The effect of distinct collaborative inventory management strategies on the Minimum Life on Receipt (MLOR) rule

The Minimum Life on Receipt (MLOR) rule is a standard retail practice that limits the age of perishable products received from suppliers. In general, this rule stipulates that food products must have at least 2/3 of their maximum shelf life remaining upon receipt (Garrone et al., 2014). The purpose of this rule is to maximize product freshness and minimize food waste for the retailer. However, the rule creates a conundrum for the supplier. If the supplier chooses to deliver the products as fresh as possible while minimizing food waste, it must produce them in small batches, which increases setup costs. Alternatively, if the supplier is more sensitive to setup costs, it may produce in larger batches and risk incurring food waste as a result of products not meeting the MLOR rule (Entrup et al., 2005).

Retailers prefer a high MLOR (i.e. a high remaining shelf life on receipt), while suppliers tend to prefer a lower MLOR, which gives them more flexibility in their production operations. Collaborating to establish an MLOR rule that benefits both retailers and suppliers can be a strategic solution to increase profits and reduce food waste (Santos et al., 2022). Therefore, setting the MLOR rule jointly between retailers and suppliers is one possible way to achieve a more sustainable food supply chain. Although promising, the literature on this topic is still in its infancy. Moreover, the few papers that address collaborative MLOR rule decisions assume one of two collaborative inventory management strategies – Vendor-Managed Inventory (VMI) or Retailer-Managed Inventory (RMI). In the VMI, the supplier manages its retailers' inventory and is responsible for waste-related costs (e.g. Akkas and Honhon (2023)). In the RMI, each entity manages its own inventory independently and is responsible for its own waste-related costs (e.g. Santos et al. (2022)). The type of collaborative inventory management strategy can be updated in retailer-supplier contracts, i.e. over time, and for a given product type, the entities may find that switching from VMI to RMI (or vice versa) is the best option.

At first glance, an MLOR rule may seem redundant in a VMI strategy, since the supplier manages its retailers' inventory. However, this rule exists and is usually fixed (Akkas and Honhon, 2023). This is because, on the one hand, suppliers do not want to ship products close to their expiry date, as they would have to reimburse the retailers for the unsold products. In addition to the refund, there are also the costs of preparing and sending the shipment, which could be avoided with an optimal issuing policy. On the other hand, retailers want to provide the best experience for their customers by offering a sufficient amount of fresh products. If they offer too many products close to their expiration date, consumers may substitute products or even switch retailers. It is therefore advisable to set up an MLOR rule that effectively benefits both the supplier and its retailers. In the context of the RMI strategy, establishing the MLOR rule jointly becomes even more important because it allows the supplier to achieve balance in an environment usually dominated by the retailer. Such a collaborative decision would provide the supplier with a better trade-off between waste-related costs and operational flexibility and promote gains for the retailer. For example, if the supplier cannot comply with the MLOR rule for one or more shipments, the retailer may receive fewer products than expected. This could lead to stockouts and lost sales, potentially resulting in loss of goodwill.

In the VMI, the MLOR is determined solely by the supplier, while in the RMI, it is determined solely by the retailer. However, the joint definition of the MLOR rule may benefit both entities, regardless of the inventory management strategy used. However, the literature lacks a comparison of the effects of the joint definition of the MLOR rule in both strategies. In this work, we take an in-depth look at the emerging research topic of collaborative MLOR rule and address the aforementioned gap in the literature. We formulate two optimization models for a food supply chain consisting of a single supplier and multiple retailers. The first model adopts a VMI strategy and the second model adopts an RMI strategy. First, we study the profits, costs, and waste generated by optimizing the MLOR rule in both collaborative inventory management strategies under a deterministic demand scenario. Then, we extend our analysis to a setting with uncertain demand, using a robust optimization approach to address the uncertainty. This approach enables us to evaluate the impact of uncertainty without making any assumptions about the demand probability density function. Interestingly, our results indicate that the optimal MLOR rule tends to be higher in the VMI strategy than in the RMI strategy. Reducing the optimal MLOR by 28 p.p. compared to the 2/3 rule decreases the average freshness of the received products by 1 p.p. in the RMI strategy. However, the same MLOR reduction decreases the freshness by about 4 p.p. in the VMI strategy. In addition, demand uncertainty has a greater impact on the RMI than on the VMI. Although the optimal MLOR increases with more conservative scenarios in the VMI, it decreases in the RMI.

The main contributions of our work are summarized below:

1. Compare the influence of different collaborative inventory management strategies on the definition of the MLOR rule in single-supplier, multiple-retailer supply chains. This has not yet been addressed in the literature.

2. Propose robust optimization models to address demand uncertainty when defining the MLOR rule. These models are under-explored in the perishable inventory literature.

3. Establish a set of managerial insights to guide practitioners in exploring the definition of a collaborative MLOR rule.

The remainder of the paper is structured as follows. Section 2 presents a literature review of papers dealing with the collaborative definition of the MLOR rule. The VMI and RMI strategies are detailed in Section 3, followed by the mathematical formulations of their deterministic and robust versions in Section 4. Subsequently, the computational experiments are reported in Section 5, and relevant managerial insights are provided in Section 6. The main conclusions and suggestions for future work are presented in Section 7.

As noted above, research on the MLOR rule is still in its early stages. To provide context for our paper and its purpose, we categorize the literature by strategy type (VMI or RMI) and MLOR setting (non-collaborative or collaborative), as shown in Table 1. The strategy type indicates whether the supplier or the retailer controls inventory in the downstream supply chain. The MLOR setting indicates whether one entity (non-collaborative) or both entities jointly set the rule (collaborative).

