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10/7/2009 9/22/2010

Evaluating the Cost-Effectiveness of Shorter, More Expensive Exponential Lead Times in

Inventory Models where Orders can Cross

(Previous title: Crashing Stochastic Lead Times)

Abstract

We use exponential lead times to demonstrate that reducing mean lead time, implying

also reducing its variance, has a secondary reduction of the variance due to order crossover, the

net effect being that of reducing the inventory cost. Now if the reduction in inventory cost

overrides the investment in lead time reduction, then the lead time reduction strategy would be

tenable.

We define lead time reduction to be the process of decreasing lead time at an increased

cost. To date, decreasing lead times has been confined to deterministic instances. Here we

examine the case where lead times are exponential, for when lead times are stochastic, deliveries

are subject to order crossover, so that we must consider effective lead times rather than the actual

lead times. The result is that the variance of these lead times is less than the variance of the

original replenishment lead times.

Here we present a two-stage procedure for reducing the mean and variance for iid lead

times, using the exponential distribution as a paradigm. We assume that the lead time is made of

one or several components and is the time between when the need of a replenishment order is

determined to the time of receipt. Our model uses exponential lead times with a deterministic

demand rate and takes advantage of order crossovers inherent in long-term ordering from a large

number of identical suppliers.

Keywords: Inventory optimization, Order crossover, Simulation

Deleted:

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Evaluating the Cost-Effectiveness of Shorter, More Expensive Exponential Lead Times in

Inventory Models where Orders can Cross

1. Introduction

Reducing the lead time has two components: reducing the mean and reducing the variance.

And with iid lead times we encounter order crossover, because what occurs in iid lead times is

that orders can cross and this order crossover transforms the original lead times into ‘effective

lead times’ whose mean is the same as that of the parent lead time but whose variance is lower

(Hayya et al., 2008). In this paper we are using an exponential lead time distribution, and, thus,

reducing mean lead time implies reducing its variance. But then there is a secondary variance

reduction due to order crossover. And building a lead time reduction model that takes account

of order crossover is a major contribution of this paper. So what we do in this paper is the

following: We use the model in Silver et al. (1998, p. 263), where the shortage penalty is applied

per unit short and where the demand rate is constant and the lead time is iid exponential. But

when lead times are iid, the variance of the effective lead times (ELT) can be written (for the

small ordering intervals characteristic of order crossover) as a linear function of that ordering

interval. This creates a new and smaller standard deviation of demand during the effective lead

time that we now substitute into the cost function, which when optimized yields a lower

inventory cost. In the optimization we cannot obtain a closed-form solution to the optimal order

order quantity; so then we use a two-stage procedure. In the first stage, we obtain closed-form

estimates that we then use as initial values in the second stage where we use

Mathematica or numerical search to obtain the answer.

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So far, research on lead time reduction has dealt only with deterministic lead times. See, for

example, Lan et al. (1999) or Hariga and Ben-Daya (1999). In these papers, the authors portray

cost as a piecewise linear function of lead time, L..

In conventional cost functions for the order quantity, reorder point inventory model,

where the transit cost of pipeline inventory is not considered, deterministic demand rate, D, plays

the following role: the standard deviation of demand during lead time (X) with L stochastic and

D constant is

x LDσ σ= . (1.1)

Using Silver and Peterson (1985, pp. 70-71), Lan et al. (1999, p. 600) give these as the

components of deterministic lead time:

• Administrative time at the stocking point (assumed to be the manufacturer or the

manufacturer’s warehouse).

• Transit time to the supplier

• Time at the supplier

• Transit time back to the stocking point of the retailer

• Time from order receipt until the item is on the shelf

But the set of lead time components can be encapsulated in the traditional definition of lead

time, which is the “interval between the time a need for a material is determined and the time for

its actual delivery” (Ammer and Ammer, 1977, p. 235). In lead time reduction, a great gain is

obtained from order crossover when lead times are iid. Magson (1979, p. 317) working at

Nohanga Consolidated Copper Mines in Zambia, noted that “The vast majority of lead times are

of course variable …”, and hence lead times are subject to order crossovers, which reduce

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inventory levels. There is now excellent literature on order crossover, starting with Zalkind

(1978) and followed by Robinson et al. (2001), Bradley and Robinson (2005), Robinson and

Bradley (2008), apart from papers by Hayya et al. (1995. 2008, 2009), Hayya and Harrison

(2010), He et al. (1998), Reizebos (2006), Riezebos and Gaalman (2009), and Srinivasan et al.

(2009).

It has been shown (Hayya et al., 2008) that the more the crossover the more the

reduction in the variance of the parent lead time, which is true for all distributions examined, and

that the degree of crossover is dependent on the periodicity of ordering, call it T = Q/D, where Q

is a constant order quantity and D is the demand rate. For those who are not familiar with this

material, we give an example similar to the one in Hayya et al. (2008). Consider Table 1.1,

where the lead times are exponential with mean 5 (thus variance 25) and where orders are placed

every period, say at t = 1, 2, …, 100,000. The order quantity is of no concern at present. What is

of concern is how, in the long run, order crossover transforms the lead time. In Table 1.1, we

only show the first ten orders. Here, the periodicity of ordering is T = 1, and the lead times in

Column C2 are random draws from an exponential with mean 5. Then, for example, order 2 with

a lead time of 0.12 would arrive at time 2.12, which is the smallest arrival time when we sort

Column C3 into Column C4. Since the order that arrives first would be at time 2.12, the effective

lead time (ELT) for the first order arrival would be at 2.12 -1 = 1.12. Similarly, the ELT for the

second order arrival (initial order 3) would be at 2.83 - 2 = 0.83. Thus, the inventory system

faces lead times not those of the parent L in Column C2, but of the ELT in Column C5.

