decision analysis

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Module 1 Week 3 (Chapters 7 and 8)
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Chapters 7 and 8Integer Programming,
Transportation and Assignment

• Types of Integer Programming Models
• Transportation Problem
• Network Representation and LP Formulation
• Assignment Problem
• Network Representation and LP Formulation

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Types of Integer Programming Models

• An LP in which all the variables are restricted
  to be integers is called an all-integer linear
  program (ILP).
• If only a subset of the variables are restricted
  to be integers, the problem is called a
  mixed-integer linear program (MILP).

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Transportation and Assignment 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.
• Transportation and assignment problems of this
  chapter are all examples of network problems.

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Example All-Integer LP

• Consider the following all-integer linear
  program
• Max 3x1 2x2
• s.t. 3x1 x2 lt
  9
• x1 3x2
  lt 7
• -x1 x2
  lt 1
• x1, x2 gt 0 and
  integer

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Example All-Integer LP

• LP Relaxation
• Solving the problem as a linear program
  ignoring the integer constraints, the optimal
  solution to the linear program gives fractional
  values for both x1 and x2. From the graph on the
  next slide, we see that the optimal solution to
  the linear program is
• x1 2.5, x2 1.5, z 10.5

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Example All-Integer LP

• LP Relaxation

x2
5
-x1 x2 lt 1
3x1 x2 lt 9
4
Max 3x1 2x2
3
LP Optimal (2.5, 1.5)
2
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
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Example All-Integer LP

• Rounding Up
• If we round up the fractional solution (x1
  2.5, x2 1.5) to the LP relaxation problem, we
  get x1 3 and x2 2. From the graph on the next
  slide, we see that this point lies outside the
  feasible region, making this solution
  infeasible.

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Example All-Integer LP

• Rounded Up Solution

x2
-x1 x2 lt 1
5
3x1 x2 lt 9
4
Max 3x1 2x2
3
ILP Infeasible (3, 2)
2
LP Optimal (2.5, 1.5)
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
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Example All-Integer LP

• Rounding Down
• By rounding the optimal solution down to x1
  2, x2 1, we see that this solution indeed is an
  integer solution within the feasible region, and
  substituting in the objective function, it gives
  z 8.
• We have found a feasible all-integer solution,
  but have we found the OPTIMAL all-integer
  solution?
• ---------------------
• The answer is NO! The optimal solution is x1
  3 and x2 0 giving z 9, as evidenced in the
  next two slides.

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Example All-Integer LP

• Complete Enumeration of Feasible ILP Solutions
• There are eight feasible integer solutions to
  this problem
• x1 x2 z
• 1. 0 0 0
• 2. 1 0 3
• 3. 2 0 6
• 4. 3 0 9
  optimal solution
• 5. 0 1 2
• 6. 1 1 5
• 7. 2 1 8
• 8. 1 2 7

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Example All-Integer LP
x2
-x1 x2 lt 1
5
3x1 x2 lt 9
4
Max 3x1 2x2
3
ILP Optimal (3, 0)
2
x1 3x2 lt 7
1
x1
1 2 3 4
5 6 7
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Example All-Integer LP

• Partial Spreadsheet Showing Problem Data

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Example All-Integer LP

• Partial Spreadsheet Showing Formulas

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Example All-Integer LP

• Partial Spreadsheet Showing Optimal Solution

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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.

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Transportation Problem

• Network Representation

1
d1
c11
1
c12
s1
c13
2
d2
c21
c22
2
s2
c23
3
d3
SOURCES
DESTINATIONS
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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

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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?
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Example BBC

• Delivery Cost Per Ton
• Northwood Westwood Eastwood
• Plant 1 24 30
  40
• Plant 2 30 40
  42

Note X11 gt Plant 1, Northwood X12 gt Plant 1,
Westwood X13 gt Plant 1, Eastwood X21 gt Plant
2, Northwood X22 gt Plant 2, Westwood X23
gt Plant 2, Eastwood
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Example BBC

• Objective Function
• 24X11 30X12 40X13 30X21 40X22 42X23
• Constraints
• X11 X12 X13 50 plant 1 capacity
• X21 X22 X23 50 plant 2 capacity
• X11 X21 25 Northwood
  demand
• X12 X22
  45 Westwood demand
• X13 X23 10 Eastwood
  demand

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Example BBC Using Excel

• Partial Spreadsheet Showing Problem Data

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Example BBC

• Partial Spreadsheet Showing Optimal Solution

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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

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Example BBC

• Partial Sensitivity Report (first half)

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Example BBC

• Partial Sensitivity Report (second half)

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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.

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Assignment Problem

• Network Representation

c11
1
1
c12
c13
AGENTS
TASKS
c21
c22
2
2
c23
c31
c32
3
3
c33
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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.

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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.

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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?
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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
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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