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Facility Location
• Logistics Management
• Factors that Affect Location Decisions
• Distance Measures
• Classification of Planar Facility Location Problems
• Planar Single-Facility Location Problems
– Minisum Location Problem with Rectilinear Distances
– Minisum Location Problem with Euclidean Distances
– Minimax Location Problem with Rectilinear Distances
– Minimax Location Problem with Euclidean Distances
• Planar Multi-Facility Location Problems
– Minisum Location Problem with Rectilinear Distances
Logistics Management
• Logistics Management can be defined as the management of the transportation and distribution of goods. The term goods includes raw materials or subassemblies obtained from suppliers as well as finished goods shipped from plants to warehouses or customers.
• Logistics management problems can be classified into three categories:– Location Problems: involve determining the location of one or more new facilities
in one or more of several potential sites. The cost of locating each new facility at each of the potential sites is assumed to be known. It is the fixed cost of locating a new facility at a particular site plus the operating and transportation cost of serving customers from this facility-site combination.
– Allocation Problems: assume that the number and location of facilities are known a priori and attempt to determine how each customer is to be served. In other words, given the demand for goods at each customer center, the production or supply capacities at each facility, and the cost of serving each customer from each facility, the allocation problem determines how much each facility is to supply to each customer center.
– Location-Allocation Problems:involve determining not only how much each customer is to receive from each facility but also the number of facilities along with their locations and capacities.
Factors that Affect Location Decisions• Proximity to source of raw materials.
• Cost and availability of energy and utilities.
• Cost, availability, skill, and productivity of labor.
• Government regulations at the federal, state, county, and local levels.
• Taxes at the federal, state, county, and local levels.
• Insurance.
• Construction costs and land price.
• Government and political stability.
• Exchange rate fluctuation.
• Export and import regulations, duties, and tariffs.
• Transportation system.
• Technical expertise.
• Environmental regulations at the federal, state, county and local levels.
• Support services.
• Community services - schools, hospitals, recreation, and so on.
• Weather.
• Proximity to customers.
• Business climate.
• Competition-related factors.
Distance Measures
• Rectilinear distance (L1 norm)
– d(X, Pi) = |x - ai| + |y - bi|
• Straight line or Euclidean distance (L2 norm)
– d(X, Pi) =
• Tchebyshev distance (L norm)
– d(X, Pi) = max{|x - ai|, |y - bi|}
(x - a ) + (y - b )i2
i2
X = (x, y)
Pi = (ai, bi)
Pi = (ai, bi)
Pi = (ai, bi)
X = (x, y)
X = (x, y)
Classification of Planar Facility Location Problems
FacilityLocation
Single-Facility
Multi-Facility
Minisum
Minimax
Rectilinear
Euclidean
Tchebyshev
Rectilinear
Euclidean
Tchebyshev
Rectilinear
Euclidean
Tchebyshev
Rectilinear
Euclidean
Tchebyshev
Minisum
Minimax
# of facilities Objectives Distance measures
Planar Single-Facility Location Formulations
• Minisum Formulation :
Min f(x) =
where X = (x, y) : location of the new facility
Pi = (ai, bi) : location of the i-th existing facility, i = 1, …, m
wi : weight associated to the i-th existing facility
For example, wi = ,
where ci : cost per hour of travel, ti : number of trips per month,
vi : average velocity.
• Minimax Formulation :
Min f(x) = Max {wi d(X, Pi)} Min z
s. t. wi d(X, Pi) z, i = 1, …, m
c i
t
vi
i
i = 1, …, m
w d X Pii
m
i1
,
Insights for the Minisum Problem with Euclidean Distance
• Majority Theorem :
When one weight constitutes a majority of the total weight, an optimal new facility
location coincides with the existing facility which has the majority weight.
w5
w1
w2
w4
w3
P1
P2
P3
P4
P5
Weight proportional to wi
String
Hole
Horizontal pegboard
Minisum Location Problem with Rectilinear Distances
Min f(x, y) =
Note that f(x, y) = f1(x) + f2(y)
where f1(x) =
f2(y) =
The cost of movement in the x direction is independent of the cost of
movement in the y direction, and viceversa.
Now, we look at the x direction.
f1(x) is convex a local min is a global min.
w [ | x a | | y b |i i ii =1
m
]
w | x a |i ii =1
m
w | y b |i ii =1
m
Minisum Location Problem with Rectilinear Distances (cont.)
• The coordinates of the existing facilities are sorted so that
a1 a2 a3 …. • Now, we consider the case of m = 3.
Case x a1 :
f1(x) = w1 |a1 - x| + w2 |a2 - x| + w3 |a3 - x|
= - (w1 + w2 + w3)x + w1 a1 + w2 a2 + w3 a3
= - W x + w1 a1 + w2 a2 + w3 a3, where W = w1 + w2 + w3
Case a1 x a2 :
f1(x) = w1 |a1 - x| + w2 |a2 - x| + w3 |a3 - x|
= (w1 - w2 - w3)x - w1 a1 + w2 a2 + w3 a3
= (- W + 2 w1) x - w1 a1 + w2 a2 + w3 a3
…
Objective Function f1(x)
a3a2a1
w3
w2
w1
w1 + w2 + w3
w1 + w2 - w3
w1 - w2 - w3
- w1 - w2 - w3
The slope changes sign
x
f1(x)
Minisum Location Problem with Rectilinear Distances (cont.)