The first literature to acknowledge the MLOR rule considered an RMI strategy. This literature examines supplier production and distribution problems in which the MLOR rule is set by retailers. Rong et al. (2011) consider two types of retailers: one imposing an MLOR of 80% and the other imposing an MLOR of 85%. Wang et al. (2016) use an MLOR input of 75% and Piva et al. (2020) an MLOR input of 80%. In all these papers, the analyses of different indicators have focused only on the supplier.

More recently, Akkas and Honhon (2023) examine the MLOR rule in a VMI context. They refer to the MLOR rule as the ship-to-trade rule and propose a Markov chain model to determine the maximum shipping age thresholds (i.e., the mix of product ages that make up a shipment, or an MLOR equivalent). They then empirically test different rule values on a real data set from a company. Although they found that changing the rule would not benefit the supplier, they note that optimizing the age composition of shipments (since a shipment can comprise products with different ages) could improve profits by 8.7% and reduce food waste by up to 14.7%.

The literature dealing with the collaborative setting of the MLOR rule has only begun to emerge in recent years. In Santos et al. (2022), the impact of optimizing the MLOR rule in a collaborative single supplier-single retailer network is evaluated and compared with the traditional case of fixing the MLOR rule a priori. The problem is modeled as a mixed-integer linear programming (MILP) with deterministic demand, and a rolling-horizon approach is then applied to address demand uncertainty. The central insight of this work is that reducing the MLOR by 12% compared to the typical 2/3 rule (i.e. a MLOR of 66%) allows for increasing profits and reducing waste in the supply chain while keeping the average product freshness almost the same. To encourage collaboration, the study also proposes a simple profit-sharing mechanism. Related rules, designated as the Maximum Remaining Shelf Life (RSL) (dictated by the production planning cycle and dispatching) and target Minimum-RSL (i.e. the minimum remaining shelf life offered to consumers), are studied in Singh and Kuthambalayan (2022). They model an integrated production and marketing problem as a two-stage stochastic nonlinear problem, where the first-stage decisions include inventory, packaging sequence, capacity allocation, maximum-RSL, and minimum-RSL. They show that the maximum RSL increases with decreasing setup time, demand variability, and shortage costs, or with increasing holding costs and packaging rate. More recently, Mohamadi et al. (2025) investigate the effect of lowering the MLOR rule in two different settings–through collaboration and through coordination. In the first setting, collaboration is structured under a reduced MLOR rule with a lower wholesale price, so that all parties can benefit while reducing waste. In the second setting, contracting mechanisms are designed, but waste reduction depends heavily on the wholesale price and the issuing policy. They conclude that coordination is more appropriate for online retailers, while collaboration is more appropriate for brick-and-mortar grocery retailers.

The collaborative setting of the MLOR rule in a VMI strategy has never been studied before. We aim to address this new setting, allowing us to infer about the effect of the rule not only at the supplier, as in Akkas and Honhon (2023), but also on retailers. More importantly, we compare for the first time the effect of the MLOR rule on two different collaborative strategies–the VMI and the RMI–and apply a robust optimization approach to deal with demand uncertainty. By studying different collaborative inventory management strategies, we aim to identify similarities and differences in their application of the MLOR rule. By using a robust optimization approach, we aim to reduce the need to assume uncertainty distribution profiles or rely on real data, which may be difficult to obtain when negotiating the MLOR rule. One of the most effective robust optimization approaches in the literature uses the budget of uncertainty set proposed by Bertsimas and Sim (2004). Such an approach is particularly useful for problems involving inventory and replenishment decisions (e.g. Alem et al. (2018), Hooshangi-Tabrizi et al. (2022), Metzker et al. (2023), Zhang et al. (2025)).

In this section, we first introduce the general single-supplier, multiple-retailer problem and then detail it under two inventory management strategies: VMI and RMI.

The general single-supplier, multiple-retailer problem is an integrated production and inventory problem. The supplier must decide how much to produce and when to meet its retailers' demand. Each entity aims to maximize its own profits, but under a collaborative strategy, the goal is to maximize the supply chain's total profits (i.e. the sum of each entity's profits). In perishable product contexts, such as the food supply chain, the risk of product spoilage due to expiration is crucial to consider. Perishable products have a maximum time during which they can be safely consumed and legally sold. Retailers must receive these products within an interval that allows them to be displayed on store shelves and consumed by consumers. Thus, retailers impose a criterion on their suppliers, requiring them to deliver products with a minimum number of days until their expiration date. This criterion defines the MLOR rule.

Our approach replaces this parameter with a joint decision between a supplier and a retailer: the collaborative MLOR rule. The collaborative MLOR is set based on trade-offs between the supplier's and retailer's goals. The lower the MLOR, the closer to the expiration date products can be shipped to retailers. However, determining the MLOR collaboratively is not straightforward. It requires balancing the benefits to the supplier and the retailer, which are often in conflict. Additionally, the MLOR rule must be balanced with production, inventory, and ordering costs, as well as salvage opportunities and waste from expired products. In general, a lower MLOR results in fewer production periods, thereby reducing the supplier's setup costs and surplus production (and thus reducing salvage opportunities). However, retailers may end up with a wide range of product ages, which is critical when consumers are sensitive to freshness. This increases the risk of product expiration. Conversely, a high MLOR requires the supplier to produce more frequently to deliver only fresh products, thereby increasing its setup costs. However, the risk of products becoming waste is lower for retailers.

