Facilities Design

1
Facilities Design

• S.S. Heragu
• Industrial Engineering Department
• University of Louisville

2
Chapter 11Basic Modelsfor theLocation Problem
3
Outline

• 11.1 Introduction
• 11.2 Important Factors in Location Decisions
• 11.3 Techniques for Discrete Space Location
  Problems
• 11.3.1 Qualitative Analysis
• 11.3.2 Quantitative Analysis
• 11.3.3 Hybrid Analysis

4
Outline Cont...

• 11.4 Techniques for Continuous Space Location
  Problems
• 11.4.1 Median Method
• 11.4.2 Contour Line Method
• 11.4.3 Gravity Method
• 11.4.4 Weiszfeld Method
• 11.5 Facility Location Case Study
• 11.6 Summary
• 11.7 Review Questions and Exercises
• 11.8 References

5
McDonalds

• QSCV Philosophy
• 11,000 restaurants (7,000 in USA, remaining in 50
  countries)
• 700 seat McDonalds in Pushkin Square, Moscow
• 60 million food plant combining a bakery,
  lettuce plant, meat plant, chicken plant, fish
  plant and a distribution center, each owned and
  operated independently at same location

6
McDonalds cont...

• Food taste must be the same at any McDonald, yet
  food must be secured locally
• Strong logistical chain, with no weak links
  between
• Close monitoring for logistical performance
• 300 in Australia
• Central distribution since 1974 with the help of
  F.J. Walker Foods in Sydney
• Then distribution centers opened in several cities

7
McDonalds cont...

• 2000 ingredients, from 48 food plants, shipment
  of 200 finished products from suppliers to DCs,
  6 million cases of food and paper products plus
  500 operating items to restaurants across
  Australia
• Delivery of frozen, dry and chilled foods twice a
  week to each of the 300 restaurants 98 of the
  time within 15 minutes of promised delivery time,
  99.8 within 2 days of order placement
• No stockouts, but less inventory

8
Entities in a Supply Chain
9
Introduction

• Design and Operation of a Supply chain
• Warehousing
• Distribution Channels
• Freight Transportation
• Freight Consolidation
• Transportation Modes

10
Introduction

• Logistics management can be defined as the
  management of transportation and distribution of
  goods.
• facility location
• transportation
• goods handling and storage.

11
Introduction Cont...
Some of the objectives in facility location
decisions (1) It must first be close as possible
to raw material sources and customers (2)
Skilled labor must be readily available in the
vicinity of a facilitys location (3) Taxes,
property insurance, construction and land prices
must not be too high (4) Utilities must be
readily available at a reasonable price
12
Introduction Cont...

• (5) Local , state and other government
  regulations must be conducive to business and
• (6) Business climate must be favorable and the
  community must have adequate support services and
  facilities such as schools, hospitals and
  libraries, which are important to employees and
  their families.

13
Introduction Cont...

• Logistics management problems can be classified
  as
• (1) location problems
• (2) allocation problems and
• (3) location-allocation problems.

14
List of Factors AffectingLocation Decisions

• Proximity to raw materials sources
• Cost and availability of energy/utilities
• Cost, availability, skill and productivity of
  labor
• Government regulations at the federal, state,
  country and local levels
• Taxes at the federal, state, county and local
  levels
• Insurance
• Construction costs, land price

15
List of Factors AffectingLocation Decisions
Cont...

• Government and political stability
• Exchange rate fluctuation
• Export, import regulations, duties, and tariffs
• Transportation system
• Technical expertise
• Environmental regulations at the federal, state,
  county and local levels
• Support services

16
List of Factors AffectingLocation Decisions
Cont...

• Community services, i.e. schools, hospitals,
  recreation, etc.
• Weather
• Proximity to customers
• Business climate
• Competition-related factors

17
11.2Important Factors in Location Decisions

• International
• National
• State-wide
• Community-wide

18
11.3.1Qualitative Analysis

• Step 1 List all the factors that are important,
  i.e. have an impact on the location decision.
• Step 2 Assign appropriate weights (typically
  between 0 and 1) to each factor based on the
  relative importance of each.
• Step 3 Assign a score (typically between 0 and
  100) for each location with respect to each
  factor identified in Step 1.

