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Unformatted text preview: Math 407 Linear Optimization 1 Introduction 1.1 What is optimization? A mathematical optimization problem is one in which some function is either maximized or minimized relative to a given set of alternatives. The function to be minimized or maximized is called the objective function and the set of alternatives is called the feasible region (or constraint region). In this course, the feasible region is always taken to be a subset of R n (real n-dimensional space) and the objective function is a function from R n to R . We further restrict the class of optimization problems that we consider to linear program- ming problems (or LPs). An LP is an optimization problem over R n wherein the objective function is a linear function, that is, the objective has the form c 1 x 1 + c 2 x 2 + + c n x n for some c i R i = 1 , . . . , n , and the feasible region is the set of solutions to a finite number of linear inequality and equality constraints, of the form a i 1 x i + a i 2 x 2 + + a in x n b i i = 1 , . . . , s and a i 1 x i + a i 2 x 2 + + a in x n = b i i = s + 1 , . . . , m. Linear programming is an extremely powerful tool for addressing a wide range of applied optimization problems. A short list of application areas is resource allocation, produc- tion scheduling, warehousing, layout, transportation scheduling, facility location, flight crew scheduling, portfolio optimization, parameter estimation, . . . . 1.2 An Example To illustrate some of the basic features of LP, we begin with a simple two-dimensional example. In modeling this example, we will review the four basic steps in the development of an LP model: 1. Identify and label the decision variables . 2. Determine the objective and use the decision variables to write an expression for the objective function . 3. Determine the explicit constraints and write a functional expression for each of them. 4. Determine the implicit constraints . 1 PLASTIC CUP FACTORY A local family-owned plastic cup manufacturer wants to optimize their production mix in order to maximize their profit. They produce personalized beer mugs and champagne glasses. The profit on a case of beer mugs is $25 while the profit on a case of champagne glasses is $20. The cups are manufactured with a machine called a plastic extruder which feeds on plastic resins. Each case of beer mugs requires 20 lbs. of plastic resins to produce while champagne glasses require 12 lbs. per case. The daily supply of plastic resins is limited to at most 1800 pounds. About 15 cases of either product can be produced per hour. At the moment the family wants to limit their work day to 8 hours. We will model the problem of maximizing the profit for this company as an LP. The first step in our modeling process is to identify and label the decision variables . These are the variables that represent the quantifiable decisions that must be made in order to determine the daily production schedule. That is, we need to specify those quantities whose valuesthe daily production schedule....
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