6.1McGraw-Hill/Irwin

P&T Company Distribution Problem

CANNERY 1 Bellingham

CANNERY 2 Eugene

WAREHOUSE 1 Sacramento

WAREHOUSE 2 Salt Lake City

WAREHOUSE 3 Rapid City

WAREHOUSE 4 Albuquerque

CANNERY 3 Albert Lea

6.2McGraw-Hill/Irwin

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

6.3McGraw-Hill/Irwin

Current Shipping Plan

Warehouse

From \ To Sacramento Salt Lake City Rapid City Albuquerque

Cannery

Bellingham 75 0 0 0

Eugene 5 65 55 0

Albert Lea 0 0 15 85

6.4McGraw-Hill/Irwin

Shipping Cost per Truckload

Warehouse

From \ To Sacramento Salt Lake City Rapid City Albuquerque

Cannery

Bellingham $464 $513 $654 $867

Eugene 352 416 690 791

Albert Lea 995 682 388 685

Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685)= $165,595

6.5McGraw-Hill/Irwin

Terminology for a Transportation Problem

P&T Company Problem

Truckloads of canned peas

Canneries

Warehouses

Output from a cannery

Allocation to a warehouse

Shipping cost per truckload from a cannery to a warehouse

General Model

Units of a commodity

Sources

Destinations

Supply from a source

Demand at a destination

Cost per unit distributed from a source to a destination

6.6McGraw-Hill/Irwin

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.

6.7McGraw-Hill/Irwin

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.

6.8McGraw-Hill/Irwin

The P&T Co. Transportation Problem

Unit Cost

Destination(Warehouse): Sacramento Salt Lake City Rapid City Albuquerque Supply

Source (Cannery)

Bellingham $464 $513 $654 $867 75

Eugene 352 416 690 791 125

Albert Lea 995 682 388 685 100

Demand 80 65 70 85

6.9McGraw-Hill/Irwin

Spreadsheet Formulation

34567891011121314151617

B C D E F G H I JUnit Cost Destination (Warehouse)

Sacramento Salt Lake City Rapid City AlbuquerqueSource Bellingham $464 $513 $654 $867

(Cannery) Eugene $352 $416 $690 $791Albert Lea $995 $682 $388 $685

Shipment Quantity Destination (Warehouse)(Truckloads) Sacramento Salt Lake City Rapid City Albuquerque Total Shipped Supply

Source Bellingham 0 20 0 55 75 = 75(Cannery) Eugene 80 45 0 0 125 = 125

Albert Lea 0 0 70 30 100 = 100Total Received 80 65 70 85

= = = = Total CostDemand 80 65 70 85 $152,535

6.10McGraw-Hill/Irwin

Network Representation

S1

S2

S3

D4

D2

D1

D3

75

125

100

80

65

70

85

Supplies Demands

SourcesDestinations

(Bellingham)

(Eugene)

(Alber t Lea)

(Sacramento)

(Salt Lake City)

(Rapid City)

(Albuquerque)

464513

654867

352 416690

791

995 682

685

388

6.11McGraw-Hill/Irwin

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 = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22

+ $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34

subject toCannery 1: x11 + x12 + x13 + x14 = 75Cannery 2: x21 + x22 + x23 + x24 = 125Cannery 3: x31 + x32 + x33 + x34 = 100Warehouse 1: x11 + x21 + x31 = 80Warehouse 2: x12 + x22 + x32 = 65Warehouse 3: x13 + x23 + x33 = 70Warehouse 4: x14 + x24 + x34 = 85

andxij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

6.12McGraw-Hill/Irwin

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.

6.13McGraw-Hill/Irwin

Location of Texago’s Facilities

Type of Facility Locations

Oil fields 1. Several in Texas2. Several in California3. Several in Alaska

Refineries 1. Near New Orleans, Louisiana2. Near Charleston, South Carolina3. Near Seattle, Washington

Distribution Centers 1. Pittsburgh, Pennsylvania2. Atlanta, Georgia3. Kansas City, Missouri4. San Francisco, California

6.14McGraw-Hill/Irwin

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.

6.15McGraw-Hill/Irwin

Production Data for Texago

Refinery

Crude OilNeeded 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

6.16McGraw-Hill/Irwin

Cost Data for Shipping to Refineries

Cost per Unit Shipped to Refinery or Potential Refinery(Millions of Dollars per Million Barrels)

New Orleans Charleston Seattle

Los Angeles Galveston St. Louis

Source

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

6.17McGraw-Hill/Irwin

Cost Data for Shipping to Distribution Centers

Cost per Unit Shipped to Distribution Center(Millions of Dollars)

Pittsburgh Atlanta Kansas City San Francisco

Refinery

New Orleans 6.5 5.5 6 8

Charleston 7 5 4 7

Seattle 7 8 4 3

Potential Refinery

Los Angeles 8 6 3 2

Galveston 5 4 3 6

St. Louis 4 3 1 5

Number of units needed 100 80 80 100

6.18McGraw-Hill/Irwin

Estimated Operating Costs for Refineries

Site Annual Operating Cost(Millions of Dollars)

Los Angeles

Galveston

St. Louis

620

570

530

6.19McGraw-Hill/Irwin

Basic Spreadsheet for Shipping to Refineries

34567891011121314151617181920

B C D E F G H I JRefineries

Unit Cost ($millions) New Orleans Charleston Seattle New SiteTexas 2 4 5

Oil California 5 5 3Fields Alaska 5 7 3

Middle East 2 3 5

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle New Site Total Shipped Supply

