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- 1. 1 Facilities DesignFacilities Design S.S. HeraguS.S. Heragu Decision Sciences and EngineeringDecision Sciences and Engineering Systems DepartmentSystems Department Rensselaer Polytechnic InstituteRensselaer Polytechnic Institute Troy NY 12180-3590Troy NY 12180-3590
- 2. 2 Chapter 13Chapter 13 Basic ModelsBasic Models for thefor the Location ProblemLocation Problem
- 3. 3 • 13.1 Introduction13.1 Introduction • 13.213.2 Important Factors in LocationImportant Factors in Location DecisionsDecisions • 13.313.3 Techniques for Discrete SpaceTechniques for Discrete Space Location ProblemsLocation Problems - 13.3.1 Qualitative Analysis13.3.1 Qualitative Analysis - 13.3.2 Quantitative Analysis13.3.2 Quantitative Analysis - 13.3.3 Hybrid Analysis13.3.3 Hybrid Analysis OutlineOutline
- 4. 4 • 13.413.4 Techniques for Continuous SpaceTechniques for Continuous Space Location ProblemsLocation Problems - 13.4.1 Median Method13.4.1 Median Method - 13.4.2 Contour Line Method13.4.2 Contour Line Method - 13.4.3 Gravity Method13.4.3 Gravity Method - 13.4.4 Weiszfeld Method13.4.4 Weiszfeld Method • 13.513.5 Facility Location Case StudyFacility Location Case Study • 13.613.6 SummarySummary • 13.713.7 Review Questions and ExercisesReview Questions and Exercises • 13.813.8 ReferencesReferences Outline Cont...Outline Cont...
- 5. 5 McDonald’sMcDonald’s • QSCV PhilosophyQSCV Philosophy • 11,000 restaurants (7,000 in USA, remaining11,000 restaurants (7,000 in USA, remaining in 50 countries)in 50 countries) • 700 seat McDonald’s in Pushkin Square,700 seat McDonald’s in Pushkin Square, MoscowMoscow • $60 million food plant combining a bakery,$60 million food plant combining a bakery, lettuce plant, meat plant, chicken plant, fishlettuce plant, meat plant, chicken plant, fish plant and a distribution center, each ownedplant and a distribution center, each owned and operated independently at same locationand operated independently at same location
- 6. 6 • Food taste must be the same at anyFood taste must be the same at any McDonald, yet food must be secured locallyMcDonald, yet food must be secured locally • Strong logistical chain, with no weak linksStrong logistical chain, with no weak links betweenbetween • Close monitoring for logistical performanceClose monitoring for logistical performance • 300 in Australia300 in Australia • Central distribution since 1974 with the helpCentral distribution since 1974 with the help of F.J. Walker Foods in Sydneyof F.J. Walker Foods in Sydney • Then distribution centers opened in severalThen distribution centers opened in several citiescities McDonald’s cont...McDonald’s cont...
- 7. 7 McDonald’s cont...McDonald’s cont... • 2000 ingredients, from 48 food plants,2000 ingredients, from 48 food plants, shipment of 200 finished products fromshipment of 200 finished products from suppliers to DC’s, 6 million cases of food andsuppliers to DC’s, 6 million cases of food and paper products plus 500 operating items topaper products plus 500 operating items to restaurants across Australiarestaurants across Australia • Delivery of frozen, dry and chilled foodsDelivery of frozen, dry and chilled foods twice a week to each of the 300 restaurantstwice a week to each of the 300 restaurants 98% of the time within 15 minutes of98% of the time within 15 minutes of promised delivery time, 99.8% within 2 dayspromised delivery time, 99.8% within 2 days of order placementof order placement • No stockouts, but less inventoryNo stockouts, but less inventory
- 8. 8 IntroductionIntroduction • Logistics management can be defined as theLogistics management can be defined as the management of transportation andmanagement of transportation and distribution of goods.distribution of goods. - facility locationfacility location - transportationtransportation - goods handling and storage.goods handling and storage.
- 9. 9 Introduction Cont...Introduction Cont... Some of the objectives in facility location decisions: (1) It must first be close as possible to raw(1) It must first be close as possible to raw material sources and customers;material sources and customers; (2) Skilled labor must be readily available in the(2) Skilled labor must be readily available in the vicinity of a facility’s location;vicinity of a facility’s location; (3) Taxes, property insurance, construction(3) Taxes, property insurance, construction andand land prices must not be too “high;”land prices must not be too “high;” (4) Utilities must be readily available at a(4) Utilities must be readily available at a “reasonable” price;“reasonable” price;
- 10. 10 Introduction Cont...Introduction Cont... • (5) Local , state and other government(5) Local , state and other government regulations must be conducive to business;regulations must be conducive to business; andand (6) Business climate must be favorable and the(6) Business climate must be favorable and the community must have adequate supportcommunity must have adequate support services and facilities such as schools,services and facilities such as schools, hospitals and libraries, which are importanthospitals and libraries, which are important to employees and their families.to employees and their families.
- 11. 11 Introduction Cont...Introduction Cont... Logistics management problems can beLogistics management problems can be classified as:classified as: (1)(1) location problems;location problems; (2)(2) allocation problems; andallocation problems; and (3)(3) location-allocation problems.location-allocation problems.
