打包
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打包问题的目标是找到将一组给定大小的项打包到具有固定packing的容器中的最佳方式。packing典型应用是高效地将箱子加载到货车上。
通常,由于容量限制,无法打包所有商品。在这种情况下,问题是找出总大小不超过最大可容纳在容器中的项的子集。
打包问题有多种类型。其中最重要的两个是“背包问题”和“装箱问题”。
背包问题
在简单的背包问题中,只有一个容器(背包)。这些项具有值和尺寸,目标是打包具有最高总价值的部分项。
对于值等于大小的特殊情况,目标是使打包商品的总大小最大化。
OR-Tools 在其算法库中为背包问题提供了多种求解器。
此外,背包问题还有更普通的版本。以下是几个示例:
请注意,您可能是针对单个背包的多维问题,也可能是只有一个维度的多维背包问题。
装箱问题
最广为人知的打包问题之一是装箱,即装箱时有多个容量相同的容器(称为装箱)。bin-packingbin-packing与多背包问题不同,箱子的数量并非固定的。相反,目标是找到容纳所有项的最小数量的分箱。
下面这个简单的示例说明了多背包问题和装箱问题之间的区别。假设一家公司有货运卡车,每辆卡车的承重能力为 18,000 磅,需要运送 130,000 磅的商品。
以下各部分介绍如何使用 OR-Tools 解决各种类型的打包问题,从背包问题开始。
如未另行说明,那么本页面中的内容已根据知识共享署名 4.0 许可获得了许可,并且代码示例已根据 Apache 2.0 许可获得了许可。有关详情,请参阅 Google 开发者网站政策。Java 是 Oracle 和/或其关联公司的注册商标。
最后更新时间 (UTC):2024-08-09。
[null,null,["最后更新时间 (UTC):2024-08-09。"],[[["\u003cp\u003ePacking problems involve finding the best way to pack items into containers, often with capacity constraints.\u003c/p\u003e\n"],["\u003cp\u003eKnapsack problems focus on maximizing the value of packed items within a single or multiple containers with limited capacity.\u003c/p\u003e\n"],["\u003cp\u003eBin packing aims to minimize the number of containers needed to pack all items, using bins of equal capacity.\u003c/p\u003e\n"],["\u003cp\u003eMultidimensional and multiple knapsack problems are variations that consider additional item properties or multiple containers.\u003c/p\u003e\n"],["\u003cp\u003eOR-Tools provides solvers and algorithms for tackling various packing problem types, including knapsack and bin packing.\u003c/p\u003e\n"]]],["Packing problems aim to pack items into containers with fixed capacities, often maximizing the total size or value of packed items. Key problem types include knapsack problems, where items have values and the goal is to maximize the total value in a single container, and bin-packing, which minimizes the number of containers needed to hold all items. Variations like multidimensional and multiple knapsack problems exist, with additional constraints or containers. OR-Tools offers solvers for these problems.\n"],null,["# Packing\n\nThe goal of *packing* problems is to find the best way to pack a set of\nitems of given sizes into containers with\nfixed *capacities*. A typical application is loading boxes onto delivery trucks\nefficiently.\nOften, it's not possible to pack all the items, due to the capacity\nconstraints. In that case, the problem is to find a subset of the items with\nmaximum total size that will fit in the containers.\n\nThere are many types of packing problems. Two of the most important are\n*knapsack problems* and *bin packing*.\n\nKnapsack problems\n-----------------\n\nIn the simple knapsack problem, there is a single container (a knapsack).\nThe items have *values* as well as sizes, and\nthe goal is to pack a subset of the items that has maximum total value.\n\nFor the special case in which value is equal to size, the\ngoal is to maximize the total size of the packed items.\n\nOR-Tools provides several solvers for knapsack problems in its\n[algorithms library](/optimization/reference/algorithms).\n\nThere are also more general versions of the knapsack problem. Here are a couple\nof examples:\n\n- *Multidimensional knapsack problems* , in which the items have\n more than one physical quantity, such as weight and volume,\n and the knapsack has a capacity for each quantity. Here,\n the term *dimension*\n does not necessarily refer to the usual spatial\n dimensions of height, length, and width.\n However, some problems might involve spatial dimensions,\n for example, finding the optimal way to pack rectangular boxes into a\n rectangular storage bin.\n\n- [*Multiple knapsack problems*](/optimization/pack/multiple_knapsack),\n in which there are multiple knapsacks, and\n the goal is to maximize the total value of the packed items in all knapsacks.\n\nNote that you can have a multidimensional problem\nwith a single knapsack, or a multiple knapsack problem with just one\ndimension.\n\nThe bin-packing problem\n-----------------------\n\nOne of the most well-known packing problems is\n*bin-packing* , in which there are multiple containers (called *bins*) of\nequal capacity. Unlike the multiple knapsack problem, the number of bins is not\nfixed. Instead, the\ngoal is to find the smallest number of bins that will hold all the items.\n\nHere's a simple example to illustrate the difference between the\nmultiple knapsack problem and the bin-packing problem. Suppose a company has\ndelivery trucks, each of which has an 18,000 pound weight capacity, and 130,000\npounds of items to deliver.\n\n- Multiple knapsack: You have five trucks and you want to load a subset of the\n items that has maximum weight onto them.\n\n- Bin packing: You have 20 trucks (more than enough to hold all the items)\n and you want to use the fewest trucks that will hold them all.\n\nThe following sections show how to solve various types of packing problems with\nOR-Tools, starting with the [knapsack problem](/optimization/pack/knapsack)."]]