THE UNDERSHOOT OF THE REORDER POINT: by · THE UNDERSHOOT OF THE REORDER POINT: TESTS OF AN...

"THE UNDERSHOOT OF THE REORDER POINT:TESTS OF AN APPROXIMATION"

byM.P. BAGANHA*

D.F. PYICE**and

G. FERRER***

95/03/TM

* Universidade Nova de Lisboa, Lisbon, Portugal.

** Amos Tuck School of Business Administration, Dartmouth College, Hanover, NewHampshire, USA.

*** Ph.D Student at INSEAD, Boulevard de Constance, 77305 Fontainebleau Cedex, France.

A working paper in the INSEAD Working Paper Series is intended as a means whereby aresearcher's thoughts and findings may be communicated to interested readers. The papershould be considered preliminary in nature and may require revision.

Printed at INSEAD, Fontainebleau, France

THE UNDERSHOOT OF THE REORDER POINT:TESTS OF AN APPROXIMATION

Manuel P. BaganhaUniversidade Nova de Lisboa

David F. PykeAmos Tuck School of Business Administration

Dartmouth College

Geraldo FerrerINSEAD

January 2, 1995

ABSTRACT

We investigate a widely used approximation for the mean and variance of the undershoot. Theapproximation is based on the limit of the excess random variable of a renewal process as theorder size approaches infinity. In the current business environment which emphasizes smallbatch sizes and frequent deliveries, many inventory systems order in batch sizes that are notlarge. We investigate the potential error that could be introduced by using the approximation fora variety of batch sizes and demand distributions.

THE UNDERSHOOT OF THE REORDER POINT:TESTS OF AN APPROXIMATION

1 - Introduction

Inventory systems with random demand divide naturally into two categories: periodic

review and continuous review. Periodic review systems take forms similar to (R,s,S), in which

inventory position is checked every R periods. (Inventory position is defined as the inventory on

hand, plus on order, minus back orders.) If the inventory position is at or below the reorder

point, s, an order is placed to bring the inventory position up to S . Periodic review systems are

somewhat more complicated than continuous review systems because, when an order is placed,

the inventory position is seldom exactly at the reorder point; rather, it is some amount below the

reorder point. This amount below the reorder point is called the "undershoot." Common

periodic review systems accommodate the undershoot in two ways: First, ordering up to S rather

than a fixed quantity; second, adjusting the order point upward for the amount of the undershoot.

Thus, the reorder point and order-up-to level are based on mean and variability of both lead time

demand and the undershoot. Choosing good values for s and S therefore depends, in part, on

accurate values for the mean and variance of the undershoot.

In continuous review inventory systems, a (Q,r) policy is often used. With this policy a

batch of size Q is ordered when the inventory position reaches a reorder point, r. Undershoots

may be observed in these systems when demand is lumpy because inventory may fall below the

reorder point by a large demand event. As in periodic review systems, inaccurate estimates of

the undershoot may result in higher costs or lower service than desired. Unfortunately, the

moments of the undershoot distribution are generally not easy to compute.

There are, however, widely used and easily computed approximations—based on

asymptotic results of renewal theory—for the mean and variance of the undershoot. In theory,

the mean and variance converge to the exact values as the order size increases, implying that

managers need not be concerned about errors in the approximation if the order size is large

enough. This then raises the issue how of large the order size should be to insure an accurate

approximation.

In this paper we test the accuracy of the widely used renewal approximation for a variety of

distributions with a variety of shapes, means and standard deviations, as well as for a variety of

order quantities, against the exact values given by the algorithm. Thus, the purpose of this paper

is twofold: To test a commonly used approximation; and to describe the circumstances when the

approximation should be avoided, even for larger order sizes. (If the recommendation is to avoid

the approximation, Baganha, Pyke, & Ferrer (1994) provide a fast and easily-implemented

algorithm for computing the exact distribution of the undershoot.)

The remainder of this paper is organized as follows: In Section 2 we present a brief review

of the literature pertaining to this problem. In Section 3 we present the approximation and

discuss our experimental design and the results of our investigation. In Section 4 we present a

summary and conclusions.

2 - Literature Review

Karlin (1958) defines the excess random variable for a renewal process and presents its

Laplace transform. The excess random variable (or the residual life) at time t is the time until the

next renewal. Likewise, the deficit random variable (or the age) at time t is the time since the

last renewal. Karlin then presents the value of the excess random variable for the case of the

exponential distribution. He applies the excess random variable to the case of the (s,S) inventory

policy, but he restricts the application to exponential demands. (In the case of exponential

demands, the undershoot is also exponential.) Karlin also notes that the excess random variable

and the deficit random variable of a renewal process are identical. Ross (1983, pp. 67ff)

discusses the excess and deficit random variables and notes their asymptotic behavior. Tijms

(1976) develops the exact and approximate distributions for the excess random variable applied

to (s,S) inventory systems for continuous demand distributions. Silver and Peterson (1985, pp.

346ff) draw on this work to present a discrete approximation of the mean and variance of the

undershoot based on the limit as the order size goes to infinity.

