Art Zhaohui

8/2/2019 Art Zhaohui

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Int. J. Production Economics 58 (1999) 147 158

The performance of two popular service measures on management

effectiveness in inventory control

Amy Zhaohui Zeng*, Jack C. Hayya

Department of Production and Decision Sciences, Cameron School of Business, University of North Carolina at Wilmington, Wilmington,

NC 28403, USA

Department of Management Science and Information Systems, 303 Beam Business Administration Building,

The Pennsylvania State University, University Park, PA 16801, USAReceived 3 July 1997; accepted 16 June 1998

Abstract

Service is one of the inventory managers concerns and is frequently incorporated into the ordering decisions. Since

there are multiple measures of service available for evaluating the efficiency of an inventory system, a comparative study

is necessary and has not been addressed in the literature. This paper evaluates two popular service measures, which are

the probability of no stockout during lead time and the fill rate, in the context of continuous inventory systems. The

performance of the two measures is examined by evaluating the tradeoff among the cost, the level of service, and the

inventory turnover ratio. 1999 Elsevier Science B.V. All rights reserved.

Keywords: Inventory control; Customer service; Optimization; Probability distribution

1. Introduction

As every organization is competing in todays

global market, it is evidenced that companies offer-

ing superior customer service remain competitive

and profitable. Since one of the key customer-basedservice measures is availability of goods, inventory

is considered practically inevitable for maintaining

good customer service. The major functions of in-

ventory can be described as: (1) to support and

provide necessary physical inputs for manufactur-

* Corresponding author. Tel.:#1 910 962 7190; fax:#1 910

962 3815; e-mail: [email protected].

ing; and (2) to protect companies against uncertain-

ties that arise from such cases as discrepancy

between demand and production, machine deterio-

ration, and human errors. Especially for finished

products, demand during lead time is regarded as

a major source of uncertainty. Nevertheless, it isalso agreed that holding inventory is extremely

costly and that inventory should be managed effi-

ciently. Regardless of the type of a firm, the man-

agement effectiveness of inventory decisions centers

on three areas: cost, service level, and turnover

ratio, which comprise a triangle as depicted in

Fig. 1. The total relevant cost involves ordering

and holding expenses; the service level is used to

control the amount of inventory needed for satisfy-

ing customers demand; and the inventory turnover

0925-5273/99/$ see front matter 1999 Elsevier Science B.V. All rights reserved

PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 00 2 1 0 - 2


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Fig. 1. A triangle of the management effectiveness of inventory

decisions.

ratio is a measure of how effectively inventories are

being used [1]. Demand during lead time, as men-

tioned earlier, is the major reason for holdinginventory and should have significant impact on

inventory decisions.It can be summarized from the literature that

two service level measures are frequently used in

inventory control (e.g., [25]). We denote these two

measures as P

probability of no stockout during

replenishment lead time and P

fraction of de-

mand satisfied directly from the shelf (also called fill

rate). Mathematical formulation of these two

measures depends on the type of inventory system

utilized by managers. While there exist a number of

control systems, in this paper we consider a single-

item, single-stage continuous (s, Q) inventory sys-

tem, where s is the reorder point and Q is the fixed

order quantity. This type of system is operated as

follows: whenever the inventory position (on hand

plus on order) drops to a reorder point, a constant

order size is placed. The advantages of this system

are numerous, including its simplicity, optimality

for most situations, and its role as a building block

for many other complicated control policies. If as-suming both reorder point (s) and order quantity

(Q) as continuous variables, the simplified math-ematical formulas for (P

, P

) found in the litera-

ture are

P"F(s)"

Q

f(x) dx, (1)

P"1!bM(s)/Q, (2)

where

bM(s)"

Q

(x!s) f(x) dx (3)

is the so-called expected number of shortages oc-

curring during lead time, and f(.) and F(.) are thedensity (pdf) and distribution (cdf) functions of

lead-time demand, respectively. Note that both

measures are valued in percentage and that a prob-

ability distribution of lead-time demand must be

assumed in order to calculate the values.

