Stochastic Process and Model

Stochastic Process Costing Models

Author(s): A. Wayne Corcoran and Wayne E. Leininger

Source: The Accounting Review, Vol. 48, No. 1 (Jan., 1973), pp. 105-114

Published by: American Accounting Association

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Stochastic Process Costing Models

A. Wayne Corcoran and Wayne E. Leininger

INTRODUCTION

HE pattern of the cost flows within a

T process costing system is a function of product flows within the corre- sponding production system. In a deter- ministic production system, all units would pass in a common sequence through the processes and exit the system as final prod- uct. However, many mass production sys- tems cannot be classified as deterministic.

Any unit entering a production system can generally be expected to exit the sys- tem in one of several forms. Even units that exit the system in the same form may not have cycled through the production processes in identical sequences. The pro- cess costing models developed in this paper are based on the premise that product flows within a mass production system can be described stochastically.

Time will be handled in two ways result- ing in two models. Standard costs will be integrated into the models for the purpose of costing output and inventories. In the final sections cost analysis based on the models will be treated.

TRANSFER PERIOD MODEL'

Assume a multiple process production system with n processes and m output states. For example, a system might have three processes and two output states. Output from the system is either final product from the third process or defective units from the second and third processes. In our example, let the probability asso-

ciated with transferring units from the first process to the second process equal one. From the second process, if the prob- ability of a unit being transferred to the third process is 0.75 and from the third process the probability of a unit exiting the system as a completed unit is 0.75, then we can represent the system as an absorbing Markov chain. The stochastic matrix describing such a system is:

0 1 G 0 0

0 0 3/4i 0 1/4 0 0 0 3/4 1/4

00 O 1 0

.0 0 0 0 1

where each element pij represents the probability of a unit being transferred from process i to process j2. The P matrix could

1 _{The transfer period model is based on material de-}

veloped by: R. M. Cyert, H. J. Davidson and G. L. Thompson, "Estimation of the Allowance for Doubtful Accounts by Markov Chains," MANAGEMENT SCIENCE

(April, 1962), pp. 287-303. The proofs of the formulas in this study are similar to the proofs of Cyert, Davidson, and Thompson.

2 _{In a stochastic matrix the values of all the elements}

are in the range O<Pij< 1. The sum of the elements of each row of the matrix equals one.

A. Wayne Corcoran is Professor of A ccount- ing at The University of Massachusetts and Wayne E. Leininger is Assistant Professor of Accounting at Virginia Polytechnic Insti- tute and State University.

106 The Accounting Review, January 1973 be relabeled as:' n m Q

IR

Elements of the Q submatrix indicate the expected transfers among the n produc- tion processes. In the R submatrix the non- zero elements indicate which production processes are expected to communicate directly with the absorbing states.4 The I submatrix is an identity matrix. Once a unit entered one of the m absorbing states (exited from the system as a finished prod- uct or a defective unit), it could not be transferred to any other state within the system. All elements of the 0 submatrix equal zero, indicating that the absorbing states do not communicate with the pro- duction processes.

The mean number of times a unit would pass through the process before exiting the system, given the entering processes were known, is:5

F = (I - Q)-1 (1) In our example, the F matrix is:

1 1 1 0]

0 1 I

_O 0 1_

-1 0 0- _ 1 0--

= 1 0 _ O 3

Units starting in the initial process, before exiting the system, on the average would pass through the first two processes once, and the third process 3 _{of a time. A similar}

interpretation can be given to the other rows of the F matrix.

Let the n element row vector k contain the new elements started into production at the beginning of each transfer period. The steady-state, in-process inventories are:

vI = kF (2)

It is possible to determine the upper bounds of the variances of the steady- state, in-process inventories. Let a vector g be the initial probability vector of a Markov chain where g= (1/E= ki)k. The upper-bound formula for the inven- tory variances is:'

Var (v) < ka(gF - gq(I - Q8q)-') (3) where a is an n element sum vector, and the subscript sq indicates that each ele- ment in the appropriate matrix is squared. Extending our example, if we assume a k vector of [100, 0, 0], then by (2) and (3) the following could be determined about the expected steady-state inventory levels of the system.

[100-

0

-

[01

B = FR (4)

contains the probability that units enter-

I _{Based on Kemeny and Snell, Finite Markov Chains}

(Van Nostrand, 1960,, p. 44). The P matrix was trans- posed so that the states would be more consistent with the way one would conceive of a production system.

