١. Dr. Samia Rouibah. Chapter 6. Transportation and Assignment Problems ... A
Case Study: The P&T Company Distribution Problem. CANNERY 1. Bellingham.
Chapter 6 Transportation and Assignment Problems
Dr. Samia Rouibah
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Introduction to Management Science
Introduction Transportation and Assignment problems fall into the category of Distribution-Network problems Transportation: many Pbs involve determining how to transport goods optimally. However some of their important application have nothing to do with transportation. Assignment: most known application involve assigning people to tasks. However, they have a variety of other applications as well.
Dr. Samia Rouibah
٢
Introduction to Management Science
A Case Study: The P&T Company Distribution Problem CANNERY 1 Bellingham WAREHOUSE 3 Rapid City
CANNERY 2 Eugene
CANNERY 3 Albert Lea
WAREHOUSE 2 Salt Lake City WAREHOUSE 1 Sacramento WAREHOUSE 4 Albuquerque
Dr. Samia Rouibah
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Introduction to Management Science
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
Dr. Samia Rouibah
٤
Introduction to Management Science
Current Shipping Plan
Warehouse To From \ Cannery
Sacramento
Salt Lake City
Rapid City
Albuquerque
Bellingham
75
0
0
0
Eugene
5
65
55
0
Albert Lea
0
0
15
85
Dr. Samia Rouibah
٥
Introduction to Management Science
Shipping Cost per Truckload
Warehouse To From \ Cannery
Sacramento
Salt Lake City
Rapid City
Albuquerque
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
Dr. Samia Rouibah
٦
Introduction to Management Science
Terminology for a Transportation Problem
P&T Company Problem
General Model
Truckloads of canned peas
Units of a commodity
Canneries
Sources (any group of supply centers)
Warehouses
Destinations (an group of receiving centers)
Output from a cannery
Supply from a source (units to distribute)
Allocation to a warehouse
Demand at a destination (units to be received)
Shipping cost per truckload from a cannery to a warehouse
Cost per unit distributed from a source to a destination
Dr. Samia Rouibah
٧
Introduction to Management Science
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. – This assumptions means that the problem is balanced: Total Supply = Total Demand
•
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.
Thus the parameters of any transportation problem are: Supplies, Demands and unit costs Dr. Samia Rouibah
٨
Introduction to Management Science
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 (next slide) 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.
Dr. Samia Rouibah
٩
Introduction to Management Science
The P&T Co. Transportation Problem
Unit Cost Destination (Warehouse):
Sacramento
Salt Lake City
Rapid City
Albuquerque
Supply
Bellingham
$464
$513
$654
$867
75
Eugene
352
416
690
791
125
Albert Lea
995
682
388
685
100
Demand
80
65
70
85
Source (Cannery)
Dr. Samia Rouibah
١٠
Introduction to Management Science
Spreadsheet Formulation
B Unit Cost
C
3 4 5 Source Bellingham 6 (Cannery) Eugene 7 Albert Lea 8 9 10 Shipment Quantity 11 (Truckloads) 12 Source Bellingham 13 (Cannery) Eugene 14 Albert Lea 15 Total Received 16 17 Demand
Dr. Samia Rouibah
D Sacramento $464 $352 $995
E F Destination (Warehouse) Salt Lake City Rapid City $513 $654 $416 $690 $682 $388
Albuquerque $867 $791 $685
Sacramento 0 80 0 80 = 80
Destination (Warehouse) Salt Lake City Rapid City 20 0 45 0 0 70 65 70 = = 65 70
Albuquerque 55 0 30 85 = 85
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G
H
I
J
Total Shipped 75 125 100
= = =
Supply 75 125 100 Total Cost $152,535
Introduction to Management Science
Network Representation De ma nds
Supplie s
Destina tions Sourc es
464 (Be llingham) 75
867 (E ugene) 125
S2
995 (Alber t Le a)100
Dr. Samia Rouibah
S3
80 (Sa cr amento)
D2
65 (Sa lt La ke City
D3
70 (Rapid City)
D4
85 (Albuquerque )
513
S1
352
D1
654
416 690
791
682
388
685
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Introduction to Management Science
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 to Cannery 1: x11 + x12 + x13 + x14 = 75 Cannery 2: x21 + x22 + x23 + x24 = 125 Cannery 3: x31 + x32 + x33 + x34 = 100 Warehouse 1: x11 + x21 + x31 = 80 Warehouse 2: x12 + x22 + x32 = 65 Warehouse 3: x13 + x23 + x33 = 70 Warehouse 4: x14 + x24 + x34 = 85 and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
Dr. Samia Rouibah
١٣
Introduction to Management Science
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.
