Abstract:

The classification of various cost categories is introduced to guarantee a differentiated analysis of the profit centers' income statements. In the chemical industry the problems of production cost determination with partially cyclic and partially coupled production processes (by-products) are handled satisfactorily through formulation of a matrix model. This model is iteratively applied to each single cost category according to a well defined number of them. Thus all different kinds of cost accounting such as full, direct, and standard costing are possible.

More detailed considerations within these objectives require refinements of the original model, which was based on the traditional Leontief theorem. Using as input (intermediate) products or recycled goods, the splitting of total cost into various cost categories does not necessarily remain constant: The cost categories may cross over into each other or partially aggregate at the separation and calculation of costs for contents and packaging - the cost categories are »oscillating«.

In the packaging area we find (intermediate) products, whose input may be classified as both »packaging« and »contents«, this depending on the characteristics of their respective input processes - moreover we find similar situations at the reuse of damaged goods in the recycling case. The solution of the problems will be an approach, that analyzes all process formulas, describes the relevant properties by 0-1-values, and keeps them in separate »packaging« matrices. Based on these results and on the former Leontief model a sequence of eliminations and transformations with the quantity coefficients of production will take place. Thus an additional set of rules within the refined framework of the reflection on production costs will be fulfilled.

The algorithms satisfy all requirements of a centralized input-output-oriented costing model for a multi-division corporation using complex loop, staged, and coupled production processes. The consequences for decentralized costing models with pre- fpost-calculations considering the specific market constraints of individual business sectors are also taken into account.

1. Introduction

In the chemical industry the problems of production cost determination with partially cyclic and partially coupled production processes (»by-products«) are handled satisfactorily through formulation of a matrix model.

At the production of n different products in n
respective cost centers the following system of
linear equations (i= 1,..., n) has to be solved
(notation in view of computer languages):

The model parameters have the following
meaning:

• k(i): Primary cost in cost center i per
unit (product).
Cost vector k is known.

• x(i): Total cost in cost center i per unit
(product).
Cost vector x has to be determined.

• a(i,j): For the production of product iin
cost center i the quantity a(i, j) of
product j is necessary.

Three matrices are introduced:

• A(n, n) : »Basic« interlacing matrix of the
a(i,j)

• I(n, n) : »Identity« matrix

• M(n, n) : »Balanced« interlacing matrix:
M = I-A

Now the interrelations of costs can be described:

The calculation of U different cost categories is
given by the solution of the equivalent number
of systems of linear equations:

For disjunct cost categories is true:

In general, the separation of total cost (xG) of
intermediate products into variable cost (xH)
and fixed cost (xL)

is transferred to a finished product, which has
as components several intermediate products:

Product 1 (Plastic bottle):

Product 2 (1 kg glue):

For the finished product 3 (1 kg glue packaged
in a plastic bottle) is true:

Two specific cost categories - variable and fixed — had been chosen for the above considerations. The results cannot be arbitrarily transferred to any set of selected cost categories. The demonstrated correlations are not valid any longer, if a framework of rules is formulated, that does not keep constant the cost categories as they were determined in a specific previous production process. The concept of »oscillating« cost categories will be illustrated by a practical example. Then the procedures and and algorithms for the solution of the problem will be developed.

2. Problem definition

The separation of total cost into costs for packaging and contents (contents as the essential component to be sold) cannot always be treated in the above described way. A set of calculation rules may cause problems, that are not solvable through the simultaneous approach:

The following considerations are mad<

Performance of separate calculation for intermediate
and finished products for both of
their components

»contents« and »packaging«

Depending on an intermediate product's input processes its production cost components »contents« and »packaging« may be transferred to the produced products (as output of these processes) in three different ways:

— separately as calculated for this input pro
duct

- as the total cost with the character »contents«

- as the total cost with the character »packaging«

The aggregated cost of all production processesmust be transferred to and shared by the finished products to be sold. Up to a certaindegree damaged products may be recycled— i.e. parts of them become input again in previous production stages. Special arrangementsfor the calculation aspects have to be made: Not only the cost of the intact parts of

such recycled products, but their total cost
are to be considered.

