Proceedings of the 2011 Industrial Engineering Research Conference T. Doolen and E. Van Aken, eds.

Zero-Modified Distributions for Inventory Control under Intermittent Demand Yasin Ünlü, Manuel D. Rossetti, Ph. D., P. E. Department of Industrial Engineering, University of Arkansas 4207 Bell Engineering Center, Fayetteville, Arkansas 72701, USA Abstract Zero-modified distributions were proposed in the literature due to the need to better characterize count data with excess number of zeros collected from economic series, agricultural surveys, manufacturing processes etc. This paper introduces the applicability of the zero-modified distributions to the lead time demand modeling for the continuous review (r, Q) inventory system. Such distributions may be especially suitable for intermittent demand due to their capability for explicitly modeling non-zero and zero demand cases. In this respect, zero-modified distributions are used for approximating the performance measures of ready-rate, the expected number of backorders and inventory on-hand levels. In the experiments, the accuracy of the approximations are compared with the results of simulation.

Keywords Zero-modified distribution, inventory performance measure, intermittent demand, continuous review (r, Q) inventory system.

1. Introduction Count data with excess zeros are common in many applications such as economic analysis, agricultural surveys, manufacturing processes etc. Zero-modified distributions were proposed in the literature due to the need to better characterize this type of data. Another name used for such distributions is known as zero-inflated distribution. [1] provides methods constructing these distributions in the context of generalized linear models. [2] presents a literature review for these distributions by providing a variety of examples from different disciplines. A number of examples for different application areas can be obtained in [3-7]. The study in this paper investigates the applicability and the potential use of zero-modified distributions in modeling lead-time demand in the face of intermittent and highly variable demand. The motivation behind considering such distributions is because these distributions are capable of explicitly modeling zero and nonzero demands, which actually is required in real life applications. The developed model should be able to determine particular service levels for a fully specified policy. It is of interest to see whether the new approach applied in this paper is promising in terms of contributing to the classical inventory management. This paper examines the use of standard (r, Q) inventory control policies under a number of lead-time demand models, namely, zero-modified lognormal (ZMLN), zero-modified Poisson (ZMP), zero-modified negative binomial (ZMNB). Three operational performances are considered: ready rate (RR), the expected number of backorders (BO) and on hand inventory levels (I). The ready rate is the fraction of time with positive stock on hand ([8], page 94). The expected number of backorders can be defined as the long-run average number of backordered demands. On hand inventory levels refer to the long-run average number of inventory units on hand. Under the assumption that F is the lead-time demand model, RRF , BF and I F represent the ready rate, expected number of backorders and on hand inventory levels, respectively. The following general formulations are used in order to evaluate the desired service levels:  1 1 GF (r) − G1F (r + Q) Q  1 2 BF = G (r) − G2F (r + Q) Q F

RRF = 1 −

IF =

1 (Q + 1) + r − µ + BF 2

(1) (2) (3)

Ünlü and Rossetti where µ, r and Q are the mean of the LTD, the re-order point and the order quantity while G1F (.) and G2F (.) are the first and second order loss functions of F, respectively. Expressions (1), (2) and (3) along with the loss functions of F are used to calculate inventory performance measures. The term 12 (Q + 1) in expression (3) is replaced with 12 Q in the case where F has a continuous distributional form. The reader is referred to [9] (page 188) for details and further explanation of the formulations. The next section provides zero-modified distributions, loss functions and parameter fitting procedure.

2. Zero-Modified Distributions Zero-modified distributions were originally derived due to the need to better characterize data sets that have an excess number of zeroes, when considering a standard distribution (e.g. Poisson) with the same mean. The analysis utilizes the degenerate distribution with all probability concentrated at the origin (zero point in the axis). Let Pj be the original discrete distribution where j = 0, 1, 2, ... and, then random variable Y is defined by the following finite-mixture distribution.  Pr [X = j] = w + (1 − w) P0 i f j = 0 Y= (4) Pr [X = j] = (1 − w) Pj if j ≥ 1 The mixture distribution (4) is referred to as a zero-modified distribution or as a distribution with added zeros [10]. 0 −P0 where f0 = Pr [X = 0] since the probability of Notice that parameter w is easily computed by the expression w = f1−P 0 observing zero is often known in experimental investigations whereas estimating the probability of non-zero is of interest. Based on the parameter w, the next section describes how we calculate the first and second order loss functions of a zero-modified distribution. 2.1 Loss Functions Let G1ZM and G2ZM be the first and second order loss functions of a zero-modified distribution, respectively. In addition, let G1O and G2O be the first and second order loss functions of the original probability distribution, respectively. Then, we calculate G1ZM and G2ZM as follows: G1ZM = w + (1 − w) G1O

