2.7 Linear 2.7 Linear ProgrammingProgramming

Linear programming - Linear programming - Certain constraints exist or are Certain constraints exist or are

placed upon the variables and some placed upon the variables and some function of these variables must be function of these variables must be maximized or minimized. The maximized or minimized. The constraints are written as a system constraints are written as a system of linear inequalities.of linear inequalities.

LP procedure:LP procedure:1.1. Define variables.Define variables.

2.2. Write the constraints as a system of Write the constraints as a system of inequalities.inequalities.

3.3. Graph the system and find the Graph the system and find the coordinates of the vertices.coordinates of the vertices.

4.4. Write an expression whose value is to be Write an expression whose value is to be maximized or minimized.maximized or minimized.

5.5. Substitute values from the coordinates Substitute values from the coordinates of the vertices into the expression.of the vertices into the expression.

6.6. Select the greatest or least result.Select the greatest or least result.

Ex 1Ex 1The profit on each set of cd’s that is The profit on each set of cd’s that is

manufactured is $8. The profit on a single manufactured is $8. The profit on a single CD is $2. Machines A and B are used to CD is $2. Machines A and B are used to produce both types of CD’s. Each set takes produce both types of CD’s. Each set takes nine minutes on Machine A and three nine minutes on Machine A and three minutes on Machine B. Each single takes minutes on Machine B. Each single takes one minute on Machine A and one minute on one minute on Machine A and one minute on Machine B. If Machine A is run for 54 Machine B. If Machine A is run for 54 minutes and Machine B is run for 42 minutes and Machine B is run for 42 minutes, determine the combination of cd’s minutes, determine the combination of cd’s that can be manufactured during the time that can be manufactured during the time period that most effectively generates profit period that most effectively generates profit within the given constraints. within the given constraints.

Different types of regionsDifferent types of regions Infeasible – when the constraints Infeasible – when the constraints

cannot be satisfied simultaneouslycannot be satisfied simultaneously

Unbounded – an optimal solution may Unbounded – an optimal solution may not exist. (may not have a max or min.)not exist. (may not have a max or min.)

Alternate optimal solutions – two or Alternate optimal solutions – two or more optimal solutions. When the more optimal solutions. When the graph of the function to be maximized graph of the function to be maximized or minimized is parallel to one side of or minimized is parallel to one side of the polygonal convex set. the polygonal convex set.

Ex 2Ex 2 A manufacturer makes widgets and A manufacturer makes widgets and

gadgets. At least 500 widgets and 700 gadgets. At least 500 widgets and 700 gadgets are needed to meet minimum daily gadgets are needed to meet minimum daily demands. The machinery can produce no demands. The machinery can produce no more than 1200 widgets and 1400 gadgets more than 1200 widgets and 1400 gadgets per day. The combined number of widgets per day. The combined number of widgets and gadgets that the packaging and gadgets that the packaging department can handle is 2300 per day. If department can handle is 2300 per day. If the company sells both widgets and the company sells both widgets and gadgets for $1.59 each, how many of each gadgets for $1.59 each, how many of each item should be produced in order to item should be produced in order to maximize profits?maximize profits?