Chapter 11 Heragu

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Facilities DesignFacilities DesignFacilities DesignFacilities Design

S.S. HeraguS.S. Heragu

Industrial Engineering DepartmentIndustrial Engineering Department

University of LouisvilleUniversity of Louisville

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Chapter 11Chapter 11

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

Chapter 11Chapter 11

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

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• 11.1 Introduction11.1 Introduction

• 11.211.2 Important Factors in Location Important Factors in Location DecisionsDecisions

• 11.311.3 Techniques for Discrete Space Techniques for Discrete Space Location ProblemsLocation Problems

- 11.3.1 Qualitative Analysis11.3.1 Qualitative Analysis

- 11.3.2 Quantitative Analysis11.3.2 Quantitative Analysis

- 11.3.3 Hybrid Analysis11.3.3 Hybrid Analysis

OutlineOutlineOutlineOutline

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• 11.411.4 Techniques for Continuous Space Techniques for Continuous Space Location ProblemsLocation Problems

- 11.4.1 Median Method11.4.1 Median Method

- 11.4.2 Contour Line Method11.4.2 Contour Line Method

- 11.4.3 Gravity Method11.4.3 Gravity Method

- 11.4.4 Weiszfeld Method11.4.4 Weiszfeld Method

• 11.511.5 Facility Location Case Study Facility Location Case Study

• 11.611.6 Summary Summary

• 11.711.7 Review Questions and Exercises Review Questions and Exercises

• 11.811.8 References References

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

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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

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• 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...

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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

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Entities in a Supply ChainEntities in a Supply ChainEntities in a Supply ChainEntities in a Supply Chain

Supplier

Supplier

Manufacturing

Plant

Manufacturing

Plant

Raw Material(s)

Assembly Plant

Central Distribution Center(s)

Regional Distribution Center(s)

Regional Distribution Center(s)

Retail Outlets

Retail Outlets

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IntroductionIntroductionIntroductionIntroduction

• Design and Operation of a Supply chainDesign and Operation of a Supply chain

- WarehousingWarehousing

- Distribution ChannelsDistribution Channels

- Freight TransportationFreight Transportation

- Freight ConsolidationFreight Consolidation

- Transportation ModesTransportation Modes

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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.

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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;

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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.

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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.

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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

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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

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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

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11.211.2Important Factors in Location Important Factors in Location DecisionsDecisions

11.211.2Important Factors in Location Important Factors in Location DecisionsDecisions

• InternationalInternational

• NationalNational

• State-wideState-wide

• Community-wideCommunity-wide

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11.3.111.3.1Qualitative AnalysisQualitative Analysis11.3.111.3.1Qualitative 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.

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11.3.111.3.1Qualitative AnalysisQualitative Analysis11.3.111.3.1Qualitative 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.

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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.

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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.

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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).

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Table 11.2. Factors and Weights Table 11.2. Factors and Weights for Three Locationsfor Three LocationsTable 11.2. Factors and Weights Table 11.2. Factors and Weights for Three Locationsfor Three 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

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Table 11.2. Cont...Table 11.2. Cont...Table 11.2. Cont...Table 11.2. 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

Click here

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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.

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Table 11.3. Weighted Scores for Table 11.3. Weighted Scores for the Three Locations in Table 11.2the Three Locations in Table 11.2Table 11.3. Weighted Scores for Table 11.3. Weighted Scores for the Three Locations in Table 11.2the Three Locations in Table 11.2

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

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Table 11.3. Cont... Table 11.3. Cont... Table 11.3. Cont... Table 11.3. 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

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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.

