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Facility Location

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Title: Facility Location


1
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

2
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 Problemsinvolve 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.

3
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.

4
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) maxx - ai, y - bi

Pi (ai, bi)
X (x, y)
Pi (ai, bi)
X (x, y)
Pi (ai, bi)
X (x, y)
5
Classification of Planar Facility Location
Problems
of facilities Objectives
Distance measures
6
Planar Single-Facility Location Formulations
  • Minisum Formulation
  • Min f(x) wi ? d(X, Pi)
  • where X (x, y) location of the new facility
  • Pi (ai, bi) location of the i-th exiting
    facility, i 1, , m
  • wi weight associated to the i-th exiting
    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

i 1, , m
7
Insights for the Minisum Problem with Euclidean
Distance
Hole
P2
P1
String
P3
P5
Horizontal pegboard
P4
Weight proportional to wi
  • 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.

8
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.

9
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 a 1 w2 a 2 w3 a 3
  • - W x w1 a 1 w2 a 2 w3 a 3, 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 a 1 w2 a 2 w3 a 3
  • (- W 2 w1) x - w1 a 1 w2 a 2 w3 a 3

10
Objective Function f1(x)
11
Minisum Location Problem with Rectilinear
Distances (cont.)
  • Slopes of f1(x)
  • M0 - W
  • M1 2 w1 M0
  • M2 2 w2 M1
  • M3 2 w3 M2 W
  • Median conditions
  • f1(x) is minimized at the point where the slope
    changes from nonpositive to nonnegative.
  • M1 w1 - w2 - w3 lt 0 ? 2 w1 lt (w1 w2
    w3) W
  • w1 lt W/2
  • M2 w1 w2 - w3 ? 0 ? 2 (w1 w2) ? (w1
    w2 w3) W
  • (w1 w2) ? W/2

12
Example 1
  • Problem Data
  • m 3
  • a1 10 a2 20 a3 40
  • w1 5 w2 6 w3 4
  • W w1 w2 w3 15
  • W/2 7.5
  • w1 5 lt 7.5
  • w1 w2 11 gt 7.5
  • ? Minimizing point a2 20
  • Solution

13
Linear Programming Formulation
  • Min f1(x) w1 a1 - x w2 a2 - x w3 a3 -
    x
  • ?
  • Min z w1 (r1 s1) w2 (r2 s2) w3 (r3 s3)
  • s. t. x - r1 s1 a1
  • x - r2 s2 a2
  • x - r3 s3 a3
  • 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 gt 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 rj
    and sj is 0, i. e., rj ? sj 0.

14
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 0
  • 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 lt yj lt 2 wj ? x aj

15
Example 1 Dual Solution
  • f1(x) 5 x - 10 6 x - 20 4 x - 40
  • W 15
  • y1 10
  • y2 5
  • y3 0
  • 0 lt y2 lt 12 ? x a2 20

16
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)
(ai, bi)
17
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.

y
contours
(ai, bi)
x
(ai, bi, 0)
x
y
18
Non-colinear Case (cont.)
  • First order optimality conditions
  • Any point where the partial derivatives are zero
    is optimal.
  • Let
  • and

? (x0, y0) is optimal
19
Non-colinear Case (cont.)
0
0
?
20
No-colinear Case (cont.)
  • If the optimal solution is in an exiting facility
    (ai, bi), then .
  • A simple way to avoid the problem of division by
    zero is to perturb the problem as follows
  • where ? gt 0 and small.
  • f(x,y) is flat near the optimum.

f(x,y)
x
(x,y)
y
21
Weiszfelds Algorithm
  • Initialization
  • Iterative step (k 1, 2, )
  • Terminating conditions

or
or
22
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 (512)/4 4.25 y0 (106)/4
4
23
Minimax Location Problem with Rectilinear
Distances
  • Possible example locating an ambulance with the
    existing facilities being the locations of
    possible accidents.

h1
h3
P3
X
P2
P4
h2
h4
24
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
  • ?
  • Min z
  • s. t. x - ai y - bi hi ? z, i 1, ,
    m

25
Minimax Location Problem with Rectilinear
Distances (cont.)
Make the intersection as small as possible with
the largest diamond as small as possible.
  • 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)

(3)
(1)
(2)
(4)
26
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
27
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)
28
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)21/2, i 1, , m
  • ? min z
  • s. t. (x - ai)2 (y - bi)21/2 ? z, i 1,
    , m
  • ? min z?
  • s. t. (x - ai)2 (y - bi)2 ? z ?, i 1,
    , m

(ai, bi)
(x, y)
29
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.

30
Elzinga-Hearn Algorithm (cont.)
Defining points BCD
Defining points ABD
Defining points CD
Defining points BD
C?
A
Defining points ACD
A
B
Defining points BD
B?
B
A?
Defining points AD
C
Defining points ABD
Defining points AD
Defining points ABD
Figure 2. Alternative C
Figure 1. Alternative B
31
Planar Multi-Facility Location Problems
P1
P4
w24
w11
X2
X1
v12
P2
w12
w23
P3
Old Facility New Facility
32
Minisum Multi-Facility Location Problem with
Rectilinear Distances
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 gt j. Weight
between new facility j and existing facility i
wji. Problem formulation Min f((x1,y1), ,
(xn, yn)) f1(x1, , xn) f2(y1, , yn)
where f1(x1, , xn) f2(y1, , yn)
33
Example 3
  • Problem Data n 2 (NF) m 3 (EF)
  • v vjk w wji
  • x xj (x1, x2) a aj (10, 20,
    40)
  • Min f1(x1, x2) 2 x1 - x2 2 x1 - 10
    x1 - 20 4 x2 - 20 5 x2 - 40 ?
  • Min f1(x1, x2) 2 (p12 q12) 2 (r11 s11)
    (r12 s12) 4 (r21 s21) 5 (r23 s23)
  • s. t. x1 - x2 - p12 q12 0
  • x1 - r11 s11 10
  • x1 - r12 s12 20
  • x2 - r21 s21 10
  • x2 - r23 s23 40
  • Relationships among variables
  • x1 - x2 p12 - q12, x1 - x2 p12 q12,
    p12, q12 ? 0
  • xi - aj rij - sij, xi - aj rij sij,
    rij, sij ? 0

