mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2008 15.1 table of contents cd chapter 15...

© The McGraw-Hill Companies, Inc., 2008

15.1McGraw-Hill/Irwin

Table of ContentsCD Chapter 15 (Transportation and Assignment Problems)

The P&T Company Distribution Problem (Section 15.1) 15.2–15.5Characteristics of Transportation Problems (Section 15.2) 15.6–15.14Variants of Transportation Problems: Better Products (Section 15.3) 15.15–15.17Variants of Transportation Problems: Nifty (Section 15.3) 15.18–15.20Applications of Transportation Problems: Metro Water (Section 15.4) 15.21–15.22Applications of Transportation Problems: Northern Airplane (Section 15.4) 15.23–15.25Applications of Transportation Problems: Middletown (Section 15.4) 15.26–15.28Applications of Transportation Problems: Energetic (Section 15.4) 15.29–15.31A Case Study: Texago Corp. Site Selection Problem (Section 15.5) 15.32–15.46Characteristics of Assignment Problems: Sellmore (Section 15.6) 15.47–15.51Variants of Assignment Problems: Job Shop (Section 15.7) 15.52-15.54Variants of Assignment Problems: Better Products (Section 15.7) 15.55Variants of Assignment Problems: Revised Middletown (Section 15.7) 15.56

Transportation & Assignment Problems (UW Lecture) 15.57–15.75These slides are based upon a lecture to second-year MBA students at the University of Washington that discusses transportation and assignment problems (as taught by one of the authors).



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



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



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



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



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



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.



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 15.5 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.



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



Spreadsheet Formulation

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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



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



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)



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.



Distribution System at Proctor and Gamble

• Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

– 50 product categories

– 60 plants

– 15 distribution centers

– 1000 customer zones

• Solved many transportation problems (one for each product category).

• Goal: find best distribution plan, which plants to keep open, etc.

• Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

• Implemented in 1996. Saved $200 million per year.

For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”



Better Products (Assigning Plants to Products)

The Better Products Company has decided to initiate the product of four new products, using three plants that currently have excess capacity.

Unit Cost

Product: 1 2 3 4CapacityAvailable

Plant

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Required production 20 30 30 40

Question: Which plants should produce which products?



Transportation Problem Formulation

Unit Cost

Destination (Product): 1 2 3 4 Supply

Source(Plant)

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Demand 20 30 30 40




345678910111213141516

B C D E F G H IUnit Cost Product 1 Product 2 Product 3 Product 4

Plant 1 $41 $27 $28 $24Plant 2 $40 $29 - $23Plant 3 $37 $30 $27 $21

ProducedDaily Production Product 1 Product 2 Product 3 Product 4 At Plant Capacity

Plant 1 0 30 30 0 60 <= 75Plant 2 0 0 0 15 15 <= 75Plant 3 20 0 0 25 45 <= 45

Products Produced 20 30 30 40= = = = Total Cost

Required Production 20 30 30 40 $3,260



Nifty Co. (Choosing Customers)

• The Nifty Company specializes in the production of a single product, which it produces in three plants.

• Four customers would like to make major purchases. There will be enough to meet their minimum purchase requirements, but not all of their requested purchases.

• Due largely to variations in shipping cost, the net profit per unit sold varies depending on which plant supplies which customer.

Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?



Data for the Nifty Company

Unit Cost

Product: 1 2 3 4CapacityAvailable

Plant

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Required production 20 30 30 40

Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?




345678910111213141516171819

B C D E F G H IUnit Profit Customer 1 Customer 2 Customer 3 Customer 4

Plant 1 $55 $42 $46 $53Plant 2 $37 $18 $32 $48Plant 3 $29 $59 $51 $35

Total ProductionShipment Customer 1 Customer 2 Customer 3 Customer 4 Production Quantity

Plant 1 7,000 0 1,000 0 8,000 = 8,000Plant 2 0 0 0 5,000 5,000 = 5,000Plant 3 0 6,000 1,000 0 7,000 = 7,000

Min Purchase 7,000 3,000 2,000 0<= <= <= <= Total Profit

Total Shipped 7,000 6,000 2,000 5,000 $1,076,000<= <= <= <=

Max Purchase 7,000 9,000 6,000 8,000



Metro Water (Distributing Natural Resources)

Metro Water District is an agency that administers water distribution in a large goegraphic region. The region is arid, so water must be brought in from outside the region.