19
11.3.1Qualitative Analysis

• Step 4 Compute the weighted score for each
  factor for each location by multiplying its
  weight with the corresponding score (which were
  assigned Steps 2 and 3, respectively)
• Step 5 Compute the sum of the weighted scores
  for each location and choose a location based on
  these scores.

20
Example 1

• A payroll processing company has recently won
  several major contracts in the midwest region of
  the U.S. and central Canada and wants to open a
  new, large facility to serve these areas. Since
  customer service is of utmost importance, the
  company wants to be as near its customers as
  possible. Preliminary investigation has shown
  that Minneapolis, Winnipeg, and Springfield,
  Ill., would be the three most desirable locations
  and the payroll company has to select one of
  these three.

21
Example 1 Cont...

• A subsequent thorough investigation of each
  location with respect to eight important factors
  has generated the raw scores and weights listed
  in table 2. Using the location scoring method,
  determine the best location for the new payroll
  processing facility.

22
Solution

• Steps 1, 2, and 3 have already been completed for
  us. We now need to compute the weighted score
  for each location-factor pair (Step 4), and these
  weighted scores and determine the location based
  on these scores (Step 5).

23
Table 11.2. Factors and Weights for Three
Locations

• Wt. Factors Location
• Minn.Winn.Spring.
• .25 Proximity to customers 95 90 65
• .15 Land/construction prices 60 60 90
• .15 Wage rates 70 45 60
• .10 Property taxes 70 90 70
• .10 Business taxes 80 90 85
• .10 Commercial travel 80 65 75

24
Table 11.2. Cont...

• Wt. Factors Location
• Minn. Winn. Spring.
• .08 Insurance costs 70 95 60
• .07 Office services 90 90 80
• Click here

25
Solution Cont...

• From the analysis in Table 3, it is clear that
  Minneapolis would be the best location based on
  the subjective information.

26
Table 11.3. Weighted Scores for the Three
Locations in Table 11.2
Weighted Score Location Minn. Winn. Spring. Prox
imity to customers 23.75 22.5 16.25 Land/construct
ion prices 9 9 13.5 Wage rates 10.5 6.75 9 Propert
y taxes 7 9 8.5 Business taxes 8 9 8.5
27
Table 11.3. Cont...
Weighted Score Location Minn. Winn. Spring. Comm
ercial travel 8 6.5 7.5 Insurance
costs 5.6 7.6 4.8 Office services 6.3 6.3 5.6
28
Solution Cont...

• Of course, as mentioned before, objective
  measures must be brought into consideration
  especially because the weighted scores for
  Minneapolis and Winnipeg are close.

29
11.3.2Quantitative Analysis
30
General Transportation Model
31
General Transportation Model

• Parameters
• cij cost of transporting one unit from
  warehouse i to customer j
• ai supply capacity at warehouse i
• bi demand at customer j
• Decision Variables
• xij number of units transported from
  warehouse i to customer j

32
General Transportation Model
33
Transportation Simplex Algorithm

• Step 1 Check whether the transportation problem
  is balanced or unbalanced. If balanced, go to
  step 2. Otherwise, transform the unbalanced
  transportation problem into a balanced one by
  adding a dummy plant (if the total demand exceeds
  the total supply) or a dummy warehouse (if the
  total supply exceeds the total demand) with a
  capacity or demand equal to the excess demand or
  excess supply, respectively. Transform all the gt
  and lt constraints to equalities.
• Step 2 Set up a transportation tableau by
  creating a row corresponding to each plant
  including the dummy plant and a column
  corresponding to each warehouse including the
  dummy warehouse. Enter the cost of transporting a
  unit from each plant to each warehouse (cij) in
  the corresponding cell (i,j). Enter 0 cost for
  all the cells in the dummy row or column. Enter
  the supply capacity of each plant at the end of
  the corresponding row and the demand at each
  warehouse at the bottom of the corresponding
  column. Set m and n equal to the number of rows
  and columns, respectively and all xij0,
  i1,2,...,m and j1,2,...,n.
• Step 3 Construct a basic feasible solution using
  the Northwest corner method.

