by Yu-Sheng ZHENG* and Xavier DE GROOTE ... - Insead

"A SENSITIVITY ANALYSIS OF STOCHAUX, 

INVENTORY SYSTEMS" 

by 

Yu-Sheng ZHENG* 

and 

Xavier DE GROOTE** 

N° 92/16/TM 

* Department of Decision Sciences, The Wharton School, University of Pennsylvania, 

Philadelphia, PA 19104, U.S.A.. 

** Assistant Professor of Operations Management, INSEAD, Boulevard de Constance, 

Fontainebleau 77305 Cedex, France. 

Printed at INSEAD, 

Fontainebleau, France

A SENSITIVITY ANALYSIS 

OF STOCHASTIC INVENTORY 

Department of Decision Sciences 

The Wharton School 

University of Pennsylvania 

Philadelphia, PA 19104-6366 

USA 

** Technology Management Area 

INSEAD 

Boulevard de Constance 

77305 Fontainebleau Cedex 

FRANCE 

SYSTEMS 

Yu-Sheng Zheng * 

Xavier de Groote ** 

October 1991 

Revised, January 1992

Abstract 

This paper derives sensitivity properties of the single item inventory model with stochastic 

demand and positive setup costs. The anaivaissumbssiSsetheeffect of changes in the variability 

of the leadtime demand. For the order-quantity/reorder-point systems, we prove that when the 

variability of the leadtime demand increases: (i) the optimal order quantity increases, but, 

somewhat contrary to the conventional wisdom, the optimal reorder point does not necessarily 

increase; (ü) the minimum cost of the inventory system increases, and it increases at a slower rate 

than and converges to the cost of the optimal base-stock policy; and (iii) the cost performance 

becomes more robust to the choice of order quantity. In addition, we also show that both the 

optimal reorder point and the optimal order-up-to level decrease (increase) as the inventory 

holding (shortage penalty) cost increases. 

1. Introduction 

It is common sense that, for the management of inventories, the variability of the leadtime demand 

has a negative impact on the precision of control and on costs. An effective way of reducing 

the variability of the leadtime demand is to shorten the leadtime itself. This reduction requises 

effort and extra cost. With the current emphasis on "just-in-time" inventory management, and on 

"time-based" competition, there are increasing needs for a more precise analysis of the effect of 

the variability of the leadtime demand ou the control and performance of inventory systems. This 

paper addresses this problem for one of the widely used stochastic inventory systems, the order 

quantity/reorder point system. 

The order quantity/reorder point system is also referred to as the (Q,r) system, where an order 

of Q units is placed as soon as the inventory position (i.e., the inventory on hand plus on order minus 

backorders) drops to the critical level r. This inventory control system is the simplest model that 

simultaneously addresses economies of scale in order quantities, and a random (leadtime) demand. 

The cost function of the (Q,r) system has been well known for about three decades, ever since the 

publication of the popular textbook by Hadley and Whitin (1963). However, it had been perceived 

to be too complex to analyse, and, until very recently, our knowledge of the optimal control of 

titis system had been limited. We refer readers to Lee and Nahmias(1989) and Federgruen and 

Zheng(1988) for recent reviews of the related literature. 

From a modeling point of view, two simple inventory models, the EOQ model and the base stock 

model, are special cases of the (Q,r) model. The EOQ model crystallizes the tradeoff between the 

1

inventory holding costs and the fixed ordering costs by assuming a deterministic demand. h is clear 

that a positive (deterministic) leadtimc does not affect the EOQ model in any essential way because 

of the predictable leadtime demand. The base stock model, on the other hand, stresses the balance 

between the inventory holding costs and the shortage penalty costs by assuming away the fixed 

ordering costs. In a base stock model, an order is placed to restore the inventory position to a fixed 

"base stock" (or "order-up-to") level whenever a demand occurs. The optimal order-up-to level 

can be determined by minimizing the expected inventory holding costs and shortage penalty costs, 

which is essentially the newsboy problem -where the demand for newspaper is interpreted-as the 

leadtime demand. The optimal order-up-to point, which is often referred to as the newsboy point, 

enjoys a closed form solution. In that case, the effect of the variability of the leadtime demand is 

relatively simple to analyze. 

With little theoretical support, it has often been argued that in a (Q,r) system, the order 

quantity should be determined by the EOQ formula in order to exploit the economies of scale, while 

the reorder point should be set to the expected leadtime demand plus some safety stock reflecting 

the variance of the leadtime demand and the service level (or the costs associated with shortage). 

While this argument is intuitive and therefore has played an important role in popularizing the 

model, it essentially dismisses the (Q,r) model by the rough functional segregation of the two control 

parameters. 

Our analysis is based on recent results of Zheng(1989), who proves for the first tinie that the 

(Q,r) model can be viewed as an aggregation of a. variant of the EOQ model and the base stock 

model. In his analysis, alter the reorder point is chosen optimally for any given order quantity, the 

(Q,r) model reduces to a problem of finding the best tradeoff between the average ordering cost 

and the average holding and penalty cost. The average cost of the model is shown to be the sum of 

two components: the cost oi the optimal base stock policy and a cost function that is very similar 

but smaller than that of the EOQ model (with backorder allowed). The analysis of these two 

cost components has led to the establishment of 2 number of properties of the stochastic inventory 

systems. In particular, it is shown in Zheng (1989) that the optimal order quantity of the (Q,r) 

model is larger than the corresponding EOQ, that the EOQ is a good heuristic when economies 

of scale prevail, that the cost of the (Q,r) model is bounded below by the maximum of the cost 

of the EOQ model and the cost of the base stock model, and bounded above by the sum of these 

costs, and that the expected total cost of the (Q,r) system is more robust to the choice of order 

quantity than the corresponding EOQ system. By comparing a system with stochastic leadtime 

demand with the corresponding deterministic EOQ system, these results provide first cut answers 

2

to the analysis of the effect of the variability of the leadtime demand. 

The results presented in this paper augment the above results by further showing how changes 

in the variability of the leadtime demand affect.the o ptimal control parameters and the system cost 

performance. Specifically, we show that when the variability of the leadtime demand is increased: 

(i) the optimal order quantity increases, but, somewhat contrary to the conventional wisdom, the 

optimal reorder point does not necessarily increase; (ii) the minimum cost of the inventory system 

increases, and it increases at a slower rate than and converges to the cost of the optimal base-stock 

policy; and (iii) the cost performance becomes more robust to the choice of order quantity. We also 

show that both the optimal reorder point and order-up-to level decrease (increase) as the inventory 

holding (shortage penalty) cost rate increises, which complement the results on the effects of the 

fixed order cost of Zheng(1989): namely, that the optimal order quantity increases and the optimal 

reorder point decreases as the fixed ordering cost increases. 

Two recent papers are relevant. Gerchack(1990) derives comparative statics for the (Q,r) model 

for both backiogging and lost sale models. For bac.klogging models, some of his results are sim- 

ilar to ours, but they appear with some qualifications which are not needed in our case. These 

qualifications are apparently due to the use of an approximate cost function, which systematically 

undercharges holding costs. Gerchack and Mossman(1988) investigate the effect of the leadtime de- 

mand variability, and conclude that the relationship between the variance of the leadtime demand 

and both the optimal reorder point and the optimal order quantity is not definit!ve. Again, this 

discrepancy appears to be due to the use of the approximate cost function. 

The paper is organized as follows. Section 2 describes the model and notation. The effect of 

changes in the cost parameters is discussed in Section 3. Section 4 is devoted to the analysis of 

changes in the variability of the leadtime demand. In section 5, we discuss the sensitivity of the 

system performance to changes of the various problem parameters and the potential application of 

the cost insensitivity to the choice of order quantity. Section 6 is a brief conclusion. 

2. Model and notation 

We consider a single-item continuous-review inventory system that faces a stationary stochastic 

demand. Unless otherwise specified, the notation and assumptions are the sanie as in Zheng(1989). 

The arrivai rate is denoted by Replenishment orders are delivered after a positive leadtime L. 