Table 1.1 An order crossover scenario with ordering periodicity of one

[L ~ Exp(1/5), n = 100,000]

C1

nT with

T =1

C2

Lead Time,

L

C3

Arrival

Time

C4

Sequenc

e of

C5

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Arrivals

Periods or Time Horizon

L~Exp(1/5) parent lead

time

C1+C2 (mixture of 2-parameter

exponentials)

Sort C3 Actual Arrival Times

Effective Lead Time (ELT)

1 1.83 2.83 2.12 1.12

2 0.12 2.12 2.83 0.83

3 1.89 4.89 4.89 1.89

4 5.79 9.79 6.01 2.01

5 1.25 6.25 6.25 1.25

6 0.01 6.01 9.62 3.62

7 5.85 12.85 9.79 2.79

8 6.24 14.24 11.14 3.14

9 0.62 9.62 12.38 3.38

10 2.38 12.38 12.85 2.85 L : lead time; Exp(1/5): exponential with mean = standard deviation = 5; C stands for column..

We see in Table 1.1 that orders are not coming in the same sequence in which they are placed,

and we call the time between receipts the ‘Effective Lead Time (ELT)’. We note that the

variance of a sequence of n integers starting at 1 in Column C1 (i.e., the variance of nT where T

= 1) is equal to ( 2n – 1)/12, It turns out that if we ordered every period, then in the long run

(Hayya et al., 2008),

V(ELT) ≈ 1,2( ) ( )E L E L− , (1.1)

where V(.) denotes ‘variance’ and 1,2)( )E L denotes the first order statistic from a random sample

of size 2 from the distribution of L. So for T = Q/D = 1, V(ELT) would be bracketed, as

1,2( ) ( )E L E L− ≤ V(ELT) ≤ Var(L). (1.2)

Thus, with L ~ Exp(1/λ ) i.e., an exponential with mean λ and variance 2λ , then for an ordering

interval T = 1,

1

2λ≤ V(ELT | L ~ Exp(1/λ ) ≤ 2λ . (1.3)

For our Table 1.1 example, we should obtain at the limit

V(ELT | L ~ Exp(1/5))≤ 2.5. (1.4)

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This is because the expected value of the first order statistic from a random sample of size 2 from

the exponential distribution is1

2λ, and substituting in Equation (1.1), we obtain V(ELT) =

1

2λ=

2.5.

The paper is organized as follows. Section 2 is a survey of recent studies on lead time

reduction and it contains a table listing the relevant articles and how they relate to each other.

Section 3 presents properties for V(ELT) and makes the argument that for a small ordering

interval, T, we could by regression analysis characterize V(ELT) as a linear function in T, with

the accuracy of the fit measured by 2R . Section 4 presents our model plus numerical examples

with data taken from Hadley and Whitin (1963, 172-174) and Ben-Daya and Raouf (1994) where

the latter authors reduce three components of a four-component lead time. Section 5 is a

summary and conclusions. Appendix A provides a sensitivity analysis for the effect on the

inventory cost of changes in the optimal ordering period, T*. Appendix B gives regression

equations for the variance and standard deviation of the effective lead times in terms of T for

various means of the parent exponential lead time. Appendix C presents, via partial derivatives, a

sensitivity analysis of the ordering interval, T, and it also provides the derivation of the value of

T* that minimizes V(ELT).

2. Literature Review

The literature we have seen on lead time reduction all deal with deterministic lead times,

and, similar to the work on setup reduction (Porteus, 1985), include a lead time cost in the

objective function. For example, Liao and Shyu (1991, p. 75) present the following cost function:

( ) ( )o xC L hz R Lσ= + , (2.1)

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where h is the holding cost in $/unit/year; oz is the safety stock factor; z ~ N(0,1);

x D Lσ σ= is

the standard deviation of demand during lead, with D denoting demand rate, and

1

1

1

360( ) { ( ) ( )}, (2.2)

i

i i j j j

j

R L c L L c b aQ

µ −

−=

= − + −∑

where R(L) denotes the lead time reduction cost, with m the daily demand rate in units per day,

Q the order quantity, aj the minimum duration of lead time component j; bj the normal duration

of lead time component j; ci the lead time reduction cost of lead time component i; and Li-1 is the

length of the lead time for component i – 1, with i = 1, 2, …, n; j = 1,2 , …, i -1. We note in the

cost Equation (2.1) the absence of expressions for the ordering and shortage costs.

Using JIT for reducing lead times, Ben-Daya and Raouf (1994, p. 580) present a model

that accounts for the ordering cost but not the shortage cost:

C(Q, L) = AD

Q+ h((Q/2) +

zoσ

x) +

D

QR(L), (2.3)

where A is the ordering cost per order, D is the constant demand rate, h is the holding cost per

unit per unit time; Q is the order quantity, R(L) is the same as Equation (2.2), and oz is the

standard normal safety stock factor. They find Equation (2.3) to be convex in Q but concave in L,

and therefore, for fixed Q, the minimum of (2.3) occurs at the end points of the interval for L.

Table 2.1 Lead Time Reduction Chronology

Authors Cost Equation Reduction Cost per cycle

Liao and

Shyu

(1991)

( ) ( )o x

C L hz R Lσ= + ,

Q constant, and C(L) concave in L

1

1

1

( ) ( ) ( )i

i i j j j

j

R L c L L c b a−

−=

= − + −∑(a piecewise linear function).

Ben-

Daya and

1

1

1

( , ) ( ) ( ( ) ( ))2

i

o x i i j j j

j

AD Q DC Q L h z c L L c b a

Q Qσ

−

−=

= + + + − + −∑1

1

1

( ) ( ) ( )i

i i j j j

j

R L c L L c b a−

−=

= − + −∑

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Raouf

(1994)

Convex in Q for fixed L; concave in L for fixed Q.

( , ) ( )2

L

o x

AD Q DC Q L h z e

Q Q

βσ α −= + + +

Convexity or concavity depends on the Hessian.

a piecewise linear function.

( ) LR L e βα −= ,

,α β constants

Ouyang

et al.

(1996)

For normally distributed LTD:

( , ) [ (1 ) ( )]2

[ (1 ) ( )] ( )

o x x o

o o

AD QMinC Q L h z G z

Q

D DG z R L

Q Q

σ β σ

π π β

= + + + −

+ + − +

Convex in Q for fixed L; concave in L for fixed Q.