Sahin (1990, Chapter 2) discusses the renewal function and its shape. Applying the

generalized cubic splining algorithm of McConalogue (1981), Sahin computes the renewal

2

function for five distributions: the Gamma, the Weibull, the truncated Normal, the Inverse

Gaussian, and the Lognormal. The algorithm approximates the convolution of the renewal

function by a cubic spline function. Sahin suggests that the accuracy of the approximation is 4 to

6 decimal places, and he notes that the renewal density may oscillate or may be monotone as it

approaches its asymptote. He reports the order size as a multiple of mean demand such that the

relative error of the exact renewal function relative to its asymptote is less than or equal to some

constant but arbitrary value. The research suggests that the accuracy of the asymptotic

approximation is a function of the number of multiples of the mean, perhaps of distribution type

and coefficient of variation of the distribution as well. There is no information on the magnitude

of the error for small order sizes; rather, information is given only about how large the order size

must be in order to make the error small. The order size varies from one-half the mean to 15

times mean demand in order for the error in the approximation to be within 1 percent. Typical

values are closer to 1-1/2 to 2-1/2 times mean demand.

Our work advances Sahin's research by specifically extending the understanding of the

errors in the commonly used renewal approximation. We do this by examining a variety of

distributions including two-mass-point distributions and by examining a wide variety of order

sizes.

Tijms and Groenevelt (1984) suggest that, if the coefficient of variation of demand over the

review period is not extremely small, the undershoot approximation is accurate if the order size is

greater than 1.5 times the mean demand. We will see below that for certain cases our resultst.

differ somewhat dramatically from the results of both Sahin and Tijms and Groenevelt.

Using an intuitive argument, Hill (1988) develops the same approximate undershoot as that

developed using renewal theory. His limited tests of its accuracy suggest that it is quite accurate.

He does suggest, however, that further tests would be valuable.

Baganha, et al. (1994) develop an algorithm for computing the undershoot distribution for

the case of discrete demand. The algorithm does not require convolutions of the demand

distribution, and can be computed easily on a spreadsheet.

3

In related work, Whitt (1984) studies two-moment approximations for the mean queue

length in GI/M/1 queues. Using the theory of complete Tchebycheff systems, he identifies the

distributions that give the largest and smallest possible mean queues (with the first two moments

of the service-time distribution fixed) as simple two-mass-point distributions. Gallego (1992)

finds that the most unfavorable distribution (fitting the first two moments of lead-time demand

and then minimizing over the parameters of a (Q,r) inventory model) is the two-mass-point

distribution.

Finally, we note that a number of authors have included the undershoot in calculations

pertaining to inventory. We list only a sample here: Silver (1970) applies the undershoot to

items having lumpy demand. Other examples include Cohen, Kleindorfer, Lee and Pyke (1992),

who apply the undershoot to multi-item (s,S) policies in logistics systems with lost sales, and

Ernst and Pyke (1992) who apply it to the problem of ordering component parts that will be

assembled into a final product. Many other examples exist in the literature.

Federgruen and Zipkin (1984) introduce an efficient algorithm for computing the optimal

(s,S) policy when demand is discrete. They prove that the algorithm converges and show how

certain one-step approximations can have high errors. A recursive computation for the renewal

function is used but no results specifically on the undershoot are presented. Zheng & Federgruen

(1991) extend this work with a simple algorithm to determine the bounds of an (s, S) policy

which outperforms the policies obtained by pervious research.

3 - Experimental Results

3.1 Introduction and Intuition

In this section we briefly discuss the application of renewal theory to the undershoot of the

reorder point in an inventory system. Then we present our experiment and results. However, we

begin by developing some intuition regarding the undershoot using Figure 1. In this figure note

that the demand distribution is of the two mass point type with demand of 4 with probability 0.2

and demand of 7 with probability 0.8. When 0 = 1 and the demand is 4, the undershoot is 3, as

4

can be seen by A = 1 column. When demand is 7 and A = 1 the undershoot is 6. Recall that A

= 1 implies that the inventory position immediately after reordering returns to one unit above the

reorder point. Then, when A = 2 and demand = 4 the undershoot is 2. When demand = 7 the

undershoot is 5. The pattern continues for A = 3 and 4. At A = 4 and demand = 4 the undershoot

is 0. When A = 5 the pattern changes slightly. Demand of 7 yields an undershoot of 2 as in the

pattern before. However, when the demand is 4, inventory position becomes one greater than the

reorder point. Then the undershoot is either 3 or 6 as in the A = 1 column. Thus, the undershoot

is 3 with probability 0.2 x 0.2 = 0.04, or 6 with probability 0.2 x 0.8 = 0.16. This pattern then

continues as can be seen from Figure 1.

For a complete development of the undershoot from the perspective of renewal theory we

refer the reader to Heyman and Sobel (1982, Chapter 5), Ross (1983, Chapter 3), Silver and

Peterson (1985, pp. 346ff), Baganha, et al. (1994), and Sahin (1990, Chapter 2). For our

purposes, we need only to present the widely used approximation. The required notation is:

X = one period demand.

II = expected value of X. = E[X]

,a` = variance of X.

cv = coefficient of variation of X = aig.

A = S - s, where S = the order up to level and s = the reorder point.

1:1 /, = approximate mean of the undershoot distribution.

^ 20-h = approximate variance of the undershoot distribution.

