2. 1 Facilities Design S.S. Heragu Industrial Engineering Department University of Louisville

3. 2 Chapter 11 Basic Models for the Location Problem

4. 3 11.1 Introduction 11.1 Introduction 11.2 Important Factors in Location Decisions 11.2 Important Factors in Location Decisions 11.3 Techniques for Discrete Space Location Problems 11.3 Techniques for Discrete Space Location Problems - 11.3.1 Qualitative Analysis - 11.3.2 Quantitative Analysis - 11.3.3 Hybrid Analysis OutlineOutline

5. 4 11.4 Techniques for Continuous Space Location Problems 11.4 Techniques for Continuous Space Location Problems - 11.4.1 Median Method - 11.4.2 Contour Line Method - 11.4.3 Gravity Method - 11.4.4 Weiszfeld Method 11.5 Facility Location Case Study 11.5 Facility Location Case Study 11.6 Summary 11.6 Summary 11.7 Review Questions and Exercises 11.7 Review Questions and Exercises 11.8 References 11.8 References Outline Cont...

6. 5 McDonald’sMcDonald’s QSCV Philosophy QSCV Philosophy 11,000 restaurants (7,000 in USA, remaining in 50 countries) 11,000 restaurants (7,000 in USA, remaining in 50 countries) 700 seat McDonald’s in Pushkin Square, Moscow 700 seat McDonald’s in Pushkin Square, Moscow $60 million food plant combining a bakery, lettuce plant, meat plant, chicken plant, fish plant and a distribution center, each owned and operated independently at same location $60 million food plant combining a bakery, lettuce plant, meat plant, chicken plant, fish plant and a distribution center, each owned and operated independently at same location

7. 6 Food taste must be the same at any McDonald, yet food must be secured locally Food taste must be the same at any McDonald, yet food must be secured locally Strong logistical chain, with no weak links between Strong logistical chain, with no weak links between Close monitoring for logistical performance Close monitoring for logistical performance 300 in Australia 300 in Australia Central distribution since 1974 with the help of F.J. Walker Foods in Sydney Central distribution since 1974 with the help of F.J. Walker Foods in Sydney Then distribution centers opened in several cities Then distribution centers opened in several cities McDonald’s cont...

8. 7 2000 ingredients, from 48 food plants, shipment of 200 finished products from suppliers to DC’s, 6 million cases of food and paper products plus 500 operating items to restaurants across Australia 2000 ingredients, from 48 food plants, shipment of 200 finished products from suppliers to DC’s, 6 million cases of food and paper products plus 500 operating items to restaurants across Australia Delivery of frozen, dry and chilled foods twice a week to each of the 300 restaurants 98% of the time within 15 minutes of promised delivery time, 99.8% within 2 days of order placement Delivery of frozen, dry and chilled foods twice a week to each of the 300 restaurants 98% of the time within 15 minutes of promised delivery time, 99.8% within 2 days of order placement No stockouts, but less inventory No stockouts, but less inventory

9. 8 IntroductionIntroduction Logistics management can be defined as the management of transportation and distribution of goods. Logistics management can be defined as the management of transportation and distribution of goods. - facility location - transportation - goods handling and storage.

10. 9 Introduction Cont... Some of the objectives in facility location decisions: (1) It must first be close as possible to raw material sources and customers; (2) Skilled labor must be readily available in the vicinity of a facility’s location; (3) Taxes, property insurance, construction and land prices must not be too “high;” (4) Utilities must be readily available at a “reasonable” price;

11. 10 Introduction Cont... (5) Local, state and other government regulations must be conducive to business; and (6) Business climate must be favorable and the community must have adequate support services and facilities such as schools, hospitals and libraries, which are important to employees and their families.

12. 11 Introduction Cont... Logistics management problems can be classified as: (1)location problems; (2)allocation problems; and (3)location-allocation problems.

