INTRODUCTORY MATHEMATICAL ANALYSIS For Business, Economics, and the Life and Social Sciences  2007 Pearson Education Asia Chapter 7 Linear Programming

 2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0.Review of Algebra 1.Applications and More Algebra 2.Functions and Graphs 3.Lines, Parabolas, and Systems 4.Exponential and Logarithmic Functions 5.Mathematics of Finance 6.Matrix Algebra 7.Linear Programming 8.Introduction to Probability and Statistics

 2007 Pearson Education Asia 9.Additional Topics in Probability 10.Limits and Continuity 11.Differentiation 12.Additional Differentiation Topics 13.Curve Sketching 14.Integration 15.Methods and Applications of Integration 16.Continuous Random Variables 17.Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS

 2007 Pearson Education Asia To represent geometrically a linear inequality in two variables. To state a linear programming problem and solve it geometrically. To consider situations in which a linear programming problem exists. To introduce the simplex method. To solve problems using the simplex method. To introduce use of artificial variables. To learn alteration of objective function. To define the dual of a linear programming problem. Chapter 7: Linear Programming Chapter Objectives

 2007 Pearson Education Asia Linear Inequalities in Two Variables Linear Programming Multiple Optimum Solutions The Simplex Method Degeneracy, Unbounded Solutions, and Multiple Solutions Artificial Variables Minimization The Dual 7.1) 7.2) 7.3) 7.4) Chapter 7: Linear Programming Chapter Outline 7.5) 7.6) 7.7) 7.8)

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables Linear inequality is written as where a, b, c = constants and not both a and b are zero.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables A solid line is included in the solution and a dashed line is not.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.1 Linear Inequalities in 2 Variables Example 1 – Solving a Linear Inequality Example 3 – Solving a System of Linear Inequalities Find the region defined by the inequality y ≤ 5. Solution: The region consists of the line y = 5 and with the half-plane below it. Solve the system Solution: The solution is the unshaded region.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 1 – Solving a Linear Programming Problem A linear function in x and y has the form The function to be maximized or minimized is called the objective function. Maximize the objective function Z = 3x + y subject to the constraints

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 1 – Solving a Linear Programming Problem Solution: The feasible region is nonempty and bounded. Evaluating Z at these points, we obtain The maximum value of Z occurs when x = 4 and y = 0.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 3 – Unbounded Feasible Region The information for a produce is summarized as follows: If the grower wishes to minimize cost while still maintaining the nutrients required, how many bags of each brand should be bought?

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.2 Linear Programming Example 3 – Unbounded Feasible Region Solution: Let x = number of bags of Fast Grow bought y = number of bags of Easy Grow bought To minimize the cost function Subject to the constraints There is no maximum value since the feasible region is unbounded.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.3 Multiple Optimum Solutions Example 1 – Multiple Optimum Solutions There may be multiple optimum solutions for an objective function. Maximize Z = 2x + 4y subject to the constraints Solution: The region is nonempty and bounded.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.4 The Simplex Method Standard Linear Programming Problem Maximize the linear function subject to the constraints where x and b are non-negative.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.4 The Simplex Method Example 1 – The Simplex Method Maximize subject to and x 1, x 2 ≥ 0. Solution: The maximum value of Z is 95.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.5 Degeneracy, Unbounded and Multiple Solutions Degeneracy A basic feasible solution (BFS) is degenerate if one of the basic variables is 0. Unbounded Solutions An unbounded solution occurs when the objective function has no maximum value. Multiple Optimum Solutions Multiple (optimum) solutions occur when there are different BFS.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.5 Degeneracy, Unbounded and Multiple Solutions Example 1 – Unbounded Solution Maximize subject to Solution: Since the 1 st two rows of the x 1 -column are negative, no quotients exist. Hence, it has an unbounded solution.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.6 Artificial Variables Example 1 – Artificial Variables Use artificial variables to handle maximization problems that are not of standard form. Maximize subject to Solution: Add an artificial variable t, Consider an artificial objective equation,

 2007 Pearson Education Asia Construct a Simplex Table, All indicators are nonnegative. The maximum value of Z is 17. Chapter 7: Linear Programming 7.6 Artificial Variables Example 1 – Artificial Variables

 2007 Pearson Education Asia Use the simplex method to maximize subject to Solution: Construct a Simplex Table, The feasible region is empty, hence, no solution exists. Chapter 7: Linear Programming 7.6 Artificial Variables Example 3 – An Empty Feasible Region

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.7 Minimization Example 1 – Minimization To minimize a function is to maximize the negative of the function. Minimize subject to Solution: Construct a Simplex Table, The minimum value of Z is −(−4) = 4.

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.8 The Dual Example 1 – Finding the Dual of a Maximization Problem Find the dual of the following: Maximize subject to where Solution: The dual is minimize subject to where

 2007 Pearson Education Asia Chapter 7: Linear Programming 7.8 The Dual Example 3 – Applying the Simplex Method to the Dual Use the dual and the simplex method to maximize subject to where Solution: The dual is minimize subject to The final Simplex Table is The minimum value of W is 11/2.