A large amount of existing literature has concen-

trated on developing optimal values of (s, Q) by

using one of the service measures as either a mana-

gerial objective or a constraint (see [6]), but few has

addressed the following questions: (1) Are the two

measures the same? (2) Under what circumstanceswould one measure outperform the other in terms

of management effectiveness? and (3) How would

lead-time demand influence the performance of

two in making inventory decisions? In addition,

through our conversations with inventory man-

agers in some leading retailing companies, we have

noticed that although practitioners indeed rely on

one of the measures to evaluate the efficiency of

their inventory decisions, they are unclear about or

confused by the managerial implications of thesemeasures. Therefore, the main objective of this

paper is to provide answers to the above questions.

To accomplish this, we rest upon the components

illustrated in Fig. 1 to investigate the interrelation-

ships between cost, service, and inventory turns. In

particular, two sets of optimization models found

in the literature are utilized to study these interre-

lationships. The first set of models considers maxi-

mization of the service level as an objective subject

to an available budget which comprises invest-ments in ordering and holding inventories. The

second set is formulated by minimizing total vari-

able costs as an objective function with a target

service level as a constraint. The analyses differ

from existing studies from a few perspectives: (1) the

focus of this research is not on how to obtain the

optimal solutions, instead, it is on the sensitivity of

the solutions; (2) numerous important results are

derived, which will provide insightful economic im-

plications for making sound inventory decisions;

148 A.Z. Zeng, J.C. Hayya/Int. J. Production Economics 58 (1999) 147158


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and (3) the effects of the lead-time demand will be

examined.

While numerous probability distributions for

lead-time demand are assumed and analyzed in the

literature, four continuous distributions will be

considered in this paper: the exponential, the nor-

mal, the special Weibull, and the gamma. The ex-ponential distribution (studied in [7,8]) possesses

appealing properties that make analysis staight-

forward and provide easy-to-interpret results; the

normal distribution always enjoys wide applica-

tions in both research and practice; the special

Weibull (assumed by [911]) can capture the

characteristic of a situation where the lead-time

demand is extremely variable; and the gamma (see

[1214]) is a more general distribution that not

only includes normal as a special case but avoids

the improper features of other distributions.The remainder of this paper is organized as fol-

lows: Section 2 relies on the first set of optimization

models to compare the performance of (P

, P

) by

assuming the four distributions individually, in par-

ticular, the conditions under which one measure

outperforms the other are identified. The results

derived from the second set of optimization models

are contained in Section 3, in which the tradeoff

among cost, service, and turnover ratios are exten-

sively examined. Finally, conclusions of this studyare summarized in Section 4.

2. Effects of lead-time demand on the optimal levels

ofP1 and P2

Prior to the beginning of a fiscal year, inventory

managers can usually estimate their available

budget for the entire year, and thus, the major

question they may be concerned with is at whatmaximum level of service they would achieve. Intu-

itively, the more budget available, the higher the

service level would be. However, would the same

amount of budget result in the same levels of

(P

, P

)? And how big would the difference be? To

answer these questions, a budget-constrained

model first developed by [15] is used to study the

dynamics of lead-time demand (LTD). First of all,

we define the following main notation that is used

throughout the paper:

A ordering cost, in $/order,

D demand per unit time, in units/yr,

K available budget per unit time, in $/yr,

v unit value of an item, in $/unit,

r carrying charge as a percentage, in %/$/unit,

mean of lead-time demand, in units,

standard deviation of lead-time demand, inunits.

The first set of models (budget-constrained

models) is given as

Model Set 1:

Maximize P

or P

(4)

s.t.: AD/Q#(Q/2#s!)vr"K. (5)

In Eq. (5), the first term represents the ordering cost

per year and the second is the expected annual

inventory holding cost. After substituting Eq. (1) to

the objective function, Eq. (4), the optimal solution

is obtained as

Q*"Q"(2AD/vr,

s*"#(K!(2ADvr)/vr, (6)

where Q

is so-called Wilsons formula for eco-

nomic order quantity in the deterministic case. The

optimal P

can be found as

P*"F(s*)"F[#K/vr!Q

], (7)

where K*(2ADvr, ensuring s!*0 (i.e., non-negative safety stock). Similarly, the optimal (s, Q) if

using P

in Eq. (2) should satisfy the following two

simultaneous equations:

Q"R(s)#(R(s)#Q

,

s"#K/vr!0.5(Q

/Q#Q), (8)

where

R(s)"bM(s)

1!F(s). (9)

It is clearly seen that the expressions of the optimal

levels of P

and P

depend on the probability

distribution of LTD. In what follows, we examine

A.Z. Zeng, J.C. Hayya/Int. J. Production Economics 58 (1999) 147158 149


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the effects of four commonly assumed LTD distri-

butions on the relationships between the optimal

levels of (P

, P

).

