Inventory Control Framework for Nonstationary, Censored Demand

Inventory Control Framework for

Nonstationary, Censored Demand

By Supervisors

J. C. MEIJER PROF. DR. K. J. ROODBERGEN

S2200244 PROF.DR. R. H. TEUNTER

Master in Econometrics, Operations Research and Actuarial Studies

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BSTRACT

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company has to decide on the inventory level of many different products. We create an inventory control framework which combines methods from inventory control and forecast-ing. For each product, the most cost effective method is selected every period to determine the inventory level. An inventory control method is employed in the case of stationary demand whereas a forecasting method predicts the underlying nonstationary demand. Moreover, censor-ing is taken into account, which is the difference between sales and demand. The framework is applicable to many types of demand patterns, the parameters are robust and the framework significantly outperforms existing methods. The generated dataset indicates 82 percent costs decrease compared to existing methods. The dataset from a Dutch retailer indicates a costs decrease up to 56 percent over the 95 products we analyzed.

Keywords:

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EDICATION AND ACKNOWLEDGEMENTS

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o my mother who is the strongest and bravest woman I know.

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n appreciation to all my colleagues at Districon, who provided an excellent working envi-ronment and have helped me with advice trough every step along the way.

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ABLE OF

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ONTENTS Page 1 Introduction 1 2 Literature Review 3 2.1 Inventory Control . . . 4 2.2 Forecasting . . . 5 3 Problem Formulation 7 4 Solution approach 11 4.1 Framework . . . 11

4.2 Unexpected Event Group . . . 13

4.2.1 Censored Unexpected Event Group . . . 14

4.2.2 Uncensored Unexpected Event Group . . . 15

4.3 Adjustments to Stationary Predictors . . . 16

4.3.1 Adjustments to Besbes . . . 16

4.3.2 Adjustments to Bisi . . . 17

4.4 Adjustments to Nonstationary Predictors . . . 18

4.4.1 Adjustments to Maximum Likelihood Estimation . . . 18

4.4.2 Percentage . . . 19

5 Numerical Experiments 21 5.1 Generated Data . . . 21

5.1.1 Parameter Tuning of Generated Data . . . 23

5.1.2 Results of Generated Data . . . 25

5.2 Retail Data . . . 31

5.2.1 Parameter Tuning of Retail Data . . . 31

TABLE OF CONTENTS

Bibliography 37

A Appendix A 39

A.1 Heuristic by Besbes . . . 39

A.2 Heuristic by Bisi . . . 40

A.3 Censored Maximum Likelihood Estimation . . . 41

A.4 MLE for Normal Distribution . . . 42

B Appendix B 43 B.1 Nonperishable Addition . . . 43

B.2 Delivery Patterns . . . 45

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H A P T E R

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NTRODUCTION

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company has to decide on the inventory level for hundreds or even thousands of products to fulfill the demand of their customers. MacKenzie et al. (2013) underline the importance of inventory control as one of the tools to reduce costs. Past sales are taken into account to set the inventory level to fulfill the demand with a predetermined probability. Demand can change from day to day due to trends, promotions, changes in customer preferences, etc. This phenomenon is known as the nonstationary behavior of demand. As a result, inventory levels that are cost effective in a specific period are not necessarily suitable over the entire time horizon.

When products are out of stock, one is at risk of losing customers. In this case, sales could fall below the actual demand, which is called demand censoring. Not all demand is observed and the amount of lost sales is unknown which makes it difficult to predict the underlying demand and set the inventory level accordingly.

CHAPTER 1. INTRODUCTION

The methods which are placed within the proposed framework can be divided into two lines of research. The first line of research, consisting of inventory control methods, takes stationary cen-sored demand into account and minimizes costs (e.g., Chen (2010), Jain et al. (2014), Besbes and Muharremoglu (2013) and Bisi et al. (2015)). Although these methods can deal with stationary censored demand, Chen and Mersereau (2015) emphasize nonstationary censored demand as an open research area.

The second line of research, the methods from demand forecasting, are considered because they deal with nonstationary demand. The overview paper by Syntetos et al. (2016) underpins the importance of nonstationary demand and cites papers such as Rostami-Tabar et al. (2015) which support nonstationarity. Forecasting methods, as suggested by Nahmias (1994) and Agrawal and Smith (1996), are able to predict censored nonstationary demand. Companies are interested in decreasing costs as opposed to estimating demand. Therefore, after estimating the nonstationary, censored demand we will set a cost effective order-up-to level.

In Chapter 2, we will discuss both lines of research in more detail, after which we formulate the problem at hand in Chapter 3. We present the framework as solution approach in Chapter 4. The results are shown in Chapter 5, after which Chapter 6 summarizes the thesis.

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H A P T E R

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L

ITERATURE

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EVIEW

CHAPTER 2. LITERATURE REVIEW

2.1

Inventory Control

The first line of research, consisting of methods from inventory control, are developed for station-ary censored demand and decide on the inventory level based on costs. Bisi et al. (2015) suggest either a parametric or a nonparametric method is used. Harpaz et al. (1982), Braden and Freimer (1991), Lariviere and Porteus (1999), Chen (2010) and Jain et al. (2014) suggest parametric meth-ods which are based on an underlying demand distribution (the Bayesian approach). Burnetas and Smith (2000), Powell et al. (2004), Besbes and Muharremoglu (2013) and Bisi et al. (2015) suggest nonparametric methods where no information of the underlying distribution is needed. These methods are developed for stationary, censored demand but not for nonstationary, censored demand. Since Chen and Mersereau (2015) and Syntetos et al. (2016) underline the importance of nonstationary demand, we will extend these methods to be able to deal with nonstationary demand.

All the above papers include the cost aspect via the repeated newsvendor problem. The costs of having too much or not enough inventory are balanced in the optimum. The inventory control heuristics are build to find this optimum for stationary demand. Most papers assume perishable products. However, Bisi et al. (2015) assume that products are not disposed (which in this field of literature is called nonperishable products). Therefore, the actual stocking level could be larger than the order-up-to level such that higher sales could be observed based on the same underlying demand.

Single exponential smoothing is often used to estimate demand (Syntetos et al. (2016)). A single exponential smoothing heuristic could be based on online convex optimization. Online convex optimization minimizes the regret, the sum of the difference between the past decision and the optimal decision, every period. Because stationarity is assumed within inventory control, Bisi et al. (2015) take all previous observations into account. Using all information helps the heuristic to converge to the stationary demand optimum faster. Although in the case of nonstationary demand, past information becomes obsolete such that not all information should be taken into account. Nevertheless, the advantage of the heuristic by Bisi et al. (2015) as depicted in Appendix A.2 is ease of applicability because no parameters require tuning.

