2.1 Challenges and opportunities of e-grocery retailing

In e-grocery, customers order groceries online, and the retailer delivers the purchase directly from local distribution warehouses to the customer. [26] provide an extensive literature review on logistics strategies in e-grocery retailing. In contrast to brick-and-mortar retailing, e-grocers face additional challenges that arise from particular aspects related to warehousing and order picking (see, e.g., [27]) as well as the management of delivery slots (see, e.g., [28]). At the same time, there are specifics regarding the inventory management process that will be discussed in the following.

Due to the additional operational processes (order picking and delivery), there is a longer interval between when a replenishment order for an SKU is placed and when it becomes available to the customer. This reduces the forecasting accuracy of crucial variables, such as the demand distribution for the period under consideration. Specifically, some features used to predict this distribution become less informative the further in advance they are considered. Moreover, the likelihood of customers cancelling the whole (virtual) shopping basket in the event of stock-outs is much higher compared to traditional brick-and-mortar retailing. The paramount convenience of online shopping, namely not having to visit a physical store, may then be outweighed by the inconvenience of needing to place a second order and the general requirement to remain at home during the delivery time slot. In addition, an out-of-stock situation that occurs after the purchase requires an effective substitution policy to mitigate customer dissatisfaction [29]. Consequently, (implicit) costs of excess demand, consisting of both short-term lost revenue and consequences of long-term customer churn, are much higher than operational costs for excess inventory [12]. This asymmetric cost structure, which accounts for customer satisfaction and long-run objectives that influence expected future sales, strongly affects the selection of strategic service levels [30]. Overcoming these challenges in e-grocery retailing necessitates applying complex forecasting methods to accurately determine an extremely right-tail quantile of the demand distribution [12].

At the same time, new types of data provide the opportunity to apply data-driven methods to tackle problems in e-grocery retailing. E-grocers differ in their policy when stock-outs become observable by the customer [31]. In the business case under consideration, customers receive in-stock information only at the final step of the ordering process, yielding uncensored demand data; this allows retailers to monitor customer preferences more accurately [12]. Furthermore, such data enables the explicit quantification of sales actually lost — something not possible with check-out data resulting from traditional brick-and-mortar retailing — leading to a more accurate calculation of costs associated with a specific replenishment order quantity. Additionally, customers can select a delivery slot up to fourteen days in advance. This provides information on known demand, the amount of customer demand for a future delivery period that is already known to the retailer when a replenishment order quantity must be determined. For example, [32] make use of such data by proposing a classification-based model selection for demand forecasting in e-grocery retailing. Previous literature also considers the case of advance demand information, suggesting that customers may exhibit a higher willingness to pay for shorter delivery lead times [17]. Translating these findings to grocery retailing, [7] analyse how demand information can reduce fulfilment costs. [18] build a subscription-based model and demonstrate that retailers can increase revenue by offering price discounts to customers willing to provide advance demand information for perishable products. By reducing uncertainty in customer demand, such data also has the potential to mitigate spoilage while maintaining a specific service level target.

Given the abundant data available in retailing, [19] discuss how retailers can focus on specific data with the highest potential to impact business. In the business case under consideration in this paper, the retailer has to deal with various sources of uncertainty in the inventory process. Therefore, an effective and targeted utilisation of data requires determining which variables warrant investments in collecting and processing distributional information. In the field of decision analysis, the enhancement in expected performance achieved through the use of distributional information is referred to as expected value of including uncertainty (EVIU). For a detailed description of EVIU, and its relation to the value of information in economics, see, e.g., [33]; a study addressing the value of information in the context of grocery retailing has been performed by [7]. In the field of stochastic programming, the concept of EVIU is commonly known as value of the stochastic solution (VSS), see, e.g., [34]. While most studies examining EVIU and VSS compare the consideration of distributions for all stochastic variables to using no distributions at all, our investigation focuses on evaluating the value of including distributions for each subset of stochastic variables. These results can be weighted against the costs associated with collecting and processing the data required in the business case to obtain distributional information for the respective stochastic variable(s). In particular, this enables the retailer to decide whether adopting a probabilistic representation is worthwhile for each source of uncertainty.

2.2 Inventory models for grocery retailing

The scientific study of inventory control problems has a long tradition that dates back to the EOQ model by [35], with its restrictive assumptions of fixed and deterministic demand as well as unlimited shelf life. Since then, extensive research has been conducted to determine optimal replenishment order quantities for different settings. To accurately represent the business environment under consideration in this paper, we need to formulate a periodic review inventory model that, firstly, considers lost sales and non-zero lead time and, secondly, applies to perishable SKUs.

