Tilburg University

A note on cost estimation errors in lot-size problems

Selen, W.J.

Publication date:1987

Document VersionPublisher's PDF, also known as Version of record

Link to publication in Tilburg University Research Portal

Citation for published version (APA):Selen, W. J. (1987). A note on cost estimation errors in lot-size problems. (pp. 1-11). (Ter Discussie FEW).Faculteit der Economische Wetenschappen.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 05. Dec. 2021

76271987

tih~ J~~~i~ ~~~~

O~,~JOO`,F. ~

Q h ~ i~ii~iiuyuiniuuiiiuu~iui~ '~ii

i (~-~.;: ~ - ---

, ~ '~i:."i: r~~ 1.~...?.

, "-,:; E3:~i~~:~ ~~r-~í-:~:'t;:,a í-ï~~ '1'~~ ti~ .r. r.j " ,. : ~ ,- íi :, . ~ ~~~-.-~.....~..~--..,..-....~.... ....-.~i

A NOTE ON COST ESTIMATION ERRORSIN LOT-SIZE PROBLEMS

Willem J. Selen

No. 87.12 ~S(p. ~~

A note on Cost Estimation Errorsin Lot-Size Problems

by

Willem J. SelenDepartment of Econometrics

Tilburg Universityp.o. Box 901535000 LE TilburgThe Netherlands

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 whenthe holding cost function is sllowed to be nonlinear. Estimation errorsfrequently occur when using average, rather than sarginal, estimates. Wedemonstrate that this cost accounting pitfall could become very costly andthat one should not blindly adhere to the traditionally believed robust-ness of the EOQ methodology.

Subject Areas: Inventory Management, Production~Operations Management, andSensitivity Analysis.

1

A Note on Cost Estimation Errors in Lot-Size Problems

Introduction

The basic economic-order-quantity (EOQ) model has been used widely withcontroversial results [1]. The parameters needed to calculate the EOQvalue are the unit cost, annual demand, order and carrying cost of theitem to be ordered, respectively. Although demand forecasts could fluctu-ate widely and may not exist as a single estimate, 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 averageinventory. However, the literature is abundant with examples of inventorymodels that incorporate nonlinear carrying cost functions, reflectingeconomies or diseconomies of scale as it relates to stocking various in-ventory levels [2], [3J, [6], [7], [10]. Recently, it was shown that undernon-linear holding costs even relatively small lot-size errors can beextremely costly to the firm [4].Another issue that recently received attention within the EOQ lot-sizingframework, is the input of marginal cost estimates for order and carryingcosts. 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 conditionwhere carrying costs are assumed to be a nonlinear function of averageinventory. It will be shown that careful attention should be given to thecollection of correct marginal estimates ín EOQ lot-sizing. Failure to doso could have very serious adverse effects, with cost penalties up to 80percent and more.

Marginal versus average cost estimation

In many situations order cost is estimated on sn averáge cost basis, usingapproximate prorations of accounts like purchasing department costs fororder placing, follow-up and receiving; stationary account; telephone,

z

telex, telegram accounts, and the like. These prorated amounts would addup to the total cost of all orders processed during the accounting period.Dividing this cost total by the number of orders placed that period yíeldsa rough estimate of average order cost per order which, however, as St.John [11] notes, is not appropriate for the EOQ formula. Instead, themarginal or incremental cost of placing an order needs to be established;how much ordering cost would increase if we placed one more order. thatis. Any fixed costs of the ordering function are irrelevant to the EOQlot-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, andsuch 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 totalaverage cost figure. This in turn could lead to severe total relevant costdeviations 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 areanalyzed for total relevant cost denoting the "controllable" portion oftotal inventory cost.