The separation of total cost xG into two cost
components »contents« (xW) and »packaging«
(xV) is considered.
Each product i has the cost relation
xG(i) = xW(i) + xV(i) or graphically:

The theoretically formulated requirements to a
cost calculation are now illustrated by three
practical examples:

Example 1

In the standard case, the separation of costs of
a product consistently is respected, whenever
this product serves as input in subsequent prc
cesses:

The separation of costs of a plastic bottle filled with glue is not changed, when this bottle is packed together with other glueing material for a do-it-yourself-kit (as a new product).

Calculated production costs of the bottl

Costs are treated at the calculation of the kit:

Example 2

A glue (product q) was calculated with total cost xG(q) and the two cost components xV(q) and xW(q). This glue may have various input conditions in subsequent production processes:

• In a filling plant (packaging process) acid is filled into barrels. Glue is used to stick the labels on the barrels (glue as packaging material). For the production cost components of the glue is true (the symbol »&« will mean, that the cost components may change at the input of the product in particular subsequent processes — the characteristic is not necessarily

The total cost (xG) of the glue are regarded as
packaging cost:

Glue as an input material brings no cost parts
for the characteristic »contents«: xW&(q) = 0.

Therefore the costs of the glue are regarded as
follows:

Calculated production costs of the glue:

Costs are treated at the calculation of the filled barrel:

The glue is used as raw material to produce
cartons. Here the glue's total cost xG(q) are
regarded as pure cost components for the
contents; the glue does not cause packaging
cost:

Calculated production costs of the glue:

Costs are treated at the calculation of the carton:

It may be pointed out that the cartons themsel-
ves are pure packaging material in subsequent
processes. The involved calculation problems
are covered by the following considerations.

Example 3

A similar situation exists, if »retoured« goods must be taken into account. Parts of finished products, that were damaged in the chain of production(packaging — transport — stocking), become input again (»recycling«): The total cost of these products (even though only parts of them can be reused) are taken into account. From the different points of view this is illustrated by the discussion of two filling processes:

Filling of acid into barrels

2% of the filled acid - finished product a -
get spoiled in storage or run out. The con-

It is not very difficult to cover the linear part of the correlations from above by mathematical procedures. Some influences, however, make the algorithms more complicate: Multi-loop, coupled production, and the fact, that the cost components of selected products do not get standardized treatment. Depending on the respective input process their two cost categories are not kept constant, they are »oscillating«, this depending on their particular use at their input. Therefore three input characteristics must be distinguished:

• (N) - Standard: Costs separated into contents
and packaging

• (W) - Contents: Input with total cost as
contents

• (V) - Packaging: Input with total cost as
packaging

Due to this definition of »oscillating« cost categories the simultaneous approach via the »batents tents- acid of value xW(a) - is lost. The barrel itself with value xV(a) is intact and can be reused as packaging material with the cost: xV&(a) = xG(a) and xW&(a) = 0.

Filling of granulated glue into bags

2% of the bags filled with granulate - finished product b - burst during transportation activities. The packaging material with value xV(b) is lost. The contents can be recollected and packed again. These 2% of the granulated glue are recycled with the cost:

lanced« (Leontief) interlacing matrix M operating
with the two submodels

cannot be successful. The necessary »manipulations«
on the input-output-matrix-model are
developed.

Additional problems arise, when certain specifications are fixed before the calculations can be started. Those requirements for intermediate and finished products may result form current marketing constraints; three types of »cost fixing« exist:

• Fixed cost:
A product is given a fixed cost value KO

• Cost bounds:
Kl as lower and K2 as upper bounds for the

• Cost relations:
Cost of product a must be lower or higher
than cost of product b

In all these cases the direct approach

fails, too.

Models are needed with features that allow and support interactive manipulations in the calculation procedures and yet fulfill the complex framework of calculation rules.

3. Solution and algorithms

Dedicated mathematical procedures must be developed to guarantee a cost calculation with the background of above described rules of calculation.

Separate calculations for the cost components »contents« and »packaging« must be performed. They have to respect the cost components of input products as calculated in previous stages of their production. Each intermediate product transfers its components to the (output) products relative to its own (input) processes. It was pointed out that according to the use of the input product the cost categories may »oscillate« — in the extreme case the same input product is to be treated in three different wavs.

