Performance measures of nonstationary inventory models for perishable products under the EWA policy

European Journal of Operational Research 303 (2022) 1137–1150 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Production, Manufacturing, Transportation and Logistics Performance measures of nonstationary inventory models for perishable products under the EWA policy Carlos Gorria a , Mikel Lezaun a , F. Javier López b , ∗ a Dpto. Matemáticas, Universidad del País Vasco - UPV/EHU, Spain b Dpto. Métodos Estadísticos and BIFI, Universidad de Zaragoza, Spain a r t i c l e i n f o Article history: Received 11 February 2021 Accepted 9 March 2022 Available online 13 March 2022 Keywords: Inventory Perishable Periodic review Stochastic demand Platelets a b s t r a c t Accurately estimating key performance indicators in inventory models for perishable items is essential in order to assess and improve the management strategy of these systems. We analyse the production of platelet concentrates at blood banks under the EWA replenishment policy. We give analytical approximations of the most important performance measures, such as the size of orders, the size of stocks, the percentage of outdating, the age distribution of stocks and the freshness of units issued, among others. The production of platelet concentrates is a prototypical example of inventory models for short life items with random demand and a weekly pattern, where a high service level is required. The methodology and the approximations presented here can be easily adapted to other inventory systems with similar characteristics. Most of the formulae in this article are new for nonstationary models under the EWA policy; indeed, formulae for the age distribution of units in stock and of units issued have not appeared in the literature even for the simpler base-stock replenishment policy. We apply our results to a real blood bank and find very close agreement between the formulae and the results of Monte Carlo simulations. The accuracy of our approximations is also tested in several scenarios, depending on the lifetime of units, safety stock levels and the probabilistic distribution of demand. © 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) 1. Introduction 1.1. Background and motivation Inventory management of perishable items is of great importance in many sectors of the economy. Food and blood products are just a couple of examples of perishable goods. The mathematical analysis of inventory models for these products is much more difficult than for nonperishable items. Also, although there are a great many research papers on models for perishable goods they are still far fewer in number than the papers and books devoted to nonperishable products; see, e.g., Silver, Pyke, & Peterson (1998) . Much of the research on inventory management for perishable products has focused on blood products, and has been published both in medical and mathematical journals; see Atkinson, Fontaine, Goodnough, & Wein (2012) , Beliën & Forcé (2012) , Civelek, Karaesmen, & Scheller-Wolf (2015) , Ensafian & Yaghoubi (2017) , Rajendran & Ravindran (2019) . Blood products are used for ∗Corresponding author. E-mail addresses: [email protected] (C. Gorria), [email protected] (M. Lezaun), [email protected] (F.J. López). transfusion in most hospitals and are seen as a scarce, precious resource. They are of vital importance for patients, so a sufficient stock must be kept in order to avoid stockouts. Different components of blood, such as red blood cells, plasma and platelets, are used for transfusion. Among them, platelet concentrates are considered as a critical product since they have a short lifetime (usually 5 or 7 days). They are also expensive (for instance, Haijema, van der Wal, & van Dijk (2007) assume a cost of more than 450 Euros per patient per treatment), and overcautious policies in keeping big stocks result in a large number of outdated units, leading to an unnecessary waste of money and ethical concerns. In this paper we focus on a periodic review model for fixed lifetime perishable goods such that stockouts must be kept to a minimum. Platelets are a clear example of such products, and we use the inventory model of the Basque Centre for Transfusion and Human Tissues (CVTTH) in Galdakao, Bizkaia, Spain, for the derivation of our formulae. This research originated in collaboration between the University of the Basque Country and the CVTTH for the implementation of a mathematical model for the management of blood products; see Pérez Vaquero, Gorria, Lezaun, López, Monge, Eguizabal, & Vesga (2016) , Gorria, Labata, Lezaun, López, Pérez Aliaga, & Pérez Vaquero (2020) . The paper focuses on this model for the https://doi.org/10.1016/j.ejor.2022.03.018 0377-2217/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ) C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 production of platelet concentrates, but our analysis can easily be adapted to other perishable goods where the service level needs to be high. 1.2. Objective of the paper In practice, it is very useful to have approximations of the performance measures of an inventory system, which can be used to validate policies and optimise the parameters of the model. When such approximations are not available, validation and optimisation must rely on simulations. This paper sets out to derive analytical approximations for the main performance measures of nonstationary models when the Estimated Withdrawal and Aging (EWA) replenishment policy is used and a high service level is needed. EWA is a modification of the base-stock replenishment policy where the inventory position is modified by subtracting an estimation of the amount of outdating, for placing a new order. We do not assume any specific form for the probabilistic distribution of the daily demand. Among others, we obtain formulae for the expected on-hand inventory, the probability of stockout, expected outdating, the age distribution of stocks and the freshness of units issued. The models that we consider are not stationary, but weekly stationary: the weekly pattern exists in the distribution of the demand and in the operational assumptions of the system (for instance, orders are placed every weekday but not on weekends). Since our model has a weekly pattern, all the formulae are obtained for each day of the week. We derive the approximations for a particular model to be described in detail in Section 2 . This may be seen as somewhat restrictive, but we see two advantages in it. First, the model is realistic, since it is very close to the operation of a real blood bank. Second, it includes a variety of situations (e.g. not all days have the same lead or review times and units arriving on Mondays have a different remaining lifetime than units arriving on other days); thus, the reasoning and derivation of the formulae for the model can be adapted without difficulty to other systems. To the best of our knowledge, this is the first paper where approximations for