' The first absorbing state will always be the final product of the system. The final product is considered to be the primary output of the system.

'For proof, see Kemeny and Snell, p. 39.

6 _{For proof, see Cyert, Davidson, and Thompson,}

p. 286.

Corcoran and Leininger: Process Costing 107 ing the system in a process will exist in a

particular state. In the example, the B matrix is:

[9/16

Therefore, a unit started in the first process would exit the system as a final product with a probability of 9/16 and as a defective unit with a probability of 7/16. The other rows of the B matrix are inter- preted in the same way.

Expected outputs from the steady-state, in-process inventories can be identified. The expected output that would be realized if production was completed on the inventories equals:

yi= vB (5) To determine the variances of the ex- pected outputs from the steady-state, in- process inventories, the n element prob- ability vector h is required where h

= (1/ En , vi)v. The variances of the ex- pected outputs from the inventories are:

Var (yj) = va(hB - (hB)aq) (6) From our example, the expected outputs, variances and standard deviations from the inventories by (5) and (6) are:

[ 168.751 =106.25]; -66.275- Var (yi) = L66.2751; 8.1- Std. Dev. (yi)

If k units are started into production each period, then the steady-state outputs

for each transfer period are:

y= kR (7)

The upper-bound formula for the vari- ances of the steady-state outputs is:

Var (y) < ka(gB - gaq( -Qsq)-1Rsq) (8) From our example, the steady-state out- puts are: -56.25- -111.17 Y I; Var (y) < 1. L43.751 Liii.ii F10.5- St. Dev. (y) <

In this system the expected unit trans- fers can be determined employing matrix algebra by altering the inventory vector; however, a more direct means using al- gebra will be specified. Let W be an n by (n+m) matrix where each element wij would equal the expected unit transfers from state i to state j. Each element of the matrix is determined by:

Zvij = _Vipij (9)

Perhaps when the W matrix is parti- tioned as shown below, the mapping of the expected transfers can be better under- stood.

n m

L Expected I Expectedi

Transfers I Output

In the example, the W matrix equals:

-0 100 l0 0

01

W= 0 0 75: 0 25

w

L

0

0

108 The Accounting Review, January 1973 transfer period equals:

x = EV' (10)

where E is a z by n matrix. Each element eij equals the amount of resource i re- quired for a unit of activity in process j, and z equals the number of resources in- puts. Let c equal a z element vector con- taining the standard costs of the resource inputs. The vector of standard costs for units of activity in the process equals:

s= cE (11)

The expected standard cost of external inputs required by process i equals:

ej= sjvj (12)

In a system with multiple entry points, units could enter the production system through various processes. Costing the outputs and inventories in such a system would be complex but not beyond the capability of the model. Unit transfer aspects of the model take into account the possibility of multiple entry points. The costing phase of the model has only been designed for a single entry system. In a multiple entry system, different transfer and output costs would be required for each entry point. Output and inventories would be costed based on the expected entry point.

The possibility of consequential trans- fers is another condition that would com- plicate the allocation of costs. Nonse- quential transfers will be considered in a later section of this paper. The costing of a system with multiple entry points and consequential transfers would result in complexities that are not essential to the objectives of this paper.

In a single entry system with sequential transfers and only the final product costed, the standard transfer costs equal:

j / i

dj= E Si! HPi+i.i (13) iX i=X

where X is the initial process. These costs are used for costing inventories and inter- process transfers. Output from the system is costed at:

Uj =?fxiss)/ f P+ii (14)

i=X i=X

These costs would be attached to the out- put of final product from process j.

Inventories of processj would be costed at:

0j = d vj (15)

If 5ij equaled the expected cost of units transferred from state i to state j, then:

i = 1 ton bj = wi1di for 1

j 1lton (16) and

3n,n+l = Wn,n+lUn (17)

Our example will be continued assum- ing the following resource input matrix E and standard cost vector c.

-2 0 0

E=[3 3 0

1 2 3

c = [$1.00, $2.00, $3.00, $1.50] By (10) the vector of resource inputs is:

20 2 0 0 100-

600 1 3 3 0 100 525 1 2 3L 75]

_275IL 1

1

1

From formula (11) the vector of standard activity costs is:

[$12.50, $13.50, $10.50]

= _{[$1.00, $2.00, $3.00, $1.50]1} 3

Corcoran and Leininger: Process Costing 109 and from formula (12) the cost of the ex-

ternal inputs is:

y = [$1,250, $1,350, $787.50] In conclusion, by (13), (14), and (15), the standard transfer costs, output costs, and inventories equal respectively:

d = [$12.50, $26.00, $45.18] u = [$12.50, $34.67, $60.22]

0 = [$1,250, $2,600, $3,388.50] The expected transfer cost matrix by (16) and (17) is:

[0 $1250 0 0 ?

b5 0 0 $26005 0

t0 0 0 $3387.50 0] A cost reconciliation is shown in Exhibit A8.