13. 12 List of Factors Affecting Location Decisions Proximity to raw materials sources Proximity to raw materials sources Cost and availability of energy/utilities Cost and availability of energy/utilities Cost, availability, skill and productivity of labor Cost, availability, skill and productivity of labor Government regulations at the federal, state, country and local levels Government regulations at the federal, state, country and local levels Taxes at the federal, state, county and local levels Taxes at the federal, state, county and local levels Insurance Insurance Construction costs, land price Construction costs, land price

14. 13 List of Factors Affecting Location Decisions Cont... Government and political stability Government and political stability Exchange rate fluctuation Exchange rate fluctuation Export, import regulations, duties, and tariffs Export, import regulations, duties, and tariffs Transportation system Transportation system Technical expertise Technical expertise Environmental regulations at the federal, state, county and local levels Environmental regulations at the federal, state, county and local levels Support services Support services

15. 14 List of Factors Affecting Location Decisions Cont... Community services, i.e. schools, hospitals, recreation, etc. Community services, i.e. schools, hospitals, recreation, etc. Weather Weather Proximity to customers Proximity to customers Business climate Business climate Competition-related factors Competition-related factors

16. 15 11.2 Important Factors in Location Decisions International International National National State-wide State-wide Community-wide Community-wide

17. 16 11.3.1 Qualitative Analysis Step 1: List all the factors that are important, i.e. have an impact on the location decision. Step 2: Assign appropriate weights (typically between 0 and 1) to each factor based on the relative importance of each. Step 3: Assign a score (typically between 0 and 100) for each location with respect to each factor identified in Step 1.

18. 17 11.3.1 Qualitative Analysis Step 4: Compute the weighted score for each factor for each location by multiplying its weight with the corresponding score (which were assigned Steps 2 and 3, respectively) Step 5: Compute the sum of the weighted scores for each location and choose a location based on these scores.

19. 18 Example 1: A payroll processing company has recently won several major contracts in the midwest region of the U.S. and central Canada and wants to open a new, large facility to serve these areas. Since customer service is of utmost importance, the company wants to be as near it’s “customers” as possible. Preliminary investigation has shown that Minneapolis, Winnipeg, and Springfield, Ill., would be the three most desirable locations and the payroll company has to select one of these three. A payroll processing company has recently won several major contracts in the midwest region of the U.S. and central Canada and wants to open a new, large facility to serve these areas. Since customer service is of utmost importance, the company wants to be as near it’s “customers” as possible. Preliminary investigation has shown that Minneapolis, Winnipeg, and Springfield, Ill., would be the three most desirable locations and the payroll company has to select one of these three.

20. 19 Example 1: Cont... A subsequent thorough investigation of each location with respect to eight important factors has generated the raw scores and weights listed in table 2. Using the location scoring method, determine the best location for the new payroll processing facility.

21. 20 Solution:Solution: Steps 1, 2, and 3 have already been completed for us. We now need to compute the weighted score for each location-factor pair (Step 4), and these weighted scores and determine the location based on these scores (Step 5).

22. 21 Table 2. Factors and Weights for Three Locations Wt.FactorsLocation Minn.Winn.Spring..25Proximity to customers959065.15Land/construction prices606090.15Wage rates704560.10Property taxes709070.10Business taxes809085.10Commercial travel806575

23. 22 Table 2. Cont... Wt.FactorsLocation Minn.Winn.Spring..08Insurance costs709560.07Office services909080 Click here Click here

24. 23 Solution: Cont... From the analysis in Table 3, it is clear that Minneapolis would be the best location based on the subjective information.

25. 24 Table 3. Weighted Scores for the Three Locations in Table 2 Weighted ScoreLocation Minn.Winn.Spring. Proximity to customers23.7522.516.25 Land/construction prices9913.5 Wage rates10.56.759 Property taxes798.5 Business taxes898.5 Weighted ScoreLocation Minn.Winn.Spring. Proximity to customers23.7522.516.25 Land/construction prices9913.5 Wage rates10.56.759 Property taxes798.5 Business taxes898.5

26. 25 Table 3. Cont... Weighted ScoreLocation Minn.Winn.Spring. Commercial travel86.57.5 Insurance costs5.67.64.8 Office services6.36.35.6 Weighted ScoreLocation Minn.Winn.Spring. Commercial travel86.57.5 Insurance costs5.67.64.8 Office services6.36.35.6

27. 26 Solution: Cont... Of course, as mentioned before, objective measures must be brought into consideration especially because the weighted scores for Minneapolis and Winnipeg are close.

