150 tonnes. Furthermore, the effluent from the production of Ackney is acid, but is neutralized by effluent from the production of Boylle; as a matter of policy, Riverside restricts the production of Ackney to not more than twice the production of Boylle. The production process of each of the products requires heat, fuelled by gas. However, Riverside is restricted in the quantity of gas it can use, the maximum being therms per week. The production of one tonne of either product requires 100 therms of gas. The production process of both Ackney and Boylle incorporates filtration, and the capacity of the filtration plant is limited to 1750 labour hours per week in normal operation. The production of one tonne of Ackney requires 7 labour hours of filtration, and of one tonne of Boylle, 5 labour hours. The company is test marketing a new product, Spotz; this requires neither additional process heat nor filtration, although it incorporates the scarce material, Zalium, which is also used in the production of Ackney. The following cost and revenue data are available concerning the three products (in per tonne): APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

25 Ackney Boylle Spotz (actual) (actual) (estimated) Variable costs: Input materials Zalium (one unit) Other Labour Filtration plant Other Process heat Total variable costs Selling price The company's fixed fixed costs are per week. You are required: (a) assuming that the new product. Spotz, is not produced, to: (i) show the feasible outputs of Ackney and Boylle, and to calculate the optimal production plan for Riverside Processors pic, (9 marks) (ii) calculate the shadow prices of Zalium, filtration plant labour, and process heat for the optimal production plan which you have calculated in (i) above, (5 marks) (iii) calculate the maximum extra quantity of Zalium (in addition to the 150 units which are available at 27 per unit) which the company should be willing to buy if it could be imported at a price of 50 per unit and if filtration plant throughput cannot be increased; (6 marks) (b) assuming that the test marketing campaign for Spotz is successful in demonstrating a sufficient demand at the proposed selling price, and also assuming that no imported Zalium is obtainable, to: {i) calculate the shadow pnce of Zalium, (2 marks) {ii) calculate the optimum contribution which Riverside Processors pic can then earn. (3 marks) (Total 25 marks) /CAEW Management Accounting July Advanced: Formulation of initial tableau and interpretation of final tableau Hint: Reverse signs in Final tableau D Electronics produces three models of satellite dishes-alpha, Beta and Gamma-which have contributions per unit of 400, 200 and 100 respectively. There is a two-stage production process and the number of hours per unit for each process are: Process 1 Process 2 Alpha 2 3 Beta 3 2 Gamma There is an upper limit on process hours of 1920 per period lor Process 1 and 2200 for Process 2. The Alpha dish was designed for a low-power satellite which is now fading and the sales manager thinks that sales will be no more than 200 per period. Fixed costs are per period. You are required to (a) formulate these data into a Linear Programming model using the following notation: x2 Xs x, z x 1 : number of Alphas x 2 : number of Betas x 3 : number of Gammas (5 marks) (b) formulate (but do not attempt to solve) the initial Simplex Tableau using x 4 : as Slack for Process 1 x 5 : as Slack for Process 2 x 6 : as Slack for any sales limit and describe the meaning of Slack; (c) interpret the final Simplex Tableau below (5 marks) x, x2 x3 X4 Xs Xs Solution (6 marks) (d) investigate the effect on the solution of each of the following: (i) an increase of 20 hours per period in Process 1, {ii) an increase of 10 units per period in the output of Alpha, (iii) receiving an order, which must be met, for 10 units of Gamma. (6 marks) (Total22 marks) CIMA Stage 3 Management Accounting Techniques November Advanced: Optimal output with a single limiting factor and interpretation of a final matrix Hint: Reverse the signs in the final matrix. (a) Corpach Ud manufactures three products for which the sales maxima, for the forthcoming year, are estimated to be: Product 1 Product 2 Product Summarized unit cost data are as follows: Product 1 Product 2 Product 3 ( ) ( ) ( ) Direct material cost Variable processing costs Fixed processing costs The allocation of fixed processing costs has been derived from last year's production levels and the figures may need revision if current output plans are different. The established selling prices are: Product 1 Product 2 Product The products are processed on machinery housed in three buildings: Building A contains type A machines on which 9800 machine QUEST10NS

