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Break-even analysis for multiple product firms
Socio-Econ.
P/on. Sci. Vol. 18, No. 3, pp. 211-217,
0038-0121/84 $3.00 + .oo Pergamon Press Ltd.
1984
Printed in the U.S.A.
BREAK-EVEN ANALYSIS FOR MULTIPLE PRODUCT FIRMS HECTOR CORREA Graduate School of Public and International Affairs, University of Pittsburgh, Pittsburgh, PA 15260,U.S.A. (Received 22 December 1982)
Abstrati-The object of the paper is to present a method to compute break-even values of production for multiple product firms that produce under conditions of joint fixed costs and product interdependence, without assuming that the ratio of the variable costs of each output to the total variable costs of the firmremainconstant.The result obtained with the methods proposed are compared with those obtained using other methods, and the implications for managerial decision-making are analyzed. 1.
stant values, at the cost of some additional mathematical and computational complexity. These are numerous papers in which the methods of Linear and Goal Programming are discussed together with those of break-even analysis. Dopuch’s paper [4] is a typical example that includes some references. However, only the papers by Charnes et al.[9] and Nalh[ll] address specifically the problem that characterizes breakeven analysis, i.e. minimization of the sales needed to cover total costs. A limitation of these approaches is described in Section 5, together with the steps needed to avoid it. The analysis in Section 5 shows that there are important similarities between the models in Sections 3 and 5. For this reason, and taking into consideration the limitations of space, beginning in Section 6 little or no attention is paid to the integer programming approach to break-even analysis. The extensions suggested here of the output index and the allocated joint fixed cost methods are compared both at a conceptual level and with sensitivity analysis in Section 6. The allocated joint fixed cost method is generalized to the case of interdependent production in Section 7, and is compared with input/output analysis in Section 8. Concluding remarks and suggestions for future research are presented in Section 9. The case of a firm that produces n goods with n + m inputs will be considered below. The following notation will be used: Qii quantity of good i used as inputs in the production of good j, where i = 1,. . . , n for raw materials produced by the firm, i = n + 1,. . , n t m for raw materials bought by the firm, and j = 1,. . , n; A total joint fixed costs; Ai part of A allocated to product i, i=l,..., n; Fi fixed cost in the production of product i = 1,. . , n; Ci cost per unit of product i used in the production process i, i = 1,. . . n t m. It will be assumed that the cost per unit of inputs bought by the firm, i.e. for i=ntl,... n + m, are known; C cost per unit of output index; Ni total output of product i, i = 1, . . . , n ; Bii = Qij/Ni, i.e. number of units of input i used to produce one unit of output j. It is assumed below that these technical coefficientsareknownfori=l,.,., ntm,j=l,..., n; N output index; PI market price of product i. These prices are assumed to be known, and P price per unit of output index.
INTRODUCTION
analysis is to specify the volumes of production and sales needed to cover total costs. The concepts of volume of output and of cost are well defined when only one product is considered. No particular problem appears in the application of break-even analysis to multiple-product firms when there are no joint fixed costs and no interdependent production. However, in practice, these conditions are not likely to occur very frequently. The object of this paper is to suggest some improvements in the methods that have been proposed to deal with these problems, namely the output index, the allocated joint fixed cost and the linear and goal programming methods. As a justification for the effort, a brief discussion of the role of break-even analysis in decision making is presented in Section 1. The presentation below will first consider break-even analysis under the assumption of joint fixed cost, but no interdependent production. The problems brought about by the relaxation of the assumption of independence will be studied later. In Section 3 the method to deal with the problems of break-even analysis in multiple product firms recommended, e.g. by Amey and Eggintonill, Bierman and Dyckman121, DeCoster and Schafer[3], Dopuch[4], Horngren[5] Korn and Boyd[6], and Matz and Usry[7] will be considered. It is based on the construction of indices of costs and of outputs for multiple product firms. For this, the volumes of output of the different products are assigned weights, which are assumed to remain constant up to the specification of the break-even value. Naturally, these weights are modified for sensitivity analysis. The method to be considered in Section 4 is that discussed by DeCoster and Schafer[3], Korn and Boyd[6], Matz and Usry[7], and Solomon and Pringle[8], among others. It is based on the allocation of joint fixed costs to the different products of the firm. This allocation is made by distributing the joint fixed cost, for instance, in proportion to the ratio of the variable costs of a product to the total variable cost of the firm. Again in this case, these ratios are assumed to remain constant. As the references above indicate, it is well known that the assumptions that the weights used to construct the indices of costs and outputs or the proportions used to distribute fixed costs remain constant are not only frequently violated in practice, but also might lead to wrong decisions. Approaches are suggested below that make it possible to eliminate these assumptions of conThe basic idea of break-even
2. THEBOLE OF BREAK-EVEN ANALYSIS IN DECISION MAKING developments of the methods of break-even analysis are justified only if it can be shown that they are likely to be useful for managerial decision making. The methods of management sciences most frequently
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H. CORREA
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used are based on the assumption that the maximization of profits or of measures of utility usually derived from profits is the only objective that managers have. However, it should be clear that, particularly in the medium and long terms, profits are just one of the elements considered by managers in their decisions. Survival of the firms is a second, and perhaps more important, objective. Some observations will be made below showing how information obtained from breakeven analysis can be used to construct indices of the ability of firms to survive. One possible form for these indices is given by the quotient (break-even sales)/(Total reference sales). Two specifications for the denominator of this fraction seem reasonable. One is the level of sales that is likely to be achieved according to information derived, say, from market surveys. The other is total sales needed to maximize profits. It seems intuitively clear that the smaller the quotient above, the more likely a firm is to survive. From this it follows that for two firms equal value of reference sales, the one with the smaller value for the index of survival is likely to be more attractive, even if it is not the one with the highest level of profits, and provided that the difference in profits between the firms compared is not too large. As a conclusion of this Section, it can be said that the observations above show that the method of break-even analysis provides information complementing that derived from the application of optimization methods, and that it is not likely that the currently available methods of management sciences, or their development in the near future, will render obsolete the methods of break-even analysis.
3.OUTPUTtNDEXMETHODOFBRBAK-EVENANALYSISWHEN INDEPENDENTOUTPUTSAREASSUMED
The assumption of independent outputs means that Qii = 0 for i = 1,. . . , n, j = 1,. . . II, i.e. the firm does not use any of its outputs as raw material for any of its products. The first problem that must be solved is the estimation of the variable costs per unit of output. The standard method reduces to the application of the formula ci = s::;
C,,Q,,JNi i = 1,. . , n
(1)
i.e. Ci=S:,:“C,,Bhi
i=l,...,n.