34
Transportation Simplex Algorithm

• Step 4 Set u10 and find vj, j1,2,...,n and ui,
  i1,2,...,n using the formula ui vj cij for
  all basic variables.
• Step 5 If ui vj - cij lt 0 for all nonbasic
  variables, then the current basic feasible
  solution is optimal stop. Otherwise, go to step
  6.
• Step 6 Select the variable xij with the most
  positive value ui vj- cij. Construct a
  closed loop consisting of horizontal and vertical
  segments connecting the corresponding cell in row
  i and column j to other basic variables. Adjust
  the values of the basic variables in this closed
  loop so that the supply and demand constraints of
  each row and column are satisfied and the maximum
  possible value is added to the cell in row i and
  column j. The variable xij is now a basic
  variable and the basic variable in the closed
  loop which now takes on a value of 0 is a
  nonbasic variable. Go to step 4.

35
Example 2

• Seers Inc. has two manufacturing plants at Albany
  and Little Rock supplying Canmore brand
  refrigerators to four distribution centers in
  Boston, Philadelphia, Galveston and Raleigh. Due
  to an increase in demand of this brand of
  refrigerators that is expected to last for
  several years into the future, Seers Inc., has
  decided to build another plant in Atlanta. The
  expected demand at the three distribution centers
  and the maximum capacity at the Albany and Little
  Rock plants are given in Table 4.

36
Table 11.4. Costs, Demand and Supply Information

• Bost. Phil. Galv. Rale. Supply
• Capacity
• Albany 10 15 22 20 250
• Little Rock 19 15 10 9 300
• Atlanta 21 11 13 6 No limit
• Demand 200 100 300 280

37
Table 11.5. Transportation Model with Plant at
Atlanta

• Bost. Phil. Galv. Rale. Supply
• Capacity
• Albany 10 15 22 20 250
• Little Rock 19 15 10 9 300
• Atlanta 21 11 13 6 880
• Demand 200 100 300 280 880
• Click here for Excel formulation
• Click here for LINGO formulation

38
Example 3

• Consider Example 2. In addition to Atlanta,
  suppose Seers, Inc., is considering another
  location Pittsburgh. Determine which of the two
  locations, Atlanta or Pittsburgh, is suitable for
  the new plant. Seers Inc., wishes to utilize all
  of the capacity available at its Albany and
  Little Rock Locations

39
Table 11.10. Costs, Demand and Supply Information

• Bost. Phil. Galv. Rale. Supply
• Capacity
• Albany 10 15 22 20 250
• Little Rock 19 15 10 9 300
• Atlanta 21 11 13 6 330
• Pittsburgh 17 8 18 12 330
• Demand 200 100 300 280

40
Table 11.12. Transportation Model with Plant at
Pittsburgh

• Bost. Phil. Galv. Rale. Supply
• Capacity
• Albany 10 15 22 20 250
• Little Rock 19 15 10 9 300
• Pittsburgh 17 8 18 12 880
• Demand 200 100 300 280 880
• Click here for Excel model
• Click here for LINDO Model
• Click here for LINGO Model

41
Min/Max Location Problem
Location
d11
d12
d1n
d21
d22
d2n
Site
dm1
dm2
dmn
42
11.3.3 Hybrid Analysis

• Critical
• Objective
• Subjective

43
Hybrid Analysis Cont...

• CFij 1 if location i satisfies critical
  factor j,
• 0 otherwise
• OFij cost of objective factor j at location i
• SFij numerical value assigned
• (on scale of 0-100)
• to subjective factor j for location i
• wj weight assigned to subjective factor
• (0lt w lt 1)

44
Hybrid Analysis Cont...
45
Hybrid Analysis Cont...

• The location measure LMi for each location is
  then calculated as
• LMi CFMi ? OFMi (1- ?) SFMi
• Where ? is the weight assigned to the objective
  factor.
• We then choose the location with the highest
  location measure LMi

46
Example 4

• Mole-Sun Brewing company is evaluating six
  candidate locations-Montreal, Plattsburgh,
  Ottawa, Albany, Rochester and Kingston, for
  constructing a new brewery. There are two
  critical, three objective and four subjective
  factors that management wishes to incorporate in
  its decision-making. These factors are
  summarized in Table 7. The weights of the
  subjective factors are also provided in the
  table. Determine the best location if the
  subjective factors are to be weighted 50 percent
  more than the objective factors.

