Glossary

6.1 Network models and terminology

3 min read•july 30, 2024

Network models are essential tools for optimizing complex systems. They use nodes and arcs to represent interconnected elements, allowing us to analyze and solve problems in transportation, supply chains, and more. Understanding these models is key to tackling real-world optimization challenges.

Network problems focus on efficiently moving resources through a system. By applying constraints like limits and flow conservation, we can solve various optimization tasks. These include finding maximum flows, minimizing costs, and determining shortest paths in networks.

Network flow model components

Nodes and arcs

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  • Nodes (vertices) represent locations, junctions, or decision points in the system
  • Arcs (edges) represent connections or pathways between nodes
  • Source node generates flow originating in the network
  • Sink node terminates flow in the network
  • Intermediate nodes facilitate flow between source and sink
  • Directed arcs allow flow in only one direction
  • Undirected arcs permit bidirectional flow

Network structure and attributes

  • Network topology characterizes the arrangement and connectivity of nodes and arcs
  • Arc attributes influence flow through the network
    • Capacity limits on an arc
    • Cost affects the expense of sending flow along an arc
    • Distance impacts the length of a path through the network
  • Mathematical representation uses matrices to describe node and arc relationships
    • Adjacency matrices show direct connections between nodes
    • Incidence matrices indicate how arcs connect to nodes

Network flow problem types

Optimization problems

  • Maximum flow problem determines highest possible flow from source to sink while respecting arc capacities
  • Minimum cost flow problem finds most cost-effective way to send specified flow through network with arc costs and capacities
  • Shortest path problem identifies path with lowest total cost or distance between two nodes
  • Minimum spanning tree problem locates subset of arcs connecting all nodes with minimum total cost or weight

Resource allocation problems

  • Transportation problem distributes goods from multiple sources to multiple destinations while minimizing total transportation costs
  • Assignment problem optimally assigns resources to tasks or workers to jobs (special case of transportation problem)

Real-world applications

  • Supply chain management optimizes product distribution (warehouses, retail stores)
  • Transportation planning improves route efficiency (delivery services, public transit)
  • Telecommunications enhances data routing (internet traffic, cellular networks)
  • Energy distribution balances power grid loads (electricity transmission, gas pipelines)
  • Project scheduling allocates resources effectively (construction projects, software development)

Network flow constraints

Capacity and flow

  • Capacity sets maximum flow limit for arc or node
  • Flow represents actual amount moving through arc or node
  • Non-negative flow constraint ensures flow values are always positive or zero
  • Capacity constraint maintains flow within arc's maximum capacity
    • Mathematically expressed as: 0≤f(i,j)≤c(i,j)0 \leq f(i,j) \leq c(i,j) for all arcs (i,j)(i,j)

Conservation of flow

  • Total flow entering node equals total flow leaving node (except source and sink)
  • Analogous to Kirchhoff's Current Law in electrical engineering
  • Mathematically expressed as: ∑(i,j)∈Ef(i,j)−∑(j,i)∈Ef(j,i)=0\sum_{(i,j) \in E} f(i,j) - \sum_{(j,i) \in E} f(j,i) = 0 for all nodes i≠s,ti \neq s,t
    • ss represents source node
    • tt represents sink node
  • Feasible flow satisfies both capacity and conservation constraints for all arcs and nodes

Network flow problem representation

Graph theoretic notation

  • Network denoted as G=(V,E)G = (V, E)
    • VV represents set of vertices (nodes)
    • EE represents set of edges (arcs)
  • Directed arcs shown as ordered pairs (i,j)(i, j) indicating flow from node ii to node jj
  • Arc capacity represented as c(i,j)c(i, j) or u(i,j)u(i, j)
  • Arc flow denoted as f(i,j)f(i, j) or x(i,j)x(i, j)
  • Residual capacity calculated as r(i,j)=c(i,j)−f(i,j)r(i, j) = c(i, j) - f(i, j)

Matrix representations

  • Incidence matrix shows how arcs connect to nodes
    • Rows represent nodes
    • Columns represent arcs
    • Entry values: +1 (arc leaves node), -1 (arc enters node), 0 (arc not connected to node)
  • Adjacency matrix indicates direct connections between nodes
    • Rows and columns both represent nodes
    • Entry values: 1 (direct connection exists), 0 (no direct connection)
  • Matrices facilitate mathematical analysis and algorithm implementation for network flow problems

Key Terms to Review (18)

