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Some Comments on 'Inventory Models with Fixed and Variable Lead Time Crash Costs

Consideration'Author(s): M. A. Hoque and S. K. GoyalSource: The Journal of the Operational Research Society, Vol. 55, No. 6 (Jun., 2004), pp. 674-676Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/4101974

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Journal of the Operational Research Society (2004) 55, 674-676 ? 2004 OperationalResearchSociety Ltd.All rightsreserved. 0160-5682/04 $30.00

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Viewpoint

Somecomments n 'Inventorymodelswith fixedandvariable

leadtime crash costs consideration'

Journalof the OperationalResearchSociety (2004) 55, 674-676.doi: 0.1057/palgrave.jors.2601762

Introduction

We would like to commenton the paper by Pan et al,1 in

which assuming reduced lead time crashing cost as a

function of both the order quantity and the reduced lead

time, they presentedtwo inventory models, the one with

known deterministicdemand and the other with unknown

demanddistribution.The solutionproceduresare illustrated

with numericalexamples.We have dealt with the determi-

nistic demand case only. This viewpoint highlightsthe misleadingbehaviour of the formulae used to obtain

the optimal order quantity. It also demonstrates the

infeasibilityof the model due to the lack of a constrainton

the orderquantity n orderto satisfythe demand n the lead

time. The modelis extendedwiththe additionof a constraint

to satisfythe demand n the lead time.An optimalsolution

technique of the extended model is presented, and a

comparativestudy of the resultsof the numericalexampleis carriedout.

Analysis of the model

They presented heir models using the followingnotations:

L the length of lead time (in weeks);Q= orderquantity;D = averagedemand per year; A = fixed orderingcost per

order;h= inventory holding cost per unit per year;r= the

reorder point; rno=the gross marginal profit per unit;

7[ = fixedpenaltycost perunitshort;1= theaveragedemandrate in units per day; a= the standard deviation of the

demand rate; k=safety factor; =the standard normal

distribution;q= the standard normal cumulativedistribu-

tion function;P= the backorderratio; n= the number of

mutually ndependent omponentsof the lead time;Ti= the

normal duration of the ith component; ti the minimum

durationof the ith component;Li= the lead timelength(in

weeks) with component i (i= 1,...,n), crashed to their

minimum values; a =unit fixed crash cost per week;

and bi= unit variable crash cost per week for the ith

component f the lead timereduced.Withthesenotations hey

formulated he expectedannualcost, EAC(Q, L), as

EAC(Q,L)=A-D

+h[Q

+ kav + (1-)avoYL(k)

+ [-E + ro(1-/)]a]V-L'P(k)Q

+ D ai(Li- L)+ a,(Tj tj)

+ D

t-

bi(Li- - L) +

-

bj(Tj1

-

tj)

(1)

where TP(k)= 0b(k)-k[1-(D(k)] and Li= E= 1T-

Z-I'(Ti-tj) for Li<L <Li-1. For correct calculation ofthe total cost bindingson L should be Li< L< Li1.

For a fixed Q, they have shown that the minimum

expectedtotal annual cost occursat one of the endpointsof

the lead time intervals.SettingOEAC(Q,L)/6Q to zero and

solvingfor Q, they obtained

Q[•2D-{A+ [7r+ to(1

-fl)]a/LT(k)i--1/2 (2)+ai(Li-I-L) +

aj(Tj

-

tJ)

The minimum and maximum of the optimal order

quantity n Equations 3) and (4) of theirpaper,respectively,are shown as

- 1/2

Qmin 2D A + [7n o(l - #)]aP(k) 1t

when none of the lead timecomponents

is crashedat all,

Qmax

-•2D

A + [ +

~-o(1

-

fl)]-•P(k)

x?

T + Ea(T,-

tt)

when all the lead time components are crashed to their

minimum imits.

Theseformulaeare not correctbecause f none of the lead

time components s crashed,the lead time is" T1

and

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Viewpoint75

in case of crashingall the lead time componentsto their

minimum imits,the lead time is,/ , tj Therefore, n the

above formulae these mathematicalexpressionsmust be

interchanged.Thusthey developed heiralgorithmbased on

erroneous formulae providing misleading solution techni-

ques. Besides, they developed their solution techniquewithout

imposing anyconstrainton the order

quantityin

order to satisfythe demand n the lead time. For thisreason,their solution techniquehas given infeasible solutions for

different values of / for the numerical example when

demand D= 5500. These optimal solutions are shown in

Table 1.