2.1. Exponential lead-time demand

Let the probability distribution of an exponential

LTD be described as f(x)"e\HV, where x*0 and'0. Then, the optimal P

based on Eq. (7) is

found to be

P*"1!eH/ exp[!1!K/vr].

Because F(s)"1!e\HQ and bM(s)"e\HQ/, R(s) ofthis exponential distribution is a constant, i.e., 1/,the optimal solutions in Eq. (8) can be simplified as

Q*"1/#(1/#Q

,

s*"#K/vr!0.5(Q

##1/

#),

where

#"1/Q*"

1#(1#(Q

),

K*0.5vr(#

Q#1/

#).

Then, the optimal P

for the exponential LTD with

a given budget K can be obtained as

P*"1!

bM(s*)

Q*

"1!(e\/)#

exp+![K/vr

!0.5(Q

##1/

#)],.

Lemma 1. For an exponential lead-time demand:

(i) when Q/(0.3344, P*'P*; (ii) whenQ

/'0.3344, P*(P*

; and (iii) the break-even

point (P*"P*

) is at Q

/"0.3344.

Proof. See Appendix A.

Lemma 1 clearly describes the impact of the

ratio, Q

/, on the performance of the optimalservice levels, and the significant value of this ratio

is found to be 0.3344 or Q"0.3344. In other

words, ifQ(0.3344, the same amount of budget

always gives a better value of P

than P

; but if

Q'0.3344, P

always outperforms P

. We illus-

trate the relationship between the optimal values of

these two measures in Fig. 2. According to Fig. 2,

one can see that the range ofQ'0.3344, is much

broader than that of Q(0.3344, implying thehigh possibility for P

to dominate P

.

2.2. Normal lead-time demand

For a normal LTD, if using an approximation

suggested in [15], the relevant properties can be

written as

f(k

)"

ab

e\@I

and 1!F

(k

)"a

e\@I

,

bM(k)"a

be\@I,

where k is the so-called safety factor and

k"(s!)/. Then, the optimal P

can be identi-

fied as

P*"1!a exp[(b/)(Q

!K/vr)].

The resulting (k*, Q*) obtained by using P

as an

objective function is given by

Q*"/b#((/b)#Q

,

k*"(1/)[K/vr!0.5(Q

,#1/

,)],

Fig. 2. The optimal P

and P

for a given budget with the

exponential LTD.



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where

,"

b/

1#(1#(Q

b/),

K*0.5vr(,

Q#1/

,).

The optimal P

for a normal LTD is then

P*"1!

a

b,

exp+!(b/)[K/vr

!0.5(Q

,#1/

,)],.

Lemma 2. For the approximate normal lead-time

demand: (i) when bQ

/(0.3344, P*'P*

; (ii)

bQ

/'0.3344, P*

(P*

; and the break-even point

is at bQ

/"0.3344.

Proof. Treating bQ

/ in the normal LTD case asQ

in the exponential distribution case, the proof

follows.

Note that Eq. (1) the performance of P

and

P

for both exponential and normal LTD distribu-

tions is related to the ratio, Q

/. It would beinteresting to examine the effect of this ratio for

other LTD distributions; and (2) for the approxi-

mated normal LTD, if substituting b"2.49, thenQ

/ will be a small value, indicating that unless thestandard deviation of LTD is significantly large

relative to Q

, P*

is unlikely to dominate P*

.