(2)
Next, an index of the total output (N), cost per unit (C), and price (P), of the index of total output, must be specified. Once this is done, the break-even formula for one product can be applied. One of the many possible ways to define the index N of total output and the values of C and P mentioned above will be described below. First, it seems reasonable to set the following conditions for every value of Ci, Pi, C, P, N,, and N
i.e. that the total cost and total revenues should be equal regardless of whether they are computed with the actual volumes of output, costs per unit, and prices, or with the index of output and the cost and price per unit of index. Since the C, and Pi can be assumed to be known. eqns (3) and (4) form a system of 2 equations with n + 3 unknowns, namely, the nNi, C, P and N. To solve the problem, n t 1 additional equations are needed. To specify them, assume that Ni=aiN
i=l,....,n
(5)
i.e. a proportional relation between the Ni and the N. Since the equations in (5) introduce the n additional unknowns ai i = 1,. . , n, they do not help to solve the problem. However, some assumptions can be made with respect to the ai. Amey and Egginton[l] and Dopuch[4], respectively, suggest making ai = N,*/S,“PiNi* and ai = N,*/S,“N,*
where N,* denotes known values of the Ni* used as terms of reference. The possibility to be considered below is ai = (F t C,N,*)/S,“(F, t CiNi*)i = 1,.
,n
(6)
It should be observed that it can be shown that with the ar defined with the formula a, = CiNi*/SI”CiNi* i = 1,.
,n
the break-even problem does not have a solution. A useful property of the ai defined in (6) is that S,“ai = 1. With this, each unit of the index of output reflects a number of units of the different products. With the definition of ai in (6) the problem of estimating the break-even point is reduced to that of solving the system formed with the following equation: N = (A t S,“F,)/S,“(P, - Ci - Ci)ai
(7)
and eqn (5). However, there is no reason why the values of the Ni* should be considered valid terms of reference in the computation of the break-even values of N and Ni. More specifically, the values of the Ni obtained with eqns (5) can be considered as new terms of reference, and the process reinitiated using them. On this basis, the following recurrent system can be specified: First, the initial value of the N, is defined as follows: N,, = 1,
i=l,...,n.
where Ni, is the output of product i according to iteration t. The definition of ai is changed to ai = CiNi,/Si”CiNi,.
S,“CiNi = CN
(3)
The definitions of C, P, and N given in eqns (3) and (4) do not change. The value of N is computed with (7). Finally, one has
S,“PiNi = PN
(4)
&,+I = aiN.
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Break-evenanalysisfor multipleproductfirms The values of N obtained with the recurrent process &fined above are bounded above and below. TO see this. observe that, making
as had to be shown. This suggests that the recurrent system defined above converges, and as a consequence, the values of the Ni are well defined.
M = maXi(Pi - Ci) m = mini(Pi - CJ it follows from (7) that (At Si”Fi)/M I N I (A + Si”FJm as had to be shown. This suggests that the recurrent process described above converges, and as a consequence, the values of N and Ni are well defined. 4. BREAK-EVEN ANALYSIS WITH THE ALLMATED JOINT F’IXFD COST METHOD
On an intuitive basis, it appears that the only difference between allocatable and non-allocatable fixed costs is in the administrative expenses needed to allocate them. These expenses are lower for the first type of fixed costs mentioned above, which means that a precise allocation of some fixed costs simply is not worthwhile, and, as a consequence, approximations can be used. One possible way to allocate A is by taking into consideration the effort that each product requires from the firm. The simplest measure of “effort” could be total costs attributed to product i, i.e.
5. PROGRAMMINGAPPROACHESTO BREAK-EVRN ANALYSIS A direct link between the optimization methods of mathematical programming and break-even analysis can be established considering that the characteristic problem of break-even analysis is the evaluation of the minimum sales needed to cover total costs. As mentioned before, Nath[ll] addresses this problem using linear programming, and Charnes et af.[9] uses goal programming. A limitation of these two approaches is that they can be applied only to firms without any fixed allocated costs, denoted with Fi above. In order to consider these costs, the method of mixed integer programming must be used. The simplest formulation of the break-even problem when independent outputs and allocatable fixed costs are considered takes the following form: Maximise Xo = Si”piNi subject to: A t SlFiyr t SlciNi = SlpiNi
Fi t CiNi.
Ni I Uiyi
As a consequence, total fixed costs can be allocated according to the proportion
and to NrO,
Ai = A(Fl + CiNi)/S1”(Fi + CrNj).
It should be observed that different indices of the “effort” attributable to each product could be used. Using allocated joint fixed costs, the specification of the break-even points for each product is reduced to the problem of solving the system of n equations PiNi=Fi-tCiN,tAi
i=l ,...,n.
(9)
In this system the break-even value for each product i depends, in a non-linear fashion, through the Ai, on the volumes of ah the other products. A recurrent approach can also be used to solve this system. For this, the following equations are used: Ni,,+, = Di,/(Pi - Ci)
(10)
where t denotes number of the iteration, Di, = Fi + A(Fi + CiNi,)/SI”(Fj t CjNj,)
(11)
Ni, = Fi/(Pi - CJ.
(12)
and
i.e. the starting point of the iterations is the break-even solution specified without considering the joint fixed costs. The values of the Nit defined with eqns (10) to (12) are bounded below and above. This can be seen observing that F&Pi - CJ % Ni, 5 (Fi t A)/(P, - Ci)
yi=O or 1 i=l,...,n
(8) where yii=l,..., n are dummy variables that take only the values 0 or 1, and Ni are arbitrary upper bounds for the Ni. According to this formulation, the minimum value of total sales is equal to total costs that include allocatable and non-allocatable fixed costs together with variable costs. However, the allocatable fixed costs considered for the determination of the minimum values of total sales are only those of the process actually being used. This is obtained with the use of the yi variables that are equal to 1 only if the corresponding Ni are greater than zero. The minimization model presented before has an important limitation. Implicit in the break-even analysis of a multi-product lirm is the assumption that there is some complementarity among the different products that the firm can produce. As a consequence of this, several values of the Ni should be other than zero. This assumption is replaced in the linear and integer programming approach with the assumption of substitutability among the products. As a consequence of this, and of the general structure of the problem, the minimum sales as specified by the integer programming model introduced above will be attained with only one Ni other than zero. The deficiency mentioned above can be eliminated by introducing two types of constraints, those affecting (a) the capacities to produce the different goods, and (b) proportions that the different outputs or sales must satisfy. The implications of introducing capacity constraints in the model above is clearly a topic that should be
214
H. COFXEA
analyzed with respect to the model above. However, it will not be discussed here, since it would take us away from break-even analysis as it is currently defined. More specifically, for a balanced treatment of this topic, capacity constraints should also be introduced in the methods of break-even analysis discussed in Sections 3 and 4, and this is clearly beyond the scope of one article. A more direct way to consider complementarities among the different goods produced by a firm in the formulation of the minimization model above is to introduce in it constraints requiring that the different outputs should be proportional. This can be done using, for instance, equations or inequalities derived from (5). In this case, and if equality constraints are used, the minimization problem above and the non-recurrent problem in Section 3 will have the same solution. From the observations above, it follows that the method in Section 3 has some advantages and some disadvantages via-a-vis the integer programming approach to break-even analysis. An advantage of the recurrent model in Section 3 is that in its solution the values of the a, are not fixed exogeneously. On the other hand, the integer programming model, particularly when the proportionality constraints take the form of inequalities, is likely to be more flexible. It is likely to make it possible to explore more possibilities. As mentioned before, limitations of space make it impossible to compare in detail the three methods of break-even analysis discussed so far. The rest of this paper refers only to the models in Sections 3 and 4.