– Sources of imported water: Colombo, Sacron, and Calorie rivers.

– Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass.

Cost per Acre Foot

Berdoo Los Devils San Go Hollyglass Available

Colombo River $160 $130 $220 $170 5

Sacron River 140 130 190 150 6

Calorie River 190 200 230 — 5

Needed 2 5 4 1.5(million

acre feet)

Question: How much water should Metro take from each river, and how much should they send from each river to each city?




34567891011121314151617

B C D E F G H IUnit Cost ($millions) Berdoo Los Devils San Go Hollyglass

Colombo River 160 130 220 170Sacron River 140 130 190 150Calorie River 190 200 230 -

Water Distribution Total(million acre-feet) Berdoo Los Devils San Go Hollyglass From River Available

Colombo River 0 5 0 0 5 <= 5Sacron River 2 0 2.5 1.5 6 <= 6Calorie River 0 0 1.5 0 1.5 <= 5Total To City 2 5 4 1.5

= = = = Total CostNeeded 2 5 4 1.5 ($million)

1,975



Northern Airplane (Production Scheduling)

Northern Airplane Company produces commercial airplanes. The last stage in production is to produce the jet engines and install them.

– The company must meet the delivery deadline indicated in column 2.

– Production and storage costs vary from month to month.

Maximum ProductionUnit Cost of

Production ($million)

Unit Costof Storage

($thousand)MonthScheduled

InstallationsRegular

Time OvertimeRegular

Time Overtime

1 10 20 10 1.08 1.10 15

2 15 30 15 1.11 1.12 15

3 25 25 10 1.10 1.11 15

4 20 5 10 1.13 1.15

Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be minimized?




3456789101112131415161718192021222324252627282930313233343536

B C D E F G H I JProduction Cost Regular Storage Cost($millions) Time Overtime ($millions per month)

Month 1 1.08 1.10 0.015Month 2 1.11 1.12Month 3 1.10 1.11Month 4 1.13 1.15

Unit Cost($millions) 1 2 3 4

1 (RT) 1.08 1.10 1.11 1.131 (OT) 1.10 1.12 1.13 1.152 (RT) - 1.11 1.13 1.14

Month 2 (OT) - 1.12 1.14 1.15Produced 3 (RT) - - 1.10 1.12

3 (OT) - - 1.11 1.134 (RT) - - - 1.134 (OT) - - - 1.15

MaximumUnits Produced 1 2 3 4 Produced Production

1 (RT) 10 5 0 5 20 <= 201 (OT) 0 0 0 0 0 <= 102 (RT) 0 10 0 0 10 <= 30

Month 2 (OT) 0 0 0 0 0 <= 15Produced 3 (RT) 0 0 25 0 25 <= 25

3 (OT) 0 0 0 10 10 <= 104 (RT) 0 0 0 5 5 <= 54 (OT) 0 0 0 0 0 <= 10

Installed 10 15 25 20= = = = Total Cost

Scheduled Installations 10 15 25 20 ($millions)77.4

Month Installed

Month Installed



Optimal Production at Northern Airplane

Month

1 (RT)

2 (RT)

3 (RT)

3 (OT)

4 (RT)

Production

20

10

25

10

5

Installations

10

15

25

0

20

Stored

10

5

5

10

0



Middletown School District

• Middletown School District is opening a third high school and thus needs to redraw the boundaries for the area of the city that will be assigned to the respective schools.

• The city has been divided into 9 tracts with approximately equal populations.

• Each school has a minimum and maximum number of students that should be assigned.

• The school district management has decided that the appropriate objective is to minimize the average distance that students must travel to school.