For the backorder case, i.e., if demand can be fulfilled in subsequent periods, an order-up-to policy is known to be optimal for non-perishable SKUs [36]. However, due to complexity considerations, there is limited knowledge about optimal replenishment policies for the more realistic case of periodic-review lost sales models, even for non-perishables (refer to the review on inventory models with lost sales by [13]). In a more recent review, [24] state that inventory models with lost sales generally cannot be solved using exact methods due to the associated extensive size of the state space. Although in specific cases and under restrictive conditions the optimal order quantity in the lost sales case can be approximated by backorder models [37], cost deviations can be as high as 30% [38].

Most SKUs in grocery retailing have a finite shelf life spanning multiple periods, leading to operational costs either for inventory holding (if units can be carried over to the following period) or spoilage and stock reductions (if they are not sold before their shelf life expires; see, e.g., [7]). These reductions in the inventory level must be considered when making replenishment order decisions. [39] provide an overview of trends in managing inventories for such perishable products; more recently, [40] explicitly discuss discrete review multi-period inventory models for this case. Most literature addressing finite shelf lives assumes that the number of sales periods is fixed and known (e.g., [41–43]). The same assumption is made by [14], who develop a periodic-review lost sales model for perishable products with non-stationary demand, relaxing the assumption of independent and identically distributed demand. [44] extend this approach to positive lead times. Both papers highlight that such models are notoriously difficult to solve due to the curse of dimensionality and thus propose approximate inventory control policies. While [45] consider the case of SKUs that decay at a constant rate, for perishable products like fruits and vegetables, it is more realistic to treat the number of sales periods as a random variable. The associated probability distribution can be estimated by modelling the decay of the SKUs over time. This decay can be described by a constant fraction of the given inventory or by a rate that changes according to an underlying function [46], such as the cumulative distribution function of an exponential distribution [47].

Regarding the demand process, some of the literature relies on relatively simple modelling approaches, such as employing a Poisson process for the arrival of customer demand (see, e.g., [7]). While such restrictive assumptions facilitate the derivation of optimal replenishment order policies, they often lack the flexibility to capture typical non-stationary demand patterns observed in real-world retailing data. In fact, [48] had already proposed using a negative binomial distribution for modelling customer demand in retailing. Concerning the non-stationary nature of the demand process, [32] highlight the importance of a case-specific estimation of the demand distribution based on data from an e-grocery business case. Specifically, the combination of high service level targets, aimed at achieving customer satisfaction, and complex demand patterns regularly observed in e-grocery retailing necessitates the use of probability distributions that can account for characteristics such as overdispersion or skewness.

Estimating more complex probability distributions suitable to quantify random demand and simultaneously integrating them into the decision-making process remains crucial in inventory management. Many data-driven approaches that leverage the availability of comprehensive data at low costs (see, e.g., [49–51]; [20]) rely on the simplifying assumptions of the newsvendor model, the classical model for stochastic customer demand [52,53], which assumes a single period and order decision. However, more complex models are required to address the challenges encountered in retail practice [21]. In this regard, [54] propose several extensions to the newsvendor model (see, e.g., [9] for the analysis of a multi-period version).

In most practical settings, retailers additionally face the risk of supply shortages, which may arise from constraints in the distribution channels. This issue is referred to as random yield in the literature. Existing literature on supply uncertainty assumes that retailers know their suppliers’ true supply distributions (see, e.g., [55–57]). [58] were among the first to address problems where both supply and demand are random, deriving the optimal order quantity for the unconstrained newsvendor problem with random yield. [59] incorporate non-stationary supply by assuming it follows a Bernoulli process, where supply can either be completely absent or fully available.

Following the literature discussed above, it appears that attempting to solve periodic review inventory control models with multi-period lead times and lost sales for perishable SKUs to optimality is not feasible for real-world problems. Moreover, the business environment considered here requires the integration of stochastic and non-stationary customer demand, shelf lives, and random yield. Consequently, the retailer faces a complex convolution of probability distributions that govern the development of inventory levels during the lead time, rendering the challenge even more complicated [60]. However, [24] emphasise the potential of approximate numerical methods, such as deep reinforcement learning, for addressing complex sequential decision problems like the one at hand. We concur with their assessment and propose an approximate dynamic programming approach for e-grocery inventory management that incorporates the multiple sources of uncertainty outlined above. Several publications address the determination of replenishment quantities for perishable SKUs – such as the work by [22], which presents a periodic-review inventory control model with non-stationary demand and deterministic shelf life. However, to the best of our knowledge, our study is the first to consider a multi-period model for perishable SKUs with lead time accounting for the complete uncertainty resulting from stochastic customer demand, stochastic shelf lives, and random yield.

