Tilburg University
A note on cost estimation errors in lot-size problems
Selen, W.J.
Publication date:
1987
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Selen, W. J. (1987). A note on cost estimation errors in lot-size problems. (pp. 1-11). (Ter Discussie FEW).
Faculteit der Economische Wetenschappen.
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A NOTE ON COST ESTIMATION ERRORS IN LOT-SIZE PROBLEMS
Willem J. Selen
A Note on Cost Estimation Errors in Lot-Size Problems Abstract
This note examines the sensitivity of the basic economic-orderquantity
(EOQ) inventory model to estimation errors in order and holdi.ng costs when the holding cost function is sllowed to be nonlinear. Estimation errors frequently occur when using average, rather than sarginal, estimates. We demonstrate that this cost accounting pitfall could become very costly and
that one should not blindly adhere to the traditionally believed
robust-ness of the EOQ methodology.
1
A Note on Cost Estimation Errors in Lot-Size Problems Introduction
The basic economic-order-quantity (EOQ) model has been used widely with controversial results [1]. The parameters needed to calculate the EOQ value are the unit cost, annual demand, order and carrying cost of the item to be ordered, respectively. Although demand forecasts could fluctu-ate widely and may not exist as a single estimfluctu-ate, the EOQ is still ac-cepted because of its well-known robustness to estimation errors [57, [8]. The EOQ model assumes that carrying costs are linearly related to average
inventory. However, the literature is abundant with examples of inventory models that incorporate nonlinear carrying cost functions, reflecting economies or diseconomies of scale as it relates to stocking various in-ventory levels [2], [3J, [6], [7], [10]. Recently, it was shown that under non-linear holding costs even relatively small lot-size errors can be extremely costly to the firm [4].
Another issue that recently received attention within the EOQ lot-sizing
framework, is the input of marginal cost estimates for order and carrying costs. Frequently costs are estimated on an average, rather than marginal,
cost basis using approximate prorations of various accounts. Cost
estima-tion errors of this kind could lead to serious deviations from optimality
[9].
This paper focuses on the sensitivity of the EOQ total cost to cost
estim-ation errors in order and holding costs under the more general condition
where carrying costs are assumed to be a nonlinear function of average
inventory. It will be shown that careful attention should be given to the
collection of correct marginal estimates ín EOQ lot-sizing. Failure to do
so could have very serious adverse effects, with cost penalties up to 80
percent and more.
Marginal versus average cost estimation
In many situations order cost is estimated on sn averáge cost basis, using approximate prorations of accounts like purchasing department costs for
z
telex, telegram accounts, and the like. These prorated amounts would add up to the total cost of all orders processed during the accounting period. Dividing this cost total by the number of orders placed that period yíelds a rough estimate of average order cost per order which, however, as St. John [11] notes, is not appropriate for the EOQ formula. Instead, the marginal or incremental cost of placing an order needs to be established; how much ordering cost would increase if we placed one more order. that is. Any fixed costs of the ordering function are irrelevant to the EOQ lot-sizing question, but would be left in an average cost estimate. Like-wise carrying costs, if calculated for the marginal unit held of a partic-ular item, must consider the warehouse building, equipment, staff, and such costs, as fixed.
Incorrectly using average cost estimates, rather than marginal cost figur-es, could result in estimation errors of a magnitude of a"few hundred" percent, depending on the relative proportion of fixed cost in the total average cost figure. This in turn could lead to severe total relevant cost deviations from the EOQ optimal solution, as will be discussed below. Since the fixed costs are the same under the two methods of cost estima-tion and are not affected by the order lot size, the cost deviations are analyzed for total relevant cost denoting the "controllable" portion of
total inventory cost. The Model
The model developed by Brown [3] is used to incorporate nonlinear carrying costs and to develop the sensitivity analysis of total cost to cost estim-ation errors in order and holding costs.
