Applications of Network Models

1
ESI 6912 Section 6129 (Fall 08)Advanced
Network Optimization

• Applications of Network Models

Ravindra K. Ahuja Professor, Industrial Systems
Engg. University of Florida, Gainesville, FL
ahuja_at_ufl.edu [phone] www.ise.ufl.edu/ahuja
2
Two Applications

• Airline Fleet Scheduling
• Radiation Therapy Treatment Planning

3
Fleet Assignment Model

• Assign planes of different types to different
  flight legs so as to minimize the cost of
  assignment ?

Different plane types and the number of
planes of each type
Assignment of a plane to each leg
Flight legs to be assigned
Cost of assigning a plane type to a flight leg

• Flight coverage and aircraft integrality
• Aircraft balance
• Fleet size

4
Fleet Assignment Model (contd.)

• Basic Question ?
• Which aircraft (fleet) type should fly each
  flight?
• Flight DL 146 Boeing 737, Boeing 767, or A320?
• Cost of Assignment
• Given expected number of passengers on flight,
• Plane too small ? lost revenue
• Plane too big ? costly and inefficient

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Time-Line Network

• Decision Variables ? xk,i Number of aircrafts
  of type k on arc i.

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Mixed Integer Programming Formulation

• Minimize ??i?L ?k?K Ck,i xk,i
• subject to
• (i) For each flight leg i ? L ?k?K xk,i
  1
• (ii) For each node p of the time-line network and
  each k ?K
• ?i?IN(p) xk,i - ?i?OUT(p) xk,i 0
• (ii) For each each plane type k?K
• ?i?Count-Time xk,i ? Nk
• xk,i ? 0 and integer

7
Problem Size

• Fleet assignment model at American Airlines
• 2,300 flights per day
• 150 cities
• 500 jets
• 10 aircraft types
• Number of variables over 50,000 integer
  variables
• Number of constraints 40,000 constraints
• One day scheduling problem can be solved in a few
  hours of time within acceptable accuracy.

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A Solution of the Fleet Assignment Model
NY
ATL
LA
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Through Assignment Problem (TAM)

• Combine two flights passing through a hub into a
  through flight.
• Through flights show up in the airline timetable
  and generate more revenues.

BOSTON
BOSTON
ATL
LA
LA
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Through Assignment Problem (contd.)

• Possibility 1

BOSTON
BOSTON

• Possibility 2

LA
LA
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Through Assignment Problem (contd.)

• Combine two flights passing through a hub into a
  through flight.
• Through flights show up in the airline timetable
  and generate more revenues.

• This problem is solved once for each hub and each
  fleet type.

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MIP Formulation of TAM
Maximize ??i?IN(o,k) ?j?OUT(o,k) pij xij subject
to (i) For each flight leg i ? IN(o, k)
?j?OUT(o, k) xij 1 (ii) For each flight leg j
? OUT(o, k) ?j?IN(o, k) xji 1 xij are
0-1 variables. where IN(o, k) is the set of
flights with plane type k arriving at station
o, and OUT(o, k) is the set of flights with plane
type k departing station o.
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Solving TAM

• In practice, there are some additional
  constraints that must also be satisfied by the
  through assignment model.
• The through assignment model is a generalization
  of the assignment problem.
• The through assignment model can be solved
  optimally in a few seconds using CPLEX.

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The Combined Through-Fleet Assignment Model
(ctFAM)

• When FAM is applied, through revenues are not
  considered.
• When TAM is applied, fleet assignment cannot be
  changed.

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The ctFAM (contd.)

• Determine the fleet assignment and also the
  through assignment to maximize the total
  contribution.

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Network for the ctFAM

• Represent flight by nodes not arcs.
• Show flight connections by arcs.
• Similar MIP formulation as earlier.

Flight nodes

• Homework Give an integer programming formulation
  of the ctFAM.

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MILP Formulation of ctFAM

• We developed a mixed integer linear programming
  (MILP) formulation of the ctFAM.
• The formulation had considerably more integer
  variables than the MILP formulation of the FAM.
• Approximately 100,800 integer variables, 18,100
  constraints and 353,000 non-zero elements in the
  constraint matrix
• The LP relaxation of the MILP formulation took
  more than 5 hours to solve on a PC.
• The airline is extending the FAM along other
  dimensions and wanted a separate module for
  incorporating the through revenues.

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Neighborhood Search Algorithms

• Local Improvement Algorithms
• Start with a feasible solution x
• Define a neighborhood of x
• Identify an improved neighbor y
• Replace x by y and repeat

Neighborhood of x1
Neighborhood of x2
Neighborhood of xk
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Time-Line Network for Airline Scheduling
Orlando
Atlanta
New York
Wash. D.C.
Cincinnati
Boston
Raleigh
Type A Plane
Type C Plane
Type B Plane
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Single A-B Swaps (Before the swap)
Orlando
Atlanta
New York
Wash. D.C.
Cincinnati
Boston
Raleigh
Type A Plane
Type B Plane
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Single A-B Swaps (After the swap)
Orlando
Atlanta
New York
Wash. D.C.
Cincinnati
Boston
Raleigh
Type B Plane
Type A Plane
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Finding Improving Changes

• Define the cost of each arc as the cost of
  switching plane types.
• Reverse the direction of arcs flown by plane type
  B.