The model as formulated requires that all transfers take place at uniform time intervals. The duration of the time inter- vals depends on the characteristics of the production system. In a school it would be the length of a semester or a quarter; in a hospital it might be a day; and in a manu- facturing system it could vary from sev- eral minutes to days, weeks, or even

months. To overcome the requirements that the expected production times in each process must be equal, dummy or holding states could be introduced to balance the system.

There is another way to handle time and employ the stochastic framework. In this case, termed the production period, the time span is assumed to be any specified length. It is necessary that the beginning inventory levels are known and that end- ing inventory levels and desired outputs are prescribed by management. To sim- plify notation, we shall not consider in- ventories in this system. This is not in- consistent with suggested input-output models of production systems. When pro- duction is completed on a unit, the unit will be transferred, and the transfers are governed by a stochastic matrix. Under these conditions, it is not possible to dis- cuss any probable inventory levels during the production because the state of the system is indeterminant. We now proceed to develop the production period model along with an example.

8 _{The outputs of the interprocess transfers in the cost}

reconciliation result from transposing the n by n portion of the a matrix.

EXHIBIT A

TRANSFER PERIOD COST RECONCILIATION

Process 1 Process 2 Process 3 Total

Inputs Beginning Inventory $1,250.00 $2,600.00 $3,388.50 $ 7,238.50 External Inputs 1,250.00 1,350.00 787.50 3,387.50 Inter-Process Transfer 0.00 1,250.00 0.00 1,250.00 Inter-Process Transfer 0.00 0.00 2,600.00 2,600.00 Inter-Process Transfer 0.00 0.00 0.00 0.00 Total Inputs $2,500.00 $5,200.00 $6,776.00 $14,476.00 Outputs Inter-Process Transfer $ 0.00 $ 0.00 $ 0.00 $ 0.00 Inter-Process Transfer 1,250.00 0.00 0.00 1,250.00 Inter-Process Transfer 0.00 2,600.00 0.00 2,600.00 Product 1 0.00 0.00 3,387.50 3,387.50 Ending Inventory 1,250.00 2,600.00 3,388.50 7,238.59 Total Outputs $2,500.00 $5,200.00 $6, 776.00 $14,476.00

110 The Accounting Review, January 1973

PRODUCTION PERIOD MODEL

The elements of row X in the funda- mental matrix F contain the expected number of units of activity necessary in each process to produce a unit of final product with a probability of bx,l given that the unit entered the system in process X. Let H be a matrix with diagonal entries bij. If a. units of final product were required during a production period, then the expected number of units that would have to be cycled through the system would equal:

k = o-gH-1 (18)

Let 7r be an n element row vector where each element would contain the expected number of units processed in the process while producing a units of final product. The wr vector would equal:

7r= kF (19)

Variances of the expected activity levels are9

Var (7r) = ka[gF(2Fd, - I) - (gF)aq] (20) and expected outputs for the production period would be:

y= rR (21)

The first element of the y vector would equal the expected output of final product (i.e., or). Variances of the expected outputs arelo

Var (y) = ka [gB - _(gB),q] (22) and expected interprocess transfers would be determined by the algebraic expression:

Wij= 7ripij (23)

The vector of expected resource require- ments for a production period equals:

x= Er (24)

Expected standard cost of the external inputs required by process j equals:

r

=

In conclusion, the standard transfer costs would be:

dj=

Formulas (19) and (20) give the ex- pected transfers and variances that equal:

-16- O

7r = 16 ; Var (r) =

Std. Dev. (ir) = [

The means and variances of the expected outputs are given by (21) and (22).

_9- _3.94]

Y =[_7]; Var (y) = _[3.94;

[1.99-

Std. Dev. (y) = 1.99

By formula (23), the expected unit trans- fers are:

0 16 OiO 01 w=j0 0 12i0 4

LO 0 0i9 31

and by (24), the vector of expected re- sources equals:

I _{For proof, refer to Cvert, Davidson, and Thompson,}

p. 296.

