Location ProblemsJohn H. Vande VateSpring 2006

Where to Locate FacilitiesRectilinear Location Problems Euclidean Location ProblemsLocation - Allocation Problems

Basic IntuitionOn the line, if the objective is to min The maximum distance traveledThe maximum distance left + rightThe distance traveled there and back to each customerThe item-miles traveled014

Rectilinear DistanceTravel on the streets and avenuesDistance = number of blocks East-West +number of blocks North-SouthManhattan Metric

Rectilinear Distance549

Locate a facility...To minimize the sum of rectilinear distancesIntuitionWhere?Why?

Solver Model

Locate a facility...To minimize the max of rectilinear distancesIntuitionWhere?Why?

Min the MaxSet Customers;Param X{Customer};Param Y{Customer};Var Xloc;var Yloc;var Xdist{Customer}>= 0;var Ydist{Customer}>= 0;var dmax;min objective: dmax;s.t. DefineMaxDist{c in Custs}: dmax >= Xdist[c] + Ydist[c];

Min the Max ContdDefineXdist1{c in Customer}:Xdist[c] >= X[c]-Xloc;DefineXdist2{c in Customer}:Xdist[c] >= Xloc-X[c];DefineYdist1{c in Customer}:Ydist[c] >= Y[c]-Yloc;DefineYdist2{c in Customer}:Ydist[c] >= Yloc-Y[c];

Solver Model

Max the MinimumLike locating a land fill or a prisonWant the nearest town as far away as possibleMust have some confined region to work in otherwiseLet say the size of the region is M x M

NIMBYWith Min the Max: Xdist[c] >= X[c] Xloc;Cant say Xdist[c]

The Model

Set Covering Models

]Sheet1

]Sheet2

]Sheet3

MBD0007DCA6.unknown

WesternAir

OutlineRectilinear Location Problems Euclidean Location ProblemsLocation - Allocation Problems

Locating a single facility Distance is not linearDistance is a convex function Local Minimum is a global Minimum

Where to Put the FacilityTotal Cost = c1d1(x,y) + c2d2(x,y) + c3d3(x,y) = c1(x1- x)2 + (y1- y)2+c2(x2- x)2 + (y2- y)2+c3(x3- x)2 + (y3- y)2

Total Cost/x = c1 (x1 - x)/d1(x,y)+c2 (x2 - x)/d2(x,y)+c3 (x3 - x)/d3(x,y)

Total Cost/x = 0 when x =[c1x1/d1(x,y)+c2x2/d2(x,y)+c3x3/d3(x,y)] [c1/d1(x,y)+c2/d2(x,y)+c3/d3(x,y)]

But that depends on x!

Iterative StrategyStart somewhere, e.g., pretend dk(x,y) = 1x = [c1x1+c2x2+c3x3]/[c1+c2+c3]y = [c1y1+c2y2+c3y3]/[c1+c2+c3]Thats the center of gravityStep 1: Calculate the real distances, i.e. the dksStep 2: Refine values of x and y

x =

y =

Repeat Steps 1 and 2. ...

Solver Model

Convex MinimizationCall on Convex Minimization ToolMinos, Interior Point Methods, Typically dont support discrete variables too

Locating Several FacilitiesFixed Number of Facilities to ConsiderSingle SourcingTwo Questions: Location: WhereAllocation: Whom to serveEach is simpleTogether they are harder

Iterative ApproachCoopers ALT Heuristic (1963)Put the facilities somewhereStep 1: Assign the Customers to the FacilitiesStep 2: Find the best location for each facility given the assignments (see previous method)Repeat Step 1 and Step 2 .

Assign Customers to FacilitiesUncapacitated (facilities can be any size)Greedy: Assign each customer to closest facilityCapacitatedUse Optimization in the assignments This is typically a simple linear integer programAssign the Customers to the Facilities to min the total distance s.t. the Facilities capacities

SummaryLocation on a LineRectilinear LocationLinear modeling techniquesEuclidean LocationConvex MinimizationContraction MapLocation AllocationCoopers ALT Heuristic

NextLocation Case StudySGM Service Parts CaseUsing the ALT Heuristic