A periodic inventory system of intermittent demand items with fixed lifetimes

Perishable items with a limited lifespan and intermittent/erratic consumption are found in a variety of industrial settings: dealing with such items is challenging for inventory managers. In this study, a periodic inventory control system is analysed, in which items are characterised by intermittent demand and known expiration dates. We propose a new inventory management method, considering both perishability and intermittency constraints. The new method is a modification of a method proposed in the literature, which uses a periodic order-up-to-level inventory policy and a compound Bernoulli demand. We derive the analytical expression of the fill rate and propose a computational procedure to calculate the optimal solution. A comparative numerical analysis is conducted to evaluate the performance of the proposed solution against the standard inventory control method, which does not take into account perishability. The proposed method leads to a bias that is only affected by demand size, in contrast to the standard method which is impacted by more severe biases driven by intermittence and periods before expiration.


Introduction
Inventory systems for perishable goods have been the focus of much attention in the academic literature, particularly for their application in common sectors of goods (e.g. food and pharmaceutical products). The assumption that an item can be stored indefinitely in warehouses does not hold for perishable goods, and this complicates their inventory control. Perishability is a broad topic in the literature, as confirmed by the extensive reviews of the relevant research, such as those of Raafat (1991), Goyal and Giri (2001), and Bakker, Riezebos, and Teunter (2012). The taxonomy drivers in the latter two reviews establish that the first key element to investigate is the lifetime of the item, which may be fixed, distributed according to certain probability distributions, or characterised by a time-inventory dependent deterioration rate (Kouki et al. 2014;White and Censlive 2015;. It is worth to remark indeed that deterioration can occur in various ways, but it must be distinguished from obsolescence, which refers to the loss of value due to technological changes or the entry of new products into the market. However, obsolescence has attracted little attention because an obsolete good is simply not reordered (Goyal and Giri 2001). In our study we regard the lifetime of goods as deterministic (i.e. known a priori). The second key element is the demand, which can be either deterministic or stochastic. Our contribution deals with a stochastic demand.
When perishable goods also exhibit intermittent consumptions, their inventory control results in a further complication due to the ineffectiveness of inventory systems for non-intermittent demand generation processes. Nevertheless, intermittency is relevant in several industrial settings. Spare parts are typical items of intermittent consumption, but intermittency could be also the consequence of batching decisions in the supply chain. Hence, food and pharmaceutical multi-echelon supply chains are contexts in which perishable goods, in particular those with fixed lifetimes (decaying products are not considered here), may also exhibit intermittency. To the best of our knowledge, no inventory models for perishable goods with intermittent demand are provided in the literature. Addressing this research gap is the objective of our study.
In this study, the periodic inventory system proposed in Teunter, Syntetos, and Babai (2010) is adapted to perishable items with fixed lifetimes. This work represents an extension of Balugani et al. (2017). Our inventory control model is validated through a two-level full factorial design experiment around the most significant variables, whereas in Balugani et al. (2017) only a scenario analysis was presented. The experimental results are statistically analysed with a linear regression, proving that the variables do not impact its performance and suggesting that the underlying model is unbiased. In addition, differently from Balugani et al., we conduct in this paper a more realistic numerical investigation where the demand distribution parameters are forecasted (using an appropriate intermittent demand forecasting method) and this is integrated in the inventory model.
The paper is organised as follows. Section 2 contains an overview of the background of two research streams, i.e. stochastic demand of perishable goods with fixed lifetimes and intermittent demand. Section 3 details the forecasting and inventory control models and their assumptions; Section 4 outlines the experiment designed to validate the model and the obtained experimental results. Section 5 provides conclusions and the research agenda.