(5)

G2ZM = w + (1 − w) G2O

(6)

where w is the previously defined parameter. Certain parameter derivations are necessary in order for a zero-modified distribution to be employed by classical inventory policies as a lead time demand model under intermittent and highly variable demand. In the the next section, we provide a methodical approach for the parameter fitting procedure. 2.2 Parameter Fitting Procedure The scope of the parameter fitting procedure is to estimate the parameters of the original distribution (i.e. θ1 , θ2 , ..., θm ) so that the first and second order loss functions of the resulting zero-modified distribution can be calculated. The procedure basically takes two steps: (1) Step 1: This step is carried out to estimate the parameter w. First, µ and σ2 are estimated by ignoring the observed frequency in the zero class. A regular procedure (e.g. method of moment matching) may be applied to estimate the original parameters of θ1 , θ2 , ..., θm . This leads to the estimation of the parameter w after estimating the probability of expecting zero value (P0 ) based on θ1 , θ2 , ..., θm and the probability of observing zero value ( f0 ) based on the given data (or other source of information). (2) Step 2: The parameters of θ1 , θ2 , ..., θm are updated by taking into account the observed frequency in the zero class. An algorithm of the parameter fitting procedure is given as follows:

Ünlü and Rossetti Algorithm 1 Parameter fitting procedure of the zero-modified distribution 1) Estimate µ, σ2 by ignoring the observed frequency in the zero class, 2) Match µ, σ2 to θ1 , θ2 , ..., θm by using method of moment matching technique, 2) Calculate P0 = f o (0|θ1 , θ2 , ..., θm ), zero count 3) Estimate f0 = , N ← total number of data points in a data series, N f0 − P0 4) Calculate w = , 1 − P0 2 5) Update µ = E [X] , σ = Var [X] by taking into account the observed frequency in the zero class, 6) Update θ1 , θ2 , ..., θm through fitting to µ and σ2 by using the method of moment matching technique.

The parameters of the original distribution can be updated by exploiting the fact that the probability generating function (pg f ) of a zero-modified distribution is derived from that of the original distribution. Let HO (z) and HZM (z) be the pg f of the original distribution and pg f of the zero-modified distribution, respectively. Then, the following holds [10]. HZM (z) = w + (1 − w) HO (z)

(7)

The expression (7) points out an important property in regard to the parameter estimation of the zero-modified distribution. Since the pg f of the original distribution is often known, one can have pg f of the zero-modified distribution after estimating the parameter w as described previously. The probability generating function of the zero-modified distribution facilitates the derivation of expressions for the moments. After these expressions are derived, the parameters of the original distribution can then be updated based on the method of moment matching. This procedure will be detailed in the next section where the zero-modified Poisson distribution is introduced. Zero-Modified Poisson Distribution (ZMP): Let λ be the parameter of the original distribution (i.e. Poisson). The random variable Y characterizing the zeromodified Poisson distribution is defined by the following finite-mixture distribution.   Pr [X = 0] = w + (1 − w) e−λ Y= (1 − w) e−λ λ j  Pr [X = j] = , j = 1, 2, ..., j! In addition, let HZMP (z) be the pg f of the zero-modified Poisson distribution. From (7), clearly, HZMP (z) = w + (1 − w) eλ(z−1)

(8)

By using (8), the expression to update λ can be derived as follows. The first raw moment of the zero-modified distribution is obtained by (1)

E [X] = HZMP (z = 1) . It follows that (1)

HZMP (z = 1) = (1 − w) λeλ(1−1) E [X] = (1 − w) λ. In most cases, λ is estimated as the average of the data. The loss functions of zero-modified Poisson distribution given in expressions (5) and (6) are then calculated using the loss functions of the Poisson distribution based on the updated parameter of µ. The parameter fitting procedure follows the similar fashion for other distributions. In what follows, we illustrate two other distributions and their fitted parameters expression.

Ünlü and Rossetti Zero-Modified Lognormal Distribution (ZMLN): The general form for the zero-modified distribution is given by ( w, if x = 0 h (x) = (1 − w) f (x) , i f x 6= 0 where h (.) denotes the probability density function (pdf) of the zero-modified distribution, f (.) is the pdf of the original distribution and w denotes the proportion of the population with the value of 0. The mean and variance of the zero-modified distribution are derived by [11] as follows: E [X] = (1 − w) µ Var [X] = (1 − w) σ2 + w (1 − w) µ2 Zero-Modified Negative Binomial Distribution (ZMNB): Zero-modified negative binomial distribution can be written as  k if x = 0  w + (1 − w)t , ! h (x) = x+k−1  t k (1 − t)x , i f x 6= 0 (1 − w) x where t =

k k+µ .