The Model

The model developed by Brown [3] is used to incorporate nonlinear carryingcosts 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 thesize of the replenishment order. The power function

~h(Q) - C(QI2)n. n ) 0 (2)

3

represents the yearly cost of holding inventory where C is the holdingcost per unit per year, Q~2 is the average level of inventory, and n isthe parameter for economies (n ( 1) or diseconomies (n ) 1) of scale. Thetotal 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 - (2nOSL nC

K(QM) - nnl ~nC(OS~2)n]lI(n;l) l5)

In order to estimate the sensitivity of total cost to cost estimationerrors, using average rather than marginal estimates, we use

VO as the marginal order cost per orderVC as the marginal holding cost per unit per yearx as the marginal order cost, expressed as a percentage of averageorder cost per ordery 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 expressedas

KA ( QA ) - (X0, Q ~ ~C, (Q, n . n ) 0 (6)

.Using well-known methods, the EOQ(QA) under average cost estimation and

sthe corresponding yearly cost (K(QA)) are given by

M Zn V~ SQA - rix(VC)

1~(n~l)

~1~(n.l)

~l f1 (VO S n 11~(n.l)K(QA) -

L1;~J LYl 2) n x (VC) J (8)

4

For x- 1 and y- 1, where marginal cost equals average cost, we obtainthe 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)ÍTCD - lt(l~n) - 1

Sensitivity Analysis and Discussion

(9)

(10)

Values of the TCD for various combinations of cost estimation errors inorder 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]

It is clear that cost estimation errors of mistakenly taking average formarginal estimates, can have severe adverse effects on the performance ofthe EOQ methodology. For example, suppose order costs have a large fixedcomponent such that the correct marginal cost component constitutes bet-ween 10 to 30 percent of the average estimate. Furthermore, assume carry-ing costs are fairly accurately estimated, within 10 percent of the cor-rect marginal estimate, that is. We then see from table 2 that for thestandard EOQ, where carrying costs are assumed to be linear, cost penal-ties between 16 and 74 percent could be incurred! The situation describedabove is not al all unrealistic, since the major holding cost componentusually used is the cost of capital or desired rate of return, which is bydefinition a marginal estimate. When the holding cost function is concaveup, or n~ 1, the situation becomes more severe with cost penalties up to85.7 percent, as can be seen in tables 3, 4 and 5. For n C 1, the casewhere economies of scale are present, the cost penalties are not as largebut cannot be ignored. We can also note that the cost penalty matrix is

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, butnot in order cost, lead to higher cost penalties than the complementarycase where carrying costs are very close to the correct marginal estimateand order costs are severely overestimated (the correct marginal estimateis 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 ordercosts with fairly accurate holding cost estimates lead to higher costpenalties 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 andcarrying cost figures to be input. Careful attention should be given tothe correct implementation and collection of relevant marginal cost estin-ates.

Summary

A sensitivity analysis on cost penalties incurred because of wrongly in-putting average, rather than marginal, order and holding cost estimates inthe basic EOQ model, was performed. Furthermore, the holding-cost curvecould be nonlinear, allowing for economies (learning effect) and disecon-omies (shrinkage, spoilage etc.) to be present. It was shown that estima-tion errors of this kind could have serious adverse effects on the optim-ality of the EOQ methodology, contradicting the traditionally believedEOQ-robustness. Careful attention should be given to the collection ofcorrect marginal estimates to be input.

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. EnglewoodCliffs, 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 deterioratingitems. AIIE Transacttons, 1977, 9, 108-112.

[11] St.John, R. The evils of lot sizing in MRP. Production and Inventory

Management, 1984, 25(4), 75-85.

7

Table 1Cost penalties for various xs and ys (z)

n - 0.5

x

y o.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 i.o

0.1 0 5.8 15.6 26 36.5 46.8 56.8 66.7 76.3 85.70.2 5 0 1.9 5.8 10.5 15.6 20.8 26 31.2 36.50.3 12.2 1.8 0 1 3 5.8 8.9 12.2 15.6 190.4 19.1 5 0.9 0 0.6 1.9 3.7 5.8 8.1 10.50.5 25.4 8.6 2.8 0.5 0 0.4 1.3 2.6 4.1 5.80.6 31.2 12.2 5 1.8 0.4 0 0.3 1 1.9 3.10.7 36.6 15.7 7.4 3.3 1.2 0.3 0 0.2 0.7 1.50.8 41.7 19.1 9.8 5 2.3 0.9 0.2 0 0.2 0.60.9 46.4 22.3 12.2 6.8 3.6 1.8 0.7 0.2 0 0.11.0 50.8 25.4 14.5 8.6 5 2.8 1.4 0.5 0.1 0