3.1 Assumptions and cost structure

The assortment of the retailer under consideration includes several thousand SKUs from various categories, such as fruits, vegetables, and meat, most of which have a very limited shelf life, ranging from one to multiple days, as is typical for grocers [40]. Our modelling approach takes the perspective of a single fulfilment centre, where we assume unlimited storage capacity. In practice, the retailer needs to effectively manage the inventory process daily for various SKUs and several fulfilment centres. Since the retailer controls the stowing and picking processes, we suppose units are picked according to the First In – First Out (FIFO) principle, meaning that the oldest SKUs are sold first.

The retailer’s objective is to control inventory levels such that total (expected) costs are minimised. These costs are related to excess inventory, incurring holding costs v per unit, spoilage costs h per unit, and costs associated with each unit of lost sales b. Although it is generally straightforward to specify cost parameters for inventory holding and spoilage, this is much more difficult for lost sales [61,62]. Thus, many retailers operate with a strategic service level target, that is, the percentage of stochastic customer demand that should be fulfilled. Mathematically, under strong model assumptions, each service level corresponds to a fixed ratio between costs for excess inventory and shortage. This allows retailers to implicitly calculate costs associated with one unit of lost sales from the service level strategically determined and known costs for excess inventory ([12]). We derive cost parameters for lost sales based on these considerations, addressing the trade-off between shortage costs and costs incurred by excess inventory (cf. [22]).

3.2 Dynamics of the inventory process and resulting cost

The inventory i_t of an SKU available at the beginning of the demand period t generally comprises units delivered at different replenishment instances. Let denote the number of units available in period t that were supplied j periods ago. Then, , where J corresponds to the maximum shelf life of the SKU. Therefore, the total inventory i_t can be expressed as a vector with elements ([7] use a similar approach). We denote the quantity supplied at the beginning of period t as q_t and demand as d_t. We assume that the replenishment order quantity to be delivered after a lead time cannot be adjusted by the retailer after its specification in . For simplicity, we assume that the intra-period dynamics can be represented by a sequence of events that occur within period t, affecting the size of the inventory i_t. Primes indicate the changes resulting from these events (refer to the timeline visualisation in Fig 1 and the numerical example in Supporting Information B). At the beginning of period t, with starting inventory i_t, the retailer must first decide on the replenishment order quantity . This is the quantity that affects supply in the future period . However, the decision needs to be taken without having information about realisations of the stochastic variables in period t. Subsequently, supply becomes known, affecting the inventory in the following manner: . Given that we assume supply to be stochastic, this quantity is uncertain and can deviate from the replenishment decision (for details, see 3.3). In our case, SKUs are picked from the inventory according to a FIFO principle. After taking the (satisfiable) demand d_t out, the new inventory is denoted as . This forms the basis for the amount affected by deterioration during period t; we denote the corresponding realisation of spoilage as z_t, leading to the new inventory . This represents the inventory at the end of period t and gives the inventory at the beginning of the following period: .

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Fig 1. Sequence of events within one demand period.

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In the case of lost sales, we obtain the inter-period dependencies as follows:

(1)

At the end of each period t, costs arise depending on demand d_t and the number of available units of the SKU . By assuming periodic replenishments, we can disregard fixed order costs in our model. Given the cost structure outlined above, costs incurred during period t can be expressed as follows with :

3.3 Modelling uncertainty in the inventory process

Retailers face uncertainty regarding the number of units requested by customers in period t, rendering demand a random variable D_t with cumulative distribution function (CDF) . Additionally, taking into account potential supply shortages, the quantity delivered by the supplier, Q_t, becomes stochastic; specifically, it depends on the quantity ordered . While we assume that the number of units delivered cannot exceed the order quantity, the retailer may experience a shortfall in the actual delivery quantity (). If the relative supply shortage were known and constant, a retailer could easily adjust the specified replenishment order quantity by adding the known percentage of shortage to derive the target order quantity. However, in retail practice, supply shortages are neither constant nor known; instead, they follow an unknown probability distribution. In our model, we consider three different supply states G_t: complete delivery (state 1), a cancellation of the total delivery (state 2), and partial delivery (state 3). These states determine the relative proportion of ordered supply that was realised in each demand period t. Since supply shortages often result from persistent issues within the supply chain, we model the sequence of supply states using a homogeneous Markov chain characterised by transition probabilities and its stationary distribution. In the case of partial delivery, we assume that the proportion of units supplied follows a beta distribution, with additional point masses at zero and one, respectively ([63]). For technical details, see Supporting Information C in S1 Appendix.

