SEO Case Study: How I Increased SEO Traffic & Ranking by 50-60% in 6 Months
Chapter 13 heragu
1. 1
Facilities DesignFacilities Design
S.S. HeraguS.S. Heragu
Decision Sciences and EngineeringDecision Sciences and Engineering
Systems DepartmentSystems Department
Rensselaer Polytechnic InstituteRensselaer Polytechnic Institute
Troy NY 12180-3590Troy NY 12180-3590
3. 3
• 13.1 Introduction13.1 Introduction
• 13.213.2 Important Factors in LocationImportant Factors in Location
DecisionsDecisions
• 13.313.3 Techniques for Discrete SpaceTechniques for Discrete Space
Location ProblemsLocation Problems
- 13.3.1 Qualitative Analysis13.3.1 Qualitative Analysis
- 13.3.2 Quantitative Analysis13.3.2 Quantitative Analysis
- 13.3.3 Hybrid Analysis13.3.3 Hybrid Analysis
OutlineOutline
4. 4
• 13.413.4 Techniques for Continuous SpaceTechniques for Continuous Space
Location ProblemsLocation Problems
- 13.4.1 Median Method13.4.1 Median Method
- 13.4.2 Contour Line Method13.4.2 Contour Line Method
- 13.4.3 Gravity Method13.4.3 Gravity Method
- 13.4.4 Weiszfeld Method13.4.4 Weiszfeld Method
• 13.513.5 Facility Location Case StudyFacility Location Case Study
• 13.613.6 SummarySummary
• 13.713.7 Review Questions and ExercisesReview Questions and Exercises
• 13.813.8 ReferencesReferences
Outline Cont...Outline Cont...
5. 5
McDonald’sMcDonald’s
• QSCV PhilosophyQSCV Philosophy
• 11,000 restaurants (7,000 in USA, remaining11,000 restaurants (7,000 in USA, remaining
in 50 countries)in 50 countries)
• 700 seat McDonald’s in Pushkin Square,700 seat McDonald’s in Pushkin Square,
MoscowMoscow
• $60 million food plant combining a bakery,$60 million food plant combining a bakery,
lettuce plant, meat plant, chicken plant, fishlettuce plant, meat plant, chicken plant, fish
plant and a distribution center, each ownedplant and a distribution center, each owned
and operated independently at same locationand operated independently at same location
6. 6
• Food taste must be the same at anyFood taste must be the same at any
McDonald, yet food must be secured locallyMcDonald, yet food must be secured locally
• Strong logistical chain, with no weak linksStrong logistical chain, with no weak links
betweenbetween
• Close monitoring for logistical performanceClose monitoring for logistical performance
• 300 in Australia300 in Australia
• Central distribution since 1974 with the helpCentral distribution since 1974 with the help
of F.J. Walker Foods in Sydneyof F.J. Walker Foods in Sydney
• Then distribution centers opened in severalThen distribution centers opened in several
citiescities
McDonald’s cont...McDonald’s cont...
7. 7
McDonald’s cont...McDonald’s cont...
• 2000 ingredients, from 48 food plants,2000 ingredients, from 48 food plants,
shipment of 200 finished products fromshipment of 200 finished products from
suppliers to DC’s, 6 million cases of food andsuppliers to DC’s, 6 million cases of food and
paper products plus 500 operating items topaper products plus 500 operating items to
restaurants across Australiarestaurants across Australia
• Delivery of frozen, dry and chilled foodsDelivery of frozen, dry and chilled foods
twice a week to each of the 300 restaurantstwice a week to each of the 300 restaurants
98% of the time within 15 minutes of98% of the time within 15 minutes of
promised delivery time, 99.8% within 2 dayspromised delivery time, 99.8% within 2 days
of order placementof order placement
• No stockouts, but less inventoryNo stockouts, but less inventory
8. 8
IntroductionIntroduction
• Logistics management can be defined as theLogistics management can be defined as the
management of transportation andmanagement of transportation and
distribution of goods.distribution of goods.
- facility locationfacility location
- transportationtransportation
- goods handling and storage.goods handling and storage.
9. 9
Introduction Cont...Introduction Cont...
Some of the objectives in facility location
decisions:
(1) It must first be close as possible to raw(1) It must first be close as possible to raw
material sources and customers;material sources and customers;
(2) Skilled labor must be readily available in the(2) Skilled labor must be readily available in the
vicinity of a facility’s location;vicinity of a facility’s location;
(3) Taxes, property insurance, construction(3) Taxes, property insurance, construction andand
land prices must not be too “high;”land prices must not be too “high;”
(4) Utilities must be readily available at a(4) Utilities must be readily available at a
“reasonable” price;“reasonable” price;
10. 10
Introduction Cont...Introduction Cont...
• (5) Local , state and other government(5) Local , state and other government
regulations must be conducive to business;regulations must be conducive to business;
andand
(6) Business climate must be favorable and the(6) Business climate must be favorable and the
community must have adequate supportcommunity must have adequate support
services and facilities such as schools,services and facilities such as schools,
hospitals and libraries, which are importanthospitals and libraries, which are important
to employees and their families.to employees and their families.
11. 11
Introduction Cont...Introduction Cont...
Logistics management problems can beLogistics management problems can be
classified as:classified as:
(1)(1) location problems;location problems;
(2)(2) allocation problems; andallocation problems; and
(3)(3) location-allocation problems.location-allocation problems.
