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8/7/2019 Inventorym odelsw ith fixed and variable
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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
Accessed: 21/11/2010 01:18
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Journal of the Operational Research Society (2004) 55, 674-676 ? 2004 OperationalResearchSociety Ltd.All rightsreserved. 0160-5682/04 $30.00
www.palgrave-journals.com/jors
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