1

1

1

( ) ( ) ( )i

i i j j j

j

R L c L L c b a−

−=

= − + −∑a piecewise linear function

Ouyang

and Wu

(1997)

For normally distributed LTD:

( , ) [ (1 ) ( )]2

o x x o

AD QMinC Q L h z G z

Qσ β σ= + + + − + ( )

DR L

Q

s.t. ( )x oG z

Q

σα≤

Does not satisfy the Kuhn-Tucker conditions.

Same as in Ouyang et al. (1996).

Ouyang

and Wu

(1998)

Distribution-free: the same as in Ouyang et al. (1996),

except that now the safety stock, r - µ L replaces o xz σ and

the ESPRC, [ ]E X r+− , replaces ( )

x oG zσ .

Moon

and Choi

(1998)

They correct a redundancy flaw (service level and a unit shortage cost in the same formulation) in Ouyang et al. (1996).

Hariga

and Ben-

Daya

(1999)

Lead time reduction models as in the above. but with full and partial information about the LTD.

Lan et al.

(1999)

Same as in Ouyang et al. (1996), but a ‘refinement’ of the solution.

Same as in Ouyang et al. (1996)

Pan et al.

(2002)

For normally distributed lead time demand,

( , , ) [ (1 ) ( )]2

x o x x o

AD QC Q L h z G z

Qπ σ β σ= + + + − + ( )

DR L

Q

The same as for Ouyang et al. (1996), except for the structure of R(L). C is convex in Q, concave in L.

1

1

1

1

( ) ( )( )

( ( ),

i i i

i

j j j j i i

j

R L a bQ L L

a b Q T t L L L

−

−

−=

= + −

+ + − ≤ ≤∑The same as in Ouyang et al.(1996), except that

j j jc a b Q= + , j j jc a b Q= +

Hoque

and

Goyal

(2004)

They argue that the formulas in Pan et al. (2002) are incorrect and they offer an alternative solution. The Pan et al. model is extended to include a constraint on Q. They extend the Pan et al. model to include a constraint on Q.

Pan and

Hsiao

(2005)

Backorder unit price discount, xπ , introduced. For normally

distributed LTD:

The same as in Pan et al. (2002) The distribution-free case is on the lines in Ouyang and Wu (1998). An algorithm used to

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( , , ) [ (1 ) ( )]2

x o x x o

AD QC Q L h z G z

Qπ σ β σ= + + + − +

[ (1 )] ( )x o x o

DG z

Qπ β π β σ+ − + ( )

DR L

Q

produce (Q*, *

xπ , L*).

Chang

(2005)

A linear programming (objective function analogous to the one in Be-Daya & Raouf, 1994) minimized subject to resource constraints.

As in Ben-Daya and Raouf (1994), except that D/Q is an integer.

C: cost; D: demand stochastic; ( )oG z : unit normal loss integral; L, deterministic lead time, but “reducible,” i.e., a deterministic piecewise step

function; Q: order quantity; i: 1, 2, …,n, index for lead time component; ia : the minimum duration of lead time component i; jb : the normal

duration of lead time component i, j = 1, 2, .. i;ic : the cost per unit time for reducing lead time component i; LTD: Lead time demand; α : a

constant; β : a constant, also fraction backordered; xσ : standard deviation of demand during lead time; π : cost per unit short; oπ : gross

marginal profit per unit; xπ : backorder price discount.

We see in Table 2.1 that Ben-Daya and Raouf (1994) extend the Liao-Shyu (1991) paper,

and then Ouyang et al. (1996) extend Ben-Daya and Raouf (1994) by adding to the model a

backorder cost captured by a fixed penalty per unit short and a lost sales cost captured by the

profit contribution per unit. But because the cost function is concave, Ouyang et al. (1996) iterate

over different values of the components of lead time, calculating a transient Q and then

calculating the transient cost, and finally choosing the (Q, L) pair that achieves minimum cost.

Here, (Ouyang et al., 1996, their Table 2, p. 832; and also in Ouyang and Wu, 1997, p. 879, and

Lan et al.,1999, their Table 2, p. 604) the minimum occurs under full backordering. Yet again in

the context of a mixture of backorders and lost sales, Ouyang and Wu (1998), depart from the

normally distributed lead time demand (LTD) setting and instead define a class of cumulative

distribution functions (cdf’s) identified only by their mean and variance, determine the worst

among these, then develop for the worst cdf an algorithm that minimizes the cost function over Q

and L. This min-max procedure is due to Scarf, (1958), and it is extended by Pan and Hsiao

(2005) to model price discounts during the backorder period.

Pan et al. (2002, p. 1048) make the lead time reduction cost not only a function of lead

time components, but also of the order quantity. Pan et al. use normal lead time demand (LTD)

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and another distribution-free LTD model and conclude that the (local) Q* depends only on the

fixed lead time reduction costs, but that the optimal cost, C*, is also a function of the variable

lead time reduction costs, “which in turn determine the global optimal order quantities” (p.

1053).

Before closing the literature review on lead time reduction, we must point out its obvious

similarities to the literature on setup time reduction. See Porteus (1985). It is clear, for example,

that setup is an element of production lead time, which is a component of total lead time (Chu,

1996). Thus, we could also consider the literature on SMED (Single-Minute Exchange of Die)

initiated by Shingo (1985) and in subsequent articles, such as McIntosh et al. (2000). We are

also induced to establish a connection between the literature on lead time reduction and that on

lead time quotation, as the later uses lead time as a decision variable for maximizing revenue by

lead time differentiation, by expediting, and by segmenting customers into delay-sensitive and

delay-insensitive. See Keskinocak et al. (2001), Plambeck (2004), Chayet et al. (2004, pp. 309-

311), and Çelic and Maglaras (2008).

Our present paper, just like those listed in Table 2.1, deals with lead time reduction.