Silver and Peterson (1985) use the asymptotic distribution of the undershoot distribution for

discrete demand as the order size goes to infinity, and derive its mean and variance:

.. a2+µ2 1Ph_ 2p. 2

5

(1)

6

.. 2 E(X 3 ) cr2 +11 2 1 2 1Ch =

311

3.2 Tests of the Approximation

In this section we test the commonly used renewal approximation presented above, against

the exact values. First, however, we note that when demand follows a geometric distribution, the

undershoot is also geometric with the same mean. Thus, if X is geometric with parameter p,

f(x)= p(1–p)x x = 0, 1, 2, ...p1 –

with mean and variance 1–p

P2

The mean of the undershoot, therefore, is 1 P The approximation,

C52

+ p.2 _[ 1–p 4. (1–p)2 ]/(2(i_p_, 1 1–p11 h 211

1 "„ P) T p

P P

2 2 2

is exact. Similar analysis shows that the approximation for the variance is exact. The

approximation for other demand distributions is not always exact. We wish to see the magnitude

of errors one may face when using the renewal approximation.

The experimental design is given in Table 1. To create the discrete version of the normal

and lognormal distributions, we applied the following technique for creating a discrete

probability distribution from a continuous one: P(X=x) = P(x-- 0.5 5_ X 5_ x + 0.5) and P(X = 0) =

P(X 0.5) Thus, the actual mean and standard deviation are slightly different from the input.

For the gamma distribution we computed the exact undershoot by numerical integration. We

include the two-mass-point distribution because Whitt (1984) and Gallego (1992) suggest that it

may provide worst-case results. Also, we wish to illustrate several cases of when the distribution

does not converge, regardless of the order size. (See Baganha, et al. (1994) for more detail on

convergence.)1

1 They show that the undershoot distribution will not converge if there exists an integer k > 1SO

such that P(X = nk) = 1, where X is the one period demand.n=0

(2)

Tables 2 and 3 contain the results for the mean and variance, respectively. Each column of

the table represents percent error in the mean (or variance) of the approximation vs. the exact,

using the exact as the base: (approximately - exact)/exact. For instance, for the normal

distribution with a mean of 10 and a standard deviation of 1, the largest error when A is larger

than t is 88 percent. The largest error when A is larger than 4 t is 34 percent. "NA" indicates

that the errors go to infinity, due to division by zero.

We highlight several points from these results. First, the lower the coefficient of variation,

cv , the higher the errors in the approximation. See Figures 2 and 3, for example. When the cv is

0.05, the errors in the mean and variance are 120 percent and 524 percent, respectively, when A

is larger than 4.t. When the cv increases to 0.1, the errors decrease to 34 percent and 73 percent,

respectively. For a cv of 0.2, the errors are less than 4.1 percent for both the mean and variance.

These results hold for the normal, lognormal, Poisson and uniform distributions, which implies

that the skew of the distribution has little effect on the errors. Clearly, the errors can be very

large, even for large batch sizes. The oscillation of the undershoot distribution changes with the

parameters of the demand distribution. Figures 4 and 5 illustrate for the case of the Gamma

distribution.

Two-mass-point distributions show significantly larger errors than standard distributions.

For example, the two-mass-point distribution with mass at 9 and 11 (and probability 0.5 at each

point) has a mean of 10 and a standard deviation of I. Errors for this distribution are 58 percent

and 77 percent for the mean and variance, respectively. when A is larger than 4g. All two-mass-

point distributions we tested had greater errors than standard distributions for like cvs. These

data would support other research which indicates that two-mass-point distributions represent the

worst case. Unlike standard distributions, there is no clear relationship between cv and

approximation errors because some of the distributions do not converge. See Figures 6 and 7 for

examples of two-mass-point errors.

Two-mass-point distributions can be illustrated by a supplier with just two customers for a

given item, each of them ordering fixed amounts. Such a supplier faces a two-mass-point

7

distribution of demand. When a warehouse faces demand of the two-mass-point type, it is

necessary to adjust the inventory policy to account for undershoots. However, if the order sizes

have a greatest common divisor greater than 1, the actual undershoot exhibits cyclical behavior

in the warehouse order size. High errors from using the approximation, therefore, are possible,

even for large warehouse order sizes.

One must use care interpreting these results. While it is true that the percentage error can

be extremely high, several comment should be considered. First, the undershoot represents only

part of the inventory system. Total relevant inventory costs are driven primarily by the demand

during the risk period -- including both the lead time demand and the undershoot. In most cases

the undershoot is a small portion of the total demand during the risk period. Thus, even large

errors in the undershoot may give rise to small cost penalties. Second, using percentage errors

can inflate small absolute differences when the absolute numbers are small. Thus, large

percentage errors may sometimes represent differences of only one or two units. On the other

hand, if items are expensive an error of one or two units can be very costly. The major lesson is

that one should not use the approximation blindly.

4 - Summary and Conclusions

In this paper we have examined a commonly used renewal approximation for the mean and

the variance of the undershoot of the reorder point. Our results indicate that for low variance

demand distributions the approximation can give extremely high errors in mean and variance.

Two-mass-point distributions, which tend to be the worst case in a number of contexts, also

create high errors for the undershoot approximation in certain cases.

8

9

References

Baganha, M.P., Pyke, D.F. and Ferrer, G., 1994. "The Residual Life of the RenewalProcess: A Simple Algorithm". The Amos Tuck School of Business Administration,Dartmouth College, Working Paper No. #294. Rev. January 10, 1995.

