# Investing in Lead-Time Variability Reduction in a Quality-Adjusted Inventory Model with Finite-Range Stochastic Lead-Time

Authors: Farrokh Nasri; Javad Paknejad; John Affisco
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

We study the impact of the efforts aimed at reducing the lead-time variability in a quality-adjusted stochastic inventory model. We assume that each lot contains a random number of defective units. More specifically, a logarithmic investment function is used that allows investment to be made to reduce lead-time variability. Explicit results for the optimal values of decision variables as well as optimal value of the variance of lead-time are obtained. A series of numerical exercises is presented to demonstrate the use of the models developed in this paper. Initially the lead-time variance reduction model (LTVR) is compared to the quality-adjusted model (QA) for different values of initial lead-time over uniformly distributed lead-time intervals from one to seven weeks. In all cases where investment is warranted, investment in lead-time reduction results in reduced lot sizes, variances, and total inventory costs. Further, both the reduction in lot-size and lead-time variance increase as the lead-time interval increases. Similar results are obtained when lead-time follows a truncated normal distribution. The impact of proportion of defective items was also examined for the uniform case resulting in the finding that the total inventory related costs of investing in lead-time variance reduction decrease significantly as the proportion defective decreases. Finally, the results of sensitivity analysis relating to proportion defective, interest rate, and setup cost show the lead-time variance reduction model to be quite robust and representative of practice.


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## Body

## 1. Introduction

The
origin of lot-size research can be traced to the development of the square root
EOQ formula in the early 20th century. This relationship is the
result of classical optimization of inventory-related costs under a series of
highly restrictive assumptions. Among these assumptions are instantaneous
replenishment, constant deterministic demand and lead-time, and perfect quality
of inventory items. More realistic cases ensue when these assumptions are
relaxed. Some of these cases that have appeared in the literature allow for
imperfect quality and variability in either demand or lead-time, or both.Gross and Soriano [1]
and Vinson [2], among others, demonstrate that lead-time variation has a major
impact on lot size and inventory costs. Furthermore, they indicate that an
inventory system is more sensitive to lead-time variation than to demand
variation. The problem of the EOQ model with stochastic lead time has been
considered by several additional authors including Liberatore [3], Sphicas
[4], and Sphicas and Nasri [5]. In this last work, the authors derive a
closed form expression for EOQ with backorders when the range of the lead-time
distribution is finite. In this
formulation, all units are assumed to be of perfect quality.Concurrently work has appeared in the literature that relaxes the perfect
quality assumption. Rosenblatt and Lee [6] have investigated the effect of
process quality on lot size in the classical economic manufacturing quantity
model (EMQ). Porteus [7] introduced a modified EMQ model that indicates a
significant relationship between quality and lot size. In both [6, 7],
the optimal lot size is shown to be smaller than that of the EMQ model. In
these works, the deterioration of the production system is assumed to follow a
random process.Cheng [8] develops a model that integrates quality considerations with
EPQ. (Economic manufacturing quantity.) The author assumes that the unit production
cost increases with increases in process capability and quality assurance
expenses. Classical optimization results in closed forms for the optimal lot
size and acceptable
optimal expected fraction. The optimal lot size is intuitively appealing since it
indicates an inverse relationship between lot size and process capability.While this previous work relaxes the perfect quality assumption, it also
considers demand to be deterministic. A number of authors have investigated the
impact of quality on lot size under conditions of stochastic demand and/or
stochastic lead-time. Moinzadeh and Lee [9] have studied the effect of
defective items on the operating characteristics of a continuous-review
inventory system with Poisson demand and constant lead time. Paknejad et al. [10] present a quality-adjusted lot-sizing model with stochastic
demand and constant lead time. Specifically, they investigate the case of
continuous-review (s,Q) models in which
an order of size Q is placed each time
the inventory position (based on nondefective items) reaches the reorder point,
s. Results indicate that as the probability of defective items increases, for a
given constant lead time, the optimal lot size and the optimal reorder point
both increase significantly. Further, for a given defective probability, the
lot size and the reorder point increase substantially as the lead time
increases.Variations in lead time can occur for purchased items and for those that
are manufactured in-house. A major factor related to these variations is
quality problems. Typically, either safety stock or safety lead time is
utilized to cushion the impact of this variability. In either case, larger
variability requires increased inventories. Heard and Plossl [11] portray high
lead-time variability as a major reason for a plant's  inability to achieve
inventory goals and to incur longer average throughput. This suggests that it
would be worthwhile to investigate the relationship between quality and
lead-time variability, and their impact on lot size and inventory cost.
Paknejad et al. [12] began to study this relationship. The authors
develop a quality-adjusted model for the case of an inventory model with
finite-range stochastic lead times first presented in [5]. This model assumes
that each lot contains a random number of defective units all of which are
discovered by the purchaser's   inspection process and returned to the vendor at
the time of the next delivery. The number of defective units in a lot is
assumed to follow a binomial distribution. Further, no crossover of orders is
allowed. Closed form results are developed for a number of decision variables
including the optimal quality-adjusted lot size and the optimal total inventory
cost. These closed forms are direct functions of the corresponding optimal
decision variables developed in [5]. In fact, when quality is perfect, the
quality-adjusted model with finite-range stochastic lead time simply reduces to
Sphicas and Nasri [5] basic model. In [12], the quality-adjusted optimal
lot-size is shown to depend directly on the variance of lead-time in addition
to the typical cost and demand parameters, as well as the proportion of defective
units in a lot. In this paper, we derive relationships for the case of
investment in reducing lead-time variability in a quality-adjusted model with
finite-range stochastic lead time. This result points to a new important line
of investigation. That is, the analytical determination of the impact of
investing to reduce lead-time variance on lot size in a nonperfect quality
environment, and ultimately inventory costs. In this paper, we derive these
important analytical results, and investigate their robustness through a series
of numerical exercises.


