Session 4b
Overview
• More Network Flow Models
• Facility Location Example
• Locating Call Centers
• Nonlinearity
Decision Models -- Prof. Juran
2
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
3
Call Center Location Example
Cost/call
Boston
New York
Charlotte
Dallas
Chicago
LA
Omaha
New
England
$1.20
$1.30
$1.50
$2.00
$2.10
$2.50
$2.20
Middle
Atlantic
$1.40
$1.00
$1.40
$1.80
$1.90
$2.10
$2.10
Southeast
$1.10
$1.30
$0.90
$1.20
$2.30
$1.90
$2.00
Decision Models -- Prof. Juran
Southwest
$2.60
$2.20
$1.90
$1.00
$1.50
$1.20
$1.30
Great
Lakes
$2.00
$1.80
$2.10
$1.70
$0.90
$1.70
$1.40
Plains
$2.20
$1.90
$2.30
$2.20
$1.30
$1.50
$0.60
Rocky
Mountains
$2.80
$2.50
$2.60
$1.80
$1.20
$1.40
$0.90
Pacific
$2.20
$2.80
$3.30
$2.70
$2.20
$1.00
$1.50
Hourly
wage
$14.00
$16.00
$11.00
$12.00
$13.00
$18.00
$10.00
Bldg cost
($MM)
$2.70
$3.00
$2.10
$2.10
$2.40
$3.60
$2.10
4
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.
Region
New England
Middle Atlantic
Southeast
Southwest
Great Lakes
Plains
Rocky Mountains
Pacific
Decision Models -- Prof. Juran
Daily Calls
1000
2000
2000
2000
3000
1000
2000
4000
5
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.
City
Boston
New York
Charlotte
Dallas
Chicago
Los Angeles
Omaha
Decision Models -- Prof. Juran
Building Cost
2.7
3.0
2.1
2.1
2.4
3.6
2.1
Hourly Wage
$14
$16
$11
$12
$13
$18
$10
6
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
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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.
Decision Models -- Prof. Juran
8
Network Representation
Sources
LAX
PA
OMA
SW
DAL
RM
ORD
PL
CLT
GL
SE
LGA
MA
BOS
NE
Destinations
Decision Models -- Prof. Juran
9
Formulation
Decision Variables
Define Vij to be the number of calls from call center i to region j.
Define Xi to be a binary variable. If a call center is built in city i, then Xi = 1;
otherwise, Xi = 0.
These Vij and Xi are the decision variables.
There are 56 + 7 = 63 decision variables here.
Objective
Define Cij to be the present value of a future call from city i to region j.
Define Bi to be the cost of building a call center in city i.
7
Minimize Z =
7
8
 X B   V C
i 1
i
i
i 1 j 1
Decision Models -- Prof. Juran
ij
ij
10
Formulation
Constraints
Define Rj to be the required number of calls to region j.
7
For every region j,
V
For every call center i,
V
i 1
8
j 1
ij
ij
 Rj
(1)
 X i 5000 
(2)
All Vij , Xi ≥ 0.
All Xi are (0, 1).
Decision Models -- Prof. Juran
11
Solution Methodology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A
Monetary summary
Annual wage cost
Annual calling cost
Building cost
PV of costs
1
1
1
1
1
1
1
B
C
D
E
$0
$0
$18,000,000
$18,000,000
F
Days per year
Minutes per call
Max calls / day
Interest rate
Years
G
250
4
5000
10%
10
H
I
J
New England
Middle Atlantic
Southeast
Southwest
Great Lakes
Plains
Rocky Mountains
Pacific
Required
0
0
0
0
0
0
0
0
>=
1000
0
0
0
0
0
0
0
0
>=
2000
0
0
0
0
0
0
0
0
>=
2000
0
0
0
0
0
0
0
0
>=
2000
0
0
0
0
0
0
0
0
>=
3000
0
0
0
0
0
0
0
0
>=
1000
0
0
0
0
0
0
0
0
>=
2000
0
0
0
0
0
0
0
0
>=
4000
Cost/call
Boston
New York
Charlotte
Dallas
Chicago
LA
Omaha
New England
$1.20
$1.30
$1.50
$2.00
$2.10
$2.50
$2.20
Middle Atlantic
$1.40
$1.00
$1.40
$1.80
$1.90
$2.10
$2.10
Southeast
$1.10
$1.30
$0.90
$1.20
$2.30
$1.90
$2.00
Southwest
$2.60
$2.20
$1.90
$1.00
$1.50
$1.20
$1.30
Great Lakes
$2.00
$1.80
$2.10
$1.70
$0.90
$1.70
$1.40
Plains
$2.20
$1.90
$2.30
$2.20
$1.30
$1.50
$0.60
Rocky Mountains
$2.80
$2.50
$2.60
$1.80
$1.20
$1.40
$0.90
Pacific
$2.20
$2.80
$3.30
$2.70
$2.20
$1.00
$1.50
Call Centers
Boston
New York
Charlotte
Dallas
Chicago
LA
Omaha
Made to region
Decision Models -- Prof. Juran
K
L
M
1. Annual wage cost is (for each center) found from the following "units" equation:
$/year = calls/day * days/year * minutes/call * hours/minute * $/hour
2. Total present value can be found from the PV function for the annual costs, but the onetime building cost must be outside the PV function.
