decision analysis

1
Lecture 13 Transportation, Assignment, and
Transshipment Problems Anderson et al 7.1,
7.2, 7.3, 7.4 For next class please read
Anderson et al Chapter 7 7.5 Homework 7 (due
April 30, 2007) Chapter 7 4 a,b (do not solve),
7 a,b (do not solve) , 21 a (do not solve), 26
a, bChapter 6 13, Chapter 8, 8a, 11a
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 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 that
  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 Protracs Transportation Problem

• Protrac has 3 manufactory plants, which are
  located in Amsterdam, Antwerp and Le Havre,
  respectively. Protrac also has 4 assembly plants
  in Europe. They are located in Nancy, Liege,
  Tilburg, Leipzig.

9
Balanced Transportation Problem

• Note that this is a balanced transportation
  problem in that the total supply of engines
  available equals the total number required.

10
Data of Cost

• Cost to Transport an Engine
• From an Origin to a Destination
• Destination
• Origin

11
Linear Programming Formulation
Min 12xA1 13xA2 4xA3 6xA4 6xB1 4xB2
10xB3 11xB4 10xC1 9xC2 12xC3 4xC4 S.t.
xA1 xA2 xA3 xA4 500 xB1 xB2 xB3 xB4
700 xC1 xC2 xC3 xC4 800 xA1 xB1
xC1 400 xA2 xB2 xC2 900 xA3 xB3 xC3
200 xA4 xB4 xC4 500 Xij ? 0 where iA, B,
C and j1,2,3,4.
12
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.

13
Assignment Problem Network Representation
c11
1
1
c12
c13
Agents
Tasks
c21
c22
2
2
c23
c31
c32
3
3
c33
14
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.

15
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.

16
Assignment Problem LP Formulation Special Cases

• 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

17
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 50 36 16
Federated 28 30 18 Goliath
35 32 20 Universal
25 25 14 How should the
contractors be assigned to minimize total
mileage costs?
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Example Who Does What? Network Representation
50
West.
A
36
16
Subcontractors
Projects
28
30
B
Fed.
18
32
35
C
Gol.
20
25
25
Univ.
14
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Example Who Does What? LP 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
20
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.

21
Transshipment Problem

• 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.

22
Transshipment Problem Network Representation
c36
3
c13
c37
6
1
s1
d1
c14
c46
c15
4
Demand
c47
Supply
c23
c56
c24
7
2
d2
s2
c25
5
c57
Destinations
Sources
Intermediate Nodes
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Transshipment Problem LP 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

24
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 slides.

25
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
5 8 Supershelf
7 4
26
Example Zeron Shelving

• The costs to install the shelving at the various
  locations are
• Zrox
  Hewes Rockrite
• Thomas 1 5
  8
• Washburn 3 4
  4

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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
WASH BURN
75
4
4
Rock- Rite
40
28
Example Zeron Shelving LP 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)

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Example Zeron Shelving LP Formulation

• Objective Function
• Minimize Overall Shipping Costs
• Min 5x13 8x14 7x23 4x24 1x35 5x36
  8x37 3x45 4x46 4x47

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Example Zeron Shelving - Constraints

• Amount Out of Arnold x13 x14 lt 75
• Amount Out of Supershelf x23 x24 lt 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 gt 0, for all
  i and j.