To capture the case of uncertain shelf lives, we model the shelf life of an SKU in days using an empirical discrete distribution. As elaborated in detail in Supporting Information D, this empirical distribution allows us to derive the conditional probability of a unit of the SKU deteriorating after a certain number of days, given that it was still saleable in the preceding period. We denote the (stochastic) total number of deteriorated units at the end of period t by Z_t. Note that in case of a positive fixed lead time , the inventory I_t at the beginning of period t is unknown at the decision instance when the order must be placed. Instead, I_t depends on demand as well as the random yield and spoilage that occur during the lead time. Consequently, the distribution of inventory I_t is a convolution of the corresponding probability distributions.

Considering all stochastic variables that affect the inventory level, we can express the expected costs for period t in terms of the specified distributions as follows:

(2)

Due to the lead time , the replenishment order decision taken in period does not affect the cost in period but only influences the cost accumulated starting in period t. In this work, we aim to simultaneously consider consecutive periods affected by the replenishment order decision [64]. Given the system dynamics in Eq (1), minimising the total expected costs over a planning horizon T,

(SDLI)

yields the periodic review stochastic dynamic lost-sales inventory model with lead time (SDLI) under consideration in this paper.

3.4 Formulating the problem as a sequential decision process

Besides proposing a solution procedure for deriving a replenishment policy for the inventory model (SDLI), supporting replenishment order decisions in e-grocery retailing particularly requires appropriately responding to relevant data that becomes available in the business environment. In our case, this data includes daily realisations of the inventory level, supply, customer demand, and spoilage. Consequently, it is necessary to rerun the solution procedure for the inventory model (SDLI) every day and for each of the various SKUs based on newly available information. This proceeding transforms (SDLI) into a decision model that is embedded within a sequential decision process, based on the model elements described in Sections 3.2 and 3.3; here, we adopt the terminology and notation conventions proposed by [65] and adapted by [66].

The core elements of a sequential decision process operating at the level of demand periods, based on the data becoming available over time, are an information model , encompassing exogenous information that is stochastic for future periods, and a decision model, in our case (SDLI). Realisations of the information model at the end of period t (demand, spoilage, and supply) are denoted by and determine the resulting costs for this period. In retail practice, parts of these realisations, along with contextual information, are fed into prediction models that provide estimates for the parameters of the underlying (non-stationary) probability distributions of the stochastic variables within the decision model. Indeed, the stochastic and dynamic decision model (SDLI) introduced here necessitates the incorporation of estimated probability distributions within . Given the availability of comprehensive data sets in the e-grocery business environment, this poses both an opportunity and a challenge for descriptive and predictive analytics [67].

The decision model utilises a state s_t and yields a decision x_t. In this context, the state s_t encompasses several components: the inventory i_t (along with the corresponding vector indicating the supply dates, as introduced in Section 3.2), the supply state from the previous period , the set of ordered (but not yet delivered) replenishment quantities , and the (estimated) probability distributions of customer demand, spoilage, and yield of future periods. Solving the decision model results in a post-decision state , incorporating the newly determined replenishment order quantity , and determines expected costs for the period under consideration. Given the state s_t, a decision x_t, and the realisation of the information model , a transition function T defines the state for the following period: . The state variable s_t+1 also includes the current supply state G_t, the set of replenishment order quantities augmented by the newly determined , and the estimated probability distributions of the various random variables that will affect the future inventory level.

The decision x_t, in our context determining a replenishment order quantity , is induced by a policy : . The goal is to identify an optimal policy , that is, a predefined reaction aimed at minimising the (expected) costs resulting from a decision over a specified planning horizon. These costs comprise the expected immediate costs, i.e., costs for period where the decision affects the inventory management process, and the expected sum of future costs depending on the post-decision state , denoted as the value of the post-decision state . Given the significance of data-driven estimation of the relevant probability distributions in the sequential decision process within the e-grocery sector, we adapt the representation proposed by [66] accordingly (see Fig 2).

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Fig 2. Representation of our sequential decision process adapted from [68].

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For a given planning horizon T and a policy , we can express the sum of expected immediate and future costs to be minimised in period t as follows:

(3)

Here, the expected immediate costs incurred by a policy that leads to the decision in period can then be written using the estimated probability distributions of the uncertain quantities (as detailed in Eq (2)); note that these estimates depend on the state .

3.5 A lookahead-based decision policy

Solving the decision models necessitates an order policy that minimises the expected cost for a given state s_t (see Eq (3)). According to [24], the most effective approach for addressing complex inventory control problems under uncertainty is to utilise approximate numerical methods. Specifically, their discussion highlights the use of deep reinforcement learning (DRL) techniques, which employ deep neural networks (DNNs) to approximate either the value of taking a particular decision in a given state or an optimal policy function. Following the policy classification proposed by [65], this approach can be categorised as a value function approximation (VFA)-based policy or as a policy that relies on policy function approximation (PFA). However, in our context, we opted against relying on learning a VFA or a PFA, as this would entail training more than 100 ML models (e.g. DNNs) for a business case. Moreover, [24] demonstrate that DRL approaches tend to perform very well for stationary problems; e-grocery inventory management typically operates in highly non-stationary environments. The non-stationarity is only partly explainable by regular effects such as seasonality. Consequently, applying a DRL-based policy would require frequent and costly re-training of the VFA/PFA models, making it less practical for our specific application.

