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Unit 5 Section 5 - 5.5 Linear Programming Objectives After...

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5.5 Linear Programming Objectives: After completing this section, you should be able to: 1. Solve linear programming problems; 2. Use linear programming to model and solve real-life problems. In Section 4, many of the problems we solved are related to the general type of problems called linear programming problems. Linear programming is a mathematical process that has been developed to help management in decision making, and it has become one of the widely used tools in a process called optimization . In this section we will study linear programming as a strategy for finding the optimal value either maximum or minimum of a quantity subject to certain constraints. We will use an intuitive graphical approach based on the techniques we discussed in Section 4 to illustrate this process for problems involving two variables. George Dantzig (1914- ), an American mathematician, was the first to formulate a linear programming problem in 1947 and introduced a solution technique, called simplex method, that finds the optimal solution of complex linear programming problems involving thousands of variables and inequalities. This method does not rely on graphing and is well-suited for computer implementation. Linear Programming Model A Linear Programming problem is concerned with finding the optimal value (maximum or minimum value) of a linear objective function of the form z = ax + by where the decision variables x and y are subject to problem constraints stated as a system of linear inequalities and to nonnegative constraints x, y ≥ 0. The set of points satisfying the system of linear inequalities and the nonnegative constraints is called the set of feasible solutions for the problem. The graph of this set is the feasible region . Any point in the feasible region that produces the optimal value of the objective function over the feasible region is called an optimal solution.
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Unit 5. Systems of Equations and Inequalities Section 5 page 2 _____________________________________________________________________________________ Example 5.5.1 Linear Programming problem Find the maximum value of z = 2 x + 3 y subject to 0 0 21 2 3 25 5 2 y x y x y x Because every point in the feasible region satisfies each constraint, the question is which of those points will yield a maximum value of z? We cannot check all of the feasible solutions to see which one results in the largest value of z. Fortunately, a simple way has been developed to find this optimal solution. Using advanced techniques, it can be shown that If the feasible region is bounded, then one or more of the corner points of the feasible region is an optimal solution to the problem. This means that you can find the maximum value of z by testing z at each of the corner points and choose the corner point that produces the largest value of z.
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