The mean and the variance of the ZMNB are derived by [6] as follows: E [X] = (1 − w) µ Var [X] = (1 − w) (1 + µ/k + wµ) µ

3. Experimental Study In order to interpret the quality of these approximations, they must be compared with the most accurate estimations. The simulation model is considered to generate the most accurate results as long as its set up is done appropriately. Also, for even more complicated demand processes, the inventory modeler can reach the most accurate estimations with a simulation model. The intermittent demand can be considered as one of the most complex demand processes due to demand transaction (i.e. occurrence) and demand size variability. This requires that the results be compared with the simulation results. In the simulation experiments, the intermittent and highly variable demand is generated through the special demand generator called a continuous time batch-on/off demand generator. The details related to the demand generator can be obtained in [12]. This special demand generator feeds the simulation by generating intermittent demand series. The simulation model processes the generated discrete demand series in a special method in order to be consistent with the underlying theory of the continuous review (r, Q) inventory system. Since the generated demand is in discrete units, we can apply the following techniques: 1) Demand splitting and 2) First-come-first-served ordering system. In doing so, the generated demand is processed in individual units even if it arrives in sizes greater than 1. Therefore, the received and backordered demands can be partially or fully satisfied from the available stock on-hand. This method provides comparable results of inventory performance measures with the analytical formulas (1), (2) and (3). The details related to the structure of the simulation model can be obtained in [12]. The simulation run-length has to be set with an appropriate run-length, warm-up period and replication numbers so that the required accuracy is gained. In the experiments, the simulation model for each test case was run for 30 replications with 100,000 time units of warm-up and 1,000,000 time units of run-time. For the estimated performance measures, the given set up provides at least 2-digit-precision on the decimal point. The performance measures of ready-rate, the expected number of backorders and inventory on-hand levels are monitored during the simulation. The results are interpreted over error results. Error in the experimental results refers to the difference between simulation results and analytical results. The associated error statistics are collected over test cases. A test case refers to a vector of (r, Q, µ, σ) and demand generator parameter values. Re-order point (r), re-order quantity (Q), the mean (µ) and the standard deviation (σ) of the LTD are given as input parameters in order to calculate the inventory performance measures through the analytical formulas. Test cases are generated based on the algorithm given in [12]. In order to

Ünlü and Rossetti capture the associated error statistics, we define the variables E Sj and RE Sj as the error and absolute relative error for test case j. Let θSj be the value of a performance measure estimated by the simulation model for test case j and let θaj be the value of a performance measure approximated for test case j under the assumption that the LTD model is Fa . Under the given notation, error statistics are obtained for each distribution by using the following expressions. E Sj = θSj − θaj S θSj −θaj RE j = θS j

a ∈ {Fa : ZMP, ZMLN, ZMNB}

(9)

a ∈ {Fa : ZMP, ZMLN, ZMNB}

(10)

The error statistics are collected across all the generated test cases through the the algorithm given in [12]. In experiments, a total of 2,200 test cases are used to record the statistics. The error results are presented in Table 1, Table 2, Table 3 and Table 4. The tables compare a number of error statistic results of the LTD models (LN, NB and P) and zero-modified versions of these models (ZMLN, ZMNB and ZMP). In addition, as pointed out by [12] the best LTD model, ADR, is also considered in the experiments in order to highlight the quality of approximations of zero-modified distributions. The reader is referred to the paper for further details of ADR. The statistics related to the probability that the absolute relative error is less than or equal to 0.10 (P(%10)), 0.05 (P(%5)) and 0.01 (P(%1)) are given in Table 1 and Table 2. The statistics of P (%) provide a reliability measure for the use of the LTD model. Table 3 and Table 4 present descriptive error statistics; namely, mean (Mean), standard deviation (Std), the minimum value (Min), first quantile (Quantiles25), median (Median), third quantile (Quantiles75) and the maximum value (Max). Table-1: P(%) Value Results for Models ADR

LN

NB

P

BO

FR

I

BO

FR

I

BO

FR

I

BO

FR

I

P(%10)

0.18

1

0.96

0.03

1

0.81

0.18

1

0.97

0.15

0.97

0.85

P(%5)

0.06

0.96

0.83

0

0.97

0.36

0.06

0.96

0.83

0.13

0.87

0.63

P(%1)