47
Table 11.13Critical, Subjective and Objective
Factor Ratings for six locations for Mole-Sun
Brewing Company, Inc.
48
Table 11.13 Cont...

• Factors

Location Albany 0 1
Kingston 1 1 Montreal 1 1 Ottawa 1 0 Plattsburgh
1 1 Rochester 1 1
Critical
Water Supply
Tax Incentives
49
Table 11.13 Cont...
Factors
Location Albany 185 80 10
Kingston 150 100 15 Montreal 170
90 13 Ottawa 200 100 15 Plattsburgh 140 75
8 Rochester 150 75 11
Critical
Objective
Labor Cost
Energy Cost
Revenue
50
Table 11.13 Cont...
Location 0.3 0.4 Albany 0.5 0.9 Kingston 0.6
0.7 Montreal 0.4 0.8 Ottawa 0.5 0.4 Plattsburgh 0.
9 0.9 Rochester 0.7 0.65
Factors
Subjective
Ease of Transportation
Community Attitude
51
Table 11.13 Cont...
Factors
Location 0.25 0.05 Albany 0.6 0.7 Kingston 0.
7 0.75 Montreal 0.2 0.8 Ottawa 0.4 0.8 Plattsburgh
0.9 0.55 Rochester 0.4 0.8
Subjective
Support Services
Labor Unionization
52
Table 11.14 Location Analysis of Mole-Sun
Brewing Company, Inc., Using Hybrid Method
53
Table 11.14 Cont...
Location Albany -95 0.7 0 Kingston -35 0.67 0.4
Montreal -67 0.53 0.53 Ottawa -85 0.45 0 Plattsbu
rgh -57 0.88 0.68 Rochester -64 0.61 0.56
Factors
LMi
Subjective
Critical
Objective
SFMi
Sum of Obj. Factors
54
11.4Techniques For Continuous Space Location
Problems
55
11.4.1 Model for Rectilinear Metric Problem

• Consider the following notation
• fi Traffic flow between new facility and
  existing facility i
• ci Cost of transportation between new facility
  and existing facility i per unit
• xi, yi Coordinate points of existing facility i

56
Model for Rectilinear Metric Problem (Cont)
The median location model is then to minimize

• Where TC is the total distribution cost

57
Model for Rectilinear Metric Problem (Cont)

• Since the cifi product is known for each
  facility, it can be thought of as a weight wi
  corresponding to facility i.

58
Median Method

• Step 1 List the existing facilities in
  non-decreasing order of the x coordinates.
• Step 2 Find the jth x coordinate in the list at
  which the cumulative weight equals or exceeds
  half the total weight for the first time, i.e.,

59
Median Method (Cont)

• Step 3 List the existing facilities in
  non-decreasing order of the y coordinates.
• Step 4 Find the kth y coordinate in the list
  (created in Step 3) at which the cumulative
  weight equals or exceeds half the total weight
  for the first time, i.e.,

60
Median Method (Cont)

• Step 4 Cont... The optimal location of the new
  facility is given by the jth x coordinate and the
  kth y coordinate identified in Steps 2 and 4,
  respectively.

61
Notes

• 1. It can be shown that any other x or y
  coordinate will not be that of the optimal
  locations coordinates
• 2. The algorithm determines the x and y
  coordinates of the facilitys optimal location
  separately
• 3. These coordinates could coincide with the x
  and y coordinates of two different existing
  facilities or possibly one existing facility

62
Example 5

• Two high speed copiers are to be located in the
  fifth floor of an office complex which houses
  four departments of the Social Security
  Administration. Coordinates of the centroid of
  each department as well as the average number of
  trips made per day between each department and
  the copiers yet-to-be-determined location are
  known and given in Table 9 below. Assume that
  travel originates and ends at the centroid of
  each department. Determine the optimal location,
  i.e., x, y coordinates, for the copiers.

