M.P. BAGANHA*by D.F. PYICE**

G. FERRER***and

95/03/TM

* Universidade Nova de Lisboa, Lisbon, Portugal.

** Amos Tuck School of Business Administration, Dartmouth College, Hanover, New Hampshire, 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 a researcher's thoughts and findings may be communicated to interested readers. The paper should 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. Baganha Universidade Nova de Lisboa

David F. Pyke

Amos Tuck School of Business Administration Dartmouth College

Geraldo Ferrer INSEAD

January 2, 1995

ABSTRACT

We investigate a widely used approximation for the mean and variance of the undershoot. The approximation is based on the limit of the excess random variable of a renewal process as the order size approaches infinity. In the current business environment which emphasizes small batch sizes and frequent deliveries, many inventory systems order in batch sizes that are not large. We investigate the potential error that could be introduced by using the approximation for a variety of batch sizes and demand distributions.

**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 g reater than 1.5 times the mean demand. We will see below that for certain cases our results t.

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.

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.

0-h ^ 2 = 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}** 1

Ph_ 2p. 2 (1)

**6**

.. 2 *E(X*

*3*

*)*

*cr*

*2*

*+11*

*2 12*

*1*Ch =

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, ...

p

with mean 1 – and variance 1–p P2

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

**C5****2 **

+ **p.**2

_[ 1–p 4. (1–p)2

]/(2(i_p_, 1 1–p

11 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 > *1

SO

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

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

**References**

Baganha, M.P., Pyke, D.F. and Ferrer, G., 1994. "The Residual Life of the Renewal Process: 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 Research*
*Logistics, *Vol. 39, pp. 561-577.

Ernst, R., and Pyke, D.F., 1992. "Component Part Stocking Policies," *Naval Research*
*Logistics, *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 *Inventory
Model," *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 of*
*the Operational Research Society, *Vol. 39, No. 2, pp. 173-181.

Karlin, S., 1958. "The Application of Renewal Theory to the Study of Inventory

Policies," in Ch. 15, Studies *in the Mathematical Theory of Inventory and Production,*
K. Arrow, S. Karlin and H. Scarf (Eds.), Stanford, California: Stanford University
Press.

McConalogue, D.J., 1981. "An Algorithm and Implementing Software for Calculating Convolution 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 and*
*Optimization, *New York: Springer-Verlag.

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

Silver, E.A., and Peterson, R., 1985. *Decision Systems for Inventory Management and*
*Production 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 in Periodic and Continuous Review (.5,․) Inventory Systems with Service Level

Constraints," *European Journal of Operations Research, *Vol. 17, pp. 175-192.

Whitt, W., 1984. "On Approximations for Queues, I: Extremal Distributions," AT&T
*Bell 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 Simple as Evaluating a Single Policy"Operwions Research, Vol. 39, No. 4, pp. 654-665.

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

**j.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, 0.9 1, 11 with probabilities 0.1, 0.9 10,21 with probabilities 0.1, 0.9 Two Point 0, 6 with probabilities 0.5, 0.5

1, 7 with probabilities 0.5, 0.5 10, 16 with probabilities 0.5, 0.5 Two Point 1, 5 with probabilities 0.25, 0.75 2, 6 with probabilities 0.25, 0.75 8, 12 with probabilities 0.25, 0.75

1, 5 with probabilities 0.75, 0.25 2, 6 with probabilities 0.75, 0.25 8, 12 with probabilities 0.75, 0.25

Two Point 7, 13 with probabilities 0.5, 0.5 9, 11 with probabilities 0.5, 0.5 9, 12 with probabilities 0.667, 0.333