Question: How many students from each tract should be assigned to each school?



Data for the Middletown School District

Distance (Miles) to School

Tract 1 2 3Number of High School Students

1 2.2 1.9 2.5 500

2 1.4 1.3 1.7 400

3 0.5 1.8 1.1 450

4 1.2 0.3 2.0 400

5 0.9 0.7 1.0 500

6 1.1 1.6 0.6 450

7 2.7 0.7 1.5 450

8 1.8 1.2 0.8 400

9 1.5 1.7 0.7 500

Minimum enrollment 1,200 1,100 1,000

Maximum enrollment 1,800 1,700 1,500




345678910111213141516171819202122232425262728293031

B C D E F G HDistance (Miles) School 1 School 2 School 3

Tract 1 2.2 1.9 2.5Tract 2 1.4 1.3 1.7Tract 3 0.5 1.8 1.1Tract 4 1.2 0.3 2Tract 5 0.9 0.7 1Tract 6 1.1 1.6 0.6Tract 7 2.7 0.7 1.5Tract 8 1.8 1.2 0.8Tract 9 1.5 1.7 0.7

Number of Total TotalStudents School 1 School 2 School 3 From Tract In Tract

Tract 1 0 500 0 500 = 500Tract 2 400 0 0 400 = 400Tract 3 450 0 0 450 = 450Tract 4 0 400 0 400 = 400Tract 5 350 150 0 500 = 500Tract 6 0 0 450 450 = 450Tract 7 0 450 0 450 = 450Tract 8 0 0 400 400 = 400Tract 9 0 0 500 500 = 500

Min Enrollment 1,200 1,500 1,350<= <= <= Total Distance

Total At School 1,200 1,500 1,350 (miles)<= <= <= 3,530

Max Enrollment 1,800 1,700 1,500



Energetic (Meeting Energy Needs)

• The Energetic Company needs to make plans for the energy systems for a new building.

• The energy needs fall into three categories:– electricity (20 units)

– heating water (10 units)

– heating space (30 units)

• The three possible sources of energy are– electricity

– natural gas

– solar heating unit (limited to 30 units because of roof size)

Question: How should Energetic meet the energy needs for the new building?



Cost Data for Energetic

Unit Cost

Energy Need: Electricity Water Heating Space Heating

Source of Energy

Electricity $400 $500 $600

Natural gas — 600 500

Solar heater — 300 400




3456789101112131415161718

B C D E F G H IEnergy Need

Unit Cost ($/day) Electricity Water Heating Space HeatingSource Electricity 400 500 600

of Natural Gas - 600 500Energy Solar Heater - 300 400

Energy Need TotalDaily Energy Use Electricity Water Heating Space Heating Used

Source Electricity 20 0 0 20of Natural Gas 0 0 10 10 Max Solar

Energy Solar Heater 0 10 20 30 <= 30Total Supplied 20 10 30

= = = Total CostDemand 20 10 30 ($/day)

24,000



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, Lousiana2. Near Charleston, South Carolina3. Near Seattle, Washington

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



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.



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



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



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



Estimated Operating Costs for Refineries

Site Annual Operating Cost(Millions of Dollars)

Los Angeles

Galveston

St. Louis

620

570

530



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



Shipping to Refineries, Including Los Angeles

34567891011121314151617181920


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







Shipping to Refineries, Including Galveston

34567891011121314151617181920


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


Middle East 2 3 5 3

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







Shipping to Refineries, Including St. Louis

34567891011121314151617181920


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


Middle East 2 3 5 4

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







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





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

34567891011121314151617181920




Los Angeles 8 6 3 2



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


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



Shipping to D.C.’s When Choose Galveston

34567891011121314151617181920




Galveston 5 4 3 6



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


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



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

34567891011121314151617181920




St. Louis 4 3 1 5



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


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



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



Sellmore Company Assignment Problem

• The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.