Finally, it is important to note that, in practice, a supplier often delivers products to multiple retailers. Under a VMI strategy, similar MLOR rules can be expected for all retailers. In an RMI strategy, such homogeneity may not occur because each retailer manages its own inventory and places its own orders. However, if consumer demand is similar, similar MLOR rules are also expected for all retailers.

In the VMI problem, the supplier manages its own inventory and that of its retailers. Therefore, there are no ordering decisions; the supplier decides when and how much to ship to its retailers. Retailer's inventory can then be stored at the supplier's facilities or at the retailers'. We assume that the supplier bears all the inventory costs, regardless of its location, and that inventory holding costs are location-agnostic. A similar assumption is found in Zhang et al. (2015).

Figure 1 illustrates the sequence of operations of the VMI problem. At the beginning of period t, the supplier receives the retailers' inventory levels (Step 2), after they determine their available inventory for the next period (Step 1), and then checks its inventory after waste disposal (Step 3). If the retailers' inventory is sufficient to meet the demand of the next period, the supplier discards only the outdated units in its inventory, and no further decisions are made for period t. Otherwise, it evaluates whether its inventory is sufficient to meet the retailers' demand for the next period. If it is, it ships the required goods. If not, it decides how many units to produce (Step 4). It then sends a given quantity of products to the retailers, which may or may not be the same age, as these products may have been depleted from existing inventory, recently produced, or both (Step 5). There is no lead time, and so the supplier can produce and deliver on the same day. At the beginning of period t + 1, the retailers receive the products from the supplier (Step 6). Consumers then withdraw products according to their preferences (Step 7). Consumers may prefer the freshest products, usually by checking the expiry date on the label (following a Last-Expire-First-Out policy–LEFO consumers), or they may choose the products upfront on the shelf, usually the oldest ones (following a First-Expire-First-Out policy–FEFO consumers). At the end of period t + 1, retailers determine the remaining stock and dispose of outdated products (Step 1), thereby initiating a similar cycle.

In the RMI model, supplier and retailers each manage only their own inventory. Unlike the VMI, each retailer must determine the quantity and timing of orders, and then pass this information to the supplier. Additionally, each retailer incurs its own inventory costs and a fixed cost per order.

Figure 2 illustrates the sequence of operations of the RMI problem. This sequence is similar to the VMI, but with new operations introduced by retailers. As in the VMI, the retailer determines the available inventory for the next period (Step 1). Based on this information, the retailer decides whether to place an order if the inventory is not sufficient (Step 2). The supplier receives the order with the quantity of units to be delivered (Step 3). The supplier then determines whether its available inventory is sufficient for the order (Step 4) and decides whether to produce new products (Step 5). At the end of the period, the supplier delivers the complete order to the retailer (Step 6), and the retailer receives it instantly (Step 7). Consumers then withdraw the products according to their preferences (Step 8). At the end of the period, the retailer checks the inventory again, discards expired products (Step 1), and initiates another cycle.

The MLOR rule is a tactical decision that, once defined, can constrain the VMI or RMI problem. Therefore, it is an exogenous decision with respect to the operational sequences depicted in Figures 1 and 2. While the MLOR rule impacts decisions such as the quantities to be discarded, production frequency, and product age composition upon delivery, it does not alter the logical sequence of operations. Therefore, the flow of operations remains the same, regardless of the established MLOR rule, but the rule can affect the feasibility and objective value of the solutions.

Similarly, the described operations are independent of the demand profile, whether deterministic or uncertain. Compared to the deterministic case, an uncertain demand will affect most decisions (e.g. moments and quantities), but the sequence of operations will remain unchanged regardless of the MLOR rule. However, the MLOR rule can affect the feasibility and the profits and costs of a solution.

When comparing the two collaborative strategies, the MLOR rule is expected to be higher (i.e. products may enter the retailer's supply chain further away from the expiry date) in the VMI than in the RMI. A higher MLOR in the VMI is due to the supplier's need to reimburse the retailers for waste that occurs at that stage. To reduce the probability of waste generation at retailers, the supplier is more likely to ship fresher products, and any surplus remaining can be salvaged through alternative channels.

It is important to note that our focus is on the influence that different collaborative inventory management systems have on the optimal MLOR rule. Therefore, regardless of the inventory strategy adopted, all models are formulated under a centralized perspective, in which the central planner is the supply chain as a whole. Thus, the objective function of all optimization models is the same: to maximize the total profits of the supply chain, which is equal to the sum of the individual profits of the supplier and retailers. While this assumption may be somewhat simplistic, it allows for a fairer comparison of the VMI and RMI strategies. Furthermore, these models and their solutions can serve as benchmarks for future collaborative problems, as the objective values represent the highest possible profits achievable.

The mathematical formulations of both problems are presented below. The deterministic VMI is detailed first (Section 4.1), followed by the RMI model (Section 4.2). Both problems are extended to their robust counterpart, where the demand is uncertain. The mathematical notation used in the models is detailed in Table 2. For both strategies, the goal is to maximize the supply chain's total profits. This assumption is similar to the works of Santos et al. (2022) and Singh and Kuthambalayan (2022). As it will be shown, the VMI and RMI models share many similarities. Both models are extensions of the one proposed by Santos et al. (2022).