• Slopes of f1(x) :
M0 = - (w1 + w2 + w3) = - W
M1 = 2 w1 + M0
M2 = 2 w2 + M1
M3 = 2 w3 + M2 = w1 + w2 + w3 = W
• Median conditions :
f1(x) is minimized at the point where the slope changes from nonpositive to
Min z = w1 (r1+ s1) + w2 (r2+ s2) + w3 (r3+ s3), Dual variables
s. t. x - r1+ s1 = a1, : y1 x
- r2+ s2 = a2, : y2
x - r3+ s3 = a3, : y3
rj, sj 0, j = 1, 2, 3.
• Relationships among variables : aj - x = rj - sj , |aj - x| = rj + sj, rj, sj 0.
• If both rj, sj > 0, we can reduce each by j = min {rj, sj}.
• This maintains feasibility and reduces z
In an optimal solution, at least one of the r j and sj is 0, i. e., rj sj = 0.
Linear Programming Formulation (cont.)• Dual Problem :
Max g = - a1y1 - a2 y2 - a3 y3 + (w1 a1 + w2 a2 + w3 a3)
s. t. y1 + y2 + y3 = w1 + w2 + w3 = W
0 yj 2 wj, j = 1, 2, 3
Min a1y1 + a2 y2 + a3 y3
s. t. y1 + y2 + y3 = W
0 yj 2 wj, j = 1, 2, 3
• Complementary slackness conditions :
0 < yj* < 2 wj x* = aj
1 2
0 y1 2 w1
a1
0 y2 2 w2
a2
a3
0 y3 2 w3
WW
a1 a2 a3
Example 1 : Dual Solution
• f1(x) = 5 |x - 10| + 6 |x - 20| + 4 |x - 40|
• W = 15
y1* = 10
y2* = 5
y3* = 0
0 < y2* < 12 x* = a2 = 20
1 2
0 y1 10
10
0 y2 12
20
40
0 y3 8
1515
Minisum Location Problem with Euclidean Distances
• Min f(x, y) =
• Colinear case : all the points are in a line
The problem reduces to minimizing f1(x), which is the rectilinear distance
problem.
(ai, bi)
The optimum location
is always in the convex
hull of
{(a1, b1), …, (am, bm)}
w [(x a ) (y b )i i2
i2
i =1
m
]1
2
(ai, bi)
Non-colinear Case
• The graph of is a cone (strictly convex function).
• f(x, y) = is strictly convex unless the
convex hull is a line segment.
(ai, bi)
contours
(ai, bi, 0)
[(x a ) (y b )i2
i2 ]
1
2
[(x a ) (y b )i2
i2 ]
1
2
y
x
yx
w [(x a ) (y b )i i2
i2
i =1
m
]1
2
Non-colinear Case (cont.)
• First order optimality conditions :
• Any point where the partial derivatives are zero is optimal.
Let
and
ii
i2
i2
(x , y)w
[(x a ) (y b )
]1
2
f(x , y)
x0
f(x , y)
y0
x
y
0
0
(x0, y0) is optimal
(x , y) (x , y)i
i
m
1
Non-colinear Case (cont.)
f(x , y)
x(x , y) (x - a )
f(x , y)
y(x , y) (y - b )
ii =1
m
i
ii =1
m
i
xa (x , y)
(x , y)
yb (x , y)
(x , y)
i ii =1
m
i ii =1
m
= 0
= 0
No-colinear Case (cont.)
• If the optimal solution is in an exiting facility (a i, bi), then .
• A simple way to avoid the problem of division by zero is to “perturb” the
problem as follows :
where > 0 and small.
f(x,y) is flat near the optimum.
M in f(x , y) w [(x a ) (y b ) ]i i2
i2
1
2
i 1
m
x
y
f(x,y)
(x*,y*)
i (x , y)
Weiszfeld’s Algorithm
Initialization :
Iterative step (k = 1, 2, …) :
Terminating conditions :
(i) (x , y )1
m(a , b )
(ii) (x , y )1
Ww (a , b ) , w h ere W = w + . . . + w
0 0i i
i 1
m
0 0i i i
i 1
m
1 m
(i) (x , y ) - (x , y )
(ii) f(x , y ) - f(x , y )
k k k -1 k -1
k -1 k -1 k k
x = a (x , y )
(x , y )
y = b (x , y )
(x , y )
ki i
k -1 k -1
i =1
m
k -1 k -1
ki i
k -1 k -1
i =1
m
k -1 k -1
or
or
Example 2
Problem Data :
m = 4
P1 = (0, 0) w1 = 1 P2 = (0, 10) w2 = 1
P3 = (5, 0) w3 = 1 P4 = (12, 6) w4 = 1
Solution :
x0 = (5+12)/4 = 4.25 y0 = (10+6)/4 = 4
k
1
2
5
10
Optimum
(x, y)
4.023, 3.116
3.949, 2.647
3.958, 2.124
3.995, 2.011
4.000, 2.000
f(x, y)
24.808
24.665
24.600
24.597
Minimax Location Problem with Rectilinear Distances
• Possible example : locating an ambulance with the existing facilities
being the locations of possible accidents.