- 12. 12 List of Factors AffectingList of Factors Affecting Location DecisionsLocation Decisions • Proximity to raw materials sourcesProximity to raw materials sources • Cost and availability of energy/utilitiesCost and availability of energy/utilities • Cost, availability, skill and productivity ofCost, availability, skill and productivity of laborlabor • Government regulations at the federal, state,Government regulations at the federal, state, country and local levelscountry and local levels • Taxes at the federal, state, county and localTaxes at the federal, state, county and local levelslevels • InsuranceInsurance • Construction costs, land priceConstruction costs, land price
- 13. 13 List of Factors AffectingList of Factors Affecting Location Decisions Cont...Location Decisions Cont... • Government and political stabilityGovernment and political stability • Exchange rate fluctuationExchange rate fluctuation • Export, import regulations, duties, and tariffsExport, import regulations, duties, and tariffs • Transportation systemTransportation system • Technical expertiseTechnical expertise • Environmental regulations at the federal,Environmental regulations at the federal, state, county and local levelsstate, county and local levels • Support servicesSupport services
- 14. 14 List of Factors AffectingList of Factors Affecting Location Decisions Cont...Location Decisions Cont... • Community services, i.e. schools, hospitals,Community services, i.e. schools, hospitals, recreation, etc.recreation, etc. • WeatherWeather • Proximity to customersProximity to customers • Business climateBusiness climate • Competition-related factorsCompetition-related factors
- 15. 15 13.213.2 Important Factors in LocationImportant Factors in Location DecisionsDecisions • InternationalInternational • NationalNational • State-wideState-wide • Community-wideCommunity-wide
- 16. 16 13.3.113.3.1 Qualitative AnalysisQualitative Analysis Step 1: List all the factors that are important,Step 1: List all the factors that are important, i.e. have an impact on the location decision.i.e. have an impact on the location decision. Step 2: Assign appropriate weights (typicallyStep 2: Assign appropriate weights (typically between 0 and 1) to each factor based on thebetween 0 and 1) to each factor based on the relative importance of each.relative importance of each. Step 3: Assign a score (typically between 0 andStep 3: Assign a score (typically between 0 and 100) for each location with respect to each100) for each location with respect to each factor identified in Step 1.factor identified in Step 1.
- 17. 17 13.3.113.3.1 Qualitative AnalysisQualitative Analysis Step 4: Compute the weighted score for eachStep 4: Compute the weighted score for each factor for each location by multiplying itsfactor for each location by multiplying its weight with the corresponding score (whichweight with the corresponding score (which were assigned Steps 2 and 3, respectively)were assigned Steps 2 and 3, respectively) Step 5: Compute the sum of the weightedStep 5: Compute the sum of the weighted scores for each location and choose ascores for each location and choose a location based on these scores.location based on these scores.
- 18. 18 Example 1:Example 1: •A payroll processing company has recentlyA payroll processing company has recently won several major contracts in the midwestwon several major contracts in the midwest region of the U.S. and central Canada andregion of the U.S. and central Canada and wants to open a new, large facility to servewants to open a new, large facility to serve these areas. Since customer service is ofthese areas. Since customer service is of utmost importance, the company wants to beutmost importance, the company wants to be as near it’s “customers” as possible.as near it’s “customers” as possible. Preliminary investigation has shown thatPreliminary investigation has shown that Minneapolis, Winnipeg, and Springfield, Ill.,Minneapolis, Winnipeg, and Springfield, Ill., would be the three most desirable locationswould be the three most desirable locations and the payroll company has to select one ofand the payroll company has to select one of these three.these three.
- 19. 19 Example 1: Cont...Example 1: Cont... A subsequent thorough investigation of eachA subsequent thorough investigation of each location with respect to eight important factorslocation with respect to eight important factors has generated the raw scores and weightshas generated the raw scores and weights listed in table 2. Using the location scoringlisted in table 2. Using the location scoring method, determine the best location for the newmethod, determine the best location for the new payroll processing facility.payroll processing facility.
- 20. 20 Solution:Solution: Steps 1, 2, and 3 have already been completedSteps 1, 2, and 3 have already been completed for us. We now need to compute the weightedfor us. We now need to compute the weighted score for each location-factor pair (Step 4), andscore for each location-factor pair (Step 4), and these weighted scores and determine thethese weighted scores and determine the location based on these scores (Step 5).location based on these scores (Step 5).
- 21. 21 Table 2. Factors and Weights forTable 2. Factors and Weights for Three LocationsThree Locations Wt.Wt. FactorsFactors LocationLocation Minn.Winn.Spring.Minn.Winn.Spring. .25.25 Proximity to customersProximity to customers 9595 9090 6565 .15.15 Land/construction pricesLand/construction prices 6060 6060 9090 .15.15 Wage ratesWage rates 7070 4545 6060 .10.10 Property taxesProperty taxes 7070 9090 7070 .10.10 Business taxesBusiness taxes 8080 9090 8585 .10.10 Commercial travelCommercial travel 8080 6565 7575
- 22. 22 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
- 23. 23 Solution: Cont...Solution: Cont... From the analysis in Table 3, it is clear thatFrom the analysis in Table 3, it is clear that Minneapolis would be the best location basedMinneapolis would be the best location based on the subjective information.on the subjective information.
- 24. 24 Table 3. Weighted Scores for theTable 3. Weighted Scores for the Three LocationsThree Locations in Table 2in Table 2 Weighted Score Location Minn. Winn. Spring. Proximity to customers 23.75 22.5 16.25 Land/construction prices 9 9 13.5 Wage rates 10.5 6.75 9 Property taxes 7 9 8.5 Business taxes 8 9 8.5
- 25. 25 Table 3. Cont...Table 3. Cont... Weighted Score Location 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
- 26. 26 Solution: Cont...Solution: Cont... Of course, as mentioned before, objectiveOf course, as mentioned before, objective measures must be brought into considerationmeasures must be brought into consideration especially because the weighted scores forespecially because the weighted scores for Minneapolis and Winnipeg are close.Minneapolis and Winnipeg are close.