The basic idea of exploration-exploitation, the concept behind another inventory control heuristic, is to obtain more information in the beginning and to use this advantage over time. The exploring behavior could be beneficial with regard to nonstationary demand because only a certain amount of past information is taken into account. The order-up-to level is fixed for a certain amount of periods, called a phase. When the phase ends, the order-up-to level is recalculated based on the observations of this phase. The method is called exploration-exploitation because we need to set a higher order-up-to level (explore) when most of the observations are censored. When enough

2.2. FORECASTING

observations are uncensored, the order-up-to level can be set similar or lower, such that the avail-able information is exploited. However, during phases, the order-up-to level is fixed and cannot be adjusted to nonstationary demand. The mathematical description of exploration-exploitation as given by Besbes and Muharremoglu (2013) is provided in Appendix A.1.

2.2

Forecasting

The second line of research, the forecasting literature, can deal with nonstationary, censored demand. These methods predict the underlying demand but do not suggest an order-up-to level based on costs. The forecasting literature can be divided into two categories: focused on un-censored or un-censored demand. The methods within the first category suggested by Barnea and Lakonishok (1980), Fliedner (1999) and Rostami-Tabar et al. (2015), are only able to deal with uncensored demand. The methods within the second category suggested by Buckley and James (1979), Lu and Burke (2005) (regression) and Nahmias (1994), Agrawal and Smith (1996) (MLE)) are able to deal with both censored and uncensored demand. The methods of the second category are employed to estimate the nonstationary, censored demand. Moreover, based on this estimate a cost effective order-up-to level will be chosen as companies are more interested in decreasing costs than understanding the underlying demand.

To the best of our knowledge, none of the methods from the forecasting literature can deal with only censored observations. However, this situation could occur when the order-up-to level is too low. Maximum likelihood estimation (MLE) might be adjusted most easily to deal with only censored observations. MLE selects the parameters of a specific distribution which fit the obser-vations most accurately. In Appendix A.3, one can find the derivation of the maximum likelihood equation as suggested by Cameron and Trivedi (2005). The censored likelihood equation gets more complicated than the uncensored version after the distribution is selected. Especially the derivations of the parameters are difficult, therefore Nahmias (1994) and Agrawal and Smith (1996) suggest simplifications.

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ROBLEM

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ORMULATION

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multiperiod inventory problem is formulated. Before every period (t = 1,2,..., T), the order-up-to level ( ˆyt∈ R+) needs to be chosen. The actual stocking level yt is either

identical to the order-up-to level or to the amount of stock left from the previous period xt:

y_{t= max{ ˆ}y_t, x_t}. (3.1)

Thereafter, the sales observation w_{t= min{Dt}, y_t}is observed. The underlying demand D_{t∈ R+} is assumed to be independently distributed. However, the underlying demand distribution F is unknown and might change over time, allowing demand to behave nonstationary.

So our formulation is based on the multiperiod newsvendor problem similar to Bisi et al. (2015). We choose this formulation to be able to deal with nonperishable products as opposed to only perishable products (newspapers). Moreover, we allow demand to behave nonstationary because Chen and Mersereau (2015) suggest this to be a further line of research.

The order-up-to level will be determined based on costs. There are two situations in which different type of costs are incurred. The first type is incurred when the actual stocking level is higher than the demand ( yt> Dt). The purchase price minus the salvage price plus the holding

costs are incurred per surplus unit, which is denoted by a nonnegative overage costs Co. Since

CHAPTER 3. PROBLEM FORMULATION

In the second situation, the actual stocking level is lower than the demand ( yt< Dt) and therefore

the observation is marked as censored. Revenue minus the purchase price plus the penalty costs are incurred per shortage unit, which is denoted by a nonnegative underage costs Cu. All unmet

demand is assumed to be unobserved and therefore lost. Table 3.1 and Equations (3.2) and (3.3) give an overview of the variables.

Table 3.1: Variables and their explanation. Variable Explanation

xt initial inventory at the beginning of period t

ˆy_t order-up-to level yt actual stocking level

r revenue, price at which a product is sold g purchase price, costs

u penalty costs when unable to sell, e.g. costs of emergency supply o _{salvage price, e.g., sold in outlet (o < r) (o < g)}

h holding costs (for a product left over after a day) Co overage costs per surplus unit

C_u underage costs per shortage unit

C_{o= h + (g − o),} (3.2)

Cu= r − g + u.

(3.3)

After deriving C_uand C_o, the costs per period C can be written as follows C( yt) = CoE(yt− Dt)++ CuE(yt− Dt)−.

(3.4)

In the above, α+_{= max{}α, 0} andα−_{= − min{}α, 0}. Notice that no costs are incurred when the actual stocking level is exactly equal to demand yt= Dt. Furthermore C can only be calculated

when the underlying distribution F is known. However C does not need to be calculated as long as Cuand Coare available to find the optimum. In the standard newsvendor optimum, Cuand

Coare balanced such that

y∗_{t = min{y}t∈ R+: F( yt) ≥β}, with β=

Cu

C_{u+ Co}. (3.5)

This optimum ( y∗_t) is still applicable in the nonstationary situation._βis fixed over time and every period has a specific demand distribution F. So for each period, y∗_t can be found based on F. F might change over time and therefore y∗_t as well. Notice that F is unknown and demand could be censored, so finding y∗_t is difficult. For this reason, our framework takes into account that in the optimum, (1 −β) of the observations are censored.

Notice that_βhas the exact same definition as the service level (Axsäter, 2007, p. 94-97). Compa-nies might find the service level more intuitive to work with than the exact overage and underage costs. The service level is the probability of no stockout per order cycle. In some cases, onlyβis known as opposed to the exact costs. Since we are interested in the ratio between Coand Cu, we

could normalize one of the costs and rewrite the other to obtain the daily costs C. If we set Co= 1

we obtain:

C( yt) = E(yt− Dt)++ β

1 −βE(yt− Dt)

−_.

(3.6)

The objective is to minimize the average costs over time by setting the order-up-to level and thereby the actual stocking level for every period, denoted by

CT_{= min} y∈RT + E " 1 T T X t=1 C( yt) # . (3.7)

Vector y = (y1 y2 · · · yT) consists of all actual stocking levels yt. Notice that demand might

be left over for the next period, even though the order-up-to levels are set for exactly one day. The time horizon T is finite, such as the length of a product life cycle. If demand behaves stationary, most heuristics eventually find the optimal order-up-to level y∗

t. When T → ∞, the

average costs will converge to the optimal costs such that different heuristics cannot be compared.

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H A P T E R

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OLUTION APPROACH

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n the one hand, inventory control methods can deal with stationary censored demand while taking costs into account. On the other hand, forecasting methods are developed for censored nonstationary demand. We combine their strong points within a framework to be able to deal with stationary as well as nonstationary censored demand. The framework decides which type of method is most cost effective for every product at every point in time.

Notice the difference between the goals of both types of methods. The methods from inven-tory control aim for the optimal order-up-to level based on the service level _β, whereas the forecasting methods predict the underlying demand. When the forecasting methods are employed, the order-up-to level is set based on the quantile function at the service levelβ. In the sequel of this chapter, we discuss our framework in detail.