performance measures are given when the model is nonstationary and the EWA policy is used (except for the fill rate, which has already been approximated in van Donselaar & Broekmeulen, 2011 ). A key point in our analysis concerns the formulae for νi t , the expected number of units ordered on day twhich are in stock at the end of day t + i , derived in Section 3.3 . These formulae enable us to give approximations for many performance measures of the model, such as expected outdating. Freshness, defined as the expected remaining lifetime of units issued, can also be easily computed from νi t . In fact, we give more comprehensive information on units issued by deriving approximations for the values w r t , the expected number of units issued on day twith rdays of remaining lifetime. We also use νi t to compute the age distribution of the stock, i.e. the expected number of units in stock with remaining lifetime 1 , . . . , m at the beginning of a day in the long run. This distribution gives full information on the behaviour of the system. We point out that age distribution of stock in nonstationary models for perishable items has not appeared in the literature even under the simpler base-stock policy. While in most instances of inventory systems demand is discrete in nature, continuous distributions are often used to model it. Throughout the paper we assume that demand is well modelled by a continuous distribution, so our formulae are expressed using integrals and probability density functions (PDF); if a discrete distribution for demand is to be used, then, PDFs must be replaced by probability mass functions and integrals by sums. 1.3. Literature review There is a great variety of mathematical models for perishable items. They differ in many characteristics, such as deterministic or random demand, fixed or random lifetime, zero or positive lead time, stationary or time-varying demand, among others. We work with a random demand, fixed lifetime model here, so we restrict ourselves to that setting for the rest of the paper. An excellent review of the research published on inventory models for perishable items can be found in Nahmias (1982) for early papers on the subject; in Raafat (1991) for papers up to 1991; in Goyal & Giri (2001) for publications from the early 90s to 20 0 0; and in Bakker, Riezebos, & Teunter (2012) , Janssen, Claus, & Sauer (2016) and Chaudhary, Kulshrestha, & Routroy (2018) for more recent work. The first mathematical studies on inventory systems for perishable items set out to find optimal solutions in terms of minimising cost functions. However, in contrast to what happens for nonperishable items, where optimal solutions are known in a wide variety of settings, researchers found that models for perishable items were much harder to analyse, at least when demand was random. Thus, optimal solutions were obtained only in a limited number of situations such as very short product lifetimes ( m = 1 , 2 ) or zero lead time; see Nahmias (1982) and references therein. One way of finding an optimal solution is by using dynamic programming, which is a suitable tool for these models, taking a state space defined by the age distribution of the stock and a stochastic transfer function (see Nahmias, 1975 ). This technique solves, at least theoretically, the problem of finding a policy which minimises the cost function subject to a service constraint. However, due to the “curse of dimensionality” of dynamic programming, the state space of the problems becomes huge even for moderate values of m (lifetime) and maximum storage capacity and the problems become unsolvable in practice. In the last few decades the increasing speed and capacity of computers have led to a reemergence of this technique, although it still needs to be combined with aggregation of states or simulation to find solutions in a reasonable time. Haijema et al. (2007) , who combine dynamic programming with simulation, work with a model whose state space is larger than 10 8 , which implies a complexity of the order of 10 13 for one week iteration, so a downscaling of four to one units is carried out. Algorithms based on aggregation of states in multiple levels are proposed in Voelkel, Sachs, & Thonemann (2020) . Another approach for finding good policies in inventory models is discrete event simulation. It consists of modelling the system and implementing it in simulation software. By running the simulation with different policies and in various settings, the performance of the policies can be compared with a view to choosing the best. Simulation has been widely used to model real blood banks. For instance, Rytilä & Spens (2006) compare different scenarios of production and distribution of blood components in Finland. Asllani, Culler, & Ettkin (2014) build a model for a blood bank centre supplying 50 health care facilities in the US to search for the best platelet production policy in the week, when platelets are differentiated by blood type. Dalalah, Bataineh, & Alkhaledi (2019) use a simulation-optimisation approach to find an optimal policy when demand is differentiated by the age of platelets, and apply it to Kuwait public hospitals. Gorria et al. (2020) use data from two blood banks in Spain to study the decrease in outdating when the lifetime of platelet concentrates is extended from 5 to 7 days via pathogen reduction technologies and analyse what days of the week are most appropriate for applying these technologies. An advantage of the simulation approach is that the model can be as realistic as desired. However, no analytical expressions of the optimal solution or the performance measures of the model are obtained, which prevents the parameters of the model from being in1138 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 terpreted directly; moreover, simulations must be run every time a change in the parameters is observed. Yet another approach is to use heuristics to find a good solution. This approach does not seek to find the best of all feasible solutions, but rather to propose reasonable, easy-to-implement policies which perform well in practice. Many of these policies are myopic, in the sense that they make period-by-period decisions, and/or involve a simplification of the state space (for instance, using a simple function of the composition of the stock instead of its complete age distribution). Nandakumar & Morton (1993) describe heuristic solutions in the case of zero lead time which are close to optimal. For positive lead time, Chiu (1995) develops a solution based on a period-by-period optimisation of an approximation of the cost function which only takes into account the size of the stock but not its age distribution. Haijema & Minner (2019) give an overview of some of the most important stock-age dependent order policies, propose new ones and compare them in a broad set of scenarios. One heuristic developed for periodic review and fixed lifetime models, which yields good results, is the EWA policy, introduced in