Corcoran and Leininger: Process Costing 111

F32]

96 x 84 L44i

The vector of the standard cost of ex- ternal inputs by (25):

7 = [$200, $216, $126]

and by (26) the vector of standard transfer costs equals:

d = [$0, $12.50, $34.67]

In conclusion, the expected transfer cost matrix by (16) and (17) equals:

0 $200 0 I 0 0-

a = 0 0 $416 1 0 j

Lo 0 0 1$542 0_ A cost reconciliation for this example is shown in Exhibit B.

COST ANALYSIS

Price and quantity cost variance anal- ysis common to most standard cost sys- tems could be employed with the stochastic models. Rather than reiterate this ma- terial, in the following subsections we will be concerned with several methods of analyzing cost information peculiar to the

stochastic models. Each subsection will be followed by an example to clarify the ma- terial in the text.

1. Nonsequential Transfers

When nonsequential transfers are per- mitted in a production system, units may enter a process from more than one source. This condition would result from the two reprocessing or recycling possibilities de- picted in Figure 1. In Figure la, a unit exiting process d would have a probability of Pdb of being recycled back into process

b. In Figure lb, the normal sequence would be that a unit would be transferred from process c to process d; however, a unit exiting from process c would have a probability of p,. of being recycled through process x before being transferred to process d.

Often the cost of reprocessing is charged to an overhead account and then allocated to the production. If this procedure is fol- lowed, then reprocessing probabilities would not be incorporated into the P matrix, and the standard costs could be determined based on the appropriate formulas from the previous sections.

However, the stochastic model would make possible cost analysis concerning

EXHIBIT B

PRODUCTION PERIOD COST RECONCILIATION

Process 1 Process 2 Process 3 Total

Inputs External Inputs $200.00 $216.00 $126.00 $ 742.00 Inter-Process Transfer 0.00 200.00 0.00 200.00 Inter-Process Transfer 0.00 0.00 416.00 416.00 Inter-Process Transfer 0.00 0.00 0.00 0.00 Total Inputs $200.00 $416.00 $542.00 $1,158.00 Outputs Inter-Process Transfer $ 0.00 $ 0.00 $ 0.00 $ 0.00 Inter-Process Transfer 200.00 0.00 0.00 200.00 Inter-Process Transfer 0.00 416.00 0.00 416.00 Product 1 0.00 0.00 542.00 542.00 Total Outputs $200.00 $416.00 $542.00 $1,158.00

112 The Accounting Review, January 1973 Pdb 1 1 1 1 -Pd,b a b c d e (a) P, 1 -*o 1 1 1 1 o -0 -H0 -*0 -0 a b c d e (b) FIGURE 1

REPROCESSING IN THE STOCHASTIC MODEL

the decision as to whether units should be recycled. The description of the analysis will be carried out by means of an ex- ample. Referring to the stochastic matrix in our example, assume the problem is to decide whether to recycle the discarded output from process three back through process two. If the recycling was under- taken, the new stochastic matrix would take the form:

0 1 0 0

01

0 0 3/4 0 1/4

P 0 1/4 0 3/4 0

0 0 0 1 0

LO 0 0 0 1_

and the F and B matrices by (1) and (2) would equal: 1 16/13 12/13- F 0 16/13 12/13

;

If we assume that the full activity cost is incurred in the second process and that the units are then transferred like all other units exiting from process two, the stan- dard unit cost with recycling can be com- pared with the unit cost without recycling. The standard output cost with the re- cycling would be determined by (Dijfxisi)/bxj to equal $56.06. Since the standard cost of a unit of final product without recycling was $60.22, by recycling the average cost of a unit of final product is reduced $4.16. The analysis would be quite similar if the question concerned the type of recycling as depicted in Figure lb. 2. Analysis of Unit Costs

In a system where nonsequential trans- fers are possible, additional information related to the unit costs of final output may be obtained by further analyzing the stochastic matrix P. The minimum cost of a unit of final product results from a unit passing through the processes in the se- quence 1, 2, 3, * * n and then exiting to state n+1. Any unit that followed this sequence would not be recycled.

The probability of a unit following the minimum cost route and exiting the system as final product would be:

P = P1,2P2,3P3,4 . . P pn-l,npn,n+l (27)

The expected unit cost of a unit passing through the system in the least cost sys- tem equals:

n

an* = E si (28) Any difference between the expected unit cost u and the minimum (superscript*) unit cost is caused by the existence of de- fects, by-products, or recycling within the system. Any combination of the aforemen- tioned characteristics causes an increase in the estimated average unit cost.

