1

Chapter 2Chapter 2

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

Chapter 2Chapter 2

Basic ModelsBasic Modelsfor thefor the

Location ProblemLocation Problem

2

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

- 2.4.1 Qualitative Analysis2.4.1 Qualitative Analysis

- 2.4.2 Quantitative Analysis2.4.2 Quantitative Analysis

- 2.4.3 Hybrid Analysis2.4.3 Hybrid Analysis

OutlineOutlineOutlineOutline

3

• 2.5 Techniques for Continuous Space 2.5 Techniques for Continuous Space Location ProblemsLocation Problems

- 2.5.1 Median Method2.5.1 Median Method

- 2.5.2 Contour Line Method2.5.2 Contour Line Method

- 2.5.3 Gravity Method2.5.3 Gravity Method

- 2.5.4 Weiszfeld Method2.5.4 Weiszfeld Method

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

4

2.4.12.4.1Qualitative AnalysisQualitative Analysis2.4.12.4.1Qualitative AnalysisQualitative Analysis

Step 1Step 1: List all the factors that are important, i.e. have an : List all the factors that are important, i.e. have an impact on the location decision.impact on the location decision.

Step 2:Step 2: Assign appropriate weights (typically between 0 Assign appropriate weights (typically between 0 and 1) to each factor based on the relative and 1) to each factor based on the relative importance of each.importance of each.

Step 3:Step 3: Assign a score (typically between 0 and 100) for Assign a score (typically between 0 and 100) for each location with respect to each factor identified in each location with respect to each factor identified in Step 1.Step 1.

Step 4:Step 4: Compute the weighted score for each factor for Compute the weighted score for each factor for each location by multiplying its weight with the each location by multiplying its weight with the corresponding score (which were assigned Steps 2 corresponding score (which were assigned Steps 2 and 3, respectively)and 3, respectively)

Step 5Step 5: Compute the sum of the weighted scores for : Compute the sum of the weighted scores for each location and choose a location based on these each location and choose a location based on these scores.scores.

5

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

6

Example 1: Cont...Example 1: Cont...Example 1: Cont...Example 1: Cont...

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

7

Solution:Solution:Solution:Solution:

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

8

Table 2.2. Factors and Weights for Table 2.2. Factors and Weights for Three LocationsThree LocationsTable 2.2. Factors and Weights for Table 2.2. 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

9

Table 2.2. Cont...Table 2.2. Cont...Table 2.2. Cont...Table 2.2. Cont...

Wt.Wt. FactorsFactors LocationLocation

Minn.Minn. Winn.Winn. Spring.Spring.

.08.08 Insurance costsInsurance costs 7070 9595 6060

.07.07 Office servicesOffice services 9090 9090 8080

Click here

10

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

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

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

11

Table 2.3. Cont... Table 2.3. Cont... Table 2.3. Cont... Table 2.3. 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

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

•From the analysis in Table 2.3, it is clear that From the analysis in Table 2.3, it is clear that Minneapolis would be the best location based Minneapolis would be the best location based on the subjective information.on the subjective information.

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

12

2.4.22.4.2Quantitative AnalysisQuantitative Analysis

2.4.22.4.2Quantitative AnalysisQuantitative Analysis

13

General Transportation ModelGeneral Transportation ModelGeneral Transportation ModelGeneral Transportation Model

14

General Transportation ModelGeneral Transportation ModelGeneral Transportation ModelGeneral Transportation Model

ParametersParameters

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

warehouse warehouse ii to customer to customer jj

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

bbii: demand at customer : demand at customer jj

Decision VariablesDecision Variables

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

warehouse warehouse ii to customer to customer jj

15

General Transportation ModelGeneral Transportation ModelGeneral Transportation ModelGeneral Transportation Model

m

i

n

jijij xcZ

1 1

Costtion Transporta Total Minimize

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

Subject to

1

n

jiij ax

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

m

ijij bx

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

16

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

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

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

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

17

Transportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex AlgorithmTransportation Simplex Algorithm

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

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

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

18

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

19

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

CapacityCapacity

AlbanyAlbany 1010 1515 2222 2020 250250

Little RockLittle Rock 1919 1515 1010 99 300300

AtlantaAtlanta 2121 1111 1313 66 No limitNo limit

DemandDemand 200200 100100 300300 280280

Table 2.4. Costs, Demand and Table 2.4. Costs, Demand and Supply InformationSupply InformationTable 2.4. Costs, Demand and Table 2.4. Costs, Demand and Supply InformationSupply Information

20

Table 2.5. Transportation Model Table 2.5. Transportation Model with Plant at Atlantawith Plant at AtlantaTable 2.5. Transportation Model Table 2.5. 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 880880

DemandDemand 200200 100100 300300 280280 880880

Click here for Excel formulation for Excel formulation

21

Example 3Example 3Example 3Example 3

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

22

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

PittsburghPittsburgh 1717 88 1818 1212 330330

DemandDemand 200200 100100 300300 280280

Table 2.10. Costs, Demand and Table 2.10. Costs, Demand and Supply InformationSupply InformationTable 2.10. Costs, Demand and Table 2.10. Costs, Demand and Supply InformationSupply Information

23

Table 2.12. Transportation Model Table 2.12. Transportation Model with Plant at Pittsburghwith Plant at PittsburghTable 2.12. Transportation Model Table 2.12. Transportation Model with Plant at Pittsburghwith Plant at Pittsburgh

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

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

Click here for Excel model for Excel model