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Multi-product break-even-analysis (Quelltext anzeigen)
Version vom 24. Januar 2012, 14:30 Uhr
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== Idea of standard break-even-analyses== | |||
The basic idea of [[Break-even-Analyse|break-even-analyses]] is <br> | The basic idea of [[Break-even-Analyse|break-even-analyses]] is <br> | ||
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== Extension of break-even-analysis to multi-product cases == | |||
An extension of the break-even-model to cases with several products (the multi-product case), or several other cost and sales drivers, has to take into account how they affect costs and sales. We distinguish specific or '''direct costs''' which are only affected by specific factors and '''joint costs''' which relate to several products or factors. Sales revenues can be handled in the same way as costs. If appropriate, individual sales of a product may be distinguished form joint sales of several products, for example when dealing with an order or a customer analysis. <br> | An extension of the break-even-model to cases with several products (the multi-product case), or several other cost and sales drivers, has to take into account how they affect costs and sales. We distinguish specific or '''direct costs''' which are only affected by specific factors and '''joint costs''' which relate to several products or factors. Sales revenues can be handled in the same way as costs. If appropriate, individual sales of a product may be distinguished form joint sales of several products, for example when dealing with an order or a customer analysis. <br> | ||
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== Simplifying the multi-product to a standard case using index numbers == | |||
Heterogeneous fixed costs and mixed capacity constraints require a complex break-even-model. An easier approach consists in the use of a '''one-dimensional driver''' for the multi-product case. The idea is to reduce the multi-product case to the standard case of break-even-analysis. This is possible if we find a measure (an index number) which is proportional to all product volumes (or other cost drivers; see Dean [Break-even-Analysis] 237 ff.). The proportions might depend on technical characteristics of the products or be determined with respect to marketing plans or strategies. Examples of possible measures are<br> | Heterogeneous fixed costs and mixed capacity constraints require a complex break-even-model. An easier approach consists in the use of a '''one-dimensional driver''' for the multi-product case. The idea is to reduce the multi-product case to the standard case of break-even-analysis. This is possible if we find a measure (an index number) which is proportional to all product volumes (or other cost drivers; see Dean [Break-even-Analysis] 237 ff.). The proportions might depend on technical characteristics of the products or be determined with respect to marketing plans or strategies. Examples of possible measures are<br> | ||
* input or throughput quantities, if there are fixed production coefficients to all products. Typical examples are joint production processes. | * input or throughput quantities, if there are fixed production coefficients to all products. Typical examples are joint production processes. | ||
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== Simplifying the multi-product case to multiple standard cases using fixed cost allocation == | |||
In some cases the sales revenues and the variable costs of several products are easy to determine, but they require some joint inputs and have to keep up with a joint fixed cost block. Examples for such a fixed cost block could be the costs of production for a common input, or even overhead costs for headquarters. If the production and sales volumes of the products are independent of the other products (for example depend on different managers or business units), the use of a pre-defined index ratio would limit the flexible reactions to changes in markets. | In some cases the sales revenues and the variable costs of several products are easy to determine, but they require some joint inputs and have to keep up with a joint fixed cost block. Examples for such a fixed cost block could be the costs of production for a common input, or even overhead costs for headquarters. If the production and sales volumes of the products are independent of the other products (for example depend on different managers or business units), the use of a pre-defined index ratio would limit the flexible reactions to changes in markets. | ||
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== Explicit formal analysis of the multi-product case == | |||
Simplification is a useful way to deal with complex problems in many situations but sometimes simple approaches neglect critical information and exclude useful solutions. A more sophisticated solution can make the difference in a competitive environment. At least, the more sophisticated solution serves as a benchmark and allows both quantitative and qualitative assessments of other approaches. We introduce the explicit analysis of the multi-product case in the following steps: | |||
* Analysis of the case of two products with joint fixed costs | |||
* Analysis of the case of two products with joint and individual fixed costs | |||
* Analysis of the two-product case with restrictions | |||
* Analysis of the multi-product case | |||
* Analysis of the multi-product case with restrictions. | |||
=== The case of two products with joint fixed costs === | |||
If we know the constant unit price (p<sub>1</sub>, p<sub>2</sub>) and unit variable cost (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>) for two products, and the joint fixed costs (K<sup>f</sup>) necessary for their production, we can put the break-even-condition with respect to the production volumes (x<sub>1</sub>, x<sub>2</sub>) in the following way:<br> | |||
E = K = p<sub>1</sub> · x<sub>1</sub> + p<sub>2</sub> · x<sub>2</sub> = k<sup>v</sup><sub>1</sub> · x<sub>1</sub> + k<sup>v</sup><sub>2</sub> · x<sub>2</sub> + K<sup>f</sup><br> | |||
Using unit contributions d<sub>1</sub> and d<sub>2</sub> respectively, and rearranging results in:<br> | |||
K<sup>f</sup> = (p<sub>1</sub>-k<sup>v</sup><sub>1</sub>) · x<sub>1</sub> + (p<sub>2</sub>-k<sup>v</sup><sub>2</sub>) · x<sub>2</sub> = d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub><br> | |||
It is obvious that the necessary contribution of one product depends on the contribution provided by the other product. In extreme cases, we concentrate on one of the products and set the volume of the other equal to zero. This leads to the individual break-even-points (or break-even-corner-points):<br> | |||
x<sub>1</sub>* = K<sup>f</sup>/d<sub>1</sub> (if x<sub>2</sub> = 0)<br> | |||
x<sub>2</sub>* = K<sup>f</sup>/d<sub>2</sub> (if x<sub>1</sub> = 0)<br> | |||
Since we assumed linear cost-volume-profit-relations (due to the constant unit prices and variable costs), a 50/50 combination of x<sub>1</sub>* and x<sub>2</sub>* will cover the fixed costs too. This can be easily seen for x<sub>1</sub> = .5* x<sub>1</sub>* and x<sub>2</sub> = .5 · x<sub>2</sub>*: <br> | |||
d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · .5 · x<sub>1</sub>* + d<sub>2</sub> · .5 · x<sub>2</sub>* = d<sub>1</sub> · .5 · K<sub>2</sub>/d<sub>1</sub> + d<sub>2</sub> · .5 * K<sup>f</sup>/d<sub>2</sub> = K<sub>2</sub>.<br> | |||
Other linear combinations allow fixed-cost-coverage in the same way. Together, they build a straight line between the break-even-corner-points, the break-even-line, as can be seen in the following three-dimensional graph (with x<sub>1</sub> and x<sub>2</sub> as horizontal axes and the profit, contribution and costs at the vertical axis.<br> | |||
t.b.c. | t.b.c. | ||