## 2. Review of Basic Models and Assumptions

The basic model
considered in this paper is the classic EOQ with constant noninterchangeable
demand, backorders, and finite-range stochastic lead time, developed by Sphicas
and Nasri [5]. Assuming that orders do not cross, the optimal values of the
decision variables, q0 and t0, the resulting optimal lot size, Q0, and the
optimal expected average cost per unit time, EAC0(q,t) are given by (2.1)(2.2)(2.3)(2.4)whereD= demand per unit time (in units),K= setup cost per setup,h=    holding cost per (nondefective) unit per unit time,p= backorder cost per (nondefective) unit per unit time,Ω=h/p,V= variance of the lead time,q=Q/D= number of time units of demand satisfied by each order,t= time differential between placing an order and the start of q  time units that will be
satisfied by a given order,EAC0(q,t)= expected average cost per unit time.Note that (2.3) is the stochastic generalization of EOQ when
backorders are allowed. Sphicas and Nasri [5] proved that in terms of the
parameters of the model, crossover may not occur if and only if k≥k2, where
(2.5)k=2K/(h+p)D,(2.6)k2=(μ-α)2/Ω-V,ifΩ≤(μ-α)/(β-μ),(2.7)k2=Ω(μ-β)2-V,ifΩ≥(μ-α)/(β-μ),whereα= lower bound of lead-time distribution,β= upper bound of lead-time distribution,μ= mean of lead-time distribution.This formulation assumes that all the units
produced by the vendor, in response to the purchaser's  order, are nondefective. Paknejad
et al. [12] relax this assumption and extend the stochastic generalization of
the EOQ model with no crossover of orders by allowing the possibility that each
lot may contain a random number of defective units and develop a
quality-adjusted model. Specifically, they assume that each lot contains
a random number of defective units. Upon arrival, the purchaser inspects the
entire lot piece by piece. The purchaser removes the defective units from the
lot and returns them to the vendor at the time of next delivery. It is assumed
that the vendor picks up the inspection cost incurred by the purchaser. The
purchaser's inventory system, however, incurs an extra cost for holding the
defective units in stock until the time they are returned to the vendor. They
use the following additional notations:h= nondefective holding cost per unit per unit time,h′= defective holding cost per unit per unit time.Assuming that the number of nondefective units in a lot of sizeQ can be described by a binomial random variable with parameters Q and (1−θ), and defining ρ, the
quality parameter, as the ratio of the probability of a defective unit to the
probability of a nondefective unit, that is,
(2.8)ρ=θ1−θ,and the expected
average cost per unit time given that a lot of size Q is ordered, is (2.9)EACadj(q,t)=K(1+ρ)q+D(1+ρ)2q(h+p)[V+(t−μ)2]+Dh(t−μ)+Dh2[ρ+Dq(1+ρ)D]+h′Dq−h′Dq1+ρ.The optimal
values for the decision variables qadj*, tadj*, the resulting optimal lot size Qadj*,
and optimal expected average cost per unit time EACadj*(q,t),
are found using calculus as follows: (2.10)(2.11)(2.12)(2.13)where
(2.14)η=[1+2h′ρ(1h+1p)]1/2,and
q0,t0,Q0, and
EAC0(q,t) are given in (2.1) through (2.4), respectively. Note that in (2.10) through (2.13), if
the quality paramete ρ=0, then quality is perfect and the
quality-adjusted model with finite-range stochastic lead time simply reduces to
Sphicas and Nasri's basic model with no crossover of orders expressed in (2.1)
through (2.4).