In the VMI, the supplier is responsible for the waste and inventory holding costs incurred at both the supplier and the retailers. The objective function in (1a) - (1f) maximizes the total profits of the supply chain. The term (1a) represents the total revenues collected by the supplier and the retailers. The term (1b) represents the inventory holding costs, where the unitary cost is the same for both the supplier and the retailer (since, in the VMI setting, the supplier bears both costs). The term (1c) is the total setup and variable production costs. The term (1d) represents the total waste disposal costs. In this case, the unitary cost at the retailers equals the supplier's unitary revenue, since the supplier has to reimburse the retailers for the expired products. The term (1e) represents the total surplus cost at the retailers at the end of the planning horizon and is also considered a cost reimbursed by the supplier. The term (1f) represents the total variable purchase costs of the retailers.

Inventory constraints at supplier: Constraints (2) describe the inventory balance (quantity and age) throughout the time horizon. Constraints (3) guarantee that the supplier cannot send more products to the retailers than it has in inventory. Constraints (4) ensure that all inventory reaching the shelf life is wasted.

Production constraints: Constraints (5) relate the inventory and the production constraints, guaranteeing that whenever production occurs, it corresponds to the total inventory of products with age 0. Constraints (6) and (7) establish that, whenever production occurs, it implies a setup and a minimum quantity to produce.

MLOR conditions: Constraints (8) establish the relationship between variables θ_a,i and variables MLOR_i. Constraints (9) can be interpreted as follows. If θ_a,i = 1, the product is fresh enough to be accepted by retailer i; if θ_a,i = 0, the age of products has surpassed the MLOR and thus they can no longer be assigned to that retailer. In fact, within a VMI, the MLOR rule is not imposed by retailers. Nonetheless, to be comparable to the other models, the MLOR conditions are here defined.

Inventory constraints at retailers: Constraints (10) describe the inventory balance at each retailer, at the end of period t. The quantity received by each retailer from the supplier is limited to the space available at each retailer, as described by Constraints (11). All inventory reaching the shelf life is waste, as ensured by Constraints (12).

LEFO and FEFO withdrawal relationships: Constraints (13) and (14) define the ratio between LEFO and FEFO consumers. Constraints (15) and (16) ensure that products of age a can only be withdrawn by LEFO and FEFO consumers if the corresponding variables $ωa,t,iL$ and $ωa,t,iF$ take the value of 1, respectively. Constraints (17) reinforce that the freshest products available are firstly withdrawn by LEFO customers, although products of the same age can be withdrawn by both LEFO and FEFO consumers. Constraints (18) ensure that products withdrawn by LEFO consumers are the freshest products available in a given period. In contrast, Constraints (19) ensure that products withdrawn by FEFO consumers are the oldest available in a given period.

To address demand uncertainty, we employ the robust optimization approach outlined in Bertsimas and Sim (2004). In robust optimization, we assume that the uncertainty does not follow a probability distribution function, but rather belongs to a specific uncertainty set. In this approach, the uncertainty set is a polyhedron, and the uncertain parameters fall within a symmetric interval. The budget of uncertainty, denoted by Γ is a user-defined parameter reflecting the risk aversion of the decision maker. In other words, the value of Γ shows how many times the worst case is allowed to occur. In our case, Γ is within the range [0, t], where Γ = 0 means no risk aversion (deterministic case, where the worst case is not allowed to happen) and Γ = t means an extreme risk aversion (the worst, most conservative case, where the worst case will happen in each period). This approach introduces the budget of uncertainty into the constraints involving demand, along with auxiliary variables that form the robust counterpart of the deterministic problem. The detailed reformulation of the deterministic model into the robust model is provided in Appendix A.

In the VMI problem, demand only appears in Constraints (13) and (14), regarding the withdrawal behaviour. Applying the robust optimization approach, these constraints are reformulated as follows:

where Γ_t,i is the budget of uncertainty of each retailer, λ and μ are the auxiliary variables of the robust reformulation and ϵ is the deviation from the expected demand.

The robust VMI problem is as follows:

In this problem, each entity's inventory is managed independently. The main difference from the VMI problem is that each retailer imposes the quantity they want to receive, and each entity is responsible for its inventory and waste costs.

The objective function combines the total profits of the supplier and the retailers. Term (23a) represents the suppliers' and retailers' revenues, term (23b) represents the suppliers' total costs, and term (23c) represents the retailers' total costs. Note that the surplus cost is not considered in this case, as it is included in the total inventory holding costs of each entity.

Inventory constraints at supplier: Constraints (2–4) apply, as they are necessary to reflect the supplier's inventory.

Production constraints: Constraints (5–7) apply, as they are necessary to reflect the supplier's production conditions.

MLOR conditions: Constraints (8–9) apply, as they reflect the relationship between product age and the MLOR rule.

Inventory constraints at retailers: Constraints (10) and (12) apply, as they reflect the inventory balance at each retailer. However, it is no longer necessary to limit the space available at the retailer to receive products, since retailers are now responsible for their own ordering decisions. Therefore, Constraints (11) are excluded.

LEFO and FEFO withdrawal relationships: Constraints (13–19) apply, as they represent consumer selection behavior based on preferences.

Ordering constraints: Constraints (24) and (25) represent the ordering requirements of the retailers. The ordering policy follows constraints (24), and whenever an order is made, β takes the value of 1 (otherwise it is 0). The total quantity ordered in a period is the sum of all units with different ages sent by the supplier in that period, as imposed by constraint (26).

