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# 数学代写|运筹学代写Operations Research代考|MATH2730 VEHICLE ROUTING PROBLEMS

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## 数学代写|运筹学代写Operations Research代考|VEHICLE ROUTING PROBLEMS

These form an important class of optimization problems that are well researched and have immense practical application. One of the early research in this problem is by Dantzig and Ramser (1959). The simplest vehicle routeing problem (VRP) is stated as:

Given a single depot and a certain number of vehicles, the problem is to meet the requirements of $n$ number of customers. Each customer requires a certain quantity that is to be delivered from the depot. There are a fixed number of vehicles available and the capacity of the vehicles are known. The distance among the customer locations as well as from the depot to the locations are known. The vehicles start from the depot and return to the depot after meeting customer requirements. The problem is to deliver the quantities to the customers such that the vehicles used travel minimum total distance.

The above problem is called the single depot vehicle routing problem. There are multiple depot vehicle routing problems where the customer can be served from any vehicle (from any depot). In this chapter we will consider only the single depot vehicle routing problem. There are three types of problems:

1. All the vehicles have to be used [It is assumed that every vehicle will have at least one city (customer) attached to it].
2. Using the minimum number of vehicles.
3. Using enough number of vehicles such that the total distance travelled is minimum.
Let us explain the algorithms and the various problems using an example.

## 数学代写|运筹学代写Operations Research代考|Optimal Solutions

We consider a 6 customer (city) VRP. The depot is denoted by 0 (zero) and the distance matrix is given in Table 9.22.
(The depot and cities are numbered in bold). The and the vehicle capacity is 10 .

Let us find the optimal solution to some problem instances. Let us assume that the vehicle capacity is sufficiently large to hold the requirements of any number of customers. The problem of finding the minimum distance

VRP for a given number of vehicles (say 4 vehicles) is given by solving a TSP created as follows:

Add as many rows and columns as the number of vehicles (in this case we get a $10 \times 10$ TSP). Rows 1 to 6 represent the cities and rows 7 to 10 represent the vehicles (depot).

Distance between cities and distance between depot and city is taken from the given table.

Distance between any entity and itself is infinity.

Distance between any two depots is infinity.
The resultant 10 -city TSP is given in Table $9.23$.

A feasible solution to the TSP is $D-2-1-3-D-4-D-5-D-6-D$ with $Z=166$.

## 数学代写|运筹学代写Operations Research代考|Optimal Solutions

TSP 的可行解决方安是 $D-2-1-3-D-4-D-5-D-6-D$ 和 $Z=166$.

## MATLAB代写

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