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Facility Location. Contributions to Location Theory Transportation Location Factors Non-transportation Location Factors Types of Location Problems Measures of Spatial Analysis Methods of Analysis Site Selection. Contributions to Location Theory. Von Thunen - Isolated State

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## Contributions to Location Theory Transportation Location Factors

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**Facility Location**Contributions to Location Theory Transportation Location Factors Non-transportation Location Factors Types of Location Problems Measures of Spatial Analysis Methods of Analysis Site Selection**Contributions to Location Theory**Von Thunen - Isolated State Weber - Locational Triangle Losch - Profit Maximization/Spatial Equilibrium Hotelling - Maximize Market Territory/Monopoly/Duopoly Hoover - Tapering Rates Greenhut - Cost, Revenue, and Personal Factors Isard - Spatial Competition and Game Theory**Von Thunen**Isolated State - Uniform Plane Purely Competitive Market Distance is the Only Advantage Thus, Minimize Transportation Cost Concentric Circles (Von Thunen's Belts) Low Value, high Weight - Near the Market Perishables - Near the Market High Value, Low Weight, Away from the Market**Alfred Weber**German Geographer Weber's Locational Triangle Differentiated Between Raw Material and Finished Goods Weight Gain in Production Weight Loss in Production Pure Materials (No Weight Gain or Loss) Localized Materials Ubiquitous Materials Effects of Labor and Agglomeration The Generalized Weber Problem**MKT**RM2 RM1 Alfred Weber Weber's Locational Triangle**Alfred Weber**German Geographer Weber's Locational Triangle Differentiated Between Raw Material and Finished Goods Weight Gain in Production Weight Loss in Production Pure Materials (No Weight Gain or Loss) Localized Materials Ubiquitous Materials Effects of Labor and Agglomeration The Generalized Weber Problem**Alfred Weber**German Geographer Weber's Locational Triangle Differentiated Between Raw Material and Finished Goods Weight Gain in Production Weight Loss in Production Pure Materials (No Weight Gain or Loss) Localized Materials Ubiquitous Materials Effects of Labor and Agglomeration The Generalized Weber Problem**Transportation Location Factors**Location of Market Location of Raw Materials Demand at the Market (Volume to be Moved) Supply of Raw Materials (Volume to be Moved) Weight Gain in Production Weight Loss in Production Rate Structure Ratio of Inbound to Outbound Rates Quality of Transportation Services Quantity of Transportation Services**Non-transportation Location Factors**Resources (Quality, Cost, Availability) Labor Management Raw Materials Financial Energy Environment Natural Climate Political Climate Government Regulation Competition Market Size Cultural (Quality of Life, etc.) Taxes Corporate Income Individual Income Real Estate Inventory Incentives**Types of Location Problems**The Allocation Problem The Single Source Problem The Multiple Source Problem The Dynamic Problem**Types of Location Problems**The Allocation Problem Also known as the Assignment Problem Not really a location problem Facilities already exist Allocate supply to demand Northwest Corner, Simplex Method, etc.**= Facility**= Customer Types of Location Problems The Allocation Problem 25 40 60 80 25 50**Customer 1**Customer 2 Customer 3 Customer 4 50 0 60 10 0 Source 1 Source 2 80 40 0 15 25 40 50 25 25 Types of Location Problems The Allocation Problem Resources are allocated to customers or Demands are assigned to supplies Use Linear Programming to Solve**Measures of Spatial Analysis**Single Space Two Space Manhattan Metric Euclidian Metric Graph Metric**Methods of Analysis**Mean vs Median Optimization Heuristics Simulation**Σ**Ci wi Mean = Σ wi Types of Location Problems The Single Source Problem Measures of Central Tendency Median = The One in the Middle Σ Aggregate Distance Ci- L wi =**wi**3 5 2 1 2 6 Ci 0 1 2 3 4 5 6 7 8 9 10 Types of Location Problems The Single Source Problem Single Space Upper level numeric values (wi) denote customer demands at those locations. Lower level numeric values (Ci) denote mile posts where customers are located; viz., i = 3, 4, 5, 7, 9, and 10.