4.1

Framework

CHAPTER 4. SOLUTION APPROACH

Figure 4.1: Flow chart of the proposed framework.

When demand behaves nonstationary, demand does not behave in the line of expectation of the stationary demand. In this case, the stationary predictors do not adjust accordingly as ex-plained above. Therefore we suggest employing methods from forecasting literature. The exact definition of an unexpected event group is defined in Section 4.2. From now on the methods such as Nahmias (1994) and Agrawal and Smith (1996) are called nonstationary predictors as they are employed when demand behaves nonstationary. In Chapter 5 we will test whether including a nonstationary predictor will make the framework more dynamic, faster and better able to deal with nonstationary demand than a separate stationary predictor.

Explaining the framework step by step, one starts by setting the initial inventory level. The initial inventory level can be based on a professional opinion. Still, if a random value is chosen, the framework will eventually find the appropriate order-up-to level. This is true because the framework is build to deal with order-up-to levels which are in the wrong order of magnitude. After each period, a new (cumulative) sales observation becomes available. One determines whether this observation belongs to an unexpected event group. If not, the demand behaves stationary, and we use a stationary predictor. If an unexpected event group is observed, demand behaves nonstationary such that a nonstationary predictor is chosen. After the order-up-to level is recalculated the next period starts and the new sales observation becomes available. The framework is dynamic because, after each period, the order-up-to level is recalculated.

4.2. UNEXPECTED EVENT GROUP

Choosing between a stationary or nonstationary predictor is our main contribution. Consequently, the definition of an unexpected event group is discussed in detail in Section 4.2. Furthermore, we will discuss why each stationary predictor is not directly applicable to nonstationary demand. Subsequently, we suggest specific adjustments to each predictor, to make them more applicable to the problem at hand. The adjustments to the stationary predictors are followed by the adjustments to the nonstationary predictors. For each predictor, the specific parameters are defined in this chapter and tuned in Chapter 5. Notice that the framework is a plug and play concept because one can plug in one desired stationary and one nonstationary predictor. In Chapter 5 the predictors which are explained in Chapter 2 are plugged in and tested.

4.2

Unexpected Event Group

At every point in time, one of the major decisions within our framework is the choice of predictor. This decision is based on the probability of an event group. An event group is defined as a number of censored observations m out of the total number of observations n. Several observations might be necessary to ensure enough reliability. The exact number of observations which are necessary (∆E) is tuned in Chapter 5. We call an event group unexpected when there is a low probability

that the current order-up-to level is correct. The definition of an unexpected event group should be carefully defined, to make sure we do not alter the order-up-to level when demand is stationary. We distinguish between two types of unexpected event groups:

CHAPTER 4. SOLUTION APPROACH

4.2.1 Censored Unexpected Event Group

A censored event group consists of at least one censored observation (m ≥ 1). The probability p represents the probability that the current order-up-to level is (still) correct compared to the observed event group. In other words, the event group is unexpected when p is low. We compare p to an threshold th_c which will be tuned in Chapter 5. If p is higher than a threshold th_c; the event group is expected and therefore the stationary predictor will be chosen. On the other hand, when the probability is lower than (or equal to) the threshold thc, the order-up-to level is

considered too low and the nonstationary predictor is chosen. In conclusion, the order-up-to level is adjusted when an unexpected event group is observed, mathematically when:

p ≤ thc.

(4.1)

The probability of an uncensored observation in the optimum F( y∗_t) is greater or equal to the service level β, as given in Equation (3.5) of Chapter 3. Sinceβis known, we use this as the probability of a certain observation when the optimal order-up-to level is chosen.βis used as the probability of an uncensored observation and 1 −βthe probability of a censored observation. The probability is calculated as follows

p = n X l=m à n l ! βn−l_{(1 −β)}n_, (4.2) where¡n

l¢, the number of different options for an event group, is included because the order of the

observations is not relevant. Furthermore, the sum is included because we are interested in the probability of an event group or worse. One would use the nonstationary predictor too often if the sum is not included.

Adjusting the heuristic too quickly is expensive, especially in the case of stationary demand. Hence, we can choose a lower threshold thc or post a restriction on the minimum number of

observations∆E to form a censored event group. thcand∆E are tuned in Chapter 5 to obtain

the lowest costs. Since the stationary predictors take many observations into account, n could become very large. In this case, an unexpected event group could be overlooked. Therefore, when two consecutive observations are censored, we reset the stationary predictor by setting n = 2. The framework would become too adaptive in the case of stationary demand if we reset after each censored observation.

4.2. UNEXPECTED EVENT GROUP

4.2.2 Uncensored Unexpected Event Group

The order-up-to level could be much higher than the sales observations. In this case, using a nonstationary predictor might be beneficial because too much stock is costly. However, dealing with censored unexpected event groups is much more costly than dealing with the uncensored unexpected event group. When one encounters a censored unexpected event group, adjusting could take many observations before one gets to the correct order of magnitude of the order-up-to level, because the order of magnitude of the order-up-to level is unknown. On the other hand, when too much stock on hand is available, we do observe all the demand such that we can make an estimate in the correct order of magnitude.

An uncensored unexpected event group is an event group of which the observations are sig-nificantly lower than the order-up-to level ˆytand does not contain any censored observations. One

calculates the cumulative probability of the current order-up-to level ˆytas if it is an observation.

The cumulative probability in the optimum would be_β. We alter the order-up-to level by using a nonstationary predictor if F( ˆyt) is significantly larger thanβ. To make sure the order-up-to level

is only changed when F( ˆyt) >>β, we suggest to use the following equation:

F( ˆyt) ≥ 1 −1 −β

thu

. (4.3)

Note thu>> 1 needs to hold such that F( ˆyt) >>βis true:

1 −1 −β thu >>β (4.4) 1 −β>>1 −β th_u (4.5) thu>> 1. (4.6)

One can use the standard maximum likelihood approach to find the parameters of the distribution based on the observations of the unexpected event group. The value of thuis tuned in Chapter 5

CHAPTER 4. SOLUTION APPROACH

4.3

Adjustments to Stationary Predictors

We suggest specific adjustments of two stationary predictors, such that they fit within the framework. Both methods that are chosen are nonparametric because Chen and Mersereau (2015) underline that nonparametric methods do not need any prior knowledge of the underlying demand distribution. In Section 4.3.1, the adjustments to Besbes and Muharremoglu (2013) will be discussed. Subsequently Section 4.3.2 covers the adjustments to Bisi et al. (2015). The basic information of the predictors can be found in Chapter 2 and Appendix A.

