McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 15.1 The Transportation Problem A common problem in logistics is how to transport goods from.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.1

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.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.2

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”

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.3

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

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15.4

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

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15.5

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

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15.6

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

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15.7

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

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.8

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.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.9

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.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.10

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

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15.11

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

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15.12

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)

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15.13

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

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15.14

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.

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.15

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”

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.16

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?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.17

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

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15.18

Spreadsheet Formulation

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

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15.19

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?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.20

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?

McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008

15.21

Spreadsheet Formulation

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

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15.22

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?

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15.23

Spreadsheet Formulation

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

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15.24

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.

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15.25

Network Representation

Assignees Tasks

Costij

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15.26

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?

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15.27

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.

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15.28

The Network Representation

A2

A1

T4A4

T3A3

T2

T1

Assignees Tasks

490

540

468

690

(Ann)

(Ian)

(Joan)

(Sean)

(Word processing)

(Graphics)

(Packets)

(Registrations)

574

378560

564

384612

507 728

559

480

765

375

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15.29

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

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15.30

Spreadsheet Formulation

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

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15.31

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?

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15.32

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

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15.33

Spreadsheet Formulation

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

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15.34

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?

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15.35

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.