2.3. Weibull lead-time demand

The study of the exponential and normal LTDs

reveals that the ratio, Q

/, affects the performance

ofP and P. If the economic order quantity, Q, issignificantly small relative to , the standard devi-ation of LTD, P

would outperform P

. We test

this conclusion for the Weibull distribution. For

the purpose of illustration, we consider a special

case of the Weibull family, which has the following

pdf:

f(x)"(1/2w)(x/w)\ exp[!(x/w)],

x*0, w*0,

where w is the scale parameter and the shape para-

meter is fixed at 1/2. The feature of this special

Weibull is that its mean is w and its standard

deviation is 2(5w, which yield the coefficient ofvariation (5. Although one may argue about thesuitability of using this distribution in inventory

control, our purpose is to examine the effect ofQ

/on service levels when LTD is extremely variable.

Other properties associated with this special

Weibull can be found as

F(s)"1! exp[!(s/w)],

bM(s)"2w[1#(s/w)] exp[!(s/w)],

which give

R(s)"2w[1#(s/w)].

Since R(s) is no longer a constant, solving (s, Q)

involves two simultaneous equations. Since no ana-

lytical result similar to the exponential and the

normal distributions can be derived explicitly, we

rely on numerical examples to investigate the rela-

tionship between the two service measures. The

following system parameters are chosen: A"10,

D"10 000, v"1, and r"0.2, then Q"1000. If

selecting the budget to be K"800, we find thatwhen w"265, i.e., "530, "1185, the two opti-mal service levels are equal: P*

"P*

"0.9740.

Moreover, it can be computed that Q

/"0.8439,suggesting that this ratio for the Weibull is much

greater than that for the exponential and that it is

more likely for this Weibull LTD to result in

P

larger than P

. The underlying reason is that

Weibull is more variable than the exponential case,

where the coefficient of variation is one.

2.4. Gamma lead-time demand

The difficulty associated with the gamma distri-

bution is the calculation of some properties arising

from inventory control theory; however, it can be

shown that with some existing mathematical soft-

ware, similar analyses to the preceding sections can

be performed, and thus, the applicability of gamma

LTD will be greatly enhanced.



6/12

Let the pdf of the gamma be described as

f(x)"aAxA\e\?V

(), x*0, a'0, '0.

Then the mean and variance are "/a, "/a,respectively. Other associated properties are

F(s)"I(as, ),

bM(s)"(/a)[1!I(as, #1)]!s[1!I(as, )],

where I(.) is the incomplete gamma function. Ap-

parently, R(s) for the gamma case is not a constant;

hence, the analysis again depends on numerical

examples. The system parameters used for the

Weibull distribution in the preceding section are

again used. The break-even conditions for the dif-

ferent mean LTDs are summarized in Table 1 whenholding the budget and the EOQ unchanged

(K"800 and Q"1000). Table 1 implies that (1)

the break-even point changes as the mean changes,

or that the critical value of the ratio, Q

/, is nolonger fixed as in the exponential and normal cases.

The underlying reason is that the gamma distribu-

tion is determined by two parameters (the scale and

the scope parameters, but for the normal case, the

approximation used in the above analyses reduces

two parameters to just one), and (2) the break-even

value of Q/ decreases as the mean and varianceincrease.

Although the analyses of the Weibull and thegamma rest on numerical examples, they support

the finding from the exponential and the normal

LTDs that the break-even condition for P*

and

Table 1

The break-even point (P*"P*

) for gamma LTD K"$800,

Q"1000

Mean of LTD Break-even

std. deviation

Break-even

ratio

Coefficient

of variation

* Q

/* */

500 1370 0.7299 2.740

1000 1889 0.5294 1.889

2000 2546 0.3928 0.273

3000 2995 0.3339 0.998

4000 3335 0.2999 0.834

5000 3620 0.2762 0.724

P*

depends solely on the ratio, Q

/. Interestinglyenough, the critical value of the ratio is fixed for the

exponential and the approximate normal cases, but

varies with respect to the mean for the special

Weibull and the gamma distributions. All four dis-

tributions indicate that the budget has no effect on

the relationship of P and P.

3. Tradeoff between the cost, turnover ratio, service

The interrelationship between the measures of

service and cost are studied extensively in previous

section. However, inventory turns, defined as the

ratio of demand per unit time to the average on-

hand inventory [2], is another issue concerned by

the inventory managers in decision-making. By thedefinition, the turnover ratio (R) can be written as

TR(s, Q)"D

IM"

D

Q/2#s!

