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数学代写|运筹学代写Operations Research代考|SOME CASE STUDIES

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数学代写|运筹学代写Operations Research代考|SOME CASE STUDIES

To give you a better perspective about the great impact linear programming can have, we now present three case studies of real applications. Each of these is a classic application, initiated in the early 1980 s, that has come to be regarded as a standard of excellence for future applications of linear programming. The first one will bear some strong similarities to the Wyndor Glass Co. problem, but on a realistic scale. Similarly, the second and third are realistic versions of the last two examples presented in the preceding section (the Union Airways and Distribution Unlimited examples).
Choosing the Product Mix at Ponderosa Industrial ${ }^{\mathbf{1}}$
Until its sale in 1988, PONDEROSA INDUSTRIAL was a plywood manufacturer based in Anhuac, Chihuahua, that supplied 25 percent of the plywood in Mexico. Like any plywood manufacturer, Ponderosa’s many products were differentiated by thickness and by the quality of the wood. The plywood market in Mexico is competitive, so the market establishes the prices of the products. The prices can fluctuate considerably from month to month, and there may be great differences between the products in their price movements from even one month to the next. As a result, each product’s contribution to Ponderosa’s total profit was continually varying, and in different ways for different products.

Because of its pronounced effect on profits, a critical issue facing management was the choice of product mix – how much to produce of each product — on a monthly basis. This choice was a very complex one, since it had to take into account the current amounts available of various resources needed to produce the products. The most important resources were logs in four quality categories and production capacities for both the pressing operation and the polishing operation.

Beginning in 1980, linear programming was used on a monthly basis to guide the product-mix decision. The linear programming model had an objective of maximizing the total profit from all products. The model’s constraints included the various resource constraints as well as other relevant restrictions such as the minimum amount of a product that must be provided to regular customers and the maximum amount that can be sold. (To aid planning for the procurement of raw materials, the model also considered the impact of the product-mix decision for the upcoming month on production in the following month.) The model had 90 decision variables and 45 functional constraints.

数学代写|运筹学代写Operations Research代考|Personnel Scheduling at United Airlines

Despite unprecedented industry competition in 1983 and 1984, UNITED AIRLINES managed to achieve substantial growth with service to 48 new airports. In 1984, it became the only airline with service to cities in all 50 states. Its 1984 operating profit reached $\$ 564$million, with revenues of$\$6.2$ billion, an increase of 6 percent over 1983 , while costs grew by less than 2 percent.

Cost control is essential to competing successfully in the airline industry. In 1982, upper management of United Airlines initiated an OR study of its personnel scheduling as part of the cost control measures associated with the airline’s 1983-1984 expansion. The goal was to schedule personnel at the airline’s reservations offices and airports so as to minimize the cost of providing the necessary service to customers.

At the time, United Airlines employed over 4,000 reservations sales representatives and support personnel at its 11 reservations offices and about 1,000 customer service agents at its 10 largest airports. Some were part-time, working shifts from 2 to 8 hours; most were full-time, working 8- or 10-hour-shifts. Shifts start at several different times. Each reservations office was open (by telephone) 24 hours a day, as was each of the major airports. However, the number of employees needed at each location to provide the required level of service varied greatly during the 24-hour day, and might fluctuate considerably from one half-hour to the next.

Trying to design the work schedules for all the employees at a given location to meet these service requirements most efficiently is a nightmare of combinatorial considerations. Once an employee begins working, he or she will be there continuously for the entire shift ( 2 to 10 hours, depending on the employee), except for either a meal break or short rest breaks every 2 hours. Given the minimum number of employees needed on duty for each half-hour interval over a 24-hour day (where these requirements change from day to day over a 7-day week), how many employees of each shift length should begin work at what start time over each 24-hour day of a 7-day week? Fortunately, linear programming thrives on such combinatorial nightmares.

Actually, several OR techniques described in this book were used in the computerized planning system developed to attack this problem. Both forecasting (Chap. 20) and queuing theory (Chaps. 17 and 18) were used to determine the minimum number of employees needed on duty for each half-hour interval. Integer programming (Chap. 12) was used to determine the times of day at which shifts would be allowed to start. However, the core of the planning system was linear programming, which did all the actual scheduling to provide the needed service with the smallest possible labor cost. A complete work schedule was developed for the first full week of a month, and then it was reused for the remainder of the month. This process was repeated each month to reflect changing conditions.

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。