63
Table 11.15 Centroid Coordinates and Average
Number of Trips to Copiers
64
Table 11.15

• Dept. Coordinates Average number of
• x y daily trips to copiers
• 1 10 2 6
• 2 10 10 10
• 3 8 6 8
• 4 12 5 4

65
Solution

• Using the median method, we obtain the following
  solution
• Step 1

Dept. x coordinates in Weights Cumulative
non-decreasing order Weights
3 8 8 8 1 10 6 14 2 10 10 24 4 12 4 28
66
Solution

• Step 2 Since the second x coordinate, namely
  10, in the above list is where the cumulative
  weight equals half the total weight of 28/2 14,
  the optimal x coordinate is 10.

67
Solution

• Step 3

Dept. y coordinates in Weights Cumulative
non-decreasing order Weights
1 2 6 6 4 5 4 10 3 6 8 18 2 10 10 28
68
Solution

• Step 4 Since the third y coordinates in the
  above list is where the cumulative weight exceeds
  half the total weight of 28/2 14, the optimal y
  coordinate is 6. Thus, the optimal coordinates
  of the new facility are (10, 6).

69
Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem

• Parameters
• fi Traffic flow between new facility and
  existing facility i
• ci Unit transportation cost between new
  facility and existing facility i
• xi, yi Coordinate points of existing
  facility i
• Decision Variables
• x, y Optimal coordinates of the new
  facility
• TC Total distribution cost

70
Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem

• The median location model is then to

71
Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem

• Since the cifi product is known for each
  facility, it can be thought of as a weight wi
  corresponding to facility i. The previous
  equation can now be rewritten as follows

72

Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem
73
Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem
74
Equivalent Linear Model for the Rectilinear
Distance, Single-Facility Location Problem
75
11.4.2 Contour Line Method
76
Algorithm for Drawing Contour Lines

• Step 1 Draw a vertical line through the x
  coordinate and a horizontal line through the y
  coordinate of each facility
• Step 2 Label each vertical line Vi, i1, 2,
  ..., p and horizontal line Hj, j1, 2, ..., q
  where Vi the sum of weights of facilities whose
  x coordinates fall on vertical line i and where
  Hj sum of weights of facilities whose y
  coordinates fall on horizontal line j

77
Algorithm for Drawing Contour Lines (Cont)
m
?

• Step 3 Set i j 1 N0 D0 wi
• Step 4 Set Ni Ni-1 2Vi and Dj Dj-1 2Hj.
  Increment i i 1 and j j 1
• Step 5 If i lt p or j lt q, go to Step 4.
  Otherwise, set i j 0 and determine Sij, the
  slope of contour lines through the region bounded
  by vertical lines i and i 1 and horizontal line
  j and j 1 using the equation Sij -Ni/Dj.
  Increment i i 1 and j j 1

i1
78
Algorithm for Drawing Contour Lines

• Step 6 If i lt p or j lt q, go to Step 5.
  Otherwise select any point (x, y) and draw a
  contour line with slope Sij in the region i, j
  in which (x, y) appears so that the line touches
  the boundary of this line. From one of the end
  points of this line, draw another contour line
  through the adjacent region with the
  corresponding slope
• Step 7 Repeat this until you get a contour line
  ending at point (x, y). We now have a region
  bounded by contour lines with (x, y) on the
  boundary of the region

79
Notes on Algorithm for Drawing Contour Lines

• 1. The number of vertical and horizontal lines
  need not be equal
• 2. The Ni and Dj as computed in Steps 3 and 4
  correspond to the numerator and denominator,
  respectively of the slope equation of any contour
  line through the region bounded by the vertical
  lines i and i 1 and horizontal lines j and j
  1