- 27. 27 13.3.213.3.2 QuantitativeQuantitative AnalysisAnalysis
- 28. 28 General Transportation ModelGeneral Transportation Model ParametersParameters ccijij: cost of transporting one unit from: cost of transporting one unit from warehouse i to customer jwarehouse i to customer j aaii: supply capacity at warehouse i: supply capacity at warehouse i bbii: demand at customer j: demand at customer j Decision VariablesDecision Variables xxijij: number of units transported from: number of units transported from warehouse i to customer jwarehouse i to customer j
- 29. 29 General Transportation ModelGeneral Transportation Model ∑∑= = = m i n j ijij xcZ 1 1 CosttionTransportaTotalMinimize i)seat warehounrestrictio(supplym1,2,...,i, Subject to 1 =≤∑= n j iij ax j)marketattrequiremen(demandn1,2,...,j, 1 =≥∑= m i jij bx ns)restrictionegativity-(nonn1,2,...,ji,,0 =≥ijx
- 30. 30 Example 2:Example 2: Seers Inc. has two manufacturing plants atSeers Inc. has two manufacturing plants at Albany and Little Rock supplying CanmoreAlbany and Little Rock supplying Canmore brand refrigerators to four distribution centersbrand refrigerators to four distribution centers in Boston, Philadelphia, Galveston and Raleigh.in Boston, Philadelphia, Galveston and Raleigh. Due to an increase in demand of this brand ofDue to an increase in demand of this brand of refrigerators that is expected to last for severalrefrigerators that is expected to last for several years into the future, Seers Inc., has decided toyears into the future, Seers Inc., has decided to build another plant in Atlanta or Pittsburgh.build another plant in Atlanta or Pittsburgh. The expected demand at the three distributionThe expected demand at the three distribution centers and the maximum capacity at thecenters and the maximum capacity at the Albany and Little Rock plants are given in TableAlbany and Little Rock plants are given in Table 4.4.
- 31. 31 Example 2: Cont...Example 2: Cont... Determine which of the two locations, AtlantaDetermine which of the two locations, Atlanta or Pittsburgh, is suitable for the new plant.or Pittsburgh, is suitable for the new plant. Seers Inc., wishes to utilize all of the capacitySeers Inc., wishes to utilize all of the capacity available at it’s Albany and Little Rockavailable at it’s Albany and Little Rock LocationsLocations
- 32. 32 Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply CapacityCapacity AlbanyAlbany 1010 1515 2222 2020 250250 Little RockLittle Rock 1919 1515 1010 99 300300 AtlantaAtlanta 2121 1111 1313 66 No limitNo limit PittsburghPittsburgh 1717 88 1818 1212 No limitNo limit DemandDemand 200200 100100 300300 280280 Table 4. Costs, Demand andTable 4. Costs, Demand and Supply InformationSupply Information
- 33. 33 Table 5. Transportation ModelTable 5. Transportation Model with Plant at Atlantawith Plant at Atlanta Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply CapacityCapacity AlbanyAlbany 1010 1515 2222 2020 250250 Little RockLittle Rock 1919 1515 1010 99 300300 AtlantaAtlanta 2121 1111 1313 66 330330 DemandDemand 200200 100100 300300 280280 880880
- 34. 34 Table 6. Transportation ModelTable 6. Transportation Model with Plant at Pittsburghwith Plant at Pittsburgh Bost.Bost. Phil.Phil. Galv.Galv. Rale.Rale. SupplySupply CapacityCapacity AlbanyAlbany 1010 1515 2222 2020 250250 Little RockLittle Rock 1919 1515 1010 99 300300 PittsburghPittsburgh 1717 88 1818 1212 330330 DemandDemand 200200 100100 300300 280280 880880
- 35. 35 Min/Max Location Problem:Min/Max Location Problem: Location d11 d12 d21 d22 d1n d2n dm1 dm2 dmn Site
- 36. 36 13.3.313.3.3 Hybrid AnalysisHybrid Analysis • CriticalCritical • ObjectiveObjective • SubjectiveSubjective
- 37. 37 Hybrid Analysis Cont...Hybrid Analysis Cont... CFCFijij = 1 if location i satisfies critical factor j,= 1 if location i satisfies critical factor j, 0 otherwise0 otherwise OFOFijij = cost of objective factor j at location i= cost of objective factor j at location i SFSFijij = numerical value assigned= numerical value assigned (on scale of 0-1)(on scale of 0-1) to subjective factor j for location ito subjective factor j for location i wwjj = weight assigned to subjective factor= weight assigned to subjective factor (0(0<< ww << 1)1)
- 38. 38 Hybrid Analysis Cont...Hybrid Analysis Cont... miSFwSFM mi OFOF OFOF OFM r j ijji q j iji q j iji q j ij q j iji i ,...,2,1, ,...,2,1, minmax max 1 11 11 == =       −      −      = ∑ ∑∑ ∑∑ = == == mi CFCFCFCFCFM p j ijipiii ,...,2,1 , 1 21 = =•••= ∏=
- 39. 39 Hybrid Analysis Cont...Hybrid Analysis Cont... The location measure LMThe location measure LMii for each location isfor each location is then calculated as:then calculated as: LMLMii = CFM= CFMii [[ αα OFMOFMii + (1-+ (1- αα) SFM) SFMii ]] WhereWhere αα is the weight assigned to theis the weight assigned to the objective factor.objective factor. We then choose the location with the highestWe then choose the location with the highest location measure LMlocation measure LMii
- 40. 40 Example 3:Example 3: Mole-Sun Brewing company is evaluating sixMole-Sun Brewing company is evaluating six candidate locations-Montreal, Plattsburgh,candidate locations-Montreal, Plattsburgh, Ottawa, Albany, Rochester and Kingston, forOttawa, Albany, Rochester and Kingston, for constructing a new brewery. There are twoconstructing a new brewery. There are two critical, three objective and four subjectivecritical, three objective and four subjective factors that management wishes to incorporatefactors that management wishes to incorporate in its decision-making. These factors arein its decision-making. These factors are summarized in Table 7. The weights of thesummarized in Table 7. The weights of the subjective factors are also provided in thesubjective factors are also provided in the table. Determine the best location if thetable. Determine the best location if the subjective factors are to be weighted 50 percentsubjective factors are to be weighted 50 percent more than the objective factors.more than the objective factors.