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11.3.211.3.2Quantitative Quantitative

AnalysisAnalysis

11.3.211.3.2Quantitative Quantitative

AnalysisAnalysis

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General Transportation ModelGeneral Transportation ModelGeneral Transportation ModelGeneral Transportation Model

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ParametersParameters

ccijij: cost of transporting one unit from : cost of transporting one unit from

warehouse warehouse ii to customer to customer jj

aaii: supply capacity at warehouse : supply capacity at warehouse ii

bbii: demand at customer : demand at customer jj

Decision VariablesDecision Variables

xxijij: number of units transported from : number of units transported from

warehouse warehouse ii to customer to customer jj

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m

i

n

jijij xcZ

1 1

Costtion Transporta Total Minimize

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

Subject to

1

n

jiij ax

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

m

ijij bx

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

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Transportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex AlgorithmStep 1: Check whether the transportation problem is balanced or

unbalanced. If balanced, go to step 2. Otherwise, transform the unbalanced transportation problem into a balanced one by adding a dummy plant (if the total demand exceeds the total supply) or a dummy warehouse (if the total supply exceeds the total demand) with a capacity or demand equal to the excess demand or excess supply, respectively. Transform all the > and < constraints to equalities.

Step 2: Set up a transportation tableau by creating a row corresponding to each plant including the dummy plant and a column corresponding to each warehouse including the dummy warehouse. Enter the cost of transporting a unit from each plant to each warehouse (cij) in the corresponding cell (i,j). Enter 0 cost for all the cells in the dummy row or column. Enter the supply capacity of each plant at the end of the corresponding row and the demand at each warehouse at the bottom of the corresponding column. Set m and n equal to the number of rows and columns, respectively and all xij=0, i=1,2,...,m; and j=1,2,...,n.

Step 3: Construct a basic feasible solution using the Northwest corner method.

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Transportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex Algorithm

Step 4:Step 4: Set Set uu11=0 and find =0 and find vvjj, , jj=1,2,...,=1,2,...,nn and and uuii, , ii=1,2,...,=1,2,...,nn using using the formula the formula uuii + + vvjj = = ccijij for all basic variables. for all basic variables.

Step 5:Step 5: If If uuii + + vvjj - - ccijij << 0 for all nonbasic variables, then the 0 for all nonbasic variables, then the current basic feasible solution is optimal; stop. Otherwise, current basic feasible solution is optimal; stop. Otherwise, go to step 6.go to step 6.

Step 6:Step 6: Select the variable Select the variable xxi*j*i*j* with the most positive value with the most positive value uui*i* + + vvj*-j*- ccij*ij*. Construct a . Construct a closed loopclosed loop consisting of horizontal consisting of horizontal and vertical segments connecting the corresponding cell in and vertical segments connecting the corresponding cell in row row i*i* and column and column j*j* to other basic variables. Adjust the to other basic variables. Adjust the values of the basic variables in this closed loop so that the values of the basic variables in this closed loop so that the supply and demand constraints of each row and column are supply and demand constraints of each row and column are satisfied and the maximum possible value is added to the satisfied and the maximum possible value is added to the cell in row cell in row i*i* and column and column j*j*. The variable . The variable xxi*j*i*j* is now a basic is now a basic variable and the basic variable in the closed loop which now variable and the basic variable in the closed loop which now takes on a value of 0 is a nonbasic variable. Go to step 4.takes on a value of 0 is a nonbasic variable. Go to step 4.

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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 brand 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 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. The expected build another plant in Atlanta. The expected demand at the three distribution centers and demand at the three distribution centers and the maximum capacity at the Albany and Little the maximum capacity at the Albany and Little Rock plants are given in Table 4. Rock plants are given in Table 4.

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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

DemandDemand 200200 100100 300300 280280

Table 11.4. Costs, Demand and Table 11.4. Costs, Demand and Supply InformationSupply InformationTable 11.4. Costs, Demand and Table 11.4. Costs, Demand and Supply InformationSupply Information

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Table 11.5. Transportation Model Table 11.5. Transportation Model with Plant at Atlantawith Plant at AtlantaTable 11.5. Transportation Model Table 11.5. Transportation Model with Plant at Atlantawith Plant at Atlanta


CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250


AtlantaAtlanta 2121 1111 1313 66 880880

DemandDemand 200200 100100 300300 280280 880880

Click here for Excel formulation for Excel formulation

Click here for LINGO formulation for LINGO formulation

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Example 3Example 3Example 3Example 3