26 hours are estimated to be available in the forthcoming year. The fixed overheads lor this building are 9800 p.a. Building B1 contains type B machines on which machine hours are estimated to be available in the forthcoming year. Building B2 also contains type B machines and again machine hours are estimated to be available in the forthcoming year. The fixed overheads for the B1 and B2 buildings are, in total, p.a. The times required lor one unit of output lor each product on each type of machine. are as follows: Type A mach1nes Type B machines Product 1 1 hour 1.5 hours Product 2 2 hours 3 hours Product3 3 hours 1 hour Assuming that Corpach ltd wishes to maximize its profits for the ensuing year. you are required to determine the optimal production plan and the profit that this should produce. {9 marks) {b) Assume that, before the plan that you have prepared in part (a) is implemented, Corpach ltd suffers a major fire which completely destroys building B2. The fire thus reduces the availability of type B machine time to hours p.a. and the estimated fixed overhead for such machines, to In all other respects the conditions set out, in part {a) to this question. continue to apply In his efforts to obtain a revised production plan the company's accountant makes use of a linear programming computer package. This package produces the following optimal tableau: z X1 X2 X3 S1 52 S3 54 ss : t t In the above: lis the total contribution, X1 is the budgeted output of product 1, X2 is the budgeted output of product 2, X3 is the budgeted output of product 3, S1 is the unsatisfied demand for product 1, S2 is the unsatisfied demand for product 2, S3 is the unsatisfied demand for product 3, S4 is the unutilized type A machine time S5 is the unutilized type B machine time. and The tableau is interpreted as follows: Optimal plan - Make 2500 units of Product 1, 1850 units of Product 2, 1200 units of Product 3, Shadow prices- Product per unit, Type A Machine Time per hour Type B Machine Time per hour. Explain the meaning of the shadow prices and cons1der how the accountant might make use of them. Calculate the profit anticipated from the revised plan and comment on its variation from the profit that you calculated in your answer to part (a). (9 marks) (c) Explain why linear programming was not necessary for the facts as set out in part (a) whereas it was required for part (b) (4 marks) (Total22 marks) GAGA Level 2 Management Accounting June Advanced: Formulation of initial tableau and interpretation of final tableau Bronx ltd is in the process of preparing its budget for the coming year. The information from the marketing department indicates that sales demand is in excess of present plant capacity. Because equipment is specialized and made to order, a lead time in excess of one year is necessary on all plant additions. Bronx Ltd produces three products and the estimated demand is as follows: Product X Product Y ProductZ units units units Production of these quantities is immediately recognized as be1ng impossible. Practical capacity is machine hours in department1 and machine hours in department2. Work study estimates of machine time for each product are as follows: X y z Department 1 30 m~nutes 15 minutes 15 minutes Department 2 15 minutes 1; minutes 45 minutes Standard costs and profits for the three products are as follows: X y z ( ) ( ) ( ) Direct materials Direct labour Factory overheads Selling and administrative overheads Profit (Loss) (4.00) Selling price Factory overhead is absorbed on the basis of direct labour cost at a rate of 250%; approximately 20% of the overheads are variable and vary with direct labour costs. Selling and administrative costs are allocated on the basis of sales at the rate of 15%; approximately one half of this is variable and varies with sales value. The estimated overhead for the period is for factory overhead and for selling and administrative overhead. The managing director has suggested that product Z should be eliminated from the product line and that the facilities be used to produce products X and Y. The sales manager objects to this solution because of the need to provide a full product line. In addition he maintains that the firm's regular customers provide, and will continue to provide, a solid base for the firm's activities and that these customers needs must be met. He has provided a list of these customers and their estimated purchases are as follows: APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