4.3.1 Adjustments to Besbes

To fit within the framework, the parameters of this stationary predictor need to be consistent instead of striving for a fast convergence to the optimum for stationary demand. We propose to fix two parameters as explained below. This adjustment will increase the ability of the heuristic by Besbes and Muharremoglu (2013) to deal with nonstationary demand. Nevertheless, the order-up-to level is fixed for a number of periods. Thus the heuristic is still not dynamic which leaves room for improvement.

The first parameter∆ (the amount of periods over which the order-up-to level is fixed) orig-inally increases over time. This leads to a faster convergence when demand is stationary, but could be problematic for nonstationary demand because the heuristic is less adaptive to changes as the phase length increases. Hence within the framework, we decide to fix the phase length over the entire time horizon as opposed to Besbes and Muharremoglu (2013).

The second parameter i is the rate at which the order-up-to level is increased when too many observations are censored. Originally i is decreased over time to improve convergence for station-ary demand. Nonetheless, this might not be preferable for nonstationstation-ary demand. For stationstation-ary demand, only the initial order-up-to level can be in the wrong order of magnitude. Whereas the order-up-to level in the case of nonstationary demand can be in the wrong order of magnitude throughout the entire lifetime of a product. When i is set too low, one needs many phases to find an appropriate order-up-to level. When chosen too high, one holds redundant inventory. Both situations are undesirable and costly. Besbes and Muharremoglu (2013) utilize i in the range of [1 : 2], we choose to tune both∆ and i in Chapter 5.

For every predictor within our framework, the minimum order-up-to level is at least one unit. The disadvantage is that products are never taken out of the assortment. However, this addition is necessary for the implementation of the predictors (Besbes and Muharremoglu (2013)).

4.3. ADJUSTMENTS TO STATIONARY PREDICTORS

4.3.2 Adjustments to Bisi

Within the single exponential smoothing heuristic based on online convex optimization, the new order-up-to level is based on all previous order-up-to levels. This results in a predictor that is not very adaptive and therefore, without adjustment, probably not suitable for nonstationary demand. To make this predictor more adaptive, we suggest restarting the heuristic after an unexpected event group. Consequently, we do not take all past information into account as Bisi et al. (2015) propose, but only the information that is still applicable.

We suggest, as a general improvement, to use the order-up-to level as the indication of cen-soring instead of the actual inventory level. Companies tend to overestimate demand and set the initial inventory level several times higher than demand as a consequence (which also occurs in the retail dataset). After a few periods, we could still have an inventory level which is higher than the order-up-to level. In this case, the observation is marked as uncensored by Bisi et al. (2015), leading to a new order-up-to level which is too low. By using our suggestion, this observation would be treated as censored, because the order-up-to level is lower than the demand. In other words, the current order-up-to level would lead to a censored observation if the order-up-to level and the inventory level are equal. This adjustment of the definition of a censored observation probably leads to a more cost-effective order-up-to level. The mathematical description of this adjustment can be found in Appendix A.2.

CHAPTER 4. SOLUTION APPROACH

4.4

Adjustments to Nonstationary Predictors

To the best of our knowledge, none of the nonstationary predictors can deal with only censored observations (n = m). Since such an event group can occur, adjusting the nonstationary predictors is important. Adjusting MLE is explained in Section 4.4.1. Adjusting regression is more difficult and therefore not suggested within this thesis. In Section 4.4.2 the method ‘Percentage’ is sug-gested as an another method to increase the order-up-to level in case of a censored unexpected event group.

A nonstationary predictor is based on just a few observations∆E and therefore the new

order-up-to level is not very accurate but merely an estimate. Since Cu>> Co, we rather have too few

censored observations as opposed to too many. If the new order-up-to level is lower than the highest observation, we propose to use the highest observation of the event group instead of the calculated order-up-to level in case of an uncensored unexpected event group.

4.4.1 Adjustments to Maximum Likelihood Estimation

One can calculate the maximum likelihood of the most appropriate distribution as explained in Appendix A.3. However, when all observations are censored, applying MLE is difficult, because the mean of the distribution with the highest probability converges to infinity. We can alter this by assuming that one of the censored observations is actually uncensored. So we interpret one observation as demand instead of sales.

Since Nahmias (1994) suggest that the normal distribution is most often assumed and used within commercial inventory control systems, we will use the normal distribution. When a normal distribution is chosen, one could also use the simplified MLE as suggested by Nahmias (1994). The simplified maximum likelihood estimators are easier to implement than the standard MLE. Nahmias (1994) show that the results of the simplification are comparable to the original formu-lation. We use the simplification since we are looking for the right order of magnitude and not the exact order-up-to level in case of an unexpected event group.

In Appendix A.4 we show how one could adjust the equations suggested by Nahmias (1994) to deal with the situation n = m. Since we assume to have one uncensored observation, the fraction of uncensored compared to the total number of observations is 1/n. The sales observation is also interpreted as the average sales, which is equal to the inventory level. It is not possible to compute the standard deviation based on only one (uncensored) observation. Therefore we force the standard deviation S to have a minimum. This minimum is based on a percentage_σminof the average sales.σ_minis tuned in Chapter 5.

4.4. ADJUSTMENTS TO NONSTATIONARY PREDICTORS

4.4.2 Percentage

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UMERICAL

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XPERIMENTS

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he framework that we introduced in Chapter 4 is implemented into the software package AIMMS. We show the difference between the inventory costs of the framework compared to the separate predictors. First, we discuss the 5300 generated demand patterns over 200 periods. Secondly, the dataset of a Dutch retailer is discussed. This dataset consists of the inventory level and sales of 95 products over 76 days in 2016. For each dataset, we first provide more information. Subsequently, the parameters of the framework are tuned and the results are shown.

5.1

Generated Data

CHAPTER 5. NUMERICAL EXPERIMENTS

Both sets have (at least) the following three type of demand scenarios: stationary, trend and jump. Trends and jumps are either up or down, which is chosen randomly. A jump occurs in one of the first 50 periods (chosen randomly) such that each method has some time to detect the jump. The value before and after the jump is randomly drawn from a range of values called ’mean’ as given in Table 5.1. To generate a trend, two random values are drawn from the same range ’mean’ which will represent the demand in the first and the last period. We extend dataset (2) by including several combinations of, for example, a jump combined with a trend or two jumps within the same demand pattern.

For fairness of testing, the means of both sets are set differently. In set (1) the mean of the demand is chosen between 20 and 50 whereas in set (2) between 1 and 200. In set (2), 25% of the stochastic stationary demand patterns are replaced by slow moving items (demand between 0 and 2) to test if the framework can deal with this type of demand. Set (1) consist of 300 demand patterns of three different type of scenarios and set (2) of 5000 demand patterns over four dif-ferent type of scenarios. Additionally, all generated values are rounded up. Moreover for each demand pattern, aβis randomly generated between [0.75 : 0.99]. Table 5.1 depicts an overview of both datasets.