28. 27 11.3.2 Quantitative Analysis

29. 28 General Transportation Model Parameters c ij : cost of transporting one unit from warehouse i to customer j c ij : cost of transporting one unit from warehouse i to customer j a i : supply capacity at warehouse i a i : supply capacity at warehouse i b i : demand at customer j b i : demand at customer j Decision Variables x ij : number of units transported from warehouse i to customer j x ij : number of units transported from warehouse i to customer j

30. 29 General Transportation Model

31. 30 Transportation Simplex Algorithm Step 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 (c ij ) 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 x ij =0, i=1,2,...,m; and j=1,2,...,n. Step 3:Construct a basic feasible solution using the Northwest corner method.

32. 31 Transportation Simplex Algorithm Step 4:Set u 1 =0 and find v j, j=1,2,...,n and u i, i=1,2,...,n using the formula u i + v j = c ij for all basic variables. Step 5:If u i + v j - c ij < 0 for all nonbasic variables, then the current basic feasible solution is optimal; stop. Otherwise, go to step 6. Step 6:Select the variable x i*j* with the most positive value u i* + v j*- c ij*. Construct a closed loop consisting of horizontal and vertical segments connecting the corresponding cell in row i* and column j* to other basic variables. Adjust the values of the basic variables in this closed loop so that the supply and demand constraints of each row and column are satisfied and the maximum possible value is added to the cell in row i* and column j*. The variable x i*j* is now a basic 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.

33. 32 Example 2: Seers Inc. has two manufacturing plants at Albany and Little Rock supplying Canmore brand refrigerators to four distribution centers in Boston, Philadelphia, Galveston and Raleigh. Due to an increase in demand of this brand of refrigerators that is expected to last for several years into the future, Seers Inc., has decided to build another plant in Atlanta or Pittsburgh. The expected demand at the three distribution centers and the maximum capacity at the Albany and Little Rock plants are given in Table 4.

34. 33 Example 2: Cont... Determine which of the two locations, Atlanta or Pittsburgh, is suitable for the new plant. Seers Inc., wishes to utilize all of the capacity available at it’s Albany and Little Rock Locations

35. 34 Bost.Phil.Galv.Rale.Supply Capacity Albany10152220250 Little Rock1915109300 Atlanta2111136No limit Pittsburgh1781812No limit Demand200100300280 Table 4. Costs, Demand and Supply Information

36. 35 Table 5. Transportation Model with Plant at Atlanta Bost.Phil.Galv.Rale.Supply Capacity Albany10152220250 Little Rock1915109300 Atlanta2111136330 Demand200100300280880 Click hereClick here for Excel formulation Click here Click hereClick here for LINGO formulation Click here

37. 36 Table 6. Transportation Model with Plant at Pittsburgh Bost.Phil.Galv.Rale.Supply Capacity Capacity Albany10152220250 Little Rock1915109300 Pittsburgh1781812330 Demand200100300280880 Click hereClick here for Excel model Click here Click hereClick here for LINDO Model Click here Click hereClick here for LINGO Model Click here

38. 37 Min/Max Location Problem: Location d 11 d 12 d 21 d 22 d 1n d 2n d m1 d m2 d mn Site

39. 38 11.3.3 Hybrid Analysis Critical Critical Objective Objective Subjective Subjective

40. 39 Hybrid Analysis Cont... CF ij = 1 if location i satisfies critical factor j, 0 otherwise OF ij = cost of objective factor j at location i SF ij = numerical value assigned (on scale of 0-1) to subjective factor j for location i w j = weight assigned to subjective factor (0< w < 1)

41. 40 Hybrid Analysis Cont...

42. 41 Hybrid Analysis Cont... The location measure LM i for each location is then calculated as: LM i = CFM i [  OFM i + (1-  ) SFM i ] Where  is the weight assigned to the objective factor. We then choose the location with the highest location measure LM i

43. 42 Example 3: Mole-Sun Brewing company is evaluating six candidate locations-Montreal, Plattsburgh, Ottawa, Albany, Rochester and Kingston, for constructing a new brewery. There are two critical, three objective and four subjective factors that management wishes to incorporate in its decision-making. These factors are summarized in Table 7. The weights of the subjective factors are also provided in the table. Determine the best location if the subjective factors are to be weighted 50 percent more than the objective factors.