6. cohll’~
OF THE INDEX AND ALLOCATED JOINT FIXED COST METHODS OF BREAK-EVEN ANALYSIS
The formulas that characterize the two methods of break-even analysis, i.e. formulas (5) and (7) for the output index method and (9) for the allocated costs method, will be compared first. This will be followed by some observations based on a study of the results obtained from the sensitivity analysis of a simple numerical example dealing with a firm that produces 3 outputs. Formulas (5) and (7) clearly show that the quotient (6) is the main determinants of the value of the Ni in the
index method of break-even analysis. On the other hand, according to formulas (IO) and (1l), the value of N, obtained with the allocated joint fixed cost method can be divided into two components. The first gives the break-even value for product i, taking into consideration only the fixed costs directly associated with its production. In this component, the quotient in (6) has no influence. The second component allocates the value A of the joint fixed costs among the different Ni, and, as a consequence, depends on the quotient in (6). The comparison above brings up a basic problem of the index method. Its conceptual basis is the need to avoid an arbitrary allocation of the joint fixed cost, i.e. an allocation based on (6) or similar formulas. However, the method ends arbitrary allocation on the basis ofthesameformulasnot onlythejointflxedcost, but an index of the total output of the firm developed on the basis of all costs and prices. This means that the arbitrariness in the determination of the N, is increased, and not decreased, as intended. On the other hand, with the allocated joint fixed cost method, the arbitrary allocation of the joint hxed cost only modifies the breakeven values that would have been obtained in the absence ofjoint tixed costs. An additional observation is needed for the interpretation of the results of the sensitivity analysis to be presented below. This observation is that, in agreement with the questions just raised, the ranking of the Ni by order or magnitude will be determined by the quotient in (6) for the index method, while the quotients on the r.h.s. of (12) will be much more inthrential in the allocated joint fixed costs method. Since the rankings of these two types of quotients do not have to be the same, the ranking of the Ni obtained with the two methods does not have to be the same, either. This shows that production decisions based on one method of break-even analysis might be diametrically opposed to those reached using the other method. As a consequence, it is particularly important to find logical and empirical reasons for selecting one of the two methods proposed. In the remainder of this Section, sensitivity analysis will be used to make a numerical study of a firm that produces three products. The formulas based on the definition of the ai in (6) will be used in this analysis.
Table 1. Data used for sensitivity analysis LOW Unequal Equal Fixed cost Non-allocatable A 10000 Allocatable 5500 F, 6000 FZ 6500 F3 Per unit variable costs 18 c1 20 cz 22 c> Prices P1 PZ P,
30 45
28 30 32
HIGH Equal Unequal
10000
50000
5OoOO
2000 6000 10000
33000 36000 39000
12000 36000 60000
10 20 30
105
110 115
50 110 160
135 150 165
70 150 230
15
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Break-even analysis for multiple product firms
The four sets of data to be used are presented in Table 1. Each of these sets includes information on fixed non-allocatable costs, fixed allocatable costs, per unit variable costs, and prices. For two of these sets, the values of all the variables are low when compared with the values of the corresponding variables in the other two sets. The sets with low values and those with high values are subdivided into two subsets. In the first, the values of fixed allocatable costs, per unit variable costs, and prices are approximately the same for the three products of the firm. The other subsets present unequal values for these variables. These four sets of data will be designated as low equal, low unequal, high equal and high unequal. In the analysis to be described below, 44=256 combinations are studied, i.e. all the possible combinations formed by taking one element from each of the four sets of data. 96 of these combinations present the special problem that one or more prices are less than the corresponding unit variable costs. As a consequence, the idea of break-even analysis is not applicable. However, it is helpful to use the methods presented above with these sets of data in order to determine whether the methods help to identify the cases in which this problem appears. A sample of the numerical results obtained is presented in Table 2. A first observation is that, in all cases in which the values of the variables are meaningful, i.e. in which all prices are larger than the respective costs, the iteration procedures presented above converged towards the values of the Ni. However, the index method required about 5 times as many iterations as the allocated joint fixed cost method to reach stable values for the Ni,. In either case, the number of iterations, i.e. 25 vs 5, does not seem to constitute a serious problem. The results obtained can be classified in two sets,
namely, results in which all the Ni present positive, i.e. acceptable values, and results in which some or all the Ni were negative. The set with some all negative Ni can be subdivided into two subsets. The first includes 64 cases in which both the index and the allocated joint fixed cost methods produced some or all negative Ni. The second includes 32 cases in which only the allocated joint fixed cost method produced negative Ni, despite that there were cases where variable costs were higher than the prices. This suggests that this method produced better results in this respect. The remaining 160 cases, with all the Ni positive, can also be subdivided into two subsets. The first includes results in which the ranking of the Ni values is the same for the two methods being analyzed ranking are different (columns (3) and (4) in Table 2). As observed, these results are a consequence of the different bases used in the two methods for specifying the values of the Ni. In Table 2 it can also be observed that the N obtained with the two methods have substantial differences in value. Two questions can be asked with respect to this problem. The first is whether the total cost of production, i.e. S13CiNi,differs for the two methods. The mean value defined in Table 2 can be used for this analysis. The results in Table 2 show that in the two comparable cases in the Table, i.e. those in columns (2) and (3) and in columns (4) and (5) differences are not large. This partial result is confirmed with the quotient of the total ,of all acceptable means for the allocated joint fixed costs, divided by the total of all acceptable means for the index method. The value of this quotient, shown as a percentage, indicates that the total cost of the break-even production specified with the first method is only 90.60% of the total costs specified using the second method.
