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Chapter 15

Chapter 15. Goal Programming. What is Goal Programming?. Mathematical model similar to Linear Programming, however it allows for multiple goals to be satisfied at the same time.

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Chapter 15

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  1. Chapter 15 Goal Programming

  2. What is Goal Programming? • Mathematical model similar to Linear Programming, however it allows for multiple goals to be satisfied at the same time. • Allows for the multiple goals to be prioritized and weighted to account for the DM’s utility for meeting the various goals.

  3. Assumptions • Similar to LP: • Non-negative variables • Conditions of certainty • Variables are independent • Limited resources • Deterministic

  4. Components • Economic Constraints • Physical • Concerned with resources • Cannot be violated • Example: # of production hours each week

  5. Components • Goal Constraints • Variable • Concerned with target values • Can be changed/modified • Example: Desire to achieve a certain level of profit

  6. Components • Objective Function • Minimizes the sum of the weighted deviations from the target values – this is ALWAYS the objective for Goal Programming • Not the same as LP (which was maximize revenue/minimize costs)

  7. Goal Programming Steps • Define decision variables • Define Deviational Variable for each goal • Formulate Constraint Equations • Economic constraints • Goal constraints • Formulate Objective Function

  8. Goal Programming Terms • Decision Variables are the same as those in LP formulations (represent products, hours worked) • Deviational Variables represent overachieving or underachieving the desired level of each goal • d+ Represents overachieving level of the goal • d- Represents underachieving level of the goal

  9. Goal Programming Constraints • Economic Constraints • Stated as <=, >=, or = • Linear (stated in terms of decision variables) • Example: 3x + 2y <= 50 hours • Goal Constraints • General form of goal constraint: - d+ + d- = Decision Variables Desired Goal Level

  10. Goal Programming Example • Microcom is a growth oriented firm which establishes monthly performance goals for its sales force • Microcom determines that the sales force has a maximum available hours per month for visits of 640 hours • Further, it is estimated that each visit to a potential new client requires 3 hours and each visit to a current client requires 2 hours

  11. Goal Programming Example • Microcom establishes two goals for the coming month: • Contact at least 200 current clients • Contact at least 120 new clients • Overachieving either goal will not be penalized

  12. Goal Programming Example • Steps Required: • Define the decision variables • Define the goals and deviational variables • Formulate the GP Model’s Parameters: • Economic Constraints • Goal Constraints • Objective Function • Solve the GP using the graphical approach

  13. Goal Programming Example • Step 1: Define the decision variables: • X1 = the number of current clients visited • X2 = the number of new clients visited • Step 2: Define the goals: • Goal 1 – Contact 200 current clients • Goal 2 – Contact 120 new clients

  14. Goal Programming Example • Step 3: Define the deviational variables • d1+ = the number of current clients visited in excess of the goal of 200 • d1- = the number of current clients visited less than the goal of 200 • d2+ = the number of new clients visited in excess of the goal of 120 • d2- = the number of new clients visited less than the goal of 120

  15. Goal Programming Example • Formulate the GP Model: • Economic Constraints: • 2X1 + 3X2 <= 640 (note: can be <, =, >) • X1, X2 => 0 • d1+, d1-, d2+, d2- => 0 • Goal Constraints: • Current Clients: X1 + d1- - d1+ = 200 • New Clients: X2 + d2- - d2+ = 120 Must be =

  16. Goal Programming Example • WebNet establishes two goals for the coming month: • Contact at least 200 current clients • Contact at least 120 new clients • Overachieving either goal will not be penalized

  17. Goal Programming Example • Objective Function: • Minimize Weighted Deviations • Minimize Z = d1- + d2-

  18. Goal Programming Example • Complete formulation: • Minimize Z = d1- + d2- • Subject to: • 2X1 + 3X2 <= 640 • X1 + d1- - d1+ = 200 • X2 + d2- - d2+ = 120 • X1, X2 => 0 • d1+, d1-, d2+, d2- => 0

  19. Goal Programming Example • Graph constraint: • 2X1 + 3X2 = 640 • If X1 = 0, X2 = 213 • If X2 = 0, X1 = 320 • Plot points (0, 213) and (320, 0)

  20. Graphical Solution X2 (0,213) 200 150 2X1 + 3X2 = 640 100 50 (320,0) X1 0 50 100 150 200 250 300 350

  21. Goal Programming Example • Graph deviation lines • X1 + d1- - d1+ = 200 (Goal 1) • X2 + d2- - d2+ = 120 (Goal 2) • Plot lines for X1 = 200, X2 = 120

  22. Goal Programming Example X2 Goal 1 (0,213) 200 d1- 2X1 + 3X2 < = 640 d1+ 150 d2+ Goal 2 (140,120) 100 d2- (200,80) 50 (320,0) X1 0 50 100 150 200 250 300 350

  23. Solving Graphical Goal Programming • Want to Minimize d1- + d2- • So we evaluate each of the candidate solution points: Optimal Point For point (140, 120) d1- = 60 and d2- = 0 Z = 60 + 0 = 60 For point (200, 80) d1- = 0 and d2- = 40 Z = 0 + 40 = 40 • Contact at least 200 current clients • Contact at least 120 new clients

  24. Goal Programming Solution • X1 = 200 Goal 1 achieved • X2 = 80 Goal 2 not achieved • d1+ = 0 d2+ = 0 • d1- = 0 d2- = 40 • Z = 40

  25. For Next Class • Complete reading Goal Programming pages (thru 727& • Do Goal Programming HWs

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