Where it differs is that we use an exponential lead time, where these papers have modeled

deterministic lead times. Also, we do not include in the objective function an expression for the

cost of lead time reduction. Rather, we calculate the savings in reducing the mean of the

exponential lead time and then see whether the savings can pay for the cost of that reduction.

3. Behavior of the Effective Lead Time (ELT) as a function of the Review Period (T)

We lay out a theoretical foundation for using a linear function for V(ELT) in T. We see

that in Figure 3.1 that it is a V-like function.

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T = Q/D

V(ELT)

50403020100

25

20

15

10

5

0

25

Figure 3.1 V(ELT) vs. T, parent L ~ Exp(1/5)

Minimum of 0.109

at T = 0.025

We show in Appendix C that the minimum for V(ELT) occurs at T > 0 and argue that the

stem of interest is that of the right section of the V-shaped curve for small T. We fit a straight

line to that stem and obtain equations of the form

( )V ELT = a + bT, (3.1)

We find that, in terms of 2R , Equation (3.1) provides nearly a perfect fit as can be seen in

Appendix B.

3.1 The Variance of the Effective Lead Time

Going back to Table 1.1, the mathematical structure is

V(ELT) = V(C4 – C1) = V (C4) + V(C1) -24, 1 4 1C C C Cρ σ σ

= 2(2 1

12

n −) +

2

1

λ - 2 4, 1C Cρ

2 2

2

1( )

12 12

n n

λ+ , (3.2)

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where n is the number of consecutive orders placed and where ρ denotes the correlation

coefficient. In Equation (3.3), n is the number of consecutive orders placed at times 1, 2, 3, …,n;

and 2

1

λ is the variance of an exponential lead time with parameterλ .

Let v =2 1

12

n −. For a large n, we could write v in terms of a continuous uniform

distribution (0, b=nT), where n is the number of orders placed and T is the time between orders.

Thus, for differentiation purposes with a large n, we could consider ν as continuous in n and

Equation (3.1) could be rewritten as

V(ELT) = 2( )

6

nT+

2

1

λ- 2 ρ

2 2

2

( ) ( ) 1

12 12

nT nT

λ

+

. (3.3)

It turns out that Eq. (3.3) is ill-conditioned (i.e., sensitive to small changes) when ρ

approaches 1.

3.2 Assumptions

1. Lead times, L, are iid exponential.

2. Reducing mean lead time, E(L), incurs a pre-determined fixed cost.

3. The shortage cost is per unit ( 2B , as in Silver et al., 1998, p. 263).

4. The demand rate, D, is deterministic.

5. Orders are placed every T = Q/D time interval; that is, for any ordering scenario, T is fixed.

6. The model is long-run, infinite horizon. .

Proposition 1. nLim→∞

1, 4C Cρ → 1 from the left, but can never be exactly 1.

Proof. Referring to Table 1.1 and since C1 and C2 are uncorrelated,

V(C4) = V(C3) = V(C1) +V(C2). (3.4)

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Then because V(C2) is very small as compared with V(C1), we could say that

V(C4) ≈ V(C1), (3.5)

so we can say

( 4)1

( 1)

V C

V C→ as .n→∞ (3.6)

and that

( 1, 4) ( 1, 1) ( 1) 0.Cov C C Cov C C V C v≈ = = ≥ (3.7)

Now using Eq. (3.6), we can write

nLim→∞

..n→∞ nLim→∞

1 1

( 1, 1)

C C

Cov C C

σ σ=

1 1

( 1, 1)

C C

Cov C C

σ σ= → 1 from the left. (3.8)

The correlation 1, 4C Cρ can never be exactly 1, because then V(ELT) = 0 as n→∞ . ■

Proposition 2. At the limit, 1, 4C Cρ approaches the left neighborhood of 1 faster than n approaches

infinity.

Proof. The easiest route is to demonstrate this phenomenon by simulation, where we see that

ln( 1, 4C Cρ ) increases quadratically or cubically as n increases linearly. Thus 1, 4C Cρ → left

neighborhood of 1 quicker than n→∞ .

■

Proposition 3. Minimum of the approximation of V(ELT) occurs at T > 0

Proof. Consider the following minimization of (3.3) with respect to T, which, when we simplify

it, yields (see the derivation in Appendix C.2):

1

* 22

6 1( ) 11

Tnλ ρ

= −−

> 0. (3.9)

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We interpret Equation (3.9) as follows: as n gets large, the numerator under the radical of the

right-hand side of (3.9) is also large (larger since, by Proposition 2, ρ approaches 1 from the left

faster than n approaches infinity) and the division under the radical sign would yield T > 0. ■

3.3 Characteristics of V(ELT)

Lemma 1. As T → 0, V(ELT) → V (L), and as T → ∞ , V(ELT) → V (L).

Proof. When T → 0, orders are placed so closely together that the effect of crossover is negated

because when the effective lead times are compared to the actual lead times (Columns C3 and C4

in Table 1.1) the difference in the sequence is small because the time between orders is small

(zero in this case). This then leads to V(ELT) → V (L). When T → ∞ , or the time between two

consecutive orders is infinitely long, the probability of order crossover approaches zero, which

also leads to V(ELT) → V(L). ■

Here are some properties of V(ELT):

1. The function (3.3) is continuous, and, by Lemma 1, it is bounded above (at large T) and

below (at T = 0) by V(L). It has a minimum at T* > 0 (Proposition 3 and Appendix C.2).

2. The function (3.3) is ill-conditioned, as denoted before (See Proposition 2).

3. V(ELT|L) is a Cauchy sequence (Binmore, 1999, pp. 50-51) in that

V(ELT|n+1) - V(ELT|n+2) < V(ELT|n) – V(ELT|n+1), (3.10)

that is V(ELT|L) keeps diminishing until it converges to min V(ELT|L). We say that nx< > is a

Cauchy sequence if, given any ε > 0, we can find an N such that for any m > N and any n > N,

| |m nx x ε− < ,

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(with ε an infinitesimally small value greater than zero), meaning that the terms of the sequence

get closer and closer together.

4. A Crossover Model

We analyze the inventory model where the shortage are charged per unit as in Silver et al.

