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<p>We address a three-dimensional packing problem encountered in the transportation of refrigerated pharmaceutical products. The input consists of a set of products, each comprising multiple boxes with specified dimensions (width, length, height), and a set of container types, each defined by internal dimensions (width, length, height), cost, and a maximum fraction of its internal volume that can be occupied by the products. The goal is to pack the boxes into containers while minimizing the total cost of the containers used, subject to several constraints. These constraints include: <OrderedList> <ListItem> <ItemNumber>(a)</ItemNumber> <ItemContent> <p>each box must be placed vertically (with some exceptions),</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(b)</ItemNumber> <ItemContent> <p>boxes must not overlap,</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(c)</ItemNumber> <ItemContent> <p>each box must be either placed directly on the container floor or have at least 70% of its base supported by other boxes,</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(d)</ItemNumber> <ItemContent> <p>the container’s maximum fillable volume must not be exceeded, and</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(e)</ItemNumber> <ItemContent> <p>each product can only be combined with other products in at most one container.</p> </ItemContent> </ListItem> </OrderedList> To solve this problem, we propose a heuristic approach that proceeds in two stages. First, dynamic programming is used to solve the mono-container and mono-product subproblem, inspired by similar existing pallet loading algorithms. This involves generating all feasible packing configurations for a single product within a single container. Subsequently, a backtracking-based procedure is employed to complete the assignment of the remaining boxes. Our intensive computational experiments for real-world scenarios show that this heuristic achieves acceptable solution times and delivers effective allocations.</p>

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A procedure for packing pharmaceutical products into refrigerated containers based on dynamic programming and backtracking techniques

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Carlos Di Masi,
  • Ivo Koch,
  • Javier Marenco,
  • Marcelo Mydlarz,
  • Leonel Vitale

摘要

We address a three-dimensional packing problem encountered in the transportation of refrigerated pharmaceutical products. The input consists of a set of products, each comprising multiple boxes with specified dimensions (width, length, height), and a set of container types, each defined by internal dimensions (width, length, height), cost, and a maximum fraction of its internal volume that can be occupied by the products. The goal is to pack the boxes into containers while minimizing the total cost of the containers used, subject to several constraints. These constraints include: (a)

each box must be placed vertically (with some exceptions),

(b)

boxes must not overlap,

 (c)

each box must be either placed directly on the container floor or have at least 70% of its base supported by other boxes,

 (d)

the container’s maximum fillable volume must not be exceeded, and

 (e)

each product can only be combined with other products in at most one container.

 To solve this problem, we propose a heuristic approach that proceeds in two stages. First, dynamic programming is used to solve the mono-container and mono-product subproblem, inspired by similar existing pallet loading algorithms. This involves generating all feasible packing configurations for a single product within a single container. Subsequently, a backtracking-based procedure is employed to complete the assignment of the remaining boxes. Our intensive computational experiments for real-world scenarios show that this heuristic achieves acceptable solution times and delivers effective allocations.