27 Product X Product Y Product Z units units 8000 units The managing director has reluctantly agreed to this proposal but has asked you, as the newly appointed accountant, to provide him with information of the maximum profit that can be obtained from the remaining capacity You decide to use mathematical programming to determine the optimal results from the uncommitted capacity and the final tableau of the linear programming model is as follows: X (Product X) z (Product Z) Slack X (Sales of X) Slack Y (Sales of Y} Slack Z (Sales of Z) Y Slack 1 Slack 2 (Producl Y) (Department 1) (Department 2) Required: (a) Prepare the first tableau of the linear programming model to which the solution is given. (8 marks) (b) What is the maximum profit for the period? Also explain how you would obtain information of how much profit is lost as a result of the imposition of the minimum sales constraints and indicate why this Information would be useful for product mix decisions. (7 marks) (c) The managing director has requested an explanation of the meaning of the final tableau. You are required to provide an explanation of the meaning of each of the terms in the final tableau and provide any other information which would be of use that can be derived from the final tableau. (10 marks) (Total 25 marks) 24.11* Advanced: Formulation of initial matrix and interpretation of final matrix using the Simple method Hint: Reverse the signs and ignore the entries of 0 and 1. A chemical manufacturer is developing three fertilizer compounds for the agricultural industry. The product codes for the three products are X1, X2 and X3 and the relevant information is summarized below: Chemical constituents: percentage make-up per tonne Nitrate Phosphate Potash Filler X X X Input prices per tonne Nitrate 150 Phosphate 60 Potash 120 Filler 10 Maximum available input in tonnes per month Nitrate 1200 Phosphate 2000 Potash 2200 Filler No limit The fertilizers will be sold in bulk and managers have proposed the following prices per tonne. X1 83 X2 81 X3 81 The manufactunng costs of each type of fertilizer, excluding materials, are 11 per tonne. You are required to: (a) formulate the above data into a linear programming model so that the company may maximize contribution; (4 marks) (b) construct the initial Simplex tableau and state what is meant by 'slack variables' (Define X4, X5, X6 as the slack variables for X1, X2, and X3 respectively); (2 marks) (c) indicate, with explanations, wh1ch will be the entering variable' and 'leaving variable' in the first iteration; (2 marks) You are not required to solve the model (d) interpret the final matrix of the simplex solution given below: Basic Variable x1 Xz X a x4 Xs Xa Solution X X X z (8 marks) (e) use the final matrix above to investigate: (i) the effect of an increase in nitrate of 100 tonnes per month; (ii) the effect of a minimum contract from an influential customer for 200 tonnes of X3 per month to be supplied. (4 marks) (Total 20 marks) CIMA Stage 3 Specimen Paper Management Accounting techniques Advanced: Formulation of an initial tableau and interpretation of a final tableau using the simplex method Hint: Reverse the signs and ignore entries of 0 and 1. The Kaolene Co. Ltd has six different products all made from fabricated steel. Each product passes through a combination of five production operations: cutting, forming, drilling, welding and coating. Steel is cut to the length required, formed into the appropriate shapes, drilled if necessary, welded together if the product is made up of more than one part, and then passed through the coating machine. Each operation is separate and independent, except for the cutting and forming operations, when, if needed, forming follows continuously after cutting. Some products do not require every production operation. The output rates from each production operations. based on a standard measure for each product, are set out 1n the tableau below, along with the total hours of work available for each operation. The contribution per unit of each product ts also given. It is estimated that three of the products have sales ceilings and these are also given at the top of page 722: QUESTIONS

