Location Problems

Location Problems John H. Vande Vate Spring 2006 1

Rectilinear Location Problems Euclidean Location Problems Location - Allocation Problems Where to Locate Facilities 2

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 Basic Intuition 1 4 0 3

Travel on the streets and avenues Distance = number of blocks East-West + number of blocks North-South Manhattan Metric Rectilinear Distance 4

Rectilinear Distance 9 5 4 5

To minimize the sum of rectilinear distances Intuition Where? Why? Locate a facility... 6

Solver Model 7

Locate a facility... • To minimize the max of rectilinear distances • Intuition • Where? • Why? 8

Set 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 9

Min the Max Cont’d • DefineXdist1{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]; 10

Solver Model 11

Like locating a land fill or a prison Want the nearest town as far away as possible Must have some confined region to work in otherwise… Let say the size of the region is M x M Max the Minimum 12

With Min the Max: Xdist[c] >= X[c] – Xloc; Can’t say Xdist[c] <= X[c] - Xloc; Want to say that either Xdist[c] <= X[c] – Xloc OR Xdist[c] <= Xloc - X[c] Find a bound on the X distance between a customer and the facility, call it M Add a variable Left[c] = 1 if X[c] > Xloc and Left[c] = 0 otherwise Xdist[c] <= Xloc – X[c] + 2*Left[c]*M Xdist[c] <= X[c] – Xloc + 2*(1-Left[c])*M NIMBY 13

The Model 14

Set Covering Models 15

WesternAir 16

Rectilinear Location Problems Euclidean Location Problems Location - Allocation Problems Outline 17

Distance is not linear Distance is a convex function Local Minimum is a global Minimum Locating a single facility 18

Total 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 = Where to Put the Facility But that depends on 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)] 19

Start somewhere, e.g., pretend dk(x,y) = 1 x = [c1x1+c2x2+c3x3]/[c1+c2+c3] y = [c1y1+c2y2+c3y3]/[c1+c2+c3] That’s the “center of gravity” Step 1: Calculate the real distances, i.e. the dk‘s Step 2: Refine values of x and y x = y = Repeat Steps 1 and 2. ... [c1x1/d1+c2x2/d2+c3x3/d3] [c1/d1+c2/d2+c3/d3] [c1y1/d1+c2y2/d2+c3y3/d3] [c1/d1+c2/d2+c3/d3] Iterative Strategy 20

Solver Model 21

Call on Convex Minimization Tool Minos, Interior Point Methods, … Typically don’t support discrete variables too… Convex Minimization 22

Fixed Number of Facilities to Consider Single Sourcing Two Questions: Location: Where Allocation: Whom to serve Each is simple Together they are “harder” Locating Several Facilities 23

Coopers ALT Heuristic (1963) 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 …. Iterative Approach 24

Uncapacitated (facilities can be any size) “Greedy”: Assign each customer to closest facility Capacitated Use Optimization in the assignments This is typically a simple linear integer program Assign the Customers to the Facilities to min the total distance s.t. the Facilities’ capacities Assign Customers to Facilities 25

Location on a Line Rectilinear Location Linear modeling techniques Euclidean Location Convex Minimization Contraction Map Location Allocation Cooper’s ALT Heuristic Summary 26

Location Case Study SGM Service Parts Case Using the ALT Heuristic Next 27

Location Problems