44. 43 Table 7: Critical, Subjective and Objective Factor Ratings for six locations for Mole-Sun Brewing Company, Inc.

45. 44 Factors Location Albany01 Kingston11 Montreal11 Ottawa10 Plattsburgh11 Rochester11 Location Albany01 Kingston11 Montreal11 Ottawa10 Plattsburgh11 Rochester11 Critical Water Supply Water Supply Tax Incentives Tax Incentives Table 7. Cont...

46. 45 Table 7. Cont... Factors Location Albany185 8010 Kingston15010015 Montreal170 9013 Ottawa20010015 Plattsburgh140 75 8 Rochester150 7511 Location Albany185 8010 Kingston15010015 Montreal170 9013 Ottawa20010015 Plattsburgh140 75 8 Rochester150 7511 Critical Labor Cost Labor Cost Energy Cost Energy Cost Objective Revenue

47. 46 Location 0.30.4 Albany0.50.9 Kingston0.60.7 Montreal0.40.8 Ottawa0.50.4 Plattsburgh0.90.9 Rochester0.70.65 Location 0.30.4 Albany0.50.9 Kingston0.60.7 Montreal0.40.8 Ottawa0.50.4 Plattsburgh0.90.9 Rochester0.70.65 Table 7. Cont... Factors Ease of Transportation Ease of Transportation Subjective Community Attitude Community Attitude

48. 47 Table 7. Cont... Factors Location 0.250.05 Albany0.60.7 Kingston0.70.75 Montreal0.20.8 Ottawa0.40.8 Plattsburgh0.90.55 Rochester0.40.8 Location 0.250.05 Albany0.60.7 Kingston0.70.75 Montreal0.20.8 Ottawa0.40.8 Plattsburgh0.90.55 Rochester0.40.8 Support Services Support Services Subjective Labor Unionization Labor Unionization

49. 48 Table 8. Location Analysis of Mole-Sun Brewing Company, Inc., Using Hybrid Method

50. 49 Location Albany-950.70 Kingston-350.670.4 Montreal-670.530.53 Ottawa-850.450 Plattsburgh-570.880.68 Rochester-640.610.56 Location Albany-950.70 Kingston-350.670.4 Montreal-670.530.53 Ottawa-850.450 Plattsburgh-570.880.68 Rochester-640.610.56 Table 7. Cont... Factors SFM i Subjective Sum of Obj. Factors Sum of Obj. Factors Critical Objective LM i

51. 50 11.4 Techniques For Continuous Space Location Problems

52. 51 11.4.1 Model for Rectilinear Metric Problem Consider the following notation: f i = Traffic flow between new facility and existing facility i c i = Cost of transportation between new facility and existing facility i per unit x i, y i = Coordinate points of existing facility i

53. 52 Model for Rectilinear Metric Problem (Cont) Where TC is the total distribution cost The median location model is then to minimize:

54. 53 Model for Rectilinear Metric Problem (Cont) Since the c i f i product is known for each facility, it can be thought of as a weight w i corresponding to facility i.

55. 54 Median Method: Step 1: List the existing facilities in non- decreasing order of the x coordinates. Step 2: Find the j th x coordinate in the list at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

56. 55 Median Method (Cont) Step 3: List the existing facilities in non- decreasing order of the y coordinates. Step 4: Find the k th y coordinate in the list (created in Step 3) at which the cumulative weight equals or exceeds half the total weight for the first time, i.e.,

57. 56 Median Method (Cont) Step 4: Cont... The optimal location of the new facility is given by the j th x coordinate and the k th y coordinate identified in Steps 2 and 4, respectively.

58. 57 NotesNotes 1. It can be shown that any other x or y coordinate will not be that of the optimal location’s coordinates 2. The algorithm determines the x and y coordinates of the facility’s optimal location separately 3. These coordinates could coincide with the x and y coordinates of two different existing facilities or possibly one existing facility

59. 58 Example 4: Two high speed copiers are to be located in the fifth floor of an office complex which houses four departments of the Social Security Administration. Coordinates of the centroid of each department as well as the average number of trips made per day between each department and the copiers’ yet-to-be-determined location are known and given in Table 9 below. Assume that travel originates and ends at the centroid of each department. Determine the optimal location, i.e., x, y coordinates, for the copiers.