In the RMI, demand appears not only in constraints related to the withdrawal behavior but also in those related to retailers' ordering policies (Li et al., 2011). Hence, it is necessary to reformulate constraints (24) as follows:

The robust RMI is as follows:

Using the MLOR rule as a constraint in supply chain planning has been standard practice for a long time. However, it is well known that this rule primarily benefits retailers in food supply chains. Recently, the literature on this topic has argued that retailers can also benefit from determining the optimal MLOR rules for a given product type. In this case, the MLOR rule is no longer a constraint, but rather a decision variable. In this section, we show how the MLOR rule can be integrated as a decision variable in both the VMI and RMI problems.

First, it is important to note that both problems optimize the MLOR rule, but, unlike all the decisions presented above, it is a tactical rather than an operational decision. In fact, based on historical data on production, inventory, and demand over a given time horizon, it is possible to find the best MLOR rule.

Second, the collaborative MLOR rule can be applied to either centralized or decentralized problems. In a centralized problem, it is assumed that all decisions are made by a central planner who has access to all relevant information from both entities, and the main objective is to maximize benefits for the whole supply chain. In contrast, a decentralized problem assumes that each entity makes its own decisions, which may or may not affect others, and the objective is to maximize individual objectives.

Finally, defining the optimal MLOR is governed by Constraints (8) and (9), and the rationale is as follows. In each period t, the age of the products sent to the retailers is tracked through variables R_a,t,i, and the “periodic” MLOR is set through variables MLOR_i. At the end of the time horizon T, the optimal MLOR is determined by the lowest MLOR found in all the periods (i.e. $MLORioptimal=mint∈TMLORi,∀i$⁠). Consider the example given in Table 3, for a product with a shelf life of 5 periods, a time horizon of 3 periods, and daily shipments to a retailer of 10 units. At time 0, there is no stock at either the retailer or the supplier, so the supplier produces a given quantity of units and delivers it to the retailers. In this case, all delivered products have an age of 0 and therefore an MLOR of 100% (or 1). In period 1, the products delivered are a mixture of products produced recently (age 0) and products held in stock from the previous period (age 1). Since the oldest product determines the MLOR, this sets the MLOR to 80% (or 0.8). In period 2, all the products shipped are from stock from previous periods, and the oldest ones have an age of 2, giving a periodic MLOR of 0.6. In the last period (t = 3), some of the products shipped have an age of 0 (i.e., they were recently produced), while others have an age of 3, yielding an MLOR of 0.4. Taking all periods into account, the minimum MLOR found was 40%. This is the optimal MLOR rule.

This section reports the results of the computational experiments conducted. First, a comparison of the deterministic VMI and RMI is performed in terms of the MLOR rule, average freshness, profits and waste. Such a comparison is also performed against the traditional case of fixing the MLOR rule a priori (i.e. setting the MLOR to 0.66, corresponding to the typical 2/3 rule). Next, we compared the deterministic solutions with those obtained from the robust versions of the problems.

The benchmark instances used in the experiments are the same as in Santos et al. (2022) for a product with a 5-period shelf life, with the only differences being that we duplicate the number of retailers (i.e. two retailers with the same parameters), and change the minimum lot size values to 25, 50, 75, and 100 for all instances. In the benchmark instances, there are five scenarios for three different demand profiles. In each scenario from Group A, demand is drawn from a uniform distribution [5,20]. In each scenario from Group B, demand is drawn from a uniform distribution [10,25]. In each scenario from Group C, demand is drawn from a uniform distribution [10,50]. The remaining parameters are as follows: r = 4, u = 5, c = 2, h = 0.1, o = 1, $ri′=5$⁠, h′ = 0.5, o′ = , f = 5 and α = 0.8. We also consider a 20-period time horizon, which is considerably longer than the products' shelf life. This allows us to cover several reorder and replenishment periods and to infer the impact of the MLOR rule. We present the results as average values and their respective standard deviations for each group.

The first analysis is of the optimal MLOR rule across different collaborative strategies. This analysis considers a deterministic setting, where the average demand is used. The individual (by retailer) and average MLOR, along with their corresponding standard deviations, are reported in Table 4.

A general observation, consistent with the findings in Santos et al. (2022), is that the MLOR decreases with larger lot sizes and increases with larger demand quantities (from group A to B) and larger variability (from group B to C). Additionally, Table 4 shows differences in average MLOR values across retailers. However, the trend is consistent: when the MLOR rule for Retailer 1 is above the fixed 2/3 rule, the same holds for Retailer 2; when the MLOR rule for Retailer 1 is below the fixed 2/3 rule, the same holds for Retailer 2. This applies to both the VMI and the RMI settings, and is an expected outcome since the retailers have similar demand, as explained in Section 3. To investigate whether more heterogeneous retailer profiles would lead to substantially different retailer-specific MLOR rules, an additional sensitivity analysis was conducted and is presented in Appendix D. In this analysis, Retailer 2 is assigned twice the demand of Retailer 1. The results confirm that retailer-specific MLOR values may be obtained, particularly under the RMI strategy. However, even under this more heterogeneous setting, the optimal MLOR values remain relatively close across retailers, suggesting that the main trends observed in the base case are not driven solely by the assumption of similar retailer demand.

