Session 4b

Session 4b

Decision Models -- Prof. Juran

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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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Cost/ call New

England Middle Atlantic Southeast Southwest

Great Lakes Plains

Rocky Mountains Pacific

Hourly wage

Bldg cost ($MM)

Boston $1.20 $1.40 $1.10 $2.60 $2.00 $2.20 $2.80 $2.20 $14.00 $2.70 New York $1.30 $1.00 $1.30 $2.20 $1.80 $1.90 $2.50 $2.80 $16.00 $3.00 Charlotte $1.50 $1.40 $0.90 $1.90 $2.10 $2.30 $2.60 $3.30 $11.00 $2.10 Dallas $2.00 $1.80 $1.20 $1.00 $1.70 $2.20 $1.80 $2.70 $12.00 $2.10 Chicago $2.10 $1.90 $2.30 $1.50 $0.90 $1.30 $1.20 $2.20 $13.00 $2.40 LA $2.50 $2.10 $1.90 $1.20 $1.70 $1.50 $1.40 $1.00 $18.00 $3.60 Omaha $2.20 $2.10 $2.00 $1.30 $1.40 $0.60 $0.90 $1.50 $10.00 $2.10


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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 Daily Calls New England 1000 Middle Atlantic 2000 Southeast 2000 Southwest 2000 Great Lakes 3000 Plains 1000 Rocky Mountains 2000 Pacific 4000


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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 Building Cost Hourly Wage Boston 2.7 $14 New York 3.0 $16 Charlotte 2.1 $11 Dallas 2.1 $12 Chicago 2.4 $13 Los Angeles 3.6 $18 Omaha 2.1 $10


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Managerial Problem DefinitionDecision VariablesThere 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.ObjectiveWe 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 DefinitionConstraintsAll 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

LAX

MA

LGAORD BOSCLTDALOMA

RMPA PL GLSW SE NE

Destinations

Sources


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FormulationD ecision V ariab les D efi n e V i j to b e th e n u m b er o f cal l s f r o m cal l cen ter i to r eg io n j . D efi n e X i to b e a b i n ary v ar iab le. I f a cal l cen ter i s b u i l t in ci ty i , th en X i = 1; o th erw i se, X i = 0. T h ese V i j an d X i are th e d eci si o n v ar iab les. T h ere ar e 56 + 7 = 63 d ecisi o n v ar iab les h er e. O b jective D efi n e C i j to b e th e p resen t v alu e o f a fu tu r e cal l f r o m ci ty i to r eg io n j . D efi n e B i to b e th e co st o f b u i ld in g a cal l cen ter in c i ty i .

M in im iz e Z = iji j

iji

ii CVBX

7

1

8

1

7

1


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FormulationConstraints

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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1ijij RV

8

15000

jiij XV


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Solution Methodology123456789101112131415161718

1920212223242526

A B C D E F G H I J K L MMonetary summary Days per year 250Annual wage cost $0 Minutes per call 4Annual calling cost $0 Max calls / day 5000Building cost $18,000,000 Interest rate 10%PV of costs $18,000,000 Years 10

Call Centers New England Middle Atlantic Southeast Southwest Great Lakes Plains Rocky Mountains Pacific Total Calls Logical Bound1 Boston 0 0 0 0 0 0 0 0 0 <= 50001 New York 0 0 0 0 0 0 0 0 0 <= 50001 Charlotte 0 0 0 0 0 0 0 0 0 <= 50001 Dallas 0 0 0 0 0 0 0 0 0 <= 50001 Chicago 0 0 0 0 0 0 0 0 0 <= 50001 LA 0 0 0 0 0 0 0 0 0 <= 50001 Omaha 0 0 0 0 0 0 0 0 0 <= 5000

Made to region 0 0 0 0 0 0 0 0>= >= >= >= >= >= >= >=

Required 1000 2000 2000 2000 3000 1000 2000 4000

Cost/call New England Middle Atlantic Southeast Southwest Great Lakes Plains Rocky Mountains Pacific Hourly wage Bldg cost ($MM)Boston $1.20 $1.40 $1.10 $2.60 $2.00 $2.20 $2.80 $2.20 $14.00 $2.70New York $1.30 $1.00 $1.30 $2.20 $1.80 $1.90 $2.50 $2.80 $16.00 $3.00Charlotte $1.50 $1.40 $0.90 $1.90 $2.10 $2.30 $2.60 $3.30 $11.00 $2.10Dallas $2.00 $1.80 $1.20 $1.00 $1.70 $2.20 $1.80 $2.70 $12.00 $2.10Chicago $2.10 $1.90 $2.30 $1.50 $0.90 $1.30 $1.20 $2.20 $13.00 $2.40LA $2.50 $2.10 $1.90 $1.20 $1.70 $1.50 $1.40 $1.00 $18.00 $3.60Omaha $2.20 $2.10 $2.00 $1.30 $1.40 $0.60 $0.90 $1.50 $10.00 $2.10

1. Annual wage cost is (for each center) found from the following "units" equation:$/year = calls/day * days/year * minutes/call * hours/minute * $/hour2. Total present value can be found from the PV function for the annual costs, but the one-time building cost must be outside the PV function.


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Solution MethodologyThe 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 B5Cells 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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Optimal Solution123456789101112131415161718

1920212223242526

A B C D E F G H I J KMonetary summary Days per year 250Annual wage cost $3,233,333 Minutes per call 4Annual calling cost $4,950,000 Max calls / day 5000Building cost $8,700,000 Interest rate 10%PV of costs $58,983,041 Years 10

Call Centers New England Middle Atlantic Southeast Southwest Great Lakes Plains Rocky Mountains Pacific Total Calls0 Boston 0 0 0 0 0 0 0 0 00 New York 0 0 0 0 0 0 0 0 01 Charlotte 1000 2000 2000 0 0 0 0 0 50001 Dallas 0 0 0 2000 0 0 0 0 20001 Chicago 0 0 0 0 3000 0 2000 0 50000 LA 0 0 0 0 0 0 0 0 01 Omaha 0 0 0 0 0 1000 0 4000 5000

Made to region 1000 2000 2000 2000 3000 1000 2000 4000>= >= >= >= >= >= >= >=

Required 1000 2000 2000 2000 3000 1000 2000 4000

Cost/call New England Middle Atlantic Southeast Southwest Great Lakes Plains Rocky Mountains Pacific Hourly wageBoston 1.2 1.4 1.1 2.6 2 2.2 2.8 2.2 $14New York 1.3 1 1.3 2.2 1.8 1.9 2.5 2.8 $16Charlotte 1.5 1.4 0.9 1.9 2.1 2.3 2.6 3.3 $11Dallas 2 1.8 1.2 1 1.7 2.2 1.8 2.7 $12Chicago 2.1 1.9 2.3 1.5 0.9 1.3 1.2 2.2 $13LA 2.5 2.1 1.9 1.2 1.7 1.5 1.4 1 $18Omaha 2.2 2.1 2 1.3 1.4 0.6 0.9 1.5 $10


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Optimal Solution

LAX

MA

LGAORD BOSCLTDALOMA

RMPA PL GLSW SE NE

Destinations

Sources


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ExtensionHow would you find the optimal solution if we only wanted to build 3 call centers?


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Nonlinear ProblemsSome 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 ProblemsThe 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

Session 4b

Documents

bdecision models

juran15decision models

juran9decision models

juran14decision models

juran12decision models

juran4decision models

juran1decision models

juran2decision models