Formal multi-product break-even-analysis - Versionsgeschichte

Wikimeister am 25. Januar 2013 um 07:36 Uhr

2013-01-25T07:36:47Z

← Nächstältere Version		Version vom 25. Januar 2013, 09:36 Uhr
Zeile 1:		Zeile 1:
	''by Clemens Werkmeister''		''by Clemens Werkmeister''

	<html><img src=""http://vg06.met.vgwort.de/na/c7a3898531b948bb8f88a959a26e3156"" width=""1"" height=""1"" alt=""""></html>		<html><img src="http://vg06.met.vgwort.de/na/c7a3898531b948bb8f88a959a26e3156" width="1" height="1" alt=""></html>

Wikimeister: /* The case of two products with joint fixed costs */

2012-01-26T20:37:15Z

The case of two products with joint fixed costs

← Nächstältere Version		Version vom 26. Januar 2012, 22:37 Uhr
Zeile 28:		Zeile 28:
	: [[Datei:x_hat_2.png]] = K<sup>f</sup>/d<sub>2</sub> (if x<sub>1</sub> = 0)<br>		: [[Datei:x_hat_2.png]] = 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> = 0.5· x<sub>1</sub>* and x<sub>2</sub> = 0.5 · x<sub>2</sub>*: <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_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 · x<sub>1</sub>* + d<sub>2</sub> · 0.5 · x<sub>2</sub>* = 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>		: 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 39:		Zeile 39:

	: [[Datei:break-even-line-set.png]]<br>		: [[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==

Wikimeister am 26. Januar 2012 um 20:07 Uhr

2012-01-26T20:07:40Z

Wikimeister: /* The case of two products with joint fixed costs */

2012-01-26T20:04:02Z

The case of two products with joint fixed costs

← Nächstältere Version		Version vom 26. Januar 2012, 22:04 Uhr
Zeile 22:		Zeile 22:
	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 ~~volume~~ of the other equal to zero. This leads to the individual break-even-points (or '''break-even-corner-points'''):<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 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>		: [[Datei:x_hat_1.png]] = K<sup>f</sup>/d<sub>1</sub> (if x<sub>2</sub> = 0)<br>

Wikimeister: /* Explicit formal analysis of the multi-product case */

2012-01-26T20:02:31Z

Explicit formal analysis of the multi-product case

← Nächstältere Version		Version vom 26. Januar 2012, 22:02 Uhr
Zeile 6:		Zeile 6:
	== 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 ~~the~~ 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>		[[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.

	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>		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>


	== The case of two products with joint fixed costs ==		== The case of two products with joint fixed costs ==

Wikimeister: /* The case of two products with individual and joint fixed costs */

2012-01-26T20:00:58Z

The case of two products with individual and joint fixed costs

← Nächstältere Version		Version vom 26. Januar 2012, 22:00 Uhr
Zeile 42:		Zeile 42:
	==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 54:		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 x<sub>1</sub>* and x<sub>2</sub>. Considering for the joint fixed cost leads to two break-even-corner-points x<sub>1</sub>* and x<sub>2</sub>**. 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 ~~volumes~~ that are necessary to cover the individual fixed costs.<br>		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 64:		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 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[[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.


	== Analysis of the two-product case with restrictions==		== Analysis of the two-product case with restrictions==

Wikimeister: /* The case of two products with joint fixed costs */

2012-01-26T19:55:08Z

The case of two products with joint fixed costs

← Nächstältere Version		Version vom 26. Januar 2012, 21:55 Uhr
Zeile 24:		Zeile 24:
	: 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 volume of the other equal to zero. This leads to the individual break-even-points (or '''break-even-corner-points'''):<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 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>		: [[Datei:x_hat_1.png]] = K<sup>f</sup>/d<sub>1</sub> (if x<sub>2</sub> = 0)<br>
			: [[Datei:x_hat_2.png]] = 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> = 0.5· x<sub>1</sub>* and x<sub>2</sub> = 0.5 · x<sub>2</sub>*: <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> = 0.5· x<sub>1</sub>* and x<sub>2</sub> = 0.5 · 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> · 0.5 · x<sub>1</sub>* + d<sub>2</sub> · 0.5 · x<sub>2</sub>* = 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>		: d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · 0.5 · x<sub>1</sub>* + d<sub>2</sub> · 0.5 · x<sub>2</sub>* = 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>
Zeile 37:		Zeile 39:

	: [[Datei:break-even-line-set.png]]		: [[Datei:break-even-line-set.png]]


	==The case of two products with individual and joint fixed costs==		==The case of two products with individual and joint fixed costs==

Wikimeister am 26. Januar 2012 um 19:52 Uhr

2012-01-26T19:52:52Z

← Nächstältere Version		Version vom 26. Januar 2012, 21:52 Uhr
Zeile 30:		Zeile 30:
	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>

