ConfirmedUser, Student, Bürokraten, Administratoren
589
Bearbeitungen
Formal multi-product break-even-analysis: Unterschied zwischen den Versionen
Zur Navigation springen
Zur Suche springen
Formal multi-product break-even-analysis (Quelltext anzeigen)
Version vom 25. Januar 2013, 09:36 Uhr
, 25. Januar 2013keine Bearbeitungszusammenfassung
Keine Bearbeitungszusammenfassung |
Keine Bearbeitungszusammenfassung |
||
| (6 dazwischenliegende Versionen desselben Benutzers werden nicht angezeigt) | |||
| Zeile 1: | Zeile 1: | ||
''by Clemens Werkmeister'' | ''by Clemens Werkmeister'' | ||
<html><img src= | <html><img src="http://vg06.met.vgwort.de/na/c7a3898531b948bb8f88a959a26e3156" width="1" height="1" alt=""></html> | ||
== Explicit formal analysis of the multi-product case == | == Explicit formal analysis of the multi-product case == | ||
[[Break-even-Analyse|Break-even-analyses]] for [[Multi-product break-even-analysis|multi-product]] cases have to deal with joint fixed costs for several products, and variable costs, fixed costs and sales revenues that can be linked directly to the products. Measuring | [[Break-even-Analyse|Break-even-analyses]] for [[Multi-product break-even-analysis|multi-product]] cases have to deal with joint fixed costs for several products (or '''common fixed costs'''), and variable costs, fixed costs and sales revenues that can be linked directly to the products '''(direct costs)'''. Measuring multiple products by a single index is one way to solve the problem; allocating (joint) fixed costs to the different products is another way.<br> | ||
Such simplifications are useful ways 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:<br> | Such simplifications are useful ways 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:<br> | ||
* Analysis of the case of two products with joint fixed costs | * Analysis of the case of two products with joint (or common) fixed costs | ||
* Analysis of the case of two products with joint and individual fixed costs | * Analysis of the case of two products with joint and individual (or direct) fixed costs | ||
* Analysis of the two-product case with capacity restrictions | * Analysis of the two-product case with capacity restrictions | ||
* Analysis of the multi-product case with capacity restrictions. | * Analysis of the multi-product case with capacity restrictions. | ||
| Zeile 23: | Zeile 23: | ||
Using '''unit contributions''' d<sub>1</sub> and d<sub>2</sub> respectively, and rearranging results in:<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> | : 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 contribution to be achieved with one product depends on the contribution provided by the other product. In extreme cases, we concentrate on one of the products and set the | It is obvious that the contribution to be achieved with one product depends on the contribution provided by the other product. In extreme cases, we concentrate on one of the products and set the number of units of the other equal to zero. This leads to the '''individual break-even-points''' (or '''break-even-corner-points'''):<br> | ||
: | |||
: | : [[Datei:x_hat_1.png]] = K<sup>f</sup>/d<sub>1</sub> (if x<sub>2</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 | : [[Datei:x_hat_2.png]] = K<sup>f</sup>/d<sub>2</sub> (if x<sub>1</sub> = 0)<br> | ||
: d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · 0.5 · | |||
Since we assumed '''linear cost-volume-profit-relations''' (due to the constant unit prices and variable costs), a 50/50-combination of [[Datei:x_hat_1.png]] and [[Datei:x_hat_2.png]] will cover the fixed costs too. This can be easily seen for x<sub>1</sub> = 0.5 · [[Datei:x_hat_1.png]] and x<sub>2</sub> = 0.5 · [[Datei:x_hat_2.png]]: <br> | |||
: d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · 0.5 ·[[Datei:x_hat_1.png]] + d<sub>2</sub> · 0.5 · [[Datei:x_hat_2.png]] = d<sub>1</sub> · 0.5 · K<sup>f</sup>/d<sub>1</sub> + d<sub>2</sub> · 0.5 · K<sup>f</sup>/d<sub>2</sub> = K<sup>f</sup>.<br> | |||
Other linear combinations enable fixed-cost-coverage in a similar way. Together, the corresponding break-even-points 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> | Other linear combinations enable fixed-cost-coverage in a similar way. Together, the corresponding break-even-points 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> | ||
| Zeile 36: | Zeile 38: | ||