8

Table 2Cost penalties for various xs and ys (X)

n - 1

x

y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

o.l 0 6.1 15.5 25 34.2 42.9 51.2 59.1 66.7 73.90.2 6.1 0 2.1 6.1 10.7 15.5 20.3 25 29.6 34.20.3 15.5 2.1 0 1 3.3 6.1 9.1 12.3 15.5 18.70.4 25 6.1 1 0 0.6 2.1 3.9 6.1 8.3 10.70.5 34.2 10.7 3.3 0.6 0 0.4 1.4 2.8 4.3 6.10.6 42.9 15-5 6.1 2.1 0.4 0 0.3 1 2.1 3.30.7 51.2 20.3 9.1 3.9 1.4 0.3 0 0.2 0.8 1.60.8 59.1 25 12.3 6.1 2.8 1.0 0.2 0 0.2 0.60.9 66.7 29.6 15.5 8.3 4.3 2.1 0.8 0.2 0 0.11.0 73.9 34.2 18.7 10.7 6.1 3.3 1.6 0.6 0.1 0

9

Table 3Cost penalties for various xs and ys (z)

n - 1.5

x

y o.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 i.o

0.1 0 5.6 13.8 21.9 29.5 36.5 43.1 49.3 55.2 60.80.2 6.1 0 1.9 5.6 9.6 13.8 17.9 21.9 25.7 29.50.3 16 2 0 1 3 5.6 8.3 11 13.8 16.50.4 26.4 6.1 1 0 0.6 1.9 3.6 5.6 7.6 9.60.5 36.6 10.9 3.3 0.6 0 0.4 1.3 2.6 4 5.60.6 46.5 16 6.1 2 0.4 0 0.3 1 1.9 30.7 56.1 21.2 9.3 3.9 1.4 0.3 0 0.2 0.7 1.50.8 65.4 26.4 12.6 6.1 2.7 1 0.2 0 0.2 0.60.9 74.4 31.5 16 8.5 4.3 z 0.8 0.2 0 0.11.0 83.1 36.6 19.4 10.9 6.1 3.3 1.6 0.6 0.l o

10

Table 4Cost penalties for various xs and ys (x)

n - 2

x

y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

o.l 0 5 12.2 19.1 25.4 31.2 36.6 41.7 46.4 50.80.2 5.8 0 1.8 5 8.6 12.2 15.7 19.1 22.3 25.40.3 15.6 1.9 0 0.9 2.8 5 7.4 9.8 12.2 14.50.4 26 5.8 1 0 0.5 1.8 3.3 5 6.8 8.60.5 36.5 10.5 3.1 0.6 0 0.4 1.2 2.3 3.6 50.6 46.8 15.6 5.8 1.9 0.4 0 0.3 0.9 1.8 2.80.7 56.8 20.8 8.9 3.7 1.3 0.3 0 0.2 0.7 1.40.8 66.7 26 12.z 5.8 2.6 1 0.2 0 0.2 0.50.9 76.3 31.2 15.6 8.1 4.1 1.9 0.7 0.2 0 0.11.0 85.7 36.5 19 10.5 5.8 3.1 1.5 0.6 0.1 0