Broekmeulen & van Donselaar (2009) . It is known (see Broekmeulen & van Donselaar, 2009; Haijema & Minner, 2019 ) that the EWA policy significantly outperforms the base-stock policy. The EWA policy has been analysed by van Donselaar & Broekmeulen (2011) , van Donselaar & Broekmeulen (2012) , Broekmeulen & van Donselaar (2019) . van Donselaar & Broekmeulen (2011) give approximations for the fill rate both when demand is stationary and when it has a weekly pattern. In the case of stationary models, van Donselaar & Broekmeulen (2012) give analytical expressions to approximate the expected outdating; these approximations are then improved by simulating a large number of scenarios and fitting a regression model. Also for stationary models, Broekmeulen & van Donselaar (2019) propose several policies for reducing waste and increasing freshness. Freshness is a very important performance measure when dealing with perishable items, since waste due to outdating occurs very frequently at customer level: e.g. households for food ( Secondi, Principato, & Laureti, 2015; van Geffen, van Herpen, & van Trijp, 2020 ) or hospitals for platelets ( Flint, McQuilten, Irwin, Rushford, Haysom, & Wood, 2020; Pérez Vaquero et al., 2016 ). Moreover, fresher units are usually preferred by customers in the case of both food products ( Li & Teng, 2018 ) and platelets, since they have better properties than older ones, Caram-Deelder, Kreuger, Jacobse, van der Bom, & Middelburg (2016) , Aubron, Flint, Ozier, & McQuilten (2018) . Broekmeulen & van Donselaar (2019) were the first to obtain an approximation of freshness in an inventory model for perishable items. To estimate freshness, they propose an expression based on Little’s formula for queuing theory; the expression uses an estimation of expected outdating, so simulations must be run and a regression model fitted as in van Donselaar & Broekmeulen (2012) in order to compute the approximation of freshness. A more comprehensive description of the performance of an inventory system for perishable items is achieved by computing the age distribution of the stock. Formulae for the age distribution of stock are challenging to obtain even in the stationary case and under the base-stock policy. They have been computed only for continuous-review models assuming that demand follows a Poisson process, using the theory of queuing networks. See Kouki, Legros, Babai, & Jouini (2020) and references therein. The rest of the paper is organised as follows. Section 2 describes the model used for developing our approximations and how the EWA policy applies to it. The formulae for the approximations are given in Section 3 . The accuracy of the approximations is assessed via comparison with Monte Carlo simulations in a real example in Section 4 . Section 5 shows the results for 72 scenarios and analyses the extent to which our approximations can be regarded as reliable. Conclusions and ideas for future work are given in Section 6 . The paper has four appendices with some additional formulae and information and a Supplementary Material file with tables related to Sections 4 and 5 . 2. The model For ease of exposition, we develop our results for a particular model, i.e. the production of platelet concentrates in the CVTTH. The characteristics of the model presented are common to many blood banks. For instance, a similar model is analysed by Haijema (2013) using dynamic programming. The formulae derived here can be easily adapted to any model with periodic review, stochastic demand, fixed lifetime and a weekly pattern. We consider a FIFO issuing policy (older items are issued first). FIFO is the most common issuing policy in the literature on perishable items, especially when dealing with blood products; it is known that the freshness of units issued is lower using a FIFO policy than with other issuing policies such as LIFO (newer items are issued first), but outdating is lower with FIFO than with LIFO; see, for instance, Cohen & Prastacos (1981) , Stanger, Yates, Wilding, & Cotton (2012) . There is daily demand from hospitals from Monday to Sunday whose distribution depends on the day of the week. Let D t be the random variable representing the demand on day t ≥1 . We take day t = 1 to be a Monday. As usual in these models, we assume that demands on different days are independent of each other. Similarly, we assume that the demand is weekly stationary, i.e. the probability distribution of D t+7 k is identical to D t for all 1 ≤t ≤7 , k ≥1 . We denote by F t the cumulative distribution function (CDF) associated with D t and its mean and variance by μt = E[ D t ] and σ2 t = V ar[ D t ] , respectively. For t, i ≥1 , the aggregated demand during the interval [ t, t + i ] is denoted by D t ,t + i = i  j=0 D t+ j . For t ≤s , let F t,s be the CDF of D t,s and μt,s = E[ D t,s ] and σ2 t,s = V ar[ D t,s ] the corresponding mean and variance. We also write F t,s with 1 ≤s < t ≤7 to denote the CDF of D t,s +7 , which has the same distribution as D t, 7 + D 1 ,s ; for instance, F 5 , 2 represents the distribution of D 5 , 9 , the demand from a Friday to the following Tuesday. Accordingly, for 1 ≤s < t ≤7 , let μt,s = E[ D t, 7 ] + E[ D 1 ,s ] and σ2 t,s = V ar[ D t, 7 ] + V ar[ D 1 ,s ] , the mean and variance of F t,s . In practice, historical data is used to fit a distribution for F t and to estimate μt and σt , t = 1 , . . . , 7 . The estimations of F t,s , μt,s and σt,s can be computed from the estimations of F t , μt and σt , t = 1 , . . . , 7 since we assume independence of the random variables D t . Platelet concentrates have a fixed lifetime of m days. We derive our formulae for general m ; when a specific value of m is needed we take m = 5 since this is the most common lifetime of platelet concentrates and the one used in Pérez Vaquero et al. (2016) . Production orders can be placed every day from Monday to Friday. This is equivalent to saying that there is a review interval of one day from Monday to Thursday, R = 1 , and of three days on Friday, R = 3 . An order means that blood is processed immediately after the order and platelet concentrates are produced during the day. If the order is placed on any day between Monday and Thursday, the concentrates are ready for use in the morning of the following day, with a remaining lifetime of m days; orders placed on Friday are ready for use on Monday morning, with a remaining lifetime of m −2 days. That means that the lead time is L = 1 for orders placed from Monday to Thursday and L = 3 for orders placed on Friday. Note that this model is not daily stationary, but weekly stationary: the distribution of demand, the review interval and the 1139 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 lead time depend on the day of the week. We assume that the service level is high, which is customary in blood banks, and that unsatisfied demand is not backlogged (in practice, stockouts are covered by an urgent request to a neighbouring bank). Orders are placed at the beginning of the day, taking into account the on-hand inventory and the arriving units but before the demand for the day is known. If a unit has not been used by the m th day of life it is disposed of; for instance, if m = 5 , concentrates ordered on Monday and not used by Saturday are discarded. The order of events each day is: (1) receive the incoming order (only for weekdays); (2) place the new order (only for weekdays); (3) observe and meet the demand; (4) dispose of outdated units. 