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# 数学代写|运筹学代写OPERATIONS RESEARCH代考|KMA255 Shortest Path in an Acyclic Network

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## 数学代写|运筹学代写Operations Research代考|Applications of the Acyclic Network

This section gives some applications of the shortest-path problem in an acyclic network. The first two applications concern planning problems in maintenance and replacement and in production and inventory control. Then, we discuss the important knapsack problem. Modeling these problems as shortest-path problems in acyclic networks requires some creativity.

A car wash is planning the purchase of a new wash system. Such a system can be used for a maximum of three years but can be traded in sooner for a new one. A new wash system costs 50 thousand euros. The residual value of a one-year-old machine is 40 thousand euros, that of a two-year-old machine is 30 thousand euros,

and that of a three-year-old machine is $24.5$ thousand euros. The maintenance cost of a newly purchased machine is 3 thousand euros the first year of operation, 4 thousand euros the second year of operation, and 6 thousand euros the third year. Due to stricter environmental requirements, the car wash will have to close down in five years’ time. What is the optimal replacement schedule for the next five years?
This problem can be formulated as a shortest-path problem in an acyclic network as follows:

• node $i$ for $i=1, \ldots, 6$ means that we are at the beginning of year $i$ and have just gotten rid of a machine (node 6 is an auxiliary node);
• an arc from node $i$ to node $j$ with $j>i$ means that at the beginning of year $i$, a new wash system is purchased that will be sold at the end of year $j-1$;
• an arc from $i$ to $j$ is assigned a total cost of $c_{i j}$ that consists of the purchase cost at the beginning of year $i$ plus the maintenance costs for years $i$ through $j-1$ minus the residual value at the end of year $j-1$.

## 数学代写|运筹学代写Operations Research代考|A Production-Stock Problem

The company Jones Chemical has entered into a contract to deliver a special type of sulfuric acid over the next six months. The agreement provides for the supply of 100 tons of sulfuric acid on July 1st, 75 tons on August 1st, 90 tons on September 1st, 60 tons on October 1st, 40 tons on November 1st, and 85 tons on December 1st. The production of sulfuric acid requires a few special measures. The production manager has therefore decided that the sulfuric acid can only be produced on the first day of the month. The production takes a negligible amount of time. The fixed setup cost of the production process is 500 euros. Each time, any desired amount of sulfuric acid can be produced. The company has sufficient storage capacity for the sulfuric acid; the storage cost is 4 euros per ton of sulfuric acid per month. At the moment, there is no stock of the product. What production plan has the lowest total cost?

This problem was solved in Section $2.2 .6$ as an ILP problem but can also be formulated as a shortest-path problem in an acyclic network. This is possible because an optimal production plan has the property that production only takes place if there is no stock and then has a size that exactly covers the demand in a yet to be determined number of periods. This property is easy to see. Suppose that production were to take place when there is a positive inventory level $V$; then if the fixed production cost remains the same, the storage costs could be reduced by producing a quantity $V$ less in the previous production run and $V$ more than originally planned in the next production run. If we number the months July through December as periods 1 through 6 , then we can formulate the problem as follows as a shortest-path problem in an acyclic network:

• node $i$ with $i=1, \ldots, 7$ means that we are at the beginning of period $i$ with no stock (node 7 is an auxiliary node);
• an arc from node $i$ to node $j$ with $j>i$ means that at the beginning of period $i$, a quantity is produced that is exactly enough to cover the demand in periods $i$ through $j-1$
• an arc from $i$ to $j$ is assigned a total cost $c_{i j}$ that consists of the fixed setup cost at the beginning of period $i$ and the storage costs for periods $i$ through $j-1$.

## 数学代写|运筹学代写OPERATIONS RESEARCH代考|APPLICATIONS OF THE ACYCLIC NETWORK

• 节点 $i$ 为了 $i=1, \ldots, 6$ 意味着我们在年初 $i$ 并且刚刚摆脱了一台机器node6isanauxiliarynode;
• 来自节点的弧 $i$ 到节点 $j$ 和 $j>i$ 意味着在年初 $i$, 购买了一昙新的洗涤系统，将在年底出售 $j-1$;
• 从一个弧 $i$ 到 $j$ 分配的总成本为 $c_{i j}$ 包括年初采购成本 $i$ 再加上多年的维修费用 $i$ 通过 $j-1$ 减去年末残值 $j-1$.

## 数学代写|运筹学代写OPERATIONS RESEARCH代考|A PRODUCTION-STOCK PROBLEM

Jones Chemical 公司已签订合同，在末来六个月内交付一种特殊类型的硫酸。协议规定7月 1 日供应 100 吨硫酸，8月 1 日供应 75 吨，9月 1 日供应 90 吨，10月1日供应60吨，11月1日供应40吨，12月1日供应85吨。硫酸的生产需要一些特殊措施。因此，生产经理决定只能在每月的第一天生产硫

• 节点和 $i=1, \ldots, 7$ 意味着我们正处于周期的开始i没有库存node7isanauxiliarynode
来自节点的弧 $i$ 到节点 $j$ 和 $j>i$ 意味着在期初 $i$ ，生产的数量恰好足以满足一段时间内的需求 $i$ 通过 $j-1$ 从一个弧 $i$ 到 $j$ 分配了总成本 $c_{i j}$ 包括期初的固定准备成本 $i$ 以及期间的存储成本 $i$ 通过 $j-1$.

## 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。