## 3. The Optimal Lead-Time Variability Model

The policy
variables in (2.9) are q and t for a fixed lead-time variance, V. In this paper, as in [13], we assume
that the option of investing to reduce V is available. There is now a cost per unit time, av(V), of changing the lead-time variance to V. Thus we consider V to be a decision variable and aim at minimizing the expected
average cost per unit time composed of investment to change V, ordering, backordering, nondefective
holding, and defective holding costs. Specifically, we seek to minimize (3.1)EACadj(q,t,V)=iav(V)+EACadj(q,t),subject to (3.2)0<V≤V0,where i is the
cost of capital av(V) is a convex and strictly decreasing function
of V, as defined before, EACadj(q,t) is the sum of inventory
related costs given in (2.9), and V0 is the original lead-time variance before any investment is made. We use
classical optimization techniques to minimize (3.1) over q,t, and V, ignoring the 0<V≤V0 restriction. Of course, if the optimal V obtained in this way does not satisfy restriction (3.2), we should not make any
investment, and the results of the quality-adjusted model of the previous
section hold. It should be pointed out that it may not always be possible to
carry out the minimization. One case where minimization is possible is that of
the logarithmic investment function.The Logarithmic Investment Function Case
This particular
function is used in previous research by Porteus [7, 14] and Paknejad et al. [10] dealing with quality improvement as well as setup cost
reduction. Paknejad et al. [13] justified its use in the context of lead-time variance
reduction based on the idea that lead-time variability reduction should exhibit
decreasing marginal return. Further, since high lead-time variability is
inevitably related to poor manufacturing, it is conceivable that the steps
taken to improve the manufacturing process through improved quality and reduced
setup time are closely analogous to that of lead-time variability reduction. In
this case, lead-time variance declines exponentially as the investment amount av increases. That is, (3.3)av(V)=1ΓlnV0Vfor0<V≤V0,where Γ is the percentage decrease in V per dollar increase in av.
Here, our main objective is to minimize EACadj(q,t,V) after substituting (3.3) and (2.9) into (3.1).Theorem 3.1.
IfV0 and Γ are strictly positive and k≥k2,
then the following hold:
(a)EACadj(q,t,V) is strictly
convex, if and only if (3.4)q>(1+ρ)(h+p)2D2V2Γ4i[2K+(h+p)DV];
(b) the optimal
values of the decision variables are given by(3.5)Vadj*=min{V0,Vimp},qadj**=min{qadj*,qimp}={qadj*ifVimp≥V0,qimpifVimp<V0,tadj**=max{tadj*,timp}={tadj*ifVimp≥V0,timpifVimp<V0,whereV0= the original