2.1. The EWA policy Ours is a typical model for perishable items with fixed lifetimes. As commented in the Introduction, there is no known optimal policy for such a model (weekly pattern, nonzero lead time, stochastic demand and lifetime greater than 2). A heuristic approach to find a reasonable solution is the base-stock policy, which is a simple order-up-to policy. The EWA policy is an improvement of that policy. We describe both these policies here. The base-stock policy has been widely used as a heuristic for stochastic demand, fixed lifetime inventory systems; see for instance Nahmias (1982) , Cooper (2001) and references therein. It has been also used as a benchmark for comparison with more complex policies: Tekin, Gürler, & Berk (2001) , Broekmeulen & van Donselaar (2009) , Duan & Liao (2013) , Haijema & Minner (2019) . The base-stock policy works like an order-up-to policy for nonperishable items. Its rationale is to have sufficient on-hand inventory to meet demand until the inventory replenishment corresponding to the next order. At each review point tan order of size Q B t = max { SS t + μt ,t + L + R −1 −IP t , 0 } (1) is placed. Here SS t is the safety stock for day t, μt ,t + L + R −1 is the expected demand from the placement of the order until the arrival of the next order and IP t is the inventory position. Note that both L and R may depend on the day of the week. When values are assigned to the subscript t + L + R −1 , the R corresponding to day tand the L corresponding to day t + R are taken. Let S t = SS t + μt ,t + L + R −1 be the order-up-to quantity of day t. This policy takes into account the inventory position for placing an order, but not its age distribution. The EWA policy, proposed by Broekmeulen & van Donselaar (2009) , works like the base-stock policy, but the inventory position is decreased by an estimation of the expected outdating between the placement of an order and day L + R −1 thereafter. The order quantity under this more sophisticated policy for day tis Q t = max { SS t + μt ,t + L + R −1 −IP t + ˆ E t , 0 } , where ˆ E t is an estimation of E t , the expected outdating during the interval [ t, t + L + R −2] . Note that the outdated quantity on day t + L + R −1 is not included because it does not affect the ability to meet demand that day. In the EWA policy, the estimation ˆ E t is computed assuming that demands during the interval [ t, t + L + R −2] are equal to their mean values. More accurate estimations of waste can be computed by using the exact distribution of demand instead of its mean, but they show little improvement and are very time-consuming (see Haijema & Minner, 2019 ). There is an irrelevant shift of the index of the days to be considered for outdating in the computation of ˆ E t , when compared to Broekmeulen & van Donselaar (2009) . In their paper the days to be considered are t + 1 , . . . , t + L + R −1 while we take t, . . . , t + L + R −2 . This is because in their study orders are placed at the end of the day but in ours they are placed at the beginning of the following day. To estimate E t , we need some notation. Let B r t , r = 1 , . . . , m be the number of units in stock (on-hand) at the beginning of day t with rdays of remaining lifetime after the arrival of new items, and B t = B 1 t + ···+ B m t . Note that B t is equal to IP t except for Saturdays and Sundays, where Friday’s order is included in IP t but not in B t . Let W r t be the number of units with rdays of remaining lifetime which are issued on day t. The following recursive formulae relate the outdated quantity O t on day tto the on-hand units B r t , the units issued W r t and the demand D t (see Broekmeulen & van Donselaar, 2009 ): O t = B 1 t −W 1 t = max B 1 t −D t , 0 , W r t = min  B r t , D t − r−1  k =1 W k t  , r = 1 , . . . , m, B r−1 t+1 = B r t −W r t + A r−1 t+1 , r = 2 , . . . , m, B m t+1 = A m t+1 , (2) where A r t is the number of units arriving on day twith rdays of remaining lifetime. The term A r−1 t+1 is not included in formula (6) of Broekmeulen & van Donselaar (2009) because in their case all units enter the system with m days of remaining lifetime. In our case, units arriving from Tuesday to Friday have m days of remaining lifetime while units arriving on Monday have m −2 . Due to the recursive nature of (2) , there is no simple way to express O t+ i as a function of B 1 t , . . . , B m t and D t , . . . , D t+ i . Moreover, since D t , ... , D t+ i are random, so are O t , . . . , O t+ i , but a deterministic value is needed for the latter in order to approximate the expected outdating E t . The EWA policy assumes that D t , . . . , D t+ L + R −2 are equal to their expected values, uses (2) to get the estimates ˆ O t , . . . , ˆ O t+ L + R −2 of O t , . . . , O t+ L + R −2 and then takes ˆ E t = ˆ O t + ···+ ˆ O t+ L + R −2 . 2.2. Application of the EWA policy to the model We first show the application of the base-stock policy to our model, which is needed for the EWA policy. Recalling (1) , and due to weekly stationarity, we need to define SS t , t = 1 , . . . , 5 , the safety stock for Mondays, Tuesdays, Wednesdays, Thursdays and Fridays, respectively (no orders are placed on Saturdays or Sundays). Note that the values of R and L depend on the day of the week. The values of R and L to be used for the order placed on day tare the time until the next order is placed ( R ) and the time between the placement of that order and its arrival ( L ). For Monday, t = 1 , the review interval is R = 1 , since a new order will be placed on Tuesday, and the lead time is L = 1 , since the order placed on Tuesday will arrive on Wednesday. Also, L = R = 1 for Tuesday and Wednesday ( t = 2 , 3 ). For Thursday, t = 4 , the review interval is R = 1 , since a new order will be placed on Friday, and L = 3 , since the order placed on Friday will arrive on Monday; for Friday, t = 5 , we have R = 3 , L = 1 . In other words, the period from tto t + L + R −1 corresponds to Mon-Tue for orders placed on Monday, Tue-Wed for orders placed on Tuesday, Wed-Thu for orders placed on Wednesday, Thu-Sun for orders placed on Thursday and Fri-Mon for orders placed on Friday. There are different ways to determine the safety stock for day t. A common option, for both nonperishable and perishable items, is to take it as a factor of the standard deviation of the demand; see Chapter 7 in Silver et al. (1998) . We take it as SS t = kσt ,t + L + R −1 + k 1 for t = 1 , 2 , 3 , kσt ,t + L + R −1 + k 2 for t = 4 , 5 , with k, k 1 , k 2 ≥0 , where the values of L, R depend on the day of the week as explained above. That is, the safety stock is proportional to the standard deviation of the demand to be covered, plus a fixed value ( k 1 or k 2 ); we allow different values of k j for Mondays, Tuesdays and Wednesdays, which cover only 2 day demand, 1140 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 Table 1 Random variables related to the inventory system. B r t t ≥1 ;r = 1 , . . . , m number of units in stock (on-hand) at the beginning of day twith rdays of remaining lifetime once the incoming order has arrived B t t ≥1 total number of units in stock (on-hand) at the beginning of day tonce the incoming order has arrived Q t t ≥1 order quantity of day t S t t ≥1 order-up-to quantity of day t O t t ≥1 number of units outdated on day t E t t ≥1 expected outdating in the interval [ t, t + L + R −2] W r t t ≥1 ;r = 1 , . . . , m number of units issued on day twith rdays of remaining lifetime V i t t ≥1 , i = 1 , . . . , m number of units ordered on day twhich are in stock (on-hand) at the end of day t + i before outdated units are discarded H t t ≥1 number of units in stock (on-hand) at the end of day tonce outdated units are discarded U t t ≥1 unsatisfied demand on day t and for Thursdays and Fridays, which cover 4 day demand. We decided to use only two parameters, k 1 and k 2 , instead of five different parameters k t , t = 1 , . . . , 5 , one for each weekday, for reasons of simplicity and efficiency. This assertion is based on the conclusions of a simulation-optimisation model for the management of platelets used in Pérez Vaquero et al. (2016) , where the optimal solutions in terms of few outdatings and freshness of units issued were found for the empirical data of 52 weeks in the CVTTH. In any event, the choice of this form of safety stocks does not affect the derivation of our formulae, and they can be straightforwardly adapted to any other form of safety stocks, such as taking a different parameter k t for t = 1 , . . . , 5 . We now turn to the application of the EWA policy. To compute ˆ E t , we consider several cases, depending on the day of the week. When day tis a Monday, from (2) , O t = (B 1 t −D t ) + , where a + = max { a, 0 } . Since the EWA policy assumes D t = μ1 , it follows that ˆ O t = (B 1 t −μ1 ) + . As L + R −2 = 0 in this case, ˆ E t = (B 1 t −μ1 ) + . The same expression is valid when day tis a Tuesday or a Wednesday (replacing μ1 by μ2 , μ3 , respectively). When day tis a Thursday or a Friday, L + R −2 = 2 . Thus, O t = B 1 t −D t + and ˆ O t = B 1 t −μt + . Also, O t+1 = B 1 t+1 −D t+1 + , with W 1 t = min B 1 t , D t , W 2 t = min B 2 t , D t −W 1 t , which yields B 1 t+1 = B 2 t −D t −B 1 t + + and O t+1 = B 2 t −D t −B 1 t + + −D t+1 + . The derivation of O t+2 = B 1 t+2 −D t+2 + is more involved. Note that B 1 t+2 = B 2 t+1 −W 2 t+1 , with B 2 t+1 = B 3 t −D t −W 1 t −W 2 t + and W 2 t = min B 2 t , D t −B 1 t + . We have B 2 t+1 = B 3 t −D t −B 1 t + −B 2 t + + , W 1 t+1 = min B 2 t −D t −B 1 t + + , D t+1 , W 2 t+1 = min B 2 t+1 , D t+1 −B 2 t −D t −B 1 t + + + . Therefore, B 1 t+2 = B 3 t −D t+1 + D t −B 1 t + −B 2 t + + . Collecting all the terms above gives O t+2 = ( B 3 t −D t+1 + D t −B 1 t + −B 2 t + + −D t+2 ) + . The value of ˆ E t is obtained by summing up O t , O t+1 and O t+2 and replacing D t , D t+1 and D t+2 by μt , μt+1 and μt+2 , respectively. The expressions are valid for general m . They are simple for Monday, Tuesday and Wednesday, but are rather complicated for Thursday and Friday. However, they get simpler when a concrete value of m is taken since some B r t are equal to 0. For instance, m = 5 gives ˆ E t = (B 3 t −μ4 , 6 ) + for Thursdays and ˆ E t = (B 2 t −μ5 , 6 ) + + (B 3 t −(μ5 , 6 −B 2 t ) + −μ7 ) + for Fridays. 3. Approximations of performance measures under the EWA policy We now derive analytical approximations of the main performance measures in the model. Table 1 summarises the random variables related to the inventory system. We use the same notation, with small instead of capital letters, for their expected values in the steady state. The model is weekly stationary, so these expected values depend on the day of the week, which means that the expected on-hand stock h t , say, is the same for all t + 7 k , k ≥0 . See Appendix A for a theoretical justification of the existence of the long-run distribution and its periodicity. In the rest of the Section we write the formulae for t = 1 , . . . , 7 only. On some occasions the subscripts in the formulae become negative or zero: for instance when day tis a Monday ( t = 1 ) and we write D t −3 ,t −1 ; in those cases the value tin the formula must be understood as t + 7 . In this section we keep m general as long as we can. When the approximations need a specific value for m , we take m = 5 . 3.1. Approximation of the order quantities Note that the order quantity of day t, Q t , is (S t −B t ) + , where S t = SS t + μt ,t + L + R −1 + ˆ O t + ···+ ˆ O t+ L + R −2 , and ˆ O t+ j is the estimation of the outdated quantity on day t + jin Section 2.2 . Approximating ˆ O t by o t gives an approximation of the order-up-to quantities: s t ∼μt ,t +1 + kσt ,t +1 + k 1 + o t , t = 1 , 2 , 3 s 4 ∼μ4 , 7 + kσ4 , 7 + k 2 + o 4 + o 5 + o 6 , s 5 ∼μ5 , 1 + kσ5 , 1 + k 2 + o 5 + o 6 + o 7 . (3) Note that, depending on the particular value of m , some of the o t are 0. For instance, if m = 5 , then o 4 = o 5 = 0 since there is no production on Saturdays or Sundays, so there is no outdating on Thursdays or Fridays. We now make two assumptions. The first is (B t −D t ) + ∼B t − D t , which is quite reasonable since the service level is high, so most days we have B t ≥D t and, even if B t < D t , the difference D t −B t = U t (unsatisfied demand on day t) is small. In fact, this assumption is common when analysing inventory systems: for instance, Silver et al. (1998) assert (p. 253) that a usual assumption for the inventory management of items with random demand is “Unit shortage costs (explicit or implicit) are so high that a practical operating procedure will always result in the average level of backorders being negligibly small when compared with the average level of the on-hand stock”. The second assumption is S t ≥B t for every weekday t, as otherwise the stock at the beginning of the day is very large and Q t = 0 , which is infrequent in many inventory models, such as models for the production of platelet concentrates in blood banks, where the size orders are positive at every review point. Now we approximate b t . For t = 2 , 3 , 4 , 5 , we have B t = ( B t−1 −D t−1 ) + + Q t−1 −O t−1 = ( B t−1 −D t−1 ) + + ( S t−1 −B t−1 ) + −O t−1 ∼S t−1 −D t−1 −O t−1 . 