Chapter 13 Heragu

Facilities DesignFacilities DesignFacilities DesignFacilities Design

S.S. HeraguS.S. Heragu

Decision Sciences and Engineering Decision Sciences and Engineering Systems DepartmentSystems Department

Rensselaer Polytechnic InstituteRensselaer Polytechnic Institute

Troy NY 12180-3590Troy NY 12180-3590

Chapter 13Chapter 13

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

Chapter 13Chapter 13

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

• 13.1 Introduction13.1 Introduction

• 13.213.2 Important Factors in Location Important Factors in Location DecisionsDecisions

• 13.313.3 Techniques for Discrete Space Techniques 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

OutlineOutlineOutlineOutline

• 13.413.4 Techniques for Continuous Space Techniques 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 Study Facility Location Case Study

• 13.613.6 Summary Summary

• 13.713.7 Review Questions and Exercises Review Questions and Exercises

• 13.813.8 References References

Outline Cont...Outline Cont...Outline Cont...Outline Cont...

McDonald’sMcDonald’sMcDonald’sMcDonald’s

• QSCV PhilosophyQSCV Philosophy

• 11,000 restaurants (7,000 in USA, remaining 11,000 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, fish lettuce plant, meat plant, chicken plant, fish plant and a distribution center, each owned plant and a distribution center, each owned and operated independently at same locationand operated independently at same location

• Food taste must be the same at any Food 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 links Strong 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 help Central distribution since 1974 with the help of F.J. Walker Foods in Sydneyof F.J. Walker Foods in Sydney

• Then distribution centers opened in several Then distribution centers opened in several citiescities

McDonald’s cont...McDonald’s cont...McDonald’s cont...McDonald’s cont...

• 2000 ingredients, from 48 food plants, 2000 ingredients, from 48 food plants, shipment of 200 finished products from shipment of 200 finished products from suppliers to DC’s, 6 million cases of food and suppliers to DC’s, 6 million cases of food and paper products plus 500 operating items to paper products plus 500 operating items to restaurants across Australiarestaurants across Australia

• Delivery of frozen, dry and chilled foods Delivery of frozen, dry and chilled foods twice a week to each of the 300 restaurants twice a week to each of the 300 restaurants 98% of the time within 15 minutes of 98% of the time within 15 minutes of promised delivery time, 99.8% within 2 days promised delivery time, 99.8% within 2 days of order placementof order placement

• No stockouts, but less inventoryNo stockouts, but less inventory

IntroductionIntroductionIntroductionIntroduction

• Logistics management can be defined as the Logistics management can be defined as the management of transportation and management of transportation and distribution of goods.distribution of goods.

- facility locationfacility location

- transportationtransportation

- goods handling and storage.goods handling and storage.

Introduction Cont...Introduction Cont...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 and land prices must not be too “high;”and 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;

Introduction Cont...Introduction Cont...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 support community must have adequate support services and facilities such as schools, services and facilities such as schools, hospitals and libraries, which are important hospitals and libraries, which are important to employees and their families.to employees and their families.

Introduction Cont...Introduction Cont...Introduction Cont...Introduction Cont...

Logistics management problems can be Logistics management problems can be classified as:classified as:

(1)(1) location problems;location problems;

(2)(2) allocation problems; and allocation problems; and

(3)(3) location-allocation problems.location-allocation problems.

List of Factors AffectingList of Factors AffectingLocation DecisionsLocation DecisionsList of Factors AffectingList of Factors AffectingLocation 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 of Cost, 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 local Taxes at the federal, state, county and local levelslevels

• InsuranceInsurance

• Construction costs, land priceConstruction costs, land price

List of Factors AffectingList of Factors AffectingLocation Decisions Cont...Location Decisions Cont...List of Factors AffectingList of Factors AffectingLocation 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

List of Factors AffectingList of Factors AffectingLocation Decisions Cont...Location Decisions Cont...List of Factors AffectingList of Factors AffectingLocation 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

13.213.2Important Factors in Location Important Factors in Location DecisionsDecisions

• InternationalInternational

• NationalNational

• State-wideState-wide

• Community-wideCommunity-wide

13.3.113.3.1Qualitative AnalysisQualitative Analysis13.3.113.3.1Qualitative AnalysisQualitative AnalysisStep 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 (typically Step 2: Assign appropriate weights (typically between 0 and 1) to each factor based on the between 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 and Step 3: Assign a score (typically between 0 and 100) for each location with respect to each 100) for each location with respect to each factor identified in Step 1.factor identified in Step 1.