60. 59 Table 9. Centroid Coordinates and Average Number of Trips to Copiers

61. 60 Table 9. Dept.Coordinates Average number of #xy daily trips to copiers 11026 2101010 3868 41254

62. 61 Solution:Solution: Using the median method, we obtain the following solution: Step 1: Dept.x coordinates inWeightsCumulative #non-decreasing orderWeights 3888 110614 2101024 412428 3888 110614 2101024 412428

63. 62 Solution:Solution: Step 2: Since the second x coordinate, namely 10, in the above list is where the cumulative weight equals half the total weight of 28/2 = 14, the optimal x coordinate is 10.

64. 63 Solution:Solution: Step 3: Dept.y coordinates inWeightsCumulative #non-decreasing orderWeights 1266 45410 36818 2101028 1266 45410 36818 2101028

65. 64 Solution:Solution: Step 4: Since the third y coordinates in the above list is where the cumulative weight exceeds half the total weight of 28/2 = 14, the optimal y coordinate is 6. Thus, the optimal coordinates of the new facility are (10, 6).

66. 65 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem Parameters f i = Traffic flow between new facility and existing facility i f i = Traffic flow between new facility and existing facility i c i = Unit transportation cost between new facility and existing facility i c i = Unit transportation cost between new facility and existing facility i x i, y i = Coordinate points of existing facility i x i, y i = Coordinate points of existing facility i Decision Variables x, y = Optimal coordinates of the new facility x, y = Optimal coordinates of the new facility TC = Total distribution cost TC = Total distribution cost

67. 66 The median location model is then to Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

68. 67 Since the c i f i product is known for each facility, it can be thought of as a weight w i corresponding to facility i. The previous equation can now be rewritten as follows Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

69. 68 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

70. 69 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

71. 70 Equivalent Linear Model for the Rectilinear Distance, Single- Facility Location Problem

72. 71 11.4.2 Contour Line Method

73. 72 Step 1: Draw a vertical line through the x coordinate and a horizontal line through the y coordinate of each facility Step 2: Label each vertical line V i, i=1, 2,..., p and horizontal line H j, j=1, 2,..., q where V i = the sum of weights of facilities whose x coordinates fall on vertical line i and where H j = sum of weights of facilities whose y coordinates fall on horizontal line j Algorithm for Drawing Contour Lines:

74. 73 m m i=1 Step 3: Set i = j = 1; N 0 = D 0 = w i Step 4: Set N i = N i-1 + 2V i and D j = D j-1 + 2H j. Increment i = i + 1 and j = j + 1 Step 5: If i < p or j < q, go to Step 4. Otherwise, set i = j = 0 and determine S ij, the slope of contour lines through the region bounded by vertical lines i and i + 1 and horizontal line j and j + 1 using the equation S ij = -N i /D j. Increment i = i + 1 and j = j + 1 Algorithm for Drawing Contour Lines (Cont) 

75. 74 Step 6: If i < p or j < q, go to Step 5. Otherwise select any point (x, y) and draw a contour line with slope S ij in the region [i, j] in which (x, y) appears so that the line touches the boundary of this line. From one of the end points of this line, draw another contour line through the adjacent region with the corresponding slope Step 7: Repeat this until you get a contour line ending at point (x, y). We now have a region bounded by contour lines with (x, y) on the boundary of the region Algorithm for Drawing Contour Lines:

76. 75 1. The number of vertical and horizontal lines need not be equal 2. The N i and D j as computed in Steps 3 and 4 correspond to the numerator and denominator, respectively of the slope equation of any contour line through the region bounded by the vertical lines i and i + 1 and horizontal lines j and j + 1 Notes on Algorithm for Drawing Contour Lines

77. 76 Notes on Algorithm for Drawing Contour Lines (Cont)

78. 77 By noting that the V i ’s and H j ’s calculated in Step 2 of the algorithm correspond to the sum of the weights of facilities whose x, y coordinates are equal to the x, y coordinates, respectively of the i th, j th distinct lines and that we have p, q such coordinates or lines (p < m, q < m), the previous equation can be written as follows Notes on Algorithm for Drawing Contour Lines (Cont)