Table 2. Sample of numerical results of the sensitivity analysis of methods to evaluate break-even points Equal ranking Index Allocat. Method Method
(1) IA IF IC IP N, N* N, Mean Variance
(2) 2 2 1 209.1: 793.50 1797.35 19725.47 662287840.0
Different ranking Index Allocat. Method Method
(3) (4) 2 3 2 1 1 1 1 3 295.61 148.97 918.16 171.82 1586.23 197.44 19527.08 3487.18 146467140.01 385572.70
Notes IZ denotes index of Z, Z = A, F, C, P defined above. IA, IF, IC, IP can take values 1 to 4 meaning 1 low equal set of data 2 low unequal set of data 3 high equal set of data 4 high unequal set of data Mean = .!L3CiNi/3,Variance = S13(CiNi- Mean)*/3.
(5) 3 1 1 3 177.59 174.36 171.72 3487.18 56298.25
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H. CORREA
The situation is substantially different with respect to variance. The total of all the variances obtained with the allocated joint fixed costs method is only 23.20% of the corresponding total for the index method.
7.BREAK-EVEN ANALYSISWITHJOINT FIXED COSTS AND INTERDEPENDENTOUTPUTS
In the analysis made so far, it has been assumed that the Qii are equal to zero for i, j = 1,. . , n; i.e. that none of the products of the firm are used as raw materials for other products. The consequences of eliminating this assumption will be studied here. The first question that will be considered is the estimation of the variable costs of the outputs produced by this firm. In the case of independent outputs, this problem is solved with eqn (2). The interdependence of inputs brings up the problem that the unknown variable costs of any of the outputs of the firm depend on the costs of the other outputs, and these other costs are also unknown. More precisely, the following system of n equations can be stated: SlnChQhi = S,“,;“ChQhi i = 1,. . . , n.
(13)
Using the equivalent expression BhiN,, instead of the Qhi, and cancelling out N,, the system (13) becomes Si”ChBhi= SrY,“ChBhi
,n.
i = 1,
(14)
Since the technical coefficients Bhi h, i = 1,. . , n are assumed to be known, system (14) has n equations in n unknowns, namely, the nC,,, h = 1,. . . , n. This solves, at least in principle, the problem of estimating the variable costs for the interdependent outputS of the firm. Once the variable costs per unit of output are estimated, it is possible to proceed with the estimation of the break-even values of the different outputs. In principle, the operations described in the system formed by eqns (5) and (7) can be performed. However, it is not acceptable to estimate the different outputs without considering that the products are used as raw materials for other processes within the firm. On the other hand, formula (9) can be modified as follows: P,N, = Fi t Ai + S,“C,,&N,
i = 1,.
,n
(15)
where the Ai retain the definition in (8). From a mathematical point of view, the systems of equations in (9) and (15) are identical. As a consequence, the conclusions reached previously with respect to the solution of (9) apply to the system in (15). 8. ACOMPABISONOFBREAK-EVENANDINPUT/OUTPUTANALYSIS The object of this Section is to call attention to the similarities between break-even analysis with joint fixed costs and interdependent production, on the one hand, and input/output analysis on the other. The basic equations of input/output analysis can be written as follows, using the notation above: S,“QiitMi=N,
i=l,....,
n
(16)
or as S,“Bi,Ni+Mi=Ni
i=l,...,n
(17)
where Mi denotes final demand, or, in the present context, the part of the output of good i that can be sold. It should be observed that in eqn (16) or (17) no attention has been paid to primary inputs, i.e. labor and capital in input/output analysis, or, in the present context, inputs not produced by the firm. The first point that should be observed when comparing break-even and input/output analysis is that the method to estimate variable costs in break-even analysis in eqns (13) and (14) is identical to the method used to estimate equilibrium prices when wages and profits are specified in input/output analysis. On the other hand, the method used to evaluate total output in break-even analysis with eqns (15) differs from that used in input/output analysis. Fixed costs play a key role in the determination of total output with eqns (15), while they are not used at all in input/output analysis. In the latter, final demand, which is not considered in break-even analysis, is the key element to estimate total outputs. From the observations above it follows that breakeven and input/output analysis are two methods that have a closely related mathematical basis, and that produce complementary information. Break-even analysis can be used to estimate the minimum outputs that a firm should produce, and, once this is done, eqns (17) can be used to evaluate the output that the firm can still sell after using some of the output in the production process. Input/output analysis can be used to estimate the amount that should be produced to satisfy internal needs and external demand. This information could be particularly useful for planning the establishment of a new firm. Once a firm is established, the application of eqns (17) could show that the current capacity of a firm is not large enough to satisfy external demand. 9.CONCLUSIONS
In summary, modifications of the index and the allocated joint fixed cost methods to compute break-even values for multiple product firms have been presented in this paper. These modifications make it possible to evaluate break-even values without the assumption that the volumes of output of the different products are fixed a priori. The comparison of the two methods suggests that the allocated joint fixed cost approach might have some advantages. This preliminary conclusion is a by-product of this paper, since a study of the problem of selecting the best method of break-even analysis for multiple product firms is not its main objective. For this, the first step must be to define precisely “better results”. Once this is done, a systematic comparison of the methods studied in this paper, among themselves and with those based on linear and goal programming, would be needed. These questions are left for future research. REFERENCES I. L. Amey and D. A. Egginton, Mana8emenfAccounting: A ConceptualApproach.Logman(1973). 2. H. Bierman and T. Dyckman,ManagerialCost Accounting (1971). 3. D. T. DeCosterand E. L. Schafer,ManagementAccounting: A DecisionEmphasis.Wiley,New York (1976). 4. N. Dopuch,Cost-volume-profit analyses.In HandbookofCost Accounting,(Editedby S. Davidsonand R. L. Weil).McGrawHill (1978). 5. C. T. Horgren, Accounting for Management Control: An
Break-even analysis for multiple product firms Introduction, 3rd. Edn. Prentice Hall, Englewood Cliffs, New Jersey (1974). 6. S. W. Korn and T. Boyd, Accounting for Management, Planning and Decision Making. Wiley, New York (1%9). 7. A. Matz and M. F. Usry, Cost Accounting: Planning and Control. South West Publishing Co. (1976).
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8. E. Solomon and J. J. Pringle, An Introduction to Financial Management, 2nd. Goodyear Publishing Co. (1980). 9. A. Charnes et al. Break-even budgeting and programming to goals. J. Accounting Res. 16-43 (1%3). 10. R. K. Jaedike, Improving B-E Analysis by linear programming techniques. N. A. A. Bull. pp. 5-12 (March l%l).