Table 5.1: Information about the generated data. set (1) set (2) Number of periods 200 200 Mean 20-50 1-200 Stationary scenarios yes yes Trend scenarios yes yes Jump scenarios yes yes Combinations scenarios no yes Number of observations 300 5000

5.1. GENERATED DATA

5.1.1 Parameter Tuning of Generated Data

First, all parameters for every implemented predictor are tuned using generated dataset (1). Since optimizing all parameters simultaneously is too time consuming, the parameters are first optimized separately. Thereafter, we test if the parameters could be optimized further given this current set of parameters. If one parameter is changed, all other parameters are tested again. One can find the results and the sensitivity analysis in Table 5.2. The range in Table 5.2 depicts the values which could have been chosen such that the average costs are not more than 5% higher compared to the average costs from the chosen value.

Table 5.2: Parameters tuned on the generated data with the range in which the average costs are not more than 5% higher than the (chosen) value, where bold depicts a large impact on the costs.

Parameter Predictor Value Range

∆ Besbes 30 _{[15 : ∞)}

i Besbes 1.4 [1 : 3.5]

∆E Unexpected event group 3 [2 : 4]

thc Censored unexpected event group 0.3 [0.01 : 0.995]

thu Uncensored unexpected event group 10 [5 : ∞)

σmin MLE 0.061 [0.053 : 0.064]

j Percentage 1.25 [1.21 : 1.27]

Tuning j,∆E and σmin has more impact on the average costs since their 5% range of these

parameters is relatively small, as depicted in Table 5.2 with a bold range. For example,∆ can be set at any value above 15, while∆E can only be 2, 3 or 4. Figure 5.1 (a) depicts the parameter

tuning of a parameter with a small 5% range whereas Figure 5.1 (b) depict the parameter tuning of a parameter with a large range. Both graphs are provided given that the other parameters are set as suggested in Table 5.2.

Tuning j has quite a large effect on the costs which could be explained by noticing that the nonstationary predictor changes the order-up-to level after each unexpected event group. When j is set too large, the order-up-to level would constantly be changed, whereas too small would imply a high cost due to lost demand over time. The explanation for the sensitivity of∆E most

CHAPTER 5. NUMERICAL EXPERIMENTS

(a) j (b) th_c

Figure 5.1: Average cost while tuning two different parameters.

Nevertheless, many parameters are robust because of their wide range, as depicted in Table 5.2, which is interesting and helpful. For example th_c, the probability limit when a censored unexpected event group is considered, has a wide range of possible values. Perhaps when∆E is

chosen carefully, the probability limit is not crucial anymore.

Since ∆ = ∞ is included, we could conclude that never resetting Besbes (which is similar to Bisi) is less than 5% worse than resetting it after 30 periods. The explanation for the robustness of i could be that the exploration option is not utilized very often because the nonstationary predictor often takes over first. Another interesting aspect is that i > j, intuitively one might expect the opposite since j is in the unexpected case and i is applied when the order-up-to level is in the right order of magnitude. Notice that j is applied more often and therefore potentially needs to be more conservative.

5.1. GENERATED DATA

5.1.2 Results of Generated Data

The results of generated dataset (2) are discussed in this section. We first discuss the general results and show that the framework outperforms the separate predictors. Thereafter, we consider the plug and play concept of the framework as we compare the methods which are implemented for both the stationary and nonstationary predictor. We finish this section by discussing the possibilities of our framework to the entire lifecycle of a product.

The different methods within the plug and play concept have different computation times. AIMMS needs about 20 minutes when Besbes is chosen as the stationary predictor and 30 minutes when Bisi is chosen. There is no significant time difference between the nonstationary predictors. During the 20 to 30 minutes, AIMMS computes the solution for 5000 demand patterns over 200 periods for one set of predictors. The average costs per scenario type for the different sets of predictors are depicted in Table 5.3. Per scenario, the predictor (combination) which yields the lowest average costs is marked bold. Italic implies that the predictor is not significantly different from the best predictor with 95% confidence.

The first two columns of Table 5.3 depict the results of the stationary predictors known in literature after which column three to six depict the results of the proposed framework. The last column depicts the solution if all information is known beforehand and stationarity is assumed, which could be interpreted as the stationary lower bound. Note that ‘Perc’ in the tables and graphs is an abbreviation of the nonstationary predictor Percentage.

To provide more information, in Figure 5.2 the average costs over time are shown. Part (a) depicts every combination of predictors and (b) provides a closer look into the predictors combined within the framework.

5.1. GENERATED DATA

Combining the predictors within the proposed framework significantly outperforms the separate predictors when we look at the 5000 generated demand patterns. After three periods, when the nonstationary predictor could be chosen for the first time, the framework indeed starts to outperform the separate predictors. The performance of the framework gets much better than the separate predictors as the time horizon increases. At T = 200, the best predictor (Besbes + MLE) yields a cost reduction of 82% compared to the best known predictor from literature (Besbes). Moreover, the stationary lower bound is even outperformed as depicted in Table 5.3 and Figure 5.2. This indicates the importance of switching between stationary and nonstationary predictors as opposed to just the assuming stationary demand.

Figure 5.3 indicates that the framework outperforms the separate predictors. The average costs of Besbes+MLE are much lower than those of Besbes, mainly because the order-up-to level is adjusted much quicker in the case of nonstationary demand. Instead of 121 periods, the framework takes only 27 periods to adjust to the jump in demand depicted in Figure 5.3. One can also observe that Besbes is having problems with deterministic demand as we observe a jumps in the order-up-to level in period 151. Besbes sets the order-up-to level exactly equal to the demand which leads to only censored demand observations and therefore an adjustment. However, as demand is not deterministic in practice, this is not an issue.

(a) Inventory level (b) Average costs Figure 5.3: Demand pattern of a jump up without noise.

Figure 5.4 (a) depicts the demand itself (black), the stationary lower bound (the horizontal line) and two of the combinations within the framework. The stationary lower bound is far off when demand behaves nonstationary, leading to a higher cost as depicted in Figure 5.4 (b). Further-more, Figure 5.4 indicates that the adjustment downwards in itself is an improvement which is supported by the fact that thuperforms better at ten compared to infinity (Table 5.2). The fact

CHAPTER 5. NUMERICAL EXPERIMENTS

(a) Inventory level (b) Average costs Figure 5.4: Demand pattern of a decreasing trend without noise.

When noise is taken into account, we observe (in table 5.3) that the framework performs signifi-cantly better for stationary demand compared to Bisi or Besbes separately. The improvement is 43% for the slow moving items (see Figure 5.5) and above 64% for the fast moving items (see Figure 5.6). So even when (most of) the demand within a company behaves quite stationary, our framework is preferred over the separate predictors for fast as well as slow moving items. This is an indication that the framework properly chooses between the stationary and nonstationary predictor, as stationary demand can still be accounted for as well.

(a) Inventory level (b) Average costs Figure 5.5: Demand pattern of a slow moving item with noise.

