小婧研学（59）运筹优化中的网络规划

摘要：网络规划是运筹学的一个重要分支，它以图论为基础，通过构建和分析由节点（也称为顶点）与边（连接节点的连线，可带有方向形成有向边，也可能无方向）组成的网络模型，来解决各类实际优化问题。网络可以直观地表示诸多现实系统，比如交通网络中的城市（节点）与道路（边）、通信网

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Today, the editor brings you "Network planning in operations research optimization"

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一、基本概念

First, the basic concept

网络规划是运筹学的一个重要分支，它以图论为基础，通过构建和分析由节点（也称为顶点）与边（连接节点的连线，可带有方向形成有向边，也可能无方向）组成的网络模型，来解决各类实际优化问题。网络可以直观地表示诸多现实系统，比如交通网络中的城市（节点）与道路（边）、通信网络中的基站（节点）及通信线路（边）、物流配送中的仓库与运输路线等。

Network planning is an important branch of operations research, which is based on graph theory to solve various practical optimization problems by constructing and analyzing network models composed of nodes (also known as vertices) and edges (connecting nodes, which can form directed edges with or without directions). The network can intuitively represent many real-world systems, such as cities (nodes) and roads (edges) in transportation networks, base stations (nodes) and communication lines (edges) in communication networks, warehouses and transportation routes in logistics and distribution, etc.

二、常见的网络规划问题类型

2. Common types of network planning problems

1. 最短路问题：在给定的网络中，求从一个指定的起点节点到另一个指定的终点节点（或者多个终点节点）之间，沿着边移动的总距离（或者总代价等可以量化衡量的指标）最短的路径。这里的距离或代价可以根据实际问题赋予不同含义，例如路程长度、运输成本、时间消耗等。

1. Shortest circuit problem: In a given network, find the shortest path along the total distance (or quantitatively measurable metrics such as total cost) along an edge from a specified starting node to another specified end node (or multiple end nodes). The distance or cost here can be given different meanings depending on the actual problem, such as the length of the journey, the cost of transportation, the consumption of time, etc.

2. 最大流问题：对于一个有向网络，存在一个源节点（供应点，有物资、流量等流出）和一个汇节点（接收点），以及连接各个节点的有向边（每条边有一定的容量限制，代表允许通过的最大流量），目标是确定从源节点到汇节点所能通过的最大流量值，同时还要给出相应的流量分配方案，即每条边上具体通过多少流量。

2. Maximum flow problem: For a directed network, there is a source node (supply point, with outflow of materials, traffic, etc.) and a sink node (receiving point), as well as a directed edge connecting each node (each edge has a certain capacity limit, representing the maximum flow allowed through), the goal is to determine the maximum flow value that can pass from the source node to the sink node, and also give the corresponding traffic distribution scheme, that is, how much traffic passes through each edge.

3. 最小费用流问题：同样是在一个有向网络里，每条边不仅有容量限制，还有单位流量的费用（例如运输成本、管道输送成本等），在满足从源节点到汇节点一定流量要求的前提下，寻求使得总费用最小的流量分配方案。它可以看作是综合考虑了流量限制以及成本因素的优化问题，是最大流问题和费用因素的结合。

3. Minimum cost flow problem: In the same directed network, each edge not only has a capacity limit, but also a unit of traffic cost (such as transportation cost, pipeline transportation cost, etc.), and on the premise of meeting a certain traffic requirement from the source node to the sink node, the traffic allocation scheme that minimizes the total cost is sought. It can be seen as an optimization problem that comprehensively considers the flow limit and the cost factor, which is a combination of the maximum flow problem and the cost factor.

4. 最小支撑树问题：对于一个无向连通网络（所有节点都能通过边相互连通），要找到一个包含所有节点且边的总权重（例如边代表的建设成本、距离等）最小的连通子图，这个子图是一棵树（无回路且连通），也就是最小支撑树。

4. Minimum support tree problem: For an undirected connected network (all nodes can communicate with each other through edges), it is necessary to find a connected subgraph that contains all nodes and the total weight of the edges (such as the construction cost, distance, etc.) of the edges is the smallest, and this subgraph is a tree (no loop and connected), that is, the minimum support tree.

三、解决网络规划问题的常用算法

3. Common algorithms for solving network planning problems

最短路算法：Dijkstra算法、Bellman - Ford算法

Shortest circuit algorithms: Dijkstra algorithm, Bellman-Ford algorithm

最大流算法：Ford - Fulkerson算法、Edmonds - Karp算法

Maximum flow algorithms: Ford-Fulkerson algorithm, Edmonds-Karp algorithm

最小费用流算法：最小费用流的单纯形算法、网络单纯形算法

Minimum cost flow algorithm: simplex algorithm and network simplex algorithm for minimum cost flow

最小支撑树算法：Prim算法、Kruskal算法

Minimum Support Tree Algorithm: Prim Algorithm and Kruskal Algorithm

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