December 4, 2015 Hanging with Harvard 4 L INEAR P ROGRAMMING

 Linear programming is a method for finding an optimal solution of an objective function (a minimum or maximum value of some quantity) given a set of constraints.  Objective function is a model of the quantity that you want to make as large or as small as possible.  Constraints are the restrictions on the variables of the objective function in a linear programming problem.  Feasible region contains all the values that satisfy the constraints of the objective function. V OCABULARY Linear programming Objective function Constraints Feasible region

Linear Programming was invented by mathematicians during World War II. After the war, the concept was published and was quickly adopted by many large businesses to manage their resources. H ISTORY...  to optimize loading of cargo on ships and planes  to manage other critical resources Many industries use linear programming as a standard tool to allocate a finite set of resources in an optimal way.  airline crew scheduling  shipping or telecommunication networks  oil refining and blending  production and sales

A Linear Programming Model... Suppose the objective function is … Subject to the constraints  Minimize C = 2x + y Graphical Solution:

Objective function… Subject to the constraints: Vertex Principle of Linear Programming Vertex Principle of Linear Programming Solution: x = 2 and y = 3 to minimize C = 7. If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region. Minimize C = 2x + y Vertex 18

You are screen-printing T-shirts and sweatshirts to sell at the Polk County Blues Festival and are working with the following constraints.  You have at most 20 hours to make shirts.  You want to spend no more than $600 on supplies.  You want to have at least 50 items to sell. How many T-shirts and how many sweatshirts should you make to maximize your profit? How much is the maximum profit? Blues Festival T-Shirts SOLUTION Let x = number of T-shirts y = number of sweatshirts P = profit T-shirts, x Sweatshirts, y TOTAL Number Minutes Cost ($) PROFIT ($) Organize information in a table.

Write the objective function and constraints. Write the objective function and constraints. Blues Festival T-Shirts Organize information in a table. Objective Function: Maximize P = 6x + 20y subject to :  You want to spend no more than $600 on supplies.  You have at most 20 hours to make shirts.  You want to have at least 50 items to sell. Constraints

Blues Festival T-Shirts Objective Function: Maximize P = 6x + 20y subject to : 10x+30y ≤ 1200 Constraints Step 1 Graph the constraints to form the feasible region. Step 2 Find the coordinates of each vertex. P = 6(50) + 20(0) = 300 A(50, 0) B(25, 25) C(75, 15) D(120, 0) P = 6(25) + 20(25) = 650 P = 6(75) + 20(15) = 750 P = 6(120) + 20(0) = 720 x+3y = 120 Step 3 Evaluate P.

A linear programming problem that uses the graphical solution,  provides linear constraints,  has an objective function,  either maximized or minimized.  expressed as a system of linear inequalities The optimal solution occurs at one or more vertices of the feasible region. Graph the system of inequalities (or constraints) to obtain the feasible region. Finally, evaluate each vertex to obtain the optimal solution.

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