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Oct 30, 2022 11:22 PM

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## Introduction

Network flow problems can be represented as a collection of nodes connected by arcs.

- Three types of nodes are: 1) Supply; 2) Demand; and 3) Transshipment.

- We use negative numbers to represent supplies and positive numbers to represent demand.

**Five types of problems**are: 1) Transshipment problem; 2) The Shortest Path Problem; 3) Minimal spanning tree problem; 4) Maximal Flow Problem; and 5) Transportation Problem.

## Rule of thumb

To formulate the constraints, all these methods are of different ways:

- For Maximal Flow Problem, all constraints, nodes = 0.

- For The Shortest Path Problem, set the supply node = -1 and terminal note = 1.

All rules of the modeling are

## 1. Outflow 就減，Inflow 就加 *(出減入加)*

例如: Note 1 outflow 去 , 所以係

相反, Node 4, 有inflow, 所以係

## 2. The balance of rules applied.

## Transshipment problem

A problem in which a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.

The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations.

### Example: **The Bavarian Motor Company**

**The Bavarian Motor Company**

- Understand the problem.

- Identify the
**decision variables**.

- State the
**objective function**as a linear combination of the decision variables.

- State the
**constraints**as linear combinations of the decision variables

## using **The Balance-of-Flow Rules.**

## One example

## All in one

Outflow 就減，Inflow 就加

*(出減入加)*例如: Note 1 outflow 去 , 所以係

相反, Node 4, 有inflow, 所以係

- Identify any
**upper or lower bounds**on the decision variables.

- Result

## The Shortest Path Problem

A special case of a transshipment problem where

- There is one supply node with a supply of - ()

- There is one demand node with a demand of + ()

- All other nodes have supply/demand of +0

## Minimal spanning tree problem

## Generalised Network Flow Problems

In some problems, a gain or loss occurs in flows over arcs.

- Applications are
- Oil or gas shipped through a leaky pipeline
- Imperfections in raw materials entering a production process
- Spoilage of food items during transit
- Theft during transit
- Interest or dividends on investments

#### Examples: Coal Bank Hollow Recycling (Explanations on YouTube is here, using **Excel**)

## Optimal Result

## How others formulate (Here)

## Maximal Flow Problem

In some network problems, the objective is to determine the maximum amount of flow that can occur through a network. The arcs in these problems have upper and lower flow limits.

Examples

- How much water can flow through a network of pipes?

- How many cars can travel through a network of streets?

Set all nodes = 0.

## Another Similar example is easier to understand from (63) Maximum Flow Problem - YouTube

## Video

## Instruction

## Transportation Problem

### Example: Lecture Review Q11

#### Our task is to formulate this solution:

**Note: You can ONLY write in this LAZY / Math way, when there are no arrows between nodes.**

**Author:**Jason Siu**URL:**https://jason-siu.com/article/a4680eaf-50ed-4701-b16e-a380a285eee7**Copyright:**All articles in this blog, except for special statements, adopt BY-NC-SA agreement. Please indicate the source!

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