lead-time variance, qadj* and tadj* are given in (2.10) and (2.11), (3.6)Vimp=2iΓhpD2η2[iΓ+i2Γ2+2η2DK(1/h+1/p)],(3.7)qimp=(1+ρ)(1/h+1/p)η2D[iΓ+i2Γ2+2η2DK(1/h+1/p)],(3.8)timp=μ-[i/Γ+i2/Γ2+2η2DK/(1/h+1/p)]η2Dp;
(c) the
resulting optimal lot size is given by(3.9)Qadj**=min{Qadj*,Qimp}={Qadj*ifVimp≥V0,QimpifVimp<V0,where Qadj* is given by (2.12) and (3.10)Qimp=(1+ρ)(1/h+1/p)η2[iΓ+i2Γ2+2η2DK(1/h+1/p)];it should be noted that Vimp, qimp, timp,
and Qimp do not depend on V0;
(d) the
resulting optimal expected average cost per unit time is given by(3.11)min{h2(ρ1+ρ)+ηEAC0(q,t),iΓlnV0Vimp+h2(ρ1+ρ)+η2DKhph+p+hpD2Vimp}.It should be pointed out that whenVimp≥V0,
we should not make any investment. In this case, V0 will be used in
place of Vimp,
and (3.7), (3.8), and (3.10) will be replaced by the results of quality-adjusted
model of Paknejad et al. [12] given in (2.10) through (2.14). Further,
when quality is perfect (i. e., ρ=0),
the results of this paper simply reduce to the optimal lead-time variability
model of Paknejad et al. [13]. Finally, when Vimp≥V0 and ρ=0,
the results of this paper reduce to the corresponding results of the basic EOQ
with constant noninterchangeable demand, backorders, and finite-range
stochastic lead time, developed by Sphicas and Nasri [5].Proof.
(a)EACadj(q,t,V) is strictly convex if all the principal minors of its Hessian determinants are
strictly positive. We proceed by producing the principal minors (3.12)|H11|=1+ρq3{2K+D(h+p)[V+(t−μ)2]}>0,|H22|=(h+p)D(1+ρ)2q4[2K+D(h+p)V]>0,|H33|=(h+p)D(1+ρ)2q4[2iKΓV2+iD(h+p)ΓV−(h+p)2D2(1+ρ)4q]>0,where |H11| and |H22| are strictly positive, and |H33|>0 if and only if the convexity condition
of part (a) of Theorem 3.1 holds.
(b) In order to
minimize (3.1), it is necessary that (3.13)∂EACadj(q,t,ρ)∂q=∂EACadj(q,t,ρ)∂t=∂EACadj(q,t,ρ)∂V=0.The solution to
these equations yields qimp, timp,
and Vimp of part (b) of Theorem 3.1. To prove that the
stationary point (qimp,timp,Vimp) is a relative minimum, it is sufficient to
show that it satisfies the convexity condition of part (a). Setting partial
derivative of EACadj(q,t,ρ) with
respect to V equal to zero and
solving, we find that (3.14)Vimp=2iqimpΓD(1+ρ)(h+p).Substituting (3.14)
into the right-hand side of the convexity condition (3.4), after extensive
simplification, the convexity condition reduces to (3.15)2(1+ρ)K+iqimp>0.Since ρ,K,i, and qimp are all nonnegative, the convexity condition
is satisfied at the point  (qimp,timp,Vimp),
and part (b) follows.
(c) This part is
the direct result of substituting the optimal values of the decision variables
into the total cost formula for the two separate cases ofVimp≥V0 and Vimp<V0.


