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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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)
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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 $170 5
Sacron River 140 130 190 150 6
Calorie River 190 200 230 — 5
Needed 2 5 4 1.5(million
acre feet)
Question: How much water should Metro take from each river, and how much should they send from each river to each city?
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
15.23
Spreadsheet Formulation
34567891011121314151617
B C D E F G H IUnit Cost ($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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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.
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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?
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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.
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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?
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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?
McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008
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.