The yearly cost of placing orders, Co(Q), is given as usual by
Co(Q) - OS~Q (1)
where S is yearly demand, 0 is the cost to place an order, and Q is the size of the replenishment order. The power function
3
represents the yearly cost of holding inventory where C is the holding cost per unit per year, Q~2 is the average level of inventory, and n is the parameter for economies (n ( 1) or diseconomies (n ) 1) of scale. The total yearly cost, K(Q), then is given by
K(Q) - ~o(Q) ; ~h(Q)
(3)
~ w
The EOQ(Q ) and corresponding yearly cost (K(Q )) are given by
Q - (2nOS L nC
K(QM) - nnl ~nC(OS~2)n]lI(n;l) l5)
In order to estimate the sensitivity of total cost to cost estimation errors, using average rather than marginal estimates, we use
VO as the marginal order cost per order
VC as the marginal holding cost per unit per year
x as the marginal order cost, expressed as a percentage of average order cost per order
y as the marginal holding cost, expressed as a percentage of aver-age holding cost per unit per year
Using average cost estimates, total yearly cost, KA(QA) can be expressed
as
KA ( QA ) - (X0, Q ~ ~C, (Q, n . n ) 0 (6)
.
Using well-known methods, the EOQ(QA) under average cost estimation and
s
the corresponding yearly cost (K(QA)) are given by
4
For x- 1 and y- 1, where marginal cost equals average cost, we obtain the same result as shown in Brown et. al [4].
The total cost deviations resulting from using average, rather than marg-inal, cost estimates are given by the ratio (TCD)
K(Q ) TCD - ~ - 1
K(Q )
Making the appropriate substitutions in (7), we obtain
(x~Y)1~(n{1)Í1.(Y~nx)Í
TCD - lt(l~n) - 1
Sensitivity Analysis and Discussion
(9)
(10)
Values of the TCD for various combinations of cost estimation errors in order and holding costs for different degrees of nonlinearity in the
car-rying cost function, are presented in Tables 1 through 5.
[take in tables 1 through 5]
5
symmetrical for n- 1, the linear case, but not for the non-linear cases. As can be seen in table 1, severe estimation errors in holding cost, but not in order cost, lead to higher cost penalties than the complementary case where carrying costs are very close to the correct marginal estimate and order costs are severely overestimated (the correct marginal estimate is only a fraction of the average cost figure used).
This pattern is reversed for n~ 1, where diseconomies of scale are pre-sent. We note from tables 3 through 5 that estimation errors in order costs with fairly accurate holding cost estimates lead to higher cost penalties than the complementary case.
It is clear that decision makers should not blindly believe in the robust-ness of the EOQ model to estimation errors, particularly the order and carrying cost figures to be input. Careful attention should be given to the correct implementation and collection of relevant marginal cost estin-ates.
Summary
6
References
[1] Adkins, A.C. EOQ in the Real World. Productton and Inventory
Manage-ment, 1984, 25(4), 50-54.
[2] Beranek, W. Financial implications of lot-size inventory models. Man-agement Science, 1967, 13, 401-408.
[3] Brown, R.M. On carrying costs and the EOQ model: A pedagogical note.
Tne Ptnanctaz xevte,~, 1985. 20. 357-360.
[4] Brown, R.M. et al. A note on holding costs and lot-size errors.
Deci-sion Sctences, 1986, 17(4), 603-608.
[5J Hadley, G., ~ Within, T.M. Analysis of [nventory systems. Englewood Cliffs, NJ: Prentice-Hall, 1963.
[6] Muth, E., ~. Spremann, K. Learning effects in economic lot sizing.
Ma-nagement Science, 1983, 29, 264-269.
[7] Naddor, E. Inventory systems (reprint). Malabar, FL: Robert E. Krieger Publishing, 1984.
[8] Peterson, R., 8~ Silver, E. Decisfon systems for inventory management
and production planning. New York: Wiley, 1979.
[9J Selen, W.J. 8~ Wood, W. Inventory cost definition in EOQ model applic-ation. Productton and Inventory Management, 1987, In Press.
[10] Shah, Y.K. An order-level lot-size inventory model for deteriorating
items. AIIE Transacttons, 1977, 9, 108-112.
[11] St.John, R. The evils of lot sizing in MRP. Production and Inventory
7
Table 1
8
Table 2
9
Table 3
10
Table 4
11
Table 5
i
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