## 4. Numerical Examples

Consider an
example where the following parameters are known:D=5200 units/year, K=$500/setup, h=$10/unit/year, p=$20/unit/year, h′=$5/unit/year, i=0.10, and Γ=0.0005. Table 1 presents the results
of calculations for the economic order quantity under three scenarios: the EOQ
with uniformly distributed lead time (EOQ-SLT) over a five-week interval, the
quality-adjusted EOQ (SLT-QA) with uniformly distributed lead time over a five-week
interval, and the EOQ including investment in lead-time variance reduction for
the case of uniformlyl distributed lead time over a five-week interval
(SLT-QA-LTVR). It is interesting to note that SLT-QA is an upper bound for total
cost and economic order quantity for the problem. Investment in lead-time
variance reduction in this problem results in a 1.48 week reduction in
lead-time interval, a 1.30 week reduction in the standard deviation of lead time,
and a 0.74 week reduction in mean lead time. Along with this, a 2.74 percent
reduction in lot size and a 0.822 percent reduction in total cost are realized.Table 1
Comparative results for a uniform numerical example.
VariableEOQ-SLTSLT-QASLT-QA-LTVRQ (units)934.75996.44969.14θ—0.20.2V (yr)——0.0003823μ
(yr)——0.033866TC ($)6,231.647,308.25—% TC Savings over
SLT-QA———K=500,D=5200,h=10,p=20,h′=5,μ0=0.048077,V0=0.0007705,i=0.1,Γ=0.0005.Table2 presents
the results of additional  calculations aimed at determining the impact of
initial lead-time variance, V0, on the model developed in this paper. Specifically, the
quality-adjusted stochastic lead-time models with and without investment in
lead-time variance reduction are compared for uniformly distributed lead-time
intervals of one through seven weeks. The value of the technical coefficient Γ is 0.0005 for the results presented in
this table. For the cases in which lead-time interval is 1, 2, or 3 weeks, it
can be observed from Table 2 that the optimal value of the variance, Vimp, is not less than the original variance, V0. (Please note that for
ease of display, values for V0 and Vimp are presented once for each lead-time interval
value.) For these cases, as indicated in (2.12), investment in lead-time variance
reduction is not warranted and the optimal lot size value remains the optimal
value for the quality-adjusted model while the optimal variance is identical to
the initial variance. For the cases where the lead-time interval is 4 through 7 weeks, investment in lead-time variability reduction is worthwhile since Vimp<V0.
For all these cases, the SLT-QA-LTVR model exhibits reduced lot sizes,
variance, and total costs when compared to the SLT-QA model. Both reduction in
lot size and lead-time variance increase as the lead-time interval, and thus,
initial lead-time variability increases. A collateral result induced by the use
of the uniform distribution is that the reduction in lead-time variance is
accompanied by a reduction in mean lead-time. This reduction also increases as
the lead-time variability increases.Table 2
Optimal value for various uniform lead-time variabilities withΓ=0.0005.
VariableQ (units)V0Vimpμ0μ*TC($)%TC Savings over SLT-QALead-time interval (weeks)1QA943.730.000030820.000382300.009615380.009615386,291.68—LTVR943.736,291.6802QA950.480.000123270.000382300.019230770.019230776,970.17—LTVR950.486,970.1703QA961.620.000277370.000382300.028846150.028846157,052.89—LTVR961.627,052.8904QA977.010.000493100.000382300.038461540.033866027,165.73—LTVR969.147,158.900.0955QA996.440.000770460.000382300.048076920.033866027,308.25—LTVR969.147,248.160.8226QA1,019.660.001109470.000382300.057692310.033866027,478.51—LTVR969.147,321.092.1057QA1,046.500.001510110.000382300.067307690.033866027,675.31—LTVR969.147,352.753.812K=500,D=5200,h=10,p=20,h'=5,i=0.1,⊖=0.2,Γ=0.0005.In Table3, we
investigate the impact of investing a greater amount in lead-time variance
reduction. This is accomplished by increasing Γ to 0.005 from 0.0005. In this situation, only for the case where
lead-time interval is one week, the optimal value of the variance, Vimp,
is greater than the original variance, V0,
and hence investment in lead-time variance reduction is not warranted. For all
the remaining cases, investment in lead-time variability reduction is
worthwhile since Vimp<V0.
Once again, for all these cases, the SLT-QA-LTVR model exhibits reduced lot
sizes, variance, and total costs when compared to the SLT-QA model. Also, both
the reduction in lot size and lead-time variance increase as the lead-time
interval, and thus, initial lead-time variability increases.Table 3
Optimal value for various uniform lead-time variabilities withΓ=0.005.