Instead of employing a learned VFA model, we propose approximating the value of an order decision using a Monte Carlo simulation with a limited lookahead horizon H. Following [65], the resulting policy can be characterised as a stochastic lookahead policy. A key advantage of this type of policy is that, according to the terminology introduced by [66], it internally utilises the information model. This allows the policy to adapt naturally to (even structural) changes in the information model without retraining an approximation model. Additionally, since the policy is based on sampling, it can be seamlessly combined with advanced and context-dependent distributional forecasting methods, such as those proposed by [32].

Given the lookahead horizon H, the order decision taken at t affects the objective function only in period , i.e., when the order is supposed to be delivered. Therefore, we set and denote the number of lookahead periods exceeding by , that is, . In case of (that is, ), we can approximate the expected cost in period for a given state s_t and replenishment order decision, induced by the policy , , by averaging over N simulated sample paths starting at period t and ending at period . For a sample path n, the cost is obtained by simulating the state-transition logic outlined in Section 3.2. This simulation begins with the initial state s_t. It utilises the given decision , along with random samples drawn from the distributions that represent supply, demand, and spoilage in each simulated period from t to . In this context, the optimisation problem to be solved in each period t can be formulated as follows:

If our lookahead horizon , that is if , we extend the sample paths described above until the final period t + H of the lookahead horizon. This extension enables us to capture the effects of the order decision on demand periods beyond more accurately. The costs incurred in the lookahead periods following are affected not only by the decision to be taken in t but also by the ’simulated’ decisions taken in periods j with that are part of the lookahead. To account for the diminished precision of forecasts for future periods and to reflect the relative decrease in the importance of the decision , we weight the expected costs by the factor , where for periods . Note that while the lookahead decisions for j > t are not implemented, they still contribute to the objective function of the decision model (SDLI). Thus, the objective used to determine the lookahead policy reads as follows:

(4)

The objective does not include costs incurred in periods before , as these costs are not affected by the decisions involved in the lookahead. To determine the replenishment order quantity to be delivered in period , we aim to find the quantity that minimises average costs over the sample paths, as outlined in Eq (4). To achieve this, we employ a widely established heuristic Nelder-Mead-based numerical optimisation approach. For , replenishment order quantities are assumed to be given; resulting costs for these periods are not part of the optimisation problem.

4.1 Simulation setup: distributions, parameters, and data

We generate an experimental data set that encompasses T consecutive demand and supply periods for an example SKU. This data includes information on demand, spoilage, and supply shortages. Considering perishable SKUs with a shelf life ranging from one to multiple periods, we employ a specific parameter vector for the data-generating process in subsequent analyses. Distributions and parameters in the data-generating processes are chosen to resemble the information structure found in the data provided by the business partner (see Supporting Information A in S1 Appendix).

Based on findings from the literature (see, e.g., [48], and for this data in particular, cf. [32]), we assume that the realisations of (uncensored) demand in period t follow a negative binomial distribution with mean and size parameter k_t, i.e.,

We reparameterise the distribution in terms of its mean and variance , where . To accommodate non-stationary demand, we draw the parameters of the demand distribution for each period as and . For the subsequent analyses, we assume and . While our model is capable of naturally adapting to structural changes in the information model, such as seasonality, with the chosen probability distributions, we simplify by avoiding such more complex structures, but consider demand to be independent across different periods. In practice, this information can be drawn from probabilistic forecasts, such as those suggested in [32]. An example realisation of simulated demand over 100 periods is shown in Fig 3(a).

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Fig 3. Realisations of demand and shortage for demand period .

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The shelf life of the SKU is generated from a distribution with probability mass function f^sl(j), as detailed in Table 1. Here, j = 1 corresponds to the situation where the unit deteriorates at the end of the delivery period (i.e., day 0). The mean shelf life implied by this distribution is three periods. The conditional probabilities of a unit deteriorating after exactly j periods, given it was still saleable at the beginning of that period, are provided in Table D1 in Supporting Information D.