"D

0.5(Q#s)#0.5s!. (10)

In this section, we investigate the tradeoff between

the minimum cost, the turnover ratio, and the ser-

vice level.

3.1. Sensitivity of the ordering policy under the

service-constrained model

In Section 2, the service level was treated as

a managerial objective in order to study the result-

ing effects. Alternatively, an optimal pair of (s, Q)

can be determined by cost minimization with a tar-

get service as a constraint. This type of model has

been used extensively in the literature and can beformulated as

Model set 2:

Minimize TC(s, Q)"AD/Q#(Q/2#s!)vr,

s.t.: P

or P"1!,

where is a small positive fraction whose value ispre-specified. The sensitivity of the optimal pair,

(s*, Q*) is summarized in the following two lemmas.



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Lemma 3. he sum of s* and Q* obtained by the

P

-constrained model is an increasing function with

respect to the value of P

.

Proof. The optimal (s, Q) when P

is used as a con-

straint is given by

Q*"Q

"(2AD/vr,

s*"F\(1!), (11)

then,

s*#Q*

"Q

#F\(1!). (12)

Eq. (12) clearly indicates that as decreases (i.e., asP

increases), s*#Q*

also increases.


P



.

Proof. See Appendix B.

Corollary 1. A high level of service and a high level

of turnover ratio cannot be achieved simultaneously

in the context of cost minimization.

Proof. First of all, let us show that s* is an increas-

ing function with respect to , which determines theservice level. Recall that the optimal reorder point

should satisfy bM(s)"Q, since bM(s)"F(s)!1(0,decreasing would increase s. Next, we considerthe turnover ratio which is computed by Eq. (10).

As the service level is increased, both s#Q and

s are increased. Since s#Q and s appear in the

denominator of turnover ratio, TR decreases as the

service level increases.

3.2. Study of the tradeoff

Having discussed the pairwise relationships of

the minimum cost, turnover ratio, and the service

level, we investigate the trade-off of these three

management concerns. The scheme of the analysis

is proceeded as follows: we use the optimal (s, Q)

obtained from the service-constrained model to

evaluate the resulting turnover ratio and the min-

imum cost, and then compare the resulting turn-

over ratios and the minimum costs.

For the sake of clarity, we use * versus the

subscripts (1, 2) to differentiate the optimal solu-

tions to the two service-constrained models. The

optimal solution obtained from P-constrainedmodel satisfies the following set of equations

(e.g., [16]):

Q*"Q

(s*

),

bM(s*

)"Q*

,

where

(s

)"

1!F(s

)

1!F(s)!2. (13)

Clearly, (s

)'1 and F(s

)(1!2; hence,Q*

would be larger than Q*

, and s*

would be

smaller than s*

(since F(s*

)"1!). Furthermore,we obtain the following two equations:

Q*!Q*

"Q

[1!(s*

)], (14)

1/Q*!1/Q*

"(1/Q

)[1!1/(s*

)]. (15)

3.2.1. By turnover ratio

Most of the inventory managers believe that the

higher the turnover ratio the better. To compare

the resulting turnover ratios, we only need to be

concerned with the expected on-hand inventory,

(Q/2#s!). When using Eq. (14), we obtain

IM"IM!IM

"0.5(Q*!Q*

)#(s*

!s*

)

"Q

[0.5!0.5(s*

)]#(s*!s*

). (16)

If P

gives a lower turnover rate, then IM'0,

which implies that

s*'s*

#Q

[0.5(s*

)!0.5]. (17)

We regard the RHS of Eq. (17) as a critical function

of s

. Let

C

(s

)"s#Q

[0.5(s

)!0.5], (18)



8/12

and denote the value ofs

that gives the equality of

Eq. (17) as a critical point, s

.

Lemma 5. For a given service level and any distribu-

tion of lead-time demand, C

(0)'0 and C

(s

) is

a strictly increasing function.

Proof. It is easy to see that C

(0)"

0.5Q

[(1!2)\!1]'0. Using a prime to de-note a derivative, one can obtain

(s

)"f(s

)[(s

)]\[1!F(s

)!2]\'0.

Then

C

(s

)"1#0.5Q

f(s

)[(s

)]\

;[1!F(s)!2]\'0,

C

(s

) is thus a monotonically increasing function.