13.3.113.3.1Qualitative AnalysisQualitative Analysis13.3.113.3.1Qualitative AnalysisQualitative AnalysisStep 4: Compute the weighted score for each Step 4: Compute the weighted score for each

factor for each location by multiplying its factor for each location by multiplying its weight with the corresponding score (which weight 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 weighted Step 5: Compute the sum of the weighted scores for each location and choose a scores for each location and choose a location based on these scores.location based on these scores.

Example 1:Example 1:Example 1:Example 1:•A payroll processing company has recently A payroll processing company has recently won several major contracts in the midwest won several major contracts in the midwest region of the U.S. and central Canada and wants region of the U.S. and central Canada and wants to open a new, large facility to serve these to open a new, large facility to serve these areas. Since customer service is of utmost areas. Since customer service is of utmost importance, the company wants to be as near importance, the company wants to be as near it’s “customers” as possible. Preliminary it’s “customers” as possible. Preliminary investigation has shown that Minneapolis, investigation has shown that Minneapolis, Winnipeg, and Springfield, Ill., would be the Winnipeg, and Springfield, Ill., would be the three most desirable locations and the payroll three most desirable locations and the payroll company has to select one of these three.company has to select one of these three.

Example 1: Cont...Example 1: Cont...Example 1: Cont...Example 1: Cont...

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

Solution:Solution:Solution:Solution:

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

Table 2. Factors and Weights for Table 2. Factors and Weights for Three LocationsThree LocationsTable 2. Factors and Weights for Table 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

Table 2. Cont...Table 2. Cont...Table 2. Cont...Table 2. Cont...

Wt.Wt. FactorsFactors LocationLocation

Minn.Minn. Winn.Winn. Spring.Spring.

.08.08 Insurance costsInsurance costs 7070 9595 6060

.07.07 Office servicesOffice services 9090 9090 8080

Solution: Cont...Solution: Cont...Solution: Cont...Solution: Cont...

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

Table 3. Weighted Scores for the Table 3. Weighted Scores for the Three Locations Three 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

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

Table 3. Cont... Table 3. Cont... Table 3. Cont... Table 3. Cont...

Minn. Winn. Spring.

Commercial travel 8 6.5 7.5

Insurance costs 5.6 7.6 4.8

Office services 6.3 6.3 5.6

Minn. Winn. Spring.

Commercial travel 8 6.5 7.5

Insurance costs 5.6 7.6 4.8

Office services 6.3 6.3 5.6

Solution: Cont...Solution: Cont...Solution: Cont...Solution: Cont...

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

13.3.213.3.2Quantitative Quantitative

AnalysisAnalysis

13.3.213.3.2Quantitative Quantitative

AnalysisAnalysis

General Transportation ModelGeneral Transportation ModelGeneral 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

General Transportation ModelGeneral Transportation ModelGeneral Transportation ModelGeneral Transportation Model

jijij xcZ

Costtion Transporta Total Minimize

i) seat warehoun restrictio(supply m1,2,...,i ,

Subject to

jiij ax

j)market at t requiremen (demandn 1,2,...,j ,1

ijij bx

ns)restrictio negativity-(nonn 1,2,...,ji, ,0 ijx

Example 2:Example 2:Example 2:Example 2:Seers Inc. has two manufacturing plants at Seers Inc. has two manufacturing plants at Albany and Little Rock supplying Canmore Albany and Little Rock supplying Canmore brand refrigerators to four distribution centers in brand refrigerators to four distribution centers in Boston, Philadelphia, Galveston and Raleigh. Boston, Philadelphia, Galveston and Raleigh. Due to an increase in demand of this brand of Due to an increase in demand of this brand of refrigerators that is expected to last for several refrigerators that is expected to last for several years into the future, Seers Inc., has decided to years into the future, Seers Inc., has decided to build another plant in Atlanta or Pittsburgh. The build another plant in Atlanta or Pittsburgh. The expected demand at the three distribution expected demand at the three distribution centers and the maximum capacity at the Albany centers and the maximum capacity at the Albany and Little Rock plants are given in Table 4. and Little Rock plants are given in Table 4.

Example 2: Cont...Example 2: Cont...Example 2: Cont...Example 2: Cont...

