# 1 Chapter 2 Basic Models for the Location Problem.

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1 Chapter 2 Basic Models for the Location Problem

2 2.4Techniques for Discrete Space Location Problems 2.4Techniques for Discrete Space Location Problems - 2.4.1 Qualitative Analysis - 2.4.2 Quantitative Analysis - 2.4.3 Hybrid Analysis OutlineOutline

3 2.5 Techniques for Continuous Space Location Problems 2.5 Techniques for Continuous Space Location Problems - 2.5.1 Median Method - 2.5.2 Contour Line Method - 2.5.3 Gravity Method - 2.5.4 Weiszfeld Method Outline Cont...

4 2.4.1 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. 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.

5 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.

6 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.

7 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).

8 Table 2.2. 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

10 Table 2.3. Weighted Scores for the Three Locations in Table 2.2 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

11 Table 2.3. 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 From the analysis in Table 2.3, it is clear that Minneapolis would be the best location based on the subjective information. From the analysis in Table 2.3, it is clear that Minneapolis would be the best location based on the subjective information. Of course, as mentioned before, objective measures must be brought into consideration especially because the weighted scores for Minneapolis and Winnipeg are close. Of course, as mentioned before, objective measures must be brought into consideration especially because the weighted scores for Minneapolis and Winnipeg are close.

12 2.4.2 Quantitative Analysis

13 General Transportation Model

14 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

15 General Transportation Model

16 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.

17 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.

18 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. The expected demand at the three distribution centers and the maximum capacity at the Albany and Little Rock plants are given in Table 4.

19 Bost.Phil.Galv.Rale.Supply Capacity Albany10152220250 Little Rock1915109300 Atlanta2111136No limit Demand200100300280 Table 2.4. Costs, Demand and Supply Information

21 Example 3 Consider Example 2. In addition to Atlanta, suppose Seers, Inc., is considering another location – Pittsburgh. 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

22 Bost.Phil.Galv.Rale.Supply Capacity Albany10152220250 Little Rock1915109300 Atlanta2111136330 Pittsburgh1781812330 Demand200100300280 Table 2.10. Costs, Demand and Supply Information