Dr. Samia Rouibah
١٤
Introduction to Management Science
Solving Transportation Problem •
Are special type of LP problem, they can be solved by the simplex method
•
Because the coefficients in the functional constraints are 0 or 1, transportation problems are solved far more quickly using transportation simplex method
•
Other distribution-network problems are solved using the network simplex method
•
A basic software package such Excel Solver is not including these two methods
Dr. Samia Rouibah
١٥
Introduction to Management Science
Completing the P&T Co. Case Study B C 3 Unit Cost 4 5 Source Bellingham 6 (Cannery) Eugene 7 Albert Lea 8 9 10 Shipment Quantity 11 (Truckloads) Source Bellingham 12 Eugene 13 (Cannery) Albert Lea 14 Total Received 15 16 Demand 17
D Sacramento $464 $352 $995
E F Destination (Warehouse) Salt Lake City Rapid City $513 $654 $416 $690 $682 $388
G Albuquerque $867 $791 $685
Sacramento 0 80 0 80 = 80
Destination (Warehouse) Salt Lake City Rapid City 20 0 45 0 0 70 65 70 = = 65 70
Albuquerque 55 0 30 85 = 85
H
I
J
Total Shipped 75 125 100
= = =
Supply 75 125 100 Total Cost $152,535
Total Cost = $152,535 a reduction of $13,060 from the current shipping plan
Dr. Samia Rouibah
١٦
Introduction to Management Science
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”, downloadable at www.mhhe.com/hillier2e/articles
Dr. Samia Rouibah
١٧
Introduction to Management Science
Modeling Variants of Transportation Problems LP problems frequently arise that are almost transportation problems (one or more features do not quite fit): – – – – –
The sum of the supplies exceeds the sum of the demands The sum of the supplies is less than the sum of the demands The destination has both a minimum demand and a maximum demand Certain source-destination combinations cannot be used for distributing units The objective is to maximize the total profit associated with distributing units
Spreadsheet models that aren’t quite transportation problems because they have at least one the above features can still be solved by the Excel Solver This is the approach we will use because we are not studying really big problems
Dr. Samia Rouibah
١٨
Introduction to Management Science
Example 1: 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 1
2
3
4
Capacity Available
1
$41
$27
$28
$24
75
2
40
29
—
23
75
3
37
30
27
21
45
Required production
20
30
30
40
Product: Plant
Question: Which plants should produce which products? Dr. Samia Rouibah
١٩
Introduction to Management Science
Transportation Problem Formulation Unit Cost Destination (Product):
1
2
3
4
Supply
1
$41
$27
$28
$24
75
2
40
29
—
23
75
3
37
30
27
21
45
Demand
20
30
30
40
Source(Plant)
It is not necessary now to use the entire supply from each source
Dr. Samia Rouibah
٢٠
Introduction to Management Science
Spreadsheet Formulation B 3 Unit Cost 4 Plant 1 5 Plant 2 6 Plant 3 7 8 9 10 Daily Production 11 Plant 1 12 Plant 2 13 Plant 3 14 Products Produced 15 16 Required Production
Dr. Samia Rouibah
C Product 1 $41 $40 $37
Product 1 0 0 20 20 = 20
D Product 2 $27 $29 $30
E Product 3 $28 $27
Product 2 30 0 0 30 = 30
Product 3 30 0 0 30 = 30
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F Product 4 $24 $23 $21
Product 4 0 15 25 40 = 40
G
H
I
Produced At Plant 60 15 45
