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Unformatted text preview: 1 1 What Is Optimization? 1.1 Introduction Optimization, or constrained optimization, or mathematical programming, is a mathematical procedure for determining optimal allocation of scarce resources. Optimization, and its most popular special form, Linear Programming (LP), has found practical application in almost all facets of business, from advertising to production planning. Transportation and aggregate production planning problems are the most typical objects of LP analysis. The petroleum industry was an early intensive user of LP for solving fuel blending problems. It is important for the reader to appreciate at the outset that the “programming” in Mathematical Programming is of a different flavor than the “programming” in Computer Programming. In the former case, it means to plan and organize (as in “Get with the program!”). In the latter case, it means to write instructions for performing calculations. Although aptitude in one suggests aptitude in the other, training in the one kind of programming has very little direct relevance to the other. For most optimization problems, one can think of there being two important classes of objects. The first of these is limited resources, such as land, plant capacity, and sales force size. The second is activities, such as “produce low carbon steel,” “produce stainless steel,” and “produce high carbon steel.” Each activity consumes or possibly contributes additional amounts of the resources . The problem is to determine the best combination of activity levels that does not use more resources than are actually available. We can best gain the flavor of LP by using a simple example. 1.2 A Simple Product Mix Problem The Enginola Television Company produces two types of TV sets, the “Astro” and the “Cosmo”. There are two production lines, one for each set. The Astro production line has a capacity of 60 sets per day, whereas the capacity for the Cosmo production line is only 50 sets per day. The labor requirements for the Astro set is 1 person-hour, whereas the Cosmo requires a full 2 person-hours of labor. Presently, there is a maximum of 120 man-hours of labor per day that can be assigned to production of the two types of sets. If the profit contributions are $20 and $30 for each Astro and Cosmo set, respectively, what should be the daily production? 2 Chapter 1 What is Optimization? A structured, but verbal, description of what we want to do is: Maximize Profit contribution subject to Astro production less-than-or-equal-to Astro capacity, Cosmo production less-than-or-equal-to Cosmo capacity, Labor used less-than-or-equal-to labor availability. Until there is a significant improvement in artificial intelligence/expert system software, we will need to be more precise if we wish to get some help in solving our problem. We can be more precise if we define: A = units of Astros to be produced per day, C = units of Cosmos to be produced per day....
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This note was uploaded on 10/24/2011 for the course ECONOMICS 30112 taught by Professor Ahfdka during the Spring '11 term at Coventry.
- Spring '11