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数学代写|运筹学代写Operations Research代考|KMA255 Clark-Wright Savings Algorithm

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数学代写|运筹学代写Operations Research代考|Clark-Wright Savings Algorithm

This method is based on calculating savings that could be availed in visiting two cities through two different routes. The first tour involves starting from origin, going to a city, coming back to origin and then starting again to visit other city. This is called as current tour. In the second route vehicle, after visiting a city from origin does not come back and directly goes to another city before coming back. This is called as joined tour.

For instance, route of Truck2 is Southfield to East Liberty to Greenburg and back to Southfield. So, two routes that it can use are shown in Figures $7.5$ and 7.6.
In the case of current tour, total distance travelled $=2^* 156+2^* 53=418$ miles. In the case of joined tour, total distance travelled $=156+160+53=369$ miles. So, joined tour has less distance to travel, so savings in distance would be $418-369=49$ miles.

This shows that it would always be better to go through a round tour rather than visiting individually. Thus, in this method, the first step is to identify cluster of cities and then routing; for this reason, this method is called cluster first route second. Now if origin city (Southfield) is denoted by city ‘ 0 ‘; Greensburg as city 1 and East Liberty by city 2 , then saving equations can be created as:
Current tour distance $=2 \mathrm{c}{01}+2 \mathrm{c}{02}$
Joined tour distance $=\mathrm{c}{01}+\mathrm{c}{12}+\mathrm{c}{20}$ If $2 \mathrm{c}{01}+2 \mathrm{c}{02}>\mathrm{c}{01}+\mathrm{c}{12}+\mathrm{c}{20}$
$$\mathrm{c}{01}+\mathrm{c}{20}-\mathrm{c}_{12}>0$$
then joined tour should be used. Solution is approached by considering the following points:

• As in this method clustering/allocation is done first and routing later, so saving values is calculated for every pair of nodes.
• These saving values are then arranged in descending order. Highest saving pairs are first allocated to trucks and taking into consideration capacity and demand comparison further allocations are made.
• It is important to make sure that while clustering vehicle capacity is not violated.

数学代写|运筹学代写Operations Research代考|SHORTEST PATH PROBLEM: DIJKSTRA’S ALGORITHM

Shortest path problem (SPP) is another network model including nodes and arcs where nods represent destinations and arcs represent distance between two destinations. The purpose of SPP is to find the shortest route out of available multiple routes. In a VRP problem, the vehicle begins its journey from a station, traverses a path moving to different stations and then comes back to origin. However, a SPP solution limits its calculations to finding the shortest path a vehicle traverses between two nodes without focusing on its backward path back to origin.
Linear programming formulation:
The purpose of the SPP is to find the shortest route out of possible multiple routes that covers the smallest distance from origin to destination passing through multiple stations on its way. Thus, the objective function would be of minimization of total distance travelled from origin to destination. Decision variable would be represented by binary value of either ‘ 1 ‘ or ‘ 0 ‘ meaning whether or not a particular city has been visited, respectively. Thus:
$$\text { Minimize } \sum_{\mathrm{i}} \sum_{\mathrm{j}} \mathrm{C}{\mathrm{ij}} \mathrm{X}{\mathrm{ij}}$$
where $\mathrm{c}{\mathrm{ij}}$ represents the distance between two nodes from $\mathrm{i}$ to $\mathrm{j}$ and $\mathrm{x}{\mathrm{ij}}$ indicated whether a city has been visited or not taking a value of 1 if it is in the shortest path and 0 if not.

数学代写|运筹学代写Operations Research代考|Clark-Wright Savings Algorithm

Current tour distance $=2 \mathrm{c} 01+2 \mathrm{c} 02$

$$\mathrm{c} 01+\mathrm{c} 20-\mathrm{c}_{12}>0$$

数学代写|运筹学代写Operations Research代考|SHORTEST PATH PROBLEM: DIJKSTRA’S ALGORITHM

SPP的目的是在可能的多条路径中找到最短路线，从起点到紟点经过多个站点的距离最短。因此，目标函数是最小化从起点到紟点 的总行驶距离。决策变量将由”1″或“0″的二进制值表示，分别表示是否访问过特定城市。因此:
$$\text { Minimize } \sum_i \sum_j C_{i j} X_{i j}$$

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。