All stockouts are backordered. The relevant costs are a fixed replenishment cost K for each order 

3

placed and proportional inventory holding (penalty) costs accumulating at constant rate h (p) per 

unit stock (backorder) per unit time. The criterion is the (long run) average total cost per unit 

time. 

We assume the demand process and the leadtime are such that, under a (Q,r) policy, the 

following three conditions are satisfied: (a) both the inventory position and inventory level processes 

have a limiting distribution; (b) the fundamental equation IL = IP - D holds and IP and D are 

independent, where IL and IP dente the steady state inventory level and inventory position 

respectively, D the leadtime demand; (c) IP is uniform between r and r Q. It is. well known 

that these conditions are satisfied if the demand process is a Poisson or a renewal process and the 

leadtime is constant. These conditions can also be shown to hold when the leadtime is stochastic, 

under certain assumptions (see Zipkin 1986), and when the demand process is a Markov-Chain- 

Driven process (see Zipkin 1991). These conditions are required for the cost function (2) below to 

hold. [Since the entire analysis of Zheng(1989) is based on (2), the assumption of constant leadtime 

in Zheng(1989) can be replaced by the above three more general conditions.) 

Let G(y) be the rate (per unit time) at which the conditional expected inventory (holding 

and penalty) costs accumulate when the inventory position equals y. Let FO be the distribution 

function (cdf) of the leadtime demand D. By condition (b) we have 

G(y) = E[h(y - p(D - y)+] = (h p) jv F(x)dx - p(y - ED) (1) 

where E denotes the expectation over the leadtime demand. It is well known that G(•) is a convex 

function. By (c), the average cost function of a (Q,r) system can be written as: 

c(Q, r) = AK r+Q G(y ) dy (2) 

Equation (2) is exact when the cumulative demand can be modeled as a nondecreasing stochastic 

process with stationary increments and continuous sample paths (see Browne and Zipkin 1991). It 

is an adequate approximation when the demand is a counting process (and Q is not too small). We 

treat both the reorder point and the order quantity as continuous decision variables. 

Let y° be the minimum point of GO. It is well known that y° = F-1(. 11-p). (Throughout 

this paper, we assume, for notational convenience, that F(•) is continuous and strictly increasing.) 

When K = 0, the base stock policies are optimal. In that case, y° is the optimal order-up-to level 

and G(y°) is the optimal average cost. In the sequel, we refer to y° and G(y°) as the newsboy point 

and the newsboy cost, respectively. 

4

Let r- and Q" denote the optimal reorder point and the optimal order quantity. The following 

lemma is established in Zheng(1989). 

Lemma 1 Q = Q" and r = if and oney 

G(r)= G(r + Q). r• < y° < r` + Q*. 

The optimality equations in the above lemma are simply the first order condition of the cost 

function. The lut part of the lemma follows from G(e) = G(r* + QI and the convexity of 

function G. 

3. Changes in the cost parameters 

It has been shown (Zheng 1989) that the optimal order quantity increases and the optimal reorder 

point decreases as the fixed ordering cost K increases. In order to obtain further comparative 

statics results, we need to differentiate varions components of the model with respect to h and 

p. For clarity, we will write expressions such as 8G(y)18h without explicitly incorporating the 

dependency of G(y) on h in the notation. We also denote dG(y)/dy by G'(y). 

Let the optimal order-up-to level be denoted by = r" + Q"). We first show that both 

the optimal reorder point and order-up-to level decrease (increase) as the holding (penalty) cost 

rate increases. Define a(y) = ERy — D)+]. Note that a(-) is a convex increasing function. 

Proposition 1 Or* 18h < 0, 8r"18p > 0, OS" 18h < 0, 8S'/8p > O. 

Proof: By totally differentiating the optimality condition G(r) = G(r + Q) = c(r,Q) with respect 

to h at r = r e, Q = Q*, we get 

0G(r) , Or OG(r + Q) Or 8Q Oc 

+ G (r) = 

Oh 

ah ah + G (1. Q)( 7971 += 

where the lut term has been simplified by recognizing that °c/& = 0 and &IN = 0 at r = r-, 

Q = Q . From (1), we have G'(y) = (h + p)F(y) — p and Oc(y)/ah = a(y). From (2), aclah = 

(g. +Q. a(y)dy)/Q". Solving for Or* 10h in (3), we get 

ar•(f,.r.+Q. a(y) dY)1(2" — a(**) 

Oh (h + p)F(r*) — p 

Since a(y) is increasing, the numerator is positive. It follows from Lemma 1, F(r") < p/(h + p), 

which implies that the denominator is negative. Thus, 8r"10h > O. The proof is similar for the 

other three cases. 

5 

(3)

4. Changes in the leadtime demand 

In this section. we study how the variability of the leadtime demand affects the optimal order 

quantity, the cost performance, and its sensitivity to the choice of order quantity. Throughout the 

analysis, we compare systems with different mean and variance of leadtime demand, while assuming 

that ail the other parameters (i.e., K, h, p and À) remain unchanged. Note that the cost function 

(2) depends on the leadtime demand only through the expected inventory cost function G. 

We start by briefiy reviewing some useful results from Zheng (1989). To focus -on -the-role of 

order quantity, we assume that the reorder point is chosen optimally for any given order quantity. 

Let r(Q) be the optimal reorder point for a given Q. The reorder-point is characterized by the 

following lemme: 

Lemme 2 For any Q > 0, r = r(Q) if and only if G(r) = G(r Q), and r(Q) < y° < r(Q) 1- Q. 

Let C(Q) de min,. min,.c(Q,r). Let H(Q) G(r(Q)),Q > 0, with //(0) G(y°). Since 

frre))+Q G(y) dy = fo H(y)dy, the optimal order quantity solves 

C(Q) = „ie.+ I Q H(y)dy)IQ. (4) 

H(Q) can be interpreted as the marginal (expected) inventory cost for the Q-th unit in a batch. 

Since C(Q) is convex, the first order condition of (4) gives 

which is simply one of the optimality equations in Lemma 1. 

C(Q. ) = H(Q"), (5) 

Let A(Q) QH(Q) — g H(y)dy. Condition (5) is then equivalent to the following lemma. 

Lemma 3 Q = Q • if and only if A(Q) = ÀK. 

Furthermore, we have 

Lemme 4 H(•) and A() are increasing convez fonction. H(Q) > eQ, // 1(0) = 0 and H'(Q) 

as Q Do. e 

6

Lemmas 3 and 4 are illustrated in Figure 1. The following lemma is geometrically obvious, therefore 

its proof is omitted. 

Lemma 5 For a > 1(< 1) and ci et;e001k4>W1H(Q). 

[Figure 1 approximately here] 

Before proceeding with the formai aaalysis, it is helpful to conjecture about our results on 

the basis of the geometrical intuition given in Figure 1. The asymptotic line of H(Q), Hd(Q) = 

hpQ/(h p), can be interpreted as the marginal inventory cost function of the system with de- 

terministic leadtime demand. It is geometrically clear that the introduction of variability of the 

leadtime demand increases the expected marginal inventory cost. It is also economically intuitive 

that a more variable leadtime demand should yield a higher H-curve. With the asymptotic line 

remaining the same, we expect a higher H-curve to be flatter. For fixed Q, A(Q) is the shaded 

area between the H-curve and the horizontal line z H(Q). At the optimal order quantity Qe, 

this area equals AK (Lemma 3). Therefore, a flatter H-curve, which makes the shaded area smaller 

for fixed Q, requires a luger Q. Consequently, our first conjecture is that Q* increases with the 

variability of the leadtime demand. 