(1998, p. 263). The ‘approximate’ cost function here is

2( , ) ( ) ( )2

o o x x o

B DAD QC Q z h z G z

Q Qσ σ= + + + , (4.1)

where A is the ordering cost per order, 2B is the shortage cost per unit, h is the holding cost per

unit per unit time, D is the demand rate, oz is the safety stock factor with z ~ N(0,1), xσ is the

standard deviation of demand during lead time, and

( ) ( ) ( )

o

o o

z

G z z z f z dz

∞

= −∫ (4.2)

is the ‘normal unit loss function.’ (Equation B.6 in Silver et al., 1998, p. 721).

Because we use stochastic (exponential) lead times, the above model would be subject to

order crossover. Hence with demand rate, D, deterministic, we let

2 2 ( )x D V ELTσ = , (4.3)

and since for small T (Appendix B) we can write

( )* 2 ( ) , 4.4o ELT

Qz R V ELT a b

Dσ = +

we can substitute the above for V(ELT) in (4.3) and subsequently substitute the V(ELT) of (4.3)

into Equation (4.1).

4.1 Optimization

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If we used use the regression of ELT

σ on T = Q/D, we would have the following cost

equation:

( )2( , ) ( ( ' ' )) ( ' ' ) ( ), 4.52

o o o

DBAD QC Q z h z a D b Q a D b Q G z

Q Q= + + + + +

where a’ and b’ are the regression estimates of ELT

σ on T (Appendix B). Since (4.1) is convex

in ( , )o

Q z (Das 1983) and consequently (4.5) is, we optimize (4.5) to yield the following closed-

form solution:

*Q = 2 2

2( ) ' ( ) /

1 2 '

o

o

EOQ a B D G z h

b z

+

+, (4.6)

and a tail probability is

2

hQ

B D. (4.7)

However if we inspect Appendix B, we see that using the regression of V(ELT) vs. T instead

would be more accurate, because that has amost perfect 2R ’s. But that approach will not yield a

closed-form solution for Q*, and we are thus resigned to use Mathematica or numerical search to

obtain the better solution. Nevertheless, we can use Eqs. (4.6) and (4.7) to give us starting points

for the search. So using V(ELT), the cost equation with order crossover is now

2

2( , ) ( / ) / ) ( )2

o o o

B DAD QC Q z h z D a bQ D a bQ D G z

Q Q= + + + + + , (4.8)

where a and b are regression estimates.

4.2 Examples

- 16 -

Suppose that we face L ~ Exp(1/10), that is an exponential lead time with mean 10, and

we want to crash that to L ~ Exp(1/5), or to a mean of 5. We then calculate the costs under both

scenarios and inspect the difference. But first we must know the parameters for the regression

Equation (4.4). We find these in Appendix B. Now we demonstrate in Table 4.1 an example

from Hadley and Whitin (1963, pp. 172 – 174), but modified for our exponential lead time.

L ~ Exp(1/10)

We begin with Eqs. (4.6) and (4.7), that is by using the regression of ELT

σ . Then we use

the results from that (in Table 4.1A) to do a Mathematica or a numerical search, using

VAR(ELT) to give us a more accurate result and put that in Table 4.1B. We see in Table 4.1A

that, with an exponential lead time with mean = 10, ‘optimality’ (with ELT

σ ) is achieved at Q* =

403.18 < EOQ = 1131.37, *

0z = 3.02, and C* = $75,568.14.

Table 4.1A. A Hadley-Whitin (1963, 172 – 174) Example, but with L ~ Exp(1/10): Using

ELTσ

(D = 1600; A = $4000; h = $10; 2B = $2,000; ELT

σ (given λ = 10) = 0.769 + 1.28 T)

ELTσ C � Q � *

oz Ordering Holding Shortage Tail Prob. T ( )

oG z

EOQ 0 11313.71 1131.37 --- 5656.85 5656.85 --- 0.00354 0.707 ---

Iterations

1.67 1.18 1.12 1.11 1.11 1.11 1.11 1.11 1.11

108203.77 77951.13 75790.63 75589.69 75570.23 75568.34 75568.15 75568.14* 75568.14

464.89 409.06 403.75 403.24 403.19 403.18 403.18 403.18* 403.18

2.69 2.98 3.02 3.02 3.02 3.02 3.02 3.02* 3.02

13766.58 15645.44 15851.22 15871.48 15873.45 15873.64 15873.66 15873.67 15873.67

74470.77 56399.50 54929.22 54790.78 54777.36 54776.05 54775.92 54775.91 54775.91

19966.42 5906.19 5010.19 4927.43 4919.42 4918.64 4918.57 4918.56 4918.56

0.00145 0.00128 0.00126 0.00126 0.00126 0.00126 0.00126 0.00126 0.00126

0.291 0.256 0.252 0.252 0.252 0.252 0.252 0.252 0.252

0.00108 0.00041 0.00036 0.00036 0.00035 0.00035 0.00035 0.00035 0.00035

*: Optimal; ELT: effective lead time; ( )o

G z : normal unit loss integral; Q*: order quantity; ELTσ : std. dev. of the

ELT; T = Q/D; *

oz : optimal safety stock factor; 2B : Shortage cost per unit; D: Demand per unit time; h : Shortage

cost per unit per unit time.

- 17 -

Then we go on to do a numerical search of the cost equation Eq. (4.8), using those intial

‘optimal’ values for Q* and *

oz . The numerical search gives Q* = 298, *

oz = 3.11, and C* ==

$77,690.

Table 4.B. A Hadley-Whitin (1963, 172 – 174) Example, but with L ~ Exp(1/10): Using

V(ELT)

(D = 1600; A = $4000; h = $10; 2B = $2,000; V(ELT (given λ = 10) = 0.124 + 4.81T) Lead Time: Mean = 10

Variance Method

Std. Dev. Method

% Diff.