Research background
Recent contributions have referred to the stochastic demand of perishable goods with fixed lifetimes. Minner and Transchel (2010) dynamically determined replenishment quantities for perishable goods with fixed lifetimes that satisfy multiple service-level constraints during a specific period, and they extended their model to non-stationary demand. Xin, Pang, and Limeng (2014) addressed a joint pricing and inventory control problem for stochastic perishable inventory systems, in which both backlogging and lost-sales cases were studied. They provided an approach able to deal with both continuous and discrete demand distributions. Similarly, Duan, Cao, and Huo (2018) dealt with the dynamic pricing and production rate for stochastic and price-dependent demand of items with fixed-lifetime in a continuous-time environment. Pauls-Worm et al. (2014) addressed the production planning of perishable products with fixed lifetimes when the demand is non-stationary; they formulated an MILP model containing a service-level constraint. Pauls-Worm et al. (2016) proposed another MILP model for a fill-rate constraint. Muriana (2016) addressed the normally distributed demand of perishable items with fixed lifetimes to reach the optimum lot size. She evaluated the probability of a product remaining in stock beyond the end of its lifetime, and determined the best order size, the time at which the inventory level drops to zero, and the cycle time minimising the expected total cost. Gutierrez-Alcoba et al. (2017) achieved the expected inventory level at different ages for the non-stationary stochastic demand of perishable items with fixed lifetimes. They also extended Silver's heuristic (Silver 1978) to deal with these conditions by means of analytical and simulation-based variants of the original heuristic. Janssen et al. (2018) adopted a periodic review system for the stochastic demand of items with fixed lifetimes, adding the closing days constraint as a typical feature of groceries.  showed the value of dual-sourcing in the context of perishable items with fixed lifetimes and a Poisson-distributed demand. They considered an age-based control with a base stock policy. Perishable items with stochastic demand and fixed lifetimes have also been studied by Kara and Dogan (2018), who proposed an aged-based replenishment policy solved by a reinforcement learning algorithm.
Intermittent demand can be characterised by two stochastic variables, the non-zero demand (i.e. demand size) and the time interval between two successive non-zero demands (i.e. the inter-demand interval). Croston's method (Croston 1972) is the seminal contribution to intermittent demand forecasting according to a normally distributed demand size and a Bernoulli probability of a demand occurrence. A simple exponential smoothing is applied to both variables when the demand occurs, and an estimator of the expected value of demand per period is then evaluated by the ratio of these estimators. Syntetos and Boylan (2005) proposed an approximately unbiased modification of Croston's method called the Syntetos-Boylan Approximation (SBA). Another modification of Croston's method was proposed by Levén and Segerstedt (2004). However, Boylan and Syntetos (2007) demonstrated that this leads to a more biased estimator than Croston's original method. Teunter and Sani (2009) compared several modifications of Croston's method, while Regattieri et al. (2005) compared other forecasting approaches for intermittent demand. Babai, Syntetos, and Teunter (2014) and Babai et al. (2018) addressed the intermittent demand forecasting issue for items with a risk of obsolescence. Machine learning techniques, in particular artificial neural networks, have also been used to forecast intermittent demand by exploiting their ability to deal with not linear processes without requiring any distributional assumptions (e.g. Gutierrez, Solis, and Mukhopadhyay 2008;Kourentzes 2013;Lolli et al. 2017). When the intermittent demand patterns also contain seasonal and trend components, Seasonal Auto Regressive Integrated Moving Average (SARIMA) modelling has shown promising results (e.g. Gamberini et al. 2010). Several researchers have recommended the use of the non-parametric bootstrapping approach to estimate the lead-time demand based on a large number of independent bootstrap replications from available data. A recent literature review on bootstrapping forecasting methods in the context of intermittent demand is presented in Hasni et al. (2018a). For a comparison between parametric and non-parametric approaches and a thorough investigation of bootstrapping, the reader can refer to Syntetos, Babai, and Gardner (2015), Sillanpää and Liesiö (2018) and Hasni et al. (2018b). Most of the above mentioned research has looked at the forecast accuracy of the forecasting methods and their inventory performance. A good discussion on the performance measures of intermittent demand forecasting methods is presented by Prestwich et al. (2014) and Petropoulos and Kourentzes (2015).
The literature provides a wide set of compound distributions for modelling the intermittent demand generation process and computing the parameters of the stock control policies. However, as emphasised by Babai, Ladhari, and Lajili (2015), as the data become more erratic the true demand size distribution may not comply with any standard theoretical distribution. Two demand generation processes are typically carried out. If time is treated as a discrete (integer) variable, demand can be assumed to be generated by a Bernoulli process, so that the inter-demand intervals are geometrically distributed. Otherwise, the Poisson demand generation is used, which leads to negative exponentially distributed intervals. When combining a Bernoulli or a Poisson demand arrival with a generic distribution of demand sizes, a compound distribution is obtained. These demand generation processes are generally conducted when modelling the re-order policies via statistical analysis. The statistical modelling of intermittent demand was conducted by Teunter, Syntetos, and Babai (2010) and Babai, Jemai, and Dallery (2011). An empirical goodness-of-fit investigation conducted by Syntetos, Babai, and Altay (2012) showed the good fit of compound Poisson distributions to thousands of spare parts characterised with intermittent demand. Lengu, Syntetos, and Babai (2014) combined issues of distributional assumptions for modelling purposes and item classification, while Syntetos and Boylan (2006) focused on the interaction between forecasting and stock control. They applied negative binomial distribution to model the demand, and completed a factorial experiment by simulating the behaviour of a periodic review system when combined with different forecasting methods (simple moving average, simple exponential smoothing, Croston's method, and SBA).
The proposed inventory model in this paper extends the Teunter, Syntetos, and Babai (2010)'s model by accounting for perishable items with fixed lifetimes. We also extend the work of Balugani et al. (2017) by conducting a numerical investigation of the inventory model where the demand distribution parameters are forecasted. Experimentally, an in-depth statistical validation is carried out to measure the method's performance.

