Session 4b

1. Session 4b

2. Overview • More Network Flow Models • Facility Location Example • Locating Call Centers • Nonlinearity Decision Models -- Prof. Juran

3. Call Center Location Example Suppose you are considering seven calling center locations: Boston, New York, Charlotte, Dallas, Chicago, Los Angeles, and Omaha. You know the average cost (in dollars) incurred if a telemarketing call is made from any these cities to any region of the country. Decision Models -- Prof. Juran

4. Call Center Location Example Decision Models -- Prof. Juran

5. Call Center Location Example Assume that an average call requires 4 minutes of labor. You make calls 250 days per year, and the average number of calls made per day to each region of the country is listed below. Decision Models -- Prof. Juran

6. Call Center Location Example The cost (in millions of dollars) of building a calling center in each possible location and the hourly wage that you must pay workers in each city is listed below. Each calling center can make up to 5000 calls per day. Decision Models -- Prof. Juran

7. Managerial Problem Definition Decision Variables There are two types of decision variables here. We need to decide where to build call centers, and we need to decide how many calls to make from each of these centers to each of 8 regions. Objective We want to minimize total costs, taking into account construction costs for the new call centers, plus the present value of calling costs from the centers to the 8 regions over a 10-year period. Decision Models -- Prof. Juran

8. Managerial Problem Definition Constraints All of the planned calls to the 8 regions must be accounted for and included in the total cost calculation. No calls are allowed from a city that has no call center. No call center can make more than 5000 calls per day. Decision Models -- Prof. Juran

9. Network Representation Sources OMA LAX DAL ORD CLT LGA BOS PA SW RM PL GL SE MA NE Destinations Decision Models -- Prof. Juran

10. Formulation Decision Models -- Prof. Juran

11. Formulation Constraints Define Rj to be the required number of calls to region j. For every region j, (1) For every call center i, (2) All Vij , Xi ≥ 0. All Xi are (0, 1). Decision Models -- Prof. Juran

12. Solution Methodology Decision Models -- Prof. Juran

13. Solution Methodology The 56 Vij decision variables are in the cells C8:J14. The 7 Xi decision variables are in the cells A8:A14. The objective function is in cell B5 Cells C15:J15 are used to keep track of constraint (1). Cells K8:K14 are used to keep track of constraint (2). Decision Models -- Prof. Juran

14. Solution Methodology Decision Models -- Prof. Juran

15. Solution Methodology Decision Models -- Prof. Juran

16. Optimal Solution Decision Models -- Prof. Juran

17. Optimal Solution Sources OMA LAX DAL ORD CLT LGA BOS PA SW RM PL GL SE MA NE Destinations Decision Models -- Prof. Juran

18. Extension How would you find the optimal solution if we only wanted to build 3 call centers? Decision Models -- Prof. Juran

19. Nonlinear Problems Some nonlinear problems can be formulated in a linear fashion (i.e. some network problems). Other nonlinear functions can be solved with our basic methods (i.e. smooth, continuous functions that are concave or convex, such as portfolio variances). However, there are many types of nonlinear problems that pose significant difficulties. Decision Models -- Prof. Juran

20. Nonlinear Problems The linear solution to a nonlinear (say, integer) problem may be infeasible. The linear solution may be far away from the actual optimal solution. Some functions have many local minima (or maxima), and Solver is not guaranteed to find the global minimum (or maximum). Decision Models -- Prof. Juran

21. Decision Models -- Prof. Juran

22. Local minima Global minimum Decision Models -- Prof. Juran

23. 3 Solvers • Simplex LP Solver • GRG Nonlinear Solver • Evolutionary Solver Decision Models -- Prof. Juran

24. Decision Models -- Prof. Juran

25. Summary • More Network Flow Models • Facility Location Example • Locating Call Centers • Nonlinearity Decision Models -- Prof. Juran