Cohen, M.A., Kleindorfer, P.R., Lee, H.L., and Pyke, D.F., 1992. "Multi-item Service-constrained (S, ․) Policies for Spare Parts Logistics Systems," Naval ResearchLogistics, Vol. 39, pp. 561-577.

Ernst, R., and Pyke, D.F., 1992. "Component Part Stocking Policies," Naval ResearchLogistics, Vol. 39, pp. 509-529.

Federgruen, A., and Zipkin, P, 1984. "An Efficient Algorithm for Computing Optimal(s,S), "Operations Research, Vol. 32, pp. 1268-1285.

Gallego, G., 1992. "A Minmax Distribution-free Procedure for the (0,0 InventoryModel," Operations Research Letters, Vol. 11, pp. 55-60.

Heyman, D. P., and Sobel, M.J., 1982. Stochastic Models in Operations Research, Vol.1, New York: McGraw-Hill Book Company.

Hill, R. M., 1988 "Stock Control and the Undershoot of the Re-order Level," Journal ofthe Operational Research Society, Vol. 39, No. 2, pp. 173-181.

Karlin, S., 1958. "The Application of Renewal Theory to the Study of InventoryPolicies," in Ch. 15, Studies in the Mathematical Theory of Inventory and Production,K. Arrow, S. Karlin and H. Scarf (Eds.), Stanford, California: Stanford UniversityPress.

McConalogue, D.J., 1981. "An Algorithm and Implementing Software for CalculatingConvolution Integrals Involving Distributions with a Singularity at the Origin," Delft:University of Technology, Department of Mathematics and Informatics, Report 81-03.

Ross, S.M., 1983. Stochastic Processes, New York: John Wiley & Sons.

Sahin, I., 1990. Regenerative Inventory Systems: Operating Characteristics andOptimization, New York: Springer-Verlag.

Silver, E.A., 1970. "Some Ideas Related to the Inventory Control of Items Having ErraticDemand Patterns," CORS Journal, Vol. 8, No. 2, July, pp. 87-100.

Silver, E.A., and Peterson, R., 1985. Decision Systems for Inventory Management andProduction Planning, 2nd Ed., New York: John Wiley & Sons.

10

Tijms, H.C., 1976. Analysis of (S, ․) Invemory Models, 2nd Edition, Mathematical Centre,Trachts, 40, Mathematich Centrum, Amsterdam.

Tijms, H.C., and Groenevelt, H. 1984. "Simple Approximations for the Reorder Point inPeriodic and Continuous Review (.5, ․) Inventory Systems with Service LevelConstraints," European Journal of Operations Research, Vol. 17, pp. 175-192.

Whitt, W., 1984. "On Approximations for Queues, I: Extremal Distributions," AT&TBell Labora/ories Technical Journal, Vol. 63, No. 1, pp. 115-138.

Zheng, Y.-S.., and Federgruen, 1991. "Finding Optimal (s,S) Policies Is about as Simpleas Evaluating a Single Policy"Operwions Research, Vol. 39, No. 4, pp. 654-665.

Normal j.t = 10 a= 1, 2 ,3, 4117)IA. =20 a= 1, 2, . . ., 5

Lognormal 1.t = 10 a = 1, 2, . . ., 5j.t = 20 a= 1, 2, ..., 5, 10

Poisson g = 1, 2, ..., 5, 10.

Gamma a = 3, 9 A. = 1, 3, 9, 27

Uniform Range = [8, 12], [7, 13], [5, 15]

Two Point 0, 10 with probabilities 0.1, 0.91, 11 with probabilities 0.1, 0.9

10,21 with probabilities 0.1, 0.9


10, 16 with probabilities 0.5, 0.5



1, 5 with probabilities 0.75, 0.252, 6 with probabilities 0.75, 0.25


Two Point 7, 13 with probabilities 0.5, 0.59, 11 with probabilities 0.5, 0.59, 12 with probabilities 0.667, 0.333

Table 1Experimental Design

Table 2Maximum I% errorl in mean (Approximate vs Exact)

Lognormal

Poisson

e= A > A> A> A > A> A> A> A >IIa c v 1 to 40 g/2 11 1.5g 24 2.5g 3µ 3.5g 4µ

10 1 0.1 173.02 173.02 88.48 88.48 62.62 62.62 45.67 45.67 34.192 0.2 53.90 53.90 21.40 20.72 10.59 8.50 5.19 3.87 2.853 0.3 45.01 22.64 12.44 6.99 4.36 2.05 0.97 0.46 0.223.16 0.32 44.49 19.94 12.00 6.45 3.65 1.51 0.70 0.31 0.13

20 1 0.05 348.51 348.51 236.34 236.34 172.84 172.84 144.56 144.56 121.742 0.1 152.60 152.60 87.00 87.00 60.83 60.83 44.54 44.54 33.693 0.15 82.29 82.29 40.98 40.98 23.49 23.49 14.15 14.15 9.124 0.2 49.90 49.90 20.50 19.70 10.40 8.49 5.09 3.83 2.455 0.25 46.70 31.75 13.39 9.29 5.31 3.43 2.39 1.45 0.90