The lemma implies that there exists only one

single value of s

that satisfies s*"s*

#

Q

[0.5(s*

)!0.5], in other words, the critical

point, or the value of s

, is unique.

3.2.2. By total relevant cost

When using P

as a constraint, denote the min-

imum cost of ordering and holding as

TC*

(s, Q)"AD/Q*#vr[0.5Q*

#s*

!].

Similarly, the minimum cost when P

is a con-

straint is

TC*

(s, Q)"AD/Q*#vr[0.5Q*

#s*

!].

Then, the difference of the two costs can be found as

TC(s, Q)"

AD(1/Q*!

1/Q*)#vr[0.5(Q*

!Q*

)#(s*

!s*

)].

Recall that 1/Q*'1/Q*

since Q*

(Q*

, and

s*'s*

, TC(s, Q) could be positive or negative;

thus it deserves some investigation. After simplifi-

cation, we obtain

TC(s, Q)"0.5vrQ

[2!(s*

)!1/(s*

)]

#vr(s*!s*

).

It is clear now that if the total cost incurred by

using P

is greater than the total cost by using P

,

i.e., TC(s, Q)'0, then

s*'s*

#Q

[0.5(s*

)#0.5/(s*

)!1]. (19)

Similarly, we regard the RHS of Eq. (19) as the

second critical function. Let

C

(s

)"s#Q

[0.5(s

)#0.5/(s

)!1], (20)

and denote the value of s

offering the equality of

Eq. (18) as the second break-even point, s

. It

can be observed that C

(s

)!C

(s

)"0.5Q

[1!

1/(s

)]'0 since (s

)'1, and both C

(s

) and

C

(s

) are functions of Q

and .

Lemma 6. For any lead-time demand distribution,

C(0)'0 and C(s) is a strictly increasing function.

Proof. It is clear that C

(0)"0.5Q

[(s

)#

1/(s

)!2]'0, because can neither be zero norone (otherwise C

(0) would be nonnegative). Using

(1/(s

))"!f(s

)[(s

)]\[1!F(s

)!2]\,

yields

C

(s

)"0.5Q

f(s

)[(s

)]\[1!F(s

)!2]\

;[(s

)!1]'0,

C

(s

) is also monotonically increasing.

Again, this lemma guarantees that s

is unique.

We now examine the consequences when evaluat-

ing the inventory system according to the specified

service levels and the resulting turnover ratio and

minimum total cost. This is accomplished by plot-

ting the critical functions, C

(s

) and C

(s

), versus

the range of s

, for a given service level. s*

can be

used for finding the two critical points on the

graph, and then one can determine the superiorityofP

or P

. We illustrate these ideas in Fig. 3. Note

that since C

(s

)'C

(s

) and both are strictly in-

creasing functions, s's

. The effect of the two

critical points on the possible relationships between

turnover ratio and total cost when using the same

value of P

and P

are described as follows:

1. If s*3[0, s

), then TR*

(s, Q)(TR*

(s, Q),

TC*

(s, Q)'TC*

(s, Q), and P

is dominant. This

case corresponds to Fig. 3a.



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Fig. 3. Possibilities when evaluating P

and P

by the turnover

ratio and the total cost for a given service level.

2. If s*3[s

, s

), then TR*

(s, Q)'TR*

(s, Q),

TC*

(s, Q)'TC*

(s, Q). There is a trade-off in

using P

or P

, since P

gives a higher turnover

ratio, but P

offers a lower minimum cost, for

a same level of service measure. This case corres-

ponds to Fig. 3(b).

3. If s*3[s

,R), then TR*

(s, Q)'TR*

(s, Q),

TC*

(s, Q)(TC*

(s, Q), and P

is dominant. This

case corresponds to Fig. 3c.

3.2.3. Normal LTD: A numerical exampleThe major question is then: when would the

three cases indicated in Fig. 3 occur? We rely on

a normal LTD as an example to answer the ques-

tion. The same system parameters used in the pre-

vious sections are chosen. Based on Eqs. (13), (18)

and (20), the normal LTD gives the following ex-

pressions:

(k)"

1!F(k)

1!