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Table 1. Distribution of the shelf life in the simulated data set.

https://doi.org/10.1371/journal.pone.0343935.t001

The sequence of delivery states – namely, complete delivery (state 1), complete shortage (state 2), and partial delivery (state 3) – across demand periods is governed by a Markov chain with the transition probability matrix (TPM)

The associated stationary distribution, which is also used as the distribution for period t = 1, is given by the transpose of . If the retailer encounters a partial supply shortage, that is, if state 3 is active, the realised relative amount of supply follows a beta distribution with shape parameters and . This results in an average relative shortage of 60% in case of partial delivery and an overall average shortage of . An example realisation of relative shortage for demand periods is illustrated in Fig 3(b).

In accordance with the strategic environment of e-grocery retailers (see Section 3.1), we assume the cost for one unit of excess demand to be b = 5, inventory costs to be v = 0.1 per unit and time period, and per-unit spoilage costs to be h = 1 for the SKU considered. This relation between excess inventory costs and shortages takes into account the high service-level target in e-grocery retailing. For the lookahead policy, the absolute values of the cost parameters are not relevant; rather, it is the relation between these parameters that influences the solution determined by the model. A lead time of days is assumed between the replenishment order decision and the delivery to the retailer’s fulfilment centre. Our evaluation is based on T = 5000 simulated periods, which corresponds to more than 15 years of data in a business case. This large number of simulated data points is used to minimise the impact of noise in the comparison. All optimisation experiments are conducted in Python, using the minimise function from the scipy.optimize package, employing the Nelder-Mead method with a tolerance of 10^-9. The program code to reproduce all simulation results can be obtained from Zenodo [70].

4.2 Myopic benchmark: Newsvendor model

In the newsvendor model, it is assumed that each unit of an SKU can be sold for only one demand period, allowing for the optimisation of replenishment order quantities for each period individually. One unit of excess inventory incurs spoilage costs h, while each unit of lost sales leads to costs of b. The optimal order quantity can be obtained at the b/(b + h)–quantile of the (estimated) CDF of demand F_D [52,53], with representing the demand forecast made in period t for period . Under these myopic assumptions, the resulting replenishment order quantity can then be calculated as:

(5)

For each period , we determine replenishment order quantities according to Eq (5). Thus, we assume that the decision-maker disregards the potential transfer of excess units at the end of a demand period, leading to an underestimation of the starting inventory at the beginning of most demand periods, i.e., if there are units transferred. Additionally, the risk of supply shortages is not taken into account when determining replenishment order quantities. At the end of each period, the realised holding of inventory, spoilage, and lost sales generate costs based on the given cost parameters (h, v, b). We then calculate average costs over the considered time horizon T.

We obtain an average order quantity of 119.03, an average inventory level of 199.42, and an average amount of spoilage of 17.52, resulting in an average per-period cost of 38.84. The retailer is able to satisfy 99.72% of customer demand, which exceeds the strategic service level target. While this increases customer satisfaction, it leads to higher costs compared to a scenario where the intended service level is precisely met, due to the additional inventory holding and spoilage costs. The deviation from the desired service level is attributed to the newsvendor models’ neglect of inter-period dependencies when determining replenishment order quantities. To balance high product availability and costs for excess inventory, more sophisticated models that account for the various stochastic factors in e-grocery retailing are necessary.

4.3 Deterministic benchmark: Expected values

We now account for the dynamic nature of the inventory management problem by considering that SKUs have an expected shelf life spanning multiple periods. Specifically, the expected shelf life is three periods, with variation as detailed in Table 1. In addition, we consider the risk of supply shortages. Due to these two additional sources of uncertainty, the newsvendor model is no longer applicable. To still derive solutions analytically, we rely on deterministic expected values for the stochastic variables that affect the inventory management process, namely demand, spoilage, and supply shortages. Consequently, we continue to ignore the stochastic variation in supply shortages and shelf life, and, compared to the newsvendor approach, also disregard uncertainty in customer demand. In period t, we calculate the expected starting inventory for period , denoted by , based on the current inventory i_t, previously determined replenishment order quantities , expected demand in the meantime , the average shelf life, and the average amount of supply shortage. In the event of a fixed relative supply shortage, the retailer could simply add this percentage to the replenishment order quantity to ensure the intended amount is delivered. The order quantity under this deterministic approach is then given by the difference between expected demand and the expected starting inventory for the period under consideration, divided by the average relative amount of units supplied (note that we consider only positive replenishment order quantities):

(6)

Applying these point forecasts results in an average order quantity of 96.33, which is approximately 19% lower compared to the newsvendor approach. At the same time, the average inventory level of 18.93 is more than 90% lower. Consequently, the amount of spoilage is also significantly reduced. While this reduces operational costs for inventory holding and spoilage, the policy results in fulfilling 93.49% of total customer demand, which falls short of the retailer’s intended service level and evokes customer dissatisfaction. Thus, compared to the newsvendor approach, lost sales occur more frequently and account for a large share of total costs. However, total costs are slightly lower. By accounting for the inter-period dependencies in the inventory management framework through point forecasts on demand, shelf life, and supply shortages, the average per-period costs are reduced by 8.5% in total (see Table 2).