- 41. 41 Table 7:Table 7: Critical, Subjective and ObjectiveCritical, Subjective and Objective Factor Ratings for six locations forFactor Ratings for six locations for Mole-Sun Brewing Company, Inc.Mole-Sun Brewing Company, Inc.
- 42. 42 FactorsFactorsLocation Albany 0 1 Kingston 1 1 Montreal 1 1 Ottawa 1 0 Plattsburgh 1 1 Rochester 1 1 Critical Water Supply Tax Incentives Table 7. Cont...Table 7. Cont...
- 43. 43 Table 7. Cont...Table 7. Cont... FactorsLocation Albany 185 80 10 Kingston 150 100 15 Montreal 170 90 13 Ottawa 200 100 15 Plattsburgh 140 75 8 Rochester 150 75 11 Critical Labor Cost Energy Cost Objective Revenue
- 44. 44 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... Factors Ease of Transportation Subjective Community Attitude
- 45. 45 Table 7. Cont...Table 7. Cont... FactorsLocation 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 Subjective Labor Unionization
- 46. 46 Table 8. Location Analysis ofTable 8. Location Analysis of Mole-Sun Brewing Company,Mole-Sun Brewing Company, Inc., Using Hybrid MethodInc., Using Hybrid Method
- 47. 47 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 7. Cont...Table 7. Cont... Factors SFMi Subjective Sum of Obj. Factors Critical Objective LMi
- 48. 48 13.413.4 Techniques ForTechniques For Continuous Space Location ProblemsContinuous Space Location Problems
- 49. 49 13.4.1 Model for Rectilinear13.4.1 Model for Rectilinear Metric ProblemMetric Problem Consider the following notation:Consider the following notation: ffii = Traffic flow between new facility and= Traffic flow between new facility and existing facility iexisting facility i ccii = Cost of transportation between new facility= Cost of transportation between new facility and existing facility i per unitand existing facility i per unit xxii, y, yii = Coordinate points of existing facility i= Coordinate points of existing facility i
- 50. 50 Model for Rectilinear MetricModel for Rectilinear Metric Problem (Cont)Problem (Cont) Where TC is the total distribution costWhere TC is the total distribution cost ∑= −+−= m i iiii yyxxfc 1 ]||||[TC The median location model is then to minimize:The median location model is then to minimize:
- 51. 51 Model for Rectilinear MetricModel for Rectilinear Metric Problem (Cont)Problem (Cont) Since the cSince the ciiffii product is known for each facility,product is known for each facility, it can be thought of as a weight wit can be thought of as a weight wii corresponding to facility i.corresponding to facility i. ∑ ∑= = −+−= m i m i iiii yywxxw 1 1 ]||[]||[TCMinimize
- 52. 52 Median Method:Median Method: Step 1: List the existing facilities in non-Step 1: List the existing facilities in non- decreasing order of the x coordinates.decreasing order of the x coordinates. Step 2: Find the jStep 2: Find the jthth x coordinate in the list atx coordinate in the list at which the cumulative weight equals orwhich the cumulative weight equals or exceeds half the total weight for the firstexceeds half the total weight for the first time, i.e.,time, i.e., ∑ ∑∑ ∑ = = − = = ≥< j i m i i i j i m i i i w w w w 1 1 1 1 1 2 and 2
- 53. 53 Median Method (Cont)Median Method (Cont) Step 3: List the existing facilities in non-Step 3: List the existing facilities in non- decreasing order of the y coordinates.decreasing order of the y coordinates. Step 4: Find the kStep 4: Find the kthth y coordinate in the listy coordinate in the list (created in Step 3) at which the cumulative(created in Step 3) at which the cumulative weight equals or exceeds half the totalweight equals or exceeds half the total weight for the first time, i.e.,weight for the first time, i.e., ∑ ∑∑ ∑ = = − = = ≥< k i m i i i k i m i i i w w w w 1 1 1 1 1 2 and 2
- 54. 54 Median Method (Cont)Median Method (Cont) Step 4: Cont... The optimal location of the newStep 4: Cont... The optimal location of the new facility is given by the jfacility is given by the jthth x coordinate and thex coordinate and the kkthth y coordinate identified in Steps 2 and 4,y coordinate identified in Steps 2 and 4, respectively.respectively.
- 55. 55 NotesNotes 1. It can be shown that any other x or y1. It can be shown that any other x or y coordinate will not be that of the optimalcoordinate will not be that of the optimal location’s coordinateslocation’s coordinates 2. The algorithm determines the x and y2. The algorithm determines the x and y coordinates of the facility’s optimal locationcoordinates of the facility’s optimal location separatelyseparately 3. These coordinates could coincide with the x3. These coordinates could coincide with the x and y coordinates of two different existingand y coordinates of two different existing facilities or possibly one existing facilityfacilities or possibly one existing facility
- 56. 56 Example 4:Example 4: Two high speed copiers are to be located in theTwo high speed copiers are to be located in the fifth floor of an office complex which housesfifth floor of an office complex which houses four departments of the Social Securityfour departments of the Social Security Administration. Coordinates of the centroid ofAdministration. Coordinates of the centroid of each department as well as the average numbereach department as well as the average number of trips made per day between each departmentof trips made per day between each department and the copiers’ yet-to-be-determined locationand the copiers’ yet-to-be-determined location are known and given in Table 9 below. Assumeare known and given in Table 9 below. Assume that travel originates and ends at the centroidthat travel originates and ends at the centroid of each department. Determine the optimalof each department. Determine the optimal location, i.e., x, y coordinates, for the copiers.location, i.e., x, y coordinates, for the copiers.