In order to facilitate algebraic formulations, we introduce the price vector (p<sub>1</sub>, p<sub>2</sub>), the variable cost vector (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>), the unit contribution vector (d<sub>1</sub>, d<sub>2</sub>) and the output vector (x<sub>1</sub>, x<sub>2</sub>). The break-even-line is defined by the following output vector: | In order to facilitate algebraic formulations, we introduce the price vector (p<sub>1</sub>, p<sub>2</sub>), the variable cost vector (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>), the unit contribution vector (d<sub>1</sub>, d<sub>2</sub>) and the output vector (x<sub>1</sub>, x<sub>2</sub>). The break-even-line is defined by the following output vector: | ||
: [[Datei:break-even-line-set.png]] | : [[Datei:break-even-line-set.png]]<br> | ||
==The case of two products with individual and joint fixed costs== | ==The case of two products with individual and joint fixed costs== | ||
Sometimes, cost-volume-profit analysis has to cope both with individual and joint fixed costs. Typical examples are '''multi-stage production processes''': Computers are often manufactured for global customers in a single plant (with joint fixed costs) and customized to local specifications (power plugs, manuals, packaging, …) in local facilities. Movie production requires a large joint cost block to get the director’s cut, but finishing, translating, synchronization, subtitles and marketing are done separately for different countries or languages and cause individual fixed costs. <br> | Sometimes, cost-volume-profit analysis has to cope both with individual and joint fixed costs. Typical examples are '''multi-stage production processes''': Computers are often manufactured for global customers in a single plant (with joint fixed costs) and customized to local specifications (power plugs, manuals, packaging, …) in local facilities. Movie production requires a large joint cost block to get the director’s cut, but finishing, translating, synchronization, subtitles and marketing are done separately for different countries or languages and cause individual fixed costs. <br> | ||
In our two-product case, we introduce K<sup>f</sup><sub>1</sub> and K<sup>f</sup><sub>2</sub> to capture the individual fixed costs of products 1 and 2 and get the following break-even-form:<br> | In our two-product case, we introduce K<sup>f</sup><sub>1</sub> and K<sup>f</sup><sub>2</sub> to capture the individual (or direct) fixed costs of products 1 and 2 and get the following break-even-form:<br> | ||
: (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> ≥ K<sup>f</sup> + K<sup>f</sup><sub>1</sub> + K<sup>f</sup><sub>2</sub><br> | : (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> ≥ K<sup>f</sup> + K<sup>f</sup><sub>1</sub> + K<sup>f</sup><sub>2</sub><br> | ||
| Zeile 53: | Zeile 54: | ||
: x<sub>2</sub> ≥ x<sub>1</sub> = K<sup>f</sup>/d<sub>2</sub> <br> | : x<sub>2</sub> ≥ x<sub>1</sub> = K<sup>f</sup>/d<sub>2</sub> <br> | ||
The solution of the two individual conditions results in two break-even-points | The solution of the two individual conditions results in two break-even-points [[Datei:x_circle_1.png]] and [[Datei:x_circle_2.png]]. Considering too for the joint fixed cost leads to two break-even-corner-points [[Datei:x_tilde_1.png]] and [[Datei:x_tilde_1.png]]. The possible break-even-points lie on the line between these corner-points. The graphical presentation of this case is similar to the previous case without individual fixed costs. However, we adjust the horizontal axes (x<sub>1</sub> and x<sub>2</sub>) for the units that are necessary to cover the individual fixed costs.<br> | ||
[[Datei:break-even-line-both.png|thumb|left]]<br> | [[Datei:break-even-line-both.png|thumb|left]]<br> | ||
| Zeile 63: | Zeile 64: | ||
: [[Datei:break-even-line-joint-set.png]] | : [[Datei:break-even-line-joint-set.png]] | ||
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 | 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[[Datei:x_tilde_1.png]] and [[Datei:x_tilde_2.png]]. Instead, it still is a linear combination of the break-even-points [[Datei:x_hat_1.png]] and [[Datei:x_hat_1.png]], corresponding to the joint fixed costs, plus the units [[Datei:x_circle_1.png]] and [[Datei:x_circle_2.png]] necessary to cover the individual fixed costs.<br> | ||
| Zeile 75: | Zeile 76: | ||
Using vector notation for the capacity restraint, he set of break-even-points is: <br> | Using vector notation for the capacity restraint, he set of break-even-points is: <br> | ||
: [[Datei:break-even-set-joint-restrictions.png]] | : [[Datei:break-even-set-joint-restrictions.png]]<br> | ||
| Zeile 88: | Zeile 89: | ||
== Evaluation of explicit multi-product break-even-analyses == | == 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. | 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 [[Controlling|controlling]] 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.<br> | ||