10 1 0.1 189.84 189.84 93.63 93.63 65.20 65.20 46.84 46.84 34.692 0.2 63.38 63.38 20.80 20.80 10.34 8.38 4.93 3.60 2.283 0.3 44.95 25.96 8.11 3.81 1.51 0.64 0.27 0.11 0.054 0.4 41.07 11.11 2.17 0.51 0.11 0.03 0.01 0.01 0.015 0.5 36.07 4.51 0.42 0.11 0.05 0.02 0.01 0.00 0.00

20 1 0.05 357.79 357.79 243.84 243.84 176.38 176.38 147.22 147.22 123.552 0.1 166.51 166.51 92.00 92.00 62.93 62.93 45.30 45.30 33.923 0.15 93.68 93.68 43.17 43.17 24.12 24.12 14.22 14.22 9.064 0.2 57.85 57.85 20.32 20.32 10.24 8.32 4.93 3.62 2.305 0.25 46.70 37.44 13.20 8.94 4.27 2.49 1.30 0.72 0.39

10 0.5 36.91 4.35 0.27 0.07 0.03 0.03 0.03 0.03 0.03

1 1 1 14.09 14.09 14.09 14.09 0.53 0.53 0.24 0.24 0.022 1.41 0.71 23.84 23.84 3.58 1.23 0.52 0.05 0.05 0.01 0.003 1.73 0.58 30.47 9.28 2.28 0.57 0.19 0.03 0.02 0.00 0.004 2 0.5 34.95 15.92 4.55 0.56 0.43 0.07 0.02 0.01 0.005 2.24 0.45 38.03 6.24 6.24 1.12 0.24 0.10 0.04 0.01 0.00

10 3.16 0.32 44.45 18.77 8.01 4.27 2.28 0.98 0.39 0.15 0.06

Distribution

Normal

Table 2 (cont'd.)Maximum I% error! in mean (Approximate vs Exact)

A = A A > A > A > A> A> A> A>Distribution a c v 1 to 40 g/2 1.5g 2g 2.5g 3µ 3.5µ 4µ

Uniform

[8,12] 10 1.41 0.14 130.00 130.00 43.75 43.75 27.78 27.78 16.40 16.40 10.07

[7,13] 10 2 0.2 56.67 56.67 21.67 16.31 9.67 6.45 4.88 3.18 2.31

[5,15] 10 3.16 0.32 44.44 12.94 12.94 12.94 3.98 1.49 1.41 0.58 0.22

2 PointProb=(0.1,0.9)

(0,10) 9 3 0.33 NA NA NA NA NA NA NA NA NA(1,11) 10 3 0.3 4,355.03 4,355.03 4,355.03 2,127.55 2,127.55 1,385.01 1,385.01 1,013.75 1,013.75(2,12) 11 3 0.27 2,334.22 2,334.22 2,334.22 1,117.26 1,117.26 711.71 711.71 509.06 509.06(3,13) 12 3 0.25 1,663.21 1,663.21 1,663.21 782.22 782.22 488.89 488.89 342.54 342.54(4,14) 13 3 0.23 1,332.54 1,332.54 1,332.54 619.32 619.32 382.61 382.61 265.07 265.07(5,15) 14 3 0.2 I 1,163.23 1,163.23 1,163.23 547.48 547.48 344.85 344.85 245.34 245.34(6,16) 15 3 0.2 1,023.08 1,023.08 1,023.08 474.06 474.06 293.06 293.06 203.90 203.90(10,20) 19 3 0.16 926.32 926.32 926.32 467.03 467.03 316.17 316.17 242.39 242.39(11,21) 20 3 0.15 872.50 872.50 872.50 434.34 434.34 289.90 289.90 218.82 218.82

Prob=(0.5,0.5)(0,6) 3 3 1 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83 95.83(1,7) 4 3 0.75 180.00 180.00 180.00 180.00 52.77 52.77 52.77 52.77 27.40(2,8) 5 3 0.6 110.91 110.91 110.91 110.91 39.03 39.03 39.03 35.72 28.85(3,9) 6 3 0.5 116.67 116.67 116.67 116.67 57.58 57.58 57.58 50.72 48.31(4,10) 7 3 0.43 61.90 61.90 61.90 42.16 42.16 42.16 33.22 33.22 33.22(5,11) 8 3 0.375 103.13 103.13 103.13 62.50 62.50 62.50 47.73 41.30 39.78(7,13) 10 3 0.3 147.50 147.50 147.50 98.00 98.00 88.57 88.57 70.32 70.32(8,14) 11 3 0.27 116.36 116.36 116.36 96.69 96.69 69.70 69.70 43.05 43.05(9,15) 12 3 0.25 200.00 200.00 200.00 140.00 140.00 128.57 128.57 125.88 125.88(10,16) 13 3 0.23 111.54 111.54 95.27 95.27 95.27 47.16 47.16 42.01 42.01

Table 2 (cont'd.)Maximum I% errorl in mean (Approximate vs Exact)

A= A 0> e> e> A> A> A> A >Distribution 11 a c v 1 to 40 g/2 1.5g 2g 2.5g 3µ 3 . 5 g 4µ