Next we focus on the results of the specific combinations of predictors. Combining Besbes with a nonstationary predictor almost always outperforms a combination with Bisi. This result could probably be assigned to the fact that Bisi takes much more historical information into account whereas Besbes restarts their predictor after a fixed amount of periods. Since demand is changing, clinging to the past is harmful. One might be able to improve Bisi by also restarting after a specific amount of periods such that the flat line of Besbes + MLE as in Figure 5.4 (a) is interrupted and restarted at a lower level.

5.1. GENERATED DATA

The only scenario where Bisi significantly outperforms Besbes, is for fast moving, stationary items (Figure 5.6 (b)). In Figure 5.6 (a) one can observe that Besbes+MLE is too adaptive, whereas Bisi+MLE is more stable. Bisi+MLE quickly coincides with the stationary lower bound which is preferable in this case. Consequently, Bisi+MLE is preferred in this situation.

(a) Inventory level (b) Average costs Figure 5.6: Demand pattern of a fast moving item with noise.

Even though the nonstationary predictor MLE is most often better than Percentage, Percentage is not statistically different. Only overall MLE is significantly better than Percentage. Figure 5.7 (b) underpins the subtle difference. For many other demand patterns MLE even outper-forms Percentage. We would still advice to use Besbes+MLE, but when ease of implementation and understanding is most important, Besbes+Perc could be preferred as performance is very comparable.

CHAPTER 5. NUMERICAL EXPERIMENTS

All previous results show that the framework can perform well in many situations. This indicates that the framework might work well throughout the entire product lifetime cycle since such a cycle might consist of all these different kind of patterns. To give an example, Figure 5.8 graphically depicts a possible demand pattern during a lifetime cycle. One can observe that the framework indeed quite closely follows this demand pattern. Nevertheless, do keep in mind that our framework does not indicate when products need to be taken out of the assortment.

Figure 5.8: Performance of Besbes + MLE for entire lifetime cycle.

5.2. RETAIL DATA

5.2

Retail Data

The retail dataset consists of the sales and the inventory level per day of one store. The data is provided by a Dutch retailer who prefers to stay anonymous. The retailer has a large number of stores throughout the Netherlands. Each store has over 10.000 products. The company buys a bulk of products which they store in their distribution centers. From there they can distribute the products over all their stores, every day.

The dataset consists of items ranging from 0 to 288 sales per day with a mean of 11 and a standard deviation of 20.92. Both slow moving items which are sold maximum twice a day and fast moving items are represented within the dataset. The inventory level of all products is high because the minimum inventory of the products is on average more than two times the maximum sales. The sales is equal to the demand because none of the items are ever sold out.

5.2.1 Parameter Tuning of Retail Data

First of all, the parameters are tuned using the same method as described in 5.1.1. We tune the parameters again to see if more costs can be saved if the parameters are tuned per company. Parameters for every implemented predictor are tuned using only the first 20 products, for the fairness of testing.

Table 5.4: Parameters tuned on the retail data with the range in which the average costs are not more than 5% higher than the (chosen) value, where bold depicts a large impact on the costs.

Parameter Predictor Value Range

∆ Besbes 10 _{[3 : ∞)}

i Besbes 1.25 [1.05 : 1.60] ∆E Unexpected event group 3 [2 : 3]

thc Censored unexpected event group 0.2 [0.01 : 0.4]

thu Uncensored unexpected event group 10 [1 : ∞)

σmin MLE 0.06 [0.025 : 0.07]

j Percentage 1.5 [1.1 : 2]

CHAPTER 5. NUMERICAL EXPERIMENTS

5.2.2 Results of Retail Data

Since the parameters are robust, we decide to use the parameters as depicted in Table 5.2. The order-up-to levels are computed using the initial inventory of the retailer. Table 5.5 depicts the average costs of inventory control of the company for different values ofβas this information was not provided to us. The average (normalized) inventory costs within the dataset, over the given period of the company, is 105.20 [77.61:132.80]. This holds for everyβsince the inventory level was always too high, which implies only incurring Co.

For every value ofβthe framework yields significantly better results than the original inventory and both separate predictors. The improvement ranges from 27 to 56 percent, the lowerβ, the higher the cost decrease of the framework. One can also observe that the differences within the framework, between the different predictors, are small forβlower than 0.9. So a company does not need to worry which predictors to use in this case. Besbes+MLE still an appropriate choice, whereas Besbes+Percentage is never far off either. So again, the importance of the framework is shown as opposed to the separate predictors. Notice that the cost improvement is lower than the generated data because the time horizon is shorter. A longer time horizon implies a higher cost decrease, because the advantage becomes larger over time.

The situation where β= 0.9 is analyzed in more detail. This gives an average improvement of 52%. Figure 5.9 shows the only product for which the framework does not outperform the separate predictor. One can possibly say that demand is so volatile that maintaining the same order-up-to level is better than constantly trying to adjust. A company that has products with only volatile demand might consider to tune the parameters to their specific situation or use the separate predictors instead of the framework.

(a) Inventory level (b) Average costs

Figure 5.9: Demand pattern of the only product for which the framework incurs higher costs.

CHAPTER 5. NUMERICAL EXPERIMENTS

5.2.3 Rounding in Practice

In practice companies often deal with discrete demand. We will have to round the estimates of the order-up-to level to account for this. We round up since F(x) ≥βneeds to be satisfied and rounding up is cheaper becauseβ>> 0.5 and therefore Cu=β/(1 −β) is bigger than Co= 1. So

since the underage costs are higher than the overage cost, rounding up is cheaper than rounding down.

Furthermore, ordering only one item is often not possible in practice. Dealing with package sizes could also be seen as a form of rounding since we potentially have to order a larger or smaller amount of items. One could round up for the same reason as above. The costs of including package sizes, for_{β= 0.9, is depicted in the last row of Table 5.5.}

When package sizes are incorporated, the framework also significantly outperforms the sep-arate predictors. Furthermore, the differences between the options within the framework are smaller. In other words, when one needs to order a certain package size, the framework is pre-ferred, and many desired predictors can be plugged in.

Nevertheless, depending on β, rounding down could potentially be cheaper when the pack-age sizes are very large and the calculated value is just above a possible order-up-to level. We test if there is a potential cost decrease when rounding up and down are combined, compared to just rounding up. As a result, the order-up-to levels are only rounded down 2.4 percent of the time, which does not have an impact on the costs. As this method is more complicated, we advise companies just to round up to the lowest possible order-up-to level when dealing with package sizes.

C

H A P T E R

6

C

ONCLUSION AND

D

ISCUSSION

A

n inventory control framework is created to set the inventory level for many different products within the same company. Demand does not always behave stationary. So volatile demand, also known as nonstationary demand, is taken into account as well. We also allow for demand censoring, which is the difference between the observed sales and the underlying demand. Ultimately, the goal is to decrease the inventory costs over time.