	[[Datei:break-even-line.png]]		[[Datei:break-even-line.png\|thumb\|left]]<br>

			<br style="clear:left;"/>

	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:
Zeile 53:		Zeile 55:
	The solution of the two individual conditions results in two break-even-points x<sub>1</sub>* and x<sub>2</sub>. Considering for the joint fixed cost leads to two break-even-corner-points x<sub>1</sub>* and x<sub>2</sub>**. 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 volumes that are necessary to cover the individual fixed costs.<br>		The solution of the two individual conditions results in two break-even-points x<sub>1</sub>* and x<sub>2</sub>. Considering for the joint fixed cost leads to two break-even-corner-points x<sub>1</sub>* and x<sub>2</sub>**. 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 volumes that are necessary to cover the individual fixed costs.<br>

	[[Datei:break-even-line-both.png]]		[[Datei:break-even-line-both.png\|thumb\|left]]<br>

			<br style="clear:left;"/>

	In a formal way this '''break-even-line''' is captured by the set:<br>		In a formal way this '''break-even-line''' is captured by the set:<br>
Zeile 71:		Zeile 75:
	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-~~with~~-restrictions.png]]		: [[Datei:break-even-set-joint-restrictions.png]]

Wikimeister am 26. Januar 2012 um 10:40 Uhr

2012-01-26T10:40:45Z

@@ Zeile 14: / Zeile 14: @@
 * 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>
@@ Zeile 60: / Zeile 60: @@
 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.

Wikimeister: Die Seite wurde neu angelegt: „''by Clemens Werkmeister'' == Explic…“

2012-01-26T10:06:49Z

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== 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 the multiple products by a single index is one way to solve the problem; allocating (joint) fixed costs to the different products is another way. 
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: 

* 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 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). 

== The case of two products with joint fixed costs ==

If we know constant '''unit prices''' (p1, p2) and unit '''variable costs''' (kv1, kv2) for two products, and the '''joint fixed costs''' (Kf) necessary for their production, we can put the break-even-condition with respect to the production volumes (x1, x2) in the following way: 
: E = K = p1 · x1 + p2 · x2 = kv1 · x1 + kv2 · x2 + Kf 
Using '''unit contributions''' d1 and d2 respectively, and rearranging results in: 
: Kf = (p1-kv1) · x1 + (p2-kv2) · x2 = d1 · x1 + d2 · x2 
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 volume of the other equal to zero. This leads to the individual break-even-points (or '''break-even-corner-points'''): 
: x1* = Kf/d1 (if x2 = 0) 
: x2* = Kf/d2 (if x1 = 0) 
Since we assumed '''linear cost-volume-profit-relations''' (due to the constant unit prices and variable costs), a 50/50-combination of x1* and x2* will cover the fixed costs too. This can be easily seen for x1 = 0.5· x1* and x2 = 0.5 · x2*: 
: d1 · x1 + d2 · x2 = d1 · 0.5 · x1* + d2 · 0.5 · x2* = d1 · 0.5 · Kf/d1 + d2 · 0.5 * Kf/d2 = Kf. 
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 x1 and x2 as horizontal axes and the profit, contribution and costs at the vertical axis. 

[[Datei:break-even-line.png]]

In order to facilitate algebraic formulations, we introduce the price vector (p1, p2), the variable cost vector (kv1, kv2), the unit contribution vector (d1, d2) and the output vector (x1, x2). The break-even-line is defined by the following output vector:

: [[Datei:break-even-line-set.png]]

==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. 
In our two-product case, we introduce Kf1 and Kf2 to capture the individual fixed costs of products 1 and 2 and get the following break-even-form: 

: (p1-kv1) · x1 + (p2-kv2) · x2 = d1 · x1 + d2 · x2 ≥ Kf + Kf1 + Kf2 

Obviously, we only accept output-quantities x1 and x2 that cover their individual fixed costs. Thus, the individual break-even-conditions hold too: 
: d1 · x1 ≥ Kf1 
: d2 · x2 ≥ Kf2 

This results in the condition: 
: x1 ≥ x1 = Kf/d1 
: x2 ≥ x1 = Kf/d2 

The solution of the two individual conditions results in two break-even-points x1* and x2*. Considering for the joint fixed cost leads to two break-even-corner-points x1** and x2**. 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 (x1 and x2) for the volumes that are necessary to cover the individual fixed costs. 

[[Datei:break-even-line-both.png]]

In a formal way this '''break-even-line''' is captured by the set: 

: [[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 x1** and x2**. Instead, it still is a linear combination of the break-even-points x1* and x2*, corresponding to the joint fixed costs, plus the quantities x1* and x2* to cover the individual fixed costs.

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	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>		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>


	== The case of two products with joint fixed costs ==		== The case of two products with joint fixed costs ==
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	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==
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	: [[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[[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.		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>


	== Analysis of the two-product case with restrictions==		== Analysis of the two-product case with restrictions==
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	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>


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	== 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>