• He is hiring four temporary employees:– Ann

– Ian

– Joan

– Sean

• Each will handle one of the following four tasks:– Word processing of written presentations

– Computer graphics for both oral and written presentations

– Preparation of conference packets, including copying and organizing materials

– Handling of advance and on-site registration for the conference

Question: Which person should be assigned to which task?



Data for the Sellmore Problem

Required Time per Task (Hours)

TemporaryEmployee

WordProcessing Graphics Packets Registrations

HourlyWage

Ann 35 41 27 40 $14

Ian 47 45 32 51 12

Joan 39 56 36 43 13

Sean 32 51 25 46 15




3456789101112131415161718192021222324252627282930

B C D E F G H I J

Required Time Word Hourly(Hours) Processing Graphics Packets Registrations Wage

Ann 35 41 27 40 $14Assignee Ian 47 45 32 51 $12

Joan 39 56 36 43 $13Sean 32 51 25 46 $15

WordCost Processing Graphics Packets Registrations

Ann $490 $574 $378 $560Assignee Ian $564 $540 $384 $612

Joan $507 $728 $468 $559Sean $480 $765 $375 $690

Word TotalAssignment Processing Graphics Packets Registrations Assignments Supply

Ann 0 0 1 0 1 = 1Assignee Ian 0 1 0 0 1 = 1

Joan 0 0 0 1 1 = 1Sean 1 0 0 0 1 = 1

Total Assigned 1 1 1 1= = = = Total Cost

Demand 1 1 1 1 $1,957

Task

Task

Task



The Model for Assignment Problems

Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.

To fit the model for an assignment problem, the following assumptions need to be satisfied:

1. The number of assignees and the number of tasks are the same.

2. Each assignee is to be assigned to exactly one task.

3. Each task is to be performed by exactly one assignee.

4. There is a cost associated with each combination of an assignee performing a task.

5. The objective is to determine how all the assignments should be made to minimize the total cost.



The Network Representation

A2

A1

T4A4

T3A3

T2

T1

Assignees Tasks

490

540

468

690

(Ann)

(I an)

(Joan)

(Sean)

(Word processing)

(Graphics)

(Packets)

(Registrations)

574

378560

564

384612

507 728

559

480

765

375



Job Shop (Assigning Machines to Locations)

• The Job Shop Company has purchased three new machines of different types.

• There are five available locations where the machine could be installed.

• Some of these locations are more desirable for particular machines because of their proximity to work centers that will have a heavy work flow to these machines.

Question: How should the machines be assigned to locations?



Materials-Handling Cost Data

Cost per Hour

Location: 1 2 3 4 5

Machine

1 $13 $16 $12 $14 $15

2 15 — 13 20 16

3 4 7 10 6 7




34567891011121314151617

B C D E F G H I JCost ($/hour) Location 1 Location 2 Location 3 Location 4 Location 5

Machine 1 13 16 12 14 15Machine 2 15 - 13 20 16Machine 3 4 7 10 6 7

TotalAssignment Location 1 Location 2 Location 3 Location 4 Location 5 Assignments Supply

Machine 1 0 0 0 1 0 1 = 1Machine 2 0 0 1 0 0 1 = 1Machine 3 1 0 0 0 0 1 = 1

Total Assigned 1 0 1 1 0<= <= <= <= <= Total Cost

Demand 1 1 1 1 1 ($/hour)31



Better Products (No Product Splitting)

3456789101112131415161718192021222324

B C D E F G H IUnit Cost Product 1 Product 2 Product 3 Product 4

Plant 1 $41 $27 $28 $24Plant 2 $40 $29 - $23Plant 3 $37 $30 $27 $21

Required Production 20 30 30 40

Cost ($/day) Product 1 Product 2 Product 3 Product 4Plant 1 $820 $810 $840 $960Plant 2 $800 $870 - $920Plant 3 $740 $900 $810 $840

TotalAssignment Product 1 Product 2 Product 3 Product 4 Assignments Supply

Plant 1 0 1 1 0 2 <= 2Plant 2 1 0 0 0 1 <= 2Plant 3 0 0 0 1 1 = 1

Total Assigned 1 1 1 1= = = = Total Cost

Demand 1 1 1 1 $3,290



Middletown School District (No Tract Splitting)

