Linear Programming

1. Linear Programming • Mathematical Technique for Solving Constrained Maximization and Minimization Problems • Assumes that the Objective Function is Linear • Assumes that All Constraints Are Linear

2. Applications of Linear Programming • Optimal Process Selection • Optimal Product Mix • Satisfying Minimum Product Requirements • Long-Run Capacity Planning • Least Cost Shipping Route(Transportation Problems)

3. Applications of Linear Programming • Airline Operations Planning • Output Planning with Resource and Process Capacity Constraints • Distribution of Advertising Budget • Routing of Long-Distance Phone Calls • Investment Portfolio Selection • Allocation of Personnel Among Activities

4. Production Processes Production processes are graphed as linear rays from the origin in input space. Production isoquants are line segments that join points of equal output on the production process rays.

5. Production Processes Isoquants Processes

6. Production Processes Optimal Solution (S) Feasible Region

7. Formulating and Solving Linear Programming Problems • Express Objective Function as an Equation and Constraints as Inequalities • Graph the Inequality Constraints and Define the Feasible Region • Graph the Objective Function as a Series of Isoprofit or Isocost Lines • Identify the Optimal Solution

8. Profit Maximization Maximize Subject to (objective function) (input A constraint) (input B constraint) (input C constraint) (nonnegativity constraint)  = $30QX + $40QY 1QX + 1QY 7 0.5QX + 1QY  5 0.5QY  2 QX, QY  0

10. Profit Maximization Multiple Optimal Solutions New objective function has the same slopeas the feasible region at the optimum

11. Profit Maximization Algebraic Solution Points of Intersection Between Constraintsare Calculated to Determine the Feasible Region

12. Profit Maximization Algebraic Solution Profit at each point of intersection between constraintsis calculated to determine the optimal point (E)

13. Cost Minimization Minimize Subject to C = $2QX + $3QY 1QX + 2QY 14 1QX + 1QY  10 1QX + 0.5QY  6 QX, QY  0 (objective function) (protein constraint) (minerals constraint) (vitamins constraint) (nonnegativity constraint)

14. Cost Minimization Optimal Solution (E) Feasible Region

15. Cost Minimization Algebraic Solution Cost at each point of intersection between constraintsis calculated to determine the optimal point (E)

16. Dual of the ProfitMaximization Problem Maximize Subject to  = $30QX + $40QY 1QX + 1QY 7 0.5QX + 1QY  5 0.5QY  2 QX, QY  0 (objective function) (input A constraint) (input B constraint) (input C constraint) (nonnegativity constraint) Minimize Subject to C = 7VA + 5VB + 2VC 1VA + 0.5VB $30 1VA + 1VB + 0.5VC  $40 VA, VB, VC  0

17. Dual of the CostMinimization Problem Minimize Subject to C = $2QX + $3QY 1QX + 2QY 14 1QX + 1QY  10 1QX + 0.5QY  6 QX, QY  0 (objective function) (protein constraint) (minerals constraint) (vitamins constraint) (nonnegativity constraint) Maximize Subject to  = 14VP + 10VM + 6VV 1VP + 1VM+ 1VV $30 2VP + 1VM + 0.5VV  $40 VP, VM, VV  0