Multi-product break-even-analysis

by Clemens Werkmeister

Idea of standard break-even-analyses

The basic idea of break-even-analyses is
(i) to identify an important factor which influences both sales revenues and costs, and
(ii) to determine the critical value of that factor that leads at least to the same sales revenues as costs. This critical value is called the break-even-point.
This critical factor can be – as in standard break-even-analyses – the quantity or volume or amount of a product. Other drivers for sales revenues and costs are possible, too, for example the number of customers, the number of product variants, the number of articles to be handled in a warehouse, critical distances for the switch from travelling by train to airplanes, work hours, … . The point is that a standard break-even-analysis focuses on one critical factor and a single fixed cost block only, and it allows the calculation of a critical value of this factor with respect to the fixed cost block in an easy way.

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.
Since variable costs in most cases can be linked directly or indirectly to a single product or factor, this distinction refers mainly to the non-variable or fixed costs (exceptions are joint variable costs in joint production processes). It can be supported by analyses of the underlying production process (see Schweitzer/Troßmann/Lawson 1992, pp. 88 ff.). This results in the following cases:

• If all costs (and sales revenues) are connected to specific products, it is possible to treat the multi-product case as multiple independent single-product cases.
• If all costs are caused by all products, we have the multi-product break-even-analysis with respect to a joint fixed block.
• If parts of the costs are caused by specific products and parts are joint costs, we need to integrate a mixed (or heterogeneous) fixed cost block into the break-even-analysis.
  

A similar distinction is possible with respect to capacity constraints. Capacity constraints may inhibit productions volumes above the break-even-point for some products. Individual capacity constraints affect specific, joint capacity restraints affect several products. In mixed forms a production volume is limited by both individual and joint constraints. Further complications of standard break-even-analysis arise in cases of multi-stage production processes (see Schweitzer/Troßmann (1992), pp. 127 ff.).

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

• input or throughput quantities, if there are fixed production coefficients to all products. Typical examples are joint production processes.
• output measures, if there is a fixed or planned relation between all outputs. This relation can be expressed using sales prices, sales volumes or other equivalence numbers.
  

Whether an input or an output index provides the more reliable and efficient measure, depends on the characteristics of the production process. The result is a product basket, with its respective sales volume and variable cost per basket. If the fixed costs consist of a joint fixed cost block, the break-even-analysis determines the critical minimum quantities of such a basket that is necessary to cover the joint fixed costs. Schweitzer/Troßmann (1998, pp. 129 ff.) explain several examples. Even if some of the products had individual fixed costs, these could be incurred to the joint fixed cost block and covered by the product baskets, too.

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 such cases allocation of the joint fixed costs among the different products allows to convert the problem into multiple standard break-even-problems. Each product has to cover a fraction of the total fixed costs. It is easy to determine a break-even-point for each product. The problem, thus, is transferred to the allocation of fixed cost. Allocation of fixed cost is always arbitrary to the extent that it doesn’t reflect actual cost behavior. But the use of allocation principles can lead to a higher acceptance among the managers affected. The most common cost allocation principles are the use of

• equal weights (just dividing the joint fixed costs by the number of products)
• physical weights (following physical or technical properties of the products)
• cost weights (using overhead factors)
• benefit weights (mainly sales, revenues, profits or contribution margins of the products).

The advantage of this approach is that it allows to include joint fixed costs in standard and easy-to-explain break-even-analysis. A shortcoming is that it limits the set of feasible and sustainable solutions since it excludes a range of combinations of production and sales volumes of the products which would allow fixed-cost-coverage, too.

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 (p₁, p₂) and unit variable cost (k^v₁, k^v₂) for two products, and the joint fixed costs (K^f) 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 = k^v₁ · x₁ + k^v₂ · x₂ + K^f
Using unit contributions d₁ and d₂ respectively, and rearranging results in:
K^f = (p₁-k^v₁) · x₁ + (p₂-k^v₂) · x₂ = d₁ · x₁ + d₂ · x₂
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):
x₁* = K^f/d₁ (if x₂ = 0)
x₂* = K^f/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 = .5* x₁* and x₂ = .5 · x₂*:
d₁ · x₁ + d₂ · x₂ = d₁ · .5 · x₁* + d₂ · .5 · x₂* = d₁ · .5 · K₂/d₁ + d₂ · .5 * K^f/d₂ = K₂.
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 x₁ and x₂ as horizontal axes and the profit, contribution and costs at the vertical axis.

t.b.c.

Break-even-Analyse

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References:
Schweitzer, M.; E. Troßmann; G. E. Lawson: Break-even-Analyses. John Wiley & Sons 1992.
Schweitzer, M.; E. Troßmann: Break-even-Analysen. 2. Aufl., Berlin 1998 (1. Aufl. 1986).