1141 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 By (3) , b t ∼μt + kσt−1 ,t + k 1 for t = 2 , 3 , 4 and b 5 = μ5 , 7 + kσ4 , 7 + k 2 + o 5 + o 6 . For Saturday, B t = (B t−1 −D t−1 −O t−1 ) + ∼S t−2 − D t −2 ,t −1 −O t−2 −O t−1 which yields b 6 ∼μ6 , 7 + kσ4 , 7 + k 2 + o 6 . For Sunday, B t = (B t−1 −D t−1 −O t−1 ) + ∼S t−3 −D t −3 ,t −1 − O t−3 −O t−2 −O t−1 and b 7 ∼μ7 + kσ4 , 7 + k 2 . Last, for Monday, B t ∼S t−3 −D t −3 ,t −1 −O t−3 −O t−2 −O t−1 , so b 1 ∼μ1 + kσ5 , 1 + k 2 . Since Q t = (S t −B t ) + ∼S t −B t , the approximations for the expected order quantities are q 1 ∼μ2 + k (σ1 , 2 −σ5 , 1 ) + k 1 −k 2 + o 1 q 2 ∼μ3 + k (σ2 , 3 −σ1 , 2 ) + o 2 q 3 ∼μ4 + k (σ3 , 4 −σ2 , 3 ) + o 3 q 4 ∼μ5 , 7 + k (σ4 , 7 −σ3 , 4 ) + k 2 −k 1 + o 4 + o 5 + o 6 q 5 ∼μ1 + k (σ5 , 1 −σ4 , 7 ) + o 7 Note that the formulae above depend on o t , t = 1 , . . . , 7 , which are unknown. We give approximations for their values in Section 3.4 . 3.2. Expected on-hand inventory Since H t = (B t −D t ) + −O t and stockouts are assumed to be uncommon, we can approximate H t ∼B t −D t −O t and the formulae in Section 3.1 give h 1 ∼kσ5 , 1 + k 2 −o 1 h t ∼kσt−1 ,t + k 1 −o t , t = 2 , 3 , 4 h 5 ∼μ6 , 7 + kσ4 , 7 + k 2 + o 6 h 6 ∼μ7 + kσ4 , 7 + k 2 h 7 ∼kσ4 , 7 + k 2 −o 7 3.3. A formula for v i t and the age distribution of the stock In this section we derive a formula for E[ V i t ] , the expected number of units ordered on day twhich are in stock at the end of day t + i before outdated units are discarded. First note that V i t = 0 for i = 1 , . . . , m when t = 6 is a Saturday or t = 7 is a Sunday; also, V 1 t = V 2 t = 0 if t = 5 is a Friday, since units arrive on Monday. For the rest of the values V i t , recall that Q t = (S t −B t ) + . Now, since the order placed on day tsets the inventory position to S t and we use a FIFO issuing policy, V i t can be approximated by (S t −D t ,t + i −(O t + ···+ O t+ i −1 )) + , with a limit of (S t −B t ) + . That is, V i t ∼min ( S t −D t ,t + i −( O t + ···+ O t+ i −1 ) ) + , (S t −B t ) + . In order to compute E[ V i t ] , we condition on B t . In what follows, we take S t as if it was deterministic, which is not true because under the EWA policy it depends on the age distribution of the stock. Let ˜ D t ,t + i = D t ,t + i + O t + ···+ O t+ i −1 , for t ≥1 , i = 1 , . . . , m . Note that if B t ≥S t , then V i t = 0 for every i = 1 , . . . , m ; if B t < S t , then E[ V i t | B t ] ∼(S t −B t ) P ( ˜ D t ,t + i ≤B t ) +  S t B t (S t −x ) f ˜ D t ,t + i (x ) dx =   { (x,y ): B t <y<S t , 0 <x<y } f ˜ D t ,t + i (x ) d xd y =  S t B t F ˜ D t ,t + i (x ) dx, where f ˜ D t ,t + i and F ˜ D t ,t + i are the PDF and CDF, respectively, of the demand in the interval [ t, t + i ] plus the outdated units in the interval [ t, t + i −1] . Thus, E[ V i t | B t ] ∼ S t B t F ˜ D t ,t + i (x ) dx if B t < S t , 0 if B t ≥S t . By the properties of conditional expectation, E V i t ∼ S t 0  S t y F ˜ D t ,t + i (x ) dx f B t (y ) dy =   { (x,y ):0 <y<x<S t } F ˜ D t ,t + i (x ) f B t (y ) d xd y =  S t 0  x 0 f B t (y ) dy F ˜ D t ,t + i (x ) dx =  S t 0 F B t (x ) F ˜ D t ,t + i ( x ) dx. (4) To apply formula (4) , we need approximations of F ˜ D t ,t + i (x ) and F B t (x ) . Since we approximate O t by its expected value o t , we have F ˜ D t ,t + i (x ) = P ( D t ,t + i + O t + ···+ O t+ i −1 ≤x ) ∼F t ,t + i (x −(o t + ···+ o t+ i −1 )) . The approximation of B t depends on the day of the week. When day tis a Monday, using the approximations in Section 3.1 , F B t (x ) ∼P (S t−3 −D t −3 ,t −1 −O t−3 −O t−2 −O t−1 ≤x ) ∼P (μ5 , 1 + kσ5 , 1 + k 2 −D t −3 ,t −1 ≤x ) = F 5 , 7 (μ5 , 1 + kσ5 , 1 + k 2 −x ) , where F (t) = 1 −F (t) . When day tis Tuesday, Wednesday or Thursday, F B t (x ) ∼P (S t−1 −D t−1 −O t−1 ≤x ) ∼P (D t−1 ≥μt−1 ,t + kσt−1 ,t + k 1 −x ) = F t−1 (μt−1 ,t + kσt−1 ,t + k 1 −x ) . with t = 2 , 3 , 4 , respectively. Analogously, for Friday: F B t (x ) = P (S t−1 −D t−1 −O t−1 ≤x ) ∼F 4 (μ4 , 7 + kσ4 , 7 + k 2 + o 5 + o 6 −x ) . The above expressions, together with (4) , yield the required approximations of v i t , t = 1 , . . . , 5 , i = 1 , . . . , m . For instance, the expected number of units that are ordered on Thursday and are in stock at the end of Sunday, v 3 4 , can be approximated by  μ4 , 7 + kσ4 , 7 + k 2 + o 4 + o 5 + o 6 0 F 3 (μ3 , 4 + kσ3 , 4 + k 1 −x ) ×F 4 , 7 (x −(o 4 + o 5 + o 6 )) dx. These formulae are explicit; however, they depend on o 1 , . . . , o 7 , the expected number of outdated units each day. In the next section we show how to estimate these quantities. Once they have been estimated, their values can be plugged into the above formulae to compute the approximations of v i t , since the distributions F t,s are known. The formulae above enable us to approximate the age distribution of the stock b r t . In fact, when day tis not a Monday, the number of units with rdays of remaining lifetime at the beginning of day t, once the incoming order has arrived, B r t , is V m −r t+ r−m −1 , for r = 1 , . . . , m −1 , and B m t = Q t−1 . When day tis a Monday, B r t = V m −r t+ r−m −1 , for r = 1 , . . . , m −3 , B m −2 t = Q t−3 and B m −1 t = B m t = 0 . The approximation of b r t is obtained by substituting the values of Qand V by the corresponding approximations of q and v . 3.4. Expected outdating In this section we set m = 5 . Derivation of the formulae when m is 4 and 6 can be found in Appendix B . Other values of m can be worked out in a similar way. 1142 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 Let m = 5 . Recall first that o 4 = o 5 = 0 since there is no outdating on Thursdays or Fridays. Now, since O t = V 5 t−5 , we use formula (4) with i = 5 , and get o 1 ∼ μ3 , 4 + kσ3 , 4 + k 1 + o 3 0 F 2 (μ2 , 3 + kσ2 , 3 + k 1 −x ) F 3 , 1 (x −o 3 −o 6 −o 7 ) dx, o 2 ∼ μ4 , 7 + kσ4 , 7 + k 2 + o 6 0 F 3 (μ3 , 4 + kσ3 , 4 + k 1 −x ) F 4 , 2 (x −o 6 −o 7 −o 1 ) dx, o 3 ∼ μ5 , 1 + kσ5 , 1 + k 2 + o 6 + o 7 0 F 4 (μ4 , 7 + kσ4 , 7 + k 2 + o 6 −x ) ×F 5 , 3 (x −o 6 −o 7 −o 1 −o 2 ) dx, o 6 ∼ μ1 , 2 + kσ1 , 2 + k 1 + o 1 0 F 5 , 7 (μ5 , 1 + kσ5 , 1 + k 2 −x ) F 1 , 6 (x −o 1 −o 2 −o 3 ) dx, o 7 ∼ μ2 , 3 + kσ2 , 3 + k 1 + o 2 0 F 1 (μ1 , 2 + kσ1 , 2 + k 1 −x ) F 2 , 7 (x −o 2 −o 3 −o 6 ) dx. (5) The formulae above are cyclical, since o s is needed to compute o t . We solve this via an iterative procedure: we set all o t = 0 , compute the formulae in (5) to get an approximation of o t and plug the new values into the formulae to get another approximation. This procedure is iterated until the changes in the o t are smaller than a tolerance value. While we have not proved analytically that this procedure converges, in all our settings below a small number of iterations (less than 8) were needed for a tolerance of 10 −3 . 