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

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CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250


AtlantaAtlanta 2121 1111 1313 66 330330

PittsburghPittsburgh 1717 88 1818 1212 330330

DemandDemand 200200 100100 300300 280280

Table 11.10. Costs, Demand and Table 11.10. Costs, Demand and Supply InformationSupply InformationTable 11.10. Costs, Demand and Table 11.10. Costs, Demand and Supply InformationSupply Information

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Table 11.12. Transportation Table 11.12. Transportation Model with Plant at PittsburghModel with Plant at PittsburghTable 11.12. Transportation Table 11.12. Transportation Model with Plant at PittsburghModel with Plant at Pittsburgh

Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250Little RockLittle Rock 1919 1515 1010 99 300300PittsburghPittsburgh 1717 88 1818 1212 880880DemandDemand 200200 100100 300300 280280 880880

Click here for Excel model for Excel modelClick here for LINDO Model for LINDO ModelClick here for LINGO Model for LINGO Model

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

Location

d11 d12

d21 d22

d1n

d2n

dm1 dm2 dmn

Site

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11.3.3 11.3.3 Hybrid AnalysisHybrid Analysis11.3.3 11.3.3 Hybrid AnalysisHybrid Analysis

• CriticalCritical

• ObjectiveObjective

• SubjectiveSubjective

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Hybrid Analysis Cont...Hybrid Analysis Cont...Hybrid Analysis Cont...Hybrid Analysis Cont...

CFCFijij = 1 if location = 1 if location ii satisfies critical factor satisfies critical factor jj, ,

0 otherwise0 otherwise

OFOFijij = cost of objective factor = cost of objective factor jj at location at location ii

SFSFijij = numerical value assigned = numerical value assigned

(on scale of 0-100) (on scale of 0-100)

to subjective factor to subjective factor jj for location for location ii

wwjj = weight assigned to subjective factor = weight assigned to subjective factor

(0(0<< ww << 1) 1)

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OFMi

max i OFijj1

q

OFij

j1

q

max i OFijj 1

q

min i OFij

j 1

q

, i 1,2,...,m

SFMi w jSFijj1

r

, i 1,2,...,m

mi

CFCFCFCFCFMp

jijipiii

,...,2,1

,1

21

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The location measure The location measure LMLMii for each location is for each location is

then calculated as:then calculated as:

LMLMii = = CFMCFMii [ [ OFMOFMii + (1- + (1- ) ) SFMSFMii ] ]

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 location measure LMLMii

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Example 4:Example 4:Example 4:Example 4: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.

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Table 11.13:Table 11.13: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.

Table 11.13:Table 11.13: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.

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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

Water

Supply

Water

SupplyTax

Incentives

Tax

Incentives

Table 11.13 Cont...Table 11.13 Cont...

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Table 11.13 Cont...Table 11.13 Cont...Table 11.13 Cont...Table 11.13 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

Labor

Cost

Labor

CostEnergy

Cost

Energy

Cost

ObjectiveObjective

RevenueRevenue

50

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 11.13 Cont...Table 11.13 Cont...Table 11.13 Cont...Table 11.13 Cont...FactorsFactors

Ease of

Transportation

Ease of

Transportation

SubjectiveSubjective

Community

Attitude

Community

Attitude

51

Table 11.13 Cont...Table 11.13 Cont...Table 11.13 Cont...Table 11.13 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


Labor

Unionization

Labor

Unionization

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Table 11.14 Location Analysis of Table 11.14 Location Analysis of Mole-Sun Brewing Company, Mole-Sun Brewing Company,

Inc., Using Hybrid MethodInc., Using Hybrid Method

Table 11.14 Location Analysis of Table 11.14 Location Analysis of Mole-Sun Brewing Company, Mole-Sun Brewing Company,

Inc., Using Hybrid MethodInc., Using Hybrid Method

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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

Table 11.14 Cont...Table 11.14 Cont...Table 11.14 Cont...Table 11.14 Cont...