Determine which of the two locations, Atlanta Determine 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 capacity Seers Inc., wishes to utilize all of the capacity available at it’s Albany and Little Rock available at it’s Albany and Little Rock LocationsLocations

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 and Table 4. Costs, Demand and Supply InformationSupply InformationTable 4. Costs, Demand and Table 4. Costs, Demand and Supply InformationSupply Information

Table 5. Transportation Model Table 5. Transportation Model with Plant at Atlantawith Plant at AtlantaTable 5. Transportation Model Table 5. Transportation Model with Plant at Atlantawith Plant at Atlanta

CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250

AtlantaAtlanta 2121 1111 1313 66 330330

DemandDemand 200200 100100 300300 280280 880880

Table 6. Transportation Model Table 6. Transportation Model with Plant at Pittsburghwith Plant at PittsburghTable 6. Transportation Model Table 6. Transportation Model with Plant at Pittsburghwith Plant at Pittsburgh

CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250

PittsburghPittsburgh 1717 88 1818 1212 330330

DemandDemand 200200 100100 300300 280280 880880

Min/Max Location Problem: Min/Max Location Problem: Min/Max Location Problem: Min/Max Location Problem:

Location

d11 d12

d21 d22

dm1 dm2 dmn

13.3.3 13.3.3 Hybrid AnalysisHybrid Analysis13.3.3 13.3.3 Hybrid AnalysisHybrid Analysis

• CriticalCritical

• ObjectiveObjective

• SubjectiveSubjective

Hybrid Analysis Cont...Hybrid Analysis Cont...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<< w w << 1) 1)

miSFwSFM

,...,2,1 ,

minmax

CFCFCFCFCFMp

jijipiii

,...,2,1

The location measure LMThe location measure LMii for each location is for each location is

then calculated as:then calculated as:

LMLMii = CFM = CFMii [ [ OFM OFMi i + (1- + (1- ) SFM) SFMii ] ]

Where Where is the weight assigned to the is the weight assigned to the objective factor.objective factor.

We then choose the location with the highest We then choose the location with the highest location measure LMlocation measure LMii

Example 3:Example 3:Example 3:Example 3:Mole-Sun Brewing company is evaluating six Mole-Sun Brewing company is evaluating six candidate locations-Montreal, Plattsburgh, Ottawa, candidate locations-Montreal, Plattsburgh, Ottawa, Albany, Rochester and Kingston, for constructing Albany, Rochester and Kingston, for constructing a new brewery. There are two critical, three a new brewery. There are two critical, three objective and four subjective factors that objective and four subjective factors that management wishes to incorporate in its decision-management wishes to incorporate in its decision-making. These factors are summarized in Table 7. making. These factors are summarized in Table 7. The weights of the subjective factors are also The weights of the subjective factors are also provided in the table. Determine the best location provided in the table. Determine the best location if the subjective factors are to be weighted 50 if the subjective factors are to be weighted 50 percent more than the objective factors.percent more than the objective factors.

Table 7:Table 7:Critical, Subjective and Objective Critical, Subjective and Objective Factor Ratings for six locations for Factor Ratings for six locations for Mole-Sun Brewing Company, Inc.Mole-Sun Brewing Company, Inc.

FactorsFactorsLocation

Albany 0 1

Kingston 1 1

Montreal 1 1

Ottawa 1 0

Plattsburgh 1 1

Rochester 1 1

Location

Albany 0 1

Kingston 1 1

Montreal 1 1

Ottawa 1 0

Plattsburgh 1 1

Rochester 1 1

CriticalCritical

Supply

Incentives

Table 7. Cont...Table 7. Cont...

FactorsFactorsLocation

Albany 185 80 10

Kingston 150 100 15

Montreal 170 90 13

Ottawa 200 100 15

Plattsburgh 140 75 8

Rochester 150 75 11

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

CriticalCritical

Energy

ObjectiveObjective

RevenueRevenue

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

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

Table 7. Cont...Table 7. Cont...Table 7. Cont...Table 7. Cont...FactorsFactors

Ease of

Transportation

Ease of

Transportation

SubjectiveSubjective

Community

Attitude

Community

Attitude

Table 7. Cont...Table 7. Cont...Table 7. Cont...Table 7. Cont...FactorsFactorsLocation

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

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

Support

Services

Support

Services

Unionization

Table 8. Location Analysis of Table 8. Location Analysis of Mole-Sun Brewing Company, Mole-Sun Brewing Company,