- 57. 57 Table 9. Centroid CoordinatesTable 9. Centroid Coordinates and Average Number of Trips toand Average Number of Trips to CopiersCopiers
- 58. 58 Table 9.Table 9. Dept.Dept. CoordinatesCoordinates Average number ofAverage number of ## xx yy daily trips to copiersdaily trips to copiers 11 1010 22 66 22 1010 1010 1010 33 88 66 88 44 1212 55 44
- 59. 59 Solution:Solution: Using the median method, we obtain theUsing the median method, we obtain the following solution:following solution: Step 1:Step 1: Dept. x coordinates in Weights Cumulative # non-decreasing order Weights 3 8 8 8 1 10 6 14 2 10 10 24 4 12 4 28
- 60. 60 Solution:Solution: Step 2: Since the second x coordinate, namelyStep 2: Since the second x coordinate, namely 10, in the above list is where the cumulative10, in the above list is where the cumulative weight equals half the total weight of 28/2 =weight equals half the total weight of 28/2 = 14, the optimal x coordinate is 10.14, the optimal x coordinate is 10.
- 61. 61 Solution:Solution: Step 3:Step 3: Dept. x coordinates in Weights Cumulative # non-decreasing order Weights 1 2 6 6 4 5 4 10 3 6 8 18 2 10 10 28
- 62. 62 Solution:Solution: Step 4: Since the third y coordinates in theStep 4: Since the third y coordinates in the above list is where the cumulative weightabove list is where the cumulative weight exceeds half the total weight of 28/2 = 14, theexceeds half the total weight of 28/2 = 14, the optimal coordinate is 6. Thus, the optimaloptimal coordinate is 6. Thus, the optimal coordinates of the new facility are (10, 6).coordinates of the new facility are (10, 6).
- 63. 63 Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem ParametersParameters ffii = Traffic flow between new facility and= Traffic flow between new facility and existing facility iexisting facility i ccii = Unit transportation cost between new= Unit transportation cost between new facility and existing facility ifacility and existing facility i xxii, y, yii = Coordinate points of existing facility i= Coordinate points of existing facility i Decision VariablesDecision Variables x, y= Optimal coordinates of the new facilityx, y= Optimal coordinates of the new facility TC = Total distribution costTC = Total distribution cost
- 64. 64 The median location model is then toThe median location model is then to ∑ ∑= = −+−= m i m i iiii yywxxw 1 1 ]||[]||[TCMinimize Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem
- 65. 65 Since the cSince the ciiffii product is known for each facility,product is known for each facility, it can be thought of as a weight wit can be thought of as a weight wii corresponding to facility i. The previouscorresponding to facility i. The previous equation can now be rewritten as followsequation can now be rewritten as follows Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem ∑ ∑= = −+−= m i m i iiii yywxxw 1 1 ]||[]||[TCMinimize
- 66. 66 Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem iii iii i ii i ii i xxxx xxxx xx xxxx x xxxx x −+ −+ − + −=− +=− ≤>−    ≤−− =    >−− = )( and 0,or0)(whetherthat,observecanWe otherwise0 0if)( otherwise0 0if)( Define
- 67. 67 Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem iii iii ii yyyy yyyy yy −+ −+ −+ −=− +=− )( and yields,ofdefinitionsimilarA
- 68. 68 ∑= −+−+ +++ n i iiiii yyxxw 1 )(Minimize ModelLineardTransforme signinedunrestrict,, n1,2,...,i0,,,, n1,2,...,i,-)( n1,2,...,i,-)( Subject to yx yyxx yyyy xxxx iiii iii iii =≥ ==− ==− −+−+ −+ −+ Equivalent Linear Model for theEquivalent Linear Model for the Rectilinear Distance, Single-Rectilinear Distance, Single- Facility Location ProblemFacility Location Problem
- 69. 69 13.4.213.4.2 Contour Line MethodContour Line Method
- 70. 70 Step 1: Draw a vertical line through the xStep 1: Draw a vertical line through the x coordinate and a horizontal line through the ycoordinate and a horizontal line through the y coordinate of each facilitycoordinate of each facility Step 2: Label each vertical line VStep 2: Label each vertical line Vii, i=1, 2, ..., p, i=1, 2, ..., p and horizontal line Hand horizontal line Hjj, j=1, 2, ..., q where V, j=1, 2, ..., q where Vii== the sum of weights of facilities whose xthe sum of weights of facilities whose x coordinates fall on vertical line i and wherecoordinates fall on vertical line i and where HHjj= sum of weights of facilities whose y= sum of weights of facilities whose y coordinates fall on horizontal line jcoordinates fall on horizontal line j Algorithm for Drawing ContourAlgorithm for Drawing Contour Lines:Lines:
- 71. 71 m i=1 Step 3: Set i = j = 1; NStep 3: Set i = j = 1; N00 = D= D00 = w= wii Step 4: Set NStep 4: Set Nii = N= Ni-1i-1 + 2V+ 2Vii and Dand Djj = D= Dj-1j-1 + 2H+ 2Hjj.. Increment i = i + 1 and j = j + 1Increment i = i + 1 and j = j + 1 Step 5: If iStep 5: If i << p or jp or j << q, go to Step 4. Otherwise,q, go to Step 4. Otherwise, set i = j = 0 and determine Sset i = j = 0 and determine Sijij, the slope of, the slope of contour lines through the region bounded bycontour lines through the region bounded by vertical lines i and i + 1 and horizontal line jvertical lines i and i + 1 and horizontal line j and j + 1 using the equation Sand j + 1 using the equation Sijij = -N= -Nii/D/Djj.. Increment i = i + 1 and j = j + 1Increment i = i + 1 and j = j + 1 Algorithm for Drawing ContourAlgorithm for Drawing Contour Lines (Cont)Lines (Cont) ∑∑
- 72. 72 Step 6: If iStep 6: If i << p or jp or j << q, go to Step 5. Otherwiseq, go to Step 5. Otherwise select any point (x, y) and draw a contour lineselect any point (x, y) and draw a contour line with slope Swith slope Sijij in the region [i, j] in which (x, y)in the region [i, j] in which (x, y) appears so that the line touches theappears so that the line touches the boundary of this line. From one of the endboundary of this line. From one of the end points of this line, draw another contour linepoints of this line, draw another contour line through the adjacent region with thethrough the adjacent region with the corresponding slopecorresponding slope Step 7: Repeat this until you get a contour lineStep 7: Repeat this until you get a contour line ending at point (x, y). We now have a regionending at point (x, y). We now have a region bounded by contour lines with (x, y) on thebounded by contour lines with (x, y) on the boundary of the regionboundary of the region Algorithm for Drawing ContourAlgorithm for Drawing Contour Lines:Lines:
- 73. 73 1. The number of vertical and horizontal lines1. The number of vertical and horizontal lines need not be equalneed not be equal 2. The N2. The Nii and Dand Djj as computed in Steps 3 and 4as computed in Steps 3 and 4 correspond to the numerator andcorrespond to the numerator and denominator, respectively of the slopedenominator, respectively of the slope equation of any contour line through theequation of any contour line through the region bounded by the vertical lines i and i +region bounded by the vertical lines i and i + 1 and horizontal lines j and j + 11 and horizontal lines j and j + 1 Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour LinesContour Lines
- 74. 74 Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) yywxxwTC yyxx i m i ii m i i −+−= == ∑∑ == 11 ,i.e.,y),(x,pointsomeatlocatedis facilitynewhen thefunction wobjectiveheConsider t
- 75. 75 By noting that the VBy noting that the Vii’s and H’s and Hjj’s calculated in’s calculated in Step 2 of the algorithm correspond to the sumStep 2 of the algorithm correspond to the sum of the weights of facilities whose x, yof the weights of facilities whose x, y coordinates are equal to the x, y coordinates,coordinates are equal to the x, y coordinates, respectively of the irespectively of the ithth , j, jthth distinct lines and thatdistinct lines and that we have p, q such coordinates or lines (pwe have p, q such coordinates or lines (p << m, qm, q << m), the previous equation can be written asm), the previous equation can be written as followsfollows Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) yyHxxVTC i q i ii p i i −+−= ∑∑ == 11
- 76. 76 Suppose that x is between the sSuppose that x is between the sthth and s+1and s+1thth (distinct) x coordinates or vertical lines (since(distinct) x coordinates or vertical lines (since we have drawn vertical lines through thesewe have drawn vertical lines through these coordinates in Step 1). Similarly, let y becoordinates in Step 1). Similarly, let y be between the tbetween the tthth and t+1and t+1thth vertical lines. Thenvertical lines. Then Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) )()()()( 1111 yyHyyHxxVxxVTC i q ti ii t i ii p si ii s i i −+−+−+−= ∑∑∑∑ +==+==
- 77. 77 Rearranging the variable and constant terms inRearranging the variable and constant terms in the above equation, we getthe above equation, we get Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) i q ti ii t i ii p si ii s i i t i q ti ii s i p si ii yHyHxVxV yHHxVVTC ∑∑∑∑ ∑ ∑∑ ∑ +==+== = +== += +−+−       −+      −= 1111 1 11 1
- 78. 78 The last four terms in the previous equationThe last four terms in the previous equation can be substituted by another constant termcan be substituted by another constant term c and the coefficients of x can be rewrittenc and the coefficients of x can be rewritten as followsas follows Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) ∑ ∑∑ ∑ = == += +−−= s i s i ii s i p si ii VVVVTC 1 11 1 Notice that we have only added andNotice that we have only added and subtracted this termsubtracted this term ∑= s i iV 1
- 79. 79 Since it is clear from Step 2 that the coefficient of x can be rewritten as Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) , 11 ∑∑ == = m i i s i i wV ∑ ∑ ∑ ∑∑ ∑∑ = = = == +== −= −=      +− s i m i ii s i p i ii s i p si i s i ii wV VVVVV 1 1 1 11 11 2 22 Similarly, the coefficient of y is ∑ ∑= = − t i m i ii wH 1 1 2
- 80. 80 cywHxwV t i m i ii s i m i ii +      −+      −= ∑ ∑∑ ∑ = == = 1 11 1 22TCThus, Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) • The NThe Nii computation in Step 4 is in factcomputation in Step 4 is in fact calculation of the coefficient of x as showncalculation of the coefficient of x as shown above. Note that Nabove. Note that Nii=N=Ni-1i-1+2V+2Vii. M. Making theaking the substitution for Nsubstitution for Ni-1i-1, we get N, we get Nii=N=Ni-2i-2+2V+2Vi-1i-1+2V+2Vii • Repeating the same procedure of makingRepeating the same procedure of making substitutions for Nsubstitutions for Ni-2i-2, N, Ni-3i-3, ..., we get, ..., we get • NNii=N=N00+2V+2V11+2V+2V22+...+2V+...+2Vi-1i-1+2V+2V11== ∑∑ == +− i k k m i i Vw 11 2
- 81. 81 Similarly, it can be verified thatSimilarly, it can be verified that Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont) ∑∑ == +−= i k k m i ii HwD 11 2 )( asrewrittenbecanwhich 22TCThus, 1 11 1 cTCx D N y cyDxN cywHxwV t s ts t i m i ii s i m i ii −+−= ++= +      −+      −= ∑ ∑∑ ∑ = == =
- 82. 82 The above expression for the total cost functionThe above expression for the total cost function at x, y or in fact, any other point in the region [s,at x, y or in fact, any other point in the region [s, t] has the form y= mx + c, where the slopet] has the form y= mx + c, where the slope m = -Nm = -Nss/D/Dtt. This is exactly how the slopes are. This is exactly how the slopes are computed in Step 5 of the algorithmcomputed in Step 5 of the algorithm Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont)
- 83. 83 3. The lines V3. The lines V00, V, Vp+1p+1 and Hand H00, H, Hq+1q+1 are required forare required for defining the “exterior” regions [0, j], [p, j], j =defining the “exterior” regions [0, j], [p, j], j = 1, 2, ..., p, respectively)1, 2, ..., p, respectively) 4. Once we have determined the slopes of all4. Once we have determined the slopes of all regions, the user may choose any point (x, y)regions, the user may choose any point (x, y) other than a point which minimizes theother than a point which minimizes the objective function and draw a series ofobjective function and draw a series of contour lines in order to get a region whichcontour lines in order to get a region which contains points, i.e. facility locations,contains points, i.e. facility locations, yielding as good or better objective functionyielding as good or better objective function values than (x, y)values than (x, y) Notes on Algorithm for DrawingNotes on Algorithm for Drawing Contour Lines (Cont)Contour Lines (Cont)