When comparing different inventory strategies, the MLOR is generally higher under VMI than under RMI. There are two reasons for this observation. The first is related to the different holding costs of each entity–the holding cost for the supplier is much lower than that for the retailer, i.e. h ≪′. Note that there is no lead time in either strategy, but the entities agree on where the inventory is held at the end of each period. If at the retailers, the products can be immediately consumed; if at the supplier, the products are shipped in the next order. Thus, in the VMI, since the supplier is responsible for all inventory costs, the cost is the same whether it is at the supplier's facility or at the retailer's. This means that in the VMI, there is more inventory at the retailers than in the RMI, and thus the MLOR can be higher. On the other hand, in RMI, it becomes less costly to keep inventory reserved with the supplier. The distribution of inventory held by the supplier and by retailers in both strategies is illustrated in Figure 3. The second reason for a higher MLOR in the VMI is the supplier's reimbursement costs for waste and surplus. The supplier prefers to send fresher products than in the RMI to avoid such huge costs.

Note that when the optimal MLOR is equal to or higher than 2/3 (or 66%), it means that no deviation from the standard practice is observed. This occurs in both problems for a minimum lot size of 25. However, lower MLOR values are found for several other cases. Thus, we compare the average freshness obtained with the fixed MLOR and the optimal MLOR rules to investigate whether the insight from Santos et al. (2022) still holds in both problems for cases with multiple retailers. To conduct this analysis, we compute the average difference between the indicators obtained with the fixed MLOR and the optimal MLOR for each group of instances. The decrease in the average freshness (ΔF), along with the decrease in the MLOR rule (ΔM), is reported in Table 5. Note that, while the average MLOR reported in Table 4 may be above 66% for each group of instances, this value can be lower for a given instance. Therefore, we report the results independently of the previously reported average MLOR. The results show that in the RMI it is possible to reduce the MLOR by up to 28 p.p., with only a slight reduction of 1 p.p. in the average freshness. Thus, in general, it follows the trend observed in Santos et al. (2022). However, in the VMI, such an MLOR reduction would result in a 4 p.p. decrease in average freshness. Therefore, the previous literature results cannot be generalized to the VMI strategy.

Let us now assess the effect of the optimal MLOR on average profits and relative waste compared with the 2/3 rule. For the sake of conciseness, Figures 4 and 5 illustrate only the differences for instance group A, whereas the remaining groups are shown in Appendix B, as the results follow the same trend. The results demonstrate the savings in profits and waste achieved by allowing MLOR optimization compared with standard practice. Such savings increase along with the minimum lot size and are more pronounced for the VMI than for the RMI. In particular, for the largest minimum lot size, applying the 2/3 rule in the VMI significantly reduces supply chain profits due to substantial waste. This result contrasts with those obtained by Akkas and Honhon (2023). However, the effect of the minimum lot size was not investigated in that work; thus, they may have assumed a small lot size, which, as shown in our work, does not affect the optimal MLOR. It is also important to note that all waste generated is at the supplier for both problems, as its associated costs are lower than if it occurred at the retailers.

Although it is demonstrated that, in general, the VMI leads to higher global profits than the RMI, the distribution of profits among players in the collaborative setting exhibits distinct behavior, as shown in Table 6. In the RMI, the supplier loses some profit when the MLOR rule changes from fixed to optimal, while the retailers always gain, and these losses and gains become more pronounced with the lotsize. This observation aligns with that of Santos et al. (2022), which is why they developed a simple profit-sharing mechanism. In opposition, in the VMI, only the supplier benefits from changing the MLOR rule definition – the supplier reduces setup, production, and holding costs, while the retailers are not affected by its decisions. In particular, for the highest lotsize, the supplier increases its profits considerably by reducing waste costs, as shown in Figure 5. Finally, it is worth noting that, since retailers are homogeneous and face the same demand, they earn the same profits. Therefore, each retailer gets half of the profits reported in Table 6.

To examine the robust versions of the VMI and RMI, we test different values of the budget of uncertainty, following the work of Alem et al. (2018). The tested values are Γ_t,i = 0.05t + 0.1, Γ_t,i = 0.1t + 0.5, Γ_t,i = 0.5t + 0.5 and Γ_t,i = t. The deviation of the expected demand is ϵ = 0.2. The cumulative protection of each budget of uncertainty considered is depicted in Figure 6 for the entire time horizon. As the budget of uncertainty represents the risk aversion of the decision-maker, the higher the Γ, the higher protection it offers over the time horizon. Considering Figure 6, the cumulative protection is higher for the first time periods and decreases over time, except for the worst-case scenario. For example, when t = 1, the budget of uncertainty Γ_t,i = 0.05t + 0.1 offers a 15% protection, the budget of uncertainty Γ_t,i = 0.1t + 0.5 offers 60% protection, and the budget of uncertainty Γ_t,i = 0.5t + 0.5 offers 100% protection. When t = 10, the protection values decrease to 6%, 15%, and 75%, respectively. However, in the worst-case scenario, when Γ_t,i = t, the protection is always 100% for any time period.

As the uncertainty stems from consumer demand, and following the robust formulations presented in Sections 4.1.1 and 4.2.1, we are only interested in cases where the actual demand is lower than expected, leading to waste. When actual demand exceeds expected demand, it leads to lost sales, which are not accounted for in our problems and thus not analyzed in this work. Nevertheless, we may infer that, in the case of high demand, the MLOR rule would probably decrease, because even if consumers tend to select the freshest products, but only the oldest are available, they would pick those rather than none. Therefore, in our setting, the MLOR rule does not affect lost sales because it does not influence the quantity shipped to the retailer but only the age composition of the shipment, and the consumers always pick products if available. Our modelling assumption is consistent with previous studies on perishable inventory management that focus on waste-related decisions under demand uncertainty without explicitly incorporating lost sales into the optimization model (e.g. Kaya and Ghahroodi (2018), Akkas and Honhon (2023), Hasiloglu-Ciftciler and Kaya (2025)). Figures 7a and 7b illustrate the relative waste generated under the different budgets of uncertainty for the VMI and RMI, respectively. We only provide results for instance group A; for the other groups, the trend is the same, though with less expression (see Appendix D). As expected, waste generated in the supply chain tends to increase with the conservatism of the budget of uncertainty and minimum lot size. However, the relative waste in the RMI is higher than in the VMI. This is because, while in the VMI, the uncertainty only appears in the withdrawal constraints, in the RMI, it also appears in the ordering constraints. Hence, the RMI becomes penalized.