Inc., Using Hybrid MethodInc., Using Hybrid Method

Table 8. Location Analysis of Table 8. Location Analysis of Mole-Sun Brewing Company, Mole-Sun Brewing Company,

Inc., Using Hybrid MethodInc., Using Hybrid Method

Location

Albany -95 0.7 0

Kingston -35 0.67 0.4

Montreal -67 0.53 0.53

Ottawa -85 0.45 0

Plattsburgh -57 0.88 0.68

Rochester -64 0.61 0.56

Location

Albany -95 0.7 0

Kingston -35 0.67 0.4

Montreal -67 0.53 0.53

Ottawa -85 0.45 0

Plattsburgh -57 0.88 0.68

Rochester -64 0.61 0.56

FactorsFactors

SFMiSFMi

Sum of

Obj. Factors

Sum of

Obj. Factors

CriticalCritical ObjectiveObjective LMi

13.413.4Techniques For Techniques For

Continuous Space Location ProblemsContinuous Space Location Problems

13.413.4Techniques For Techniques For

Continuous Space Location ProblemsContinuous Space Location Problems

13.4.1 Model for Rectilinear 13.4.1 Model for Rectilinear Metric ProblemMetric Problem13.4.1 Model for Rectilinear 13.4.1 Model for Rectilinear Metric ProblemMetric ProblemConsider the following notation:Consider the following notation:

ffi i = 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

Model for Rectilinear Metric Model for Rectilinear Metric Problem (Cont)Problem (Cont)Model for Rectilinear Metric Model for Rectilinear Metric Problem (Cont)Problem (Cont)

Where TC is the total distribution costWhere TC is the total distribution cost

iiiii yyxxfc

]||||[ TC

The median location model is then to minimize:The median location model is then to minimize:

Model for Rectilinear Metric Model for Rectilinear Metric Problem (Cont)Problem (Cont)Model for Rectilinear Metric Model 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.

iiiii yywxxw

]||[]||[ TC Minimize

Median Method:Median Method: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 at x coordinate in the list at which the cumulative weight equals or which the cumulative weight equals or exceeds half the total weight for the first exceeds half the total weight for the first time, i.e.,time, i.e.,

1 1 2 and

Median Method (Cont)Median Method (Cont)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 list y 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 total weight equals or exceeds half the total weight for the first time, i.e.,weight for the first time, i.e.,

1 1 2 and

Median Method (Cont)Median Method (Cont)Median Method (Cont)Median Method (Cont)

Step 4: Cont... The optimal location of the new Step 4: Cont... The optimal location of the new facility is given by the jfacility is given by the jthth x coordinate and the x coordinate and the kkthth y coordinate identified in Steps 2 and 4, y coordinate identified in Steps 2 and 4, respectively.respectively.

NotesNotesNotesNotes

1. It can be shown that any other x or y 1. It can be shown that any other x or y coordinate will not be that of the optimal coordinate will not be that of the optimal location’s coordinateslocation’s coordinates

2. The algorithm determines the x and y 2. The algorithm determines the x and y coordinates of the facility’s optimal location coordinates of the facility’s optimal location separatelyseparately

3. These coordinates could coincide with the x 3. These coordinates could coincide with the x and y coordinates of two different existing and y coordinates of two different existing facilities or possibly one existing facilityfacilities or possibly one existing facility

Example 4:Example 4:Example 4:Example 4:

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

Table 9. Centroid Coordinates Table 9. Centroid Coordinates and Average Number of Trips to and Average Number of Trips to

CopiersCopiers

Table 9. Centroid Coordinates Table 9. Centroid Coordinates and Average Number of Trips to and Average Number of Trips to

CopiersCopiers

Table 9.Table 9.Table 9.Table 9.

Dept.Dept. Coordinates Coordinates Average number ofAverage number of

## xx yy daily trips to copiers daily trips to copiers

11 1010 22 66

22 1010 1010 1010

33 88 66 88

44 1212 55 44

Using the median method, we obtain the Using the median method, we obtain the following solution:following solution:

Step 1:Step 1:

Dept. x coordinates in Weights Cumulative Dept. x coordinates in Weights Cumulative # non-decreasing order Weights# non-decreasing order Weights

3 8 8 8

1 10 6 14

2 10 10 24

4 12 4 28

3 8 8 8

1 10 6 14

2 10 10 24

4 12 4 28

Step 2: Since the second x coordinate, namely Step 2: Since the second x coordinate, namely 10, in the above list is where the cumulative 10, 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.

