In this study, a period t in the model is a day at the store, from opening until closing time. In the retail practice of perishable products, mostly the order quantity Q t of today is delivered the next day, so the lead time is L = 1 day. The used symbols are presented in Table 1. In the model the sequence of events is as follows: 1.

Store opening;

The demand is independently Poisson distributed with expectation μ t for day t. When demand is higher than the inventory level, sales are lost and the inventory level will be zero. Moreover, we consider a mixed LIFO-FIFO withdrawal, where the number of customers buying LIFO is binomially distributed.

The general reorder decision problem can be formulated as a stochastic optimization model that minimizes purchasing cost and fulfils the service level constraint. For this specific model, minimization of cost and minimization of waste coincide. (1) min { E ( T C ) : = ∑ t = 1 T E ( c Q t ) } . Let x + : = max { x , 0 }. The total inventory balance for all ages is given by (2) ∑ b = 1 M I b t = ( ∑ b = 1 M − 1 I b , t − 1 + Q t − 1 − d t ) + , where t − 1 = 0 corresponds to T = 7. Period t starts with the inventory levels at the end of period t − 1 of ages b = 1 , … , M − 1. Items of age M are waste. The starting inventory is added to the delivered quantity Q t − 1 and demand in period t is subtracted to give the end inventory. The service level requirement is put on the probability that sufficient material is available to fulfil demand: (3) P ( d t ⩽ ∑ b = 1 M − 1 I b , t − 1 + Q t − 1 ) ⩾ α . This means that we consider a minimal service level constraint as studied by Chen and Krass (2001), where for every day, on average, the service level has to be met.

The company we cooperated with in this study viewed demand d t for this type of fresh products as Poisson distributed. A part of the customers pick the freshest items first. The distribution of demand into LIFO and FIFO follows a binomial distribution B ( d t , λ ), with a fraction 0 ⩽ λ ⩽ 1 of customers that choose the items according to LIFO. This means that the LIFO inventory dynamics is followed for LIFO demand binomially drawn from B ( d t , λ ) and the rest of the customers follow the FIFO dynamics, where demand is fulfilled first by the oldest items before the fresher items. LIFO dynamics is modelled as (4) I 1 t = ( Q t − 1 − d t ) + , t = 1 , … , T and (5) I b t = ( I b − 1 , t − 1 − ( d t − Q t − 1 − ∑ j = 1 b − 2 I j , t − 1 ) + ) + , t = 1 , … , T , b = 2 , … , M . In the FIFO dynamics, first the older items are picked: (6) I 1 t = Q t − 1 − ( d t − ∑ b = 1 M − 1 I b , t − 1 ) + , t = 1 , … , T and (7) I b t = ( I b − 1 , t − 1 − ( d t − ∑ j = b M − 1 I j , t − 1 ) + ) + , t = 1 , … , T , b = 2 , … , M . The model should keep track of the end of the week balance, where (8) I b 0 = I b T and Q 0 = Q T . Moreover, order quantities and inventory cannot be negative: (9) Q t ⩾ 0 , t = 1 , … , T , (10) I b t ⩾ 0 , t = 1 , … , T , b = 1 , … , M .

Broekmeulen and Van Donselaar (2009) proposed the so-called Estimated Withdrawal and Ageing (EWA) heuristic, where knowledge of the number of oldest items, which is going to expire during lead time, is known as part of the total inventory. Notice that the lead time is L = 1. So in that case, the number of items to expire during lead time is I M − 1 , t − 1 . The order quantity is determined by an order-up-to level containing a safety stock, which is corrected for expected waste. Broekmeulen and Van Donselaar (2009) used a fixed safety stock, optimized by simulation. Kiil et al. (2018) determine the safety stock using the standard deviation of forecast errors during lead time and replenishment cycle times a safety factor. Their modified EWA_ss policy uses a smaller buffer stock consisting of the maximum of either the safety stock, or the expected waste. In our implementation, we calculate the safety stock SS in the order-up-to level by (13) S S t = S t − μ t − μ t + 1 .

Due to the lead time of one day, the optimal order quantity Q ∗ can be determined if the number I M − 1 , t − 1 of oldest items in stock at the beginning of the day is known. This means that we can derive a table Q ∗ ( Q t − 1 + ∑ b = 1 M − 2 I b , t − 1 , I M − 1 , t − 1 ) of optimal order quantities. For the ease of notation, consider the fresh inventory X = Q t − 1 + ∑ b = 1 M − 2 I b , t − 1 and old inventory Z = I M − 1 , t − 1 . The optimal order quantity Q ∗ ( X , Z , t ) follows from minimizing order quantity Q t , such that the leftover x of the amount in stock at the end of the day plus the ordered quantity Q t fulfils the demand d t + 1 with a required service level α: (14) ∑ x = 0 X P x ( x | X , Z , t ) × P ( d t + 1 ⩽ x + Q t ) ⩾ α . To find this quantity requires deriving the probability function of the conditional loss function (15) P x ( x | X , Z , t ) = P ( ∑ b = 1 M − 1 I b t = x | X , Z , t ) . This is not straightforward, as one not only has to take the probability distribution of d t into account, but also the binomial distribution of LIFO demand. Consider a realization D of demand d t and y the LIFO demand. The boundary of the probability mass function of the loss is the probability that there is no left-over (16) P x ( 0 | X , Z , t ) = P ( d t ⩾ X + Z ) + ∑ D = X X + Z − 1 ∑ y = X D D y λ y ( 1 − λ ) ( D − y ) P ( d t = D ) . The probability mass for obtaining a left-over of 1 ⩽ x ⩽ X is (17) P x ( x | X , Z , t ) = P ( y = X − x ) = ∑ D = X − x X + Z − x D X − x λ ( X − x ) ( 1 − λ ) ( D − X + x ) P ( d t = D ) . A practical way to derive the values of the probability distribution of x is to enumerate all events of demand realizations d t = 0 , … , X + Y − 1 and of the binomial event y of the LIFO demand with their probability of occurrence. Then one accumulates the probability on the event on the outcome x. After generating the corresponding probability distribution and minimizing Q t in (14), one obtains the optimal daily ordering table Q ∗ ( X , Z , t ) for the case that the retailer not only knows the total inventory level X + Z, but also how many items Z = I M − 1 , t − 1 expire at the end of the day.