3456789101112131415161718192021222324252627282930

B C D E F G H I J KDistance Number of Cost(Miles) School 1 School 2 School 3 Students (Miles) School 1 School 2 School 3

Tract 1 2.2 1.9 2.5 500 Tract 1 1100 950 1250Tract 2 1.4 1.3 1.7 400 Tract 2 560 520 680Tract 3 0.5 1.8 1.1 450 Tract 3 225 810 495Tract 4 1.2 0.3 2 400 Tract 4 480 120 800Tract 5 0.9 0.7 1 500 Tract 5 450 350 500Tract 6 1.1 1.6 0.6 450 Tract 6 495 720 270Tract 7 2.7 0.7 1.5 450 Tract 7 1215 315 675Tract 8 1.8 1.2 0.8 400 Tract 8 720 480 320Tract 9 1.5 1.7 0.7 500 Tract 9 750 850 350

TotalAssignment School 1 School 2 School 3 Assignments Supply

Tract 1 0 1 0 1 = 1Tract 2 1 0 0 1 = 1Tract 3 1 0 0 1 = 1Tract 4 0 1 0 1 = 1Tract 5 1 0 0 1 = 1Tract 6 0 0 1 1 = 1Tract 7 0 1 0 1 = 1Tract 8 0 0 1 1 = 1Tract 9 0 0 1 1 = 1

Total Assigned 3 3 3= = = Total Distance

Demand 3 3 3 (Miles)3560



The Transportation Problem

• A common problem in logistics is how to transport goods from a set of sources (e.g., plants, warehouses, etc.) to a set of destinations (e.g., warehouses, customers, etc.) at the minimum possible cost.

• Given– a set of sources, each with a given supply,

– a set of destinations, each with a given demand,

– a cost table (cost/unit to ship from each source to each destination)

• Goal– Choose shipping quantities from each source to each destination so as to minimize

total shipping cost.



The Network Representation

Sources Destinations

Supply1

Supply2

Supply3

Demand1

Demand2

Demand3

Demand4CostijShipment Quantityij



Transportation Problem Example

A company has two plants (in Seattle and Atlanta) producing a certain product that is to be shipped to three distribution centers (in Sacramento, St. Louis, and Pittsburgh).

– The unit production costs are the same at the two plants, and the shipping costs per unit are shown in the table below.

– Shipments are made once per week.

– During each week, each plant produces at most 60 units and each distribution center needs at least 40 units.

Unit Shipping Cost Distribution Center

Sacramento St. Louis Pittsburgh

PlantSeattle $2 $6 $8

Atlanta $7 $5 $3

Question: How many units should be shipped from each plant to each distribution center?



Spreadsheet Solution

3

45678

9

101112131415

B C D E F G HSacramento St. Louis Pittsburgh

Cost Dist. Center Dist. Center Dist. CenterSeattle Plant $2 $6 $8Atlanta Plant $7 $5 $3

Shipment Sacramento St. Louis Pittsburgh

Quantities Dist. Center Dist. Center Dist. Center Shipped AvailableSeattle Plant 40 20 0 60 <= 60Atlanta Plant 0 20 40 60 <= 60

Shipped 40 40 40 Cost = $420>= >= >=

Needed 40 40 40



Shipping from D.C.’s to Customers

The same company ships one of its products from its three distribution centers to four different customers

– The shipping costs per unit are shown in the table below.

– Shipments are made once per week.

– During each week, each distribution center has received 40 units.

– Customer demand is also shown in the table below.

Unit Shipping Cost Customer

1 2 3 4

DistributionCenter

Sacramento $8 $10 $7 $11

St. Louis $12 $11 $9 $6

Pittsburgh $10 $9 $15 $10

Customer Demand 40 30 25 25

Question: How many units should be shipped from each distribution center to each customer?