Each period, the framework chooses the appropriate predictor for different kind of demand patterns. When demand behaves stationary, the framework selects an inventory control method. On the other hand, when demand behaves nonstationary a forecasting method will be employed to estimate the changing demand after which the cost effective order-up-to level can be determined. We test different stationary and nonstationary predictors within this plug and play framework.

CHAPTER 6. CONCLUSION AND DISCUSSION

Additionally, the framework itself is widely applicable since the framework can deal with many types of demand patterns. The combination of Besbes and Muharremoglu (2013) as stationary predictor and MLE as nonstationary predictor performs best. The framework improves the inventory for stationary and different type of nonstationary demand as well as for fast and slow moving items. This indicates that the framework could apply to the entire lifecycle of a product. However, our framework is not designed to detect when a product is obsolete, which could be a further line of research.

One might be worried about running out of stock and thereby disappointing people. Let us note that this framework is not applicable if the only goal is to fulfill all demand. Nevertheless, when the service level (β) is chosen very close to 1, the number of lost customers will be small. Nonetheless, the framework does not take into account that stock might also serve another purpose. When there are more products in store, this in itself might generate a higher demand. A slight adjustment is adding a minimum required stock, but this does not completely solve the question. So one might look into this topic for further research.

Concerning lead time, two practical situations could be investigated as a further line of re-search. First of all, considering a stochastic lead time, which could not simply be solved by adding safety stock. Notice that the translation is not one on one with regard to a standard (R, Q) or (s, S) policy because these policies have a decreasing inventory level over some consecutive periods. As we strive for the optimum inventory level per period, we also do not take quantity discounts into account. Secondly, specific delivery days or delivery patterns ask for a different approach as well. In this case, the inventory needs to be sufficient for several periods. Higher holding costs are incurred, but censoring will certainly be avoided on specific days. A start of developing such an approach is given in Appendix B.2.

B

IBLIOGRAPHY

Agrawal, N. and S. A. Smith (1996). Estimating negative binomial demand for retail inventory management with unobservable lost sales. Naval Research Logistics (NRL) 43(6), 839–861. Axsäter, S. (2007). Inventory control, Volume 90. Springer Science & Business Media.

Barnea, A. and J. Lakonishok (1980). An analysis of the usefulness of disaggregated accounting data for forecasts of corporate performanc. Decision Sciences 11(1), 17–26.

Besbes, O. and A. Muharremoglu (2013). On implications of demand censoring in the newsvendor problem. Management Science 59(6), 1407–1424.

Bisi, A., K. Kalsi, and G. Abdollahian (2015). A non-parametric adaptive algorithm for the censored newsvendor problem. IIE Transactions 47(1), 15–34.

Braden, D. J. and M. Freimer (1991). Informational dynamics of censored observations. Manage-ment Science 37(11), 1390–1404.

Buckley, J. and I. James (1979). Linear regression with censored data. Biometrika 66(3), 429–436. Burnetas, A. N. and C. E. Smith (2000). Adaptive ordering and pricing for perishable products.

Operations Research 48(3), 436–443.

Cameron, A. C. and P. K. Trivedi (2005). Microeconometrics: methods and applications. Cambridge university press.

Chen, L. (2010). Bounds and heuristics for optimal bayesian inventory control with unobserved lost sales. Operations research 58(2), 396–413.

BIBLIOGRAPHY

Harpaz, G., W. Y. Lee, and R. L. Winkler (1982). Learning, experimentation, and the optimal output decisions of a competitive firm. Management Science 28(6), 589–603.

Jain, A., N. Rudi, and T. Wang (2014). Demand estimation and ordering under censoring: Stock-out timing is (almost) all you need. Operations Research 63(1), 134–150.

Lariviere, M. A. and E. L. Porteus (1999). Stalking information: Bayesian inventory management with unobserved lost sales. Management Science 45(3), 346–363.

Lu, X. and M. D. Burke (2005). Censored multiple regression by the method of average derivatives. Journal of multivariate analysis 95(1), 182–205.

MacKenzie, I., C. Meyer, and S. Noble (October, 2013). How retailers can keep up with consumers. Retrieved from http://www.mckinsey.com/industries/retail/our-insights/how-retailers-can-keep-up-with-consumers.

Nahmias, S. (1994). Demand estimation in lost sales inventory systems. Naval Research Logistics 41(6), 739–758.

Powell, W., A. Ruszczy ´nski, and H. Topaloglu (2004). Learning algorithms for separable approxi-mations of discrete stochastic optimization problems. Mathematics of Operations Research 29(4), 814–836.

Rostami-Tabar, B., M. Z. Babai, Y. Ducq, and A. Syntetos (2015). Non-stationary demand forecasting by cross-sectional aggregation. International Journal of Production Economics 170, 297–309.

Syntetos, A. A., Z. Babai, J. E. Boylan, S. Kolassa, and K. Nikolopoulos (2016). Supply chain forecasting: Theory, practice, their gap and the future. European Journal of Operational Research 252(1), 1–26.

A

P P E N D I X

A

A

PPENDIX

A

A.1

Heuristic by Besbes

The heuristic based on exploration-exploitation as given by Besbes and Muharremoglu (2013) is given below. First of all, notice that the same definitions as in Chapter 3 and 4 are used. After a certain amount of periods, the phase, the quantile function

q = inf ( k :∆−1 t X τ=t−∆+11{min{Dτ , y_τ} ≤ k} ≥β ) (A.1)

over the last∆ observations is computed.

If q < min(yτ) holds, whereτ= {t −∆+1,...,∆}, the new inventory level will be based on q:

ˆy_{t+1= max {1, q} .} exploitation (A.2)

Otherwise the inventory level is increased:

ˆy_{t+1= max {1, i ˆy}t} . exploration

APPENDIX A. APPENDIX A

A.2

Heuristic by Bisi

The heuristic of Bisi et al. (2015), using the same definitions as given in this thesis in Chapter 3 and 4, is given by the following lines:

ˆyt= max ½ 1, Pat−1 S µ_b t−1 a_t−1 ¶¾ (A.4) (A.5)

The costs are incorporate via

H_τ( y_{τ) ≡ Hτ}(D_τ, y_{τ) =}    Co if Dτ< yτ −Cu if Dτ≥ yτ, (A.6)

after which they are plugged into Pat−1 S (v) = argmin u∈S {_{(u − v)}2a_t−1} (A.7) a_t−1= t−1 X τ=1 H2_τ( y_τ) (A.8) b_t−1= t−1 X τ=1 · H2_τ( y_τ) y_τ−1 βHτ( yτ) ¸ , (A.9)

to calculate the order up to level.