We developed three heuristics for the order quantity based on the knowledge of the total inventory for the case that the age of the items in stock is not known. •

An easy way to deal with lack of knowledge of the oldest items in stock is to simulate the system using the table of Q ∗ and derive the corresponding average rounded value for order-up-to levels S t . Basically, we simulate the system with knowledge of the oldest items in stock and measure the realizations X t and Z t of the inventory system and average observations of the revealed order-up-to level S t = Q t + X t + Z t using Q t = Q ∗ ( X , Z , t ) for each day t of the week. The computational procedure requires simulating the optimal Q ∗ with the known age distribution only once. We call this heuristic S for Total estimated Inventory of Perishables, STIP.

All approaches are evaluated in a rolling horizon simulation of 10,000 weeks using pseudo random samples from the Poisson – and the binomial distribution. The expected demand μ t varies during the week. A base demand pattern is taken from observed data in a practical retail case regarding iceberg lettuce. To evaluate the effect of larger numbers, a double base pattern has been designed. A pattern with higher peaks on Wednesday and Saturday provides more challenge to fulfil the service level and has therefore been included. The target service level is taken as α = 90 %. The evaluated three demand patterns are depicted in Fig. 1. Table 2 provides the exact numbers of the expected values in order to be able to repeat the experiments.

The variable purchasing cost is c = 1 per unit. As a result, the average total cost per week is equal to the average order quantity per week. The practitioners in retail we consulted, estimate a LIFO fraction of about 0.4 to be realistic. For the base demand pattern, we vary the fraction of LIFO demand λ ∈ { 0 , 0.4 , 0.6 } for all schedules. To show the effect of partly LIFO demand on the order policies and the average amount of waste, also the situation of only FIFO withdrawal and a LIFO fraction of 0.6 are investigated for the base demand pattern. The other demand patterns are evaluated for a LIFO fraction of λ = 0.4, for all schedules.

Table 3 summarizes the results for the base demand pattern for all described approaches. It shows the average total cost (avgTC) and the average total waste (avgTW) per week. To observe the feasibility of the policy with respect to the minimal service level (SL) constraint, the attained SL is measured for each day. We provide the minimum of these values over the days of the week (minSL) in the table. The policies using information about the oldest items in stock are supposed to do better than the heuristic policies that do not use this information. As expected, one can observe that the costs and the number of wasted items increase when the fraction of LIFO demand increases.

Table 4 compares all approaches for the three demand patterns with a LIFO fraction of λ = 0.4. From the measured performance indicators in Tables 3 and 4, it is surprising to observe that the policies STIP, SEW and S_augmented, which do not require age information, perform even better than the EWA policy. In this case, EWA_ss has also lower costs and waste than the EWA approach, validating the claim of Kiil et al. (2018). However, in case of the double base demand, the EWA approach outperforms the EWA_ss approach, which does not consistently meet the service level constraint due to a lowest attained SL of 0.88. Q ∗ gives the optimal order policy when age information is available. One might expect a reached SL of 0.90 in this approach, but the measured SL is higher due to the small numbers of discrete demand. However, the over achievement may also be related to the conditional character of the order policy as described by Pauls-Worm and Hendrix (2015).

The SEW approach quickly generates easy-to-calculate order quantities with feasible results. The only exception occurs for the peak demand pattern where a minimum average SL of 0.89 is reached for one of the days. The STIP approach, which corresponds to lowest costs and waste and reasonable service levels, fails to reach the target SL of 90% for the double base demand and peaks demand patterns, reaching an average SL of 0.89 for one of the days, which is still close to the target level.

The evaluated approaches include four new approaches to determine order policies. The existing policies – EWA Broekmeulen and Van Donselaar (2009) and EWA_ss Kiil et al. (2018) policy – require knowledge of the oldest items in stock. As a benchmark, we evaluated the newly derived optimal quantity Q ∗ policy.

The EWA, EWA_ss and SEW approaches offer easy and fast calculation rules to determine the order quantity. Like the determination of table Q ∗ , they require computation based on probability theory. Basically, the complexity would increase linearly with the considered horizon T. However, in the practical retail setting, the stationary behaviour captures one week. The STIP approach follows from simulating the Q ∗ policy only once. However, the complexity of the simulation-optimization S_augmented approach depends also on the number of iterations of increments to reach a solution which is feasible in terms of service level. Computing time is hard to compare among the implemented methods, because different software and processors were used. The computational speed also depends on the way of programming.

For the investigated experiments, determination of Q ∗ requires for all experiments about 7.5 s. The S_augmented heuristic needs 4.5 s to 9.0 s. These experiments were performed in Matlab, on an Intel Core i7-4770 CPU, 3.40 GHz desktop processor.