Texas 0 0 0 0 0 = 80Oil California 0 0 0 0 0 = 60

Fields Alaska 0 0 0 0 0 = 100Middle East 0 0 0 0 0 = 120

Total Received 0 0 0 0= = = = Total Cost

Demand 100 60 80 120 ($millions)0

6.20McGraw-Hill/Irwin

Shipping to Refineries, Including Los Angeles

34567891011121314151617181920

B C D E F G H I JRefineries

Unit Cost ($millions) New Orleans Charleston Seattle Los AngelesTexas 2 4 5 3

Oil California 5 5 3 1Fields Alaska 5 7 3 4

Middle East 2 3 5 4

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle Los Angeles Total Shipped Supply

Texas 40 0 0 40 80 = 80Oil California 0 0 0 60 60 = 60

Fields Alaska 0 0 80 20 100 = 100Middle East 60 60 0 0 120 = 120

Total Received 100 60 80 120= = = = Total Cost

Demand 100 60 80 120 ($millions)880

6.21McGraw-Hill/Irwin

Shipping to Refineries, Including Galveston

34567891011121314151617181920

B C D E F G H I JRefineries

Unit Cost ($millions) New Orleans Charleston Seattle GalvestonTexas 2 4 5 1

Oil California 5 5 3 3Fields Alaska 5 7 3 5

Middle East 2 3 5 3

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle Galveston Total Shipped Supply

Texas 20 0 0 60 80 = 80Oil California 0 0 0 60 60 = 60

Fields Alaska 20 0 80 0 100 = 100Middle East 60 60 0 0 120 = 120

Total Received 100 60 80 120= = = = Total Cost

Demand 100 60 80 120 ($millions)920

6.22McGraw-Hill/Irwin

Shipping to Refineries, Including St. Louis

34567891011121314151617181920

B C D E F G H I JRefineries

Unit Cost ($millions) New Orleans Charleston Seattle St. LouisTexas 2 4 5 1

Oil California 5 5 3 4Fields Alaska 5 7 3 7

Middle East 2 3 5 4

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle St. Louis Total Shipped Supply

Texas 0 0 0 80 80 = 80Oil California 0 20 0 40 60 = 60

Fields Alaska 20 0 80 0 100 = 100Middle East 80 40 0 0 120 = 120

Total Received 100 60 80 120= = = = Total Cost

Demand 100 60 80 120 ($millions)960

6.23McGraw-Hill/Irwin

Basic Spreadsheet for Shipping to D.C.’s

34567891011121314151617181920

B C D E F G H I JDistribution Center

Unit Cost ($millions) Pittsburgh Atlanta Kansas City San FranciscoNew Orleans 6.5 5.5 6 8

Refineries Charleston 7 5 4 7Seattle 7 8 4 3

New Site

Shipment Quantity Distribution Center(millions of barrels) Pittsburgh Atlanta Kansas City San Francisco Total Shipped Supply

New Orleans 0 0 0 0 0 = 100Refineries Charleston 0 0 0 0 0 = 60

Seattle 0 0 0 0 0 = 80New Site 0 0 0 0 0 = 120

Total Received 0 0 0 0= = = = Total Cost

Demand 100 80 80 100 ($millions)0

6.24McGraw-Hill/Irwin

Shipping to D.C.’s When Choose Los Angeles

34567891011121314151617181920

B C D E F G H I JDistribution Center

Unit Cost ($millions) Pittsburgh Atlanta Kansas City San FranciscoNew Orleans 6.5 5.5 6 8

Refineries Charleston 7 5 4 7Seattle 7 8 4 3

Los Angeles 8 6 3 2

Shipment Quantity Distribution Center(millions of barrels) Pittsburgh Atlanta Kansas City San Francisco Total Shipped Supply

New Orleans 80 20 0 0 100 = 100Refineries Charleston 0 60 0 0 60 = 60

Seattle 20 0 0 60 80 = 80Los Angeles 0 0 80 40 120 = 120

Total Received 100 80 80 100= = = = Total Cost

Demand 100 80 80 100 ($millions)1,570

6.25McGraw-Hill/Irwin

Shipping to D.C.’s When Choose Galveston

34567891011121314151617181920

B C D E F G H I JDistribution Center

Unit Cost ($millions) Pittsburgh Atlanta Kansas City San FranciscoNew Orleans 6.5 5.5 6 8

Refineries Charleston 7 5 4 7Seattle 7 8 4 3

Galveston 5 4 3 6

Shipment Quantity Distribution Center(millions of barrels) Pittsburgh Atlanta Kansas City San Francisco Total Shipped Supply

New Orleans 100 0 0 0 100 = 100Refineries Charleston 0 60 0 0 60 = 60

Seattle 0 0 0 80 80 = 80Galveston 0 20 80 20 120 = 120

Total Received 100 80 80 100= = = = Total Cost

Demand 100 80 80 100 ($millions)1,630

6.26McGraw-Hill/Irwin

Shipping to D.C.’s When Choose St. Louis

34567891011121314151617181920

B C D E F G H I JDistribution Center

Unit Cost ($millions) Pittsburgh Atlanta Kansas City San FranciscoNew Orleans 6.5 5.5 6 8

Refineries Charleston 7 5 4 7Seattle 7 8 4 3

St. Louis 4 3 1 5

Shipment Quantity Distribution Center(millions of barrels) Pittsburgh Atlanta Kansas City San Francisco Total Shipped Supply

New Orleans 100 0 0 0 100 = 100Refineries Charleston 0 60 0 0 60 = 60

Seattle 0 0 0 80 80 = 80St. Louis 0 20 80 20 120 = 120

Total Received 100 80 80 100= = = = Total Cost

Demand 100 80 80 100 ($millions)1,430

6.27McGraw-Hill/Irwin

Annual Variable Costs

Site

Total Costof ShippingCrude Oil

Total Costof Shipping

Finished Product

Operating Costfor NewRefinery

TotalVariable

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