P/on. Sci. Vol. 18, No. 3, pp. 211-217,
0038-0121/84 $3.00 + .oo Pergamon Press Ltd.
1984
Printed in the U.S.A.
BREAK-EVEN ANALYSIS FOR MULTIPLE PRODUCT FIRMS HECTOR CORREA Graduate School of Public and International Affairs, University of Pittsburgh, Pittsburgh, PA 15260,U.S.A. (Received 22 December 1982)
Abstrati-The object of the paper is to present a method to compute break-even values of production for multiple product firms that produce under conditions of joint fixed costs and product interdependence, without assuming that the ratio of the variable costs of each output to the total variable costs of the firmremainconstant.The result obtained with the methods proposed are compared with those obtained using other methods, and the implications for managerial decision-making are analyzed. 1.
stant values, at the cost of some additional mathematical and computational complexity. These are numerous papers in which the methods of Linear and Goal Programming are discussed together with those of break-even analysis. Dopuch’s paper [4] is a typical example that includes some references. However, only the papers by Charnes et al.[9] and Nalh[ll] address specifically the problem that characterizes breakeven analysis, i.e. minimization of the sales needed to cover total costs. A limitation of these approaches is described in Section 5, together with the steps needed to avoid it. The analysis in Section 5 shows that there are important similarities between the models in Sections 3 and 5. For this reason, and taking into consideration the limitations of space, beginning in Section 6 little or no attention is paid to the integer programming approach to break-even analysis. The extensions suggested here of the output index and the allocated joint fixed cost methods are compared both at a conceptual level and with sensitivity analysis in Section 6. The allocated joint fixed cost method is generalized to the case of interdependent production in Section 7, and is compared with input/output analysis in Section 8. Concluding remarks and suggestions for future research are presented in Section 9. The case of a firm that produces n goods with n + m inputs will be considered below. The following notation will be used: Qii quantity of good i used as inputs in the production of good j, where i = 1,. . . , n for raw materials produced by the firm, i = n + 1,. . , n t m for raw materials bought by the firm, and j = 1,. . , n; A total joint fixed costs; Ai part of A allocated to product i, i=l,..., n; Fi fixed cost in the production of product i = 1,. . , n; Ci cost per unit of product i used in the production process i, i = 1,. . . n t m. It will be assumed that the cost per unit of inputs bought by the firm, i.e. for i=ntl,... n + m, are known; C cost per unit of output index; Ni total output of product i, i = 1, . . . , n ; Bii = Qij/Ni, i.e. number of units of input i used to produce one unit of output j. It is assumed below that these technical coefficientsareknownfori=l,.,., ntm,j=l,..., n; N output index; PI market price of product i. These prices are assumed to be known, and P price per unit of output index.
INTRODUCTION
analysis is to specify the volumes of production and sales needed to cover total costs. The concepts of volume of output and of cost are well defined when only one product is considered. No particular problem appears in the application of break-even analysis to multiple-product firms when there are no joint fixed costs and no interdependent production. However, in practice, these conditions are not likely to occur very frequently. The object of this paper is to suggest some improvements in the methods that have been proposed to deal with these problems, namely the output index, the allocated joint fixed cost and the linear and goal programming methods. As a justification for the effort, a brief discussion of the role of break-even analysis in decision making is presented in Section 1. The presentation below will first consider break-even analysis under the assumption of joint fixed cost, but no interdependent production. The problems brought about by the relaxation of the assumption of independence will be studied later. In Section 3 the method to deal with the problems of break-even analysis in multiple product firms recommended, e.g. by Amey and Eggintonill, Bierman and Dyckman121, DeCoster and Schafer[3], Dopuch[4], Horngren[5] Korn and Boyd[6], and Matz and Usry[7] will be considered. It is based on the construction of indices of costs and of outputs for multiple product firms. For this, the volumes of output of the different products are assigned weights, which are assumed to remain constant up to the specification of the break-even value. Naturally, these weights are modified for sensitivity analysis. The method to be considered in Section 4 is that discussed by DeCoster and Schafer[3], Korn and Boyd[6], Matz and Usry[7], and Solomon and Pringle[8], among others. It is based on the allocation of joint fixed costs to the different products of the firm. This allocation is made by distributing the joint fixed cost, for instance, in proportion to the ratio of the variable costs of a product to the total variable cost of the firm. Again in this case, these ratios are assumed to remain constant. As the references above indicate, it is well known that the assumptions that the weights used to construct the indices of costs and outputs or the proportions used to distribute fixed costs remain constant are not only frequently violated in practice, but also might lead to wrong decisions. Approaches are suggested below that make it possible to eliminate these assumptions of conThe basic idea of break-even
2. THEBOLE OF BREAK-EVEN ANALYSIS IN DECISION MAKING developments of the methods of break-even analysis are justified only if it can be shown that they are likely to be useful for managerial decision making. The methods of management sciences most frequently
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used are based on the assumption that the maximization of profits or of measures of utility usually derived from profits is the only objective that managers have. However, it should be clear that, particularly in the medium and long terms, profits are just one of the elements considered by managers in their decisions. Survival of the firms is a second, and perhaps more important, objective. Some observations will be made below showing how information obtained from breakeven analysis can be used to construct indices of the ability of firms to survive. One possible form for these indices is given by the quotient (break-even sales)/(Total reference sales). Two specifications for the denominator of this fraction seem reasonable. One is the level of sales that is likely to be achieved according to information derived, say, from market surveys. The other is total sales needed to maximize profits. It seems intuitively clear that the smaller the quotient above, the more likely a firm is to survive. From this it follows that for two firms equal value of reference sales, the one with the smaller value for the index of survival is likely to be more attractive, even if it is not the one with the highest level of profits, and provided that the difference in profits between the firms compared is not too large. As a conclusion of this Section, it can be said that the observations above show that the method of break-even analysis provides information complementing that derived from the application of optimization methods, and that it is not likely that the currently available methods of management sciences, or their development in the near future, will render obsolete the methods of break-even analysis.
3.OUTPUTtNDEXMETHODOFBRBAK-EVENANALYSISWHEN INDEPENDENTOUTPUTSAREASSUMED
The assumption of independent outputs means that Qii = 0 for i = 1,. . . , n, j = 1,. . . II, i.e. the firm does not use any of its outputs as raw material for any of its products. The first problem that must be solved is the estimation of the variable costs per unit of output. The standard method reduces to the application of the formula ci = s::;
C,,Q,,JNi i = 1,. . , n
(1)
i.e. Ci=S:,:“C,,Bhi
i=l,...,n.