VariableQ (units)V0V*μ0μ*TC($)%TC Savings over SLT-QALead-time interval (weeks)1QA943.730.000030820.000037250.009615380.009615386,291.68—LTVR943.736,291.6802QA950.480.000123270.000037250.019230770.010570706,970.17—LTVR944.206,949.070.3033QA961.620.000277370.000037250.028846150.010570707,052.89—LTVR944.206965.291.2424QA977.010.000493100.000382300.038461540.010570707,165.73—LTVR944.206,976.802.6375QA996.440.000770460.000037250.048076920.010570707,308.25—LTVR944.206,985.724.4136QA1,019.660.001109470.000037250.057692310.010570707,478.51—LTVR944.206,993.026.4927QA1,046.500.001510110.000037250.067307690.010570707,675.31—LTVR944.206,999.188.809K=500,D=5200,h=10,p=20,h′=5,i=0.1,⊖=0.2,Γ=0.005.Comparing Tables2 and 3 results shows that the increase in technical coefficient, Γ, from 0.0005 to 0.005 results in an
increase in lead-time variance reduction, when the lead-time interval is 7
weeks, from 74.53 to 97.52 percent (3.049 to 3.983 weeks2).
Interestingly, this increased lead-time variance reduction is accompanied by an
increased reduction in lot size from 7.39 to 9.78 percent (77.36 units to 102.3 units). Interestingly, the total cost also decreases from $7,382.75 to
$6,999.18, a reduction of 5.2%. This indicates that the decrease in lead-time
variance along with the resulting synergistic impact on the lot size clearly
improves the overall performance of the production and inventory system.Table4 presents
similar results, for lead-time intervals from 1 to 5 weeks, when lead time
follows a normal distribution which is truncated at ±3σ. Similar to the uniform case, only for the
case where lead-time interval is one week, is the optimal value of the
variance, Vimp, is greater than the original variance, V0, and hence investment in
lead-time variance reduction is not warranted. For all the remaining cases,
investment in lead-time variability reduction is worthwhile since Vimp<V0.
Once again, for all these cases, the SLT-QA-LTVR model exhibits reduced lot
sizes, variance, and total costs when compared to the SLT-QA model. Also, both
the reduction in lot size and lead-time variance increase as the lead-time
interval, and thus the initial lead-time variability increases.Table 4
Optimal value for various
normal lead-time variabilities.
Lead-time interval (weeks)Variable12345QALTVRQAImpQAImpQAImpQAImpQ (units)942.73942.23944.48944.20948.23944.20953.46944.20960.44944.20V00.000010270.000041090.000092460.000164370.00025682Vimp0.000037250.000037250.000037250.000037250.00003725TC($)6,910.646,910.646,927.206,927.096,954.726,943.326,993.066,954.827,041.056,963.75% TC Savings over SLT-QA—0—0.159—0.164—0.976—1.098K=500,D=5200,h=10,p=20,h′=5,i=0.1,⊖=0.2,Γ=0.005.For all these
results, the proportion defective is 0.20. Equations (3.6) and (3.10) indicate
that the optimal lot size and optimal variance for the SLT-QA-LTVR model both
depend on the proportion defective, θ.
Therefore, a logical question to ask is exactly what is the impact of this
parameter? Table 5 presents results for
defect proportion values from 0.025 to 0.30 for the situation where lead-time
follows a uniform distribution with a 7-week lead-time interval and the
technical parameter Γ=0.005. Note
that a lead-time variance reduction on the order of 97% is realized in all
cases. There is slightly less of an impact of lead-time variance reduction for a
smaller-proportions defective. We also observe a decrease in lot size as the
proportion defective decreases. On a relative basis, this decrease is on the
order of 5.2 percent when compared to the lot size generated by the solution to
the SLT-QA model, increasing slightly on a percentage basis as θ increases. The total inventory-related cost of
implementing SLT-QA-LTVR decreases significantly as the proportion defective
decreases. These results provide some preliminary evidence for the concept that
programs directed at simultaneously improving quality, and reducing lead-time
variability will have a synergistic impact on the performance of
production-inventory systems.Table 5
Optimal values for various values of⊖.
VariableQVimpμ*TC ($)⊖̲0.025891.860.000042880.011341666,091.390.050897.930.000042060.011233386,208.250.075904.430.000041250.011124626,330.000.100911.380.000040450.011015306,456.870.125918.790.000039640.010905356,589.080.150926.710.000038840.010794666,726.900.175935.170.000038040.010683146,870.640.200944.200.000037250.010570697,020.650.225953.840.000036450.010457227,177.320.250964.150.000035660.010342607,341.110.275975.170.000034860.010226717,512.520.300986.960.000034070.010109417,692.15K=500,D=5200,h=10,p=20,h′=5,i=0.1, LT Interval =7 weeks, Γ=0.005.