11

Table 5Cost penalties for various xs and ys (z)

n - 3

x

y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

o.l 0 4.1 9.7 14.9 19.6 23.9 27.8 31.4 34.7 37.80.2 5.1 0 1.5 4.1 6.9 9.7 12.4 14.9 17.3 19.60.3 14 1.7 0 0.7 2.3 4.1 5.9 7.8 9.7 11.50.4 23.7 5.1 0.8 0 0.5 1.5 2.7 4.1 5.5 6.90.5 33-7 9.4 2.7 0.5 0 0.3 1 1.9 3 4.10.6 43.8 14 5.1 1.7 0.3 0 0.2 0.7 1.5 2-30.7 53.7 18.8 7.9 3-2 1.1 0.2 0 0.2 0.6 1.10.8 63.5 23-7 10.9 5-1 2.3 0.8 0.2 0 0.1 0.50.9 73.2 28-7 14 7-2 3-6 1.7 0.6 o.i 0 0.1i.o 82.8 33.7 17.2 9.4 5.1 2.7 1.3 0.5 o.i o

i

IN 1986 REEDS VERSCHENEN

O1 F. van der PloegMonopoly Unions, Investment and Employment: Benefits of ContingentWage Contracts

02 J. van MierGewone differentievergelijkingen met niet-constante coëfficiënten enpartiéle differentievergelijkingen (vervolg R.T.D. no. 84.32)

03 J.J.A. MoorsHet Bayesiaanse Cox-Snell-model by accountantscontroles

04 G.J. van den BergNonstationarity in job search theory

05 G.J. van den BergSmall-sample properties of estimators of the sutocorrelation coeffi-cient

06 P. KooremanHuishoudproduktie en de analyse van tijdsbesteding

07 R.J. CasimirDSS, Information systems and Management Games

08 A.J. van ReekenDe ontwikkeling van de informatiesysteemontwikkeling

09 E. BernsFilosofie, economie en macht

10 Anna Hara~SczykThe Comparative Analysis of the Social Development of Cracow, Bratis-lava, and Leipzig, in the period 1960-1985

11 A.J. van ReekenOver de relatie tussen de begrippen: offer, resultaat, effíciëntie,effectiviteit, produktiviteit, rendement en kwaliteit

12 A.J. van ReekenGroeiende Index van Informatiesysteemontwikkelmethoden

13 A.J. van ReekenA note on Types of Information Systems

14 A.J. van ReekenHet probleem van de Componentenanalyse in ISAC

15 A. Kapteyn, P. Kooreman, R.J.M. WillemseSome methodological issues in the implementation of subjective pover-ty definitions

16 I. WoittiezPreference Interdependence and Habit Formation in Family Labor Supply

11

1~ A.J. van ReekenA new concept for allocation of joint costs: Stepwise reduction ofcosts proportional to joint savings

18 A.J. van ReekenNear een andere aanpak in de systemering

19 J.G. de Boer, N.J.W. GrevelingInformatieplanning met behulp van referentie-informatiemodellen 1.Totstandkoming bedrijfsinformatiemodellen

20 J.G. de Boer, N.J.W. GrevelingInformatieplanning met behulp van referentie-informatiemodellen 2.Een methode voor informatieplanning

21 W. ReijndersDirect Marketing: "Van tactiek naar strategie"

22 H. GremmenA four economy computer simulation game

iii

IN 198~ REEDS VERSCHENF.N

O1 J.J.A. MoorsAnalytical Properties of Bayesian Cox-Snell Bounds in Auditing

02 H.P.A. Mulders, A.J. van ReekenDATAAL - een hulpmiddel voor onderhoud van gegevensverzamelingen

03 Drs. A.J. van ReekenInformatisering en de beloning van arbeid

04 P.C. van Batenburg, J. KriensBayesian Discovery Sampling: a simple model of Bayesian Inference inAuditing. ~

05 Prof.Dr. J.P.C. KleijnenSimulatie

06 Rommert J. CasimirCharacteristics and implementation of decision support systems

07 Rommert J. CasimirInfogame, the model

08 J.J.A. MoorsA Quantile Alternative for Kurtosis

09 Rommert J. CasimirOntwerpen van Bedrijfsspelen

10 Prof. Drs. J.A.M. OonincxInformatiesystemen en het gebruik van 4e generatie talen

11 R. Heuts, J. van den BerghProductieplanning met stochastische vraagpatronen en simultane be-schouwing van regelmatige en onregelmatige productieprogramma's: eenanalyse van het éénperiodeprobleem

i u ~u iiiW r~u~~ iu