Facility Location

Single-Facility Rectilinear Distance Location Problem Locating a new facility among n existing facilities 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 Locate the new facility to minimize a weighted sum of rectilinear distances measured from the new facility to the existing facilities 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 Existing facilities are located at points (a 1, b 1 ), (a 2, b 2 ), …, (a n, b n ) Existing facilities are located at points (a 1, b 1 ), (a 2, b 2 ), …, (a n, b n ) Find values of x and y (the location of the new facility) to minimize Find values of x and y (the location of the new facility) to minimize Weights (w i ) are included to allow for different traffic rates between the new facility and the existing facilities Weights (w i ) are included to allow for different traffic rates between the new facility and the existing facilities

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

Some examples Two existing locations (5, 10) and (20, 30) and a weight of 1 applied to each facility 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) Four existing locations (3, 3), (6, 9), (12, 8), and (12, 10) and a weight of 1 applied to each facility 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, 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 Four existing machines in a job shop, (3, 3), (6, 9), (12, 8), and (12, 10) 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 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 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) 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.) x locations in increasing order 3, 3, 6, 6, 6, 6, 12, 12, 12, 12 – the median location is x = 6 (g 1 (x) = 30) x locations in increasing order 3, 3, 6, 6, 6, 6, 12, 12, 12, 12 – the median location is x = 6 (g 1 (x) = 30) y locations in increasing order 3, 3, 8, 8, 8, 9, 9, 9, 9, 10 - median location is any value of y on the interval [8, 9] (g 2 (y) = 16) y locations in increasing order 3, 3, 8, 8, 8, 9, 9, 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 Compute the cumulative weights - then determine the location or locations corresponding to half of the cumulative weights Compute the cumulative weights - then determine the location or locations corresponding to half of the cumulative weights Machiney CoordinateWeightCumulative Wght one322 three835 two949 four101 Machinex CoordinateWeightCumulative Wght one322 two646 three1239 four12110

Example problem University of the Far West has purchased equipment that permits faculty to prepare cd’s of lectures. The equipment 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 equipment 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. University of the Far West has purchased equipment that permits faculty to prepare cd’s of lectures. The equipment 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 equipment 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 Equipment 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 Minimize electrical cable when locating power- generation facilities or reaching the greatest number of customers with cellphone tower locations Minimize electrical cable when locating power- generation facilities or reaching the greatest number of customers with cellphone tower locations Objective is to minimize straight-line (Euclidean) distance 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) Determining the optimal solution mathematically is more difficult than for either rectilinear or squared straight-line distance (gravity problem)

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

The Straight-line Distance Problem Iterative solution process Iterative solution process Use (x *, y * ) calculated from gravity problem to determine initial g i (x, y) Use (x *, y * ) calculated from gravity problem to determine initial g i (x, y) Recompute g i (x, y) using the new values of x and y Recompute g i (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) 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 Physical model – supported map with holes, ring, strings, weights University of the Far West example University of the Far West example