XP3
P4
P2
P1
h3
h4h2
h1
Hospital
Hospital
Hospital
Hospital :
Poss. Accident :
Ambulance :
Minimax Location Problem with Rectilinear Distances (cont.)
• Notation
EF (existing facilities) locations : Pi = (ai, bi), i = 1, …, m
NF (new facility = ambulance) location : X = (x, y)
Travel distance from EF i to the nearest hospital = hi, i = 1, …, m
Travel distance from NF to EF i = |x - ai| + |y - bi|
• Formulation :
Min g(x, y)
where g(x, y) = max {|x - ai| + |y - bi| + hi : i = 1, …, m}
Min z
s. t. |x - ai| + |y - bi| + hi z, i = 1, …, m
Minimax Location Problem with Rectilinear Distances (cont.)
• We want to make the inequalities linear
|x - ai| + |y - bi| z - hi
x - ai + y - bi z - hi (1)
ai - x + bi - y z - hi (2)
ai - x + y - bi z - hi (3)
x - ai + bi - y z - hi (4)
Make the
intersection as small
as possible with the
largest diamond as
small as possible.
(2) (4)
(3) (1)
Minimax Location Problem with Rectilinear Distances (cont.)
Min z
s. t. x + y - z ai + bi - hi, i = 1, ..., m
x + y + z ai + bi + hi, i = 1, ..., m
- x + y - z - ai + bi - hi, i = 1, ..., m
- x + y + z - ai + bi + hi, i = 1, ..., m
Min z
s. t. x + y - z c1 where c1 = Min {ai + bi - hi}
x + y + z c2 c2 = Max {ai + bi + hi}
- x + y - z c3 c3 = Min {- ai + bi - hi}
- x + y + z c4 c4 = Max {- ai + bi + hi}
i = 1, ..., m
i = 1, ..., m
i = 1, ..., m
i = 1, ..., m
Minimax Location Problem with Rectilinear Distances (cont.)
Min z
s. t. - x - y + z - c1
x + y + z c2
x - y + z - c3
- x + y + z c4
c5 = Max {c2 - c1, c4 - c3} z = c5/2
Optimal solution :
(x1, y1, z1) = 1/2 (c1 - c3, c1 + c3 + c5, c5)
(x2, y2, z2) = 1/2 (c2 - c4, c2 + c4 - c5, c5)
The line segment joining (x1, y1) and (x2, y2) is the set of optimal NF locations
z (c2 - c1)/2 (lower bound)
z (c4 - c3)/2 (lower bound)
Minimax Location Problem with Euclidean Distances
• Examples : helicopter in an emergency unit,
radio transmitter
EF : (ai, bi), i = 1, …, m
NF : (x, y)
min g(x, y)
where g(x, y) = max {[(x - ai)2 + (y - bi)2]1/2, i = 1, …, m}
min z
s. t. [(x - ai)2 + (y - bi)2]1/2 z, i = 1, …, m
min z
s. t. (x - ai)2 + (y - bi)2 z , i = 1, …, m
(ai, bi)(x, y)
Elzinga-Hearn Algorithm (1971)
Step 1. Choose any two points and go to Step 2.
Step 2. Find the minimum covering circle for the chosen points*. Discard from the set of chosen points those points not defining the minimum covering circle, and go to Step 3.
Step 3. If the constructed circle contains all the points, then the center of the circle is a minimax location, so stop. Otherwise, choose some point outside the circle, add it to the set of points defining the circle, and go to Step 2.
* Find the minimum covering circle for the chosen points :
A. If there are two points, let the two points define the diameter of the circle.
B. If there are three points defining a right or obtuse triangle, let the two points opposite to the right or obtuse angle define the diameter of the circle. Otherwise, construct a circle through the three points (see Figure 1).
C. If there are four points, construct a circle using as defining points those indicated in Figure 2.
Elzinga-Hearn Algorithm (cont.)
Defining points :
BD
Defining points :
ABD
Defining points :
ABD
Defining points :
AD
A
B
Defining points :
BCD
Defining points :
ACD
Defining points :
ABD
Defining points :
AD
A
B
A
Defining points :
CD
Defining points :
BD
Figure 1. Alternative B Figure 2. Alternative C
B
C
C
Planar Multi-Facility Location Problems
Old Facility :
New Facility :
X2
X1
P4
P3
P2
P1
v12
w24
w23
w12
w11
Minisum Multi-Facility Location Problem with Rectilinear Distances
ij
n
1j
m
1iji
nkj1kjjk axwxxv
Location of new facilities: Xj = (xj, yj), j = 1, …, n.
Location of existing facilities: Pi = (ai, bi), i = 1, …, m.
Weight between new facilities j and k: vjk, where k > j.
Weight between new facility j and existing facility i: wji.