12. 12
List of Factors AffectingList of Factors Affecting
Location DecisionsLocation Decisions
• Proximity to raw materials sourcesProximity to raw materials sources
• Cost and availability of energy/utilitiesCost and availability of energy/utilities
• Cost, availability, skill and productivity ofCost, availability, skill and productivity of
laborlabor
• Government regulations at the federal, state,Government regulations at the federal, state,
country and local levelscountry and local levels
• Taxes at the federal, state, county and localTaxes at the federal, state, county and local
levelslevels
• InsuranceInsurance
• Construction costs, land priceConstruction costs, land price
13. 13
List of Factors AffectingList of Factors Affecting
Location Decisions Cont...Location Decisions Cont...
• Government and political stabilityGovernment and political stability
• Exchange rate fluctuationExchange rate fluctuation
• Export, import regulations, duties, and tariffsExport, import regulations, duties, and tariffs
• Transportation systemTransportation system
• Technical expertiseTechnical expertise
• Environmental regulations at the federal,Environmental regulations at the federal,
state, county and local levelsstate, county and local levels
• Support servicesSupport services
14. 14
List of Factors AffectingList of Factors Affecting
Location Decisions Cont...Location Decisions Cont...
• Community services, i.e. schools, hospitals,Community services, i.e. schools, hospitals,
recreation, etc.recreation, etc.
• WeatherWeather
• Proximity to customersProximity to customers
• Business climateBusiness climate
• Competition-related factorsCompetition-related factors
15. 15
13.213.2
Important Factors in LocationImportant Factors in Location
DecisionsDecisions
• InternationalInternational
• NationalNational
• State-wideState-wide
• Community-wideCommunity-wide
16. 16
13.3.113.3.1
Qualitative AnalysisQualitative Analysis
Step 1: List all the factors that are important,Step 1: List all the factors that are important,
i.e. have an impact on the location decision.i.e. have an impact on the location decision.
Step 2: Assign appropriate weights (typicallyStep 2: Assign appropriate weights (typically
between 0 and 1) to each factor based on thebetween 0 and 1) to each factor based on the
relative importance of each.relative importance of each.
Step 3: Assign a score (typically between 0 andStep 3: Assign a score (typically between 0 and
100) for each location with respect to each100) for each location with respect to each
factor identified in Step 1.factor identified in Step 1.
17. 17
13.3.113.3.1
Qualitative AnalysisQualitative Analysis
Step 4: Compute the weighted score for eachStep 4: Compute the weighted score for each
factor for each location by multiplying itsfactor for each location by multiplying its
weight with the corresponding score (whichweight with the corresponding score (which
were assigned Steps 2 and 3, respectively)were assigned Steps 2 and 3, respectively)
Step 5: Compute the sum of the weightedStep 5: Compute the sum of the weighted
scores for each location and choose ascores for each location and choose a
location based on these scores.location based on these scores.
18. 18
Example 1:Example 1:
•A payroll processing company has recentlyA payroll processing company has recently
won several major contracts in the midwestwon several major contracts in the midwest
region of the U.S. and central Canada andregion of the U.S. and central Canada and
wants to open a new, large facility to servewants to open a new, large facility to serve
these areas. Since customer service is ofthese areas. Since customer service is of
utmost importance, the company wants to beutmost importance, the company wants to be
as near it’s “customers” as possible.as near it’s “customers” as possible.
Preliminary investigation has shown thatPreliminary investigation has shown that
Minneapolis, Winnipeg, and Springfield, Ill.,Minneapolis, Winnipeg, and Springfield, Ill.,
would be the three most desirable locationswould be the three most desirable locations
and the payroll company has to select one ofand the payroll company has to select one of
these three.these three.
19. 19
Example 1: Cont...Example 1: Cont...
A subsequent thorough investigation of eachA subsequent thorough investigation of each
location with respect to eight important factorslocation with respect to eight important factors
has generated the raw scores and weightshas generated the raw scores and weights
listed in table 2. Using the location scoringlisted in table 2. Using the location scoring
method, determine the best location for the newmethod, determine the best location for the new
payroll processing facility.payroll processing facility.
20. 20
Solution:Solution:
Steps 1, 2, and 3 have already been completedSteps 1, 2, and 3 have already been completed
for us. We now need to compute the weightedfor us. We now need to compute the weighted
score for each location-factor pair (Step 4), andscore for each location-factor pair (Step 4), and
these weighted scores and determine thethese weighted scores and determine the
location based on these scores (Step 5).location based on these scores (Step 5).
21. 21
Table 2. Factors and Weights forTable 2. Factors and Weights for
Three LocationsThree Locations
Wt.Wt. FactorsFactors LocationLocation
Minn.Winn.Spring.Minn.Winn.Spring.
.25.25 Proximity to customersProximity to customers 9595 9090 6565
.15.15 Land/construction pricesLand/construction prices 6060 6060 9090
.15.15 Wage ratesWage rates 7070 4545 6060
.10.10 Property taxesProperty taxes 7070 9090 7070
.10.10 Business taxesBusiness taxes 8080 9090 8585
.10.10 Commercial travelCommercial travel 8080 6565 7575
23. 23
Solution: Cont...Solution: Cont...
From the analysis in Table 3, it is clear thatFrom the analysis in Table 3, it is clear that
Minneapolis would be the best location basedMinneapolis would be the best location based
on the subjective information.on the subjective information.