We discuss an intermediate product q, which is input for the three production processes R, S, T, that have the output products r, s, t respectively. One of the three basic input characteristics is found in each process:

This table illustrates again (»xV(q) + xW(q)«), that multi-loop production cycles and »retoured« goods (recycling) do not allow the standard approach. A stepwise calculation, that determines in the first step

Cost components »contents«:

and in the second step

Cost components »packaging«:

cannot be successful,

When calculating the cost components »contents« (see process S) the cost component »packaging« is to be used, although not yet determined at this point. Starting the calculation with »packaging« the same problem arises - vice versa.

The essential starting point for the solution is
the interpretation of the consequences of the
two new input »characteristics«

(W): Input with total cost as »contents«
(V) : Input with total cost as »packaging«

The cost separation into components of contents and packaging has to be neglected for several intermediate products in selected processes. The first requirement (»Calculate separately contents and packaging !«) seems to be in conflict with the second one (»Use the sum of the two components as input value for either the contents or the packaging component !«). And this applies not uniformly for all processes: For the same product q we find both characteristics (W) and (V) - and in addition the standard case (N), of course.

The starting point for the development of the
algorithms is based on the demand, that the

sum of the two components is used as input: The procedure starts with the determination of the total cost (xG). The second step will then arbitrarily calculate one of the two cost categories, for example the packaging parts. Choosing this succession the cost component »contents« will then be the balance:

It was explained, that the packaging cost cannot
be calculated via the direct approach

The calculations only can be continued after certain »manipulations« have been performed in the matrix model. Nature and scope of the necessary operations will be developed from the framework of cost calculations as it is applied to product q in the processes R, S, T.

It will be evident, that the algorithms must be based on the »basic« interlacing matrix A. The »balanced« Leontief matrix M contains already a »mixture« of the influences and does not allow the necessary separation of costs. The arithmetic interrelations will be transformed into generalized expressions, which allow easy handling by electronic data processing.

Process R - Characteristic (N)

At the calculation of product r the two cost components xV(q) und xW(q) of intermediate product q are »added« to the components xV(r) and xW(r) respectively.

xV(r) is an unknown quantity in the calculation
step »packaging« : a(r, q) X xV(r) is part of the
calculation.

Process S - Characteristic (W)

At the production of product s the intermediate product q is pure raw material; q does not influence the calculation of the packaging components.

During this step of calculation :a(s,q) = 0.

Process T— Characteristic (V)

At the production of product t the intermediate product q is used as pure packaging material, this with the cost value xV&(q) = xG(q) from the first calculation step »total cost«.

Thus a(t, q) X xV(q) = a(t, q) X xV&(q) is a known quantity and from the calculation's point of view it only does represent pseudo-secondary cost. This cost component is balanced with the corresponding primary cost kV(t) — the also known quantity of the right hand side of the linear equations' system. Then the calculation is continued, as now a(t, q) = 0, because of the correction of kV(t) via a(t,q) X xV&(q).

The three typical production processes R, S, T show, what kind of manipulations are performed in the step »packaging«, when the input of product q takes place. This discussion allows the development of the formalism, which is the precondition for an algorithmic treatment of the allied input-output-problem.

The bilateral input-output-relations have to be analyzed for all (intermediate) products being involved in the production processes and in the coherent cost calculation. The results are described by 0-1-relations and mapped by incidence

For any possible relation (i, j= 1,..., n)

»Input of product j at the production of
product i«

two incidence coefficients d(i, j) and e(ij) together with their related incidence matrices D(n, n) and E(n, n) have to be established. Their values are derived from the following table:

For each element a(i,j) of the interlacing matrix A(n, n) two multiplications with these incidence coefficients are defined: At first a(i, j) is to be multiplied by d(i, j), then separately by e(i,j). Two matrices describe the results:

Now the following transformations must be
performed:

Interlacing matrix A —» »Packaging matrix« B
Primary cost kV -^kV + CxxG

The respective deductions from the interlacing matrix A together with their resulting linear equations' systems realize the cost considerations from above. Within the step of packaging calculation each product is guaranteed to be treated in the following way:

No cost contribution at its input as pure
component of contents (coefficient = 0).