Furthermore, since Q" depends only on the shape but not on the absolute height of the H- 

curve, it is conceptually helpful to decompose HO in two parts: 

H(Q) = G(e) + 1-1.(Q)• 

Let C* be the optimal average cost of the (Q,r) system, i.e., C" = minQ>o C(Q). Then, (4) can be 

rewritten as 

C'=G(y°)+ Cô 

de 

with C: = minc».0C.(Q) =) (AI( j Ho(y)dy)IQ. H.( .) can be interpreted as the part of the 

marginal inventory cost that can be used to trade off the ordering cost through the choice of order 

quantity, and therefore is referred to as the controllable inventory cost. Note that H. is a 

vertical shift of H. Therefore, a more variable leadtime demand would be associated with a flatter 

H.-curve. Since H.(0) = 0, a flatter Hecurve means 11.(Q) is smaller. Hence, for any Q > 0 fixed, 

110(Q) should decrease as the variability of the leadtime demand increases. This leads us to more 

conjectures. First, as variability of the leadtime demand increases, the controllable inventory cost 

110(Q) is diminishing, so is C. Therefore, the optimal average cost of the system, which is the 

7

sum of the newsboy cost and C:, should grow more slowly than the newsboy cost when variability 

of the leadtim, demand increases, and it should be close to the newsboy cost when the variability 

is high. Second, for more variable leadtime denund, the total average cost should be less sensitive 

to the choice of order quantity because Co(Q) is smatier and it is the only part of the total cost 

that is affected by Q. We proceed to prove these conjectures analytically. 

To proceed with the analysis, we need a dear definition of the variability of the leadtime demand. 

We assume in the sequel that the leadtime demand distribution takes the form of F(z)= 

where p and a are its mean and standard deviation respectively, and 4i is strictly increasing. Then, 

the variability of the leadtime demand can be unambiguously represented by We will show how 

the optimal control parameters and the optimal colts change as o changes. The above assumption is 

certainly satisfied when the leadtime demand is Normal, which is often an adequate approximation 

when the (mean) leadtime demand is not too small. The above construction is also a special case 

of the general notion of convex-ordering; see Rothschild and Stiglitz (1970) for a discussion. 

We note that the variance of the leadtime demand is determined by both the demand process and 

the leadtime. When the demand process is given exogenously, the variance of the leadtime demand 

is affected only by the leadtime. For deterministic leadtimes, a longer leadtime typically leads to 

a larger variance of the leadtime demand. When a Normal distribution is used to approximate 

the leadtime demand, o is often assumed to be linear to f.t. When the leadtime is stochastic, 

the variance of the leadtime demand is determined by the variability of the demand process and 

by the mean and the variance of the leadtime. More specifically, o can be computed by _using 

the conditional variance formula (interested readers are referred to Zipkin 1991 for a detailed 

discussion). 

The G-function can now be written as 

G(y) = (h + p) 41( z- )dz - p(y - 

-00 o 

= a[(h p) 4,(z)dz - - 

-00 

We observe that a change of p in (6) would only cause a horizontal shift of the G(•)-function, which 

does not affect the newsboy cost nor the H(•)-function. The reexamination of equation (4) yields 

the following proposition. 

Proposition 2 For a given demand process, the optimal order quantity and the minimum cost of 

the (Q,r) system are independent of the mean of the leadtime demand. 

8 

(6)

Therefore, in the sequel, we set = 0 in (6) for notational simplicity. Moreover, we add the 

subscript a to all the notation to indicate its dependency upon the leadtime demand variance 

(subscript d instead of 0 is used to indicate zero variance of the leadtime demand, which can be 

true either when the demand process dateininittic or the leadtime is zero). 

It follows from (6) that 

G.(0= aG1(Y10 ) (7) 

where Gt (y) = (h + p) poe 4(y)dy— py is the (shifted) expected inventory cost function when the 

variance of the leadtime demand is 1. 

Let e be the minimum point of Ga ( .). The following result is immediate. 

Proposition 3 The newsboy cost grows linearly in a, Ge( g ) = crGi(e)• 

Proof: By (7), e = agi. 

The following lemma is crucial: 

Lemma 6 110(Q) = all1(Q1e) 

Proof: By definition of H, we need to show 

Ga(ro(Q)) = eGi(ri(Q1cr))• (8) 

Since by (7) GrGi(ri(Q/a)) = Ga(ari (Q1a)), (8) is equivalent to Ga(r„(Q)) = Gcr(ar1(Q1a)). 

Thus, it suffices to show ra (Q)= ari (Q1a), which follows from the following equations 

G«,(ar1(Q1a)) = aGi( rt(Q la)) = GrGi(rt(Q/a)+ Q/a) 

= G 5( 49r1(Q/a)+ Q) 

where the first and the third equations are due to (7), and the second is by Lemma I. 

Now, we are able to show 

Corollary 1 For any flzed Q > 0, Hc,(Q) is increasing in o. 

9

Proof: It suffices to prove 11„(Q)> H1 (Q) for any o > 1. By Lemma 5, 111 (Q/al> H i (Q)la for 

> 1. By Lemma 6, II„(Q)= a I 1 i (Q1a). Therefore. //e(Q) > HI(Q). 

q 

Corollary 2 For any fixed Q > 0, 11,;(Q) and are deLrèosing in a; Hf, (Q) j 0 as a 1‘ oo. 

Proof: Differentiating the equation in Lemma 6, we have II,,(Q)= I4(Q la). Fix Q > O. The fact 

that II:,(Q) is decreasing in a follows from the convexity of H1 . 110,e(Q) is decreasing in a because 

H:,,,„(Q)(= 14(Q)) is decreasing and 1/,,,„(0) = O. The rest of the corollary is due to 14(0) = O. 

Corollary 3 For fized Q > 0, A„(Q) is decreasing in a; A,,(Q) j 0 as cr j oo. 

Proof: It follows from Corollary 2 and A,(Q) = Q11,1,(Q) that AgQ) j 0 as a j oo. The corollary 

then follows from A,,(0) = 0 and A(Q) = g A'(y) dy. q 

The effect of variability of the leadtime demand on the optimal order quantity is now clear. 

The following proposition is a direct implication of Lemma 4 and Corollary 3. 

Proposition 4 The optimal order quantity Q; increases in cr. Moreover Qa" j oo as a I oo. 

We note for contrast that the reorder point is not necessarily mcreasing in o., particularly when 

p is not very large relative to h. Zheng(1989) provides counter examples which show that the 

reorder point of a stochastic (Q,r) system can be lower than that of the corresponding EOQ system 

= 0). 

Although the optimal order quantity grows with a, the following proposition shows that it grows 

at a rate slower than o.. 

Proposition 5 is decreasing in a, and I 0 as a j oo. 

Proof: 

ak = A0(C2;) = Q;14(Q;) — .10 He(Y)(1Y 

= (72 [ 4-2-1 0 Hi(Q; / a ) — tnia Hi(Y)cild 

= a2A1(Q;/a)• 

10

The proposition follows from the fact that A 1 is increasing and A i (0) = O. q 

Then the following result is also immediate. 

Proposition 6 (a )The optimal conteellable cost C",;„„ is decreasing in cr, and C;0 1, 0 as o j oo; 

(b) The optimal mienne cost C; is increasing in a, but the increase is slower than the newsboy 

cost, and C; crGi(yî) as a j oo. 

Proof: (a) Since 110,,,(y) is decreasing in o, so is the controllable cost function Co(Q) and its 

minimum By Lemma 8 in Zheng(1989), we have Co,,Q; < 2À/C, thus 1 0 as ex î oo 

because Q; j oo as cr j oo. 

(b) The cost function (4) and Corollary 1 imply that C; is increasing in o. C; ti aGi(e) because 

C; = Go(g) Co,,,, 0 as a j oo and Proposition 4. q 

Fin*, we show that the total cost is more robust to the choice of order quantity when the 

demand variance is larger. More specifically, we show that ea(a) c"ca • 77) decreases as o in- 

creases. This result augments the inequality ea(a)< ed(a) (where ea(a) = 1(a + 1/a)) established 

in Zheng(1989). 

Proposition 7 For a > 0, ea(a) is decreasing in a, and eo(a) j 1 as o j oo. 

Proof: 

Co(aQ;) )tk + Ha(y)dy fe: ii,(y)dy 

eo(a)= — 

C; aQ;C; 

rce; Ha(Y)dY 

;" Q;Hu(Q7,) 

1 fia Ha(13Q;)d,C1 ) 

-(1 + 

a Ho(Q;) 

1 r HHi 0Q;,a) (zicy) di3) 

+ J1 

where the third equation is due to (5) and the fifth Lemma 6. In view of Proposition 5, it suffices 

to show that 

clef Hi(AQ) 

H1(Q) 

11

is increasing in Q for > 1 and decreasing for d < 1. We show the former, the latter is similar. 

