Chapter 7 Transportation, Assignment, and Transshipment Problems

1
Chapter 7Transportation, 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 lt si for
  each origin i
• j
• ?xij dj for
  each destination j
• i
• xij gt 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 gt Lij
• Maximum route capacity from i to j
• xij lt 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 -- 25 tons, Westwood -- 45
tons, and Eastwood -- 10 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 1 24 30
  40
• Plant 2 30 40
  42

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 5 120
• Plant 1 Westwood 45
  1,350
• Plant 2 Northwood 20
  600
• Plant 2 Eastwood 10
  420
• 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
1
1
c12
c13
AGENTS
TASKS
c21
c22
2
2
c23
c31
c32
3
3
c33
16
Assignment Problem

• LP Formulation
• Min ??cijxij
• i j
• s.t. ?xij 1
  for each agent i
• j
• ?xij 1
  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 lt 1 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 lt 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 50 36 16
Federated 28
30 18 Goliath
35 32 20
Universal 25 25 14 How
should the contractors be assigned to minimize
total costs?
20
Example Hungry Owner

• Network Representation

50
West.
A
36
16
Subcontractors
Projects
28
30
B
Fed.
18
32
35
C
Gol.
20
25
25
Univ.
14
21
Example Hungry Owner

• Linear Programming Formulation
• Min 50x1136x1216x1328x2130x2218x23
• 35x3132x3220x3325x4125x4214x43
• s.t. x11x12x13 lt 1
• x21x22x23 lt 1
• x31x32x33 lt 1
• x41x42x43 lt 1
• x11x21x31x41 1
• x12x22x32x42 1
• x13x23x33x43 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
c37
6
1
s1
d1
c14
c46
c15
4
c47
Demand
Supply
c23
c56
c24
7
2
d2
s2
c25
5
c57
INTERMEDIATE NODES
SOURCES
DESTINATIONS
24
Transshipment Problem

• Linear Programming Formulation
• xij represents the shipment from node i to node
  j
• Min ??cijxij
• i j
• s.t. ?xij lt si
  for each origin i
• j
• ?xik - ?xkj 0 for
  each intermediate
• i j
  node k
• ?xij dj
  for each destination j
• i
• xij gt 0
  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 5 8
Supershelf 7
4 The costs to install the shelving at the
various locations are
Zrox Hewes Rockwright Thomas
1 5 8
Washburn 3 4 4
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Example Transshipping

• Network Representation

Zrox
ZROX
50
1
5
Thomas
Arnold
ARNOLD
75
5
8
8
Hewes
60
HEWES
3
7
Super Shelf
Wash- Burn
4
WASH BURN
75
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 lt
  75
• Amount Out of Supershelf x23 x24 lt 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 gt 0, for all
  i and j.

30
Example Transshipping

• Optimal Solution (from The Management Scientist )
• Objective Function Value 1150.000
• Variable Value
  Reduced Costs
• X13 75.000
  0.000
• X14
  0.000 2.000
• X23
  0.000 4.000
• X24
  75.000 0.000
• X35
  50.000 0.000
• X36
  25.000 0.000
• X37
  0.000 3.000
• X45
  0.000 3.000
• X46
  35.000 0.000
• X47
  40.000 0.000

31
Example Transshipping

• Optimal Solution

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