Facility Location Facility Location

Single-Facility Rectilinear Single-Facility Rectilinear Distance Location ProblemDistance Location Problem

Locating a new facility among n existing Locating a new facility among n existing facilities facilities – locating a warehouse that distributes merchandise to locating a warehouse that distributes merchandise to

a number of retail outletsa number of retail outlets– locating a supplier that provides parts to a number of locating a supplier that provides parts to a number of

different facilitiesdifferent facilities– locating a new piece of equipment that processes locating a new piece of equipment that processes

parts that are subsequently sent downstream to a parts that are subsequently sent downstream to a number of different workstations number of different workstations

Locate the new facility to minimize a weighted Locate the new facility to minimize a weighted sum of rectilinear distances measured from the sum of rectilinear distances measured from the new facility to the existing facilitiesnew facility to the existing facilities

Setting up the problem Setting up the problem mathematicallymathematically

Existing facilities are located at points (aExisting facilities are located at points (a11, , bb11), (a), (a22, b, b22), …, (a), …, (ann, b, bnn))

Find values of x and y (the location of the Find values of x and y (the location of the new facility) to minimizenew facility) to minimize

Weights (wWeights (wii) are included to allow for ) are included to allow for different traffic rates between the new different traffic rates between the new facility and the existing facilitiesfacility and the existing facilities

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Setting up the problem Setting up the problem mathematically (cont.)mathematically (cont.)

The values of x and y can be determined The values of x and y can be determined separatelyseparately

There is always an optimal solution with x There is always an optimal solution with x equal to some value of aequal to some value of aii and y equal to and y equal to some value of bsome value of bii (there may be other optimal (there may be other optimal solutions as well)solutions as well)

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Some examplesSome examples

Two existing locations (5, 10) and (20, 30) and a Two existing locations (5, 10) and (20, 30) and a weight of 1 applied to each facilityweight of 1 applied to each facility– x can assume any value between 5 and 20 (gx can assume any value between 5 and 20 (g11(x) = 15)(x) = 15)

– y can assume any value between 10 and 30 (gy can assume any value between 10 and 30 (g22(y) = 20)(y) = 20)

Four existing locations (3, 3), (6, 9), (12, 8), and Four existing locations (3, 3), (6, 9), (12, 8), and (12, 10) and a weight of 1 applied to each facility(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 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 – 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 any value of x between 6 and 12 is a median value and is optimal (goptimal (g11(x) = 15)(x) = 15)

– The median y value – 3, 8, 9, 10 – any value of y between The median y value – 3, 8, 9, 10 – any value of y between 8 and 9 is a median value and is optimal (g8 and 9 is a median value and is optimal (g22(y) = 8)(y) = 8)

The effect of weightsThe effect of weights

Four existing machines in a job shop, (3, 3), (6, Four existing machines in a job shop, (3, 3), (6, 9), (12, 8), and (12, 10)9), (12, 8), and (12, 10)

Locate a new machine to minimize the total Locate a new machine to minimize the total distance traveled to transport material between distance traveled to transport material between this fifth machine and the existing onesthis fifth machine and the existing ones

Assume there are on average 2, 4, 3, and 1 Assume there are on average 2, 4, 3, and 1 materials handling trips per hour, respectively, materials handling trips per hour, respectively, from the existing machines to the new machinefrom the existing machines to the new machine

This is equivalent to one trip but with 2 This is equivalent to one trip but with 2 machines at location (3, 3), 4 machines at machines at location (3, 3), 4 machines at location (6, 9), 3 machines at location (12, 8) location (6, 9), 3 machines at location (12, 8) and the one machine at location (12, 10)and the one machine at location (12, 10)

The effect of weights (cont.)The effect of weights (cont.)

x locations in increasing order 3, 3, 6, 6, 6, 6, 12, 12, x locations in increasing order 3, 3, 6, 6, 6, 6, 12, 12, 12, 12 – the median location is x = 6 (g12, 12 – the median location is x = 6 (g11(x) = 30)(x) = 30)

y locations in increasing order 3, 3, 8, 8, 8, 9, 9, 9, 9, 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 10 - median location is any value of y on the interval [8, 9] (g[8, 9] (g22(y) = 16)(y) = 16)

