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CHAPTER. 6s. Linear Programming. Linear Programming. Used to obtain optimal solutions to problems that involve restrictions or limitations, such as: Materials Budgets Labor Machine time. Linear Programming.

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  1. CHAPTER 6s Linear Programming

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

  3. 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

  4. Linear Programming Model • Objective: the goal of an LP model is maximization or minimization • 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

  5. 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

  6. 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

  7. 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

  8. Linear Programming Example

  9. Linear Programming Example

  10. Linear Programming Example Inspection Storage Assembly Feasible solution space

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

  12. 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

  13. 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

  14. 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

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

  16. MS Excel Worksheet Solution Figure 6S.17

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