Step 3: Step 3:

Dept. x coordinates in Weights Cumulative Dept. x coordinates in Weights Cumulative # non-decreasing order Weights# non-decreasing order Weights

1 2 6 6

4 5 4 10

3 6 8 18

2 10 10 28

1 2 6 6

4 5 4 10

3 6 8 18

2 10 10 28

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

Equivalent Linear Model for the Equivalent 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

The median location model is then toThe median location model is then to

iiiii yywxxw

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 previous corresponding to facility i. The previous equation can now be rewritten as followsequation can now be rewritten as follows

iiiii yywxxw

0, or 0)( whether that,observecan We

otherwise 0

0 if )(

otherwise 0

0 if )(

Define

yields , of definitionsimilar A

iiiii yyxxw1

)( Minimize

ModelLinear dTransforme

signin edunrestrict ,,

n1,2,...,i 0, ,,,

n1,2,...,i ,-)(

Subject to

13.4.2 13.4.2 Contour Line MethodContour Line Method

Step 1: Draw a vertical line through the x Step 1: Draw a vertical line through the x coordinate and a horizontal line through the y coordinate 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 x the sum of weights of facilities whose x coordinates fall on vertical line i and where coordinates 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 Contour Algorithm for Drawing Contour Lines:Lines:Algorithm for Drawing Contour Algorithm for Drawing Contour Lines:Lines:

i=1i=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-1 i-1 + 2V+ 2Vii and D and 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 i Step 5: If i << p or j p 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 by contour lines through the region bounded by vertical lines i and i + 1 and horizontal line j vertical 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 Contour Algorithm for Drawing Contour Lines (Cont)Lines (Cont)Algorithm for Drawing Contour Algorithm for Drawing Contour Lines (Cont)Lines (Cont)

Step 6: If i Step 6: If i << p or j p or j << q, go to Step 5. Otherwise q, go to Step 5. Otherwise select any point (x, y) and draw a contour line select 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 the boundary appears so that the line touches the boundary of this line. From one of the end points of this of this line. From one of the end points of this line, draw another contour line through the line, draw another contour line through the adjacent region with the corresponding slopeadjacent region with the corresponding slope

Step 7: Repeat this until you get a contour line Step 7: Repeat this until you get a contour line ending at point (x, y). We now have a region ending at point (x, y). We now have a region bounded by contour lines with (x, y) on the bounded by contour lines with (x, y) on the boundary of the regionboundary of the region

Algorithm for Drawing Contour Algorithm for Drawing Contour Lines:Lines:Algorithm for Drawing Contour Algorithm for Drawing Contour Lines:Lines:

1. The number of vertical and horizontal lines 1. The number of vertical and horizontal lines need not be equalneed not be equal

2. The N2. The Nii and D and Djj as computed in Steps 3 and 4 as computed in Steps 3 and 4

correspond to the numerator and correspond to the numerator and denominator, respectively of the slope denominator, respectively of the slope equation of any contour line through the equation 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 Drawing Notes on Algorithm for Drawing Contour LinesContour LinesNotes on Algorithm for Drawing Notes on Algorithm for Drawing Contour LinesContour Lines

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

yywxxwTC

, i.e., y),(x,point someat located is

facility new hen thefunction w objective heConsider t

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 sum Step 2 of the algorithm correspond to the sum of the weights of facilities whose x, y of 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 that distinct lines and that we have p, q such coordinates or lines (p we have p, q such coordinates or lines (p << m, q m, q << m), the previous equation can be written as m), the previous equation can be written as followsfollows

yyHxxVTC i

Suppose that x is between the sSuppose that x is between the sthth and s+1 and s+1thth (distinct) x coordinates or vertical lines (since (distinct) x coordinates or vertical lines (since we have drawn vertical lines through these we have drawn vertical lines through these coordinates in Step 1). Similarly, let y be coordinates in Step 1). Similarly, let y be between the tbetween the tthth and t+1 and t+1thth vertical lines. Then vertical lines. Then

)()()()(1111

yyHyyHxxVxxVTC i

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

yHyHxVxV

yHHxVVTC

1 11 1

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

siii VVVVTC

1 11 1

Notice that we have only added and Notice that we have only added and subtracted this termsubtracted this term