0.01

0.61

0.13

0

0.74

0

0.01

0.6

0.13

0.05

0.37

0.06

Table-2: P(%) Value Results for Zero-Modified Models ZMP

ZMLN

ZMNB

BO

FR

I

BO

FR

I

BO

FR

I

P(%10)

0.86

0.22

0.96

0.72

0.26

0.93

0.91

0.32

0.95

P(%5)

0.65

0.14

0.88

0.32

0.12

0.83

0.69

0.22

0.86

P(%1)

0.07

0.03

0.48

0

0.04

0.44

0.13

0.04

0.53

Table-3: Descriptive Statistical Results for Models ADR

LN

NB

P

Statistics

BO

FR

I

BO

FR

I

BO

FR

I

BO

FR

I

Mean

-0.077

0.009

0.361

-0.157

0.005

0.781

-0.077

0.01

0.361

0.145

-0.01

0.905

Std

0.096

0.01

0.271

0.143

0.01

0.276

0.096

0.01

0.271

0.316

0.026

0.319

Min

-0.469

-0.002

-0.064

-0.817

-0.015

0.171

-0.462

-0.002

-0.064

-0.367

-0.111

0.477

Quantiles25

-0.092

0.003

0.188

-0.201

0

0.616

-0.091

0.004

0.187

0.012

-0.026

0.688

Median

-0.039

0.006

0.279

-0.119

0.004

0.741

-0.04

0.007

0.279

0.063

-0.005

0.794

Quantiles75

-0.016

0.013

0.474

-0.059

0.008

0.92

-0.017

0.012

0.473

0.161

0.006

1.053

Max

0.002

0.069

1.532

-0.005

0.063

1.908

0.002

0.068

1.531

2.087

0.083

1.918

Table-4: Descriptive Statistical Results for Zero-Modified Models ZMP

ZMLN

ZMNB

Statistics

BO

FR

I

BO

FR

I

BO

FR

I

Mean

0.116

-0.007

0.554

-0.033

-0.019

0.905

0.035

-0.014

0.473

Std

0.194

0.024

0.451

0.172

0.025

0.319

0.195

0.023

0.34

Min

-0.116

-0.087

0.005

-0.739

-0.101

0.477

-0.486

-0.1

0.003

Quantiles25

0.011

-0.017

0.275

-0.078

-0.03

0.688

-0.029

-0.023

0.237

Median

0.051

-0.004

0.429

-0.041

-0.014

0.794

-0.004

-0.006

0.354

Quantiles75

0.134

0.004

0.677

-0.015

0

1.053

0.027

0.002

0.626

Max

1.052

0.083

2.178

0.783

0.017

1.918

1.108

0.018

1.673

Ünlü and Rossetti

The results indicate that in almost all the cases under ADR, LN, NB and P, the error values associated with BO are higher than those of RR and I. In fact, these models produce poor results in terms of approximating BO. As can be seen from Table 3 and Table 4, in most of the cases, the performance measure of BO is overestimated by all the models except P. P(%) statistics show that these models yield much better results in terms of approximating the performance measures of RR and I as compared to BO. For the same performance measures, they also produce better results than zero-modified distributions. In order to discern the quality of zero-modified distributions in approximating performance measures, the results should be read by comparing Table 1 and Table 2 or Table 3 and Table 4. The tables reveal the superiority of zero-modified distributions in approximating BO. One can see a distinct improvement in the associated BO results by comparing P(%) values in Table 1 and Table 2. For example, the best LTD model, ADR, proposed in [12], gives only %18 for P(%10) statistics. This means that only %18 of the time absolute error results will be within 10% of the true performance. However, for the same statistic measure, it is possible to improve the BO results up to %91 through ZMNB. In addition, as far as BO results are concerned, P(%) results indicate that zero-modified versions of distributions increase the performance of P, LN and NB. For example, LN gives only 0.03% for P(%10) statistic while ZMLN increases the same results to 72%.

3. Conclusion In this paper, we propose a method for approximating inventory performance measures under zero-modified distributions for the continuous review (r, Q) inventory systems. The method can easily be extended to other inventory systems under appropriate approximation formulas instead of (1), (2) and (3). Since zero-modified distributions are capable of modeling zeros and non-zeros, the proposed method is especially suitable for intermittent demand. The experiments reveal the following key result: the approximation quality of BO is improved to a great extent compared to the best results given in a previous study [12]. In this respect, we think that the study in this paper can be considered as a complementary work for [12]. The future research is to investigate other zero-modified distributions under a different inventory system (e.g. (r, NQ)) in order to further improve the approximation results.

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