1. Chapter 7 Transportation, Assignment, and Transshipment Problems
Transportation Problem
Assignment Problem
The Transshipment Problem

2. Transportation, Assignment, and Transshipment Problems
A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

3. Transportation, Assignment, and Transshipment Problems
Each of the three models of this chapter (transportation, assignment, and transshipment models) can be formulated as linear programs and solved by general purpose linear programming codes.
For each of the three models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables.
However, there are many computer packages (including The Management Scientist) which contain separate computer codes for these models which take advantage of their network structure.

4. 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.
The network representation for a transportation problem with two sources and three destinations is given on the next slide.

5. Transportation Problem
Network Representation 1 d1
c11 1
c12 s1
c13 2 d2
c21
c22 2 s2
c23 3 d3
SOURCES
DESTINATIONS

6. Transportation Problem
LP Formulation
The LP formulation in terms of the amounts shipped from the origins to the destinations, xij , can be written as:
Min cijxij
i j
s.t. xij < si for each origin i j
xij = dj for each destination j i
xij > 0 for all i and j

7. Transportation Problem
LP Formulation Special Cases
The following special-case modifications to the linear programming formulation can be made:
Minimum shipping guarantee from i to j:
xij > Lij
Maximum route capacity from i to j:
xij < Lij
Unacceptable route:
Remove the corresponding decision variable.

8. Example: BBC
Building Brick Company (BBC) has orders for 80 tons of bricks at three suburban locations as follows: Northwood tons, Westwood tons, and Eastwood tons. BBC has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown on the next slide.
How should end of week shipments be made to fill the above orders?

9. Example: BBC Delivery Cost Per Ton Northwood Westwood Eastwood
Plant
Plant

10. Example: BBC
Partial Spreadsheet Showing Problem Data

11. Example: BBC
Partial Spreadsheet Showing Optimal Solution

12. Example: BBC Optimal Solution From To Amount Cost
Plant 1 Northwood
Plant 1 Westwood ,350
Plant 2 Northwood
Plant 2 Eastwood
Total Cost = $2,490

13. Transportation Simplex Method
The transportation simplex method requires that the sum of the supplies at the origins equal the sum of the demands at the destinations.
If the total supply is greater than the total demand, a dummy destination is added with demand equal to the excess supply, and shipping costs from all origins are zero. (If total supply is less than total demand, a dummy origin is added.)
When solving a transportation problem by its special purpose algorithm, unacceptable shipping routes are given a cost of +M (a large number).

14. 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.

15. Assignment Problem Network Representation c11 c12 c13 c21 c22 c23 c31
AGENTS
TASKS
c21
c22 2 2
c23
c31
c32 3 3
c33

16. Assignment Problem LP Formulation Min cijxij i j
s.t. xij = for each agent i j
xij = for each task j i
xij = 0 or 1 for all i and j
Note: A modification to the right-hand side of the first constraint set can be made if a worker is permitted to work more than 1 job.

17. Assignment Problem LP Formulation Special Cases
Number of agents exceeds the number of tasks:
xij < for each agent i j
Number of tasks exceeds the number of agents:
Add enough dummy agents to equalize the
number of agents and the number of tasks.
The objective function coefficients for these
new variable would be zero.

18. Assignment Problem LP Formulation Special Cases (continued)
The assignment alternatives are evaluated in terms of revenue or profit:
Solve as a maximization problem.
An assignment is unacceptable:
Remove the corresponding decision variable.
An agent is permitted to work a tasks:
xij < a for each agent i j

19. Example: Hungry Owner
A 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 costs?

20. Example: Hungry Owner Network Representation A Subcontractors Projects50
West. A 36 16
Subcontractors
Projects 28
Fed. 30 B 18 35 32
Gol. C 20 25 25
Univ. 14

21. Example: Hungry Owner 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

22. Transshipment Problem
Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes)before reaching a particular destination node.
Transshipment problems can be converted to larger transportation problems and solved by a special transportation program.
Transshipment problems can also be solved by general purpose linear programming codes.
The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

23. Transshipment Problem
Network Representation
c36 3
c13 1
c37 6 s1 d1
c14
c46
c15 4
c47
Demand
Supply
c23
c24
c56 7 2 d2 s2
c25 5
c57
SOURCES
INTERMEDIATE
NODES
DESTINATIONS

24. Transshipment Problem
Linear Programming Formulation
xij represents the shipment from node i to node j
Min cijxij
i j
s.t. xij < si for each origin i j
xik - xkj = for each intermediate
i j node k
xij = dj for each destination j i
xij > for all i and j

25. Example: Transshipping
Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Rockwright) 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 Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers.
Additional data is shown on the next slide.

26. Example: Transshipping
Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:
Thomas Washburn
Arnold
Supershelf
The costs to install the shelving at the various locations are:
Zrox Hewes Rockwright
Thomas
Washburn

27. Example: Transshipping
Network Representation
Zrox
ZROX 50 1
Arnold 5
Thomas 75
ARNOLD 5 8 8
Hewes 60
HEWES 3 7
Super
Shelf
Wash-
Burn 4 75
WASH
BURN 4 4
Rock-
Wright 40

28. Example: Transshipping
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 (Thomas), 4 (Washburn)
k = 5 (Zrox), 6 (Hewes), 7 (Rockwright)
Objective Function Defined
Minimize Overall Shipping Costs:
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37
+ 3x45 + 4x46 + 4x47

29. Example: Transshipping
Constraints Defined
Amount Out of Arnold: x13 + x14 < 75
Amount Out of Supershelf: x23 + x24 < 75
Amount Through Thomas: x13 + x23 - x35 - x36 - x37 = 0
Amount Through Washburn: x14 + x24 - x45 - x46 - x47 = 0
Amount Into Zrox: x35 + x45 = 50
Amount Into Hewes: x36 + x46 = 60
Amount Into Rockwright: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.

30. Example: Transshipping
Optimal Solution (from The Management Scientist )
Objective Function Value =
Variable Value Reduced Costs X X X X X X X X X X

31. Example: Transshipping
Optimal Solution
Zrox
ZROX 50 50 75 1
Arnold 5
Thomas 75
ARNOLD 5 25 8 8
Hewes 60 35
HEWES 3 7 4
Super
Shelf
Wash-
Burn 40 75
WASH
BURN 4 75 4
Rock-
Wright 40