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Linear Programming

6S. Linear Programming. 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

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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 Inspection 2X1 + 1X2 <= 22 hours Storage 3X1 + 3X2 <= 39 cubic feet X1, X2 >= 0

  9. Linear Programming Example

  10. Linear Programming Example

  11. Linear Programming Example Inspection Storage Assembly Feasible solution space

  12. Linear Programming Example Z=900 Z=300 Z=600

  13. Solution • The intersection of inspection and storage • Solve two equations in two unknowns 2X1 + 1X2 = 22 3X1 + 3X2 = 39 X1 = 9 X2 = 4 Z = $740

  14. 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

  15. 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.

  16. Slack and Surplus • Surplus: when the optimal values of decision variables are substituted into a greater than or equal to constraint and the resulting value exceeds the right side value • Slack: when the optimal values of decision variables are substituted into a less than or equal to constraint and the resulting value is less than the right side value

  17. Simplex Method • Simplex: a linear-programming algorithm that can solve problems having more than two decision variables

  18. MS Excel Worksheet for Microcomputer Problem Figure 6S.15

  19. MS Excel Worksheet Solution Figure 6S.17

  20. Sensitivity Analysis • Range of optimality: the range of values for which the solution quantities of the decision variables remains the same • Range of feasibility: the range of values for the fight-hand side of a constraint over which the shadow price remains the same • Shadow prices: negative values indicating how much a one-unit decrease in the original amount of a constraint would decrease the final value of the objective function

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