Session 4b

Decision Models -- Prof. Juran2 Overview More Network Flow Models Facility Location Example Locating Call Centers Nonlinearity

Decision Models -- Prof. Juran3 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. Juran4 Call Center Location Example

Decision Models -- Prof. Juran5 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. Juran6 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. Juran7 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. Juran8 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. Juran9 Network Representation LAX MA LGAORDBOSCLTDAL OMA RMPAPLGLSWSENE Destinations Sources

Decision Models -- Prof. Juran10 Formulation

Decision Models -- Prof. Juran11 Formulation Constraints Define R j to be the required number of calls to region j. For every region j, (1) For every call center i, (2) All V ij, X i ≥ 0. All X i are (0, 1).

Decision Models -- Prof. Juran12 Solution Methodology

Decision Models -- Prof. Juran13 Solution Methodology The 56 V ij decision variables are in the cells C8:J14. The 7 X i 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).

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Decision Models -- Prof. Juran15 Optimal Solution

Decision Models -- Prof. Juran16 Optimal Solution LAX MA LGAORDBOSCLTDAL OMA RMPAPLGLSWSENE Destinations Sources

Decision Models -- Prof. Juran17 Extension How would you find the optimal solution if we only wanted to build 3 call centers?

Decision Models -- Prof. Juran18 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. Juran19 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. Juran20

Decision Models -- Prof. Juran21 Local minima Global minimum

Decision Models -- Prof. Juran22 3 Solvers Simplex LP Solver GRG Nonlinear Solver Evolutionary Solver

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Decision Models -- Prof. Juran24 Summary More Network Flow Models Facility Location Example Locating Call Centers Nonlinearity