28 Products x, x2 x3 x4 Xs Xa Contribution per unit ( ) Output rate per hour: Cutting Forming Drilling Welding Coating Maximum sales units (000) Cutting Forming Drilling Welding Coating Production hours available The production and sales for the year were found using a linear programming algorithm. The final tableau is given below: X, x2 x3 x4 Xs Xa x1 X a Xg Variables X 7 to X 11 are the slack variables relating to the production constraints, expressed in the order of production. Variables X 12 to X 14 are the slack variables relating to the sales ceilings of X 3, X 5 and X 6 respectively. After analysis of the above results, the production manager believes that further mechanical work on the cutting and forming machines costing 200 can improve their hourly output rates as follows: Cutting Forming x, X Xs X a The optimal solution to the new situation indicates the shadow prices of the cutting, drilling and welding sections to be 59.3, 14.2 and 71.5 per hour respectively. Requirements: (a) Explain the meaning of the seven items ringed in the final tableau. (9 marks) (b) Show the range of values within which the following variables or resources can change without changing the optimal mix indicated in the final tableau (i) c 4 : contribution of X 4 (ii) bs: available coating time. (4 marks) (c) Formulate the revised linear programming problem taking note of the revised output rates for cutting and forming. (5 marks) (d) Determine whether the changes in the cutting and forming rates will increase profitability. (3 marks) Variable in basic Value of variable in basic X,o X11 x,2 x,3 x,4 solution solution l x, units J X a hoursj X units Xs units x,, j hoursj x, units x, units x, units (.2;- C,) l J (e) Using the above information discuss the usefulness of linear programming to managers in solving this type of problem. (4 marks) (Total 25 marks) /CAEW P2 Management Accounting December Formulation of initial tableau and interpretation of final tableau using the simplex method (a) The Argonaut Company makes three products. Xylos, Yoyos and Zicons. These are assembled from two components. Agrons and Bovons which can be produced internally at a variable cost of 5 and 8 each respectively. A limited quantity of each of these components may be available for purchase from an external supplier at a quoted price which varies from week to week. The production of Agrons and Bovons is subject to several limitations. Both components require the same three production processes (l, M and N), the first two of which have limited availabilities of 9600 minutes per week and 7000 minutes per week respectively. The final process (N) has effectively unlimited availability but for technical reasons must produce at least one Agron for each Bovon produced. The processing times are as follows: Process Time (mins) required to produce 1 Agron 1 Bovon L 6 8 M 5 5 N APPUCATION OF UNEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

29 The component requirements of each of the three final products are: Product Xylo Yo-yo Zicon Number of components required Agrons 1 3 Bovons 2 2 The ex-factory selling prices of the final products are given below, together with the standard direct labour hours involved in their assembly and details of other assembly costs incurred: Product Selling price Direct labour hours used Other assembly costs Xylo Yo-yo Zicon The standard direct labour rate is 5 per hour. Factory overhead costs amount to 4350 per week and are absorbed to products on the basis of the direct labour costs incurred in their assembly. The current production plan is to produce 100 units of each of the three products each week. Requirements (i) Present a budgeted weekly profit and loss account, by product, lor the factory. (4 marks) (ii) Formulate the production problem facing the factory manager as a linear program: (1) assuming there is no external availability of Agrons and Bovons; (5 marks) and (2) assuming that 200 Agrons and 300 Bovons are available at prices of 10 and 12 each respectively. (4 marks) (b) In a week when no external availability of Agrons and Bovons was expected, the optimal solution to the linear program and the shadow prices associated with each constraint were as follows: Production of Xylos Production of Yo-yos Production of Zicons Shadow price associated with: Process L Process M Process N Agron availability Bovon availability 50 units 0 units: shadow price units per minute per minute per minute 9.50 each each If sufficient Bovons were to become available on the external market at a price of 12 each, a revised linear programming solution indicated that only Xylos should be made. Requirement Interpret this output from the linear program in a report to the factory manager. Include calculations of revised shadow prices in your report and indicate the actions the manager should take and the benefits that would accrue if the various constrains could be overcome (12 marks) (Total 25 marks) /CAEW P2 Management Accounting July Advanced: Single and multi-period capital rationing Schobert Ltd is a retailing company which operates a small chain of outlets. The company is currently (i.e. December 1988) finalising its capital budgets for the years to 31 December 1989 and 31 December Budgets lor existing trading operations have already been prepared