24. 24
Table 3. Weighted Scores for theTable 3. Weighted Scores for the
Three LocationsThree Locations
in Table 2in Table 2
Weighted Score Location
Minn. Winn. Spring.
Proximity to customers 23.75 22.5 16.25
Land/construction prices 9 9 13.5
Wage rates 10.5 6.75 9
Property taxes 7 9 8.5
Business taxes 8 9 8.5
26. 26
Solution: Cont...Solution: Cont...
Of course, as mentioned before, objectiveOf course, as mentioned before, objective
measures must be brought into considerationmeasures must be brought into consideration
especially because the weighted scores forespecially because the weighted scores for
Minneapolis and Winnipeg are close.Minneapolis and Winnipeg are close.
28. 28
General Transportation ModelGeneral Transportation Model
ParametersParameters
ccijij: cost of transporting one unit from: cost of transporting one unit from
warehouse i to customer jwarehouse i to customer j
aaii: supply capacity at warehouse i: supply capacity at warehouse i
bbii: demand at customer j: demand at customer j
Decision VariablesDecision Variables
xxijij: number of units transported from: number of units transported from
warehouse i to customer jwarehouse i to customer j
29. 29
General Transportation ModelGeneral Transportation Model
∑∑= =
=
m
i
n
j
ijij xcZ
1 1
CosttionTransportaTotalMinimize
i)seat warehounrestrictio(supplym1,2,...,i,
Subject to
1
=≤∑=
n
j
iij ax
j)marketattrequiremen(demandn1,2,...,j,
1
=≥∑=
m
i
jij bx
ns)restrictionegativity-(nonn1,2,...,ji,,0 =≥ijx
30. 30
Example 2:Example 2:
Seers Inc. has two manufacturing plants atSeers Inc. has two manufacturing plants at
Albany and Little Rock supplying CanmoreAlbany and Little Rock supplying Canmore
brand refrigerators to four distribution centersbrand refrigerators to four distribution centers
in Boston, Philadelphia, Galveston and Raleigh.in Boston, Philadelphia, Galveston and Raleigh.
Due to an increase in demand of this brand ofDue to an increase in demand of this brand of
refrigerators that is expected to last for severalrefrigerators that is expected to last for several
years into the future, Seers Inc., has decided toyears into the future, Seers Inc., has decided to
build another plant in Atlanta or Pittsburgh.build another plant in Atlanta or Pittsburgh.
The expected demand at the three distributionThe expected demand at the three distribution
centers and the maximum capacity at thecenters and the maximum capacity at the
Albany and Little Rock plants are given in TableAlbany and Little Rock plants are given in Table
4.4.
31. 31
Example 2: Cont...Example 2: Cont...
Determine which of the two locations, AtlantaDetermine which of the two locations, Atlanta
or Pittsburgh, is suitable for the new plant.or Pittsburgh, is suitable for the new plant.
Seers Inc., wishes to utilize all of the capacitySeers Inc., wishes to utilize all of the capacity
available at it’s Albany and Little Rockavailable at it’s Albany and Little Rock
LocationsLocations
32. 32
Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply
CapacityCapacity
AlbanyAlbany 1010 1515 2222 2020 250250
Little RockLittle Rock 1919 1515 1010 99 300300
AtlantaAtlanta 2121 1111 1313 66 No limitNo limit
PittsburghPittsburgh 1717 88 1818 1212 No limitNo limit
DemandDemand 200200 100100 300300 280280
Table 4. Costs, Demand andTable 4. Costs, Demand and
Supply InformationSupply Information
33. 33
Table 5. Transportation ModelTable 5. Transportation Model
with Plant at Atlantawith Plant at Atlanta
Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply
CapacityCapacity
AlbanyAlbany 1010 1515 2222 2020 250250
Little RockLittle Rock 1919 1515 1010 99 300300
AtlantaAtlanta 2121 1111 1313 66 330330
DemandDemand 200200 100100 300300 280280 880880
34. 34
Table 6. Transportation ModelTable 6. Transportation Model
with Plant at Pittsburghwith Plant at Pittsburgh
Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply
CapacityCapacity
AlbanyAlbany 1010 1515 2222 2020 250250
Little RockLittle Rock 1919 1515 1010 99 300300
PittsburghPittsburgh 1717 88 1818 1212 330330
DemandDemand 200200 100100 300300 280280 880880
37. 37
Hybrid Analysis Cont...Hybrid Analysis Cont...
CFCFijij = 1 if location i satisfies critical factor j,= 1 if location i satisfies critical factor j,
0 otherwise0 otherwise
OFOFijij = cost of objective factor j at location i= cost of objective factor j at location i
SFSFijij = numerical value assigned= numerical value assigned
(on scale of 0-1)(on scale of 0-1)
to subjective factor j for location ito subjective factor j for location i
wwjj = weight assigned to subjective factor= weight assigned to subjective factor
(0(0<< ww << 1)1)
39. 39
Hybrid Analysis Cont...Hybrid Analysis Cont...
The location measure LMThe location measure LMii for each location isfor each location is
then calculated as:then calculated as:
LMLMii = CFM= CFMii [[ αα OFMOFMii + (1-+ (1- αα) SFM) SFMii ]]
WhereWhere αα is the weight assigned to theis the weight assigned to the
objective factor.objective factor.