80
Notes on Algorithm for Drawing Contour Lines
(Cont)
81
Notes on Algorithm for Drawing Contour Lines
(Cont)

• By noting that the Vis and Hjs calculated in
  Step 2 of the algorithm correspond to the sum of
  the weights of facilities whose x, y coordinates
  are equal to the x, y coordinates, respectively
  of the ith, jth distinct lines and that we have
  p, q such coordinates or lines (p lt m, q lt m),
  the previous equation can be written as follows

82
Notes on Algorithm for Drawing Contour Lines
(Cont)

• Suppose that x is between the sth and s1th
  (distinct) x coordinates or vertical lines (since
  we have drawn vertical lines through these
  coordinates in Step 1). Similarly, let y be
  between the tth and t1th vertical lines. Then

83
Notes on Algorithm for Drawing Contour Lines
(Cont)

• Rearranging the variable and constant terms in
  the above equation, we get

84
Notes on Algorithm for Drawing Contour Lines
(Cont)

• The last four terms in the previous equation can
  be substituted by another constant term c and the
  coefficients of x can be rewritten as follows

Notice that we have only added and subtracted the
term
85
Notes on Algorithm for Drawing Contour Lines
(Cont)
Since it is clear from Step 2 that
the coefficient of x can be rewritten as
Similarly, the coefficient of y is
86
Notes on Algorithm for Drawing Contour Lines
(Cont)

• The Ni computation in Step 4 is in fact
  calculation of the coefficient of x as shown
  above. Note that NiNi-12Vi. Making the
  substitution for Ni-1, we get NiNi-22Vi-12Vi
• Repeating the same procedure of making
  substitutions for Ni-2, Ni-3, ..., we get
• NiN02V12V2...2Vi-12V1

87
Notes on Algorithm for Drawing Contour Lines
(Cont)

• Similarly, it can be verified that

88
Notes on Algorithm for Drawing Contour Lines
(Cont)

• The above expression for the total cost function
  at x, y or in fact, any other point in the region
  s, t has the form y mx c, where the slope
  m -Ns/Dt. This is exactly how the slopes
  are computed in Step 5 of the algorithm

89
Notes on Algorithm for Drawing Contour Lines
(Cont)

• 3. The lines V0, Vp1 and H0, Hq1 are required
  for defining the exterior regions 0, j, p,
  j, j 1, 2, ..., p, respectively)
• 4. Once we have determined the slopes of all
  regions, the user may choose any point (x, y)
  other than a point which minimizes the objective
  function and draw a series of contour lines in
  order to get a region which contains points, i.e.
  facility locations, yielding as good or better
  objective function values than (x, y)

90
Example 6

• Consider Example 5. Suppose that the weight of
  facility 2 is not 10, but 20. Applying the
  median method, it can be verified that the
  optimal location is (10, 10) - the centroid of
  department 2, where immovable structures exist.
  It is now desired to find a feasible and
  near-optimal location using the contour line
  method.

91
Solution

• The contour line method is illustrated using the
  figure below

92
Solution

• Step 1 The vertical and horizontal lines V1,
  V2, V2 and H1, H2, H2, H4 are drawn as shown. In
  addition to these lines, we also draw line V0, V4
  and H0, H5 so that the exterior regions can be
  identified
• Step 2 The weights V1, V2, V2, H1, H2, H2, H4
  are calculated by adding the weights of the
  points that fall on the respective lines. Note
  that for this example, p3, and q4

93
Solution
Step 3 Since
set N0 D0 -38 Step 4 Set N1 -38 2(8)
-22 D1 -38 2(6) -26 N2 -22 2(26)
30 D2 -26 2(4) -18 N3 30 2(4) 38
D3 -18 2(8) -2 D4 -2 2(20)
38 (These values are entered at the bottom of
each column and left of each row in figure 1)
94
Solution

• Step 5 Compute the slope of each region.
• S00 -(-38/-38) -1 S14 -(-22/38) 0.58
• S01 -(-38/-26) -1.46 S20 -(30/-38)
  0.79
• S02 -(-38/-18) -2.11 S21 -(30/-26)
  1.15
• S03 -(-38/-2) -19 S22 -(30/-18) 1.67
• S04 -(-38/38) 1 S23 -(30/-2) 15
• S10 -(-22/-38) -0.58 S24 -(30/38)
  -0.79
• S11 -(-22/-26) -0.85 S30 -(38/-38) 1
• S12 -(-22/-18) -1.22 S31 -(38/-26)
  1.46
• S13 -(-22/-2) -11 S32 -(38/-18) 2.11

95
Solution

• Step 5 Compute the slope of each region.
• S33 -(38/-2) 19
• S34 -(38/38) -1
• (The above slope values are shown inside each
  region.)