Spreadsheet Solution

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9

10111213141516

B C D E F G H I

Cost Customer 1 Customer 2 Customer 3 Customer 4Sacramento DC $8 $10 $7 $11

St. Louis DC $12 $11 $9 $6Pittsburgh DC $10 $9 $15 $10

ShipmentQuantities Customer 1 Customer 2 Customer 3 Customer 4 Shipped Available

Sacramento DC 30 0 10 0 40 <= 40St. Louis DC 0 0 15 25 40 <= 40

Pittsburgh DC 10 30 0 0 40 <= 40Shipped 40 30 25 25 Cost = $965

>= >= >= >=Needed 40 30 25 25



Managing the Whole Supply Chain(Plant to D.C. to Customer)

3

45678

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1819

202122232425

26

272829303132333435

B C D E F G H ISacramento St. Louis Pittsburgh

Cost Dist. Center Dist. Center Dist. CenterSeattle Plant $2 $6 $8Atlanta Plant $7 $5 $3

Shipment Sacramento St. Louis Pittsburgh


Shipped 60 25 35 Cost = $350

Distribution from DC's to Customers


St. Louis DC $12 $11 $9 $6Pittsburgh DC $10 $9 $15 $10


Sacramento DC 35 0 25 0 60 <= 60St. Louis DC 0 0 0 25 25 <= 25

Pittsburgh DC 5 30 0 0 35 <= 35Shipped 40 30 25 25 Cost = $925

>= >= >= >=Needed 40 30 25 25

Total Cost = $1,275



Site Selection

• The lease is up on their distribution center in St. Louis. They now must decide whether to sign a new lease in St. Louis, or move the distribution center to a new location.

• One possible new location is Omaha, Nebraska, which is offering a better deal on the lease.

Question: Should they move their distribution center to Omaha?



Spreadsheet Solution to Site Selection

123

45678

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26

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A B C D E F G H I

Distribution from Plants to DC's

Sacramento Omaha PittsburghCost Dist. Center Dist. Center Dist. Center

Seattle Plant $2 $5 $8Atlanta Plant $7 $6 $3

Shipment Sacramento Omaha Pittsburgh


Shipped 60 0 60 Cost = $300

Distribution from DC's to Customers


Omaha DC $13 $10 $8 $8Pittsburgh DC $10 $9 $15 $10


Sacramento DC 35 0 25 0 60 <= 60Omaha DC 0 0 0 0 0 <= 0

Pittsburgh DC 5 30 0 25 60 <= 60Shipped 40 30 25 25 Cost = $1,025

>= >= >= >=Needed 40 30 25 25

Total Cost = $1,325



Distribution System at Proctor and Gamble

• Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

– 50 product categories

– 60 plants

– 15 distribution centers

– 1000 customer zones

• Solved many transportation problems (one for each product category).

• Goal: find best distribution plan, which plants to keep open, etc.

• Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

• Implemented in 1996. Saved $200 million per year.

For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”



The Assignment Problem

• The job of assigning people (or machines or whatever) to a set of tasks is called an assignment problem.

• Given– a set of assignees

– a set of tasks

– a cost table (cost associated with each assignee performing each task)

• Goal– Match assignees to tasks so as to perform all of the tasks at the minimum possible

cost.



Network Representation

Assignees Tasks

Costij



Assignment Problem Example

The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below.

Backstroke Breaststroke Butterfly Freestyle

Carl 37.7 43.4 33.3 29.2

Chris 32.9 33.1 28.5 26.4

David 33.8 42.2 38.9 29.6

Tony 37.0 34.7 30.4 28.5

Ken 35.4 41.8 33.6 31.1

Question: How should the swimmers be assigned to make the fastest relay team?



Algebraic Formulation

Let xij = 1 if swimmer i swims stroke j; 0 otherwisetij = best time of swimmer i in stroke j

Minimize Time = ∑ i ∑ j tij xij

subject to

each stroke swum: ∑ i xij = 1 for each stroke j

each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i

andxij ≥ 0 for all i and j.