Methodology
In this section, we first present the forecasting and inventory control assumptions considered, followed by the proposed inventory replenishment model.

Forecasting and inventory control assumptions
The standard compound Bernoulli intermittent demand model we consider in this study was proposed in Croston (1972). The demand d i of a period i is defined as: where p is the probability that a positive demand takes place in a period and φ(d i , 1) is the probability that a positive demand with size d i occurs during a single period. More generally, we denote by φ(x, y) the distribution function that a positive demand x occurs during y periods. The model is quite general and, as such, the demand size can assume any positive probability distribution, the most common ones being truncated Normal and Gamma. Given the model expressed in Equation (1), the Teunter, Syntetos, and Babai (2010) model, which is also considered in our work, estimates the demand using SBA (Syntetos and Boylan 2005) as follows: whereẑ i is the estimated size of a positive demand, after the positive demand z i−1 in period i − 1, and in i is the estimate inter-arrival between positive demands, calculated from the last inter-arrival in i−1 between positive demands. Both forecasts are used in Equation (4) to produce an estimate of the demandd i . Equations (2), (3) and (4) are updated only after a positive demand using the smoothing parameter α ∈ (0, 1) defines how much new data has an effect on the previous estimates. From Equation (3), an estimate of the probability p, denoted byp i , can also be obtained as follows: To compute an expected mean squared error MSE(z i ) for the positive demand, an exponential smoother is applied to the squared error, as in Teunter, Syntetos, and Babai (2010): where the smoothing parameter β ∈ (0, 1) does not necessarily equal the smoothing parameter α.
As in Teunter, Syntetos, and Babai (2010), the inventory replenishment model considered in this study is the periodic order-up-to (T,S) policy where the order-up-to-level S is calculated to satisfy a target fill rate service level.
In the following subsection we propose a solution to calculate the order-up-to-level by taking into account the perishability constraint.

Development of the inventory replenishment model
To derive the fill rate expression when considering the perishability constraint, we first present the fill rate model that is considered. An order can be placed every t periods, collectively defining the constant review time, and requires a fixed lead time l to arrive, with l ≤ t. At the beginning of a review time in this scenario, a stock s ≥ 0 is available and an order o can be placed. The total amount s + o is expected to cover the demand of t periods after the lead time, and a new order can in fact be placed only after t periods and requires l periods to arrive. The performance measure associated with this model is the fill rate over the specified t periods, defined as the probability of a positive demand taking place in one of those periods that is satisfied by o + s and thus generates no stock-out.
Given a stock s and order quantity o, the fill-rate fr is: where (x, y) is the cumulative distribution function that a positive demand x occurs during y periods while p is the probability a period yields a positive demand. The replenishment model that takes into account the perishability constraint divides the stock s into two separate stocks: • s e the amount of goods that will expire at the end of one of the t periods after the lead time l.
• s ne the amount of goods that will not expire in the given time frame.
These quantities are updated as in the Teunter, Syntetos, and Babai (2010)'s model at the beginning of each period, before the order o is placed. The expired stock is discarded and the expiring stock is moved from s ne to s e . The stock s e is assumed to expire at the end of period t e , calculated from the update period before the lead time, while the ordered quantity o is assumed not to expire in the time frame.
Given a hypothetical positive demand d i at period i, two mutually exclusive cases can arise: • The period i occurs before the expiration date.
• The period i occurs after the expiration date.
In the first case, the perishability has no effect, thus Equation (7) is used. In the second case s e has expired and a different Equation is required. As in Section 2.1, given a positive demand d i in period i after the lead time l, all the possible demands in the previous periods i+l−1 k=0 d k must be considered. This leads to two scenarios: • The demands before the expiration date partially or totally consumed the expiring stock, i.e.
• The demands before the expiration date consumed more than the expiring stock, i.e. s e < t e +l−1 The fill rate f r i1 of the first scenario is the probability that the demands before t e are satisfied by s e and the demands after t e including d are satisfied by s ne + o: where: is a notation shortcut for the probability that x periods over y present a positive demand, and: as in absence of positive demands no stock out can occur.
The fill rate f r i2 of the second scenario is the probability that the demands d be before t e are satisfied by o + s and the demands after t e including d i are satisfied by the remaining stock o + s − d be with d be > s e : These scenarios are mutually exclusive, thus the fill rate f r i of period i is: The overhaul fill rate fr accounts for the individual fill rates of t periods after the lead time, as in Equation (7): This methodology expands that defined at the beginning of this section, by considering a portion s e of the stock as perishable. The calculations above refer to a single expiration date, but similar considerations can be applied to address multiple expiration dates in the frame of analysis. From a computational perspective, the proposed methodology is more demanding than the original, and the calculation of f r i2 requires an analysis of o + s ne demands before t e . This calculation is necessary as the last component of Equation f r i2 , defining the probability a demand after t e does not produce a stock out, and requires the number of units o + s − d be left in stock.