We then choose the location with the highestWe then choose the location with the highest
location measure LMlocation measure LMii
40. 40
Example 3:Example 3:
Mole-Sun Brewing company is evaluating sixMole-Sun Brewing company is evaluating six
candidate locations-Montreal, Plattsburgh,candidate locations-Montreal, Plattsburgh,
Ottawa, Albany, Rochester and Kingston, forOttawa, Albany, Rochester and Kingston, for
constructing a new brewery. There are twoconstructing a new brewery. There are two
critical, three objective and four subjectivecritical, three objective and four subjective
factors that management wishes to incorporatefactors that management wishes to incorporate
in its decision-making. These factors arein its decision-making. These factors are
summarized in Table 7. The weights of thesummarized in Table 7. The weights of the
subjective factors are also provided in thesubjective factors are also provided in the
table. Determine the best location if thetable. Determine the best location if the
subjective factors are to be weighted 50 percentsubjective factors are to be weighted 50 percent
more than the objective factors.more than the objective factors.
41. 41
Table 7:Table 7:
Critical, Subjective and ObjectiveCritical, Subjective and Objective
Factor Ratings for six locations forFactor Ratings for six locations for
Mole-Sun Brewing Company, Inc.Mole-Sun Brewing Company, Inc.
49. 49
13.4.1 Model for Rectilinear13.4.1 Model for Rectilinear
Metric ProblemMetric Problem
Consider the following notation:Consider the following notation:
ffii = Traffic flow between new facility and= Traffic flow between new facility and
existing facility iexisting facility i
ccii = Cost of transportation between new facility= Cost of transportation between new facility
and existing facility i per unitand existing facility i per unit
xxii, y, yii = Coordinate points of existing facility i= Coordinate points of existing facility i
50. 50
Model for Rectilinear MetricModel for Rectilinear Metric
Problem (Cont)Problem (Cont)
Where TC is the total distribution costWhere TC is the total distribution cost
∑=
−+−=
m
i
iiii yyxxfc
1
]||||[TC
The median location model is then to minimize:The median location model is then to minimize:
51. 51
Model for Rectilinear MetricModel for Rectilinear Metric
Problem (Cont)Problem (Cont)
Since the cSince the ciiffii product is known for each facility,product is known for each facility,
it can be thought of as a weight wit can be thought of as a weight wii
corresponding to facility i.corresponding to facility i.
∑ ∑= =
−+−=
m
i
m
i
iiii yywxxw
1 1
]||[]||[TCMinimize
52. 52
Median Method:Median Method:
Step 1: List the existing facilities in non-Step 1: List the existing facilities in non-
decreasing order of the x coordinates.decreasing order of the x coordinates.
Step 2: Find the jStep 2: Find the jthth
x coordinate in the list atx coordinate in the list at
which the cumulative weight equals orwhich the cumulative weight equals or
exceeds half the total weight for the firstexceeds half the total weight for the first
time, i.e.,time, i.e.,
∑ ∑∑ ∑ = =
−
= =
≥<
j
i
m
i
i
i
j
i
m
i
i
i
w
w
w
w
1 1
1
1 1 2
and
2
53. 53
Median Method (Cont)Median Method (Cont)
Step 3: List the existing facilities in non-Step 3: List the existing facilities in non-
decreasing order of the y coordinates.decreasing order of the y coordinates.
Step 4: Find the kStep 4: Find the kthth
y coordinate in the listy coordinate in the list
(created in Step 3) at which the cumulative(created in Step 3) at which the cumulative
weight equals or exceeds half the totalweight equals or exceeds half the total
weight for the first time, i.e.,weight for the first time, i.e.,
∑ ∑∑ ∑ = =
−
= =
≥<
k
i
m
i
i
i
k
i
m
i
i
i
w
w
w
w
1 1
1
1 1 2
and
2
54. 54
Median Method (Cont)Median Method (Cont)
Step 4: Cont... The optimal location of the newStep 4: Cont... The optimal location of the new
facility is given by the jfacility is given by the jthth
x coordinate and thex coordinate and the
kkthth
y coordinate identified in Steps 2 and 4,y coordinate identified in Steps 2 and 4,
respectively.respectively.
55. 55
NotesNotes
1. It can be shown that any other x or y1. It can be shown that any other x or y
coordinate will not be that of the optimalcoordinate will not be that of the optimal
location’s coordinateslocation’s coordinates
2. The algorithm determines the x and y2. The algorithm determines the x and y
coordinates of the facility’s optimal locationcoordinates of the facility’s optimal location
separatelyseparately
3. These coordinates could coincide with the x3. These coordinates could coincide with the x
and y coordinates of two different existingand y coordinates of two different existing
facilities or possibly one existing facilityfacilities or possibly one existing facility
56. 56
Example 4:Example 4:
Two high speed copiers are to be located in theTwo high speed copiers are to be located in the
fifth floor of an office complex which housesfifth floor of an office complex which houses
four departments of the Social Securityfour departments of the Social Security
Administration. Coordinates of the centroid ofAdministration. Coordinates of the centroid of
each department as well as the average numbereach department as well as the average number
of trips made per day between each departmentof trips made per day between each department
and the copiers’ yet-to-be-determined locationand the copiers’ yet-to-be-determined location
are known and given in Table 9 below. Assumeare known and given in Table 9 below. Assume
that travel originates and ends at the centroidthat travel originates and ends at the centroid
of each department. Determine the optimalof each department. Determine the optimal
location, i.e., x, y coordinates, for the copiers.location, i.e., x, y coordinates, for the copiers.