(2)
Next, an index of the total output (N), cost per unit (C), and price (P), of the index of total output, must be specified. Once this is done, the break-even formula for one product can be applied. One of the many possible ways to define the index N of total output and the values of C and P mentioned above will be described below. First, it seems reasonable to set the following conditions for every value of Ci, Pi, C, P, N,, and N
i.e. that the total cost and total revenues should be equal regardless of whether they are computed with the actual volumes of output, costs per unit, and prices, or with the index of output and the cost and price per unit of index. Since the C, and Pi can be assumed to be known. eqns (3) and (4) form a system of 2 equations with n + 3 unknowns, namely, the nNi, C, P and N. To solve the problem, n t 1 additional equations are needed. To specify them, assume that Ni=aiN
i=l,....,n
(5)
i.e. a proportional relation between the Ni and the N. Since the equations in (5) introduce the n additional unknowns ai i = 1,. . , n, they do not help to solve the problem. However, some assumptions can be made with respect to the ai. Amey and Egginton[l] and Dopuch[4], respectively, suggest making ai = N,*/S,“PiNi* and ai = N,*/S,“N,*
where N,* denotes known values of the Ni* used as terms of reference. The possibility to be considered below is ai = (F t C,N,*)/S,“(F, t CiNi*)i = 1,.
,n
(6)
It should be observed that it can be shown that with the ar defined with the formula a, = CiNi*/SI”CiNi* i = 1,.
,n
the break-even problem does not have a solution. A useful property of the ai defined in (6) is that S,“ai = 1. With this, each unit of the index of output reflects a number of units of the different products. With the definition of ai in (6) the problem of estimating the break-even point is reduced to that of solving the system formed with the following equation: N = (A t S,“F,)/S,“(P, - Ci - Ci)ai
(7)
and eqn (5). However, there is no reason why the values of the Ni* should be considered valid terms of reference in the computation of the break-even values of N and Ni. More specifically, the values of the Ni obtained with eqns (5) can be considered as new terms of reference, and the process reinitiated using them. On this basis, the following recurrent system can be specified: First, the initial value of the N, is defined as follows: N,, = 1,
i=l,...,n.
where Ni, is the output of product i according to iteration t. The definition of ai is changed to ai = CiNi,/Si”CiNi,.
S,“CiNi = CN
(3)
The definitions of C, P, and N given in eqns (3) and (4) do not change. The value of N is computed with (7). Finally, one has
S,“PiNi = PN
(4)
&,+I = aiN.
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Break-evenanalysisfor multipleproductfirms The values of N obtained with the recurrent process &fined above are bounded above and below. TO see this. observe that, making
as had to be shown. This suggests that the recurrent system defined above converges, and as a consequence, the values of the Ni are well defined.
M = maXi(Pi - Ci) m = mini(Pi - CJ it follows from (7) that (At Si”Fi)/M I N I (A + Si”FJm as had to be shown. This suggests that the recurrent process described above converges, and as a consequence, the values of N and Ni are well defined. 4. BREAK-EVEN ANALYSIS WITH THE ALLMATED JOINT F’IXFD COST METHOD
On an intuitive basis, it appears that the only difference between allocatable and non-allocatable fixed costs is in the administrative expenses needed to allocate them. These expenses are lower for the first type of fixed costs mentioned above, which means that a precise allocation of some fixed costs simply is not worthwhile, and, as a consequence, approximations can be used. One possible way to allocate A is by taking into consideration the effort that each product requires from the firm. The simplest measure of “effort” could be total costs attributed to product i, i.e.
5. PROGRAMMINGAPPROACHESTO BREAK-EVRN ANALYSIS A direct link between the optimization methods of mathematical programming and break-even analysis can be established considering that the characteristic problem of break-even analysis is the evaluation of the minimum sales needed to cover total costs. As mentioned before, Nath[ll] addresses this problem using linear programming, and Charnes et af.[9] uses goal programming. A limitation of these two approaches is that they can be applied only to firms without any fixed allocated costs, denoted with Fi above. In order to consider these costs, the method of mixed integer programming must be used. The simplest formulation of the break-even problem when independent outputs and allocatable fixed costs are considered takes the following form: Maximise Xo = Si”piNi subject to: A t SlFiyr t SlciNi = SlpiNi
Fi t CiNi.
Ni I Uiyi
As a consequence, total fixed costs can be allocated according to the proportion
and to NrO,
Ai = A(Fl + CiNi)/S1”(Fi + CrNj).
It should be observed that different indices of the “effort” attributable to each product could be used. Using allocated joint fixed costs, the specification of the break-even points for each product is reduced to the problem of solving the system of n equations PiNi=Fi-tCiN,tAi
i=l ,...,n.
(9)
In this system the break-even value for each product i depends, in a non-linear fashion, through the Ai, on the volumes of ah the other products. A recurrent approach can also be used to solve this system. For this, the following equations are used: Ni,,+, = Di,/(Pi - Ci)
(10)
where t denotes number of the iteration, Di, = Fi + A(Fi + CiNi,)/SI”(Fj t CjNj,)
(11)
Ni, = Fi/(Pi - CJ.
(12)
and
i.e. the starting point of the iterations is the break-even solution specified without considering the joint fixed costs. The values of the Nit defined with eqns (10) to (12) are bounded below and above. This can be seen observing that F&Pi - CJ % Ni, 5 (Fi t A)/(P, - Ci)
yi=O or 1 i=l,...,n
(8) where yii=l,..., n are dummy variables that take only the values 0 or 1, and Ni are arbitrary upper bounds for the Ni. According to this formulation, the minimum value of total sales is equal to total costs that include allocatable and non-allocatable fixed costs together with variable costs. However, the allocatable fixed costs considered for the determination of the minimum values of total sales are only those of the process actually being used. This is obtained with the use of the yi variables that are equal to 1 only if the corresponding Ni are greater than zero. The minimization model presented before has an important limitation. Implicit in the break-even analysis of a multi-product lirm is the assumption that there is some complementarity among the different products that the firm can produce. As a consequence of this, several values of the Ni should be other than zero. This assumption is replaced in the linear and integer programming approach with the assumption of substitutability among the products. As a consequence of this, and of the general structure of the problem, the minimum sales as specified by the integer programming model introduced above will be attained with only one Ni other than zero. The deficiency mentioned above can be eliminated by introducing two types of constraints, those affecting (a) the capacities to produce the different goods, and (b) proportions that the different outputs or sales must satisfy. The implications of introducing capacity constraints in the model above is clearly a topic that should be
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analyzed with respect to the model above. However, it will not be discussed here, since it would take us away from break-even analysis as it is currently defined. More specifically, for a balanced treatment of this topic, capacity constraints should also be introduced in the methods of break-even analysis discussed in Sections 3 and 4, and this is clearly beyond the scope of one article. A more direct way to consider complementarities among the different goods produced by a firm in the formulation of the minimization model above is to introduce in it constraints requiring that the different outputs should be proportional. This can be done using, for instance, equations or inequalities derived from (5). In this case, and if equality constraints are used, the minimization problem above and the non-recurrent problem in Section 3 will have the same solution. From the observations above, it follows that the method in Section 3 has some advantages and some disadvantages via-a-vis the integer programming approach to break-even analysis. An advantage of the recurrent model in Section 3 is that in its solution the values of the a, are not fixed exogeneously. On the other hand, the integer programming model, particularly when the proportionality constraints take the form of inequalities, is likely to be more flexible. It is likely to make it possible to explore more possibilities. As mentioned before, limitations of space make it impossible to compare in detail the three methods of break-even analysis discussed so far. The rest of this paper refers only to the models in Sections 3 and 4.