Inventory replenishment model: computational solution
The model presented in Section 2.2 aims to define the order quantity o min at the beginning of lead time l. o min is the minimum order capable of achieving a target fill rate f r target for t periods after the lead time l. In contrast, Equation (13) calculates the fill rate fr of t periods after the lead time l given a predefined order quantity o. Equation (13) is not easy to invert, thus no direct equation is available to solve the problem at hand. A common solution in the relevant literature involves a stepwise search: Step 1. Start assuming an order quantity o = 0 .
Step 2. Calculate fr for the value of o under analysis.
Step 3. If fr ≥ f r target then stop, o min = o.
Step 4. If fr < f r target then increment o by one unit and go to Step 2. This procedure is feasible if the computational cost for the fill rate calculation is limited. In our case such cost is significant and increases with o, thus the algorithm reactivity decreases as it goes on.
An alternative procedure, based on the secant method, is proposed to decrease the amount of calculations involved. The optimum is formally defined as: where fr(o, s e , s ne ) is the fill rate relative to the order quantity o and the stocks s e and s ne . As for fixed stocks s e and s ne the fill rate can grow only if o increases, Equation (14) can be rewritten as: Two properties of the fill rate, as calculated in Equation (13), provide two extremes o sup and o inf to initialise the secant method. This initialisation requires no initial calculation of Equation (13) itself: • Ceteris paribus, a decrease in s e reduces fr.
• Ceteris paribus, substituting part of s e with stock not expiring in t increases fr.
From these properties, two quantities can be defined: with the property: In Equation (16), starting from the optimum order quantity as defined in Equation (15), the elimination of s e reduces fr. From this point, to achieve fr(o, 0, s ne ) ≥ f r target fixed s ne , the order quantity now defined as o sup increases. A similar effect takes place in Equation (17) where the expiring stock is fully substituted by non-expiring stock. The substitution increases the fill rate, and for this new configuration the initial order quantity is no longer the minimum required to achieve fr(o, 0, s ne ) ≥ f r target . The order quantity now defined as o inf decreases to reach the required minimum fill rate.
Equations (16) and (17) contain no expiring stock, and thus the computationally expensive calculations of Section 2.2 are not required. Equation (7) is iteratively applied to define both o sup and o inf .
To apply the bisection method, the fill rate expressed in Equation (13) is shifted by f r target : The fill rate strictly increases, as a function of o, if s e and s ne are fixed. If Equation (19)  Given o sup and o inf the calculation of their fill rate using Equation (13)

Probability distribution and estimations
To calculate Equation (13), both the distribution function φ(x, y) and the cumulative distribution function (x, y) of a positive demand x during y periods must be known. The probability p is also required to be known, in which a positive demand occurs during a period. These three components of Equation (13) vary across time and must be indirectly forecast from the item time series. As suggested in Teunter, Syntetos, and Babai (2010), the positive demand distribution (both cumulative and not cumulative) is hard to determine over an arbitrary number of periods, unless the multiple periods distribution can be defined from the single period distribution. This experimental analysis assumes that the positive demand during a single period follows a negative binomial distribution. The sum of independent negative binomial distributions is a negative binomial distribution itself, with different parameters depending on the number of random variables added. In our case, the number of random variables is the number of periods. The use of a discrete random variable, instead of a continuous one as in Teunter, Syntetos, and Babai (2010), is coherent with Equation (11) where d be moves through integer values.
To estimate the single period parameters of the negative binomial distribution, a time series analysis is required. The methodology used for this experimental analysis is the same applied in Teunter, Syntetos, and Babai (2010). The forecasting technique described in Section 2.1 is applied to definep, z i and MSE(z i ) and the parameter α is optimised over the initial warmup periods. From z i and MSE(z i ) the negative binomial distribution parameters are derived using the method of moments.