- 84. 84 Example 5:Example 5: Consider Example 4. Suppose that the weightConsider Example 4. Suppose that the weight of facility 2 is not 10, but 20. Applying theof facility 2 is not 10, but 20. Applying the median method, it can be verified that themedian method, it can be verified that the optimal location is (10, 10) - the centroid ofoptimal location is (10, 10) - the centroid of department 2, where immovable structuresdepartment 2, where immovable structures exist. It is now desired to find a feasible andexist. It is now desired to find a feasible and “near-optimal” location using the contour line“near-optimal” location using the contour line method.method.
- 85. 85 Solution:Solution: The contour line method is illustrated usingThe contour line method is illustrated using Figure 1Figure 1 Step 1: The vertical and horizontal lines VStep 1: The vertical and horizontal lines V11, V, V22,, VV22 and Hand H11, H, H22, H, H22, H, H44 are drawn as shown. Inare drawn as shown. In addition to these lines, we also draw line Vaddition to these lines, we also draw line V00, V, V44 and Hand H00, H, H55 so that the “exterior regions can beso that the “exterior regions can be identifiedidentified Step 2: The weights VStep 2: The weights V11, V, V22, V, V22, H, H11, H, H22, H, H22, H, H44 areare calculated by adding the weights of the pointscalculated by adding the weights of the points that fall on the respective lines. Note that forthat fall on the respective lines. Note that for this example, p=3, and q=4this example, p=3, and q=4
- 86. 86 Solution:Solution: Step 3: Since set N0 = D0 = -38 Step 4: Set N1 = -38 + 2(8) = -22; D1 = -38 + 2(6) = -26; N2 = -22 + 2(26) = 30; D2 = -26 + 2(4) = -18; N3 = 30 + 2(4) = 38; D3 = -18 + 2(8) = -2; D4 = -2 + 2(20) = 38; (These values are entered at the bottom of each column and left of each row in figure 1) 38 4 1 =∑=i iw
- 87. 87 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;
- 88. 88 Solution:Solution: Step 5: Compute the slope of each region.Step 5: Compute the slope of each region. SS3333 = -(38/-2) = 19;= -(38/-2) = 19; SS3434 = -(38/38) = -1;= -(38/38) = -1; (The above slope values are shown inside each(The above slope values are shown inside each region.)region.)
- 89. 89 Solution:Solution: Step 6: When we draw contour linesStep 6: When we draw contour lines through point (9, 10), we get thethrough point (9, 10), we get the region shown in figure 1.region shown in figure 1. Since the copiers cannot be placed at theSince the copiers cannot be placed at the (10, 10) location, we drew contour lines(10, 10) location, we drew contour lines through another nearby point (9, 10).through another nearby point (9, 10). Locating anywhere possible within thisLocating anywhere possible within this region give us a feasible, near-optimalregion give us a feasible, near-optimal solution.solution.
- 90. 90 13.4.313.4.3 Single-facility Location Problem withSingle-facility Location Problem with Squared Euclidean DistancesSquared Euclidean Distances
- 91. 91 La Quinta Motor InnsLa Quinta Motor Inns Moderately priced, oriented towards businessModerately priced, oriented towards business travelerstravelers Headquartered in San Antonio TexasHeadquartered in San Antonio Texas Site selection - an important decisionSite selection - an important decision Regression Model based on locationRegression Model based on location characteristics classified as:characteristics classified as: - Competitive, Demand Generators,Competitive, Demand Generators, Demographic, Market Awareness, andDemographic, Market Awareness, and PhysicalPhysical
- 92. 92 La Quinta Motor Inns (Cont)La Quinta Motor Inns (Cont) Major Profitability Factors - Market awareness,Major Profitability Factors - Market awareness, hotel space, local population, lowhotel space, local population, low unemployment, accessibility to downtown officeunemployment, accessibility to downtown office space, traffic count, college students, presencespace, traffic count, college students, presence of military base, median income, competitiveof military base, median income, competitive ratesrates
- 93. 93 Gravity Method:Gravity Method: As before, we substitute wAs before, we substitute wi = f= fii ccii, i = 1, 2, ..., m, i = 1, 2, ..., m and rewrite the objective function asand rewrite the objective function as [ ]∑= −+−= m i iiii yyxxfc 1 22 )()(TCMinimize 2 11 2 )()(TCMinimize yywxxw i m i i m i ii −+−= ∑∑ == The cost function isThe cost function is
- 94. 94 Since the objective function can be shown toSince the objective function can be shown to be convex, partially differentiating TC withbe convex, partially differentiating TC with respect to x and y, setting the resulting tworespect to x and y, setting the resulting two equations to 0 and solving for x, y provides theequations to 0 and solving for x, y provides the optimal location of the new facilityoptimal location of the new facility Gravity Method (Cont)Gravity Method (Cont) ∑∑ ∑∑ == == =∴ =−= ∂ ∂ m 1i m 1i m 1i m 1i 022 x TC iii iii wxwx xwxw
- 95. 95 Similarly,Similarly, Gravity Method (Cont)Gravity Method (Cont) ∑∑ ∑∑ == == =∴ =−= ∂ ∂ m 1i m 1i m 1i m 1i 022 y TC iii iii wywy ywyw Thus, the optimal locations x and y are simplyThus, the optimal locations x and y are simply the weighted averages of the x and y coordinatesthe weighted averages of the x and y coordinates of the existing facilitiesof the existing facilities
- 96. 96 Example 6:Example 6: Consider Example 4. Suppose the distanceConsider Example 4. Suppose the distance metric to be used is squared Euclidean.metric to be used is squared Euclidean. Determine the optimal location of the newDetermine the optimal location of the new facility using the gravity method.facility using the gravity method.