and these indicate that the company will have cash available of on 1 January 1989 and on 1 January This cash will be available for the payment of dividends and/or for the financing of new projects. Seven new capital projects are currently being considered by Schobert Ltd. Each is divisible but none is repeatable. Relevant data for each of the investments for the periods to 31 December 1991 are provided below: Project A B c D E F G Cash Flows Year to 31 December ( 000) ( 000) ( 000) (80) (30) 50 (70) (55) (60) (140) 55 (80) 70 (100) 30 Net Present Value at 10% per annum Internal Rate of Return ( 000) 39 36% 35 38% 5 20% 15 28% 14 19% 32 46% 24 29% The financial director of Schobert Ltd predicts that no new external sources of capital will become available during the period from 1 January 1989 to 31 December 1990, but believes conditions will improve in 1991, when the company would no longer expect capital to be rationed. The objective of the directors of Schobert Ltd is to maximize the present value of the company's ordinary shares, assuming that the value of ordinary shares is determined by the dividend growth model. The company's cost of capital is 10% per annum, and all cash surpluses can be invested elsewhere to earn 8% per annum. A dividend of at least is to be paid to 1 January 1989, and the company's policy is to increase its annual dividend by at least 5% per annum. Requirements: (a) Formulate, but do not solve, the company's capital rationing problem as a linear programme. (8 marks) (b) Assuming that the results of the linear programme show a dual price of cash in 1989 of 0.25 and 1990 of 0, and a range of cash amounts for which the dual price is relevant of to , explain their significance to the directors of Schobert Ltd. (7 marks) (c) Discuss the circumstances under which capital might be QUESTIONS

30 rationed, and the problems these present for capital budgeting decisions. (10 marks) (Total25 marks) Note: Ignore taxation. IGAEW P2 Anancial Management December * Multi-period capital rationing and minimum profit constraints Details of projects available to Glaser Ltd, a wholly owned subsidiary of a publicly quoted company, are: Accounting profit in year Cash flows at time NPV IRR to time ( M) ( M) ( M) ( M) ( M) (%) ( M) ( M) ( M) Project A (7) B (6) c (2) 4.0 (0.5) (2.0) (0.6) (0.1) (0.2) (0.3) D (8.0) (2.0) (1.0) (0.3) (0.2) E (3.0) Each of the projects have cash flows which extend beyond time 3 and this has been reflected in the NPV and IRR calculations. The investment shown is the maximum possible for each project but partial investment in a project is possible and this would result in strictly proportional cash flows, accounting profits and NPV figures. The timing of the start of each project cannot be altered and, if started, a project must run for its whole life. External funds available for investment are: Time 0 up to 12M, of which 5 M is a loan to be repaid with interest at 10% at time 1. Time 1 a new equity injection from Glaser's holding Time 2 and 3 nil. company of 7.5 M. Funds generated from investment in the five available projects will also be available for further investment by Glaser. None of the funds generated by Glaser's existing activities are available for investment in projects A to E or for the payment of interest, or principal, relating to the loan. At time 0, Glaser's holding company will require the surrender of any cash not used for investment. With effect from time 1 confiscation of surplus cash will cease and excess funds can be put on deposit to earn the competitive risk free interest rate of 8% per year. the gross amount then being available for investment. After time 3 Glaser will be free to seek funds from the capital market. The holding company requires Glaser to produce accounting profits from projects (i.e. ignoring interest payments or receipts) which are always at least 1 0% higher than those of the previous year. Existing projects will produce profits of Year to time Profits ( M) Required: (a) Provide an appropriate linear programming formulation which is capable of assisting in, or indicating the impossibility of, deriving a solution to the problem of selecting an optimum mix of projects within the constraints imposed on Glaser. Glaser's objective is to maximize the economic well being of the shareholders of the holding company. Clearly specify how this objective is to be incorporated in the programming fonnulation. Specify the meaning of each variable and describe the purpose of every constraint used. You are required to fonnulate the problem, you are not required to attempt a solution. Ignore tax. (10 marks) (b) Briefly explain the circumstances under which it may be