Since it is clear from Step 2 thatSince it is clear from Step 2 that

the coefficient of x can be rewritten asthe coefficient of x can be rewritten as

1 11 11

Similarly, the coefficient of y isSimilarly, the coefficient of y is

iii wH

cywHxwVt

1 11 1

22TC Thus,

• The NThe Nii computation in Step 4 is in fact calculation computation in Step 4 is in fact calculation

of the coefficient of x as shown above. Note that of the coefficient of x as shown above. Note that NNii=N=Ni-1i-1+2V+2Vii. Making the substitution for N. Making the substitution for Ni-1i-1, we get , we get

NNii=N=Ni-2i-2+2V+2Vi-1i-1+2V+2Vii

• Repeating the same procedure of making Repeating 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==

Similarly, it can be verified thatSimilarly, it can be verified that

iii HwD

asrewritten becan which

22TC Thus,1 11 1

cywHxwV

The above expression for the total cost function The above expression for the total cost function at x, y or in fact, any other point in the region at x, y or in fact, any other point in the region [s, t] has the form y= mx + c, where the slope [s, t] has the form y= mx + c, where the slope m = -N m = -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

3. The lines V3. The lines V00, V, Vp+1p+1 and H and H00, H, Hq+1q+1 are required for are 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 all 4. 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 the other than a point which minimizes the objective function and draw a series of objective function and draw a series of contour lines in order to get a region which contour 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 function yielding as good or better objective function values than (x, y)values than (x, y)

Consider Example 4. Suppose that the weight Consider Example 4. Suppose that the weight of facility 2 is not 10, but 20. Applying the of facility 2 is not 10, but 20. Applying the median method, it can be verified that the median method, it can be verified that the optimal location is (10, 10) - the centroid of optimal location is (10, 10) - the centroid of department 2, where immovable structures department 2, where immovable structures exist. It is now desired to find a feasible and exist. 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.

The contour line method is illustrated using The 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, V, V22

and Hand H11, H, H22, H, H22, H, H44 are drawn as shown. In addition are drawn as shown. In addition

to these lines, we also draw line Vto these lines, we also draw line V00, V, V44 and H and H00, H, H55

so that the “exterior regions can be identifiedso that the “exterior regions can be identified

Step 2: The weights VStep 2: The weights V11, V, V22, V, V22, H, H11, H, H22, H, H22, H, H44 are are

calculated by adding the weights of the points calculated by adding the weights of the points that fall on the respective lines. Note that for this that fall on the respective lines. Note that for this example, p=3, and q=4example, p=3, and q=4

Step 3: SinceStep 3: Since

set N0 = D0 = -38

Step 4: SetN1 = -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)

set N0 = D0 = -38

Step 4: SetN1 = -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)

Solution:Solution:Solution:Solution:Step 5: Compute the slope of each region.Step 5: Compute the slope of each region.

SS0000 = -(-38/-38) = -1; = -(-38/-38) = -1; SS1414 = -(-22/38) = 0.58; = -(-22/38) = 0.58;

SS0101 = -(-38/-26) = -1.46; = -(-38/-26) = -1.46; SS2020 = -(30/-38) = 0.79; = -(30/-38) = 0.79;

SS0202 = -(-38/-18) = -2.11; = -(-38/-18) = -2.11; SS2121 = -(30/-26) = 1.15; = -(30/-26) = 1.15;

SS0303 = -(-38/-2) = -19; = -(-38/-2) = -19; SS2222 = -(30/-18) = 1.67; = -(30/-18) = 1.67;

SS0404 = -(-38/38) = 1; = -(-38/38) = 1; SS2323 = -(30/-2) = 15; = -(30/-2) = 15;

SS1010 = -(-22/-38) = -0.58; = -(-22/-38) = -0.58; SS2424 = -(30/38) = -0.79; = -(30/38) = -0.79;

SS1111 = -(-22/-26) = -0.85; = -(-22/-26) = -0.85; SS3030 = -(38/-38) = 1; = -(38/-38) = 1;

SS1212 = -(-22/-18) = -1.22; = -(-22/-18) = -1.22; SS3131 = -(38/-26) = 1.46; = -(38/-26) = 1.46;

SS1313 = -(-22/-2) = -11; = -(-22/-2) = -11; SS3232 = -(38/-18) = 2.11; = -(38/-18) = 2.11;

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.)