- 97. 97 Solution - Table 10Solution - Table 10 Department i xi yi wi wixi wiyi 1 10 2 6 60 12 2 10 10 10 100 100 3 8 6 8 64 48 4 12 5 4 48 20 Total 28 272 180 4.628180and7.928272 thatconcludewe10,tableFrom ==== yx
- 98. 98 Example 6. Cont...Example 6. Cont... If this location is not feasible, we only need toIf this location is not feasible, we only need to find another point which has the nearestfind another point which has the nearest Euclidean distance to (9.7, 6.4) and is a feasibleEuclidean distance to (9.7, 6.4) and is a feasible location for the new facility and locate thelocation for the new facility and locate the copiers therecopiers there
- 99. 99 13.4.413.4.4 WeiszfeldWeiszfeld MethodMethod
- 100. 100 Weiszfeld Method:Weiszfeld Method: As before, substituting wAs before, substituting wii=c=ciiffii and taking theand taking the derivative of TC with respect to x and y yieldsderivative of TC with respect to x and y yields )y(y)x(xfcTCMinimize m 1i iiii 22 ∑= −+−= The objective function for the single facilityThe objective function for the single facility location problem with Euclidean distance canlocation problem with Euclidean distance can be written as:be written as:
- 101. 101 Weiszfeld Method:Weiszfeld Method: [ ] ∑ ∑ ∑ = = = = −+− − −+− = −+− − = ∂ ∂ m 1i ii i m 1i ii ii m 1i ii ii 0 )y(y)x(x xw )y(y)x(x xw )y(y)x(x )x2(xw 2 1 x TC 22 22 22
- 102. 102 Weiszfeld Method:Weiszfeld Method: )y(y)x(x w )y(y)x(x xw x m 1i ii i m 1i ii ii 22 22 ∑ ∑ = = −+− −+− =∴
- 103. 103 Weiszfeld Method:Weiszfeld Method: [ ] ∑ ∑ ∑ = = = = −+− − −+− = −+− − = ∂ ∂ m 1i ii i m 1i ii ii m 1i ii ii 0 )y(y)x(x yw )y(y)x(x yw )y(y)x(x )y2(yw 2 1 y TC 22 22 22
- 104. 104 Weiszfeld Method:Weiszfeld Method: ∑ ∑ = = −+− −+− =∴ m 1i ii i m 1i ii ii 22 22 )y(y)x(x w )y(y)x(x yw y
- 105. 105 Weiszfeld Method:Weiszfeld Method: Step 0: Set iteration counter k = 1; ∑ ∑ ∑ ∑ = = = = == m m m m 1i i 1i ii k 1i i 1i ii k w yw y; w xw x
- 106. 106 Weiszfeld Method:Weiszfeld Method: Step 1: Set )y(y)x(x w )y(y)x(x xw x m 1i k i k i i m 1i k i k i ii 1k 22 22 ∑ ∑ = =+ −+− −+− =∴
- 107. 107 )y(y)x(x w )y(y)x(x xw x m 1i k i k i i m 1i k i k i ii 1k 22 22 ∑ ∑ = =+ −+− −+− =∴ Weiszfeld Method:Weiszfeld Method: • Step 2: If xStep 2: If xk+1k+1 = x= xkk and yand yk+1k+1 = y= ykk , Stop., Stop. Otherwise, set k = k + 1 and go to Step 1Otherwise, set k = k + 1 and go to Step 1
- 108. 108 Example 7:Example 7: Consider Example 5. Assuming the distanceConsider Example 5. Assuming the distance metric to be used is Euclidean, determine themetric to be used is Euclidean, determine the optimal location of the new facility using theoptimal location of the new facility using the Weiszfeld method. Data for this problem isWeiszfeld method. Data for this problem is shown in Table 11.shown in Table 11.
- 109. 109 Table 11.Table 11. Coordinates and weights forCoordinates and weights for 4 departments4 departments
- 110. 110 Table 11:Table 11: Departments # xi yi wi 1 10 2 6 2 10 10 20 3 8 6 8 4 12 5 4
- 111. 111 Solution:Solution: Using the gravity method, the initial seed canUsing the gravity method, the initial seed can be shown to be (9.8, 7.4). With this as thebe shown to be (9.8, 7.4). With this as the starting solution, we can apply Step 1 of thestarting solution, we can apply Step 1 of the Weiszfeld method repeatedly until we find thatWeiszfeld method repeatedly until we find that two consecutive x, y values are equal.two consecutive x, y values are equal.

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