1. Transportation, Transshipment, and Assignment Problems
Chapter 6

2. The Transportation Model The Transshipment Model The Assignment Model
Chapter Topics
The Transportation Model
Computer Solution of a Transportation Problem
The Transshipment Model
Computer Solution of a Transshipment Problem
The Assignment Model
Computer Solution of an Assignment Problem
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3. Overview
Transportation, Transshipment, and Assignment models are part of a larger class of LP problems known as network flow models.
These models have special mathematical features that permit management scientists to develop very efficient, unique solution methods (variations of traditional simplex procedure).
Detailed description of methods is contained on the companion website.
Text focuses on model formulation and solution with Excel and QM for windows.
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4. The Transportation Model: Characteristics
A product is transported from a number of sources to a number of destinations at the minimum possible cost.
Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product.
The linear programming model has constraints for supply at each source and demand at each destination.
All constraints are equalities in a balanced transportation model where supply equals demand.
Constraints contain inequalities in unbalanced transportation models where supply does not equal demand.
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5. Transportation Model Example Problem Definition and Data
How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation?
Grain Elevator Supply Mill Demand
1. Kansas City 150 A. Chicago 200
2. Omaha B. St. Louis 100
3. Des Moines C. Cincinnati 300
Total tons Total tons
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6. Transportation Model Example Transportation Network Routes
Figure Network of Transportation Routes for Wheat Shipments
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7. Transportation Model Example Model Formulation
xij = tons of wheat transported from grain elevator i, ( i = 1, 2, 3)
to mill j, (j = A, B, C).
Minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C x3A + 5x3B + 12x3C
subject to:
x1A + x1B + x1C = 150
x2A + x2B + x2C = 175
x3A + x3B + x3C = 275
x1A + x2A + x3A = 200
x1B + x2B + x3B = 100
x1C + x2C + x3C = 300
xij  0
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8. Transportation Model Example
Computer Solution with QM for Windows (1 of 3)
Exhibit 6.7
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9. Transportation Model Example
Computer Solution with QM for Windows (2 of 3)
Exhibit 6.8
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10. Transportation Model Example
Computer Solution with QM for Windows (3 of 3)
Exhibit 6.9
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11. The Transshipment Model Characteristics
Extension of the transportation model.
Intermediate transshipment points (distribution centers or warehouses) are added between the sources and destinations.
Items may be transported from:
Sources through transshipment points to destinations
One source to another
One transshipment point to another
One destination to another
Directly from sources to destinations
Some combination of these S1 S2 D1 T1 T2
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12. Transshipment Model Example Problem Definition and Data
Suppose wheat is harvested at two farms in:
Nebraska (300 tons)
Colorado (300 tons)
Before being shipped to three mills (200, 100, 300 tons), the wheat is shipped to three grain elavators in:
Kansas City
Omaha
Des Moines
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13. Transshipment Model Example Problem Definition and Data
Extension of the transportation model in which intermediate transshipment points are added between sources and destinations.
Shipping Costs
1. Nebraska
2. Colorado
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14. Formulating the LP Model
Now, objective function will also include the shipping costs from farms to grain elavators.
There will be 3 groups of constranits:
Supply constraints for the farms.
Demand constraints for the mills.
Conservation of flow constraints: at each transshipment point, the amount of grain shipped IN must also be shipped OUT.
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15. Figure 6.3 Network of Transshipment Routes
Transshipment Model Example
(Farms, Grain Elavators, Mills)
Figure Network of Transshipment Routes
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16. Transshipment Model Example Model Formulation
xij = amount transported from i to j , i = 1, 2, 3
Minimize Z = $16x x x x x24
+ 17x25 + 6x36 + 8x x38 + 7x x47
+ 11x48 + 4x56 + 5x x58
subject to:
x13 + x14 + x15 = 300
x23 + x24 + x25 = 300
x36 + x46 + x56 = 200
x37 + x47 + x57 = 100
x38 + x48 + x58 = 300
x13 + x23 - x36 - x37 - x38 = 0
x14 + x24 - x46 - x47 - x48 = 0
x15 + x25 - x56 - x57 - x58 = 0
xij  0
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17. Transshipment Model Example Computer Solution (3 of 3)
Figure 6.4
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18. The Assignment Model Characteristics
Special form of linear programming model similar to the transportation model.
Supply at each source and demand at each destination is limited to one unit.
In a balanced model supply equals demand.
In an unbalanced model supply does not equal demand.
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19. Assignment Model Example Problem Definition and Data
Problem: Assign four teams of officials to four games in a way that will minimize the total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game.
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20. Assignment Model Example Model Formulation
xij = 1 if i is assigned to j, 0 otherwise. i= A, B, C, D, j=R, A, C, D
Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR +70xBA
+ 130xBD + 200xBC + 175xCR + 105xCA +140xCD
+ 170xCC + 80xDR + 65xDA + 105xDD + 120xDC
subject to:
xAR + xAA + xAD + xAC = 1 xij : 0 or 1
xBR + xBA + xBD + xBC = 1
xCR + xCA + xCD + xCC = 1
xDR + xDA + xDD + xDC = 1
xAR + xBR + xCR + xDR = 1
xAA + xBA + xCA + xDA = 1
xAD + xBD + xCD + xDD = 1
xAC + xBC + xCC + xDC = 1
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21. Assignment Model Example
Computer Solution with QM for Windows (1 of 2)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 6.16