57. 57
Table 9. Centroid CoordinatesTable 9. Centroid Coordinates
and Average Number of Trips toand Average Number of Trips to
CopiersCopiers
58. 58
Table 9.Table 9.
Dept.Dept. CoordinatesCoordinates Average number ofAverage number of
## xx yy daily trips to copiersdaily trips to copiers
11 1010 22 66
22 1010 1010 1010
33 88 66 88
44 1212 55 44
59. 59
Solution:Solution:
Using the median method, we obtain theUsing the median method, we obtain the
following solution:following solution:
Step 1: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
60. 60
Solution:Solution:
Step 2: Since the second x coordinate, namelyStep 2: Since the second x coordinate, namely
10, in the above list is where the cumulative10, in the above list is where the cumulative
weight equals half the total weight of 28/2 =weight equals half the total weight of 28/2 =
14, the optimal x coordinate is 10.14, the optimal x coordinate is 10.
62. 62
Solution:Solution:
Step 4: Since the third y coordinates in theStep 4: Since the third y coordinates in the
above list is where the cumulative weightabove list is where the cumulative weight
exceeds half the total weight of 28/2 = 14, theexceeds half the total weight of 28/2 = 14, the
optimal coordinate is 6. Thus, the optimaloptimal coordinate is 6. Thus, the optimal
coordinates of the new facility are (10, 6).coordinates of the new facility are (10, 6).
63. 63
Equivalent Linear Model for theEquivalent Linear Model for the
Rectilinear Distance, Single-Rectilinear Distance, Single-
Facility Location ProblemFacility Location Problem
ParametersParameters
ffii = Traffic flow between new facility and= Traffic flow between new facility and
existing facility iexisting facility i
ccii = Unit transportation cost between new= Unit transportation cost between new
facility and existing facility ifacility and existing facility i
xxii, y, yii = Coordinate points of existing facility i= Coordinate points of existing facility i
Decision VariablesDecision Variables
x, y= Optimal coordinates of the new facilityx, y= Optimal coordinates of the new facility
TC = Total distribution costTC = Total distribution cost
64. 64
The median location model is then toThe median location model is then to
∑ ∑= =
−+−=
m
i
m
i
iiii yywxxw
1 1
]||[]||[TCMinimize
Equivalent Linear Model for theEquivalent Linear Model for the
Rectilinear Distance, Single-Rectilinear Distance, Single-
Facility Location ProblemFacility Location Problem
65. 65
Since the cSince the ciiffii product is known for each facility,product is known for each facility,
it can be thought of as a weight wit can be thought of as a weight wii
corresponding to facility i. The previouscorresponding to facility i. The previous
equation can now be rewritten as followsequation can now be rewritten as follows
Equivalent Linear Model for theEquivalent Linear Model for the
Rectilinear Distance, Single-Rectilinear Distance, Single-
Facility Location ProblemFacility Location Problem
∑ ∑= =
−+−=
m
i
m
i
iiii yywxxw
1 1
]||[]||[TCMinimize
66. 66
Equivalent Linear Model for theEquivalent Linear Model for the
Rectilinear Distance, Single-Rectilinear Distance, Single-
Facility Location ProblemFacility Location Problem
iii
iii
i
ii
i
ii
i
xxxx
xxxx
xx
xxxx
x
xxxx
x
−+
−+
−
+
−=−
+=−
≤>−
≤−−
=
>−−
=
)(
and
0,or0)(whetherthat,observecanWe
otherwise0
0if)(
otherwise0
0if)(
Define
67. 67
Equivalent Linear Model for theEquivalent Linear Model for the
Rectilinear Distance, Single-Rectilinear Distance, Single-
Facility Location ProblemFacility Location Problem
iii
iii
ii
yyyy
yyyy
yy
−+
−+
−+
−=−
+=−
)(
and
yields,ofdefinitionsimilarA
70. 70
Step 1: Draw a vertical line through the xStep 1: Draw a vertical line through the x
coordinate and a horizontal line through the ycoordinate and a horizontal line through the y
coordinate of each facilitycoordinate of each facility
Step 2: Label each vertical line VStep 2: Label each vertical line Vii, i=1, 2, ..., p, i=1, 2, ..., p
and horizontal line Hand horizontal line Hjj, j=1, 2, ..., q where V, j=1, 2, ..., q where Vii==
the sum of weights of facilities whose xthe sum of weights of facilities whose x
coordinates fall on vertical line i and wherecoordinates fall on vertical line i and where
HHjj= sum of weights of facilities whose y= sum of weights of facilities whose y
coordinates fall on horizontal line jcoordinates fall on horizontal line j
Algorithm for Drawing ContourAlgorithm for Drawing Contour
Lines:Lines:
71. 71
m
i=1
Step 3: Set i = j = 1; NStep 3: Set i = j = 1; N00 = D= D00 = w= wii
Step 4: Set NStep 4: Set Nii = N= Ni-1i-1 + 2V+ 2Vii and Dand Djj = D= Dj-1j-1 + 2H+ 2Hjj..