A.3. CENSORED MAXIMUM LIKELIHOOD ESTIMATION

A.3

Censored Maximum Likelihood Estimation

We cannot simply use the standard maximum likelihood as we encounter censoring. Let us derive the log likelihood function (Cameron and Trivedi, 2005, p. 533), such that we find the estimators most appropriate to the (censored) observations. Let k_{t= 1 when we encounter a} censored observation (Dt≥ yt) and kt= 0 otherwise. The following equations depict the idea

behind the censored maximum likelihood:

wt=    D_t if Dt< yt yt if Dt≥ yt (A.11) f∗(wt) =    f (wt) if wt< yt 1 − F(yt) if wt= yt (A.12) f∗(w_{t) = f (wt})1−kt_{(1 − F(y} t))kt (A.13) ln L(_{θ) =} s X t=1 {(1 − kt) ln [ f (wt|θ)] + ktln [1 − F(yt|θ)]} . (A.14)

Equation A.11 depicts the relation between demand Dt, inventory level yt and sales wt. The

distribution that deals with censoring f∗ _{based on the underlying distribution f is depicted}

in Equation A.12 and combined with the censoring indicator kt in Equation A.13. For ease of

APPENDIX A. APPENDIX A

A.4

MLE for Normal Distribution

Nahmias (1994) propose to use the following equations to calculate the mean and standard deviation of a normal distribution when dealing with censored demand:

a =n − m n (A.15) z =Φ−1(a) (A.16) w = 1 n − m n−m X i=1 wi (A.17) S2₌ 1 n − m − 1 n−m X i=1 (wi− w)2 (A.18) e σ2 = S 2 1 − zφ(z)/a − [φ(z)/a]2 (A.19) e µ= w +σφe (z) a . (A.20)

Notice that the same definitions are used as in Chapter 3 and 4. a depicts the fraction of uncen-sored observation out of the total number of observations, z the quantile function of the standard normal distributionΦ of this fraction. Moreover, ¯w and S2 are the mean and standard deviation of the uncensored observations. Whereasσ_e2andµ_edepict the mean and standard deviation when the uncensored observations are taken into account as well.

We suggest to use the following adjustment, as explained in Chapter 4, to be able to deal with only censored observations within an event

a = max ½₁ n, n − m n ¾ (A.21) z =Φ−1(a) (A.22) w = min ( wi, 1 n − m n−m X i=1 wi ) (A.23) S2= max ( σminw, 1 n − m − 1 n−m X i=1 (wi− w)2 ) (A.24) e σ2 = S 2 1 − zφ(z)/a − [φ(z)/a]2 (A.25) e µ= w +σφe (z) a . (A.26)

The inventory level is set, based upon the quantile function of the standard normal distribution, using the following equation

ˆy_{t+1= max{1,µe+σe}Φ−1(_β)}. _{τ= {t + 1 . . . t + s}.} (A.27)

A

P P E N D I X

B

A

PPENDIX

B

B.1

Nonperishable Addition

Within the basic newsvendor model and the algorithm by Besbes and Muharremoglu (2013), it is assumed that the product is perishable just like a newspaper. So products are bought for one period and cannot be sold the next period. In this paper, we also deal with nonperishable products of which left over stock can be used the next day. Notice thatβcould be slightly different in this case, since the overage costs C_o do not consist of the purchase price c and the salvage price s because the products can still be sold in the next period. We do still have the holding costs h. In other words, it is cheaper to have ’too many’ products compared to the perishable case.

Mathematically we implement the idea proposed by Bisi et al. (2015) into the algorithm of Besbes and Muharremoglu (2013), combining the stock on hand xtand the order up to level ˆytto

find the actual stoking level y_t. This makes sure that we can use the stock that is left over from the day before, mathematically given by:

y_{t= max{ ˆ}y_t, x_t}. (B.1)

Besbes and Muharremoglu (2013) indicate that the efficiency of the algorithm (in case of a discrete distribution) they propose in the censored demand case is:

Kcl o gM[Ml o gM + logT].

(B.2)

APPENDIX B. APPENDIX B

The reasoning is that more information is available (it is known if the demand is higher then the current stock or exactly equal) which leads to a good decision within a shorter time span. In this same line of reasoning, we expect that our proposed adjustment will also lead to a faster rate of convergence, which could be explained by the following example.

Assume a nonperishable product with an actual stoking level of y_{t= 25, an order up to level of} ˆyt= 20 and the stock on hand of xt= 25. In the case of a perishable product, all 25 products, left

over from the day before, have to be thrown away and 20 new products are ordered. In this case only a sales of maximum 20 can be observed. Whereas, in the case of a nonperishable product, all products would be kept and a maximum sales of 25 could be observed. So in the end, less information will be lost.

On the other hand, one can not easily conclude that the nonperishable case is always bet-ter then the partially censored case. It could be that there is no inventory left from the day before, such that we have exactly the order up to level. In this case the partially censored case outperforms the nonperishable case.

We could identify several cases as given in Table B.1, where ‘+’ indicates that all information about the demand is gathered and ‘-’ indicates that some of the information is lost.

Table B.1: Information loss for every different case. Stocking level Censored Partially Censored Nonperishable

Dt< ˆyt + + + Dt= ˆyt + + + Dt= yt+ 1 - + -Dt> yt+ 1 - - -yt≥ Dt= ˆyt+ 1 - + + yt≥ Dt> ˆyt+ 1 - - +

Table B.1 supports the line of reasoning that the nonperishable case will have a faster convergence then the censored case, because we could obtain more information. It does not give any indication about the difference in rate of convergence between the partially censored and the nonperishable case. We can conclude that the algorithm will have to balance the (extra) benefits of more information against the costs from having too much stock.

B.2. DELIVERY PATTERNS

B.2

Delivery Patterns

Our assumption is that we can place an order every single period such that the new inventory will arrive in the next period. We denote this by b = 1. When we can only receive new inventory every other period, b = 2, the optimum could drastically change. Then we have to take into account that all inventory of the second day will encounter holding cost. In other words, one has to include the underage and overage costs over all periods until the next order arrives. This will change the cost function to t+b X τ=t+1 C( y_{τ) =} t+b X τ=t+1 CoE(yτ− Dτ)++ CuE(yτ− Dτ)−. (B.4)

For b = 2, we can write

t+2

X

τ=t+1

C( y_{τ) = C}oE(yt+1− Dt+1)++ CuE(yt+1− Dt+1)−+ CoE(yt+2− Dt+2)++ CuE(yt+2− Dt+2)−

(B.5)

= CoE(yt+1− Dt+1)++ CuE(yt+1− Dt+1)−+ CoE((yt+1− Dt+1)+− Dt+2)++ CuE((yt+1− Dt+1)+− Dt+2)−

(B.6)

= Co©E(yt+1− Dt+1)++ E((yt+1− Dt+1)+− Dt+2)+ª + Cu©E(yt+1− Dt+1)−+ E((yt+1− Dt+1)+− Dt+2)−ª

(B.7)

using that the inventory of the second day is the inventory of the first day minus the demand of the first day, namely y_{t+2= (yt+1− Dt+1})+_{. One could suggest the simplification that D}

t+1< yt+1