Experiment settings
The experimental analysis consists of two experiments, carried out with different parameters, where the proposed methodology is tested on a generated series and compared against the case where the standard order-up-to-level (T,S) policy is applied without taking into account the perishability constraint. In these simulations both methodologies consume the stock following a FIFO policy, which is in line with a fixed number of periods before expiration. Intermittent demands following Equation (1) are generated, and their positive demand size follows a negative binomial distribution, while the probability p that a positive demand occurs is time invariant. If the demand size distribution yields a null demand it is still considered a positive demand to avoid uncontrolled changes in p.
The fixed and variable parameters, varying in each simulation, for experiments 1 and 2 are summarised in Table 1. In both experiments the values for n e are greater than those for t to avoid expirations before the end of a single cycle. All the possible combinations of the second set of parameters are tested 10 times to assess the method performance in different contexts.
• Low lumpiness ( The aim of these experiments was to identify a structure in the algorithms' behaviour in order to be able to define outperforming regions for the two methods.
The results are collected for each simulation period after the first lead time, when the first order has already arrived. Using this precaution, no initial level of backorders and stock is required to make the first measurements fair.

Performance metrics
Performance measurements are recorder after each simulation period. If the period presents a positive demand, then the total number of positive demand in the simulation is updated. Simultaneously, the performance record keeps track of the number of positive demand that have been satisfied from the stock, not adding to the backlog. The ratio between these two raw measurements is the fill-rate of the simulation f r sim . At the beginning of each cycle the optimal order quantity o min is defined. In this context parameters s e or s ne cannot be changed and only positive values of o min are produced. The fill rate is thus set to achieve fr(o min , s e , s ne ) ≥ f r obj . This goalsetting leads to difficulties when comparing f r sim and f r obj as, by construction, on average f r sim ≥ f r obj thus the difference f r sim − f r obj is designed to be ≥ 0. To avoid this unfair comparison the values of fr(o min , s e , s ne ) are collected in each simulation as they are generated, and their average is compared with f r sim instead of f r obj : To measure the benefit of using the proposed method rather than the standard method that does not take into account perishability, we also calculate f r New , which is expressed using Equation (23). Note that by the standard method we mean the order-up-to-level (T,S) inventory policy without taking into account the perishability constraint as described in Teunter, Syntetos, and Babai (2010). Obviously the standard method is expected to be biased and to underachieve the target fill rate when the perishability is not taken into account. This is analysed in the following subsection.
where f r pr is the fr of the proposed method and f r st is the fr of the standard (T,S) method.

Results
Tables 2 and 3 summarise the average f r pr , f r st and f r New results obtained for each combination of parameters. The values for E(z) and √

MSE(z) E(z)
differ from those provided in Section 4.2 since the negative binomial the generated distribution, used for data generation, requires one of its parameters to be a natural number. Not any combination of E(z) and √

MSE(z) E(z)
is allowed and their values end up changed when the parameter is rounded.
The standard (T,S) method is biased, leading to fill-rates that are always lower than the target ( f r st < 0) in experiment 1 (Table 2), and lower than the target 93.7% of the times in experiment 2 (Table 3). In the proposed method, this bias is corrected and, as a result, the f r pr values are affected only by random error and thus characterised by varying signs. On average the new method produces a 9.78% fill-rate performance increase for perishable items in experiment 1 and a 5.53% increase in experiment 2.
A linear regression is fitted over the non-averaged simulation results using p, E(z), and n e , in the first experiment, and p, E(z), , n e , t and l, in the second experiment, as features to separately predict fr and f r New . The values of used as features are obtained from the negative binomial distribution parameters used for data generation, they differ from those listed in Tables 2 and 3 as one of the distribution parameters must be an integer and is rounded when calculated from the original E(z) and are correlated (r = 0.64), while the others are not. The way this issue is handled is outline below. The linear regressions coefficients and their t-tests are summarised in Table 4.
The results of experiment 1 indicate that the performance of the proposed method is not affected by the simulation parameters, while experiment 2 shows that E(z) and together influence f r pr . The results of experiment 1 are highlighted by the t-tests in Table 4, which show no significance (a significant result would yield a p-value lower than 0.05). The correlation between E(z) and could hide a significant model behind non-significant t-tests. Consequently, to assess this scenario an F-test was performed which resulted in a non-significant p-value of 0.454. The results for experiment 2 are in line with findings from experiment 1. No t-test in experiment 2 reached a significant p-value, while the F-test in experiment 2 yielded a significant p-value of 0.005. Performing a PCA on standardised E(z) and √