Policy: Q*; *

oz 298; 3.11 403; 3.02 -35.2%; 2.9%

Ordering Cost $21,476 $15,874 26.1%

Holding Cost 51,742 54,776 -5.9%

Shortage Cost 4,472 4,918 -10%

Total Cost, C* $77,690 $75,568 2.7%

Ordering Interval, T*=Q/D 0.186 0.252 -35.5%

ELTσ 1.01 1.11 -9.9%

L ~ Exp(1/5)

We now use the same example, but with a lead time mean of 5, that is we have crashed

the mean lead time from 10 to 5. We present the results in Tables 4.2A and 4.2B..

Table 4.2A. A Hadley-Whitin (1963, 172 – 174) Example, but with L ~ Exp(1/5): Using

ELTσ

(D = 1600; A = $4000; h = $10; 2B = $2,000;

ELTσ (givenλ = 5) = 0.405 + 0.973 T)

- 18 -

ELTσ C � Q � *

oz Ordering Holding Shortage Tail Prob. T ( )

oG z

EOQ 0 11313.71 1131.37 --- 5656.85 5656.85 0 --- 0.707 ---

Iterations

1.09 0.70 0.68 0.68 0.68 0.68 0.68 0.68 0.68* 0.68

74937.90 74937.90 53303.98 52097.63 52008.67 52001.92 52001.40 52001.37 52001.36* 52001.36

490.98 450.28 447.25 447.02 447.00 447.00 447.00 447.00 447.00* 447.00

2.69 2.96 2.99 2.99 2.99 2.99 2.99 2.99 2.99* 2.99*

13035.28 14213.32 14309.80 14317.19 14317.75 14317.79 14317.80 14317.80 14317.80 14317.80

49559.07 35581.49 34682.34 34615.33 34610.24 34609.86 34609.83 34609.82 34609.82 34609.82

12343.55 3509.17 3105.48 3076.15 3073.93 3073.76 3073.74 3073.74 3073.74 3073.74

0.00354 0.00153 0.00141 0.00140 0.00140 0.00140 0.00140 0.00140 0.00140 0.00140

0.307 0.281 0.280 0.279 0.279 0.279 0.279 0.279 0.279* 0.279

0.00108 0.00044 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040 0.00040

*: Optimal; ELT: effective lead time; ( )o

G z : normal unit loss integral; Q*: order quantity; ELTσ : std. dev. of the

ELT; T = Q/D; *

oz : optimal safety stock factor

In Table 4.2A, we see that the ‘optimal’ solution under L ~ Exp(1/5) is Q* = 447 <

EOQ = 1131.37, *

0z = 2.99, and C* = $52,001.36. If we do a numerical search, using V(ELT) we

would have Q* =383, *

0z = 3.00, and C* = $60,377. We put these results in Table 4.2B, where

we make a comparison with the ELT

σ method.

Table 4.2B. A Hadley-Whitin (1963, 172 – 174) Example, but with L ~ Exp(1/5): Using V(ELT)

(D = 1600; A = $4000; h = $10; 2B = $2,000; V(ELT (given λ = 5) = 0.0603 + 2.33T)

Lead Time: Mean 5

Variance Method

Std. Dev. Method

% Diff.

Policy: Q*; *

oz 383; 3.0 447; 2.99 -16.7%; 0.3%

Ordering Cost $16,710 14,318 14.31%

Holding Cost 39,651 34,610 12.71%

Shortage Cost 4,016 3,073 23.48%

Total Cost, C* $60,377. $52,001 13.87%

Ordering Interval, T=Q/D 0.239 0.279 16.74%

ELTσ 1.13 0.68 39.82%

For the above numerical example, the savings in the total inventory cost when reducing

E(L) = 10 to E(L) = 5 is

- 19 -

$77,690 - $60,377.36 = $17,313,

or about 22% of the original cost. Now if the cost of lead time reduction is less than or equal to

this amount, then it pays to reduce exponential the lead time from a mean of 10 to a mean of 5.

We now go on to illustrate an example from Ben-Daya and Raouf (1994, p. 581). In that

example,

Ordering cost A = $200 per order

Demand, D ~ N(600, 26 )

Holding cost, h = $20/unit/unit time

Safety stock factor, k = 2.3 (i.e., Probability of stockout = 0.0107)

Number of lead time components, n = 3.

Also the lead time data given are as in Table 4.3.

Table 4.3 The Ben-Daya and Raouf Lead Time Data (Ben-Daya and Raouf, 1994, p. 581) .

Lead time component

Normal duration (days)

Minimum duration (days)

Unit reduction cost,

ic ($/day)

1 16 2 0.40

2 16 2 1.20

3 10 3 5.00

The summary of their solution procedure is in Table 4.4, where the the lead timeiL , i = 0, 1, 2, 3,

is in weeks, whereas it is in days in Table 4.3. Thus, the i

L in weeks in Table 5.2 is computed

from the normal durations and the minimum duration in days in Table 4.3; for example,

oL = (16 + 16+ 10)/7 = 6 weeks,

1L = (16 + 10 +2)/7 = 4 weeks, after reducing component 1 to its minimum,

2L = (10 + 2+ 2)/7 = 2 weeks, after reducing components 1 and 2 to their minima,

- 20 -

3L = (2 +2 + 3)/7 = 1 week, after reducing components 1, 2, and 3 to their minima.

So if we reduced all components to their minima, we would reduce the lead time from 6 weeks

(42 days) to one week (7 days).

Table 4.4 The Ben-Daya and Raouf Solution Procedure (Ben-Daya and Raouf, 1994, 581): the deterministic case

i iL in weeks R(L) per

inventory cycle Q in units Annual cost,

K(Q,L)

0 6 0 110 $2,866.95

1 4 5.6 111 2,773.35

2 2 22.4 116 2,700.65*

3 1 57.4 125 2,761.48

*Minimum

In their analysis, Ben-Daya and Raouf assume that the shortages are negligible, which for

i = 0 leads to Q = EOQ = 109.54 ≈ 110 in the first row in Table 4.4, and using the equation

( , ) ( )2

o

D QK Q L A h z k L

Qσ= + + (4.8)

gives a cost K(Q,L) = 2,866.95, which is C(EOQ) = 2,190.89 plus the cost of stochasticity,

hoz Lσ = 676.06, with L = 6. (The number of units short per unit time, not considered in the

analysis, is (2.3)D

GQσ = 0.118, with G(

oz ) the unit normal loss function. See Silver et al.