Note that the demandperweekis 110 and hence none of

the optimal ot sizesmeetsthedemand or thecorresponding

optimal lead time. Thus the solution technique eads to an

infeasiblesolution.

An alternative solution technique

In order to satisfy the demandin the lead time, the order

quantity must be greater than or equal to the requireddemand n the lead time, that is,

Q~ DL/50 + kav-L (assuming50weeks na year) (3)

For each value of i, the optimalvalue of Qdenotedby Q*can be determinedas follows:

Q*= Max(Q0, DL/50 + ka/L) (4)

Table1 Infeasible ptimal ot sizes of theexample1 fordifferent aluesof P when D= 5500

Valueof # 0.0 0.5 0.8 1.0Optimal ot size 573 510 454 412Optimal ead time(inweeks) 6 8 8 8

whereQ? s thevalue of Qcalculated romEquation 2).The

total cost calculated rom(1) for Q= Q*withLi< L < Li- is

the correspondingminimal cost. The minimum of the

minimal costs thus calculated for all i gives the final

minimum cost. The value of Q associated with the final

minimum cost is the minimal order quantity. Following

this solution approach,the numericalexample 1 solved byPan et all is solved and comparativeresults are given in

Table2 (in the table,PHL denotesthe methoddevelopedby

Pan, Hsiao and Lee and HG denotesthe methoddeveloped

by us):For the same lot size and lead time, the total minimal

costs calculatedfollowing our approachare always found

to be different from the correspondingone they put in

their Table 4. These may be due to the limitation

Li<L<~Li_1 in their formulation of the total cost.

For D=5500, the lot sizes found out for minimal

total costs by Pan et al's method do not satisfy the

demandfor the lead time and hence the solutions are

infeasible, whereas our approach provides minimal cost

solutionssatisfying he demand n the lead time plus safetystock.

Concludingremarks

This viewpoint highlights the erroneous formulae pro-

posed by Pan et al. It also demonstrates hat the model

proposed by them leads to an infeasible solution because

of the lack of constraint on the order quantity in order

to satisfy the demand in the lead time plus safety shock.

To satisfy the demand in the lead time, the model isextended to include a constraint on the order quantity.An optimal solution technique of the extended model

is proposed, which is able to provide minimal cost

solution of the problem considering feasibility. The

potentiality of our method is shown by a comparative

study of the results of the numerical example theyillustrated.

Table2 Comparativeesultsof example with deterministicemand or different aluesof PValuesof / Solution by D = 250 units/year D = 600 units/year D = 5500 units/year

Q, L EAC(Q,L) Q, L EAC(Q,L) Q, L EAC(Q,L)

0.0 PHL 115,3 2587.81 178,3 3931.24 573,6 12087.29aHG 115,3 2591.50 178,3 3936.88 542,4 12335.06

0.5 PHL 103,3 2344.99 158,4 3553.96 510,8 10554.44aHG 104,3 2247.52 159,4 3558.49 480,4 11087.741

0.8 PHL 93,4 2165.93 145,4 3269.75 454,8 9420.76aHG 93,4 2167.99 145,4 3272.89 452,4 10246.04

1.0 PHL 87,4 2031.39 136,8 3058.24 412,8 8580.80aHG 87,4 2032.74 136,8 3061.17 452,4 9673.22

alInfeasible olutionbecausehe lot size doesnot satisfy hedemand or the leadtimeplus safetystock.

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676 Journalf heOperationalesearchocietyol. 5,No.

Acknowledgements--Weacknowledge that the authors of the originalpaper pointed out a typo in Equation (1) and the authors of the view

point are grateful to them.

References

1 PanJC-H,HsiaoY-C and Lee C-J(2002).Inventorymodelswith

fixed andvariable ead time crashcostsconsiderations. OplResSoc 53: 1048-1053.

UniversitySains Malaysia MA Hoqueand

Concordia University, SK GoyalCanada

Editor's note: The authors of the orginal paper have seen

and agreed to the publication of this viewpoint.

T Williams