Assignment model: An assignment model is a type of optimization model used to determine the most efficient way to assign resources or tasks to agents in a way that minimizes costs or maximizes efficiency. This model typically involves finding the best one-to-one correspondence between two sets, such as jobs and workers, ensuring that each task is assigned to one agent and vice versa. Understanding the assignment model helps in solving various practical problems where resources must be allocated optimally.
Bipartite Graph: A bipartite graph is a type of graph where the set of vertices can be divided into two distinct and independent sets such that no two graph vertices within the same set are adjacent. This structure is useful for modeling relationships between two different groups, allowing connections only between members of different sets. The bipartite nature helps in solving various problems involving network flow, transportation, and assignment scenarios effectively.
Capacity: Capacity refers to the maximum amount of flow that a network link can handle without exceeding its limits. This concept is critical in understanding how networks function, as it dictates the feasible flow of resources from one point to another. Capacity plays a key role in optimizing network performance, influencing both the design of the network and the solutions to flow-related problems.
Conservation of Flow: Conservation of flow is a principle that states that the total amount of flow entering a network at any point must equal the total amount of flow leaving that point, ensuring that resources or quantities are neither created nor destroyed within the system. This concept is fundamental in network models, where it connects to how nodes and edges interact, making sure that the flow is balanced throughout the entire network.
Demand constraint: A demand constraint is a limitation that specifies the maximum amount of goods or services that can be consumed or demanded within a certain period. It plays a crucial role in optimization problems, particularly in network models, where the flow of resources must adhere to certain limits set by consumer needs or market demands. Understanding demand constraints helps in effectively allocating resources and optimizing supply chain operations.
Dijkstra's Algorithm: Dijkstra's Algorithm is a popular method used to find the shortest path from a starting node to all other nodes in a weighted graph. It effectively uses a greedy approach, continually selecting the closest node and updating the distance to neighboring nodes until the shortest paths are determined. This algorithm is crucial in network models, as it helps in optimizing routes and managing resources efficiently.
Directed Graph: A directed graph, or digraph, is a set of vertices connected by edges, where each edge has a direction associated with it. This means that the edges are ordered pairs, indicating a one-way relationship from one vertex to another. Directed graphs are particularly useful in modeling scenarios where relationships are not reciprocal, such as in network flow problems, shortest path calculations, and cost flow analysis.
Edge: An edge is a fundamental component of a graph in network models, representing a connection between two vertices (or nodes). In the context of network optimization, edges can signify various relationships or pathways, such as roads in transportation networks or communication links in data networks. The attributes of edges, such as their weights or capacities, play a crucial role in determining the efficiency and effectiveness of network flow and routing problems.
Flow: Flow refers to the quantity of a commodity that passes through a network or system over a specified period of time. It is crucial in understanding the dynamics of network models, as it helps in analyzing how resources like goods, information, or services are transported from one node to another within a network structure.
Ford-Fulkerson Method: The Ford-Fulkerson Method is an algorithm used to compute the maximum flow in a flow network. It operates by finding augmenting paths from the source to the sink and increasing the flow along these paths until no more augmenting paths can be found. This method is fundamental in network optimization, as it helps determine the most efficient way to send materials or information through networks.
Integer Programming: Integer programming is a type of optimization where some or all of the decision variables are required to take on integer values. This concept is crucial in many real-world applications where discrete choices are necessary, such as assigning resources or scheduling tasks. Understanding integer programming helps in modeling complex problems with constraints and objectives that can be represented mathematically.
Linear Programming: Linear programming is a mathematical method used for optimizing a linear objective function, subject to linear equality and inequality constraints. This technique is widely utilized in various fields to find the best possible outcome under given constraints, making it essential for decision-making processes in resource allocation and optimization.
Maximum Flow: Maximum flow is a concept in network theory that refers to the greatest possible flow of a commodity from a source to a sink in a flow network, considering the capacities of the edges that connect the nodes. This concept is crucial in understanding how resources, information, or materials can be efficiently transported through a network without exceeding any constraints. It involves various algorithms that help determine the optimal way to allocate flow and is deeply connected to both theoretical and practical applications in numerous fields.
Supply Chain Optimization: Supply chain optimization is the process of improving the efficiency and effectiveness of a supply chain, which encompasses the flow of goods, information, and finances from suppliers to customers. This involves mathematical modeling techniques to identify the best ways to minimize costs, reduce lead times, and improve service levels while ensuring that resources are used efficiently. By optimizing the supply chain, organizations can enhance their operational performance and respond better to market demands.
Supply Constraint: A supply constraint refers to the limitations on the quantity of resources available to meet demand in a network model. These constraints are critical in optimizing resource allocation, determining the maximum capacities at nodes and edges, and influencing the overall efficiency of the system. Recognizing these constraints helps in developing effective strategies for transportation, logistics, and network flows.
Traffic Routing: Traffic routing is the process of determining the most efficient paths for data packets or vehicles through a network or transportation system. This involves analyzing various routes, considering factors such as distance, traffic conditions, and constraints to optimize flow and minimize delays. It plays a crucial role in network models and their terminology by helping to efficiently manage resources and improve overall performance.
Transportation Model: The transportation model is a type of linear programming used to determine the most efficient way to transport goods from several suppliers to several consumers while minimizing transportation costs. This model helps organizations make optimal shipping decisions by analyzing supply, demand, and transportation costs, which are represented in a network structure of nodes and arcs.
Vertex: A vertex is a point where two or more edges meet in a graph or a corner point in geometric shapes. It plays a critical role in defining the structure of graphs in network models and represents potential solutions in optimization problems, particularly as extreme points in linear programming. Understanding vertices helps analyze the connectivity and flow in networks as well as determine optimal solutions in various mathematical scenarios.
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