Solution:Solution:Solution:Solution:Step 6: When we draw contour lines Step 6: When we draw contour lines

through point (9, 10), we get the through point (9, 10), we get the region shown in figure 1.region shown in figure 1.

Since the copiers cannot be placed at the Since 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 this Locating anywhere possible within this region give us a feasible, near-optimal region give us a feasible, near-optimal solution.solution.

13.4.313.4.3Single-facility Location Problem with Single-facility Location Problem with

Squared Euclidean DistancesSquared Euclidean Distances

13.4.313.4.3Single-facility Location Problem with Single-facility Location Problem with

Squared Euclidean DistancesSquared Euclidean Distances

La Quinta Motor InnsLa Quinta Motor InnsLa Quinta Motor InnsLa Quinta Motor Inns

Moderately priced, oriented towards business Moderately 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 location Regression Model based on location characteristics classified as:characteristics classified as:

- Competitive, Demand Generators, Competitive, Demand Generators, Demographic, Market Awareness, and Demographic, Market Awareness, and PhysicalPhysical

La Quinta Motor Inns (Cont)La Quinta Motor Inns (Cont)La Quinta Motor Inns (Cont)La Quinta Motor Inns (Cont)

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

Gravity Method:Gravity Method:Gravity Method:Gravity Method:

As before, we substitute wAs before, we substitute wi = f= fii c cii, i = 1, 2, ..., m , i = 1, 2, ..., m

and rewrite the objective function asand rewrite the objective function as

iiiii yyxxfc

22 )()( TC Minimize

2 )()( TC Minimize yywxxw i

The cost function isThe cost function is

Since the objective function can be shown to Since the objective function can be shown to be convex, partially differentiating TC with be convex, partially differentiating TC with respect to x and y, setting the resulting two respect to x and y, setting the resulting two equations to 0 and solving for x, y provides the equations 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)Gravity Method (Cont)Gravity Method (Cont)

Similarly,Similarly,

Gravity Method (Cont)Gravity Method (Cont)Gravity Method (Cont)Gravity Method (Cont)

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

Consider Example 4. Suppose the distance Consider 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 new Determine the optimal location of the new facility using the gravity method.facility using the gravity method.

Solution - Table 10Solution - Table 10Solution - Table 10Solution - Table 10

Department i xi yi wi wixi wiyiDepartment 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

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 180Total 28 272 180

4.628180 and 7.928272

thatconclude we10, tableFrom

Example 6. Cont...Example 6. Cont...Example 6. Cont...Example 6. Cont...

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

13.4.413.4.4WeiszfeldWeiszfeldMethodMethod

Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:

As before, substituting wAs before, substituting wii=c=ciiffii and taking the and taking the

derivative of TC with respect to x and y yieldsderivative of TC with respect to x and y yields

)y(y)x(xfc TC Minimizem

1iiiii

The objective function for the single facility The objective function for the single facility location problem with Euclidean distance can location problem with Euclidean distance can be written as:be written as:

0)y(y)x(x

)y(y)x(x

)x2(xw

)y(y)x(x

0)y(y)x(x

)y(y)x(x

)y2(yw

)y(y)x(x

Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:Step 0: Set iteration counter k = 1; Step 0: Set iteration counter k = 1;

Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:Weiszfeld Method:Step 1: SetStep 1: Set

)y(y)x(x

• Step 2: If xStep 2: If xk+1k+1 = x = xkk and y and yk+1k+1 = y = ykk, Stop. Otherwise, set k , Stop. Otherwise, set k = k + 1 and go to Step 1= k + 1 and go to Step 1

Consider Example 5. Assuming the distance Consider Example 5. Assuming the distance metric to be used is Euclidean, determine the metric to be used is Euclidean, determine the optimal location of the new facility using the optimal location of the new facility using the Weiszfeld method. Data for this problem is Weiszfeld method. Data for this problem is shown in Table 11.shown in Table 11.

Table 11.Table 11.Coordinates and weights forCoordinates and weights for

4 departments4 departments

Table 11.Table 11.Coordinates and weights forCoordinates and weights for

4 departments4 departments

Table 11:Table 11:Table 11:Table 11:

Departments # xi yi wiDepartments # xi yi wi

1 10 2 6

2 10 10 20

3 8 6 8

4 12 5 4

1 10 2 6

2 10 10 20

3 8 6 8

4 12 5 4

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

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