3456789

10

111213141516171819

B C D E F G H I

Best Times Backstroke Breastroke Butterfly FreestyleCarl 37.7 43.4 33.3 29.2Chris 32.9 33.1 28.5 26.4David 33.8 42.2 38.9 29.6Tony 37.0 34.7 30.4 28.5Ken 35.4 41.8 33.6 31.1

Assignment Backstroke Breastroke Butterfly FreestyleCarl 0 0 0 1 1 <= 1Chris 0 0 1 0 1 <= 1David 1 0 0 0 1 <= 1Tony 0 1 0 0 1 <= 1Ken 0 0 0 0 0 <= 1

1 1 1 1 Time = 126.2= = = =1 1 1 1



Bidding for Classes

• In the MBA program at a prestigious university in the Pacific Northwest, students bid for electives in the second year of their program.

• Each of the 10 students has 100 points to bid (total) and must take two electives.

• There are four electives available:– Quantitative Methods

– Finance

– Operations Management

– Accounting

• Each class is limited to 5 students.

Question: How should students be assigned to the classes?



Points Bid for Electives

Electives

StudentQuantitative

Methods FinanceOperations

Management Accounting

George 60 10 10 20

Fred 20 20 40 20

Ann 45 45 5 5

Eric 50 20 5 25

Susan 30 30 30 10

Liz 50 50 0 0

Ed 70 20 10 0

David 25 25 35 15

Tony 35 15 35 15

Jennifer 60 10 10 20



Spreadsheet Solution(Maximizing Total Points)

34567891011121314

15

1617181920212223242526272829

B C D E F G H I J K

Points QMETH Finance Op Mgt. AccountingGeorge 60 10 10 20

Fred 20 20 40 20Ann 45 45 5 5Eric 50 20 5 25

Susan 30 30 30 10Liz 50 50 0 0Ed 70 20 10 0

David 25 25 35 15Tony 35 15 35 15


Total Classes StudentAssignment QMETH Finance Op Mgt. Accounting Classes to Take Points

George 1 0 0 1 2 = 2 80Fred 0 0 1 1 2 = 2 60Ann 1 1 0 0 2 = 2 90Eric 0 1 0 1 2 = 2 45

Susan 0 1 1 0 2 = 2 60Liz 1 1 0 0 2 = 2 100Ed 1 0 1 0 2 = 2 80

David 0 1 1 0 2 = 2 60Tony 0 0 1 1 2 = 2 50

Jennifer 1 0 0 1 2 = 2 805 5 5 5

<= <= <= <= Total Points = 705Capacity 5 5 5 5



Spreadsheet Solution(Maximizing the Minimum Student Point Total)

34567891011121314

15

161718192021222324252627282930

B C D E F G H I J K L M

Points QMETH Finance Op Mgt. AccountingGeorge 60 10 10 20

Fred 20 20 40 20Ann 45 45 5 5Eric 50 20 5 25

Susan 30 30 30 10Liz 50 50 0 0Ed 70 20 10 0

David 25 25 35 15Tony 35 15 35 15


Total Classes MinAssignment QMETH Finance Op Mgt. Accounting Classes to Take Points Points

George 1 0 0 1 2 = 2 80 >= 50Fred 0 1 1 0 2 = 2 60 >= 50Ann 0 1 0 1 2 = 2 50 >= 50Eric 1 0 1 0 2 = 2 55 >= 50

Susan 0 1 1 0 2 = 2 60 >= 50Liz 0 1 0 1 2 = 2 50 >= 50Ed 1 0 0 1 2 = 2 70 >= 50

David 0 1 1 0 2 = 2 60 >= 50Tony 1 0 1 0 2 = 2 70 >= 50

Jennifer 1 0 0 1 2 = 2 80 >= 505 5 5 5

<= <= <= <= Total Points = 635Capacity 5 5 5 5

Min Points = 50

mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2008 15.1 table of contents cd chapter 15...

Documents

sellmore section

nifty section

energetic section

mcgrawhill companies

better products section

revised middletown section

site selection problem

metro water section