(1998, p. 730)).

Now we make the lead times in this example exponentially distributed, instead of

deterministic, as as before we use a two-stage procedure. In the first stage we arrive at closed-

form solution for Q* and *

oz , and in the second stage, we do a numerical search using these

solutions as initial values.

- 21 -

So if the lead time were exponentially distributed, the total cost in the first stage,

including shortage and lead time reduction cost, would have been

2( , ) [ ( ' ' )] ( ' ' ) ( )2

o o o

DBAD QC Q z h z a D b Q a D b Q G z

Q Q= + + + + + + ( )

DR L

Q, (4.9)

where a’ and b’ (in Tablre B.1) are the estimates for the regression of ELT

σ on T = Q/D. Then Eq.

(4.9) leads to leads to

2* 22[ ( ) ( )

(1 2 )

o

o

AD aB D G z DR LQ

h z b

+ +=

+, (4.10)

( )o

P z z> =2

hQ

B D, (4.11)

However, if, as in Ben-Daya and Raouf, we ignored the shortages and kept the service

criterion at oz = 2.3, Equation (4.9) would be reduced to

( ) ( 2.3( ' ' ))2

AD QC Q h a D b Q

Q= + + + + ( )

DR L

Q, (4.12)

with

* 2[ ( )]

(1 4.6 ')

AD DR LQ

h b

+=

+. (4.13)

Using the data from the Ben-Daya and Raouf example, we simplify the optimal order quantity to

12,000 60 ( )*

1 4.6 '

R LQ

b

+=

+, (4.14)

where b’ is the slope of ELTσ . (from Table B.1). The cost equation (4.12) becomes

** *

*

120,000( ) 20( 1380 2.3 ) ( )

2i

Q DC Q a bQ R L

Q Q= + + + + , (4.15)

and yields the results in Table 4.5 for i = 0, 1, 2, 3.Now we apply our first-stage procedure to

Eqs. (4.11) – (4.13). The first column in the table refers to components; the second to the

- 22 -

componrnt lead times (from Table 4.4); the third to ELT

σ taken from Table B.1; the fourth to the

Ben-Daya and Raouf lead-time reduction cost (from Table 4.4); the fifth, our Q* is from Eq.

(4.14); and our cost, k(q,l) is computed from Eq. (4.12). Then if the lead time were exponentially

distributed, Table 4.5 would be a stochastic analog of the deterministic Table 4.4.

Table 4.5 The Ben-Daya and Raouf (Ben-Daya and Raouf, 1994, 581) Example modified:

the exponential lead time case

(A = $200 per order, D = 600units/yr., h = $20/unit/yr., b is the slope of ELTσ )

Q* K(Q,L) i

E(i

L )

ELTσ

R(L) Table 4.4

Ours Eq. (4.14)

Ben-Daya & Raouf

(Table 4.4)

Ours Eq. (4.12)

Ben-Daya and Raouf Table (4.4)

0 6 0.532+1.02T 0 45.92 110 $19,910.21 $2866.95

1 4 0.394+0.853T 5.6 50.05 111 15,803.50 2773.35

2 2 0.275+0.579T 22.4 60.35 116 12, 012.00 2700.65*

3 1 0.149+0.396T 57.4 73.98 125 8287.41* 2761.48

*Minimum; E(i

L ): mean lead time for component i in weeks; R(L): lead time reduction cost; Q*:

optimal order quantity; K(Q,L): inventory cost in Ben-Daya and Raouf (1994, p. 581).

In comparing the stochastic results (ours) and the deterministic ones (Ben-Daya and Raouf’s) in

the last two columns of Table 4.5, we see that the stochastic case would reduce the lead time of

all three components, whereas the deterministic case would do that only for the first two

components.

5. Summary and Conclusions

There is an extensive literature on the reduction of deterministic lead time, but nothing for

stochastic lead times. In this paper, we undertake the latter task, using the exponential

distribution to characterize lead times. In reducing the lead times we take advantage of order

crossover. With order crossover, the lead times are transformed to effective lead times whose

- 23 -

mean is the same as that of the original lead time, but whose variance is smaller than that of the

original variance. Now when we introduce the resulting variance (due to order crossover) in the

optimization it would naturally lead to a lower cost, because of lower safety stocks. So we could

do a trade-off analysis and evaluate whether reducing the mean would recover the lead time

reduction cost.

The two-stage model we present is reduced to a simple form in that the demand rate is

constant, the lead time reduction cost can be calculated, and the shortages negligible. The model

itself requires numerical search (or Mathmatica) to yield an optinmal solution, because it does

not directly yield closed-forms. So we then use a first stage to obtain closed-form estimates that

we then use as initial values for the second stage.

- 24 -

Appendix A: Sensitivity Analysis of the Optimal Ordering Period, T*

To see the effect of a change in the optimal T on the total inventory cost C, we augment T

by %δ± , recalculate the order quantity Q, the safety stock factor, oz , and the ordering, holding,

and shortage costs to arrive at the total cost.. Using input data from Table 4.2 in the text, we

produce Figure A.1.

1.00.50.0-0.5-1.0

175000

150000

125000

100000

75000

50000

delta T

The Cost, C

$52,001.40

Optimal Cost =

Figure A.1 Scatterplot of C vs delta T

mean = 5

Exponential lead time

h = 10

D = 1600

B2 = 2,000

A=4,000

From Table 4.2:

We see that the effect of a decrease from the optimal T (delta = 0) is more significant

than an increase, no doubt because the increase in the ordering costs (with delta negative)

overwhelms the corresponding decrease in the holding and shortage costs, as seen in Table A.1.