An easier way to determine the An easier way to determine the median locationmedian location

Compute the cumulative weights - then determine the location Compute the cumulative weights - then determine the location or locations corresponding to half of the cumulative weightsor locations corresponding to half of the cumulative weights

Machine y Coordinate Weight Cumulative Wght

one 3 2 2

three 8 3 5

two 9 4 9

four 10 1 10

Machine x Coordinate Weight Cumulative Wght

one 3 2 2

two 6 4 6

three 12 3 9

four 12 1 10

Example problemExample problem

University of the Far West has purchased University of the Far West has purchased equipment that permits faculty to prepare cd’s of equipment that permits faculty to prepare cd’s of lectures. The equipment will be used by faculty lectures. The equipment will be used by faculty from six schools on campus: business, education, from six schools on campus: business, education, engineering, humanities, law, and science. The engineering, humanities, law, and science. The coordinates of the schools and the number of coordinates of the schools and the number of faculty that are anticipated to use the equipment faculty that are anticipated to use the equipment are shown on the next slide. The campus is laid are shown on the next slide. The campus is laid out with large grassy areas separating the out with large grassy areas separating the buildings and walkways are mainly east-west or buildings and walkways are mainly east-west or north-south, so that distances between buildings north-south, so that distances between buildings are rectilinear. The university planner would like are rectilinear. The university planner would like to locate the new facility so as to minimize the to locate the new facility so as to minimize the total travel time of all faculty planning to use it.total travel time of all faculty planning to use it.

Example problem (cont.)Example problem (cont.)

SchoolSchool Campus Campus LocationLocation

Number of Number of Faculty Using Faculty Using

EquipmentEquipment

BusinessBusiness (5, 13)(5, 13) 3131

EducationEducation (8, 18)(8, 18) 2828

EngineeringEngineering (0, 0)(0, 0) 1919

HumanitiesHumanities (6, 3)(6, 3) 5353

LawLaw (14, 20)(14, 20) 3232

ScienceScience (10, 12)(10, 12) 4141

Single-Facility Straight-line Single-Facility Straight-line Distance Location ProblemDistance Location Problem

Minimize electrical cable when locating power-Minimize electrical cable when locating power-generation facilities or reaching the greatest generation facilities or reaching the greatest number of customers with cellphone tower number of customers with cellphone tower locationslocations

Objective is to minimize straight-line Objective is to minimize straight-line (Euclidean) distance (Euclidean) distance

Determining the optimal solution Determining the optimal solution mathematically is more difficult than for either mathematically is more difficult than for either rectilinear or squared straight-line distance rectilinear or squared straight-line distance (gravity problem)(gravity problem)

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The Gravity ProblemThe Gravity Problem

The objective is to minimize the square of the straight-line The objective is to minimize the square of the straight-line distancedistance

Differentiating and setting the partial derivatives equal to Differentiating and setting the partial derivatives equal to zero zero

Physical model - map, weights, "balance point" on mapPhysical model - map, weights, "balance point" on map University of the Far West example University of the Far West example

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The Straight-line Distance The Straight-line Distance ProblemProblem

Iterative solution processIterative solution process Use (xUse (x**, y, y**) calculated from gravity problem to determine initial ) calculated from gravity problem to determine initial

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Recompute gRecompute gii(x, y) using the new values of x and y(x, y) using the new values of x and y Continue to iterate until the values of the coordinates converge Continue to iterate until the values of the coordinates converge

(procedure yields optimal solution as long as (x, y) at each (procedure yields optimal solution as long as (x, y) at each iteration does not converge to an existing location)iteration does not converge to an existing location)

Physical model – supported map with holes, ring, strings, weightsPhysical model – supported map with holes, ring, strings, weights University of the Far West exampleUniversity of the Far West example

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