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Ardavan Asef-Vaziri Systems and Operations Management Engineering

Introduction to Linear Programming. Ardavan Asef-Vaziri Systems and Operations Management Engineering. Goals, Aims, and Requirements. Goals and aims To introduce Linear Programming To find a knowledge on graphical solution for LP problems

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Ardavan Asef-Vaziri Systems and Operations Management Engineering

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  1. Introduction to Linear Programming Ardavan Asef-Vaziri Systems and Operations Management Engineering

  2. Goals, Aims, and Requirements Goals and aims To introduce Linear Programming To find a knowledge on graphical solution for LP problems To solve linear programming problems using excel. Requirements To drop all your mental luggage outside the class To stop me immediately if you feel any difficulty

  3. The Lego Production Problem You have a set of logos 8 small bricks 6 large bricks These are your “raw materials”. You have to produce tables and chairs out of these logos. These are your “products”.

  4. The Lego Production Problem Weekly supply of raw materials: 8 Small Bricks 6 Large Bricks Products: Table Chair Profit = $20/Table Profit = $15/Chair

  5. Problem Formulation X1 is the number of Chairs X2 is the number of Tables Large brick constraint X1+2X2 <= 6 Small brick constraint 2X1+2X2 <= 8 Objective function is to Maximize 15X1+20 X2 X1>=0 X2>= 0

  6. Graphical Solution to the Prototype Problem Tables 5 4 3 2 X1 + 2 X2 = 6 Large Bricks 1 0 Chairs 1 2 3 4 5 6

  7. Graphical Solution to the Prototype Problem Tables 5 4 2 X1 + 2 X2 = 8 Small Bricks 3 2 1 0 Chairs 1 2 3 4 5 6

  8. Graphical Solution to the Prototype Problem Tables 5 4 2 X1 + 2 X2 = 8 Small Bricks 3 2 X1 + 2 X2 = 6 Large Bricks 1 0 Chairs 1 2 3 4 5 6

  9. Graphical Solution to the Prototype Problem Tables 5 4 3 X1 + 2 X2 = 6 Large Bricks 2 2 X1 + 2 X2 = 8 Small Bricks 1 0 Chairs 1 2 3 4 5 6

  10. The Objective Function Z = 15 X1 + 20 X2 Lets draw it for 15 X1 + 20 X2 = 30 In this case if # of chair = 0, then # of table = 30/20 = 1.5 if # of table = 0, then # of chair = 30/15 = 2

  11. Graphical Solution to the Prototype Problem Tables 5 4 3 X1 + 2 X2 = 6 Large Bricks 2 2 X1 + 2 X2 = 8 Small Bricks 1 0 Chairs 1 2 3 4 5 6

  12. A second example • We can make Product1 and or Product2. • There are 3 resources; Resource1, Resource2, Resource3. • Product1 needs one unit of Resource1, nothing of Resource2, and three units of resource3. • Product2 needs nothing from Resource1, two units of Resource2, and two units of resource3. • Net profit of product 1 and Product2 are 3 and 5, respectively. • Formulate the Problem • Solve it graphically • Solve it using excel.

  13. Problem 2 : Original version Product 1 needs 1 unit of resource 1, and 3 units of resource 3. Product 2 needs 2 units of resource 2 and 2 units of resource 3 There are 4 units of resource 1, 12 units of resource 2, and 18 units of resource 3 Objective Function Z = 3 x1 +5 x2 Constraints Resource 1 x1 4 Resource 2 2x2  12 Resource 3 3 x1 + 2 x2  18 Nonnegativity x1  0, x2  0

  14. Problem 2 : Original version x 2 10 9 8 7 6 5 4 3 2 1 x 1 1 2 3 4 5 6 7 8 9 10 Max Z = x1 + x2 Subject to x1 4 2x2  12 3 x1 + 2 x2  18 x1  0, x2  0

  15. Problem 2 x 2 10 9 8 7 6 5 4 3 2 1 x 1 1 2 3 4 5 6 7 8 9 10 Max Z = x1 + x2 Subject to x1 4 2x2  12 3 x1 + 2 x2  18 x1  0, x2  0

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