## 5. Sensitivity Analysis

In this section,
we turn our attention to an investigation of the conditions under which
investment in lead-time variance reduction is worthwhile. Specifically, we
assume both the probabilistic conditions presented in (2.6) and (2.7) and the
convexity condition (3.4) are satisfied. Under this scenario, investment is
warranted if and only if Vimp<V0, which is the equivalent of requiring the optimal lead-time variance to be
strictly less than the original lead-time variance, V0. By substituting (3.6) for Vimp in this relationship, we may solve for critical points for various parameters of
interest in order to perform sensitivity analysis. These derived relationships
can provide the manager with a yardstick to determine if investment in
lead-time variance reduction would be worthwhile.Following the
procedure outlined above, the critical point forρ is (5.1)ρ>2i2h'(h+p)[2K+V0D(h+p)(h+p)V02Γ2D3−hp].Further, since ρ=θ/(1−θ), the critical point for θ is (5.2)θ>ρ1+ρ.Thus when θ is greater than the right-hand side of (5.2),
it pays to invest. Similarly, the critical point for interest rate is (5.3)i<[V02Γ2D3η2hp4[2K/(h+p)+V0D]]1/2.Thus when the
interest rate is less than the right-hand side of (5.3), it pays to invest.
Finally, the critical point for the original setup cost is
(5.4)K<(h+p)V0D2[V0Γ2D2η2hp4i2-1].Thus when the
setup cost is less than the right-hand side of (5.4), it pays to invest. For each of
these relationships, we examine their sensitivity to a single parameter,
holding all others constant. We examined a number of cases for which all
parameter values are the same as in the sample problem whose results are
presented in Table 2. Critical points for proportion defective show the lead-time
variability reduction model to be quite robust. As the uniformly distributed
lead-time interval increases from two to seven weeks, the lower bound on
proportion defective for which the model remains optimal decreases from 0.05 to,
essentially, zero. For the following situations, we used a uniformly
distributed lead time over a five-week interval. Whether setup cost, K, increases from $250 to $2,000 per setup,
annual demand increases from 4 000 to 15 000 units, or interest rate increases from
2.5 to 20 percent, investment in lead-time reduction is warranted for the
smallest possible values of proportion
defective. This indicates that even if quality is perfect, there is a benefit
to be gained for the performance of the production-inventory system from
reducing lead-time variance. Similar results are obtained for variations in the
per-unit inventory costs, such as defective holding cost, h′,
nondefective holding cost, h, and
backorder cost, p.In terms of the
critical point for interest rate, as the uniformly distributed lead-time
interval increases from one to seven weeks, the upper bound on interest rate
for which the model remains optimal increases from 1 to 28.5 percent. As
proportion defective,θ,
increases from 0.05 to 0.40, the upper bound on interest rate for which the
model remains optimal increases from 14 to 19 percent. In both cases, this is
in the correct direction since we would expect to be willing to pay more to
reduce lead-time variability when variability is higher or quality is poorer.
As the setup cost, K, increases from $250 to $2,000 per setup, the upper bound
on interest rate decreases from 29 to 4.3 percent. This makes intuitive sense
since high setup cost means large lot sizes and hence a fewer number of
