Session 4b

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Session 4b
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Overview

• More Network Flow Models
• Facility Location Example
• Locating Call Centers
• Nonlinearity

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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.
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Call Center Location Example
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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.
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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.
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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.
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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.
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Network Representation
Sources
LAX
LGA
ORD
BOS
CLT
DAL
OMA
MA
RM
PA
PL
GL
SW
SE
NE
Destinations
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Formulation
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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).
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Solution Methodology
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Solution Methodology
The 56 Vij decision variables are in the cells
C8J14. The 7 Xi decision variables are in the
cells A8A14. The objective function is in cell
B5 Cells C15J15 are used to keep track of
constraint (1). Cells K8K14 are used to keep
track of constraint (2).
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Solution Methodology
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Solution Methodology
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Optimal Solution
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Optimal Solution
Sources
LAX
LGA
ORD
BOS
CLT
DAL
OMA
MA
RM
PA
PL
GL
SW
SE
NE
Destinations
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Extension
How would you find the optimal solution if we
only wanted to build 3 call centers?
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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.
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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).
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Local minima
Global minimum
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3 Solvers

• Simplex LP Solver
• GRG Nonlinear Solver
• Evolutionary Solver

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Summary

• More Network Flow Models
• Facility Location Example
• Locating Call Centers
• Nonlinearity