irwin/mcgraw-hill © the mcgraw-hill companies, inc., 2003 6.1 chapter 6 transportation and...

© The McGraw-Hill Companies, Inc., 20036.1Irwin/McGraw-Hill

Chapter 6

Transportation and Assignment Problems

© The McGraw-Hill Companies, Inc., 20036.2McGraw-Hill/Irwin

Transportation Problem

• The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.


Network Representation of Transportation Problem


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


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 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.


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.


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


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?


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?


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?


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


Assignment Problem

• An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij.

• It assumes all workers are assigned and each job is performed.

• An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs.

• The network representation of an assignment problem with three workers and three jobs is shown on the next slide.


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.


Network Representation of Assignment Problem


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


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


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?


Spreadsheet Formulation

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

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