Increment i = i + 1 and j = j + 1Increment i = i + 1 and j = j + 1
Step 5: If iStep 5: If i << p or jp or j << q, go to Step 4. Otherwise,q, go to Step 4. Otherwise,
set i = j = 0 and determine Sset i = j = 0 and determine Sijij, the slope of, the slope of
contour lines through the region bounded bycontour lines through the region bounded by
vertical lines i and i + 1 and horizontal line jvertical lines i and i + 1 and horizontal line j
and j + 1 using the equation Sand j + 1 using the equation Sijij = -N= -Nii/D/Djj..
Increment i = i + 1 and j = j + 1Increment i = i + 1 and j = j + 1
Algorithm for Drawing ContourAlgorithm for Drawing Contour
Lines (Cont)Lines (Cont)
∑∑
72. 72
Step 6: If iStep 6: If i << p or jp or j << q, go to Step 5. Otherwiseq, go to Step 5. Otherwise
select any point (x, y) and draw a contour lineselect any point (x, y) and draw a contour line
with slope Swith slope Sijij in the region [i, j] in which (x, y)in the region [i, j] in which (x, y)
appears so that the line touches theappears so that the line touches the
boundary of this line. From one of the endboundary of this line. From one of the end
points of this line, draw another contour linepoints of this line, draw another contour line
through the adjacent region with thethrough the adjacent region with the
corresponding slopecorresponding slope
Step 7: Repeat this until you get a contour lineStep 7: Repeat this until you get a contour line
ending at point (x, y). We now have a regionending at point (x, y). We now have a region
bounded by contour lines with (x, y) on thebounded by contour lines with (x, y) on the
boundary of the regionboundary of the region
Algorithm for Drawing ContourAlgorithm for Drawing Contour
Lines:Lines:
73. 73
1. The number of vertical and horizontal lines1. The number of vertical and horizontal lines
need not be equalneed not be equal
2. The N2. The Nii and Dand Djj as computed in Steps 3 and 4as computed in Steps 3 and 4
correspond to the numerator andcorrespond to the numerator and
denominator, respectively of the slopedenominator, respectively of the slope
equation of any contour line through theequation of any contour line through the
region bounded by the vertical lines i and i +region bounded by the vertical lines i and i +
1 and horizontal lines j and j + 11 and horizontal lines j and j + 1
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour LinesContour Lines
74. 74
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
yywxxwTC
yyxx
i
m
i
ii
m
i
i −+−=
==
∑∑ == 11
,i.e.,y),(x,pointsomeatlocatedis
facilitynewhen thefunction wobjectiveheConsider t
75. 75
By noting that the VBy noting that the Vii’s and H’s and Hjj’s calculated in’s calculated in
Step 2 of the algorithm correspond to the sumStep 2 of the algorithm correspond to the sum
of the weights of facilities whose x, yof the weights of facilities whose x, y
coordinates are equal to the x, y coordinates,coordinates are equal to the x, y coordinates,
respectively of the irespectively of the ithth
, j, jthth
distinct lines and thatdistinct lines and that
we have p, q such coordinates or lines (pwe have p, q such coordinates or lines (p << m, qm, q
<< m), the previous equation can be written asm), the previous equation can be written as
followsfollows
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
yyHxxVTC i
q
i
ii
p
i
i −+−= ∑∑ == 11
76. 76
Suppose that x is between the sSuppose that x is between the sthth
and s+1and s+1thth
(distinct) x coordinates or vertical lines (since(distinct) x coordinates or vertical lines (since
we have drawn vertical lines through thesewe have drawn vertical lines through these
coordinates in Step 1). Similarly, let y becoordinates in Step 1). Similarly, let y be
between the tbetween the tthth
and t+1and t+1thth
vertical lines. Thenvertical lines. Then
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
)()()()(
1111
yyHyyHxxVxxVTC i
q
ti
ii
t
i
ii
p
si
ii
s
i
i −+−+−+−= ∑∑∑∑ +==+==
77. 77
Rearranging the variable and constant terms inRearranging the variable and constant terms in
the above equation, we getthe above equation, we get
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
i
q
ti
ii
t
i
ii
p
si
ii
s
i
i
t
i
q
ti
ii
s
i
p
si
ii
yHyHxVxV
yHHxVVTC
∑∑∑∑
∑ ∑∑ ∑
+==+==
= +== +=
+−+−
−+
−=
1111
1 11 1
78. 78
The last four terms in the previous equationThe last four terms in the previous equation
can be substituted by another constant termcan be substituted by another constant term
c and the coefficients of x can be rewrittenc and the coefficients of x can be rewritten
as followsas follows
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
∑ ∑∑ ∑ = == +=
+−−=
s
i
s
i
ii
s
i
p
si
ii VVVVTC
1 11 1
Notice that we have only added andNotice that we have only added and
subtracted this termsubtracted this term
∑=
s
i
iV
1
79. 79
Since it is clear from Step 2 that
the coefficient of x can be rewritten as
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
,
11
∑∑ ==
=
m
i
i
s
i
i wV
∑ ∑
∑ ∑∑ ∑∑
= =
= == +==
−=
−=
+−
s
i
m
i
ii
s
i
p
i
ii
s
i
p
si
i
s
i
ii
wV
VVVVV
1 1
1 11 11
2
22
Similarly, the coefficient of y is