79. 78 Suppose that x is between the s th and s+1 th (distinct) x coordinates or vertical lines (since we have drawn vertical lines through these coordinates in Step 1). Similarly, let y be between the t th and t+1 th vertical lines. Then Notes on Algorithm for Drawing Contour Lines (Cont)

80. 79 Rearranging the variable and constant terms in the above equation, we get Notes on Algorithm for Drawing Contour Lines (Cont)

81. 80 The last four terms in the previous equation can be substituted by another constant term c and the coefficients of x can be rewritten as follows Notes on Algorithm for Drawing Contour Lines (Cont) Notice that we have only added and subtracted the term

82. 81 Since it is clear from Step 2 that the coefficient of x can be rewritten as Notes on Algorithm for Drawing Contour Lines (Cont) Similarly, the coefficient of y is

83. 82 Notes on Algorithm for Drawing Contour Lines (Cont) The N i computation in Step 4 is in fact calculation of the coefficient of x as shown above. Note that N i =N i-1 +2V i. Making the substitution for N i-1, we get N i =N i-2 +2V i-1 +2V i The N i computation in Step 4 is in fact calculation of the coefficient of x as shown above. Note that N i =N i-1 +2V i. Making the substitution for N i-1, we get N i =N i-2 +2V i-1 +2V i Repeating the same procedure of making substitutions for N i-2, N i-3,..., we get Repeating the same procedure of making substitutions for N i-2, N i-3,..., we get N i =N 0 +2V 1 +2V 2 +...+2V i-1 +2V 1 = N i =N 0 +2V 1 +2V 2 +...+2V i-1 +2V 1 =

84. 83 Similarly, it can be verified that Notes on Algorithm for Drawing Contour Lines (Cont)

85. 84 The above expression for the total cost function at x, y or in fact, any other point in the region [s, t] has the form y= mx + c, where the slope m = -N s /D t. This is exactly how the slopes are computed in Step 5 of the algorithm Notes on Algorithm for Drawing Contour Lines (Cont)

86. 85 3. The lines V 0, V p+1 and H 0, H q+1 are required for defining the “exterior” regions [0, j], [p, j], j = 1, 2,..., p, respectively) 4. Once we have determined the slopes of all regions, the user may choose any point (x, y) other than a point which minimizes the objective function and draw a series of contour lines in order to get a region which contains points, i.e. facility locations, yielding as good or better objective function values than (x, y) Notes on Algorithm for Drawing Contour Lines (Cont)

87. 86 Example 5: Consider Example 4. Suppose that the weight of facility 2 is not 10, but 20. Applying the median method, it can be verified that the optimal location is (10, 10) - the centroid of department 2, where immovable structures exist. It is now desired to find a feasible and “near-optimal” location using the contour line method.

88. 87 Solution:Solution: The contour line method is illustrated using Figure 1 Step 1: The vertical and horizontal lines V 1, V 2, V 2 and H 1, H 2, H 2, H 4 are drawn as shown. In addition to these lines, we also draw line V 0, V 4 and H 0, H 5 so that the “exterior regions can be identified Step 2: The weights V 1, V 2, V 2, H 1, H 2, H 2, H 4 are calculated by adding the weights of the points that fall on the respective lines. Note that for this example, p=3, and q=4

89. 88 Solution:Solution: Step 3: Since set N 0 = D 0 = -38 Step 4: Set N 1 = -38 + 2(8) = -22;D 1 = -38 + 2(6) = -26; N 2 = -22 + 2(26) = 30;D 2 = -26 + 2(4) = -18; N 3 = 30 + 2(4) = 38; D 3 = -18 + 2(8) = -2; D 4 = -2 + 2(20) = 38; (These values are entered at the bottom of each column and left of each row in figure 1) set N 0 = D 0 = -38 Step 4: Set N 1 = -38 + 2(8) = -22;D 1 = -38 + 2(6) = -26; N 2 = -22 + 2(26) = 30;D 2 = -26 + 2(4) = -18; N 3 = 30 + 2(4) = 38; D 3 = -18 + 2(8) = -2; D 4 = -2 + 2(20) = 38; (These values are entered at the bottom of each column and left of each row in figure 1)