FactorsFactors

SFMiSFMi


Sum of

Obj. Factors

Sum of

Obj. Factors

CriticalCritical ObjectiveObjective LMi

54

11.411.4Techniques For Techniques For

Continuous Space Location ProblemsContinuous Space Location Problems

11.411.4Techniques For Techniques For

Continuous Space Location ProblemsContinuous Space Location Problems

55

11.4.1 Model for Rectilinear 11.4.1 Model for Rectilinear Metric ProblemMetric Problem11.4.1 Model for Rectilinear 11.4.1 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 existing facility ii

ccii = Cost of transportation between new facility = Cost of transportation between new facility

and existing facility and existing facility ii per unit per unit

xxii, , yyii = Coordinate points of existing facility = Coordinate points of existing facility ii

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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

m

iiiii yyxxfc

1

]||||[ TC

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

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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 Since the cciiffii product is known for each facility, product is known for each facility,

it can be thought of as a weight it can be thought of as a weight wwii

corresponding to facility corresponding to facility ii. .

m

i

m

iiiii yywxxw

1 1

]||[]||[ TC Minimize

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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 decreasing order of the xx coordinates. coordinates.

Step 2: Find the Step 2: Find the jjthth xx coordinate in the list at 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.,

j

i

m

i

ii

j

i

m

i

ii

ww

ww

1 1

1

1 1 2 and

2

59

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 decreasing order of the yy coordinates. coordinates.

Step 4: Find the Step 4: Find the kkthth yy coordinate in the list 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.,

k

i

m

i

ii

k

i

m

i

ii

ww

ww

1 1

1

1 1 2 and

2

60

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 facility is given by the jjthth xx coordinate and the coordinate and the kkthth yy coordinate identified in Steps 2 and 4, coordinate identified in Steps 2 and 4, respectively.respectively.

61

NotesNotesNotesNotes

1. It can be shown that any other 1. It can be shown that any other xx or or yy 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 2. The algorithm determines the xx and and yy coordinates of the facility’s optimal location coordinates of the facility’s optimal location separatelyseparately

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

62

Example 5:Example 5:Example 5:Example 5:

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., location, i.e., xx, , yy coordinates, for the copiers. coordinates, for the copiers.

63

Table 11.15 Centroid Coordinates Table 11.15 Centroid Coordinates and Average Number of Trips to and Average Number of Trips to

CopiersCopiers

Table 11.15 Centroid Coordinates Table 11.15 Centroid Coordinates and Average Number of Trips to and Average Number of Trips to

CopiersCopiers

64

Table 11.15Table 11.15Table 11.15Table 11.15

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

65


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

66


Step 2: Since the second Step 2: Since the second xx coordinate, namely 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 14, the optimal xx coordinate is 10. coordinate is 10.

67


Step 3: Step 3:

Dept. y coordinates in Weights Cumulative Dept. y 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

68


Step 4: Since the third Step 4: Since the third yy coordinates in the 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 optimal yy coordinate is 6. Thus, the optimal coordinate is 6. Thus, the optimal coordinates of the new facility are (10, 6).coordinates of the new facility are (10, 6).

69

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 existing facility ii

ccii = Unit transportation cost between new = Unit transportation cost between new facility and existing facility facility and existing facility ii

xxii, , yyii = Coordinate points of existing facility = Coordinate points of existing facility ii

Decision VariablesDecision Variables xx, , yy = Optimal coordinates of the new facility = Optimal coordinates of the new facility TC = Total distribution costTC = Total distribution cost