For fi > 1, 

where the inequality follows from 

;11(Q) = HuQ)[0e1(PQ)111(Q)— 

HI(PC2) Ht(AQ) 

14(Q) Ait(Q) 

egh(RQ)] > 

where the first inequality is due to the convexity of H1 and the second follows from Lemma 

5. Sensitivity of the System Cost 

In this section, we briefly discuss how the problem parameters affect the cost performance of a 

(Q,r) system. It is well known that, due to the square root formula, the cost of the EOQ model 

is robust to change in all its parameters (K, h, p, A). It would be interesting to know whether this 

robustness carries over to the cost of the (Q,r) system. 

First, since the ordering cost K affects only the cost function C., which is smaller than that 

of the EOQ and diminishing as o increases, the system cost is robust, and it is more robust when 

the leadtime demand is more variable. This can be verified formally as follows. Differentiating 

(AI( fe; He(y)dy)IQ; with respect to K yields dC;IdK = XIQ;. Since Q; increases in o, 

dC;IdK decreases in o. 

The effect of change in the holding cost h or the penalty cost p is more involved because the 

change affects both the newsboy cost and Since the newsboy model does not enjoy the sanie 

degree of insensitivity as the EOQ model, we do not expect that the average cost of the (Q,r) 

system to be as robust to change in h or p, unless the effect of economies of scale dominates. 

We can dso quantify, at least approximately, the effect of a change in the leadtime on the cost 

performance of the (Q,r) system. A change in the leadtime affects both the mean and standard 

deviation of the leadtime demand. Since the mean does not have any impact on the optimal average 

cost, the system's cost would be affected only through the change of o. Assuming that the leadtime 

demand is constant and o is linear in ft-, we conclude, by Proposition 6, that the change in the 

system cost is sublinear in ././, and approximately linear in -IL when L (and hence o) is large. 

When the leadtime is stochastic, not only its mean, but also its standard deviation would have an 

impact on o. The reader is referred to Zipkin(1991) for a discussion. 

12

If a change in À does not cause a change in the leadtime demand, thon its effect on the system 

performance is identical to that of K because À and K are symmetric in the cost function Ciee (2)). 

This assumption, however, is very ite t practice. Typically, a change in the demand rate 

induces a change in o. In this caue,thei0bal effect is more complex. Assuming that the leadtime 

L is constant and the demand process is Poisson (but approximately Normal), we have o= 

Therefore, the Newsboy cost changes linearly in f (Proposition 3); C c,`,,, is affected by change of 

both A and o. The effect of A on CZ, is sublinear in N/X. Since the effect of a on CZ„ is in the 

opposite direction as indicated by Proposition 6(a), the total average cost C:„." must be sublimer to 

VI In other words, the average cost of the (Q,r) system is more robust to change in the demand 

rate than that of the EOQ model. 

Finally, we comment on Proposition 7, which states that a larger variance in the leadtime de- 

mand makes the average cost more robust to the choice of order quantity assuming the monter point 

is always chosen optimally. Care is needed in interpreting this insensitivity result. In particular, 

we cannot condude from this proposition that the average cost is insensitive to inaccurate data. 

The reason is that when a suboptimal order quantity is chosen because of the inaccuracy ofsome 

pazameter estimates, the reorder point is unlikely to be set optimally (with respect to the chosen 

order quantity). 

This proposition, and Theorem 4 in Zheng(1989), are motivated by their potentiel nsefulness 

for analyzing multi-stage stochastic systems. For multi-stage systems with constant demand, such 

an insensitivity property of the EOQ model has led to remarkable worst case performance bounds 

for the power-of-two heuristic policies (Roundy 1985, 1986). Here, we briefly discuss how-there 

insensitivity results for the (Q,r) model could be useful for similar analysis on some plausible policies 

for multi-stage stochastic inventory systems. For multi-stage inventory systems where demands are 

stochastic and where there are fixed costs for inventory replenishment at each stage, a plausible 

control policy is to use an order-quantity/reorder-point policy at each stage. Since upstream stages 

are the suppliers for their downstream stages, the order quantities of different stages should be 

coordinated. For example, it is plausible to use the "integer ratio policy," under which the order 

quantity of an upstream stage is restricted to an integer multiple of that of a downstream stage 

(if the downstream stage is the only customer of the upstream stage); see e.g. De Bodt and 

Graves(1985). Such a quantity coordination across different stages imposes a constraint on the 

choice of order quantities, but not on the reorder points; that is, while the order quantity at a stage 

may be required, for the global coordination, to deviate from its own "optimal" value the reorder 

point remains "local", and can be chosen optimally for the choice of order quantity. Our insensitivity 

13

esults indicate that such a quantity coordination would not cause a significant cost increase from 

each stage's minimum cost. For systems with deterministic demands, Roundy(1986) showed by his 

lower bounding theorem that the sum of the singiereage minimum cost of all the stages (with some 

additional minor constraints on the order quantities i ni a. sumer vound on the system-wide average 

cost, which in turn leads a very tight worst case performance of the power-of-two policies (which 

are a subset of the integer ratio policies). For stochastic systems, a parallel lower bounding theorem 

seems significantly more difficult to formulate. However, even in the absence of such a theorem, the 

heuristic policies that require quantity coordination across different stages are much encouraged by 

ourInsensitivity results for the single-item (Q,r) system. For a serial system with Poisson demand 

at the lowest stage, an accurate cost evaluation procedure has been devised (Chen and Zheng 1991 

a) for echelon stock order-quantity/reorder point policies. Numerical studies(Chen and Zheng 1991 

b) show that for a fairly wide range of data these policies are indeed close to optimal. Research on 

worst case analysis of heuristic policies for multi-stage systems is in progress. 

6. Conclusion 

Combining the results of this paper with those of Zheng(1989), we now have a more complete 

understanding of the behavior of (Q,r) systems. By assuming that the leadtime demand is (apprcximately) 

Normal, a (Q,r) system can be fully characterized by the cost parameters K,h, p and 

the demand parameters o. It is shown in Zheng(1989) that when K is large relative to cr, the 

(Q,r) model behaves like the EOQ model in every respect, and that the optimal control parameters 

and the optimal average cost of the EOQ model (with backorder allowed) are very good approxi- 

mations of those of the (Q,r) model. The results of this paper show that when a is large relative 

to K, on the other hand, the expected marginal inventory cost function Ha is relatively flat. In 

that case. we can use a large order quantity to offset the ordering cost without incurring much 

increase in the average inventory cost; consequently, the total average cost of the system is close to 

the newsboy cost. 

Strictly speaking, the (Q,r) model only applies to continuous review inventory systems with 

unit demand. When the inventory system is reviewed periodically, or when the demand size is 

random, the reorder point can be "overshot," in which case the control policy must be revised. A 

well known inventory system that accommodates the overshot is the (s,S) system, where an order 

is placed to increase the inventory position to S as soon as the inventory position drops to or below 

s. Here s denotes the reorder point and S the order-up-to level. The order quantity is no longer

(S-s), the difference between the order-up-to level and reorder point, but that plus the overshot, 

which varies from order to order. Another variation that can accommodate overshot is the so 

called (nQ, r) system, where an order is placed whenever the inventory position drops to or below 

the reorder point r. The order quantity is chop te be an integer (n) multiple of Q such that the 

inventory position is raised to between r+1 and r+Q. For recent analyses on these systems, see 

Zheng(1990), Zheng and Federgruen(1991) and Zheng and Chen(1990). Although the (Q,r) model 

appears as a special case of the more general (s,S) and (nQ,r) modela, the qualitative properties 

derived for the former shed light on these general stochastic inventory systems. It should especially 

be emphasized that when the overshot is not significant relative to the order quantity, the behavior 

of (s,S) and (nQ,r) systems is similar to that of (Q,r) systems. 