**Table 1**

**Experimental Design**

**Table 2**

**Maximum 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 34.19

2 0.2 53.90 53.90 21.40 20.72 10.59 8.50 5.19 3.87 2.85

3 0.3 45.01 22.64 12.44 6.99 4.36 2.05 0.97 0.46 0.22

3.16 0.32 44.49 19.94 12.00 6.45 3.65 1.51 0.70 0.31 0.13

20 1 0.05 348.51 348.51 236.34 236.34 172.84 172.84 144.56 144.56 121.74

2 0.1 152.60 152.60 87.00 87.00 60.83 60.83 44.54 44.54 33.69

3 0.15 82.29 82.29 40.98 40.98 23.49 23.49 14.15 14.15 9.12

4 0.2 49.90 49.90 20.50 19.70 10.40 8.49 5.09 3.83 2.45

5 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 34.69

2 0.2 63.38 63.38 20.80 20.80 10.34 8.38 4.93 3.60 2.28

3 0.3 44.95 25.96 8.11 3.81 **1.51** 0.64 0.27 0.11 0.05

4 0.4 41.07 11.11 2.17 0.51 0.11 0.03 0.01 0.01 0.01

5 0.5 36.07 4.51 0.42 0.11 0.05 0.02 0.01 0.00 0.00

20 1 0.05 357.79 357.79 243.84 243.84 176.38 176.38 147.22 147.22 123.55

2 0.1 166.51 166.51 92.00 92.00 62.93 62.93 45.30 45.30 33.92

3 0.15 93.68 93.68 43.17 43.17 24.12 24.12 14.22 14.22 9.06

4 0.2 57.85 57.85 20.32 20.32 10.24 8.32 4.93 3.62 2.30

5 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 0.02

2 1.41 0.71 23.84 23.84 3.58 1.23 0.52 0.05 0.05 0.01 0.00

3 1.73 0.58 30.47 9.28 2.28 0.57 0.19 0.03 0.02 0.00 0.00

4 2 0.5 34.95 15.92 4.55 0.56 0.43 0.07 0.02 0.01 0.00

5 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**

**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 Point**

**Prob=(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

1,459.53 332.57 112.36 42.82 21.09 0.34

1,459.53
332.57
112.36
42.82
16.82
0.13
**Lognormal**

e= e>_ e>_e>_

ila ^{cv} ^{1 to 40}^{g/2} _{il} **1.5g**

10 1 0.1 711.27 711.27 305.89 305.89

2 0.2 154.88 154.88 43.39 43.39

3 0.3 55.85 55.85 13.97 8.94

3.16 0.32 49.10 49.10 12.43 7.01
20 1 0.05 3,019.16 3,019.16 1,459.54 1,459.54
**2** 0.1 764.32 764.09 332.32 332.32
**3** 0.15 315.97 315.97 113.59 113.59
**4** 0.2 159.79 159.79 45.27 45.27

**5** 0.25 91.38 91.38 21.88 19.62

10 I 0.1 712.20 712.19 306.15 306.15

2 0.2 155.04 155.04 40.69 40.69

3 0.3 48.78 48.78 12.18 6.16

4 0.4 16.42 16.42 2.92 0.75

5 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 0.24

2 1.41 0.71 16.09 16.09 1.65 1.65 0.31 0.08

3 1.73 0.58 15.44 4.54 4.54 0.35 0.35 0.04

4 2 0.5 11.60 8.10 5.70 0.90 0.20 0.12

5 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 4µ

**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 **Point**

**Prob=(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

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 = **12

0 0 0 0 () 0 0.2 0 0 0.8 0.04 0 0 0.32 0.008,

1 0 1 0 0 0.2 0 0 0.8 0.04 0 0 0.32 0.008 0

2 0 2 0 0.2 0 0 0.8 0.04 0 0 0.32 0.008 0 0.64

3 0 3 0.2 0 0 0.8 0.04 0 0 0.32 0.008 0 0.64, 0.096

4 **0.2** 4 0 0.8 0 0 0 0.16 0 0 0.64 0.032 0

5 0 5 0 0.8 0 0 0 0.16 0 0 0.64 0.032 0 0

6 0 6 0.8, 0 0 0 0.16 0 0 0.64 0.032 0 0 0.256

7 **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}

Normal (Mean = 10, Standard Deviation = 1)

8

### --1

### 7 —1 1

^{It}

1 ii

### I \I

I°\ A### 6 — I I ' I ■ / 1 t"

### 1 1 1 1 1 / 1 1 \ /\% iN p

### vi 5 ""..

1 I I 1 11 j 1 I / \ i kN / I.\ /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 • -

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### I 1 1 I / 1 1 I % I - / % %%0/ ‘i

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### 11 LI

### 2 — %I

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CD CD 1- CO 1- (1) •-• **CO** **(D** **CD**

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

### Order Size

### Approximate

### Exact

**Approximate**

--- **Exact**
iN

**Figure 3:**

**Approximate vs. Exact Mean**

**Normal (Mean = 10, Standard Deviation = 3)**

**1** **0**
**9 --**
**8 --i**1

• 7 —1I

**in**

**"--c**
**...**

m 6

**5**

— II1
**4 —**

N //

N"

**3 —**
**2 —**
**1 —**

co ^{CD }^{IP}^{.}** ^{' }**(0 •-• CO v- CD /- (C) V". CD 1- VD

**.-**CD v- CD N N 0") (v)

**V' It**LO U) CD CD N- f■ CO CO 0) 0)

**Order Size**

2.25 —

**2**.2 —

2.15 —

4..U)

*c- 2.1 —

## n

2.05 —

2 ^{.INNINI.}

ii- Approximate

—C3-- Exact

1.95 I ^{1}^{ } i

## i i

0 2 4 6 8 10

**Order Size**

a II — • 11..11 •

■ ■.■ S.. • •••• -.41

■

250

200

150

U)

••••■

mc

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**