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Table 2. Comparison of the lookahead policy to the myopic and deterministic approach.

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4.4 Evaluation of the lookahead policy

In our setting with multiple sources of uncertainty, using point forecasts reduces average per-period costs for perishable SKUs compared to the more myopic newsvendor model, which only addresses demand stochasticity. However, the approach presented in the previous section results in a higher level of unfulfilled demand than intended by the strategic service level of the e-grocery retailer. Therefore, we continue to seek a policy that effectively balances the trade-off between product availability and operational costs associated with excess inventory. For this purpose, we now apply the lookahead policy introduced in Section 3.5 to our simulated data set and evaluate the policy in detail. Since the outcome of any policy is highly dependent on the specific business case, this is followed by a discussion on the sensitivity of our results with respect to the underlying parameter values, thereby generalising to other inventory management settings.

In practice, a retailer must determine order quantities for all SKUs in the assortment daily within a limited time frame, which limits the computing power and time available for individual SKUs. To tackle this, we parameterise the policy based on a set of initial experiments, addressing the trade-off between computation time and stability of the simulation results. We use N = 1000 sample paths (simulation runs) for the stochastic lookahead while considering additional periods with a weighting factor . In each period, the retailer determines the replenishment order quantity according to the lookahead policy and the (known) probability distributions for each source of uncertainty. At the end of a period, we again use realised inventory holding, spoilage, and lost sales to calculate average per-period costs for the evaluation period.

Table 2 summarises the results of our analysis and compares them to those obtained under the newsvendor model and the deterministic approach. We find an average order quantity of 103.05, which is 7.0% higher than when relying solely on expected values. Meanwhile, the average inventory level is more than three times higher, although it remains only about 30% of the level observed under the newsvendor model. On average, 98.47% of customer demand is fulfilled, aligning with the retailer’s intended service level. The average per-period costs, when accounting for uncertainty in all three sources, are reduced by 52.0% compared to using expected values and even by 56.1% compared to the newsvendor model. This demonstrates that incorporating probabilistic information yields more accurate replenishment order quantities than both the myopic and deterministic approaches, successfully balancing customer satisfaction and operational costs associated with excess inventory.

In practice, retailers must accurately estimate the underlying probability distributions from often multidimensional historical data or features before they are able to make replenishment order decisions based on probabilistic information. These estimations require data collection, data preparation, and data analysis, incurring operational effort and costs for retailers, which must be considered. Addressing the research gap on targeted data utilisation identified by previous literature [19], we evaluate the benefit of applying distributional information in terms of EVIU, i.e., cost reductions achieved through precise distributional information, for each of the different sources of uncertainty (while limiting information on the other two sources to point forecasts). Table 3 presents information on the various settings compared to probabilistic information for all sources of uncertainty and also provides savings relative to the newsvendor model and the setting of point forecasts.

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Table 3. Analysis of the expected value of including uncertainty (NV: newsvendor model; PF: point forecasts).

https://doi.org/10.1371/journal.pone.0343935.t003

Incorporating distributional information solely for demand already results in a substantial reduction in total costs compared to point forecasts (−51.6%). To accommodate demand variability, the retailer increases replenishment order quantities and maintains a significantly higher safety stock. Consequently, the average inventory level and amount of spoilage increase more than threefold compared to the situation of point forecasts. However, because of the asymmetric cost structure, savings due to the enhanced service level and resulting customer satisfaction surpass additional operational expenditures associated with spoilage and inventory holding. Cost improvements are also observed when including the shelf life’s probability distribution only, albeit with a much smaller effect — a total cost reduction of 6.8% compared to using a policy based on point forecasts — due to the low probability of spoilage within the first two sales periods. In contrast, incorporating a probability distribution only for supply shortages results in a decrease in summary statistics on average order quantity, inventory level, and percentage of fulfilled demand. Specifically, the average order quantity of 95.63 (see Table 3) is lower than the corresponding value of 96.33 under the deterministic approach (see Table 2), where the retailer adds a fixed percentage to each order to account for expected supply shortages (see Eq (6)). Overall, the deterministic approach overcompensates for stochastic supply shortages in most periods, resulting in a 13% increase in the average inventory level. Consequently, the higher inventory level, due to less information on supply shortages, leads to increased holding and spoilage costs. However, the increased inventory level also has a positive effect, as indicated by the comparatively higher percentage of fulfilled demand in the deterministic case: it unintentionally increases customer satisfaction due to reduced lost sales, decreasing associated costs incurred by disregarding demand variation in the deterministic setting. Given the asymmetry of cost parameters, average total costs are lower for the less informed decision-maker. The results presented in this section are derived from simulation-based analyses and therefore depend on the underlying assumptions, such as demand following a negative binomial distribution. In practice, the magnitude of cost savings achievable through probabilistic information will vary with the validity of these assumptions as well as the accuracy of data collection and preparation processes. Accordingly, while our findings illustrate the potential benefits under controlled experimental conditions, the exact cost reductions realised in practice may differ depending on the retailer’s data environment and operational setting.