Acknowledgment 

The research of the first author was supported in part by NSF grant DDM 9111183. The anthors 

would also like to thank Professor Paul Zipkin for sharing part of his recent manuscript (Chapter 6 

of Foundation of Inventory Management, Draft Version 0.1, November, 1991), in which proposition 

3 of this paper is independently observed. 

References 

[1] Browne S. end P. Zipkin 1989 Inventory Models with Continuons, Stochastic Demands, Grad- 

uate School of Business, Columbia University. 

[2] Chen F. and Y.-S. Zheng(1991a) Evaluating the Performance of a Serial Inventory System 

with Stochastic Demand, Working paper, Department of Decision Sciences, Wharton School, 

University of Pennsylvania. 

[3] Chen F. and Y.-S. Zheng(1991b) A Lower Bounding Framework for Stochastic Inventory Systems 

with Setup Costs Working paper, Department of Decision Sciences, Wharton School. 

University of Pennsylvania. 

[4] deBodt, M. A. and S. C. Graves(1985) Continuous Review Policies for a Multi-Echelon In- 

ventory Problems with Stochastic Demand, Management Science, 31, 1286-1295. 

15

[5] Federgruen A. and Y.-S. Zheng 1988 A Simple and Efficient Algorithm for Computing Optimal 

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[8] Hadley, G. and T. M. Whitin 1963 Analysis of Inventory Systems, Prentice-Hall, Englewood 

Cliffs, NJ. 

[9] Lee, H. and Nahmias, S. 1989 Single product, single-location models, Chapter 2 in Handbook in 

Operations Research and Management Science, Vol. 4: Logistics of Production and Inventory 

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[10] Rothschild, M. and J.E. Stiglitz 1970 Increasing Risk: a Definition, Journal of Economic 

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[11] Roundy, R. 1985 98% Effective Integer-Ratio Lot- Sizing for One Warehouse Multi-retailer 

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[12] Roundy, R. 1986 A 98% Effective Lot-Sizing Rule for a Multi-Product, Multi-Stage Production 

Inventory Systems, Mathematics of Operations Research, 11, 699-727. 

[13] Zheng Y. S. 1989 On Properties of Stochastic Inventory Systems, Management Science, to 

appear. 

[14] Zheng Y.-S. 1990 A Simple Proof for Optimality of (s, S) Policies in Infinite Horizon Inventory 

Systems Journal of Applied Probability, to appear. 

[15] Zheng Y.-S. and F. Chen 1990 Inventory Policies with Quantized Ordering, Naval Research 

Logistics, to appear. 

[16] Zheng Y.-S. and A. Federgruen, 1991 Finding Optimal (s,S) Policies is About as Sim ple as 

Evaluating a Single Policy Operations Research 39, 654-665. 

[17] Zipkin. P. 1986 Stochastic Leadtimes in Continuous-rime Inventory Models, Navel Research 

Logistics Quarterly, 33 763-774. 

16

[18] Zipkin. P. 1991 Foundation of Inuentory Management, Version 1.0, Columbia University, 

Manuscript to be published. 

17

Ce = H(Q* 

0 Q * 

Figure 1 

HO, A() — functions and the optimality condition

1988 

88/01 

88/02 

8/1/03 

/04/84 

88/88 

88/06 

88187 

88/08 

101/09 

88/10 

88/11 

Miehael LAWRENCE and 

Spyros MAKRIDAKIS 

Spyms MAKRIDAKIS 

larmes TEBOUL 

Susan SCHNEIDER 

Chartes wYrws7 

Reinhanl ANGELMAR 

Ingemar DIERICKX 

and Karci COOL 

Reinhard ANGELMAR 

and Susan SCHNEIDER 

Bernard SINCLAIR- 

DESGAGNÉ 





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(TM) 

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AC, FIN) 

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1990 

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TM/F,P/AC 

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EP 


TM 

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approach", November 1989. 

Compiexity of simulation gondele: A araph 

thenretit apprentie, November 1989. 

''MARS: A merl:ers and acquisitions 

manade" systtune, November 1989. 

*On the reaulation of proceoreenent 

November 1989. 

"Market miormignecture 'Mets of 

Revenante« intervention in the foreleg 

vienne market•, Dota aber 1989. 

lhoavaidabie Moclauldlige, January 1990. 

fflepeopenotie ColneolIllan, Conte of 

Adjudant«, and thelbiloarionr of European 

Manufacturing Fantileneent", lanuary 1990. 

•Managegoeut of Counnunkatina in 

International lemme and Development', 

lanuary 1990. 

"The Trnagronnitign or the %mno 

Financial Services Industry: Fon. 

Fragmentation to botegration*, lanuary 1990. 

.E•rtell111 bey *latine: Tenant 1992 

and %your, Ume 1990.

90/06 Gabriel HAWAWINI and "Integration of European Equity Markets: 

FIN/EP Erie RAIENDRA beplicatimm of Sine:nal Change fer Key 90/17 Nathalie DIERKENS "Information Asymmetry and F.qvity 

Market Participants tel and Beyond 1992". 

FIN Reviaed lanuary 1990. 

lanuary 1990. 

90/18 Wilfried VANHONACKER "Magagerial Draisine Raies and the 

90/07 Gabriel IlAWAWINI "Stock Market A/101111dieli and the Pricing of MKT F.alimadon of Dynainit Sales Restes» 

FIN/EP Fsmity on the Tokyo Stock Exrhange", 

lanuary 1990. 

Model •, Revised lanuary 1990. 

90/19 %th JONES and "Tbe KIWI of Computer Inlemention and 

90/08 

TM/EP 

Tawfik JELASSI and 

B. SINCLAIR-DESGAGNÉ 

"Modelling *ah MCDSS: What »Mn« 

Effiles?". January 1990. 

TM Tawfik JELASSI Tati: Saurante ou Rariaining Unicorne", 

February 1990. 

90/09 Albedo GIOVANNINI "Capital Controls and International Trade 90/20 Tawfik JELASSI, "An introdection ta Croup Draisine »d 

EP/I1N and lac WON PARK Finance", lanuary 1990. TM Gregory KERSTEN and Nrantiation Support", Fehnary 1990. 

90/10 Joyce BRYFR and "The Impact of Language Theorim on MS 

Stanley ZIONTS 

TM 'nova JELASSI Dialog", ;gentry 1990. 90/21 Roy SMITH and "Recollait:radium of Global Sec:trille* 

9011I Enver YUCF.SAN *An Overview of Frequency Demain 

FIN Ingo WALTER Industry in the 199re.Fenaary 1990. 

TM Mea:odelette for Simulation Sensitivity 90/22 logo WALTER •Farepaan Pinamitibligration mut Ils 

Analysis", Januar, 1990. FIN Impticatiomt for the Uilltd States•, February 

90/12 Miehael BARDA "Feellaral Change. Unemployment Renefits 

and Iligh Unemployment: A U.S.-European 90/23 Damien NEVEN "EEC lotettratioe Mem& 1992: Some 

Comparium", lanuary 1990. EP/SM Distrbutineal Aspacts", Revimd December 

1929 

90/13 Soumitra DUITA and •Apprandetate %menine about Temporal 

1/11 Shashi SHEKHAR Commtraints in Real Time Planning and 90/24 Lars Tyge NIELSEN "Positive Prime in cArm". lanuary 1990. 

90/14 

TM 

90/15 

TM 

Albert ANGEHRN and 

Hana-Jakoh LÜTH1 

Search", /anuary 1990. FIN/EP 

•Ireseal Interactive Modelling and Intelligent 

DSS: Petting llwory hen Practice". January 

1990. 

90/25 

FIN/EP 

1990. 

Lars Tyge NIELSEN "Elisa.» et Fmailihrione in CAPM", 

January 1990. 

90/26 Charles KADUSHIN and •Wily netwericiam Faits: Double lands »d 

Arnaud DE MEYER, 

"The Intered Technological Renewal of a 

OB/BP Miehael BRIMM the Limilatiosta of Shed» Networke, 

Dirk DESCHOOLMEESTER, %miens Unit with a Mature Technology". 

February 1990. 