Under some circumstances it may be advantageous to segregate estimates of the

Corcoran and Leininger: Process Costing 113 unit costs. The expected number of mini-

mum cost units would equal p*k and they could assume a unit cost of uO. The re- maining (o--p*k) units of final product could be assigned an average unit cost of (sr-ku*)/(o--p*k).

The foregoing will now be demonstrated by assuming the stochastic matrix that was used in the nonsequential transfer example. The probability of a unit follow- ing the minimum cost route would equal 9/16 from (27). The minimum unit cost would equal $36.50 from (28). If v equaled 9, then kh would equal 13 by (18) and the total cost of producing 9 units would equal $504.54 by (14). The segregated unit costs would equal:

7.3125 units at $ 36.50 1.6875 units at $140.82

The impact of a change in the cost of a resource input upon the estimated unit cost of final product may be obtained di- rectly from the stochastic costing model. Assume the cost of resource j shifts Acj; then the change in the estimated unit of final product equals:

n

Un fxE fjiejiAcj/^bi (29)

i=1

In the example with recycling, if the cost of resource 2 was increased by one dollar, the estimated unit output cost by (29) would increase by $9.67.

3. Cost A analysis Employing Conditional Probabilities

Employing conditional probabilities, it would be possible to determine all of the probabilities for a unit given that the unit exited from the production system in a particular state. Assume that the the unit exists from the system as final product; let H be a matrix with diagonal entries

bl,k. The conditional Q matrix would equal :

Q= H-'QH (30)

The conditional fundamental matrix would be:

F = H-VFH (31)

With the conditional fundamental ma- trix, the conditional standard cost of a unit of final product could be determined. This cost would be less than the nonconditional cost because there is only one possible way of exiting from the system. The difference between the two standard unit cost is the maximum amount a person would be willing to pay for information concerning the outcome of a unit entering the system. Employing the stochastic matrix with recycling, by formula (30) we find that the conditional stochastic matrix would be:

0 1 01 0

]

0 0 1 0

0 3/16 0 13/16

LO 0 O1 _

The conditional fundamental matrix would equal:

1 16/13 16/13- F 0 16/13 16/13

LO

By formula (14) the conditional unit cost would equal $42.10. The difference be- tween the expected cost of $56.06 and con- ditional cost of $42.10 is the maximum that one would be willing to pay for the information that a unit entering the pro- duction system would exit as final product.

Additional information may also be ob- tained from the conditional fundamental matrix. By formula (19), the conditional activity levels in each process could be determined. The difference between the conditional activity levels and those nor-

114 The Accounting Review, January 1973 mally estimated would indicate the pro-

duction activity that would be com- mitted to units that would not exit the system as final product. In the example, the two expected activity levels would equal:

or= 16 ;X= 144/13 _12_ _144/13_j

The difference between the two of:

64/13 L12/131

equals the activity on units that would not exit as final product.

SUMMARY

The relationship between the Leontief input-output model and mass production systems has been shown elsewhere."2 The stochastic models developed in this paper are similarly related and offer many of the same advantages. These models are cap-

able of achieving the same results as traditional costing systems. The same models could also be employed in planning to determine the expected activity levels and resource requirements.

The added advantage of the stochastic models is that variations in the outputs and activity levels and therefore costs could be accounted for based on analysis of the stochastic transition matrix. This information should prove of value to management when making operational control decisions.

In conclusion, the formalization in mathematical terms of accounting systems perhaps offers the best means of providing more information to decision makers.

12 _{Some examples are found in: Trevor E. Gambling,}

"A Technological Model for Use in Input-Output Anal- ysis and Cost Accounting," MANAGEMENT ACCOUNTING

(December, 1968), pp. 33-8; Yuji Ijiri, "An Application of Input-Output Analysis to Some Problems in Cost Accounting," MANAGEMENT ACCOUNTING, Vol. XLIX (April, 1968), pp. 49-61; John Leslie Livingstone, "Input-Output Analysis for Cost Accounting, Planning and Control," THE ACCOUNTING REVIEW, Vol. XLIV (January, 1969), pp. 48-69; Gerald A. Feltham, "Some Quantitative Approaches to Planning for Multiproduct

Production Systems," THE ACCOUNTING REvIEw, Vol.