34
Example 3 (Dual Problem)
Max (- 10 u11 - 20 u12 - 10 u21 - 40 u23)
(10?2 20?1 10?4 40?5) ? Min 10 u11
20 u12 10 u21 40 u23 s. t. z12
u11 u12 5 - z12 u21
u23 7 0 ? z12 ? 4 0 ? u11 ?
4 0 ? u12 ? 2 0 ? u21 ? 8 0 ? u23
? 10
35
Equivalent Network Flow Problem
(10)
u11 ? 4
(0)
Cap ?
u12 ? 2
(0)
(0)
12
Cap ?
(20)
z12 ? 4
u21 ? 8
(0)
Cap ?
u23 ? 10
(40)
(0)
After drawing the network, the solution can be
usually obtained by inspection.
36
Equivalent Network Flow Problem (cont.)
Complementary slackness conditions 1. 0 lt
zjk ? xk ? xj zjk lt 2 vjk ? xj ?
xk In particular, 0 lt zjk lt 2 vjk
? xj xk 2. 0 lt uji ? ai ? xj
uji lt 2 wji ? xj ? ai In particular,
0 lt uji lt 2 wji ? xj ai In Example
3, 0 lt z12 lt 2 v12 ? x1 x2, 0 lt
u12 2 w12 ? x2 a1 10. If the network
is not connected, then the problem decomposes
into independent problems, one for each component.
37
Example 4
  • Four hospitals located within a city are
    cooperating to establish a centralized blood-bank
    facility that will serve the hospitals. The new
    facility is to be located such that the (total)
    distance traveled is minimized. The hospitals
    are located at the following coordinates
    P1(5,10), P2(7,6), P3(4,2), and P4(16,3).
    The number of deliveries to be made per week
    between the blood-bank facility and each hospital
    is estimated to be 3, 8, 2, and 10, respectively.
    Assuming rectilinear travel, determine the
    optimum location.
  • m 4 P1 (5, 10) w1 3 P2 (7, 6) w2 8
  • P3 (4, 2) w3 2 P4 (16, 3) w4 10 W
  • Computation of x
  • a3 4 w3 2 w3 2
  • a1 5 w1 3 w3 w1 5
  • a2 7 w2 8 w3 w1 w2 13 ? x
  • a4 16 w4 10
  • Computation of y
  • b3 2 w3 2 w3 2
  • b4 3 w4 10 w3 w4 12 ? y
  • b2 6 w2 8
  • b1 10 w1 3

38
Example 5
(10, 16)
16 14 12 10 8 6
  • Find the optimal location of an ambulance with
    respect to four (known) possible accident
    locations which coordinates are P1(6,11),
    P2(12,5), P3(14,7), and P4(10,16). The
    objective is to minimize the maximum distance
    from the ambulance location to an accident
    location and from the accident location to its
    closest hospital. The distances from the
    accident locations to their closest hospitals are
    h110, h216, h314, and h411. Assume that
    distances are rectilinear. If multiple optima
    exist, find all optimal solutions.

h4 11
(6, 11)
h1 10
(12, 9)
(14, 7)
(10, 7)
h3 14
(12, 5)
h2 16
(5, 4) 6 8
10 12 14
39
Example 5 Solution
  • m 4 P1 (6, 11) h1 10 P2 (12, 5) h2 16
  • P3 (14, 7) h3 14 P4 (10, 16) h4 11
  • c1 min ai bi - hi min 611-10, 125-16,
    147-14, 1016-11
  • c2 max ai bi hi min 61110, 12516,
    14714, 101611
  • c3 min -ai bi - hi min -611-10,
    -125-16, -147-14, -1016-11
  • c4 min -ai bi hi min -61110,
    -12516, -14714, -101611
  • c5 max (c2 - c1, c4 - c3 max -
    , -
  • Optimal objective value
  • z
  • Set of optimal solutions line segment defined
    by the following end points
  • (x1, y1) (c1 - c3, c1 c3 c5)
    ( - , ) (
    , )
  • (x2, y2) (c2 - c4, c2 c4 - c5)
    ( - , - ) (
    , )

40
Example 6
  • Given five existing facilities located at points
    P1, P2, P3, P4, and P5 as shown below, determine
    the optimum location for a new facility which
    will minimize the maximum distance to the
    existing facilities.
  • Elzinga-Hearn algorithm
  • Figure 1 Initial set of points P1, P2, P3
    center C1.
  • Figure 2 2nd set of points P1, P2, P4
    center C2.
  • Figure 3 3rd set of points P2, P4, P5
    center C3 (optimal location).

41
Figure 1
P3 ?
P2
P5
P1
C1
P1 ?
P2?
P3
P4
42
Figure 2
P4?
P2
P1
P5
C2
P3
P1?
P2?
P4
43
Figure 3
P2
P4?
P5
P1
C3
P3
P5?
P1?
P4
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