F(k)!

2

,

C

(k

)"k#(Q

/)[0.5(k

)!0.5],

C

(k

)"k#(Q

/)[0.5(k

)!0.5/(k

)!1].

Note that the normal distribution is not approxi-

mated and that the critical functions given in

Eqs. (18) and (20) for the normal LTD are depen-

dent on Q

/ and .The system parameters (A"100, D"10000,

v"10, r"0.2) lead to Q"1000. By varying the

standard deviation of LTD, we choose Q/"10,4, 2, 1, 0.67 (correspondingly, "100, 250, 500,1000, 1500). The two critical points, k

and k

,

and k*

and k*

are obtained by a mathematical

software package and the results are summarized in

Table 2. The results show that for all five values of

the standard deviation, k*

is less than both critical

points, k

and k

, which implies that the fill rate,

P

, always outperforms the probability of no stock-

out, P

, in all five cases.

4. Concluding remarks

Customer service has become a key factor in

making various managerial decisions in every or-

ganization. Apart from improving service, inven-

tory managers are also concerned with reducing

cost and maintaining an acceptable ratio of inven-

tory turns. With the recognition of these three areas

as the components of the management effectiveness



10/12

Table 2Critical points: a numerical illustration for normal LTD,A"100; D"10 000; v"10; r"0.2; Q

"1000

k*

k

k

k*

Q

/"100.01 2.3263 1.4553 1.9790 0.8782

0.02 2.0537 1.0998 1.6669 0.45050.03 1.8808 0.8637 1.4641 0.1622

Q

/"40.01 2.3263 1.6789 1.9846 1.30580.02 2.0537 1.3507 1.6478 0.95080.03 1.8808 1.1362 1.4720 0.71670.04 1.7507 0.9708 1.3173 0.53490.05 1.6449 0.8331 1.1891 0.38260.06 1.5548 0.7133 1.0782 0.24930.07 1.4758 0.6061 0.9794 0.1293

Q

/"10.01 2.3263 1.9055 2.0040 1.78680.02 2.0537 1.5977 1.6970 1.47960.03 1.8808 1.3985 1.4983 1.28100.04 1.7507 1.2462 1.3463 1.12930.05 1.6449 1.1204 1.2207 1.00410.06 1.5548 1.0118 1.1123 0.89600.07 1.4758 0.9153 1.0159 0.80000.08 1.4051 0.8277 0.9283 0.71290.09 1.3408 0.7469 0.8476 0.63270.10 1.2816 0.6716 0.7723 0.55780.11 1.2265 0.6006 0.7014 0.48720.12 1.1750 0.5332 0.6340 0.42020.13 1.1264 0.4687 0.5695 0.3562

Q

/"20.01 2.3263 1.8078 1.9923 1.5735

0.02 2.0537 1.4923 1.6832 1.24880.03 1.8808 1.2874 1.4828 1.03770.04 1.7507 1.1302 1.3293 0.87550.05 1.6449 1.0001 1.2023 0.74120.06 1.5548 0.8875 1.0925 0.62480.07 1.4758 0.7871 0.9948 0.52110.08 1.4051 0.6958 0.9060 0.42660.09 1.3408 0.6115 0.8240 0.33920.10 1.2816 0.5326 0.7475 0.25740.11 1.2265 0.4581 0.6752 0.18010.12 1.1750 0.3872 0.6065 0.10640.13 1.1264 0.3193 0.5407 0.0356

Q

/"0.67

0.01 2.3263 1.9491 2.0123 1.88210.02 2.0537 1.6441 1.7067 1.57980.03 1.8808 1.4469 1.5091 1.38450.04 1.7507 1.2962 1.3578 1.23450.05 1.6449 1.1719 0.2330 1.11250.06 1.5548 1.0647 1.1253 1.00640.07 1.4758 0.9694 1.2096 0.91220.08 1.4051 0.8829 0.9427 0.82670.09 1.3408 0.8033 0.8627 0.74800.10 1.2816 0.7290 0.7880 0.67460.11 1.2265 0.6591 0.7177 0.60550.12 1.1750 0.5928 0.6510 0.53990.13 1.1264 0.5293 0.5872 0.4772