The known quantity »total cost« from the previous calculation step is the cost contribution at its input as pure packaging material. Then the respective coefficient is modified.

In the standard case no interference into the
system of linear equations takes place.

The calculation of production costs — according to the cost components »contents«, »packaging«, »total« - is performed in these steps:

Transformation of the interlacing matrix A: For each product its input relations at the production of other products are analyzed. Based on the results of this analysis the matrix A is transformed into the two matrices B andC.

Calculation of the total cost xG:

Calculation of the packaging cost xV:

Calculation of the cost of contents xW:

4. Example

An example will show the consequences of the
defined cost calculation framework; moreover
it will verify the derived procedures.

Six production processes (n = 6) with cyclic interlacings are chosen. Their interrelations are described in a table. In addition a chart will make the tracing of the costs much easier.

The following chart will show the interlacing of
the processes:

That means for the figures of the example:

5. Application and experience

The demand for solutions of the described
problems comes from an existing production
and calculation world in the chemical field.

These procedures were developed and successfully
implemented on a large data processing
unit.

The specific production conditions such as

coupled production

production loops

alternative processes

together with a product range of about 8000
items lead to an input-output-model of high dimension

The realization and the application of this model do not cause performance problems on the computer. This was achieved by decomposition and iteration algorithms in combination with powerful routines checking the consistency of all production processes according to their quantity and value figures.

This calculation instrument is used regularly - global cost calculations for all divisions of the enterprise are normally performed in each quarter of the year. In addition there are separate calculations in the planning or inventory periods; moreover specific divisions may get extra treatment.

Besides this centralized calculation system several decentralized calculation models were developed. They take care of the very specific conditions of selected divisions. These systems allow flexible reactions to changing marketing situations of the corresponding business sectors.

Decentralized calculations supply the centralized system with preliminary information and pre-calculated values. Based on this information the model parameters and the interlacing coefficients can be fixed — this is necessary for the consistency of the formulation of the centralized »overall calculation«, that covers the whole product range of the company.

6. References

C. G. Cullen: Matrices and Linear Transformation
London 1966.

W. Duck, M. Bliefernich: Operationsforschung Bd. 3,
Berlin 1972.

S. E. Elmaghraby: A note on the »Explosion« and »Netting« Problems in the Planning of Material Requirements. Operations Research, Vol. 11, p. 530-535. Baltimore (Maryland) 1963.

L. R. Ford, D. R. Fulkerson: Flows in networks, Princeton
University Press, Princeton (N.J.) 1962.

E. Heinen: Industriebetriebslehre - Entscheidungen
Industriebetrieb. Wiesbaden, 1972.

* J. Kornai: Mathematical Planning of Structural Deci
sions, Vol. 45 of the Series »Contributions to Economii
Analysis«, ed. by Johnson a.0., Amsterdam 1967.

H. Langer, E.-M. Borst: Die Ermittlung der Produktionsmengen
bei verbundener Produktion. Ablauf und
Planungsforschung Bd. 4 Munchen, Wien 1965.

W. Leontief: Input-Output Economics, New York 1966

H. Miinstermann: Vcrrechung innerbetrieblicher Lei stungen mit Hilfe des Matrizenkalkiils. In: Beitråge zui Lehre von der Unternehmung (Festschrift fur Kar Schafer). Zurich 1968.

W. H. Micrnyk: The Elements of Input-Output Analy
sis, New York 1966.

* O. Pichler: Kostenrechnung und Matrizenkalkiil. Ablauf
- und Planungsforschung Bd. 2, S. 29-46. Miinchen,
Wienl96l.

* A. Vazsonyi: Scientific Programming In Business And
Industry. New York, 1958.

* M. Welscheid: ..Online-calculation in Multi-loop and
Multi-stage Production-cycles« in »EURO VII - Se-

* G. Wohe: Einfuhrung in die Allgemeine Betriebswirtschaftslehre.
Munchen, 1986.

* J. Yamada: Theory and Application of interindustry
analysis, Kinokuniya Bookstore Comp., Ltd., Tokio
1961.