Location Problems

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Location Problems

• John H. Vande Vate
• Spring 2005

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Where to Locate Facilities

• Rectilinear Location Problems
• Euclidean Location Problems
• Location - Allocation Problems

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Basic Intuition

• On the line, if the objective is to min
• The maximum distance traveled
• The maximum distance left right
• The distance traveled
• there and back to each customer
• The item-miles traveled

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4
0
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Rectilinear Distance

• Travel on the streets and avenues
• Distance
• number of blocks East-West
• number of blocks North-South
• Manhattan Metric

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Rectilinear Distance
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5
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Locate a facility...

• To minimize the sum of rectilinear distances
• Intuition
• Where?
• Why?

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Solver Model
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Locate a facility...

• To minimize the max of rectilinear distances
• Intuition
• Where?
• Why?

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Min the Max

• Set Customers
• Param XCustomer
• Param YCustomer
• Var Xloc
• var Yloc
• var XdistCustomergt 0
• var YdistCustomergt 0
• var dmax
• min objective dmax
• s.t. DefineMaxDistc in Custs
• dmax gt Xdistc Ydistc

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Min the Max Contd

• DefineXdist1c in Customer
• Xdistc gt Xc-Xloc
• DefineXdist2c in Customer
• Xdistc gt Xloc-Xc
• DefineYdist1c in Customer
• Ydistc gt Yc-Yloc
• DefineYdist2c in Customer
• Ydistc gt Yloc-Yc

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Solver Model
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Locate a facility...

• To minimize the max of rectilinear distances
• Intuition
• Where?
• Why?

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Finding the Center(s)
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NIMBY

• Maximize the Minimum Distance
• Cant say Xdistc lt Xc - Xloc
• Want to say that either
• Xdistc lt Xc Xloc OR
• Xdistc lt Xloc - Xc
• Find a bound on the X distance between a customer
  and the facility, call it M
• Add a variable
• Leftc 1 if Xc gt Xloc and Leftc 0
  otherwise
• Xdistc lt Xc Xloc 2LeftcM
• Xdistc lt Xloc Xc 2(1-Leftc)M

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The Model
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Outline

• Rectilinear Location Problems
• Euclidean Location Problems
• Location - Allocation Problems

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Locating a single facility

• Distance is not linear
• Distance is a convex function
• Local Minimum is a global Minimum

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Where to Put the Facility

• Total Cost S ckdk(x,y)
• S ck?(xk- x)2 (yk- y)2
• ?Total Cost/?x S ck (xk - x)/dk(x,y)
• ?Total Cost/?x 0 when
• x Sckxk/dk(x,y)/Sck/dk(x,y)
• y Sckyk/dk(x,y)/Sck/dk(x,y)
• But dk(x,y) changes with location...

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Iterative Strategy

• Start somewhere, e.g.,
• x Sckxk/Sck
• y Sckyk/Sck
• as though dk 1.
• Step 1 Calculate values of dk
• Step 2 Refine values of x and y
• x Sckxk/dk/Sck/dk
• y Sckyk/dk/Sck/dk
• Repeat Steps 1 and 2. ...

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Solver Model
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Convex Minimization

• Call on Convex Minimization Tool
• Minos, Interior Point Methods,
• Typically dont support discrete variables too

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Locating Several Facilities

• Fixed Number of Facilities to Consider
• Single Sourcing
• Two Questions
• Location Where
• Allocation Whom to serve
• Each is simple
• Together they are harder

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Iterative Approach

• Put the facilities somewhere
• Step 1 Assign the Customers to the Facilities
• Step 2 Find the best location for each facility
  given the assignments (see previous method)
• Repeat Step 1 and Step 2 .

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Assign Customers to Facilities

• Uncapacitated (facilities can be any size)
• Greedy Assign each customer to closest
  facility
• Capacitated
• Use Optimization

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Dealers sourced by multiple ramps
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An Allocation Model
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Allocation Model

• Var xCusts, Facs binary
• minimize AllocationCost
• sumc in Custs, f in Facs Cc,fxc,f
• s.t. AssignEachCustc in Custs
• sumf in Facs xc,f 1
• s.t. FacilityCapacityf in Facssumc in
  CustsDccc,f lt Capf

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Set Covering Models
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WesternAir
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The Rest of the Story

• If there is
• Value Added E.g., BMW Assembly Plant
• High Value items E.g., Intel EU distribution
  center
• If there is labor content
• Competition
• Service vs Cost...