Rudy MOENAERT and 

January 1990. 

Jan BARBE 90/27 Abbas FOROUGH1 and •NSS Sol:Mun Io Meer N'inaction 

1111 Tswfik IELASSI Sheaddim Mets', February 1990. 

90/16 Richard LEVICH and "Tas-Driven Regolatory Drag: European 

FIN logo WALTER Financial Center: in the 1990's", lemme 90/211 Anmud DE MEYER 'Tic Manufacturing Contstiontiou te 

1990. TM Innovation", February 1990.

90/40 Manfred KETS DE VRIES "Leaders « the Coach: The case of Roberto 

90/29 Nathalie DIERKENS "A Discuesion of Correct Mesures of OB Calvi', April 1990. 

I1N/AC Information Asymmetry", January 1990. 

9n/41 Gabriel HAWAV/INI, 

"Capital Market Reaction to the 

90/30 Lies Type NIELSEN "Te Expetted U$Ifty of Portfolios of 

FIN/EP 

117115k SWARY and 

Anortarscement of Internat, Rauking 

FIN/EP Assets", Mare 1990. lk HWAN JANG Legidatinn", Mach 1990. 

90/31 David GAUTSCIII and "What Devinions U.S. Retail Manias?". 90/42 Joel STECKEL and "Crocs-Vandatind Remmena Marker in 

MKT/EP Roger BETANCOURT Fehmary 1990. MKT Wilfried VANHONACKER Marke/MI Research". (Revised April 1990). 

90/32 Srinivamn BALAK- "Informatima Asymmetry, Adverse Selection 90/43 Ruben KORA1CZYK and "FAmity Ride Prends and the Peint of 

SM RISHNAN and and John-Ventures: Throry and Evidence". FIN Claude VIALLET Foreign Excluante Risk', May 1990. 

Mitchell KOZA Revised, la cary 1990. 

90/33 Carco SIEHL, 'The Rale of Rites of lei:ration in Service 90/44 Gilles AMADO, .01191dienould Chante nad Cultural 

OB David BOWEN and Deliver'''. Match 1990. 011 Claude FAUCHEUX and Rendes: Franco-American Contres", April 

Christine PEARSON 

André LAURENT 

1990. 

90134 

t1N/EP 

Jean DERMINE "The Cakes fana Fitmpean Rankine 

Inter:ration, a Cali for a Pro-Active 

Compention Poney". April 1990. 

90/45 Soumitra DUTTA and 

elategratint Case RIO« and Rode Rend 

TM Piero BONISSONE Reasoning: The tàoaulhstic Correction", 

May 1990. 

90/46 Soma MAKRIDAKIS "Exponentiel Surmandswinerra Met of 

90/35 Jae Won PARK •Changing Uncertahtty and the Time- TM and Michèle HIBON Initial Values and Leva Falletioes an Post- 

EP 

Varying Risk Premier in the Tenu Structure 

of Nominal Inter'« Rates". Decenther 1989, 

Semple Foreca tien Ancem". 

Revised March 1990. 90/47 Lydia PRICE and 

Improper Samprame in Idifferil 

MKT Wilfried VANHONACKER Experimente: Linitalianme the Use of 

90/36 Arnaud DE MEYER "An %Wied lame:tee of Meta-Arniyais Reonita9119191‘18 

TM Masurfacturing Stratégies in European 

Industry", April 1990. 

Updating", Reviaed May 1990. 

90/4* lm WON PARK "The Infornation in the Tenu Structure of 

90/37 William CATS-BARIL "Executive Information Systems: Developing 

EP latere* Rates: Out-andinsole Forecading 

TM/013/SM 

as Approach Io Open the Possibles", April 

1990. 

Performance*, lune 1990. 

90/49 Soumitra DUTTA "Approenate Remet ky /valloRY 

90/38 Wilfried VANHONACKER "Managerial bectent >barioler and the TM Answer Na Quinine', lune 1990. 

MKT Estimation of Dynamic Sales Respmrse 

Modela", (Revised Fentary 1990). 90/50 Daniel COHEN and 

"Price and Trade Fifeets et Exchange Rates 

EP 

chartes WYPLOSZ 

Flactuations and die Nana of Poney 

90/39 Louis LE BLANC and 

'An Englue« and Section Metbodology 

Coonfonation", April 1990. 

TM Tawfik JELASSI 

for Expert System Shelh", May 1990.

90/51 Michael BURDA and "Grass Labeur Market Flows in Vampe: 90/63 Sumwers GHOSHAL and "Organising Coussinet« Analyais Systeme, 

EP Chartes WYPLOSZ Some Stylized Falls". lune 1990. SM Eleanor WESTNEY Augias 1990 

90/52 Lars Tyge NIELSEN "The Mie of Inlimite Menus". lune 1990. 90/64 Sumantra GHOSHAL "Intentai Differentiation and Corporate 

FIN SM Performance: Cam of the Multinational 

Corporation", Angle 1990 

90/53 Michael Borda "The Consequences of German &momie 

EP and Mnnelary Union". /une 1990. 90/65 Chades WYPLOSZ "A Note ou the Real Exclunme Rate Effect of 

EP German Unification", Anon 1990 

90/54 Damien NEVEN and "Rumens Financial Regulation: 

El' Colin MEYER Framework for Pulky Analysis". (Reviaed 90/66 Snumitra DUTTA and "Computer Support for Strategic and Tartkal 

90/55 Michael BURDA and "Intertempond Prices and the US Trade 

EP 

Stefan GERLACH 

Balance". (Reviaed JuIy t990). 

90/56 

Damien NEVEN and 

May 1990). TM/SE/11N Nem BONISSONE Munia" in Mergero and Acquisitions", 

*The Structure and Determistants of Fast-West 

90/67 

TNI/SE/F1N 

Sombra DUTTA and 

Piero BONISSONE 

September 1990 

ilategratkig Prier Canes and Expert Knowledge ln 

a Menem and Ammiabitomn Re:domina System', 


EP Lars-Hendrik RÔLLER Trade: A Preliminary Analysis of the 

Manuficturbse Serine". lut), 1990 90/68 Saumura DUTTA P'renseumult amileilhodology 

for Enlumina the 

TM/SE Business Impact et Atticial Intelligence 

90/57 fars Tyge NIELSEN Comma Knowledge of a Multivariate Aggregate ApOications", Sepresmar 1090 

FIN/EP/ Statistie". July 1990 

TM 90/69 Sonmitra DUTTA "A Medd Bir Temporal Remanie« kt Medical 

TM Expert &Siffle% Sesternber 1990 

90/511 Lam Tyge NIELSEN Cannna Knowledge of Price and Expected Cosi 

FIN/EP/TM in an OliRopelktic Market". August 1990 90/70 Albert ANGEHRN "'Triple C': A Mena Interactive MCDSS', 

TM 


90/59 Lean DERMINE and "Feemornies of Seule and 

FIN Lam-Hendrik RÔLLER Steve in the French Mutuel Fumés (SICAV) 90/71 Philip PARKER and "Campedtive Efforts in Diffistion Modela: An 

Industry". Auguu 1990 MKT Hubert GATIGNON Empirical Analysée'. September 1990 

90/60 Pen 1Z and 'An Interactive Group Occitan Aid for 90/72 Enver YÛCESAN 'Andy* tif Marli« Chia Ugine 5anndation 

TM Tintin JELASSI Muttiohjective Proldents: An Empirical TM Grapk Modela", October 1990 

Autrement", September 1990 

90/61 Pankaj CHANDRA and "Modela for the Emilasittion of Manufacturing 

TM Mihkel TOMBAK %adule, August 1990 

90/73 

TM 

Arnoud DE MEYER and 

Kasra FERDOWS 

"Remaniant the Bardera in Manufacturing", 

October 1990 

90/62 Damien NEVEN and "Public relief Towards TV Break:Ming in the 90/74 Summum GHOSHAL and "Rognidte Cemplexity: O. Headquartem- 

FP Menno VAN DUK Netherlands", August 1990 SM Nitin NORRIA Subeithary Rebtirms in MNICs". October 1990

9ons 

MKT 

Roser BETANCOURT and 

David GAUTSCHI 

'Ibo Orrai et Relia Artividem Coarekals, 

Mensorment ma Evillente. October 1990 

9047 

FIN/Er 

Lam Tyne NIELSEN 

911/70 Wilfried VANHONAC/CER 'Men Bond Doride* Behavione and the Yetinution 9011011 Susan C. SCHNEIDER and 

MKT of Dreemdaelta gemma» Modeae. 00/MKT Reinhard ANGELMAR 

Revive/ October 1990 

tern Wilfried VANHONACKER nada; Ihe Kart &Bembo ai Site Respire as OB 

MKT Advertisima: Am Aarettalisn-ludemendent 

90/18 Micked BURDA and *Embaume Rate Drineirs and Cauvery 

EIP Stefan GERLACH Unification: The Ostmark - DM Rate. 