70

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

m

i

m

iiiii yywxxw

1 1




71

Since the Since the cciiffii product is known for each facility, product is known for each facility,

it can be thought of as a weight it can be thought of as a weight wwii

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



m

i

m

iiiii yywxxw

1 1


72



iii

iii

i

iii

iii

xxxx

xxxx

xx

xxxxx

xxxxx

)(

and

0, or 0)( whether that,observecan We

otherwise 0

0 if )(

otherwise 0

0 if )(

Define

73



iii

iii

ii

yyyy

yyyy

yy

)(

and

yields , of definitionsimilar A

74

n

i

iiiii yyxxw1

)( Minimize

ModelLinear dTransforme

signin edunrestrict ,,

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

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

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

Subject to

yx

yyxx

yyyy

xxxx

iiii

iii

iii



75

11.4.2 11.4.2 Contour Line MethodContour Line Method

11.4.2 11.4.2 Contour Line MethodContour Line Method

76

Step 1: Draw a vertical line through the Step 1: Draw a vertical line through the xx coordinate and a horizontal line through the coordinate and a horizontal line through the yy coordinate of each facilitycoordinate of each facility

Step 2: Label each vertical line Step 2: Label each vertical line VVii, , ii=1, 2, ..., =1, 2, ..., pp

and horizontal line and horizontal line HHjj, , jj=1, 2, ..., =1, 2, ..., qq where where VVii= =

the sum of weights of facilities whose the sum of weights of facilities whose xx coordinates fall on vertical line coordinates fall on vertical line ii and where and where HHjj= sum of weights of facilities whose = sum of weights of facilities whose yy

coordinates fall on horizontal line coordinates fall on horizontal line jj

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

77

mm

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)

78

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:

79

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

80

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

yyxx

i

m

iii

m

ii

11

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

facility new hen thefunction w objective heConsider t

81

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

q

iii

p

ii

11

82

Suppose that Suppose that xx is between the is between the ssthth and and ss+1+1thth (distinct) (distinct) xx coordinates or vertical lines (since 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 between the ttthth and and tt+1+1thth vertical lines. Then vertical lines. Then


TC Vi(i1

s

x x i) Viis1

p

(x i x)

Hi(i1

t

y y i) Hiit 1

q

(y i y)

83

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


i

q

tiii

t

iii

p

siii

s

ii

t

i

q

tiii

s

i

p

siii

yHyHxVxV

yHHxVVTC

1111

1 11 1

84

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 the coefficients of xx can be rewritten as follows can be rewritten as follows


s

i

s

iii

s

i

p

siii VVVVTC

1 11 1

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

s

iiV

1

85

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


,11

m

ii

s

ii wV

s

i

m

iii

s

i

p

iii

s

i

p

sii

s

iii

wV

VVVVV

1 1

1 11 11

2

22

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

t

i

m

iii wH

1 1

2

86

cywHxwVt

i

m

iii

s

i

m

iii

1 11 1

22TC Thus,


• The NThe Nii computation in Step 4 is in fact computation in Step 4 is in fact calculation of the coefficient of x as shown calculation of the coefficient of x as shown above. Note that Nabove. Note that Nii=N=Ni-1i-1+2V+2Vii. Making the . Making 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 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==

i

kk

m

ii Vw

11

2

87

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


i

kk

m

iii HwD

11

2

)(

asrewritten becan which

22TC Thus,1 11 1

cTCxD

Ny

cyDxN

cywHxwV

t

s

ts

t

i

m

iii

s

i

m

iii

88

The above expression for the total cost function The above expression for the total cost function at at xx, , yy or in fact, any other point in the region or in fact, any other point in the region [[ss, , tt] has the form ] has the form yy= = mxmx + + cc, where the slope , where the slope mm = -N = -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


89

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)


90


Consider Example 5. Suppose that the weight Consider Example 5. 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.

91


The contour line method is illustrated using The contour line method is illustrated using the figure belowthe figure below

92


Step 1: The vertical and horizontal lines VStep 1: The vertical and horizontal lines V11, V, V22, ,

VV22 and H and H11, H, H22, H, H22, H, H44 are drawn as shown. In are 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 be so 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 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 that fall on the respective lines. Note that for this example, p=3, and q=4this example, p=3, and q=4

93


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)

384

1

i

iw

94

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;

95


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.)

