5.3 Linear Programming in Two Dimensions: A Geometric Approach

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Lauren Lester
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1 : A Geometric Approach A Linear Programming Problem Definition (Linear Programming Problem) A linear programming problem is one that is concerned with finding a set of values of decision variables x 1, x 2,..., x n such that f(x 1, x 2,..., x n ) = a 1 x 1 + a 2 x a n x n objevtive function attains its optimal value (maximum or minimum), and such that b 11 x 1 + b 12 x b 1n x n B 1... b m1 x 1 + b m2 x b mn x n B m problem constraints c 11 x 1 + c 12 x c 1n x n C 1... c p1 x 1 + c p2 x c pn x n C p x nonnegative constraints x n 0 are satisfied. The set of points satisfying both the problem constraints and the nonnegative constraints is called the feasible region for the problem. Any point in the feasible region that produces the optimal value of the objective function over the feasible region is called an optimal solution. Note: both objective function and problem constraints are linear.

2 Example 1 A manufacturing plant makes two types of inflatable boatsa twoperson boat and a four-person boat. Each two-person boat requires 0.9 labor-hour from the cutting department and 0.8 laborhour from the assembly department. Each four-person boat requires 1.8 labor-hours from the cutting department and 1.2 laborhours from the assembly department. The maximum labor-hours available per month in the cutting department and the assembly department are 864 and 672, respectively. The company makes a profit of $25 on each two-person boat and $40 on each four-person boat. (a) Identify the decision variables. (b) Summarize all relevant information given in a form of a table. (c) Write the objective function P. (d) Write the problem constraints and nonnegative constraints. (e) Graph the feasible region. Include graphs of the objective function for P = $5,000, P = $10,000, P = $15,000, and P = $21,600. (f) From the graph and constant-profit lines, determine how many boats should be manufactured each month to maximize the profit. What is the maximum profit? 2

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6 Fundamental Results Theorem 1 (Fundamental Theorem of Linear Programming) If the optimal value of the objective function in a linear programming problem exists, then that value must occur at one or more of the corner points of the feasible region. Theorem 1 provides a simple procedure for solving a linear programming problem, provided that the problem has an optimal solutionnot all do. In order to use Theorem 1, we must know that the problem under consideration has an optimal solution. Theorem 2 provides some conditions that will ensure that a linear programming problem has an optimal solution. Theorem 2 (Existence of Optimal Solutions) (a) If the feasible region for a linear programming problem is bounded, then both the maximum value and the minimum value of the objective function always exist. (b) If the feasible region is unbounded and the coefficients of the objective function are positive, then the minimum value of the objective function exists but the maximum value does not. (c) If the feasible region is empty (that is, there are no points that satisfy all the constraints), then both the maximum value and the minimum value of the objective function do not exist. 6

Geometric Method for Solving Linear Programming Problems The preceding discussion leads to the following procedure for the geometric solution of linear

7 Geometric Method for Solving Linear Programming Problems The preceding discussion leads to the following procedure for the geometric solution of linear programming problems with two decision variables: Example 2 Solve a linear programming problem Maximize P = 20x + 10y subject to 3x + 3y 30 2x + y 26 2x + 5y 34 x, y 0 7

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10 Example 3 Solve a linear programming problem Minimize P = 20x + 10y subject to 3x + 3y 30 2x + y 26 2x + 5y 34 x, y 0 10

11 Example 4 Solve a linear programming problem Minimize and maximize z = 10x + 20y subject to 6x + 2y 36 2x + 4y 32 y 20 x, y 0 11

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Applications of Systems of Linear Inequalities

Applications of Systems of Linear Inequalities Finite Math 26 April 2017 Finite Math Applications of Systems of Linear Inequalities 26 April 2017 1 / 17 Quiz What does it mean for a feasible region to