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Formal multi-product break-even-analysis (Quelltext anzeigen)
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* Analysis of the multi-product case with capacity restrictions. | * Analysis of the multi-product case with capacity restrictions. | ||
A detailed analysis of theses cases is available in Schweitzer/Troßmann (1998) and Schweitzer/Troßmann/Lawson (1992).<br> | A detailed analysis of theses cases is available in Schweitzer/Troßmann (1998) and Schweitzer/Troßmann/Lawson (1992). The [[Accounting exercises 4: Break-even-Analysis|exercise pages]] offer useful examples.<br> | ||
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Note, that different to the previous case without individual fixed costs (and different to the formula proposed in Schweitzer/Troßmann (1998, p. 177), in this case the set of break-even-points is not just a linear combination of the two break-even-corner-points x<sub>1</sub>** and x<sub>2</sub>**. Instead, it still is a linear combination of the break-even-points x<sub>1</sub>* and x<sub>2</sub>*, corresponding to the joint fixed costs, plus the quantities x<sub>1</sub>* and x<sub>2</sub>* to cover the individual fixed costs. | Note, that different to the previous case without individual fixed costs (and different to the formula proposed in Schweitzer/Troßmann (1998, p. 177), in this case the set of break-even-points is not just a linear combination of the two break-even-corner-points x<sub>1</sub>** and x<sub>2</sub>**. Instead, it still is a linear combination of the break-even-points x<sub>1</sub>* and x<sub>2</sub>*, corresponding to the joint fixed costs, plus the quantities x<sub>1</sub>* and x<sub>2</sub>* to cover the individual fixed costs. | ||
== Analysis of the two-product case with restrictions== | |||
If our production process suffers from limited capacity (for example limited availability of inputs, limited market demand for some products, and so on), not all of the points on the break-even-line presented in the unrestricted case are feasible. For our product analysis it is important whether each product has a specific capacity constraint (individual restrictions) or whether one or several capacity constraints affect several products simultaneously (joint restrictions). <br> | |||
Individual restrictions are rather easy to handle. A direct restriction B<sub>j</sub> to the volume of product j (caused for example by limited market size) can be amended to the break-even-line-set. The same is possible if we face an indirect constraint B<sub>ij</sub> of factor i (for example skilled labour) for product j. In this case, a production coefficient a<sub>ij</sub> can capture the relation between product volume and necessary labor. The corresponding restrictions are:<br> | |||
: x<sub>j</sub> ≤ B<sub>j</sub> or <br> | |||
: x<sub>j</sub> ≤ B<sub>j</sub>/a<sub>ij</sub>.<br> | |||
To be sure, for a<sub>ij</sub> = 1, the former case is the same as the latter.<br> | |||
Joint restrictions are more difficult to handle. They occur if several products need the same, limited input factors or try to sell at the same market. In the linear case, with limitational production factors, the sum of inputs i needed for the several products j has to stay below the available capacity B<sub>i</sub>. Since this is the general case which covers the former cases with direct or indirect product-specific restrictions too, the set of break-even-points is given just for this case. | |||
Using vector notation for the capacity restraint, he set of break-even-points is: <br> | |||
: [[Datei:break-even-set-with-restrictions.png]] | |||
== The multi-product case == | |||
Since we are familiar with the two-product case, it is straightforward to generalize it to the multi-product case with n products (j = 1, …, J). Using again the unit prices p<sub>j</sub>, the variable costs k<sup>v</sup><sub>j</sub>, the product-related fixed costs K<sup>f</sup><sub>j</sub> and joint fixed costs K<sup>f</sup>, and the capacity constraints B<sub>i</sub> and production coefficients a<sub>ij</sub>, we get the following set of break-even-points in the multi-product case with joint restrictions.<br> | |||
: | |||
[[Datei:break-even-multi-product-general-set.png ]] | |||
A break-even-line consists of the possible quantities of two products which cover joint fixed costs and linear sales revenue and variable cost functions. In the case of three products we get a '''break-even-plane.''' The combinations of four or more products result in '''break-even-hyper-planes''' which are difficult to present graphically. For such cases, the analysis is restricted to the vector form of the set of break-even-points (as presented above; and in more detail by Schweitzer/Troßmann (1998, pp. 173 ff.; 203 ff.) or Schweitzer/Troßmann/Lawson (1992), Chapter 4).<br> | |||
== Evaluation of explicit multi-product break-even-analyses == | |||
A large number of managerial decisions deal with multi-product or similar problems (for example product or regional market differentiation, or aiming at economies of scope). They combine both individual and joint production processes, and hence, cost. In many cases, the exact outputs of those products or volumes of the markets are unknown. Thus, a break-even- model that provides information about critical outputs or market volumes, is useful. Break-even-corner-points and intuitive graphical representations of the cost-volume-profit-problem support the decision making processes and offer valuable additional information compared to the simplifying use of single-product or single-index break-even approaches. However, the more products we include in the analysis, the more contingent are our break-even-hyper-planes. Whether a certain quantity of a product is sufficient or not, depends on the outputs of (too) many other products. In such situations, other approaches may provide better guidelines for decision-making. Apart from the simplifying break-even-analyses already mentioned, this could be the identification of typical output combinations and the calculation of their profit. | |||
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References:<br/> | |||
[[Break-even-Analyse]]<br> | |||
[[Multi-product break-even-analysis]]<br> | |||
[[Accounting exercises 4: Break-even-Analysis]]<br> | |||
Schweitzer, M.; E. Troßmann; G. E. Lawson: Break-even-Analyses. John Wiley & Sons 1992.<br/> | |||
Schweitzer, M.; E. Troßmann: Break-even-Analysen. 2. Aufl., Berlin 1998 (1. Aufl. 1986).<br/> | |||