96
Solution

• Step 6 When we draw contour lines through point
  (9, 10), we get the region shown in the previous
  figure.
• Since the copiers cannot be placed at the (10,
  10) location, we drew contour lines through
  another nearby point (9, 10). Locating anywhere
  possible within this region give us a feasible,
  near-optimal solution.

97
11.4.3Single-facility Location Problem with
Squared Euclidean Distances
98
La Quinta Motor Inns

• Moderately priced, oriented towards business
  travelers
• Headquartered in San Antonio Texas
• Site selection - an important decision
• Regression Model based on location
  characteristics classified as
• Competitive, Demand Generators, Demographic,
  Market Awareness, and Physical

99
La Quinta Motor Inns (Cont)

• Major Profitability Factors - Market awareness,
  hotel space, local population, low unemployment,
  accessibility to downtown office space, traffic
  count, college students, presence of military
  base, median income, competitive rates

100
Gravity Method
The cost function is

• As before, we substitute wi fi ci, i 1, 2,
  ..., m and rewrite the objective function as

101
Gravity Method (Cont)

• Since the objective function can be shown to be
  convex, partially differentiating TC with respect
  to x and y, setting the resulting two equations
  to 0 and solving for x, y provides the optimal
  location of the new facility

102
Gravity Method (Cont)

• Similarly,

Thus, the optimal locations x and y are simply
the weighted averages of the x and y coordinates
of the existing facilities
103
Example 7

• Consider Example 5. Suppose the distance metric
  to be used is squared Euclidean. Determine the
  optimal location of the new facility using the
  gravity method.

104
Solution - Table 11.16
Department i xi yi wi wixi wiyi
1 10 2 6 60 12 2 10 10 10 100 100 3 8 6 8 64 48
4 12 5 4 48 20
Total 28 272 180
105
Example 6. Cont...

• If this location is not feasible, we only need to
  find another point which has the nearest
  Euclidean distance to (9.7, 6.4) and is a
  feasible location for the new facility and locate
  the copiers there

106
11.4.4WeiszfeldMethod
107
Weiszfeld Method
The objective function for the single facility
location problem with Euclidean distance can be
written as

• As before, substituting wicifi and taking the
  derivative of TC with respect to x and y yields

108
Weiszfeld Method
109
Weiszfeld Method
110
Weiszfeld Method
111
Weiszfeld Method
112
Weiszfeld Method
Step 0 Set iteration counter k 1
113
Weiszfeld Method
Step 1 Set Step 2 If xk1 xk and
yk1 yk, Stop. Otherwise, set k k 1 and go
to Step 1
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Example 8

• Consider Example 6. Assuming the distance metric
  to be used is Euclidean, determine the optimal
  location of the new facility using the Weiszfeld
  method. Data for this problem is shown in Table
  11.

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Table 11.17Coordinates and weights for4
departments
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Table 11.17
Departments xi yi wi
1 10 2 6 2 10 10 20 3 8 6 8 4 12 5 4
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Solution

• Using the gravity method, the initial seed can be
  shown to be (9.8, 7.4). With this as the
  starting solution, we can apply Step 1 of the
  Weiszfeld method repeatedly until we find that
  two consecutive x, y values are equal.

118
Summary Methods for Single-Facility, Continuous
Space Location Problems

• Problem
• Rectilinear
• Squared Euclidean
• Euclidean

• Method
• Median
• Gravity
• Weiszfeld

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Facility Location Case Study

• See Section 11.5