Prob=(0.25,0.75)(1,5) 4 1.73 0.43 471.43 471.43 471.43 192.09 192.09 192.09 101.34 101.34 57.89(2,6) 5 1.73 0.35 308.89 308.89 308.89 131.58 131.58 76.80 76.80 76.80 52.45(3,7) 6 1.73 0.29 214.29 214.29 214.29 73.40 73.40 47.16 47.16 40.57 40.57(4,8) 7 1.73 0.25 328.57 328.57 328.57 174.29 174.29 128.10 128.10 108.36 108.36(5,9) 8 1.73 0.22 268.75 268.75 268.75 126.92 126.92 80.84 80.84 66.05 66.05(6,10) 9 1.73 0.19 233.33 233.33 233.33 100.50 100.50 70.67 66.93 66.93 66.93(7,11) 10 1.73 0.17 210.00 210.00 210.00 83.70 83.70 78.74 67.66 67.66 67.66(8,12) 11 1.73 0.16 193.51 193.51 193.51 91.12 88.92 88.92 77.75 77.75 77.75

Prob=(0.75,0.25)(1,5) 2 1.73 0.87 25.00 25.00 23.08 23.08 16.15 16.15 16.15 6.49 6.49(2,6) 3 1.73 0.58 60.00 60.00 60.00 60.00 60.00 57.38 51.22 51.22 51.22(3,7) 4 1.73 0.43 87.50 87.50 87.50 87.50 50.00 30.43 30.43 21.57 20.83(4,8) 5 1.73 0.35 206.67 206.67 206.67 206.67 188.63 188.63 188.63 187.57 187.57(5,9) 6 1.73 0.29 175.00 175.00 100.00 100.00 100.00 79.59 79.59 52.38 52.38(6,10) 7 1.73 0.25 221.43 221.43 65.90 65.90 65.90 64.57 64.57 52.66 52.66(7,11) 8 1.73 0.22 268.75 268.75 84.38 84.38 55.26 55.26 55.26 53.25 53.25(8,12) 9 1.73 0.19 316.67 316.67 108.33 108.33 70.94 70.94 60.64 60.64 57.56

Prob=(0.5,0.5)(7,13) 10 3 0.3 147.50 147.50 147.50 98.00 98.00 88.57 88.57 70.32 70.32(9,11) 10 1 0.1 355.00 355.00 127.50 127.50 102.22 102.22 73.33 73.33 58.26

Prob=(0.667,0.333)(9,12) 10 1.41 0.14 360.41 360.41 130.20 130.20 55.17 55.17 55.17 54.60 44.09

1,459.53332.57112.3642.8221.090.34

1,459.53332.57112.3642.8216.820.13

Table 3Maximum I% errorl in variance (Approximate vs Exact)

Lognormal

e= e>_ e>_ e>_

ila cv 1 to 40 g/2 il 1.5g

10 1 0.1 711.27 711.27 305.89 305.892 0.2 154.88 154.88 43.39 43.393 0.3 55.85 55.85 13.97 8.943.16 0.32 49.10 49.10 12.43 7.01

20

1

0.05 3,019.16 3,019.16 1,459.54 1,459.542

0.1 764.32 764.09 332.32 332.323

0.15 315.97 315.97 113.59 113.594

0.2 159.79 159.79 45.27 45.275

0.25 91.38 91.38 21.88 19.62

10 I 0.1 712.20 712.19 306.15 306.152 0.2 155.04 155.04 40.69 40.693 0.3 48.78 48.78 12.18 6.164 0.4 16.42 16.42 2.92 0.755 0.5 6.47 6.47 0.30 0.30

A � A> A> A> A >

2g 2.5g 311 3.5g 4g

172.44 172.44 108.68 108.68 73.63

17.10 16.40 8.63 6.79 4.07

3.60 1.82 1.09 0.68 0.38

2.83 1.45 1.03 0.55 0.26

939.68 939.68 679.76 679.76 523.82

190.02 190.02 121.61 121.61 83.37

54.05 54.05 29.40 29.40 17.11

17.44 17.03 9.07 7.32 4.41

8.41 5.58 2.92 1.75 0.96

172.19 172.19 108.48 108.48 73.29

16.02 14.88 7.90 6.14 3.71

2.36 1.02 0.42 0.18 0.07

0.16 0.16 0.16 0.16 0.16

0.10 0.04 0.01 0.00 0.00

Distribution

Normal

20 I 0.05 3,019.35 3,019.35

2 0.1 765.04 765.02

3 0.15 317.74 317.74

4 0.2 159.82 159.82

5 0.25 87.71 87.71

10 0.5 5.88 5.86

Poisson

939.61 939.61 679.66 679.66 523.74

189.73 189.73 121.33 121.33 83.11

53.01 53.01 28.66 28.66 16.62

16.94 15.90 8.47 6.65 4.03

7.26 4.44 2.26 1.27 0.68

0.13 0.12 0.10 0.08 0.06

1 1 1 11.79 11.79 11.79 11.79 0.24 0.242 1.41 0.71 16.09 16.09 1.65 1.65 0.31 0.083 1.73 0.58 15.44 4.54 4.54 0.35 0.35 0.044 2 0.5 11.60 8.10 5.70 0.90 0.20 0.125 2.24 0.45 11.64 11.64 4.93 1.35 0.36 0.07

