The container relocation problem (CRP) is concerned with finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers, while respecting a given order of retrieval. However, the assumption of knowing the full retrieval order of containers is particularly unrealistic in real operations. This paper studies the stochastic CRP, which relaxes this assumption. A new multistage stochastic model, called the batch model, is introduced, motivated, and compared with an existing model (the online model). The two main contributions are an optimal algorithm called Pruning-Best-First-Search (PBFS) and a randomized approximate algorithm called PBFS-Approximate with a bounded average error. Both algorithms, applicable in the batch and online models, are based on a new family of lower bounds for which we show some theoretical properties. Moreover, we introduce two new heuristics outperforming the best existing heuristics. Algorithms, bounds, and heuristics are tested in an extensive computational section. Finally, based on strong computational evidence, we conjecture the optimality of the “leveling” heuristic in a special “no information” case, where, at any retrieval stage, any of the remaining containers is equally likely to be retrieved next. The online appendix is available at https://doi.org/10.1287/trsc.2018.0828 .

Retailers have become increasingly interested in personalizing their products and services such as promotions. For this, we need new personalized demand models. Unfortunately, social data is not available to many retailers for cost and/or privacy issues. Thus, we focus on the problem of detecting customer relationships from transactional data, and using them to target promotions to the right customers.Academic / Practical Relevance: From an academic point of view, this paper solves the novel problem of jointly detecting customer trends and using them for optimal promotion targeting. Notably, we estimate the causal customer-to-customer trend effect solely from transactional data, and target promotions for multiple items and time periods. In practice, we provide a new tool for Oracle Retail clients that personalizes promotions. Methodology:We develop a novel probabilistic demand model distinguishing between a base purchase probability, capturing factors such as price and seasonality, and a customer trend probability, capturing customer-to-customer trend effects. The estimation procedure is based on regularized bounded variables least squares and instrumental variable methods. The resulting customer trend estimates feed into the dynamic promotion targeting optimization problem, formulated as a non-linear mixed-integer optimization model.Though it is NP-hard, we propose an adaptive greedy algorithm.Results: We prove our customer-to-customer trend estimates are statistically consistent, and the adaptive greedy algorithm is provably good. Having access to Oracle Retail fashion client data, we show that our demand model reduces the WMAPE by 11% on average. Also, we provide evidence of the causality of our estimates. Finally, we demonstrate that the optimal policy increases profits by 3-11%.Managerial Implications: The demand model with customer trend and the optimization model for targeted promotions form a decision support tool for promotion planning. Next to general planning, it also helps to find important customers and target them to generate additional sales.

The Container Relocation Problem (CRP) involves finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers in a given order. In this paper, we focus on average case analysis of the CRP when the number of columns grows asymptotically. We show that the expected minimum number of relocations converges to a simple and intuitive lower-bound for which we give an analytical formula.

Problem definition: Retailers have become increasingly interested in personalizing their products and services such as promotions. For this, we need new personalized demand models. Unfortunately, social data are not available to many retailers because of cost and privacy issues. Thus, we focus on the problem of detecting customer relationships from transactional data and using them to target promotions to the right customers. Academic/practical relevance: From an academic point of view, this paper solves the novel problem of jointly detecting customer trends and using them for optimal promotion targeting. Notably, we estimate the causal customer-to-customer trend effect solely from transactional data and target promotions for multiple items and time periods. In practice, we provide a new tool for Oracle Retail clients that personalizes promotions. Methodology: We develop a novel customer trend demand model distinguishing between a base purchase probability, capturing factors such as price and seasonality, and a customer trend probability, capturing customer-to-customer trend effects. The estimation procedure is based on regularized bounded variables least squares and instrumental variable methods. The resulting customer trend estimates feed into the dynamic promotion targeting optimization problem, formulated as a nonlinear mixed-integer optimization model. Though it is nondeterministic polynomial-time hard, we propose a greedy algorithm. Results: We prove that our customer-to-customer trend estimates are statistically consistent and that the greedy optimization algorithm is provably good. Having access to Oracle Retail fashion client data, we show that our demand model reduces the weighted-mean absolute percentage error by 11% on average. Also, we provide evidence of the causality of our estimates. Finally, we demonstrate that the optimal policy increases profits by 3%–11%. Managerial implications: The demand model with customer trend and the optimization model for targeted promotions form a decision-support tool for promotion planning. Next to general planning, it also helps to find important customers and target them to generate additional sales.

The Container Relocation Problem (CRP) is concerned with finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers, while respecting a given order of retrieval. However, the assumption of knowing the full retrieval order of containers is particularly unrealistic in real operations. This paper studies the stochastic CRP (SCRP), which relaxes this assumption. A new multi-stage stochastic model, called the batch model, is introduced, motivated, and compared with an existing model (the online model). The two main contributions are an optimal algorithm called Pruning-Best-First-Search (PBFS) and a randomized approximate algorithm called PBFS-Approximate with a bounded average error. Both algorithms, applicable in the batch and online models, are based on a new family of lower bounds for which we show some theoretical properties. Moreover, we introduce two new heuristics outperforming the best existing heuristics. Algorithms, bounds and heuristics are tested in an extensive computational section. Finally, based on strong computational evidence, we conjecture the optimality of the "Leveling" heuristic in a special "no information" case, where at any retrieval stage, any of the remaining containers is equally likely to be retrieved next.