6. cohll’~
OF THE INDEX AND ALLOCATED JOINT FIXED COST METHODS OF BREAK-EVEN ANALYSIS
The formulas that characterize the two methods of break-even analysis, i.e. formulas (5) and (7) for the output index method and (9) for the allocated costs method, will be compared first. This will be followed by some observations based on a study of the results obtained from the sensitivity analysis of a simple numerical example dealing with a firm that produces 3 outputs. Formulas (5) and (7) clearly show that the quotient (6) is the main determinants of the value of the Ni in the
index method of break-even analysis. On the other hand, according to formulas (IO) and (1l), the value of N, obtained with the allocated joint fixed cost method can be divided into two components. The first gives the break-even value for product i, taking into consideration only the fixed costs directly associated with its production. In this component, the quotient in (6) has no influence. The second component allocates the value A of the joint fixed costs among the different Ni, and, as a consequence, depends on the quotient in (6). The comparison above brings up a basic problem of the index method. Its conceptual basis is the need to avoid an arbitrary allocation of the joint fixed cost, i.e. an allocation based on (6) or similar formulas. However, the method ends arbitrary allocation on the basis ofthesameformulasnot onlythejointflxedcost, but an index of the total output of the firm developed on the basis of all costs and prices. This means that the arbitrariness in the determination of the N, is increased, and not decreased, as intended. On the other hand, with the allocated joint fixed cost method, the arbitrary allocation of the joint hxed cost only modifies the breakeven values that would have been obtained in the absence ofjoint tixed costs. An additional observation is needed for the interpretation of the results of the sensitivity analysis to be presented below. This observation is that, in agreement with the questions just raised, the ranking of the Ni by order or magnitude will be determined by the quotient in (6) for the index method, while the quotients on the r.h.s. of (12) will be much more inthrential in the allocated joint fixed costs method. Since the rankings of these two types of quotients do not have to be the same, the ranking of the Ni obtained with the two methods does not have to be the same, either. This shows that production decisions based on one method of break-even analysis might be diametrically opposed to those reached using the other method. As a consequence, it is particularly important to find logical and empirical reasons for selecting one of the two methods proposed. In the remainder of this Section, sensitivity analysis will be used to make a numerical study of a firm that produces three products. The formulas based on the definition of the ai in (6) will be used in this analysis.
Table 1. Data used for sensitivity analysis LOW Unequal Equal Fixed cost Non-allocatable A 10000 Allocatable 5500 F, 6000 FZ 6500 F3 Per unit variable costs 18 c1 20 cz 22 c> Prices P1 PZ P,
30 45
28 30 32
HIGH Equal Unequal
10000
50000
5OoOO
2000 6000 10000
33000 36000 39000
12000 36000 60000
10 20 30
105
110 115
50 110 160
135 150 165
70 150 230
15
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Break-even analysis for multiple product firms
The four sets of data to be used are presented in Table 1. Each of these sets includes information on fixed non-allocatable costs, fixed allocatable costs, per unit variable costs, and prices. For two of these sets, the values of all the variables are low when compared with the values of the corresponding variables in the other two sets. The sets with low values and those with high values are subdivided into two subsets. In the first, the values of fixed allocatable costs, per unit variable costs, and prices are approximately the same for the three products of the firm. The other subsets present unequal values for these variables. These four sets of data will be designated as low equal, low unequal, high equal and high unequal. In the analysis to be described below, 44=256 combinations are studied, i.e. all the possible combinations formed by taking one element from each of the four sets of data. 96 of these combinations present the special problem that one or more prices are less than the corresponding unit variable costs. As a consequence, the idea of break-even analysis is not applicable. However, it is helpful to use the methods presented above with these sets of data in order to determine whether the methods help to identify the cases in which this problem appears. A sample of the numerical results obtained is presented in Table 2. A first observation is that, in all cases in which the values of the variables are meaningful, i.e. in which all prices are larger than the respective costs, the iteration procedures presented above converged towards the values of the Ni. However, the index method required about 5 times as many iterations as the allocated joint fixed cost method to reach stable values for the Ni,. In either case, the number of iterations, i.e. 25 vs 5, does not seem to constitute a serious problem. The results obtained can be classified in two sets,
namely, results in which all the Ni present positive, i.e. acceptable values, and results in which some or all the Ni were negative. The set with some all negative Ni can be subdivided into two subsets. The first includes 64 cases in which both the index and the allocated joint fixed cost methods produced some or all negative Ni. The second includes 32 cases in which only the allocated joint fixed cost method produced negative Ni, despite that there were cases where variable costs were higher than the prices. This suggests that this method produced better results in this respect. The remaining 160 cases, with all the Ni positive, can also be subdivided into two subsets. The first includes results in which the ranking of the Ni values is the same for the two methods being analyzed ranking are different (columns (3) and (4) in Table 2). As observed, these results are a consequence of the different bases used in the two methods for specifying the values of the Ni. In Table 2 it can also be observed that the N obtained with the two methods have substantial differences in value. Two questions can be asked with respect to this problem. The first is whether the total cost of production, i.e. S13CiNi,differs for the two methods. The mean value defined in Table 2 can be used for this analysis. The results in Table 2 show that in the two comparable cases in the Table, i.e. those in columns (2) and (3) and in columns (4) and (5) differences are not large. This partial result is confirmed with the quotient of the total ,of all acceptable means for the allocated joint fixed costs, divided by the total of all acceptable means for the index method. The value of this quotient, shown as a percentage, indicates that the total cost of the break-even production specified with the first method is only 90.60% of the total costs specified using the second method.