10 3.16 0.32 42.63 42.63 10.34 4.85 1.74 0.72

0.24 0.24 0.01

0.04 0.01 0.01

0.04 0.00 0.00

0.02 0.01 0.00

0.05 0.01 0.00

0.37 0.21 0.10

Table 3 (cont'd.)Maximum I% errorl in variance (Approximate vs Exact)

A= e >_ A>

A>_ A>_ A> A> A>

A>_Distribution 11 a c v 1 to 40 p/2 11 1.5p 2p 2.5p 3p 3.54 4 µ

Uniform

[8,12] 10 1.41 0.14 362.00 362.00 131.00 131.00 59.27 59.27 31.37 31.37 18.60

[7,13] 10 2 0.2 155.25 155.25 36.83 36.83 14.33 11.33 6.43 4.61 2.79

[5,15] 10 3.16 0.32 30.00 30.00 12.66 9.95 5.21 2.28 1.74 0.51 0.30

2 PointProb=(0.1,0.9)

(0,10) 9 3 0.33 NA NA NA NA NA NA NA NA NA(1,11) 10 3 0.3 8,120.67 8,120.67 8,119.80 4,010.59 4,010.32 2,640.83 2,639.99 1,955.86 1,955.84(2,12) II 3 0.27 2,384.01 2,384.01 2,361.26 1,155.11 1,136.54 748.09 728.74 545.84 545.83(3,13) 12 3 0.25 1,262.74 1,262.74 1,201.64 610.45 561.10 394.23 350.55 285.77 285.76(4,14) 13 3 0.23 866.91 866.90 748.01 395.98 377.26 239.42 239.40 172.44 172.44(5,15) 14 3 0.21 709.08 709.08 562.34 343.24 252.35 217.25 152.26 152.16 112.20(6,16) 15 3 0.20 581.80 581.80 421.59 288.89 189.49 189.49 138.41 138.41 138.41(10,20) 19 3 0.16 305.14 305.14 305.14 143.33 143.33 92.14 92.14 68.43 68.43(11,21) 20 3 0.15 305.55 305.55 268.68 145.79 121.93 95.47 75.81 72.22 59.55

Prob=(0.5,0.5)(0,6) 3 3 1 80.84 80.84 80.84 80.84 80.84 80.84 80.84 44.68 44.68(1,7) 4 3 0.75 226.14 226.14 226.14 208.60 89.07 89.07 89.07 72.59 44.51(2,8) 5 3 0.6 93.70 93.70 93.70 69.38 27.09 27.09 27.09 14.01 14.01(3,9) 6 3 0.5 52.31 52.31 52.31 52.31 22.62 22.62 22.62 14.69 14.69(4,10) 7 3 0.43 70.25 70.25 70.25 23.30 23.30 23.30 8.85 8.85 8.85(5,11) 8 3 0.375 71.52 71.52 71.52 37.22 37.22 37.22 27.37 27.37 22.90(7,13) 10 3 0.3 47.62 47.62 47.62 39.42 39.42 36.80 36.80 33.74 33.74(8,14) II 3 0.27 59.25 59.25 44.23 44.23 44.23 30.71 30.71 22.14 22.14(9,15) 12 3 0.25 61.73 61.73 61.73 41.14 41.14 37.55 37.55 36.73 36.73(10,16) 13 3 0.23 104.22 104.22 42.07 42.07 42.07 20.53 20.53 11.35 11.35

Table 3 (cont'd.)Maximum I% errorl in variance (Approximate vs Exact)

e= A> A> A> A> A> A> A >Distribution a c v 1 to 40 g/2 1.5g 21.1 2.5g 3µ 3.5µ 4µ

Prob=(0.25,0.75)(1,5) 4 1.73 0.43 471.43 462.50 445.45 203.77 203.77 169.13 124.57 124.57 83.47(2,6) 5 1.73 0.35 221.07 221.07 154.31 85.60 85.60 41.92 41.92 36.72 25.78(3,7) 6 1.73 0.29 168.06 168.06 168.06 81.30 81.30 51.62 51.62 36.30 36.30(4,8) 7 1.73 0.25 112.04 112.04 112.04 55.95 55.95 41.56 41.56 36.34 36.34(5,9) 8 1.73 0.22 115.49 115.49 72.40 64.19 49.00 49.00 40.65 40.65 40.65(6,10) 9 1.73 0.19 163.89 163.89 89.23 89.23 59.61 59.61 41.62 41.62 40.06(7,11) 10 1.73 0.17 217.58 217.58 109.54 109.54 64.91 64.91 41.22 41.22 32.86(8,12) 11 1.73 0.16 276.65 276.65 124.76 124.76 64.76 64.76 36.23 36.23 36.23

Prob=(0.75,0.25)(1,5) 2 1.73 0.87 27.08 27.08 27.08 24.90 24.90 24.90 7.33 7.33 7.33(2,6) 3 1.73 0.58 21.79 21.79 21.79 21.79 21.79 11.43 11.43 11.43 11.43(3,7 ) 4 1.73 0.43 38.19 38.19 38.19 38.19 18.45 17.93 17.93 12.76 9.24(4,8) 5 1.73 0.35 56.31 56.31 56.31 56.31 49.27 49.27 49.27 48.86 48.86(5,9) 6 1.73 0.29 56.25 56.25 56.25 56.25 53.85 47.47 47.47 43.78 39.46(6,10) 7 1.73 0.25 91.33 91.33 45.77 43.64 43.64 27.61 27.61 18.28 14.35(7,11) 8 1.73 0.22 132.16 132.16 32.66 28.09 28.09 23.82 23.82 18.81 18.81(8,12) 9 1.73 0.19 178.70 178.70 39.35 39.35 22.80 22.80 19.06 19.06 18.02