Table E1 in Supporting Information E presents additional results on scenarios where distributional information is applied for two sources of uncertainty while using expected values for the third source. Our findings indicate that the EVIU varies across the different model components, and that also the sequence of including distributional information matters. For instance, including the probability distribution for supply is beneficial only when the retailer also accounts for uncertainty in demand.

4.5 Sensitivity analysis

The simulation-based analysis above demonstrates how retailers can reduce total costs by using probability distributions instead of expected values for each source of uncertainty, assuming the underlying probability distributions are known. However, the results are likely to be highly dependent on the exact specification of the distributions of the random variables associated with demand, supply, and shelf life (and, of course, also on the cost structure). Therefore, we provide additional sensitivity analyses, considering variation in location and dispersion of the underlying probability distributions and parameters. For each of the three sources of uncertainty, we compare the results obtained using the deterministic approach (referred to as Scenario 1) with those obtained using full probabilistic information (referred to as Scenario 8) within the same simulation setting.

Given that our costs are asymmetric, with lost sales being more expensive than inventory and spoilage, we expect the benefit of incorporating the demand distribution to increase as its variance increases. We continue to allow for non-stationary demand but vary the parameter in the variance-generating Poisson distribution, , while keeping constant. Fig 4 illustrates that average costs increase substantially when using expected values only (Scenario 1), while the per-period costs only slightly increase when incorporating full distributional information for all sources of uncertainty (Scenario 8). As expected, the importance of integrating information on the demand distribution increases with its variance.

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Fig 4. Resulting per-period costs depending on the variance of demand.

https://doi.org/10.1371/journal.pone.0343935.g004

For the sensitivity of the shelf life (for details see Supporting Information F in S1 Appendix), we find that including information on the shelf life distribution is most beneficial for distributions with high variance or a small mean (corresponding to a high risk of spoilage in early periods). The relevance of incorporating probabilistic information on potential supply shortages (for details see Supporting Information G in S1 Appendix) depends not only on the associated risk but also on the persistence of the corresponding process (i.e., whether shortages tend to occur in several consecutive periods).

Finally, we examine changes in the cost structure for lost sales, inventory holding, and spoilage. As previously mentioned, costs in e-grocery retailing are generally asymmetric due to the economic impact of lost sales. Therefore, the benefits of incorporating additional information on probability distributions are expected to diminish if cost parameters become more symmetric. We test this hypothesis by changing the relationship between cost parameters. While maintaining a constant relationship between inventory costs v = 0.1 and spoilage costs h = 1, we vary the costs for one unit of lost sales. In the first analysis, we assume that lost sales costs are equal to inventory costs, leading to b₁ = 0.1. In addition, we consider b₂ = 0.5, b₃ = 1 (where the costs of lost sales and spoilage are identical), b₄ = 2, b₅ = 5, and b₆ = 10.

Fig 5 illustrates average per-period costs for the deterministic approach (Scenario 1, red solid line) and full probabilistic information (Scenario 8, blue dashed line) across various unit costs for lost sales. When these costs fall between the unit costs for inventory holding and spoilage, the difference is negligible. However, incorporating probability distributions becomes more significant if the cost structure is asymmetric.

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Fig 5. Resulting average per period costs depending on unit costs for lost sales.

https://doi.org/10.1371/journal.pone.0343935.g005

This result is further supported by Fig 6, which shows the relative difference in average per-period costs. With b = 0.5, i.e., the cost per unit lost is half that of the cost per unit of spoilage, incorporating distributional information yields a saving of only 2.6%. In contrast, for the business case of e-grocery retailing with an asymmetric cost structure (and the corresponding high service-level targets), savings are significantly larger. As mentioned earlier, with costs per unit lost set at b = 5, including information on the distribution of demand, spoilage, and supply shortages reduces costs by more than 50%, and potential savings are even greater for b = 10.

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Fig 6. Relative difference in resulting average per period costs between Scenarios 1 and 8 depending on unit costs for lost sales.

https://doi.org/10.1371/journal.pone.0343935.g006