96

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

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.

97

11.4.311.4.3Single-facility Location Problem with Single-facility Location Problem with

Squared Euclidean DistancesSquared Euclidean Distances

11.4.311.4.3Single-facility Location Problem with Single-facility Location Problem with

Squared Euclidean DistancesSquared Euclidean Distances

98

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

99

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

100

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

As before, we substitute As before, we substitute wwi = = ffii c cii, i = 1, 2, ..., m , i = 1, 2, ..., m

and rewrite the objective function asand rewrite the objective function as

Minimize TC c i f i (x i x )2 (yi y )2 i 1

m

2

11

2 )()( TC Minimize yywxxw i

m

ii

m

iii

The cost function isThe cost function is

101

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 respect to xx and and yy, setting the resulting two , 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)

m

1i

m

1i

m

1i

m

1i

022 x

TC

iii

iii

wxwx

xwxw

102

Similarly,Similarly,

Gravity Method (Cont)Gravity Method (Cont)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 Thus, the optimal locations xx and and yy are simply are simply the weighted averages of the the weighted averages of the xx and and yy coordinates coordinates of the existing facilitiesof the existing facilities

103


Consider Example 5. Suppose the distance Consider Example 5. 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.

104

Solution - Table 11.16Solution - Table 11.16Solution - Table 11.16Solution - Table 11.16

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

yx

105

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

106

11.4.411.4.4WeiszfeldWeiszfeldMethodMethod

11.4.411.4.4WeiszfeldWeiszfeldMethodMethod

107

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

As before, substituting As before, substituting wwii==cciiffii and taking the and taking the

derivative of TC with respect to derivative of TC with respect to xx and and yy yields yields

)y(y)x(xfc TC Minimizem

1iiiii

22

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:

108


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

109


)y(y)x(x

w

)y(y)x(x

xw

x m

1i ii

i

m

1i ii

ii

22

22

110


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

111


m

1i ii

i

m

1i ii

ii

22

22

)y(y)x(x

w

)y(y)x(x

yw

y

112

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

m

m

m

m

1ii

1iii

k

1ii

1iii

k

w

ywy ;

w

xwx

113

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

Step 2: If Step 2: If xxkk+1+1 = = xxkk and and yykk+1+1 = = yykk, Stop. Otherwise, , Stop. Otherwise, set set kk = = kk + 1 and go to Step 1 + 1 and go to Step 1

Step 1: Set

Step 2: If Step 2: If xxkk+1+1 = = xxkk and and yykk+1+1 = = yykk, Stop. Otherwise, , Stop. Otherwise, set set kk = = kk + 1 and go to Step 1 + 1 and go to Step 1

m

iii

i

m

iii

ii

k

yyxx

w

yyxx

xw

x

122

122

1

m

iii

i

m

iii

ii

k

yyxx

w

yyxx

yw

y

122

122

1

114


Consider Example 6. Assuming the distance Consider Example 6. 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.

115

Table 11.17Table 11.17Coordinates and weights forCoordinates and weights for

4 departments4 departments

Table 11.17Table 11.17Coordinates and weights forCoordinates and weights for

4 departments4 departments

116

Table 11.17:Table 11.17:Table 11.17:Table 11.17:

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

117


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 two consecutive xx, , yy values are equal. values are equal.

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Summary: Methods for Single-Summary: Methods for Single-Facility, Continuous Space Facility, Continuous Space Location ProblemsLocation Problems

Summary: Methods for Single-Summary: Methods for Single-Facility, Continuous Space Facility, Continuous Space Location ProblemsLocation Problems• ProblemProblem- RectilinearRectilinear

- Squared Squared EuclideanEuclidean

- EuclideanEuclidean

• MethodMethod- MedianMedian

- GravityGravity

- WeiszfeldWeiszfeld

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Facility Location Case StudyFacility Location Case StudyFacility Location Case StudyFacility Location Case Study

• See Section 11.5See Section 11.5

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Chapter 11 Heragu