PT Company Distribution Problem Mc GrawHillIrwin 1 Shipping

P&T Company Distribution Problem Mc. Graw-Hill/Irwin 1

Shipping Data Cannery Output Warehouse Allocation Bellingham 75 truckloads Sacramento 80 truckloads Eugene 125 truckloads Salt Lake City 65 truckloads Albert Lea 100 truckloads Rapid City 70 truckloads Total 300 truckloads Albuquerque 85 truckloads Total 300 truckloads Mc. Graw-Hill/Irwin 2

Current Shipping Plan Warehouse To Sacramento Salt Lake City Rapid City Albuquerque Bellingham 75 0 0 0 Eugene 5 65 55 0 Albert Lea 0 0 15 85 From Cannery Mc. Graw-Hill/Irwin 3

Shipping Cost per Truckload Warehouse To Sacramento Salt Lake City Rapid City Albuquerque Bellingham $464 $513 $654 $867 Eugene 352 416 690 791 Albert Lea 995 682 388 685 From Cannery Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165, 595 Mc. Graw-Hill/Irwin 4

Terminology for a Transportation Problem P&T Company Problem General Model Truckloads of canned peas Units of a commodity Canneries Sources Warehouses Destinations Output from a cannery Supply from a source Allocation to a warehouse Demand at a destination Shipping cost per truckload from a cannery to a warehouse Cost per unit distributed from a source to a destination Mc. Graw-Hill/Irwin 5

Characteristics of Transportation Problems • The Requirements Assumption – Each source has a fixed supply of units, where this entire supply must be distributed to the destinations. – Each destination has a fixed demand for units, where this entire demand must be received from the sources. • The Feasible Solutions Property – A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands. • The Cost Assumption – The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed. – This cost is just the unit cost of distribution times the number of units distributed. Mc. Graw-Hill/Irwin 6

The Transportation Model Any problem (whether involving transportation or not) fits the model for a transportation problem if 1. It can be described completely in terms of a table like Table 6. 5 that identifies all the sources, destinations, supplies, demands, and unit costs, and 2. satisfies both the requirements assumption and the cost assumption. The objective is to minimize the total cost of distributing the units. Mc. Graw-Hill/Irwin 7

The P&T Co. Transportation Problem Unit Cost Destination (Warehouse): Sacramento Salt Lake City Rapid City Albuquerque Supply Bellingham $464 $513 $654 $867 75 Eugene 352 416 690 791 125 Albert Lea 995 682 388 685 100 Demand 80 65 70 85 Source (Cannery) Mc. Graw-Hill/Irwin 8

Spreadsheet Formulation Mc. Graw-Hill/Irwin 9

Network Representation Mc. Graw-Hill/Irwin 10

The Transportation Problem is an LP Let xij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4) Minimize Cost = $464 x 11 + $513 x 12 + $654 x 13 + $867 x 14 + $352 x 21 + $416 x 22 + $690 x 23 + $791 x 24 + $995 x 31 + $682 x 32 + $388 x 33 + $685 x 34 subject to Cannery 1: x 11 + x 12 + x 13 + x 14 = 75 Cannery 2: x 21 + x 22 + x 23 + x 24 = 125 Cannery 3: x 31 + x 32 + x 33 + x 34 = 100 Warehouse 1: x 11 + x 21 + x 31 = 80 Warehouse 2: x 12 + x 22 + x 32 = 65 Warehouse 3: x 13 + x 23 + x 33 = 70 Warehouse 4: x 14 + x 24 + x 34 = 85 and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4) Mc. Graw-Hill/Irwin 11

Integer Solutions Property As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables. Therefore, it is not necessary to add constraints to the model that restrict these variables to only have integer values. Mc. Graw-Hill/Irwin 12

Location of Texago’s Facilities Type of Facility Locations Oil fields 1. Several in Texas 2. Several in California 3. Several in Alaska Refineries 1. Near New Orleans, Louisiana 2. Near Charleston, South Carolina 3. Near Seattle, Washington Distribution Centers 1. Pittsburgh, Pennsylvania 2. Atlanta, Georgia 3. Kansas City, Missouri 4. San Francisco, California Mc. Graw-Hill/Irwin 13

Potential Sites for Texago’s New Refinery Potential Site Main Advantages Near Los Angeles, California 1. Near California oil fields. 2. Ready access from Alaska oil fields. 3. Fairly near San Francisco distribution center. Near Galveston, Texas 1. Near Texas oil fields. 2. Ready access from Middle East imports. 3. Near corporate headquarters. Near St. Louis, Missouri 1. Low operating costs. 2. Centrally located for distribution centers. 3. Ready access to crude oil via the Mississippi River. Mc. Graw-Hill/Irwin 14

Production Data for Texago Refinery Crude Oil Needed Annually (Million Barrels) Oil Fields Crude Oil Produced Annually (Million Barrels) New Orleans 100 Texas 80 Charleston 60 California 60 Seattle 80 Alaska 100 New site 120 Total 240 Total 360 Needed imports = 360 – 240 = 120 Mc. Graw-Hill/Irwin 15

Cost Data for Shipping to Refineries Cost per Unit Shipped to Refinery or Potential Refinery (Millions of Dollars per Million Barrels) New Orleans Seattle Los Angeles Charleston Galveston St. Louis Texas 2 4 5 3 1 1 California 5 5 3 1 3 4 Alaska 5 7 3 4 5 7 Middle East 2 3 5 4 3 4 Source Mc. Graw-Hill/Irwin 16

Cost Data for Shipping to Distribution Centers Cost per Unit Shipped to Distribution Center (Millions of Dollars) Pittsburgh Atlanta Kansas City San Francisco 6. 5 5. 5 6 8 Charleston 7 5 4 7 Seattle 7 8 4 3 Los Angeles 8 6 3 2 Galveston 5 4 3 6 St. Louis 4 3 1 5 100 80 80 100 Refinery New Orleans Potential Refinery Number of units needed Mc. Graw-Hill/Irwin 17

Estimated Operating Costs for Refineries Site Mc. Graw-Hill/Irwin Annual Operating Cost (Millions of Dollars) Los Angeles 620 Galveston 570 St. Louis 530 18

Basic Spreadsheet for Shipping to Refineries Mc. Graw-Hill/Irwin 19

Shipping to Refineries, Including Los Angeles Mc. Graw-Hill/Irwin 20

Shipping to Refineries, Including Galveston Mc. Graw-Hill/Irwin 21

Shipping to Refineries, Including St. Louis Mc. Graw-Hill/Irwin 22

Basic Spreadsheet for Shipping to D. C. ’s Mc. Graw-Hill/Irwin 23

Shipping to D. C. ’s When Choose Los Angeles Mc. Graw-Hill/Irwin 24

Shipping to D. C. ’s When Choose Galveston Mc. Graw-Hill/Irwin 25

Shipping to D. C. ’s When Choose St. Louis Mc. Graw-Hill/Irwin 26

Annual Variable Costs Total Cost of Shipping Crude Oil Total Cost of Shipping Finished Product Operating Cost for New Refinery Total Variable Cost Los Angeles $880 million $1. 57 billion $620 million $3. 07 billion Galveston 920 million 1. 63 billion 570 million 3. 12 billion St. Louis 960 million 1. 43 billion 530 million 2. 92 billion Site Mc. Graw-Hill/Irwin 27