22. Assignment Model Example
Computer Solution with QM for Windows (2 of 2)
Exhibit 6.17
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23. Example Problem Solution Transportation Problem Statement
A concrete company transports concrete from three plants to three construction sites. Determine the linear programming model formulation and solve using QM:
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24. Example Problem Solution Model Formulation
Minimize Z = $8x1A + 5x1B + 6x1C + 15x2A + 10x2B + 12x2C
+3x3A + 9x3B + 10x3C
subject to:
x1A + x1B + x1C = 120
x2A + x2B + x2C = 80
x3A + x3B + x3C = 80
x1A + x2A + x3A  150
x1B + x2B + x3B  70
x1C + x2C + x3C  100
xij  0
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25. Transportation Problem
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26. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

27. Example: Tina’s Tailoring
Tina's Tailoring has five idle tailors and four custom garments to make. The estimated time (in hours) it would take each tailor to make each garment is shown in the next slide. (An 'X' in the table indicates an unacceptable tailor-garment assignment.)
Tailor
Garment
Wedding gown
Clown costume X
Admiral's uniform X 9
Bullfighter's outfit X

28. Example: Tina’s Tailoring
Formulate an integer program for determining the tailor-garment assignments that minimize
the total estimated time spent making the four garments.
No tailor is to be assigned more than one garment and each garment is to be worked on by only one tailor.

29. Example: Tina’s Tailoring
Define the decision variables
xij = 1 if garment i is assigned to tailor j
= 0 otherwise.
Number of decision variables =
[(number of garments)(number of tailors)]
- (number of unacceptable assignments)
= [4(5)] - 3 = 17

30. Example: Tina’s Tailoring
Define the objective function
Minimize total time spent making garments:
Min 19x x x x x x x x x x x x x x x x x45

31. Example: Tina’s Tailoring
Define the Constraints
Exactly one tailor per garment:
1) x11 + x12 + x13 + x14 + x15 = 1
2) x21 + x22 + x24 + x25 = 1
3) x31 + x32 + x33 + x35 = 1
4) x42 + x43 + x44 + x45 = 1

32. Example: Tina’s Tailoring
Define the Constraints (continued)
No more than one garment per tailor:
5) x11 + x21 + x31 < 1
6) x12 + x22 + x32 + x42 < 1
7) x13 + x33 + x43 < 1
8) x14 + x24 + x44 < 1
9) x15 + x25 + x35 + x45 < 1
Nonnegativity: xij > 0 for i = 1, . . ,4 and j = 1, . . ,5

33. Example: Zeron Shelving
The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.
Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers.
Additional data is shown on the next slide.

34. Example: Zeron Shelving
Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:
Zeron N Zeron S
Arnold
Supershelf
The costs to install the shelving at the various locations are:
Zrox Hewes Rockrite
Thomas
Washburn

35. Example: Zeron Shelving
Network Representation
ZROX
Zrox 50 1 5
Arnold
Zeron N 75
ARNOLD 5 8 8
Hewes 60
HEWES 3 7
Super
Shelf
Zeron S 4 75
WASH
BURN 4 4
Rock-
Rite 40

36. Example: Zeron Shelving
Linear Programming Formulation
Decision Variables Defined
xij = amount shipped from manufacturer i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Arnold), 2 (Supershelf)
j = 3 (Zeron N), 4 (Zeron S)
k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)
Objective Function Defined
Minimize Overall Shipping Costs:
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37
+ 3x45 + 4x46 + 4x47

37. Example: Zeron Shelving
Constraints Defined
Amount Out of Arnold: x13 + x14 = 75
Amount Out of Supershelf: x23 + x24 = 75
Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0
Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0
Amount Into Zrox: x35 + x45 = 50
Amount Into Hewes: x36 + x46 = 60
Amount Into Rockrite: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.

38. Example: Zeron Shelving
Optimal Solution
Zrox
ZROX 50 50 75 1 5
Arnold
Zeron N 75
ARNOLD 5 25 8 8
Hewes 60 35
HEWES 3 7 4
Super
Shelf
Zeron S 40 75
WASH
BURN 4 75 4
Rock-
Rite 40

39. Example: Who Does What?
An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.
Projects
Subcontractor A B C
Westside
Federated
Goliath
Universal
How should the contractors be assigned to minimize total mileage costs?

40. Example: Who Does What? Network Representation A Subcontractors50
West. A 36 16
Subcontractors
Projects 28
Fed. 30 B 18 35 32
Gol. C 20 25 25
Univ. 14

41. Example: Who Does What? Linear Programming Formulation
Min 50x11+36x12+16x13+28x21+30x22+18x23
+35x31+32x32+20x33+25x41+25x42+14x43
s.t. x11+x12+x13 < 1
x21+x22+x23 < 1
x31+x32+x33 < 1
x41+x42+x43 < 1
x11+x21+x31+x41 = 1
x12+x22+x32+x42 = 1
x13+x23+x33+x43 = 1
xij = 0 or 1 for all i and j
Agents
Tasks

42. Example: Who Does What? The optimal assignment is:
Subcontractor Project Distance
Westside C
Federated A
Goliath (unassigned)
Universal B
Total Distance = 69 miles