3.5. Remaining lifetime of units issued and freshness For a given day t, the number of units issued with a remaining lifetime of rdays, W r t can be expressed as W r t = V m −r t+ r−m −1 − V m +1 −r t+ r−m −1 , for r = 1 , . . . , m −1 and W m t = Q t−1 −V 1 t−1 . This formula has two exceptions related to the weekend: when tis a Monday and r = m −2 , W m −2 t = Q t−3 −V 3 t−3 , and when tis a Saturday and r = m , W m t = 0 . Thus, the values w r t can be approximated by substituting Qand V by their approximations in Sections 3.1 and 3.3 , respectively. Also, freshness of units issued on day t, t = 1 , . . . , 7 can be approximated by  m r=1 rw r t  m r=1 w r t . Note that extending the formula derived by Broekmeulen & van Donselaar (2019) for freshness in stationary models to the present situation is difficult, since the distribution of the demand, the values of the lead time L and the review interval R , and the remaining lifetime of units when they enter the inventory depend on the day of the week. Thus, there seems to be no easy way of using Little’s formula in our context to find the freshness of units delivered each day of the week. Appendix C shows how that approach can be used to get a formula for the freshness of units without differentiating by the day of issue, although it still requires the approximations in Section 3.4 . 3.6. Expected shortage We are assuming that stockouts are rare, but they may still occur, so it is important to have an approximation of the expected shortage, u t , where U t = (D t −B t ) + . We begin when day tis a Tuesday, Wednesday or Thursday. Using the approximations for B t , S t and O t in the previous sections, U t ∼(D t −S t−1 + D t−1 + O t−1 ) + , so u t ∼E (D t−1 ,t −(μt−1 ,t + kσt−1 ,t + k 1 )) +  =  ∞ μt−1 ,t + kσt−1 ,t + k 1 (x −(μt−1 ,t + kσt−1 ,t + k 1 )) f t−1 ,t (x ) dx =  ∞ μt−1 ,t + kσt−1 ,t + k 1 F t−1 ,t (x ) dx, for t = 2 , 3 , 4 . When tis a Friday, U t ∼(D t −S t−1 + D t−1 + O t−1 ) + and u 5 ∼ ∞ μ4 , 7 + kσ4 , 7 + k 2 + o 5 + o 6 F 4 , 5 (x ) dx. When tis a Saturday, U t ∼(D t −S t−2 + D t −2 ,t −1 + O t−2 + O t−1 ) + , so u 6 ∼ ∞ μ4 , 7 + kσ4 , 7 + k 2 + o 6 F 4 , 6 (x ) dx. For Sunday, U t ∼(D t −S t−3 −D t −3 ,t −1 + O t−3 + O t−2 + O t−1 ) + and u 7 ∼ ∞ μ4 , 7 + kσ4 , 7 + k 2 F 4 , 7 (x ) dx. Lastly, when tis a Monday, U t ∼(D t −S t−3 −D t −3 ,t −1 + O t−3 + O t−2 + O t−1 ) + , so u 1 ∼ ∞ μ5 , 1 + kσ5 , 1 + k 2 F 5 , 1 (x ) dx. The fill rate of day tcan be approximated by 100(1 −u t /μt ) . 3.7. Probability of on-hand inventory being lower than a threshold The probability of the on-hand inventory at the end of day t (before outdated units are discarded) being less than a is P [ D t > B t −a ] , which can be approximated in a similar way to the previous section, obtaining: P (D t > B t −a ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ F 5 , 1 (μ5 , 1 + kσ5 , 1 + k 2 −a ) if day t is a Monday , F t−1 ,t (μt−1 ,t + kσt−1 ,t + k 1 −a ) if day t is a Tuesday , Wednesday or Thursday , F 4 , 5 (μ4 , 7 + kσ4 , 7 + k 2 + o 5 + o 6 −a ) if day t is a Friday , F 4 , 6 (μ4 , 7 + kσ4 , 7 + k 2 + o 6 −a ) if day t is a Saturday , F 4 , 7 (μ4 , 7 + kσ4 , 7 + k 2 −a ) if day t is a Sunday . In particular, taking a = 0 gives an approximation of the probability of a stockout, i.e. 1 −service level. 4. Application to the CVTTH data We assess the accuracy of our approximations by comparing them with the values obtained by Monte Carlo simulations in a real example. For the distributions of the daily demand we choose those fitted to the data of the CVTTH for 2012 (which were considered by Pérez Vaquero et al. (2016) ), i.e. (discretised) normal distributions with means and standard deviations as shown in Table 2 . The lifetime of platelet concentrates is m = 5 . The normality assumption was checked using the Shapiro test for normality. Significant p−values were found for Saturday and Sunday. Note however that the distributions of the demand on those days are never used on their own in our formulae; instead they are used at least together with Friday (for Saturday) and with Friday and Saturday (for Sunday). Therefore, what needs to be checked is not the normality of the demand on Saturdays and on Sundays but the normality on Friday + Saturday and Friday + Saturday + Sunday. Table 3 shows that the normality assumption is reasonable. We remark that the hypothesis of normality is not necessary for the derivation of our formulae: they can be applied with any other distribution, including the empirical distribution of historical data if available. Regarding independence of the daily demand, we performed the (Pearson) correlation test for each pair of consecutive days. Two of the pairs were found to be significant, namely Tue-Wed ( p−value 0,045) and Thu-Fri ( p−value 0,004). An analysis of the 1143 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 Table 2 Means and standard deviation for daily demand of platelet concentrates in CVTTH in 2012. Monday Tuesday Wednesday Thursday Friday Saturday Sunday μt 27,75 23,71 24,57 22,16 29,39 13,29 11,82 σt 6,85 5,65 7,86 6,90 7,81 4,89 4,38 Table 3 p−values of the Shapiro test of normality for demand of platelet concentrates in CVTTH in 2012. Mon Tue Wed Thu Fri Fri + Sat Fri + Sat + Sun 0,05 0,32 0,95 0,22 0,50 0,59 0,41 Table 4 p−values of the Pearson correlation test for independence of demand on consecutive days in CVTTH in 2012. Mon-Tue Tue-Wed Wed-Thu Thu-Fri Fr-Sat Sat-Sun Sun-Mon 0,10 0,06 0,38 0,11 0,67 0,07 0,79 scatterplots of these two pairs reveals the existence of influential points which correspond to very low demand (under 10 units) on public holidays over the year. Once these points are removed the p−values are greater than 0,05, so independence can be assumed: see Table 4 . We compare our approximations with the estimations obtained from simulations for different values of parameters k, k 1 , k 2 . Namely, we take all combinations where k ranges in { 1 . 5 , 2 , 2 . 5 , 3 } and (k 1 , k 2 ) is (0,0) or (10,5). For the Monte Carlo simulations, we set 10 0 0 runs and a run length of 520 weeks (10 years), where the first 52 weeks of each simulation run are taken as the warm-up period. The number of simulations and their length are chosen to give a small simulation error. This can be checked in Table D.8 in Appendix D which shows, for the case k = 1 . 5 , k 1 = k 2 = 0 , the standard deviation of the estimations using simulation together with their relative errors, measured as the ratio of the half-width of the 95 % confidence interval over the sample mean. In almost every instance the relative error is smaller than 2% , with the only exceptions being quantities with a very small sample mean. Increasing the length of the simulation runs or their number (and thus increasing the simulation time) produces almost no changes in the sample means. The computation of our formulae in Section 3 for the setting k = 1 . 5 , k 1 = 0 , k 2 = 0 takes 4 ms. on an Intel(R) Core(TM) i5 (3.30 GHz), while simulation takes 1130 ms. 4.1. Results for the CVTTH data Table 5 shows the results of our formulae and the simulation for k = 1 . 