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Linear Programming

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Linear Programming

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  1. 6S Linear Programming

  2. Learning Objectives • Describe the type of problem tha would lend itself to solution using linear programming • Formulate a linear programming model from a description of a problem • Solve linear programming problems using the graphical method • Interpret computer solutions of linear programming problems • Do sensitivity analysis on the solution of a linear progrmming problem

  3. Linear Programming • Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: • Materials • Budgets • Labor • Machine time

  4. Linear Programming • Linear programming (LP) techniques consist of a sequence of steps that will lead to an optimal solution to problems, in cases where an optimum exists

  5. Linear Programming Model • Objective Function: mathematical statement of profit or cost for a given solution • Decision variables: amounts of either inputs or outputs • Feasible solution space: the set of all feasible combinations of decision variables as defined by the constraints • Constraints: limitations that restrict the available alternatives • Parameters: numerical values

  6. Linear Programming Assumptions • Linearity: the impact of decision variables is linear in constraints and objective function • Divisibility: noninteger values of decision variables are acceptable • Certainty: values of parameters are known and constant • Nonnegativity: negative values of decision variables are unacceptable

  7. Graphical Linear Programming • Set up objective function and constraints in mathematical format • Plot the constraints • Identify the feasible solution space • Plot the objective function • Determine the optimum solution Graphical method for finding optimal solutions to two-variable problems

  8. Linear Programming Example • Objective - profit Maximize Z=60X1 + 50X2 • Subject to Assembly 4X1 + 10X2 <= 100 hours Materials 3X1 + 3X2 <= 39 cubic feet X1, X2 >= 0

  9. Solution

  10. Constraints • Redundant constraint: a constraint that does not form a unique boundary of the feasible solution space • Binding constraint: a constraint that forms the optimal corner point of the feasible solution space

  11. Solutions and Corner Points • Feasible solution space is usually a polygon • Solution will be at one of the corner points • Enumeration approach: Substituting the coordinates of each corner point into the objective function to determine which corner point is optimal.

  12. Example No.1 GM operates a plant that assembles and finishes cars and trucks. It takes 5 man-days to assemble a truck and 2 man-days to assemble a car. It takes 3 man-days to finish each type of vehicle. Because of man-power limitations, assembly can take no more than 180 man-days per week and finishing no more than 135 man-days per week. If the profit on each truck is $300 and $200 on each car, how many of each should they produce to maximize profit?

  13. Flair Furniture Co. Data Example • Other Limitations: • Make no more than 450 chairs • Make at least 100 tables