- 25 -

Table A.1 Percent change in T (δ ) vs. increase in total inventory cost

δ Ordering

Cost

Holding

Cost

Shortage

Cost

Total Cost % above

optimal

-90% $143,177 $25,350 $1,687 $170,214 227%

-70% 47,726 26,671 2,030 76,427 47

-50% 28,635 28,772 2,336 59,743 15

-20% 17,897 32,235 2,780 52,912 2

0% 14,318* 34,610* 3,074* 52,001* 0

20% 11,931 37,001 3,368 52,380 0.6

50% 9,545 40,589 3,811 53,944 4

70% 8,422 42,973 4,107 55,503 7

90% 7,536 45,348 4406 57,289 10

- 26 -

Appendix B: V (ELT), ELTσ vs. T

These results were obtained by simulation.

Table B.1. Variances and standard deviations of ELT for small T = Q/D, given D constant

Exponential mean V(ELT) vs. T

ELTσ vs. T Range of T=Q/D R

2-Adj.

1

V(ELT) = 0.0171+0.360T

ELTσ = 0.149 +0.396T

0.004 ≤T ≤2.0 0.004 ≤T ≤2.0

99.1% 93.1%

2.0

V(ELT)=0.0264+0.851T

ELTσ =0.275 + 0.579T

0.01 ≤T ≤2.0 0.01 ≤T ≤2.0

99.8% 94.5%

2.5

V(ELT) = 0.036 + 1.10T

ELTσ .= 0.348+0.636T

0.0125 ≤T ≤ 2.0 0.0125 ≤T ≤2.0

99.8% 94.3%

3

V(ELT) = 0.0298+ 1.35T

ELTσ = 0.337 + 0.731T

0.01 ≤T ≤2.0 0.01 ≤T ≤2.0

99.9% 95.6%

4

V(ELT) = 0.0375+ 1.84T

ELTσ = 0.394 + 0.853T

0.02 ≤T ≤2.0 0.02 ≤T ≤2.0

99.9% 95.9%

5 V(ELT) = 0.0603 + 2.33T

ELTσ = 0.405 + 0.973T

0.025 ≤T ≤2 0.0125 ≤T ≤2

99.9% 97.3%

6 V(ELT) = 0.0630+ 2.83T

ELTσ = 0.532 + 1.02T

0.03 ≤T ≤2.0 0.03 ≤T ≤2.0

99.9% 96.1%

7.5

V(ELT) = 0.0642 + 3.61T

ELTσ = 0.598 + 1.41 T

0.05 ≤T ≤1 0.05 ≤T ≤1

99.9% 97.4%

10 V(ELT) = 0.124 + 4.81T

ELTσ = 0.769 + 1.28T

0.05 ≤ T ≤ 2 0.05 ≤ T ≤ 2

99.9% 97.6%

12.5

V(ELT) = 0.160 + 6.05T

ELTσ = 0.819 + 1.79T

0.05 ≤T ≤1 0.05 ≤T ≤1

99.9% 97.9%

16 V(ELT) = 0.390+7.60T

ELTσ = 1.07 + 1.59T

0.07 ≤ T ≤ 2 0.07 ≤ T ≤ 2

99.9% 97.2%

- 27 -

Appendix C: The ordering interval, T

C.1. Sensitivity Analysis for T in terms of the Problem Parameters: To do that we take

partial derivatives of T with respect to the problem parameters (a, b, A, 2B , D, ( )o

G z .Since T =

Q/D, we can from Eq. (5.3) write

2

2( )

(1 2 )

o

o

AaB G z

DTh bz

+=

+. (C.1)

To fathom the influence of the problem parameters (a, A, 2B , ( )oG z , h, b) on T, we take partial

derivatives, obtaining:

T

A

∂∂=

2

2

(1 2 )

2(1 2 ) ( )

o

o o

h bz

AD h bz B G z

D

+

+ +

, (C.2)

and similar to the EOQ, T is increasing concave with respect to the ordering cost A.

T is also directly proportional to the unit shortage cost2B :

2

T

B

∂∂

=

2

( ) 1 2

22 (1 2 )( ( )

o o

o o

aG z bz

Ah bz aB G z

D

+

+ +

. (C.3)

But it is inversely proportional to the demand rate, D:

T

D

∂∂

22

2( )

(1 2 )(1 2 )

o

o

o

A

AB G z

DD h bzh bz

= −

++

+

. (C.4)

- 28 -

As for the normal unit loss integral, T is directly proportional to it:

( )o

T

G z

∂∂

= 2

2

1 2

22 (1 2 ) ( )

o

o o

aB bz

Ah bz aB G z

D

+

+ +

.. (C.5)

Similarly for the regression intercept a:

2

2

( )

22 (1 2 )( ( )

o

o o

B G zT

a Ah bz aB G z

D

∂=

∂+ +

. (C.6)

And also for the unit holding cost, h:

2

1.5

2( )

2 (1 2 )

o

o

AaB G z

T D

h h bz

+∂

= −∂ +

. (C.7)

But T is inversely proportional to the regression slope parameter b and the safety stock factor oz :

2

0.5 1.5

2( )

(1 2 )

o o

o

Az aB G z

T D

b h bz

+∂

= −∂ +

, (C.8)

o

T

z

∂∂

= 2

0.5 1.5

2( )

(1 2 )

o

o

Ab aB G z

D

h bz

+−

+. (C.9)

- 29 -

C.2. Derivation of the value of T that minimizes V(ELT): We start with the representation of

V(ELT) given by (3.2):

V(ELT) = 2( )

6

nT+

2

1

λ- 2 ρ

2 2

2

( ) ( ) 1.

12 12

nT nT

λ

+

Differentiating with respect to T yields:

V’(ELT) = n2T

3−

ρn4T 3

6+ 2n2T

1

λ2+n2T 2

12

2 3 n2T 2 1

λ2+n2T 2

12

Solving this for T yields the critical points. The following value for T* minimizes V(ELT).

T * → 6 −λ2n2 −1+ ρ2( )+ −λ 4n4 −1+ ρ2( )

λ 4n4 −1+ ρ2( )

- 30 -

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