Prob=(0.5,0.5)(7,13) 10 3 0.3 47.62 47.62 47.62 39.42 39.42 36.80 36.80 33.74 33.74(9,11) 10 1 0.1 774.75 774.75 337.38 337.38 191.58 191.58 118.69 118.69 77.30

Prob=(0.667,0.333)(9,12) 10 1.41 0.14 365.46 365.46 132.73 132.73 55.15 55.15 32.08 32.08 19.32

Figure 1: Sample Undershoot Probabilities

Dmdl Prob. U I A = 11 A = 21 A = 31 A = 41 A = 51 A = 61 A = 71 A = 81 A = 91 A = 101 A = 111 A = 120 0 0 0 () 0 0.2 0 0 0.8 0.04 0 0 0.32 0.008,

01 0 1 0 0 0.2 0 0 0.8 0.04 0 0 0.32 0.0082 0 2 0 0.2 0 0 0.8 0.04 0 0 0.32 0.008 0 0.643 0 3 0.2 0 0 0.8 0.04 0 0 0.32 0.008 0 0.64, 0.0964 0.2 4 0 0.8 0 0 0 0.16 0 0 0.64 0.032 05 0 5 0 0.8 0 0 0 0.16 0 0 0.64 0.032 0 06 0 6 0.8, 0 0 0 0.16 0 0 0.64 0.032 0 0 0.2567 0.8 7 0 0 0 0 0 0 0 0 0 0 0 08 0 8 0 0 0 0 0 0 0 0 0 0 0

9 T

Figure 2:Approximate vs. Exact Mean

Normal (Mean = 10, Standard Deviation = 1)

8 --11

7 —1 It1 ii

I \I I°\ A6 — I I ' I ■ / 1 t"

1 1 1 1 1 / 1 1 \ /\% iN pvi 5

"".. 1 I I 1 11 j 1 I k I./ \ i N / \ /N.... 1 1 1 I i i / •.-

c I 1 i 1 X I i / 1k / \ I\4-1 I 1 1 1 1 %/ xi, x 1 s.... v • -

/ /

I1 1 I / 1 % I %%0/ ‘i

I - /1 %3 — 1 i / /

11 LI2 — %I

■

—

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C■I CI) co Ti* LI) CD CD N. F■ CO CO 0) 0)

Order Size

Approximate

Exact

Approximate

--- ExactiN


Normal (Mean = 10, Standard Deviation = 3)

1 0

9 --

8 --i1

•7 —1

I

in"--c...

m

6

5

— II1

4 —/

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3 —

2 —

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co CD IP.' (0 •-• CO v- CD /- (C) V". CD 1- VD .- CD v- CDN N 0") (v) V' It LO U) CD CD N- f■ CO CO 0) 0)

Order Size

2.25 —

2 .2 —

2.15 —

U)4..*c- 2.1 —

n

2.05 —

2 .INNINI.

ii- Approximate

—C3-- Exact

Figure 4:Approximate vs. Exact Mean: Gamma (a = 3, X = 1)

1.95 I

1

i i i

0 2

4

6

8 10

Order Size

a

II — • 11..11 •

■ ■.■ S.. • •••• -.41

■

250

200

150U)

••••■cm

100

50

Figure 5:

Approximate vs. Exact Mean: Gamma (a = 9, A. = 27)

6- Approximate

-°---- Exact

0 100 200 300 400 500 600

Order Size


Two Mass Point (6 w.p. 0.5,12 w.p. 0.5)

8 T

7 —1 Ik I I 1 I I1 1 1 jkI I 11 - 1 11 it 1 PI11 II 11 i it

6 _ 1III 1 I 1

III

11

II1

1I I

Iit II II

IIIII I

11I

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1 I 11 1 1

1 1 1

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11 I

1 I 11I I

1 1 1 11, 1 1 t1 I 1 1

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— I

1 1 I

1 It 11 I

1 11 I

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111 1 1 I I 1

1 I 11 1 1 1 1 I

1 1 1 1 I 1 1

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1 1 I 1 I

11 1 Id It III 1

1 1 1 1 I11 1 11 III I 1 1 1 1 1

3 — I 1 11 11 i• t I I 1 1 11 11 1 1I II 11 ' 1 11 1 1 II 1 1

II I V 1 1 1 1J I VI I 1 1 1 1 1 I I 1 i 1 i I I2 —1(

Approximate

---- Exact

1 MOM

CD T- (C) (f) (j) (C) (0

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Order Size


Two Mass Point (1 w.p. 0.5, 7 w.p. 0.5)

A "I % / ‘ P \ N '"*- .... ...- ....... ....• -

I V/ --..

Approximate

— — — Exact

0.5 —

1". (0 T CO T. (0 •.. CO 1' CO w' (0 •-- CD .- co .- (D 1,-.. CD1- N CV (1) (Y) wt lzt to (0 CO (0 f`■ N-

CO CO 0) CY)

Order Size

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