Aeletendoliiee T. October 1990 90190 Philip PARKER 

MKT 

October 1990 

90/79 Mil GARA •Interencen ai* an Ustimown Noise Lied in a 

TM Bernes% haceur, October 1990 

90/1111 Ale GARA and •Gaine Sarre Data I.• laterearet about hercha» 

TM Robert WIN/CUIR Rellanionet Omober 1990 1991 

Tarvfd 1ELASSI "Dm niant ar INItnr: Bien et Orientations des 

»Wied F.R. KETS DE VRIES 

TM Brevet Interactifs d'Aide II In Decisbee 9101 Luk VAN WASSENHOYE. 

901113 Otaries WYPLOSZ Ideeetery Union rd Fiscal Poliry Diodes'," 

October 1990 ni/St Leonard FORTUIN and 

Paul VAN BEFJC 

EP November 1990 91/113 

7M/5111 

Lut VAN WASSF.NHOVE, 

Leonard FORTU1N and 

9063 Nathalie DIUICENS and noiormatien Myestretry and Corporate Paul VAN BEEK 

nNrni Bernard SINCLAIR-DESGAGNE Commaitatiem Remdts of • Pilet Sonde, 

Noverdber 1990 91103 Pekka HIETALA ad 

FIN Tien ILOYMNIEMI 

9014 Phifip M. PARKER "The Effort d Advertitina on Price and Quality: 

MKT Ibo Ormaehie Industry Rerithed• 91104 Lars Tyge NIELSEN 

December 1990 FIN 

Avijit GHOSH end •Opetri Tiadas and Lerma be Competitire 91105 Srs SCHNEIDER 

MKT Varna TIBREWALA lelerkeee November 1990 OB 

*MM 

EPITM 

Olivier CADOT and 

Bernard SINCLAIR-DESGAONE 

°nyder, and Stoss in roides; November 1990 MAK 

OB 

Minfœtt Kers DE VRIES, 

Daum MILLER and 

Alain NOEL 

•Faisten•• of EssualBwinas in CAMA: M'aber 

Nasadif. Decendier 1990 

`Cognai» in Organientioeud Andria: 11bo's 

Mien the Semer Revient December 1990 

CE0 Celle* Ti gratta and Other 

Tes fier the Bard Rama; December 1990 

'hire ihraticity Dr:amies mer the Adoption 

Literie: Am Empiriud Study; December 1910 

'0perational Itamtemitffla Mont fer Marge« 

Thon %y Mie 

lermary 1991 

*Oparediond Remo* enrovirommote 

hinurop 1991 

Dividend herease in Rem Issue 

Thur, »I Bedonne towesery.1991 

"Tweetued Separallma Parme Structure ami 

Relsemesse lantary 1191 

9Mlegettiag beneterke ia Or .e 

kami PM 

gledeeilledies the Leolereretep Welter« 

Applicage, et tbe Sieste* Relideeelip imervire 

Mine Imam 1990 09111. mvised Apal 19901

91/07 Olivier CADOT "levain to Inmadvent Countriem: A farallosical 

Er Sem," lanuary 1991 91/19 

MKT 

Vikas TIBREWALA end • Aggregate Test of Purtimse Regularite, 

Bruce BUCHANAN March 1991 

91/08 Charles WYPLX/SZ "Pola-Refone Emt and West: Capital 

Er Aemennlotioe and the Labour Mubility 91/20 Darius SABAVALA and Monitoring Short-Rome Changes in Purrhasing 

Construisit," lanuary 1991 MKT Vika* TIBREWALA Behaviour", March 1991 

91/09 Spyroe MAKRIDAKIS •What eau we Leoni from navrer, Felnuary 1991 91/21 Sumatra GHOSHAL, Internât Conammiration within biNns: 

TM SM Harry KORINE and lemme of Formel Structure Verses lalegrative 

Gabriel SZULANSKI Promues*, April 1991 

91/10 Luc Van WASSENHOVE end "Integrating Schedeing with Batching and 

TM C. N. P01111 Let-Seing: A Revive or Alliamilkulla and 91/22 David GOOD, "EC lekgratian nad Me Stream of the Fromm- 

Causplenity•, Febnery 1991 EP Lers-Hendrik RÔLLER and American Aies* ladmetries: Implications fer 

Robin S1CKLFS Efficient" and Wren*, April 1991 

91/11 Luc VAN WASSENHOVE et al. "Muid-Item in Caparitated Mugi-Stage 

TM Serial System", February 1991 91/23 Spyros MAKRIDAKIS and "Kemmel SeeldRiagtlie Effect of Initial 

TM Michèle HEION Vaine" and Lem Fewetimature Foot-Semple 

91/12 Albert ANGEHRN •Interpretadve Computer Intelâmame: A Link Forweerting Areemey;111.111 1991 (Revision of 

TM between Umm, Modela and Metbods in ose. 90146) 

91/13 

EP 

February 1991 

Michael BURDA •Ulm and 'Muet Markets à Czechariovidia and 

the E."-GDR: A Twin Stedy", February 1991 

91/14 Roger BETANCOURT and 'ne Output of Rata At-livides: Fumai Shi/7m Charles CORBEIT 

MKT David GAUTSCHI Evident •, February 1991 

98/15 

OB 

Manfred F.R. KETS DE VRIES •Expérang the Mytb about Ratinant Organiusiows 

and Emcmives", March 1991 

91/24 Louis LE BLANC and •A. %Orient ArresmarMWChoice Modela for 

1M Tawfik IELASS1 Software Eveleaden anifillkoMnr, May 1991 

91/25 Luk N. VAN WASSENHOVE and "Tradaelle What Trtieothr April 1991 

91/26 Luk N. VAN WASSENHOVE and "Sangle Media- Sdrifi bilai tenaillée Total Lat. 

1M C.N. POTTS %kWh• April 1991 

91/27 Nathalie DIERKENS •A Dimmion of Cornet Mandes of Informatisa 

91/16 Arnaud DE MEYER and •Faeteties of the Vehme: L'endive Senweary of FIN Asyremetry: 1be Lime or Myes and ISIMInf`s 

TM East. FERDOWS et.al. the 1990 Interea6unal Manufacturing Futures 

Survey", March 1991 

Maint or Ibe Importance of the Amr( Sawcture of 

the Me. May 1991 

91/17 Dirk CATTRYSSE, "Ileuriaties for Ibe Marrie letvizing and 91/20 Philip M. PARKER •A Note au 'Aéreribia« and the Prim lad Quality 

TM Roda KUIK, 

Marc SALOMON and 

&Madelin Noble. laids Setup Times•, March 1991 MKT of Oplometrie Services', hume 1991 

Luk VAN WASSENHOVE 91/29 Tswfik JELASSI and 'An t'aphte] Moly of an Interactive, Seamiam 

TM Abbas FOROUGH1 Orieded Compoterised %sertirai« Support System 

91/18 C.N. POTTS end •Approximnalion Algorithmes fer Scinda« a Sm* (NSS)", lune 1991 

1111 Luk VAN WASSENHOVE Machine M Milhatime Total Lute Work•, 

Match 1991

91/30 Wilfried R. VANHONAC10ER and MM« Meta-Anallysis Rimes in Bayeniatt Updating: 