90. 89 Solution:Solution: Step 5: Compute the slope of each region. S 00 = -(-38/-38) = -1;S 14 = -(-22/38) = 0.58; S 01 = -(-38/-26) = -1.46; S 20 = -(30/-38) = 0.79; S 02 = -(-38/-18) = -2.11; S 21 = -(30/-26) = 1.15; S 03 = -(-38/-2) = -19; S 22 = -(30/-18) = 1.67; S 04 = -(-38/38) = 1;S 23 = -(30/-2) = 15; S 10 = -(-22/-38) = -0.58; S 24 = -(30/38) = -0.79; S 11 = -(-22/-26) = -0.85; S 30 = -(38/-38) = 1; S 12 = -(-22/-18) = -1.22; S 31 = -(38/-26) = 1.46; S 13 = -(-22/-2) = -11; S 32 = -(38/-18) = 2.11;

91. 90 Solution:Solution: Step 5: Compute the slope of each region. S 33 = -(38/-2) = 19; S 34 = -(38/38) = -1; (The above slope values are shown inside each region.)

92. 91 Solution:Solution: Step 6: When we draw contour lines through point (9, 10), we get the region shown in figure 1. Since the copiers cannot be placed at the (10, 10) location, we drew contour lines through another nearby point (9, 10). Locating anywhere possible within this region give us a feasible, near-optimal solution.

93. 92 11.4.3 Single-facility Location Problem with Squared Euclidean Distances

94. 93 La Quinta Motor Inns Moderately priced, oriented towards business travelers Headquartered in San Antonio Texas Site selection - an important decision Regression Model based on location characteristics classified as: - Competitive, Demand Generators, Demographic, Market Awareness, and Physical

95. 94 La Quinta Motor Inns (Cont) Major Profitability Factors - Market awareness, hotel space, local population, low unemployment, accessibility to downtown office space, traffic count, college students, presence of military base, median income, competitive rates

96. 95 Gravity Method: As before, we substitute w= f i c i, i = 1, 2,..., m and rewrite the objective function as As before, we substitute w i = f i c i, i = 1, 2,..., m and rewrite the objective function as The cost function is

97. 96 Since the objective function can be shown to be convex, partially differentiating TC with respect to x and y, setting the resulting two equations to 0 and solving for x, y provides the optimal location of the new facility Gravity Method (Cont)

98. 97 Similarly, Gravity Method (Cont) Thus, the optimal locations x and y are simply the weighted averages of the x and y coordinates of the existing facilities

99. 98 Example 6: Consider Example 4. Suppose the distance metric to be used is squared Euclidean. Determine the optimal location of the new facility using the gravity method.

100. 99 Solution - Table 10 Department ix i y i w i w i x i w i y i 110266012 2101010100100 38686448 412544820 110266012 2101010100100 38686448 412544820 Total28272180

101. 100 Example 6. Cont... If this location is not feasible, we only need to find another point which has the nearest Euclidean distance to (9.7, 6.4) and is a feasible location for the new facility and locate the copiers there

102. 101 11.4.4 Weiszfeld Method

103. 102 Weiszfeld Method: As before, substituting w i =c i f i and taking the derivative of TC with respect to x and y yields The objective function for the single facility location problem with Euclidean distance can be written as:

104. 103 Weiszfeld Method:

105. 104 Weiszfeld Method:

106. 105 Weiszfeld Method:

107. 106 Weiszfeld Method:

108. 107 Weiszfeld Method: Step 0: Set iteration counter k = 1;

109. 108 Weiszfeld Method: Step 1: Set Step 2: If x k+1 = x k and y k+1 = y k, Stop. Otherwise, set k = k + 1 and go to Step 1 Step 1: Set Step 2: If x k+1 = x k and y k+1 = y k, Stop. Otherwise, set k = k + 1 and go to Step 1

110. 109 Example 7: Consider Example 5. Assuming the distance metric to be used is Euclidean, determine the optimal location of the new facility using the Weiszfeld method. Data for this problem is shown in Table 11.

111. 110 Table 11. Coordinates and weights for 4 departments

112. 111 Table 11: Departments #x i y i w i 11026 2101020 3868 41254 11026 2101020 3868 41254

113. 112 Solution:Solution: Using the gravity method, the initial seed can be shown to be (9.8, 7.4). With this as the starting solution, we can apply Step 1 of the Weiszfeld method repeatedly until we find that two consecutive x, y values are equal.