Systèmes d’aide à la décision
Integer Programming, Formulation
M2 GSI, 2011-2012
Ayse AKBALIK & Christophe RAPINE
1
Course outline
0-1 Integer Programs
Plan
1
Course outline
2
Some 0-1 integer programs
2
Course outline
0-1 Integer Programs
Integer programming - Outline
Theory of integer programming through real world-based
problems
How to reformulate these problems to obtain better results ?
How one can use mixed integer programs more effectively ?
Which commercial softwares are available in this domain and
how to use them efficiently ?
3
Course outline
0-1 Integer Programs
Integer programming - Outline
Modeling part :
Various problems modeling with IP
Exercises on lot sizing problem extensions
Production planning problems, cutting problems,
telecommunication and network problems, assignment,
knapsack and bin-packing problems, traveling salesman
problem, lot sizing extensions, etc...
4
Course outline
0-1 Integer Programs
Integer programming - Outline
Software requirements
Xpress optimization suite
IBM ILOG Cplex Optimization Studio (free trial version for 90
days)
5
Course outline
0-1 Integer Programs
FICO Xpress Mosel
6
Course outline
0-1 Integer Programs
Some real problems optimized with FICO Xpress
American Airlines reduces costs and increases its revenues
National Football League of America optimizes the playing schedule
Avis Europe maximizes the use of each car in its fleet
Honeywell process solutions proposes different optimization
solutions by embedding FICO Xpress
7
Course outline
0-1 Integer Programs
IBM ILOG Cplex
8
Course outline
0-1 Integer Programs
Some real problems optimized with IBM ILOG Cplex
AirFrance optimizes the plane schedules
Auction system for school food services for Chilean national school
lunch program
Deutsche Fussball Liga, Making Soccer Available to More Fans
Optimization of power generation by Powel ASA, calculates optimal
production schedules for each generator
Supply chain master planning solution for Plastic Goods Maker
Planning and scheduling forestry operations by Latin American
University
9
Course outline
0-1 Integer Programs
Textbooks
1
Integer Programming, L.A. Wolsey, Wiley-Interscience
2
Production planning by mixed integer programming, Y.Pochet
and L.A.Wolsey.
3
Integer and combinatorial optimization, G.L.Nemhauser and
L.A.Wolsey.
10
Course outline
0-1 Integer Programs
Plan
1
Course outline
2
Some 0-1 integer programs
11
Course outline
0-1 Integer Programs
What is an integer program ?
Suppose a linear program :
max{cx : Ax ≤ b, x ≥ 0}
Mixed integer program : if some but not all variables x are
integer
Pure integer program : if all variables are integer
0-1 (or Binary) integer program : all variables are 0 or 1
12
Course outline
0-1 Integer Programs
Some 0-1 integer programs
Assignment problem
0-1 Knapsack problem
Bin Packing problem
Set covering problem
Traveling salesman problem
13
Course outline
0-1 Integer Programs
Assignment problem
a set of people N = {1, 2 . . . n}
a set of jobs N = {1, 2 . . . n}
an estimated cost cij if person i is
assigned to job j
find the minimum cost assignment
14
Course outline
0-1 Integer Programs
Assignment formulation
variables
xij = 1 if i is assigned to j, 0 else
P
objective
min i∈N,j∈N cij xij
P
constraints
x = 1, ∀i
Pj∈N ij
i∈N xij = 1, ∀j
xij ∈ {0, 1}
∀i, ∀j ∈ N
15
Course outline
0-1 Integer Programs
Assignment applications in industry
Assignment of aircraft to the flights over one period (cost on
arcs)
Assignment of crews to the flights over one period
(preferences)
Truck - inbound crossdock assignment
16
Course outline
0-1 Integer Programs
0-1 Knapsack problem
a set of objects N = {1, 2 . . . n}
each object has a
value ui and
weight wi
the capacity of the knapsack to respect is
W
which objects to put in the knapsack to
maximize the total value ?
17
Course outline
0-1 Integer Programs
0-1 Knapsack formulation
variables
xi = 1 if object i is chosen, 0 else
P
objective
max i∈N ui xi
P
constraints
i∈N wi xi ≤ W
xi ∈ {0, 1}
∀i ∈ N
18
Course outline
0-1 Integer Programs
Knapsack applications in industry
They can be either standalone knapsack problems or some
subproblems of more complex programming models.
Transportation, logistics (capacity to manage and a priority on
customer demand)
Investment, budget allocation, financial portfolios (an initial
cost and a return value)
Merkle Hellman knapsack cryptosystem
19
Course outline
0-1 Integer Programs
Bin Packing problem
a set of objects N = {1, 2 . . . n}
infinite number of boxes of capacity C
each object i has a
size si
what is the minimum number of boxes to
pack all the objects ?
20
Course outline
0-1 Integer Programs
Bin Packing formulation
variables
yj = 1 if the box j is used, 0 else
xij = 1 if object i is put into box j
P
objective
min j∈N yj
P
constraints
∀i ∈ N
j∈N xij = 1
P
∀j ∈ N
i∈N si xij ≤ Cyj
yj , xij ∈ {0, 1}
∀i, j ∈ N
21
Course outline
0-1 Integer Programs
Bin Packing applications
Filling up containers
Loading trucks with weight capacity
22
Course outline
0-1 Integer Programs
Set covering problem
a set of regions M = {1, .., m}
a set of potential centers N = {1, .., n}
cj the cost of installing center j
aij = 1 if center j services region i
where to install centers to cover all the
regions with min cost ?
23
Course outline
0-1 Integer Programs
Set covering formulation
variables
xj = 1 if the center j is selected, 0 else
P
objective
min j∈N cj xj
P
constraints
∀i ∈ M
j∈N aij xj ≥ 1
xj ∈ {0, 1}
∀j ∈ N
24
Course outline
0-1 Integer Programs
Set covering applications
A very general form appearing in many complex problems
Anti-virus program design
Installation of emergency centers serving many regions
25
Course outline
0-1 Integer Programs
Traveling salesman problem, TSP
a salesman must visit each of n cities
(N = {1, .., n}) exactly once and then
return to its starting point
cij , time (or cost) to travel from city i to
city j
find the less costly tour
26
Course outline
0-1 Integer Programs
TSP, 1962, Procter&Gamble, 33 cities
http ://www.tsp.gatech.edu/index.html
27
Course outline
0-1 Integer Programs
TSP formulation
variables
xij = 1 if the salesman goes directly from i to j,
0 else
P
P
objective
min i∈N j∈N cij xij
P
constraints
x =1
∀i ∈ N
Pj∈N,j6=i ij
x
=
1
∀j
∈N
Pi∈N,i6=j ij
i∈S,j∈S xij ≤ |S| − 1, ∀S ⊂ N, 2 ≤ |S| ≤ n − 1
xij ∈ {0, 1}
∀i, j ∈ N, i 6= j
28
Course outline
0-1 Integer Programs
TSP applications
Transportation, logistics problem
Robotic manufacturing processes
Material handling, picking
29