Table 2. Sample of numerical results of the sensitivity analysis of methods to evaluate break-even points Equal ranking Index Allocat. Method Method
(1) IA IF IC IP N, N* N, Mean Variance
(2) 2 2 1 209.1: 793.50 1797.35 19725.47 662287840.0
Different ranking Index Allocat. Method Method
(3) (4) 2 3 2 1 1 1 1 3 295.61 148.97 918.16 171.82 1586.23 197.44 19527.08 3487.18 146467140.01 385572.70
Notes IZ denotes index of Z, Z = A, F, C, P defined above. IA, IF, IC, IP can take values 1 to 4 meaning 1 low equal set of data 2 low unequal set of data 3 high equal set of data 4 high unequal set of data Mean = .!L3CiNi/3,Variance = S13(CiNi- Mean)*/3.
(5) 3 1 1 3 177.59 174.36 171.72 3487.18 56298.25
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The situation is substantially different with respect to variance. The total of all the variances obtained with the allocated joint fixed costs method is only 23.20% of the corresponding total for the index method.
7.BREAK-EVEN ANALYSISWITHJOINT FIXED COSTS AND INTERDEPENDENTOUTPUTS
In the analysis made so far, it has been assumed that the Qii are equal to zero for i, j = 1,. . , n; i.e. that none of the products of the firm are used as raw materials for other products. The consequences of eliminating this assumption will be studied here. The first question that will be considered is the estimation of the variable costs of the outputs produced by this firm. In the case of independent outputs, this problem is solved with eqn (2). The interdependence of inputs brings up the problem that the unknown variable costs of any of the outputs of the firm depend on the costs of the other outputs, and these other costs are also unknown. More precisely, the following system of n equations can be stated: SlnChQhi = S,“,;“ChQhi i = 1,. . . , n.
(13)
Using the equivalent expression BhiN,, instead of the Qhi, and cancelling out N,, the system (13) becomes Si”ChBhi= SrY,“ChBhi
,n.
i = 1,
(14)
Since the technical coefficients Bhi h, i = 1,. . , n are assumed to be known, system (14) has n equations in n unknowns, namely, the nC,,, h = 1,. . . , n. This solves, at least in principle, the problem of estimating the variable costs for the interdependent outputS of the firm. Once the variable costs per unit of output are estimated, it is possible to proceed with the estimation of the break-even values of the different outputs. In principle, the operations described in the system formed by eqns (5) and (7) can be performed. However, it is not acceptable to estimate the different outputs without considering that the products are used as raw materials for other processes within the firm. On the other hand, formula (9) can be modified as follows: P,N, = Fi t Ai + S,“C,,&N,
i = 1,.
,n
(15)
where the Ai retain the definition in (8). From a mathematical point of view, the systems of equations in (9) and (15) are identical. As a consequence, the conclusions reached previously with respect to the solution of (9) apply to the system in (15). 8. ACOMPABISONOFBREAK-EVENANDINPUT/OUTPUTANALYSIS The object of this Section is to call attention to the similarities between break-even analysis with joint fixed costs and interdependent production, on the one hand, and input/output analysis on the other. The basic equations of input/output analysis can be written as follows, using the notation above: S,“QiitMi=N,
i=l,....,
n
(16)
or as S,“Bi,Ni+Mi=Ni
i=l,...,n
(17)
where Mi denotes final demand, or, in the present context, the part of the output of good i that can be sold. It should be observed that in eqn (16) or (17) no attention has been paid to primary inputs, i.e. labor and capital in input/output analysis, or, in the present context, inputs not produced by the firm. The first point that should be observed when comparing break-even and input/output analysis is that the method to estimate variable costs in break-even analysis in eqns (13) and (14) is identical to the method used to estimate equilibrium prices when wages and profits are specified in input/output analysis. On the other hand, the method used to evaluate total output in break-even analysis with eqns (15) differs from that used in input/output analysis. Fixed costs play a key role in the determination of total output with eqns (15), while they are not used at all in input/output analysis. In the latter, final demand, which is not considered in break-even analysis, is the key element to estimate total outputs. From the observations above it follows that breakeven and input/output analysis are two methods that have a closely related mathematical basis, and that produce complementary information. Break-even analysis can be used to estimate the minimum outputs that a firm should produce, and, once this is done, eqns (17) can be used to evaluate the output that the firm can still sell after using some of the output in the production process. Input/output analysis can be used to estimate the amount that should be produced to satisfy internal needs and external demand. This information could be particularly useful for planning the establishment of a new firm. Once a firm is established, the application of eqns (17) could show that the current capacity of a firm is not large enough to satisfy external demand. 9.CONCLUSIONS
In summary, modifications of the index and the allocated joint fixed cost methods to compute break-even values for multiple product firms have been presented in this paper. These modifications make it possible to evaluate break-even values without the assumption that the volumes of output of the different products are fixed a priori. The comparison of the two methods suggests that the allocated joint fixed cost approach might have some advantages. This preliminary conclusion is a by-product of this paper, since a study of the problem of selecting the best method of break-even analysis for multiple product firms is not its main objective. For this, the first step must be to define precisely “better results”. Once this is done, a systematic comparison of the methods studied in this paper, among themselves and with those based on linear and goal programming, would be needed. These questions are left for future research. REFERENCES I. L. Amey and D. A. Egginton, Mana8emenfAccounting: A ConceptualApproach.Logman(1973). 2. H. Bierman and T. Dyckman,ManagerialCost Accounting (1971). 3. D. T. DeCosterand E. L. Schafer,ManagementAccounting: A DecisionEmphasis.Wiley,New York (1976). 4. N. Dopuch,Cost-volume-profit analyses.In HandbookofCost Accounting,(Editedby S. Davidsonand R. L. Weil).McGrawHill (1978). 5. C. T. Horgren, Accounting for Management Control: An
Break-even analysis for multiple product firms Introduction, 3rd. Edn. Prentice Hall, Englewood Cliffs, New Jersey (1974). 6. S. W. Korn and T. Boyd, Accounting for Management, Planning and Decision Making. Wiley, New York (1%9). 7. A. Matz and M. F. Usry, Cost Accounting: Planning and Control. South West Publishing Co. (1976).
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8. E. Solomon and J. J. Pringle, An Introduction to Financial Management, 2nd. Goodyear Publishing Co. (1980). 9. A. Charnes et al. Break-even budgeting and programming to goals. J. Accounting Res. 16-43 (1%3). 10. R. K. Jaedike, Improving B-E Analysis by linear programming techniques. N. A. A. Bull. pp. 5-12 (March l%l).