**wi**wi 3 5 2 1 2 6 = 19 Ci 0 1 2 3 4 5 6 7 8 9 10 Median Mean Σ Ci wi Mean = 3x3 +5x4 + 2x5 + 1x7 + 2x9 +6x10 = 124 = Σ wi 124/19 = 6.5 Types of Location Problems The Single Source Problem Mean vs Median Median = The one in the middle, (19/2) = 9.5 or 10. There are 9 on either side, and the 10th unit is delivered to C5. Thus, the median (M) = C5 (or mile post 5).**Σ**Aggregate Distance Ci- L wi (Note absolute values) = Σ Aggregate Distance Ci- X wi (Mean) = 3|3-6.5 | + 5|4-6.5| + 2|5-6.5| + 1|7-6.5| + 2|9-6.5| + 6|10-6.5| = 52.5 Σ Aggregate Distance Ci- M wi = (Median) 3|3-5| + 5|4-5| + 2|5-5| + 1|7-5| + 2|9-5| + 6|10-5| = 51 Types of Location Problems The Single Source Problem Mean vs Median**wi**wi 3 5 2 1 2 16 = 29 Ci 0 1 2 3 4 5 6 7 8 9 10 Median Types of Location Problems The Single Source Problem Majority Theorem Anytime a majority of the demand is a one location, you will locate there.**N**10 10 15 25 5 5 15 5 10 15 15 20 20 5 S E W Types of Location Problems The Single Source Problem The Generalized Weber Problem Numeric values denote customer demands at those locations**Σ** wi dij = (Y2-Y1)2 + (X2-X1)2 Minimize Types of Location Problems The Single Source Problem The Generalized Weber Problem Pythagorean Theorem: The square of the hypotenuse is equal to the sum of the two sides squared.**7**10 10 15 25 6 5 5 15 5 5 4 10 3 15 15 2 20 20 5 1 0 0 1 2 3 4 5 6 7 8 9 10 Types of Location Problems The Single Source Problem The Generalized Weber Problem**7**6 5 4 (d) (1) 3 (2) 2 1 0 Types of Location Problems The Single Source Problem The Generalized Weber Problem 10 10 15 25 5 15 5 5 10 15 15 20 20 5 0 1 2 3 4 5 6 7 8 9 10**** d = d = d = d = (Y2-Y1)2 + (X2-X1)2 (4 - 3)2 + (8 - 6)2 5 (1)2 + (2)2 d = 2.24 Types of Location Problems The Single Source Problem The Generalized Weber Problem**Σ** wi dij = (Y2-Y1)2 + (X2-X1)2 Minimize Types of Location Problems The Single Source Problem The Generalized Weber Problem Problem – You cannot isolate either X or Y. That is, anytime you solve for X in terms of Y, X is still on the right hand side. Ditto, Y. Thus, a solution is not trivial. This problem has generated enormous attempts to find a solution.**7**10 10 15 25 6 5 5 5 5 15 4 10 3 15 15 2 20 20 5 1 0 Types of Location Problems The Single Source Problem Manhattan Matrix 0 1 2 3 4 5 6 7 8 9 10**7**10 10 15 25 60 6 5 5 5 5 5 15 25 4 10 3 10 15 15 2 30 20 20 5 1 45 0 10 25 20 20 20 20 15 45 Types of Location Problems The Single Source Problem Manhattan Matrix 0 1 2 3 4 5 6 7 8 9 10**7**10 10 15 25 6 5 60 5 5 5 15 5 4 25 10 3 15 15 10 2 20 20 5 30 1 45 0 175 M = 88 10 25 20 20 20 20 15 45 Types of Location Problems The Single Source Problem Manhattan Matrix 0 1 2 3 4 5 6 7 8 9 10**Types of Location Problems**The Allocation Problem The Single Source Problem The Multiple Source Problem The Dynamic Problem**Types of Location Problems**The Multiple Source Problem Locate – Allocate, Locate – Allocate, Locate – Allocate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Locate – Allocate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Locate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Allocate**10**15 5 10 15 5 Types of Location Problems The Multiple Source Problem Locate**10**15 5 10 15 5 Types of Location Problems The Multiple Source Problem Locate**10**25 5 5 Types of Location Problems The Multiple Source Problem Locate**10**25 5 5 Types of Location Problems The Multiple Source Problem Locate**15**15 20 20 Types of Location Problems The Multiple Source Problem Locate**15**15 20 20 Types of Location Problems The Multiple Source Problem Locate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Locate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Allocate**10**15 5 5 5 10 15 15 5 Types of Location Problems The Multiple Source Problem Locate**10**15 5 5 5 10 15 15 5 Types of Location Problems The Multiple Source Problem Locate**10**25 15 Types of Location Problems The Multiple Source Problem Locate**10**25 15 Types of Location Problems The Multiple Source Problem Locate**20**20 Types of Location Problems The Multiple Source Problem Locate**20**20 Types of Location Problems The Multiple Source Problem Locate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Locate**10**10 15 25 5 5 15 5 10 15 15 20 20 5 Types of Location Problems The Multiple Source Problem Allocate Solution is stable when no customers are reassigned

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