∑ ∑= =
−
t
i
m
i
ii wH
1 1
2
80. 80
cywHxwV
t
i
m
i
ii
s
i
m
i
ii +
−+
−= ∑ ∑∑ ∑ = == = 1 11 1
22TCThus,
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
• The NThe Nii computation in Step 4 is in factcomputation in Step 4 is in fact
calculation of the coefficient of x as showncalculation of the coefficient of x as shown
above. Note that Nabove. Note that Nii=N=Ni-1i-1+2V+2Vii. M. Making theaking the
substitution for Nsubstitution for Ni-1i-1, we get N, we get Nii=N=Ni-2i-2+2V+2Vi-1i-1+2V+2Vii
• Repeating the same procedure of makingRepeating the same procedure of making
substitutions for Nsubstitutions for Ni-2i-2, N, Ni-3i-3, ..., we get, ..., we get
• NNii=N=N00+2V+2V11+2V+2V22+...+2V+...+2Vi-1i-1+2V+2V11== ∑∑ ==
+−
i
k
k
m
i
i Vw
11
2
81. 81
Similarly, it can be verified thatSimilarly, it can be verified that
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
∑∑ ==
+−=
i
k
k
m
i
ii HwD
11
2
)(
asrewrittenbecanwhich
22TCThus,
1 11 1
cTCx
D
N
y
cyDxN
cywHxwV
t
s
ts
t
i
m
i
ii
s
i
m
i
ii
−+−=
++=
+
−+
−= ∑ ∑∑ ∑ = == =
82. 82
The above expression for the total cost functionThe above expression for the total cost function
at x, y or in fact, any other point in the region [s,at x, y or in fact, any other point in the region [s,
t] has the form y= mx + c, where the slopet] has the form y= mx + c, where the slope
m = -Nm = -Nss/D/Dtt. This is exactly how the slopes are. This is exactly how the slopes are
computed in Step 5 of the algorithmcomputed in Step 5 of the algorithm
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
83. 83
3. The lines V3. The lines V00, V, Vp+1p+1 and Hand H00, H, Hq+1q+1 are required forare required for
defining the “exterior” regions [0, j], [p, j], j =defining the “exterior” regions [0, j], [p, j], j =
1, 2, ..., p, respectively)1, 2, ..., p, respectively)
4. Once we have determined the slopes of all4. Once we have determined the slopes of all
regions, the user may choose any point (x, y)regions, the user may choose any point (x, y)
other than a point which minimizes theother than a point which minimizes the
objective function and draw a series ofobjective function and draw a series of
contour lines in order to get a region whichcontour lines in order to get a region which
contains points, i.e. facility locations,contains points, i.e. facility locations,
yielding as good or better objective functionyielding as good or better objective function
values than (x, y)values than (x, y)
Notes on Algorithm for DrawingNotes on Algorithm for Drawing
Contour Lines (Cont)Contour Lines (Cont)
84. 84
Example 5:Example 5:
Consider Example 4. Suppose that the weightConsider Example 4. Suppose that the weight
of facility 2 is not 10, but 20. Applying theof facility 2 is not 10, but 20. Applying the
median method, it can be verified that themedian method, it can be verified that the
optimal location is (10, 10) - the centroid ofoptimal location is (10, 10) - the centroid of
department 2, where immovable structuresdepartment 2, where immovable structures
exist. It is now desired to find a feasible andexist. It is now desired to find a feasible and
“near-optimal” location using the contour line“near-optimal” location using the contour line
method.method.
85. 85
Solution:Solution:
The contour line method is illustrated usingThe contour line method is illustrated using
Figure 1Figure 1
Step 1: The vertical and horizontal lines VStep 1: The vertical and horizontal lines V11, V, V22,,
VV22 and Hand H11, H, H22, H, H22, H, H44 are drawn as shown. Inare drawn as shown. In
addition to these lines, we also draw line Vaddition to these lines, we also draw line V00, V, V44
and Hand H00, H, H55 so that the “exterior regions can beso that the “exterior regions can be
identifiedidentified
Step 2: The weights VStep 2: The weights V11, V, V22, V, V22, H, H11, H, H22, H, H22, H, H44 areare
calculated by adding the weights of the pointscalculated by adding the weights of the points
that fall on the respective lines. Note that forthat fall on the respective lines. Note that for
this example, p=3, and q=4this example, p=3, and q=4
86. 86
Solution: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)
38
4
1
=∑=i
iw
88. 88
Solution:Solution:
Step 5: Compute the slope of each region.Step 5: Compute the slope of each region.
SS3333 = -(38/-2) = 19;= -(38/-2) = 19;
SS3434 = -(38/38) = -1;= -(38/38) = -1;
(The above slope values are shown inside each(The above slope values are shown inside each
region.)region.)
89. 89
Solution:Solution:
Step 6: When we draw contour linesStep 6: When we draw contour lines
through point (9, 10), we get thethrough point (9, 10), we get the
region shown in figure 1.region shown in figure 1.
Since the copiers cannot be placed at theSince the copiers cannot be placed at the
(10, 10) location, we drew contour lines(10, 10) location, we drew contour lines
through another nearby point (9, 10).through another nearby point (9, 10).
Locating anywhere possible within thisLocating anywhere possible within this
region give us a feasible, near-optimalregion give us a feasible, near-optimal
solution.solution.