However, while in the VMI, the total waste generated in the supply chain occurs solely at the supplier, in the RMI, the highest share of waste occurs at retailers. Due to our robust formulation, the robust RMI model introduces a bullwhip effect of demand uncertainty, as the uncertainty also appears in ordering decisions (introduced by Constraints (27)), which forces the retailer to possibly order more than the effective withdrawal decisions (introduced by Constraints (20) and (21)). As the budget of uncertainty increases, so does the bullwhip effect. The share of waste at the suppliers and retailers is reported in Table 7.

Another insight obtained from these results is that, independently of the collaborative strategy, VMI or RMI, when the minimum lot size is small (e.g., 25), the total relative waste is extremely low. Therefore, reducing the minimum lot size to the minimum may be a good strategy to avoid problems arising in retailer-supplier relationships for perishable products (note that the MLOR rule is imposed to reduce the number of possible expired products at retailers).

Next, the optimal MLOR in the robust versions of the problems is compared with their deterministic counterparts (i.e. when Γ = 0). Table 8 shows these results. A general observation is that, while in the VMI the optimal MLOR tends to increase with the conservatism of the budget of uncertainty, in the RMI it tends to decrease. Under the VMI strategy, the supplier must reimburse retailers for waste and surplus generated at the retailers. To avoid these costs, the supplier delivers fresher products at the expense of higher holding costs for retailers. Such costs, however, are much smaller than the reimbursement. In the RMI strategy, the supplier is no longer responsible for the costs at the retailers. Thus, it will profit from all the products sent, even if they result in outdated or surplus products. On the other hand, retailers' waste costs are much lower than the revenue generated from selling products to the final consumer. Hence, the MLOR can be reduced to lower the supplier's revenue losses, while the retailers' revenue losses remain roughly the same as in the VMI strategy. In Table 9, we observe the revenue losses for the supplier and retailers under the worst-case solutions compared to their deterministic counterparts.

The models developed in this paper take a centralized perspective, in which a planner chooses production, shipment, and inventory decisions to maximize total supply chain profit for a single supplier and multiple retailers. The MLOR level is endogenous rather than exogenous: it is implicitly defined by the age profile of the units shipped in the optimal solution. Consequently, the managerial insights below should be interpreted as guidelines; they are based on how the implied MLOR, profits, waste, and freshness would behave if the supplier and retailers could perfectly coordinate under VMI or RMI.

The computational study shows systematic patterns in the implied MLOR across strategies, lot sizes, and demand profiles (Table 4). For fixed demand profile and lot size, the implied MLOR under VMI is generally higher than under RMI, especially for medium and large minimum lot sizes (75–100 units). For small lot sizes (25 units), both strategies produce MLOR values close to or above the traditional 2/3 rule, and the difference between VMI and RMI is negligible. In practice, when the supplier carries most of the waste and holding cost risk (a VMI-like setting), one should expect contracts or implicit operating rules to induce higher MLOR levels than when retailers bear those costs (an RMI-like setting). The models provide a quantitative upper bound on the size of that difference.

Comparisons between the fixed 2/3 MLOR and the implied optimal MLOR show that relaxing the rule can generate non-trivial economic gains, particularly when minimum lot sizes are large. For small lot sizes (25 units), the optimal policy naturally respects or exceeds the 2/3 rule in both strategies, and profits, freshness, and waste are essentially unchanged when MLOR is fixed at 2/3 versus implicitly optimized. But, as lot size increases, enforcing the 2/3 rule becomes costly, especially under VMI: at large lot sizes (e.g. 100 units), fixing MLOR at 2/3 substantially reduces total profit and increases relative waste compared with the optimal MLOR solution. All waste occurs at the supplier because its disposal costs are lower than those at the retailer.

Regarding freshness, the numerical results show that in RMI, it is possible to reduce the implied MLOR by as much as 28% points relative to 2/3, with only a 1 percentage-point decrease in average freshness. And in VMI, the same reduction in MLOR results in an average freshness loss of about 4% points.

From a prescriptive perspective, relaxing MLOR below the standard 2/3 rule can deliver significant cost savings and waste reductions, but the freshness–cost trade-off is steeper in VMI than in RMI. Hence, in environments closer to VMI, managers should be more cautious in lowering MLOR, particularly when consumer demand is strongly freshness-sensitive.

Across both strategies and across deterministic and robust settings, the minimum production lot size emerges as a key driver of performance. Again, when the minimum lot size is small (25 units), relative waste is essentially zero under all demand profiles and uncertainty levels, and the optimal policy is compatible with the traditional 2/3 MLOR. As lot size increases, both waste and the sensitivity of profits to the MLOR rule increase sharply, especially under VMI, where large lots combined with a high MLOR generate significant scraps at the supplier.