Total Calls
0
0
0
0
0
0
0
<=
<=
<=
<=
<=
<=
<=
Logical Bound
5000
5000
5000
5000
5000
5000
5000
Hourly wage
$14.00
$16.00
$11.00
$12.00
$13.00
$18.00
$10.00
Bldg cost ($MM)
$2.70
$3.00
$2.10
$2.10
$2.40
$3.60
$2.10
12
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
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Decision Models -- Prof. Juran
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Optimal Solution
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
A
Monetary summary
Annual wage cost
Annual calling cost
Building cost
PV of costs
0
0
1
1
1
0
1
B
C
D
E
New England
Middle Atlantic
Southeast
Required
0
0
1000
0
0
0
0
1000
>=
1000
0
0
2000
0
0
0
0
2000
>=
2000
0
0
2000
0
0
0
0
2000
>=
2000
Cost/call
Boston
New York
Charlotte
Dallas
Chicago
LA
Omaha
New England
1.2
1.3
1.5
2
2.1
2.5
2.2
Middle Atlantic
1.4
1
1.4
1.8
1.9
2.1
2.1
Southeast
1.1
1.3
0.9
1.2
2.3
1.9
2
$3,233,333
$4,950,000
$8,700,000
$58,983,041
Call Centers
Boston
New York
Charlotte
Dallas
Chicago
LA
Omaha
Made to region
Decision Models -- Prof. Juran
F
Days per year
Minutes per call
Max calls / day
Interest rate
Years
G
250
4
5000
10%
10
H
I
J
K
Southwest
Great Lakes
Plains
Rocky Mountains
Pacific
0
0
0
2000
0
0
0
2000
>=
2000
0
0
0
0
3000
0
0
3000
>=
3000
0
0
0
0
0
0
1000
1000
>=
1000
0
0
0
0
2000
0
0
2000
>=
2000
0
0
0
0
0
0
4000
4000
>=
4000
Total Calls
0
0
5000
2000
5000
0
5000
Southwest
2.6
2.2
1.9
1
1.5
1.2
1.3
Great Lakes
2
1.8
2.1
1.7
0.9
1.7
1.4
Plains
2.2
1.9
2.3
2.2
1.3
1.5
0.6
Rocky Mountains
2.8
2.5
2.6
1.8
1.2
1.4
0.9
Pacific
2.2
2.8
3.3
2.7
2.2
1
1.5
Hourly wage
$14
$16
$11
$12
$13
$18
$10
15
Optimal Solution
Sources
LAX
PA
OMA
SW
DAL
RM
ORD
PL
CLT
GL
SE
LGA
MA
BOS
NE
Destinations
Decision Models -- Prof. Juran
16
Extension
How would you find the optimal solution
if we only wanted to build 3 call centers?
Decision Models -- Prof. Juran
17
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
18
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
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Decision Models -- Prof. Juran
20
Local minima
Global minimum
Decision Models -- Prof. Juran
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3 Solvers
• Simplex LP Solver
• GRG Nonlinear Solver
• Evolutionary Solver
Decision Models -- Prof. Juran
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Decision Models -- Prof. Juran
23
Summary
• More Network Flow Models
• Facility Location Example
• Locating Call Centers
• Nonlinearity
Decision Models -- Prof. Juran
24