91. 91
La Quinta Motor InnsLa Quinta Motor Inns
Moderately priced, oriented towards businessModerately priced, oriented towards business
travelerstravelers
Headquartered in San Antonio TexasHeadquartered in San Antonio Texas
Site selection - an important decisionSite selection - an important decision
Regression Model based on locationRegression Model based on location
characteristics classified as:characteristics classified as:
- Competitive, Demand Generators,Competitive, Demand Generators,
Demographic, Market Awareness, andDemographic, Market Awareness, and
PhysicalPhysical
92. 92
La Quinta Motor Inns (Cont)La Quinta Motor Inns (Cont)
Major Profitability Factors - Market awareness,Major Profitability Factors - Market awareness,
hotel space, local population, lowhotel space, local population, low
unemployment, accessibility to downtown officeunemployment, accessibility to downtown office
space, traffic count, college students, presencespace, traffic count, college students, presence
of military base, median income, competitiveof military base, median income, competitive
ratesrates
93. 93
Gravity Method:Gravity Method:
As before, we substitute wAs before, we substitute wi = f= fii ccii, i = 1, 2, ..., m, i = 1, 2, ..., m
and rewrite the objective function asand rewrite the objective function as
[ ]∑=
−+−=
m
i
iiii yyxxfc
1
22
)()(TCMinimize
2
11
2
)()(TCMinimize yywxxw i
m
i
i
m
i
ii −+−= ∑∑ ==
The cost function isThe cost function is
94. 94
Since the objective function can be shown toSince the objective function can be shown to
be convex, partially differentiating TC withbe convex, partially differentiating TC with
respect to x and y, setting the resulting tworespect to x and y, setting the resulting two
equations to 0 and solving for x, y provides theequations to 0 and solving for x, y provides the
optimal location of the new facilityoptimal location of the new facility
Gravity Method (Cont)Gravity Method (Cont)
∑∑
∑∑
==
==
=∴
=−=
∂
∂
m
1i
m
1i
m
1i
m
1i
022
x
TC
iii
iii
wxwx
xwxw
95. 95
Similarly,Similarly,
Gravity Method (Cont)Gravity Method (Cont)
∑∑
∑∑
==
==
=∴
=−=
∂
∂
m
1i
m
1i
m
1i
m
1i
022
y
TC
iii
iii
wywy
ywyw
Thus, the optimal locations x and y are simplyThus, the optimal locations x and y are simply
the weighted averages of the x and y coordinatesthe weighted averages of the x and y coordinates
of the existing facilitiesof the existing facilities
96. 96
Example 6:Example 6:
Consider Example 4. Suppose the distanceConsider Example 4. Suppose the distance
metric to be used is squared Euclidean.metric to be used is squared Euclidean.
Determine the optimal location of the newDetermine the optimal location of the new
facility using the gravity method.facility using the gravity method.
97. 97
Solution - Table 10Solution - Table 10
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
4.628180and7.928272
thatconcludewe10,tableFrom
==== yx
98. 98
Example 6. Cont...Example 6. Cont...
If this location is not feasible, we only need toIf this location is not feasible, we only need to
find another point which has the nearestfind another point which has the nearest
Euclidean distance to (9.7, 6.4) and is a feasibleEuclidean distance to (9.7, 6.4) and is a feasible
location for the new facility and locate thelocation for the new facility and locate the
copiers therecopiers there
100. 100
Weiszfeld Method:Weiszfeld Method:
As before, substituting wAs before, substituting wii=c=ciiffii and taking theand taking the
derivative of TC with respect to x and y yieldsderivative of TC with respect to x and y yields
)y(y)x(xfcTCMinimize
m
1i
iiii
22
∑=
−+−=
The objective function for the single facilityThe objective function for the single facility
location problem with Euclidean distance canlocation problem with Euclidean distance can
be written as:be written as:
101. 101
Weiszfeld Method:Weiszfeld Method:
[ ]
∑
∑
∑
=
=
=
=
−+−
−
−+−
=
−+−
−
=
∂
∂
m
1i ii
i
m
1i ii
ii
m
1i ii
ii
0
)y(y)x(x
xw
)y(y)x(x
xw
)y(y)x(x
)x2(xw
2
1
x
TC
22
22
22
103. 103
Weiszfeld Method:Weiszfeld Method:
[ ]
∑
∑
∑
=
=
=
=
−+−
−
−+−
=
−+−
−
=
∂
∂
m
1i ii
i
m
1i ii
ii
m
1i ii
ii
0
)y(y)x(x
yw
)y(y)x(x
yw
)y(y)x(x
)y2(yw
2
1
y
TC
22
22
22
108. 108
Example 7:Example 7:
Consider Example 5. Assuming the distanceConsider Example 5. Assuming the distance
metric to be used is Euclidean, determine themetric to be used is Euclidean, determine the
optimal location of the new facility using theoptimal location of the new facility using the
Weiszfeld method. Data for this problem isWeiszfeld method. Data for this problem is
shown in Table 11.shown in Table 11.
111. 111
Solution:Solution:
Using the gravity method, the initial seed canUsing the gravity method, the initial seed can
be shown to be (9.8, 7.4). With this as thebe shown to be (9.8, 7.4). With this as the
starting solution, we can apply Step 1 of thestarting solution, we can apply Step 1 of the
Weiszfeld method repeatedly until we find thatWeiszfeld method repeatedly until we find that
two consecutive x, y values are equal.two consecutive x, y values are equal.
