Facility Location SingleFacility Rectilinear Distance Location Problem n

Facility Location

Single-Facility Rectilinear Distance Location Problem n Locating a new facility among n existing facilities – locating a warehouse that distributes merchandise to a number of retail outlets – locating a supplier that provides parts to a number of different facilities – locating a new piece of equipment that processes parts that are subsequently sent downstream to a number of different workstations n Locate the new facility to minimize a weighted sum of rectilinear distances measured from the new facility to the existing facilities

Setting up the problem mathematically n n n Existing facilities are located at points (a 1, b 1), (a 2, b 2), …, (an, bn) Find values of x and y (the location of the new facility) to minimize Weights (wi) are included to allow for different traffic rates between the new facility and the existing facilities

Setting up the problem mathematically (cont. ) n The values of x and y can be determined separately n There is always an optimal solution with x equal to some value of ai and y equal to some value of bi (there may be other optimal solutions as well)

Some examples n Two existing locations (5, 10) and (20, 30) and a weight of 1 applied to each facility – x can assume any value between 5 and 20 (g 1(x) = 15) – y can assume any value between 10 and 30 (g 2(y) = 20) n Four existing locations (3, 3), (6, 9), (12, 8), and (12, 10) and a weight of 1 applied to each facility – The median x value (half the x values lie above it and half the x values lie below it) – in increasing order 3, 6, 12 – any value of x between 6 and 12 is a median value and is optimal (g 1(x) = 15) – The median y value – 3, 8, 9, 10 – any value of y between 8 and 9 is a median value and is optimal (g 2(y) = 8)

The effect of weights n n Four existing machines in a job shop, (3, 3), (6, 9), (12, 8), and (12, 10) Locate a new machine to minimize the total distance traveled to transport material between this fifth machine and the existing ones Assume there are on average 2, 4, 3, and 1 materials handling trips per hour, respectively, from the existing machines to the new machine This is equivalent to one trip but with 2 machines at location (3, 3), 4 machines at location (6, 9), 3 machines at location (12, 8) and the one machine at location (12, 10)

The effect of weights (cont. ) n n x locations in increasing order 3, 3, 6, 6, 12, 12 – the median location is x = 6 (g 1(x) = 30) y locations in increasing order 3, 3, 8, 8, 8, 9, 9, 10 median location is any value of y on the interval [8, 9] (g 2(y) = 16)

An easier way to determine the median location n Compute the cumulative weights - then determine the location or locations corresponding to half of the cumulative weights Machine x Coordinate Weight Cumulative Wght one 3 2 2 two 6 4 6 three 12 3 9 four 12 1 10 Machine y Coordinate Weight Cumulative Wght one 3 2 2 three 8 3 5 two 9 4 9 four 10 1 10

Example problem n A university is in the process of deciding on the site for its new faculty tech support facility. The facility will be used by faculty from six schools on campus: business, education, engineering, humanities, law, and science. The coordinates of the schools and the number of faculty that are anticipated to use the facility each semester are shown on the next slide. The campus is laid out with large grassy areas separating the buildings and walkways are mainly east -west or north-south, so that distances between buildings are rectilinear. The university planner would like to locate the new facility so as to minimize the total travel time of all faculty planning to use it.

Example problem (cont. ) School Campus Location Number of Faculty Using Facility Business (5, 13) 31 Education (8, 18) 28 Engineering (0, 0) 19 Humanities (6, 3) 53 Law (14, 20) 32 Science (10, 12) 41

Single-Facility Straight-line Distance Location Problem n n n Minimize electrical cable when locating powergeneration facilities or reaching the greatest number of customers with cellphone tower locations Objective is to minimize straight-line (Euclidean) distance Determining the optimal solution mathematically is more difficult than for either rectilinear or squared straight-line distance (gravity problem)

The Gravity Problem n The objective is to minimize the square of the straight-line distance n Differentiating and setting the partial derivatives equal to zero n Physical model - map, weights, "balance point" on map University example n

The Straight-line Distance Problem n n Iterative solution process Use (x*, y*) calculated from gravity problem to determine initial g i(x, y) n n n Recompute gi(x, y) using the new values of x and y Continue to iterate until the values of the coordinates converge (procedure yields optimal solution as long as (x, y) at each iteration does not converge to an existing location) Physical model – supported map with holes, ring, strings, weights University example