rational for a finn to undertake a project with a negative NPV, such as project C. (3 marks) (c) Outline the main merits and deficiencies of mathematical programming and mathematical modelling in practical financial planning. (7 marks) (Total 20 marks) GAGA P3 Financial Management December Advanced: Single and multi-period capital rationing Raiders Ltd is a private limited company which is financed entirely by ordinary shares. Its effective cost of capital, net of tax, is 1 0% per annum. The directors of Raiders Ltd are considering the company's capital1nvestment programme for the next two years, and have reduced their initial list of projects to four. Details of the projects are as follows: After one Immediately year ( 000) ( 000) ProJect A B c D Cash nows (net of tax) After After two three Net present value years years (at10%) ( 000) ( 000) ( 000) Internal rate of return (to nearest 1%) None of the projects can be delayed. All projects are divisible: 26% 25% 23% 50% outlays may be reduced by any proportion and net inflows will then be reduced in the same proportion. No project can be undertaken more than once Ra1ders Ltd is able to invest surplus funds in a bank deposit account y1elding a return of 7% per annum. net of tax. You are required to: (a) prepare calculations showing which projects Raiders Ltd should undertake if capital for immediate investment is limited to , but is expected to be available without limit at a cost of 10% per annum thereafter: (5 marks) (b) prov1de a mathematical programming fonnulation to assist the directors of Raiders Ltd In choosing investment projects if capital available immediately is limited to , capital available after one year is limited to , and capital is APPLICATION OF LINEAR PROGRAMMING TO MANAGEMENT ACCOUNTING

31 available thereafter without limit at a cost of 10% per annum; (8 marks) (c) ouuine the limitations of the formulation you have provided in (b); (6 marks) (d) comment briefly on the view that in practice capital is rarely limited absolutely, provided that the borrower is willing to pay a sufficiently high price, and in consequence a technique for selecting investment projects which assumes that capital is limited absolutely. is of no use. (6 marks) (Total25 mar,ks) ICAEW Anancial Management July Advanced: Capital rationing and beta analysis The directors of Anhang pic are considering how best to invest in four projects, details of which are given below. Net present value ( 000) Beta factor of project Initial payment ( 000) Project Project ProJect Proiect I II Ill IV The net present values of the projects have been calculated using specific. risk-adjusted discount rates. The director's choice is complicated because Anhang pic has only currently available for investment in new projects. Each project must start on the same date and cannot be deferred. Acceptance of any one project would not affect acceptance of any other and all projects are divisible. The directors at a recent board meeting were unable to agree upon how best to invest the A summary of the views expressed at the meeting follows: (i) Wendling argued that as the presumed objective of the company was to maximize shareholder wealth, project Ill should be undertaken as this project produced the highest net present value. (ii) Ramm argued that as funds were 1n short supply investment should be concentrated in those projects with the lowest in~ial outlay, that is in projects I and II. (iii) Ritter suggested that project Ill should be accepted on the grounds of risk reduction. Project Ill has the lowest beta, and by its acceptance the risk of the company (the company's present beta is 1.0) would be reduced. Ritter also cautioned against acceptance of project IV as it was the most risky project; he pointed out that its high net present value was, in part, a reward for its higher level of associated risk. (iv) Punto argued against accepting project Ill, stating that if the project were discounted at the company's cost of capital, its net present value would be greatly reduced. Requirements: (a) Write a report to the directors of Anhang pic advising them how best to invest the 90000, assuming the restriction on capital to apply for one year only. Your report should address the issues raised by each of the four directors. (17 marks) (b) Explain why the criteria you have used in (a) above to determine the best allocation of capital may be inappropriate if funds are rationed for a period longer than one year. (4 marks) (c) Describe the procedures available to a company for the selection of projects when capital is rationed in more than one penod. (4 marks) (Total25 marks) /CAEW Financial Management December 1986 QUESTIONS