Graduate Program in Business Information Systems

1. Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer

2. Integer Program • Structurally like the LP models where the solutions must be integers! Ex: • Production order quantities • Number of items sent from plants to warehouses • Binary decisions • produce or do not produce • Assign staff i to job j or not • Locate facility i in location j or not BIS 517-Aslı Sencer

3. Ex:Redwood Furniture Model (revisited) Optimal solution: BIS 517-Aslı Sencer

4. The Branch and Bound Method • An effective search procedure that involves solving a succession of carefully formulated LP’s. • If the optimal solution of the parent program includes fractions, then two descendent LP’s are developed and solved to force the noninteger variable to be an integer. • The optimal objective function of the parent program constitues a bound on the optimal objective function value of the descendent LP’s. BIS 517-Aslı Sencer

5. Graphical Solution Xc 15 Problem 2 Problem1 11 Problem 3 Xt 10 22 4 5 BIS 517-Aslı Sencer

6. Branch and Bound Solution Problem 4 Xt=4 Xc=9 P=96 Problem 2 Xt=4 Xc=9.3 P=98.4 Problem 1 Xt=4.2 Xc=9.2 P=98.8 OPTIMAL SOLUTION Problem 5 Xt=2.6 Xc=10 P=95.6 Problem 3 Xt=5 Xc=8 P=94 BIS 517-Aslı Sencer

7. When do we round off the LP? • For problems having small number of variables, IP is solved in a short time. • Bigger problems require a lot of time. • If no integer solution has been found, a rounded LP may be used. • Which solution to be used is judgemental. BIS 517-Aslı Sencer

8. Solving Integer Programs with a Spreadsheet • When using Excel Spreadsheet, it is sufficient to add a new constraint and define the decision variables as integers. • You can also use WinQSB or QuickQuant to solve LP and IP problems. BIS 517-Aslı Sencer

9. Ex: Maui Miser Car Rentals • Budget $500,000. • Three types of cars: Economy vans, compact cars, large cars • Income/vehicle: $15,000, $7,600, $10,600 • Costs/vehicle: $25,000, $15,000, $21,000 • At least 25 vehicles • At least 5 vans, at least 5 large cars, at least 12 passenger cars • # large cars can not exceed the #compacts • Average daily rental rate should be less than $39 where Daily rental prices/vehicle: $50, $30, $40 BIS 517-Aslı Sencer

10. Ex: Formulation BIS 517-Aslı Sencer

11. Optimal Solution BIS 517-Aslı Sencer

12. Goal Programming Maximize profit OR Minimize cost Single Goal Sometimes we have Multiple Goals. Minimize Cost AND Maximize Customer Satisfaction Maximize Return AND Minimize Risk What is the optimal balance in attaining the conflicting goals? BIS 517-Aslı Sencer

13. Basic Idea • It is impossible to meet all goals simultaneously. • The focus aims at achieving certain targets for each goal. • The overall objective is to find the solution that collectively minimizes deviations from these targets. BIS 517-Aslı Sencer

14. Ex:Goal Programming in Redwood Furniture Co. Objectives • Maximize Profit • Maximize Revenue to maintain marketshare growth • Maximize training time to increase work-force productivity Each LP has a different optimal solution. An idea is to set targets for the goals. BIS 517-Aslı Sencer

15. LP with Goal Targets Idea 1: We can treat targets as the lower bounds on the goals Warning: When these constraints are added to the Redwood Product Mix Formulation there may be no feasible solution satisfying the minimum requirements of all goals! What should we do then? Any comments? Answer: Try to lower the targets BIS 517-Aslı Sencer

16. Goal Deviations Idea 2: We can allow some goals to be partially met! • Compared to LP, goal programming is flexible. Optimal balance between goals can be achieved in several ways. • Goal Deviation: Amount of deviation from the goal target is a variable to be minimized. BIS 517-Aslı Sencer

17. Defining Goal Deviations Yp: The amount by which profit deviates from the target level When Yp is positive, When Yp is negative BIS 517-Aslı Sencer

18. Goal Deviation Constraints Regular Constrains: BIS 517-Aslı Sencer

19. Goal Programming Objective • It costs $1 to fall a dollar below the profit target. However exceeding the profit target is desirable. • Falling below revenue target lowers market share, lost earnings is $2 for a dollar below revenue target. • Training time upto 30hrs enhances future efficiency. It costs $0.50 per hour obtained. Beyond 30hrs. training effect is negligable. BIS 517-Aslı Sencer

20. Ex: Optimal Solution Profit and Training levels are $6 and 6hrs. below their targets. Revenue target is met. BIS 517-Aslı Sencer

21. Ex: Maui Miser Case (revisited) • Achieve annual rental income of at least $300,000 from fleet addition • Spend as close to as possible to $500,000 • Do not exceed average daily rental of $39 for the vehicles in the fleet addition. Penalty Costs: $1 for each dollar violation of goals 1 and 2. $10,000 for each dollar violation of goal 3. BIS 517-Aslı Sencer

22. Ex: Maui Miser Case GP Formulation BIS 517-Aslı Sencer

23. Ex: Maui Miser Case GP Optimal Solution BIS 517-Aslı Sencer