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Finding Improving Changes (contd.)

• A negative cost directed cycle in the AB-network
  gives an improving swap.

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Double AB Swaps
B
A
B
B
A
A
A
Reverse arcs of type B
A
A
B
B
B
B
A
A
B
B
AB-Network
A
A
A
B
A
B
B
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Double AB Swaps (contd.)
B
A
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A
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A
After the swap
B
A
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A
B
Before the swap
B
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A
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B
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Multi AB Swaps
After the swap
B
Before the swap
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Neighborhood Search for FAM

• Start with a feasible fleeting assignment, select
  some pair of fleet types A and B, and apply the
  following procedure
• procedure Improve(A, B)
• begin
• construct the AB-improvement graph
• while there is some negative cost cycle in AB-
  improvement graph do
• begin
• identify a negative cost cycle in the
  AB-improvement graph
• change the fleet assignment
• update the AB-improvement graph
• end
• end

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Neighborhood Search for the ctFAM

• Start with a feasible fleeting and connection
  assignment, select some pair of fleet types A and
  B, and apply the following procedure
• procedure Improve(A, B)
• begin
• construct the AB-improvement graph
• while there is some negative cost constrained
  cycle in AB-improvement graph do
• begin
• identify a negative cost constrained cycle in
  the AB-improvement graph
• change the fleet and through assignment
• update the AB-improvement graph
• end
• end

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Computational Results on ctFAM
REFERENCE Talluri, K.T. 1996. Swapping
applications in a daily fleet assignment.
Transportation Science 31, 237-248.
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Two Applications

• Airline Fleet Scheduling
• Radiation Therapy Treatment Planning

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Overview

• We consider the problem of minimizing the beam-on
  time to realize a given intensity matrix.
• We show this problem can be formulated as a
  network flow problem and can be solved very
  efficiently.

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Introduction

• We have I a1S1 a2S2 a3S3 a4S4

I

• Beam-on Time 2 3 1 5 11
• Setup Time ?(S1,S2) ?(S2,S3) ?(S3,S4)
• Delivery Time Beam-on Time Setup Time

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A Simplification
0
0
0
0
0
0
1
1
0
1
0
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0

• We can determine the aperture settings for each
  row of MLC separately, and combine the row
  apertures to construct matrix apertures.

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A Simplification (contd.)
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
0

• We can determine the aperture settings of each
  row of MLC separately.
• We can combine the row apertures to construct
  matrix apertures. The total beam-on time is the
  maximum of beam-on times of row apertures.

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Problem Statement

• Given
• An intensity row I
• Determine
• Row apertures R1, R2, , Rk
• Their beam-on times a1, a2, , ak
• Such that I a1R1 a2R2 akRk
• Minimize ?i1,kai

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An Example
1 4 3 0 a11 a12
a13 a14
a22 a23
a24 a33
a34 a44
0
1
0
0

• The optimization problem is to determine
• The times a11, a12, a13, a14, a22, a23, a24,
  a33, a34, a44 such that their sum is minimum

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Using Columns Instead of Rows

• Minimize a11 a12 a13 a14 a22 a23
  a24 a33 a34 a44
• Subject to

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Using Columns Instead of Rows (contd.)
Minimize a11 a12 a13 a14 a22 a23
a24 a33 a34 a44 Subject to
a11, a12, a13, a14, a22, a23, a24, a33, a34, a44
0

• Observe that each column has all 1s in
  consecutive positions.
• This LP can be transformed to a network flow
  problem.

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Add a Row of Zero
Minimize a11 a12 a13 a14 a22 a23
a24 a33 a34 a44 Subject to
a11, a12, a13, a14, a22, a23,
a24, a33, a34, a44 0
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Subtract Each Row from the Next Row
Minimize a11 a12 a13 a14 a22 a23
a24 a33 a34 a44 Subject to
a11, a12, a13, a14, a22, a23, a24, a33,
a34, a44 0

• Each column has one 1, one 1, and other
  elements are zero.
• This LP is the formulation of a network flow
  problem.

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The Corresponding Network Flow Problem
1
3
-1
-3

• Each arc cost is 1.
• The network is acyclic.
• It is a complete network.
• Arcs have infinite capacity.

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Solving the Network Flow Problem

• The O(n) Time Sweep Algorithm
• Select the least-index supply node and the
  least-index demand node.
• Send flow from the source node to the demand
  node.
• Repeat until all demand it met by available
  supplies.

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Extensions

• The approach applies when we allow only a subset
  of row apertures but not all row apertures.
• The approach applies when different apertures
  have different weights.