This indicates that many of the MLOR-related inefficiencies observed in practice may be driven by production batching constraints rather than by the MLOR rule itself. Consequently, before devoting managerial effort to renegotiating MLOR levels, firms should examine whether reducing minimum lot sizes—e.g. by investing in changeover reduction or more flexible lines—would yield comparable gains with simpler contractual changes. When lot sizes can be kept small, the marginal value of optimizing MLOR becomes limited.

The robust optimization experiments introduce a budget of uncertainty Γ, which measures how conservative the planner is with respect to demand shortfalls. Higher Γ values protect against low realizations of demand but also tend to increase waste and reduce profit.

Results show that, for a given lot size and Γ, total relative waste increases with Γ in both VMI and RMI. For the same Γ, RMI generates more waste than VMI. From another perspective, in VMI, the implied MLOR tends to increase with Γ, especially for smaller and medium lot sizes (e.g. for a lot size of 25, from 80% in the deterministic case to 96% when Γ = t). And, in RMI, the implied MLOR tends to decrease or remain low as Γ increases (e.g. for lot size 25, from 92% to 40% when Γ = t).

From a modeling standpoint, this is intuitive: under VMI, higher risk aversion leads the centralized planner to ship fresher products, because the supplier is financially exposed to expiration at retailers. Under RMI, the planner responds to uncertainty by allowing older products to be shipped and held downstream, since the supplier is less exposed to retailer waste.

In summary, demand uncertainty does not justify a uniform “raise MLOR” or “lower MLOR” policy. Under uncertainty, VMI-like systems should couple higher MLOR requirements with investments in production flexibility and explicit recognition of supplier waste exposure. And RMI-like systems should address uncertainty primarily through order and contract design, using MLOR primarily as a supporting parameter rather than as the main instrument of risk management.

The Minimum Life on Receipt (MLOR) rule establishes the minimum remaining shelf life perishable products must have upon reaching retailers. The higher the MLOR rule, the higher the remaining shelf life upon receipt. This rule is usually fixed for any type of product and serves best retailers aiming to minimize waste resulting from expired products. However, this rule forces suppliers to either produce in small batches, thereby increasing their setup costs, or incur waste costs from products that do not comply with the MLOR rule. Recently, some literature has examined how this MLOR rule can be collaboratively defined by suppliers and retailers to benefit all parties.

This work complements this emergent literature by comparing the effect of a collaborative-defined MLOR rule across two types of collaborative inventory management strategies – vendor-managed inventory (VMI) and retailer-managed inventory (RMI). In the VMI, the supplier manages its own inventory and retailers' inventory, is responsible for all inventory costs, and reimburses retailers for all expired products. In the RMI, each entity manages its own inventory and is responsible for its own costs. This work also contributes to the literature by addressing demand uncertainty through a robust optimization approach rather than traditional stochastic optimization. With robust optimization, there is no need to rely on the probability distribution function of the uncertainty. Rather, the uncertainty is represented by an uncertainty set, focused on the context of having less demand than predicted, which can promote waste generation, and only the decision maker's risk aversion profile is considered. Specifically, we use the budget of uncertainty approach, one of the most widely applied in the literature.

Our results indicate that the collaborative MLOR is generally higher for the VMI than for the RMI strategy. This shows that when the supplier is responsible for reimbursing retailers for expired products, it prefers to send fresher products rather than older ones. Furthermore, larger quantities of fresh products can be held in stock at the retailer at a lower cost (because inventory costs are the same at the retailer and the supplier) and sold to consumers more rapidly than if they are held at the supplier. In contrast, the RMI distinguishes between retailers' and suppliers' inventory costs, and the former are, in general, higher. Hence, from a global supply chain perspective, it is more profitable to hold stock in the supplier than in the retailer. In addition, the impact of demand uncertainty, specifically the amount of relative waste, is also more expressive in the RMI than in the VMI. This is because the RMI accounts for the propagation of demand uncertainty not only from consumer withdrawal policies but also from retailers' ordering decisions, which are not accounted for in the VMI. Finally, our results also demonstrate that the optimal MLOR tends to increase with the decision-maker's risk aversion in the VMI but decreases in the RMI.

Based on our results, we can provide two main managerial insights. The first relates to the importance of the minimum lot size definition. In all experiments, considering demand uncertainty or not, and independently of the collaborative strategy used, VMI or RMI, the lower the minimum lot size, the less interesting it becomes to define a collaborative MLOR rule. Moreover, reducing the minimum lot size improves all indicators, namely profits, average freshness, and relative waste. The second insight from our work concerns the collaborative strategy used. It became evident that adopting the VMI is a highly beneficial strategy for all entities in the food supply chain. The VMI can generate higher profits for retailers and suppliers, reduce food waste, and increase the average freshness of perishable products to a greater extent than the RMI.

A possible direction for future work is to compare the results from robust optimization with those from a traditional stochastic optimization approach. This would allow us to evaluate how conservative the proposed solutions are. In addition, other approaches to handling uncertainty could be explored, including hybrid methods that combine robust and stochastic optimization. Our robust formulations consider only downward deviations from nominal demand, as the focus of this work is on uncertainty scenarios that generate surplus inventory and food waste. Future research could extend the proposed formulations to also account for upward demand deviations, enabling the explicit modelling of stockouts and lost sales and, therefore, a more comprehensive analysis of the trade-offs between waste generation, inventory surplus, and service performance (Buisman et al., 2020). Finally, this work could also benefit from empirical validation supported by a real-world case study.

The supplementary material for this article can be found online.