MSE(z) E(z)
and re-fitting the regression model revealed that those features in combination are the only ones that influence f r pr . The findings of experiments 1 and 2 are not inconsistent. In fact the tests in experiment 2 leverage more simulations, resulting in a higher power and, as a result, the conclusions drawn from experiment 2 are more accurate. Figure 1 highlights the coherence between experiments 1 and 2 and plots the confidence intervals (one std) for p, E(z), and n e coefficients in experiment 1 (x) and experiment 2 (*). The coefficients for the most precise experiment are included in those for the least precise one.
According to experiment 1 (Table 4), the standard (T,S) is unaffected by demand size or variability as the p-values obtained are significantly higher than 0.05. The parameters affecting f r st , in addition to the intercept, are intermittence and periods before expiration. Experiment 2 confirms these findings, moreover re-fitting both experiments regression models after a PCA on standardised E(z) and √

MSE(z) E(z)
does not highlight any impact of such features over f r st . The coefficients in experiments 1 and 2 are less coherent on their impact on f r st than those for f r pr depicted in Figure 1, suggesting more complex phenomena that cannot be defined with a simple linear model. As a robustness test, the impact of the intermittent demand assumption on the proposed method is measured by rerunning experiment 1 while using a SES and erroneously assuming p = 1. The results are found to be reliant on the intermittence assumption as the f r pr obtained in this scenario is even higher than the f r st achieved in experiment 1.

Conclusions and further research
Managing the inventories of perishable items is a key lever that enables inventory costs to be reduced by reducing waste, thus increasing the level of customer service. Managing the inventories of perishable items becomes a more challenging task when the demand for such items is intermittent. This study provides the first attempt in the literature to overcome this challenge. We have proposed a new methodology that modifies the standard order-up-to-level (T,S) policy for intermittent demand (Teunter, Syntetos, and Babai 2010), which analytically derives the target fill rate for a compound binomial demand generation process in order to take into account the perishability constraint as well. We have also proposed an analytical expression of the fill rate under the new method, and due to the computational complexity to calculate the fill rate we have developed a procedure to obtain the optimal solution. We conducted a simulation experiment to analyse the performance of the standard and the proposed methods.
The results of this study show that when a proportion of the stock is affected by perishability, the proposed methodology leads to a considerable benefit by reducing the bias in the fill rate, unlike the standard method. The experiments reported in Section 3 demonstrate that the proposed methodology bias is only affected by demand size. On the other hand, intermittence, lumpiness, or number of periods before expiration do not impact its performance. The standard method is also proven to be unaffected by lumpiness, its effectiveness is only dictated by the number of periods before expiration and intermittence. From a computational standpoint the new methodology is significantly more expensive than the old one, as a combinatorial number of cases must be analysed, so practitioners are advised to apply the new methodology to scenarios characterised by high intermittence and low demand size. The use of simulation techniques to manage multiple expiration dates is also advised, to overcome the difficulty of determining analytical solutions in this case, since this can provide reliable results in exchange for a reasonable computational effort. Further research efforts are expected to gauge the effectiveness of simulation techniques and compare them with the available analytical solutions. Alongside this research avenue, efforts will be directed towards the characterisation of different compound Bernoulli distributions, with the aim of encompassing both integer and continuous positive demand sizes (Syntetos, Babai, and Altay 2012;Syntetos, Lengu, and Babai 2013). Other distributions such as Compound Poisson distributions (Babai, Jemai, and Dallery 2011;Lengu, Syntetos, and Babai 2014) or Compound Erlang distributions (Saidane et al. 2013) have also been used to model intermittent demand and can be considered in future research. Once this characterisation is achieved, comparisons between different distributions can take place and the effect of incorrectly selecting the demand size distribution can be quantified. The choice of an incorrect demand size distribution could seriously impact the performance of the inventory system. Another interesting avenue for further research would be to analyse the combined service and cost efficiency of the proposed methodology when compared to the standard one. Finally, it would be interesting to empirically show the benefits of the proposed model through an empirical investigation with real data, as it has been done in Teunter, Syntetos, and Babai (2010).