5 , k 1 = 0 , k 2 = 0 . The tables for the rest of the cases are available in Section 1 of the Supplementary Material file. In the tables there are two rows for each quantity: the upper row (plain font) is the result of the application of the formulae in Section 3 ; the lower row (italic font) is the result obtained by averaging the results over the simulation runs. We explain Table 5 in detail, and the rest of the tables have the same structure. The first part of the table gives the results for each day of the week. The first column shows the values of b t ; for instance b 1 , the expected number of units in stock at the beginning of a Monday is 46,2 by our formula in Section 3.1 and 46,8 by the simulation. Below we write first the result of our formula and then, in brackets, the result of simulation, i.e. b 1 is 46,2 (46,8). The second column is q t , the expected order size; thus, for instance, the expected order size on Mondays is 18,6 (18,3). The next column gives o t , the expected quantity outdated on day t: so there are, on average, 0,17 (0,14) units outdated on Wednesdays. The next column is h t , the Fig. 1. Efficient Frontier relating waste with the fill rate (solid curve) for m = 4 (blue), m = 5 (green) and m = 6 (red). Dotted curves represent the freshness of units issued in each optimal configuration. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) expected number of units on-hand at the end of day t. Column u gives the value of u t , the expected shortage on day t. The following two columns give the service level and the probability of the onhand stock being below a threshold at the end of the day; we use 5 units as the threshold in all settings. Thus, for instance, the service level on Tuesdays is 94,0 % (95,2 % ) and 14,7 % (14,1 % ) of Tuesdays end with an on-hand stock lower than 5 units. The following columns b(1) , ... , b(5) are the values of b r t , the expected number of units with a remaining lifetime of rdays at the beginning of day t; for instance, the expected number of units with 2 days of remaining lifetime at the beginning of a Monday is 18,8 (18,9). The following five columns w (1) , ... , w (5) are w r t , the number of units issued on day twith remaining lifetime equal to rdays; for instance, the expected number of units issued on Monday with 2 days of remaining lifetime is 16,7 (16,7). The last column is freshness, the expected remaining lifetime of units issued on day t; for instance, units issued on Wednesday have an expected remaining lifetime of 4,14 (4,09). The second part of the table, below the “Week” line, summarises the values over the whole week, computing sums, averages or percentages over the 7 days where appropriate. The first column gives the average of the b t values, i.e. the average number of units on-hand at the beginning of a day: 43,9 (44,1). q is the total number of units ordered over the week: 152,9 (151,7), which correspond to 100,2 % (99,4 % ) of the demand. The next column shows that 0,25 (0,22) units go out of date over the week, i.e. 0,16 % (0,15 % ) of the units ordered. The average on-hand inventory at the end of the day is 22,0 (22,4) units. There are, on average, 1,426 (1,243) units not served over the whole week, i.e. 0,93 % (0,81 % ) of the demand, so the fill rate is 99,07 % (99,19 % ). The following two columns show that the service level is 95.6 % (96,2 % ); i.e. 4,4 % (3,8 % ) of the days have a stockout, and 9,6 % (9,5 % ) of the days end with less than 5 units in stock. Columns b(1) , ... , b(5) 1144 C. Gorria, M. Lezaun and F.J. López European Journal of Operational Research 303 (2022) 1137–1150 Table 5 Performance measures for the CVTTH data in Section 4 : approximations by formulae (plain font) and simulation (italic font). k = 1 . 5 ; k 1 = 0 ; k 2 = 0 . Day b q o h u s.l. P(u.t.) b(1) b(2) b(3) b(4) b(5) w(1) w(2) w(3) w(4) w(5) freshness Monday 46,2 18,6 0,00 18,4 0,328 0,938 0,120 0,0 18,8 27,7 0,0 0,0 0,0 16,7 11,1 0,0 0,0 2,40 46,8 18,3 0,00 19,3 0,226 0,954 0,110 0,0 18,9 27,9 0,0 0,0 0,0 16,7 10,8 0,0 0,0 2,39 Tuesday 37,0 25,9 0,08 13,2 0,229 0,940 0,147 2,2 16,6 0,0 0,0 18,6 2,1 13,6 0,0 0,0 8,0 2,93 37,6 25,4 0,08 14,0 0,197 0,952 0,141 2,2 17,1 0,0 0,0 18,3 2,1 13,9 0,0 0,0 7,5 2,87 Wednesday 39,1 23,5 0,17 14,4 0,252 0,940 0,139 3,1 0,0 0,0 10,5 25,9 2,9 0,0 0,0 9,6 12,1 4,14 39,5 22,6 0,14 15,0 0,238 0,947 0,138 3,2 0,0 0,0 10,8 25,4 3,1 0,0 0,0 9,8 11,5 4,09 Thursday 37,8 57,3 0,00 15,7 0,275 0,939 0,132 0,0 0,0 0,9 13,7 23,5 0,0 0,0 0,9 11,9 9,3 4,38 37,5 57,5 0,00 15,6 0,254 0,944 0,141 0,0 0,0 1,0 14,0 22,6 0,0 0,0 1,0 12,2 8,8 4,36 Friday 73,0 27,7 0,00 43,6 0,000 1,000 0,000 0,0 0,0 1,8 14,2 57,3 0,0 0,0 1,8 13,2 14,7 4,43 73,1 27,9 0,00 43,7 0,000 1,000 0,000 0,0 0,0 1,8 13,8 57,5 0,0 0,0 1,8 12,9 14,6 4,44 Saturday 43,6 0,0 0,00 30,3 0,013 0,996 0,011 0,0 0,0 1,0 42,6 0,0 0,0 0,0 0,8 12,4 0,0 3,93 43,7 0,0 0,00 30,4 0,011 0,997 0,011 0,0 0,0 0,9 42,8 0,0 0,0 0,0 0,8 12,5 0,0 3,94 Sunday 30,3 0,0 0,00 18,5 0,329 0,938 0,120 0,0 0,1 30,2 0,0 0,0 0,0 0,1 11,4 0,0 0,0 2,99 30,4 0,0 0,00 18,9 0,317 0,942 0,124 0,0 0,1 30,3 0,0 0,0 0,0 0,1 11,4 0,0 0,0 2,99 Week Average 43,9 22,0 0,956 0,096 0,8 5,1 8,8 11,6 17,9 44,1 22,4 0,962 0,095 0,8 5,2 8,8 11,6 17,7 Sum 152,9 0,25 1,426 5,0 30,4 25,9 47,2 44,2 3,62 151,7 0,22 1,243 5,2 30,7 25,8 47,4 42,4 3,60 Percentage 100,2% 0,16% 0,93% 3,3% 19,9% 17,0% 30,9% 28,9% 99,4% 0,15% 0,81% 3,4% 20,3% 17,0% 31,3% 28,0% Table 6 Performance measures for the CVTTH data in Section 4 : approximations by formulae (plain font) and simulation (italic font). Aggregated weekly results. Safety stock b q o h u s.l. P(u.t.) b(1) b(2) b(3) b(4) b(5) w(1) w(2) w(3) w(4) w(5) freshness k = 1,5, k1 = 0, k2 = 0 43,9 100,2% 0,16% 22,0 0,93% 0,956 0,096 0,8 5,1 8,8 11,6 17,9 3,3% 19,9% 17,0% 30,9% 28,9% 3,62 44,1 99,4% 0,15% 22,4 0,81% 0,962 0,095 0,8 5,2 8,8 11,6 17,7 3,4% 20,3% 17,0% 31,3% 28,0% 3,60 k = 1,5, k1 = 10, k2 = 5 51,0 100,4% 0,36% 29,1 0,19% 0,990 0,024 1,3 6,3 10,7 14,8 17,9 5,5% 23,2% 20,0% 36,9% 14,4% 3,31 51,0 100,1% 0,31% 29,2 0,18% 0,991 0,025 1,3 6,4 10,7 14,8 17,8 5,6% 23,3% 20,1% 36,8% 14,2% 3,31 k = 2, k1 = 0, k2 = 0 49,4 100,4% 0,43% 27,5 0,26% 0,985 0,038 1,4 6,5 10,1 13,6 18,0 6,0% 23,4% 16,4% 34,1% 20,1% 3,39 49,5 100,1% 0,36% 27,6 0,25% 0,987 0,039 1,4 6,5 10,1 13,6 17,8 6,1% 23,5% 16,5% 34,4% 19,5% 3,38 k = 2, k1 = 10, k2 = 5 56,6 100,9% 0,86% 34,6 0,04% 0,997 0,007 2,2 7,8 12,4 16,2 18,0 9,1% 25,7% 21,2% 35,4% 8,6% 3,09 56,5 100,7% 0,72% 34,5 0,05% 0,998 0,008 2,2 7,8 12,5 16,1 17,9 9,2% 25,7% 21,4% 35,3% 8,3% 3,08 k = 2,5, k1 = 0, k2 = 0 55,0 101,0% 1,00% 33,0 0,06% 0,996 0,012 2,3 7,9 11,6 15,2 18,1 9,8% 25,4% 16,8% 35,1% 13,0% 3,16 54,8 100,7% 0,82% 32,8 0,06% 0,996 0,013 2,3 7,8 11,5 15,2 18,0 9,8% 25,4% 16,9% 35,3% 12,7% 3,16 k = 2,5, k1 = 10, k2 = 5 62,2 101,8% 1,80% 40,0 0,01% 0,999 0,002 3,4 9,2 14,2 17,2 18,3 13,6% 26,7% 22,8% 32,1% 4,7% 2,88 61,8 101,5% 1,47% 39,7 0,01% 1,000 0,002 3,3 9,1 14,2 17,1 18,1 13,6% 26,5% 23,5% 31,8% 4,5% 2,87 k = 3, k1 = 0, k2 = 0 60,6 102,1% 2,02% 38,4 0,01% 0,999 0,003 3,5 9,2 13,1 16,6 18,3 14,1% 25,8% 18,1% 34,1% 7,9% 2,96 60,6 101,7% 1,69% 38,4 0,01% 0,999 0,004 3,5 9,1 13,1 16,6 18,3 14,5% 25,6% 18,1% 34,0% 7,8% 2,95 k = 3, k1 = 10, k2 = 5 67,8 103,4% 3,28% 45,2 0,00% 1,000 0,000 4,7 10,5 15,9 18,1 18,6 18,3% 26,5% 24,5% 28,2% 2,5% 2,70 67,7 102,9% 2,78% 45,2 0,00% 1,000 0,000 4,7 10,4 15,9 18,1 18,6 18,7% 26,1% 25,5% 27,4% 2,4% 2,69 1145