Linear Programming Essay -- Computer Science

Linear Programming Part A Introduction â€Å"Linear programming was developed by George B. Dantzig in 1947 as a technique for planning the diversified activities of the U.S Air Force.† Linear programming is a powerful mathematical technique that can be used to deal with the problem of allocating limited facilities and resources among many alternative uses in order to find out the optimal benefits. The main objective of the linear programming problem in management is to maximize profit or minimize cost. Linear programming has a wide variety of applications. It is used by oil companies to determine the best mixture of ingredients for blending gasoline. It is also plays an important part in making the optimal schedules for transportation, production, and construction. In addition, linear programming is a flexible problem-solving tool for portfolio selection in finance, budgeting advertising expenditures in marketing, assigning personnel in human resources management. Applications One of the most important applications of linear programming is the formulation of blends. Blending problems appear whenever a manager must decide how to blend tow or more recourse in order to produce one or more products. In these situations, the recourses often contains one or more essential components that must be mixed in a given pattern and the final product will contains specific percentage of the essential components. In most of these applications management then has to decide how much of each recourse to purchase in order to satisfy product specification and produce demand at minimum cost. Blending problems occur frequently in the petroleum industry( such as blending crude oil to produce different octane gasoline), chemical industry( such as blending chemicals to produce fertilizers, week killers, and so on), and food industry( such as blending input ingredients to product soft drinks, soups, and so on). Linear programming is also a very useful tool that can be used to deal with problems in manufacturing industry, such as the product-mix problem. In this situation, the objective of the manager is to determine the production levels that will allow the company to meet the product demand requirements, given limitations on labor capacity, machine hour’s capacity and so on, at the same time, to make the cost of production to minimum. The... ...simultaneously, we get X=2/3, Y=7, plug them into objective function 40X+20Y we get a profit of  £166.67. The difference between this profit and the original max-profit is 166.67-160= £6.67, which means the dual price for increasing/decreasing in purchasing 1 pig, is  £6.67. Economic meaning for shadow price The economics meaning of shadow price is the improvement in the optimal value of the objective function per unit increase in the right-hand side of the constraint. In a profit maximization problem, the dual price is the same as the shadow price. Managers could get information from the performance of each constraint and therefore make decisions on any changes in a particular input factor or resource in order to increase profit. In this case, to get more profit, the farmer is recommended to increase the number of bushels rather than increase the amount of pigs. Reference: ‘The Quantitative methods for business decision with casesÂ’, Lawrence L. Lapim, 6th Edition, Dryden, Chapter 9. ‘An introduction to management science-quantitative approaches to decision makingÂ’, David R.Anderson, Dennis J. Sweeney, Thomas A. Williams. . 6Th Edition, West, Chapter 4.