MKT Lydie J. PRICE lbe Erne Cal Profilent", lune 1991 91143 Sumatra OHOSHAL and %W in Trammieend Combattiez: The 

SM Christopher BARTLETT Managemeat Challenge', September 1991 

91131 Reraul KARJR and "leader Tratfieg Ruinaient muid the Stock 

FIN Theo VERMAELEN Mana% Jase 1991 91/44 Summar* GHOSHAL and Inabilated Innevation I. the IfIfferantioded 

SM Nilin NOHRIA Network . MnItinetietur, September 1991 

91/32 Stman C. SCHNEIDER 'Orgamisadonel Sememaking: 1992 • , lune 1991 

011 91/43 Philip M. PARKER "The F/feet et Adverfidad es Frite and Quality: 

MKT As Fmapirical Study of Eye Examinalions, Sweet 

91/33 Miction' C. BURDA and *Germe Trade Usinas gifler Umilication Third Lama muid Self-Decevers", September 1991 

EP Michael FUNKE Dere, Wace Marin/mea Mossopolistarr 

lutte 1991 91/46 Philip M. PARKER *Prelog Santals in Mante whi g tem* 

MKT Flatikifies". October 1991 

91/34 Jean DERMINE *The BIS Noeud fer the Memerenleat of Interest 

FIN Rate Reit, Some ?Wells", lune 1991 91/47 Philip M. PARKER Sady of Price Ehiseleky Dynamite Usai 

91/33 

FIN 

Jean DERMINE "The Radeau of Finacial Servies in the F.C, 

Centrai:sten er %kraal Antemear lune 1991 

MKT Parsianonions Redmeneent/Naltiode Fareham 

Diffames Modela•, October 1991 

91/414 H. tandis LABEL and 9Mourgerial Ineentive amillhaviromantal 

91/36 Albert ANGEHRN •Suppertiat Modelaient Decktie« Mana: New EP/TM Bernant SINCLAIR-DESOAONE Cooludiemee. Oember I991 

TM Perspeedves mad New Systems", Auget 1991 

91/49 Bernard SINCLAIR-DESGAGNB "Me Net-Onler Afflua» MmItiendu 

91/37 Mao WALTER and 'The latrodactim of Universel Bakin in Canada: TM FradgekAgent TreblearttOmber 1991 

ET Hugh THOMAS An Emme Study", Auget 1991 

91/30 Luk VAN WASSENHOVE end Mea Gien h Yeer blegataring Strelfear 

911311 Ingo WALTER and "Natimml Ilad Global Canspetitivemeas of New York SM/TM Charles CORBETT October 1991 

EP Anthony SAUNDERS City as a %amis! Center', Mea 1991 

91/31 Piao M. PARKER Mimais Aatatlf Made Modela Sarre 

91/39 Mg° WALTER and 

Anthony SAUNDERS 

"Reeantiguratien of Rankine and Capital Markets 

hi Eastern Emme. Augura 1991 

MKT 

Empiried Goddelinee, Odober 1991 

91/32 Michel BURDA and 'Mana Caplet Imeatieent and Menti« lia an 

91140 Luk VAN WASSENHOVE. 'A Set Fartitieming Renne fer the Gemerefued EP Charles WYPLOSZ Integrated Fitropee, October 1991 

TM Dirk CATTRYSSE and Awdeament Profiles", August 1991 

Marc SALOMON 91/53 Michael BURDA end • aimer Marna and German Integration: Sente 

91/41 

TM 

EP Charles WYPLOSZ Ifiemette •, Ocaber 1991 

Luk VAN WASSENHOVE, 'A nie Pelyeandal Approwienation Stigma for 

M.Y. KOVALYOU and 

Schee/Mg a Male Maitise to Minimise Total 91/34 Albert ANGEHRN •Sanie Adage: An Alternative Frimaire* for 

C.N. POTTS Weigilted Lite Work", Annal 1991 TM Caentleter-Aided Raban Maldnor. October 1991 

91/42 R. WEfTZ and "Solving A Mani-Ctiteria Allocation Freedom.: 

TM Tawfik IELASS1 A DIPCfRi011 Support System Apprends", 

August 1991

91/55 Robin HOGARTH, 

"Leenevity of Business Firmus A Four-Stase 

92/03 Manfred F.R. KETS DE VRIES 'Ibn Fansily Film An Owen', Maenso•, 

EP/SM Claude MICHAUD, 

Yves DOZ and 

Freinework for Analysis", November 1991 

OB 

lanuary 1992 

Ludo VAN DER HEYDEN 92/04 Philippe HASPESLAGH and "Maine Acquisitions Work", /anuary 1992 

SM David MM/SON 

91/56 Bernard SINCLAIR-DESGAGNE 

TM/EP 

"Aspirations an Emmen» Development', 

91/57 Lydda PRICE 'Ibo Indirect Efforts of Neptive Information en 

November 1991 92/05 Xavier DE GROOTE "Fledhally and Prend Dives* in Loteniap 

TM Modela". lamas, 1992 (revised) 

MKT Attitude ange", November 1991 92/06 Theo VERMAELEN and "Enencial Innevatien: Self Tender Offen I.• the 

FIN Kees COOLS January 1992 

91/58 Manfred F. R. KETS DE VRIES "Leaders Who Go Cran", November 1991 

OB 92/07 Xavier DE GROOTE "lb. Fleoldlity of Production Prenne A 

TM Gemmi Framework", January 1992 (rened) 

91/59 Paul A. L. EVANS 

OB 

"Managemeat Development as Glue Technology", 

November 1991 

92/08 

TM 

Luk VAN WASSENHOVE, 

Leo KROON and 

91/60 Xavier DE GROOTE 

`Flesildity nad MarketiegIblannfaceung 

Marc SALOMON 

TM Coordintin", November 1991 (revised) 

92/09 Luk VAN WASSENHOVE, 

91/61 Arnaud DE MEYER "Prend Development in ne Textile Markimery 

1M 

Roelof KUBC and 

TM Industry". November 1991 Mare SALOMON 

•End and Approdemlion Ateritleas for the 

Operatiend Fimed ber ved Sebaddin Profilent", 

Jarasary 1992 

"Statistired Se r» MIllikne for lAtaing 

Prolden", lennety 

91/62 Philip PARICER and *Sperifying Cempatitive Efforts in Diffusion 92/10 Yves DOZ and •Repainap Ermandlinness: A Proroge of 

MET Hubert GATIGNON Mn»: An Empirerai Anoblie, November 1991 SM Heinz THANHEISER Orgemisntiond Renewe •, Jamary 1992 

91/63 Michael BURDA 'Sem New Indes ou the Interindmstry Wein 92/11 Enver YUCESAN and •Che the hernetability of Vesifying Stretteral 

EP Sentare frein the Gemme Soeioneesesnir Pond• 

December 1991 

17S1 &bridon JACOBSON ?reperdes et pinne End Simulation Models", 

February 1992 

91/64 Jean DERMINE "InerrationnEntion of FInamen1 Mark». 92/12 Gabriel HAWAWINI "Valentin of Cie-Bender Meilen end 

FIN Ef&ioscy and Stanley', December 1991 FIN Andeilimer, Penny 1992 

12,1 92/13 Spyros MAICRIDAICIS and ail. IdeCenepotiliseu A Budget Related 

1M Michèle HIBON et.al. Fapieieal Fonmens Study", Felnuary 1992 

92/01 Wilfried VANHONACKER "CONPRO4DOGIT: A New Brand amoire Medd 

111KT/EPI7AI Imerporating a Coasideration Set Formation 92/14 Lydda PR/CE "Inde.' Cluster Ondin with NORMIX 

92/02 Wilfried VANHONACKER "Ibn Drin» et tbe Considoration Set Foreudem 

MICT/EP/TM Promu A Radient Medelling Penipartivo 

Pronee, Jonas, 1992 MKT Population Atembenialp Prembalities", 

Febtuary 1992 

Sun Nemosical Rente", lersdary 1992

by Yu-Sheng ZHENG* and Xavier DE GROOTE ... - Insead

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