replenishment cycles. Thus one would be less willing to pay higher investment
costs to reduce lead-time variability when fewer replenishment cycles are
experienced by the system. These results suggest that a program of quality
improvement should logically be accompanied by one aimed at reducing lead-time
variability. The same argument can be made for the case of increasing demand.
The results show that as demand increases from 4 000 to 15 000 units, the upper
bound on the interest rate increases from 11.1 to 65.4 percent. Hence, for a
given lot size, larger demand induces more replenishment cycles and a
willingness to pay more to reduce lead-time variability. Further, improvement
in quality will result in smaller lot sizes and, in turn, more replenishment
cycles indicating that an accompanying reduction in lead-time variability is in
order. An investigation of Γ indicates the upper bound on interest rate increases from 1.6 to 32.1 percent
as Γ increases from 0.000025 to 0.01
indicating that the greater the impact of investment in lead-time variance
reduction, the greater the upper bound of what one would be willing to pay to
fund such a program. These results indicate that the model is robust with
respect to interest rate since, in general, the upper bounds are above the
prevailing cost of capital.Some final
interesting results are obtained from an investigation of the critical point
for original setup cost,K0.
As lead-time interval increases from two to seven weeks, the upper bound on
original setup cost increases from $342 to $826 549 per setup. This indicates
that it is essentially always worthwhile to reduce lead-time variability
regardless of the setup cost. Further, the setup cost essentially is no
restriction when the lead-time variability is large. When the proportion
defective, θ,
increases from 0.05 to 0.40, the upper bound on original setup cost increases
from $168 796 to $312 941 per setup, indicating that investment in lead-time
variability reduction is worthwhile for all cases since actual setup cost for
the vast majority of situations are lower than the upper bound. These results
indicate that there is certainly significant room for reducing setup costs and
lead-time variability while simultaneously engaging in a program of quality
improvement.


## 6. Conclusion

This paper presents an extension of
the quality-adjusted EOQ model with finite-range stochastic lead-times in which
investment in lead-time variance reduction is considered. Specifically, a
quality-adjusted lead-time variance reduction model is developed in which the
lead-time variance is treated as a decision variable. This model assumes
investment in lead-time variance reduction proceeds according to a logarithmic
investment function. Relationships for economic lot size, optimal total cost,
optimal number of time units of demand satisfied by each order, and the optimal
time differential between placing an order and the start ofq time units that
will be satisfied by the given order, as well as optimal lead-time variance,
are derived. Results of numerical examples indicate that savings can be
realized by investing in lead-time variance reduction. The results of
sensitivity analysis relating to  proportion defective, interest rate, and setup
cost show the lead-time variance reduction model to be quite robust and
representative of practice.



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*Source: 795869-2008-04-21.xml*