in inventory control, this paper aims to examine the

performance of two popular service measures in

achieving an efficient balance among these three

components. Specifically, lead-time demand is

treated as the major exogenous impact on the man-

agement effectiveness and a single-item, single-

stage continuous inventory system is considered.By assuming four widely used lead-time demand

distributions and relying on two sets of optimiza-

tion models, we have studied the following two

popular service measures that have wide applica-

tions in research and practice: the probability of no

stockout during lead time and the fill rate. The

results suggest that (1) the condition that one

measure outperforms the other depends on the

ratio of economic order quantity to the variance of

lead-time demand; and (2) the two service measures

yield different levels of the total inventory cost andthe turnover ratio. These indications provide im-

portant guidelines for inventory managers to make

sound decisions. In addition, the managers should

be aware of the differences of the cost, the level of

service, and the turnover ratio resulted from using

each service measure and should seek the balance

among the three elements based on their desired

managerial objectives.

Appendix A.

Lemma 1. For an exponential lead-time demand: (i)

when Q

/(0.3344, P*'P*

; (ii) when Q

/'

0.3344, P*(P*

; and (iii) the break-even point

(P*"P*

) is at Q

/"0.3344.

Proof. P!P

"G[exp [0.5( Q

##1/

#)]

!exp(Q

) (1#(1#(Q

))], where

G"exp (!1!K/vr)

1#(1#(Q

).

Note that G is positive and can be ignored in the

following analysis. Furthermore,

#

Q"

Q

1#(1#(Q

)

"(1/)[(1#(Q

)!1],



11/12

and

1/#"(1/)[1#(1#(Q

)].

Then, after some simplification,

P!P

Jexp [(1#(Q

)]

!eH/[1#(1#(Q

)]. (A.1)

Letting y"Q

yields

P!P

JeW[exp ((1#y!y)

!(1#(1#y)].

Moreover, let

z"exp((1#y!y)!(1#(1#y).

Note that when zeW"0, P!P"0; ze

W(0, orz(0, P

!P

(0; and zeW'0, or z'0,

P!P

'0. Thus, one needs to focus only on the

function, z, to determine the sign of P!P

. Since

the first derivative of z is

z"(y/(1#y!1) exp((1#y!y)

!y/(1#y(0,

and when y"0, z"W"e!2, one can conclude

that the function, z, is a decreasing function with

respect to y, and will reach zero at some value of y.Letting Eq. (A.1) be zero, one can find that

y"Q"0.3344. The plot ofzeW versus y is shown

in Fig. 2. The illustration demonstrates that

1. when y"Q"0.3344: zeW"0NP

!P

"0

NP"P

;

2. when 0(y"Q(0.3344: zeW'0NP

!P

'0NP

'P

;

3. when y"Q'0.3344: zeW(0NP

!P

(0NP

(P

.

Using "1/, the results summarized in thelemma then follow.

Appendix B.


P



.

Proof. The P

-constrained model is

Minimize AD/Q#vr(Q/2#s!),

s.t. bM(s)).

The Lagrangian function can be written as

(s, Q, M)"AD

Q#vr

Q

2#s!#M

bM(s)

Q!.

(B.1)

The first-order conditions are

j

jQ"!

AD

Q#

vr

2!

M

QbM(s)"0, (B.2)

j

js"vr#M[F(s)!1]/Q"0, (B.3)

j

jM"bM(s)/Q!"0.

The optimal (s, Q) obtained by these conditions

should be functions of, and we denote the optimalsolution as s*() and Q*(). Furthermore, Eq. (B.3)ensures that M is positive, and Eq. (B.2) can be

written as

AD

Q

#M

Q

bM(s)"0.5Qvr. (B.4)

It is evident that the total cost increases as the

service level increases. For a specified value of ,minimizing Eq. (B.1) is equivalent to

Minimize ()"AD

Q#vr

Q

2#s!#

MbM(s)

Q.

(B.5)

Therefore, substituting Eq. (B.4) to Eq. (B.5) yields

()"vr[Q*(#s*()!]. Since we have shownthat () increases as the service level increases, theconclusion in the lemma follows.

References

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[2] L.A. Johnson, D.C. Montgomery, Operations Research in

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12/12

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