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Introduction to Integer Programming
Eduardo Camponogara
Department of Automation and Systems EngineeringFederal University of Santa Catarina
October 2016
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Summary
Introduction
Rounding and Integer Programming
Applications
Modeling Strategies
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Introduction
Summary
Introduction
Rounding and Integer Programming
Applications
Modeling Strategies
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Introduction
Integer Problem
Integer Linear Problem (IP):
PL : max cTx
s.t. Ax ≤ b,
x ∈ Zn
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Introduction
Mixed-Integer (Linear) Problem
Mixed-Integer Linear Problem (MILP):
IP : max cTx
s.t. Ax ≤ b,
x = (xC, xI)
xC ≥ 0
xI ∈ Zn
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Introduction
Integer Problems
There are several classes of integer problems
I Integer (Linear) Problem
I Mixed-Integer (Linear) Problem
I Linear Binary Problem
I Combinatorial Optimization Problem
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Rounding and Integer Programming
Summary
Introduction
Rounding and Integer Programming
Applications
Modeling Strategies
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Rounding and Integer Programming
Rounding and Integer Programming
Questao: Why not use Linear Programming?
I We could disregard the constraints on binary variables.
I Obtain an optimal solution x? for the resulting linear program.
I And then round x? such as to obtain a solution to the integerprogram.
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Rounding and Integer Programming
Rounding and Integer Programming
Questao: Why not use Linear Programming?
I We could disregard the constraints on binary variables.
I Obtain an optimal solution x? for the resulting linear program.
I And then round x? such as to obtain a solution to the integerprogram.
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Rounding and Integer Programming
Rounding and Integer Programming
Issue
I The rounding strategy does not work.
I The following counter-example clarifies the issue:
max x1 + 0.6x2s.t. : 50x1 + 31x2 ≤ 250
3x1 − 2x2 ≥ −4
with being x1, x2 ≥ 0 and integer.
I An optimal solution to LP, x = (376193 ,950193) = (1.94, 4.92),
could be rounded to obtain the solution x = (2, 4).
I But this solution is quite “far” from the optimal solutionx? = (5, 0).
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Rounding and Integer Programming
Rounding and Integer Programming
Issue
I The rounding strategy does not work.
I The following counter-example clarifies the issue:
max x1 + 0.6x2s.t. : 50x1 + 31x2 ≤ 250
3x1 − 2x2 ≥ −4
with being x1, x2 ≥ 0 and integer.
I An optimal solution to LP, x = (376193 ,950193) = (1.94, 4.92),
could be rounded to obtain the solution x = (2, 4).
I But this solution is quite “far” from the optimal solutionx? = (5, 0).
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Rounding and Integer Programming
Rounding and Integer Programming
Issue
I The rounding strategy does not work.
I The following counter-example clarifies the issue:
max x1 + 0.6x2s.t. : 50x1 + 31x2 ≤ 250
3x1 − 2x2 ≥ −4
with being x1, x2 ≥ 0 and integer.
I An optimal solution to LP, x = (376193 ,950193) = (1.94, 4.92),
could be rounded to obtain the solution x = (2, 4).
I But this solution is quite “far” from the optimal solutionx? = (5, 0).
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Rounding and Integer Programming
Rounding and Integer Programming
1 2 3 4 5
1
2
3
4
5
(376/193, 950/193)
Fecho convexo
dos pontos inteiros
x2
x*
x1
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Applications
Summary
Introduction
Rounding and Integer Programming
Applications
Modeling Strategies
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Applications
Fundamentals of Integer Programming
Applications
I Several problems of academic and practical relevance can beformulated in integer programming.
I Examples:
I combinatorial problems found in graph theory (set covering,maximum clique);
I problems in logic (satisfiability problem); andI problems in logistics.
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Applications
Fundamentals of Integer Programming
Applications
I Several problems of academic and practical relevance can beformulated in integer programming.
I Examples:
I combinatorial problems found in graph theory (set covering,maximum clique);
I problems in logic (satisfiability problem); andI problems in logistics.
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Applications
Fundamentals of Integer Programming
Airline Crew Scheduling
I Allocation of flight crews subject to physical, temporal, andwork-related constraints.
I High economic impact on airline companies.
I Given flight legs for a type of airplane, the problem is toallocate weekly crews to cyclic flight routes.
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Applications
Travel Salesman Problem
Background
I Choose an order for a travel salesman to leave his home city,let us say city 1, visit the remaining n− 1 cities precisely once,and then return to the home city.
I The distance traveled should be as short as possible.
I We are given a set of n cities.
I cij is the cost (distance) to travel from city i to city j .
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Applications
Travel Salesman Problem
Background
I Choose an order for a travel salesman to leave his home city,let us say city 1, visit the remaining n− 1 cities precisely once,and then return to the home city.
I The distance traveled should be as short as possible.
I We are given a set of n cities.
I cij is the cost (distance) to travel from city i to city j .
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Applications
Travel Salesman Problem
Background
I The problem is to find the shortest route (circuit) that visitseach city precisely once and whose travel distance is minimum.
I Applications are found in vehicle routing, welding of electroniccircuits, and garbage collection.
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Applications
Travel Salesman Problem
Defining variables
xij =
{1 if salesman travels from city i to city j0 otherwise
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Applications
Travel Salesman Problem
Defining constraints
a) The salesman departs from city i exactly once:
n∑j=1
xij = 1 i = 1, . . . , n
b) The salesman arrives at city j exactly once:
n∑i=1
xij = 1 j = 1, . . . , n
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Applications
Travel Salesman Problem
Defining constraints
c) Connectivity constraints:∑i∈S
∑j /∈S
xij ≥ 1 ∀S ⊂ N, S 6= ∅
or subtour elimination:∑i∈S
∑j∈S, j 6=i
xij ≤ |S | − 1 ∀S ⊆ N, 2 ≤ |S | ≤ n − 1
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Applications
Travel Salesman Problem
S N − S
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Applications
Travel Salesman Problem
Defining the objective
minn∑
i=1
n∑j=1
cijxij
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Modeling Strategies
Summary
Introduction
Rounding and Integer Programming
Applications
Modeling Strategies
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Modeling Strategies
Modeling Fixed Costs
Modeling Fixed Cost
We wish to model the nonlinear function given by:
h(x) =
{f + px if 0 < x ≤ c0 if x = 0
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Modeling Strategies
Modeling Fixed Costs
Modeling Fixed Cost
f
c
h(x)
x
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Modeling Strategies
Modeling Fixed Costs
Modeling Fixed Cost
I Variables:
y =
{1 if x > 00 if x = 0
I Constraints and objective function:
h(x) = fy + pxx ≤ cy
y ∈ {0, 1}
I Model valid only for minimization.
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
I A promising area in theory and practice is disjunctiveprogramming, that is, models and algorithms based ondisjunctions.
I To understand disjunctive programming, suppose that x ∈ Rn
satisfies:0 ≤ x ≤ u and
(aT1 x ≤ b1) or (aT2 x ≤ b2)(1)
I x must satisfy one of the linear constraints, not necessarilyboth constraints.
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
I A promising area in theory and practice is disjunctiveprogramming, that is, models and algorithms based ondisjunctions.
I To understand disjunctive programming, suppose that x ∈ Rn
satisfies:0 ≤ x ≤ u and
(aT1 x ≤ b1) or (aT2 x ≤ b2)(1)
I x must satisfy one of the linear constraints, not necessarilyboth constraints.
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
The feasible region of a disjunction with two constraints: noticethat the feasible region is nonconvex.
x2
x1
a2Tx = b2Regiao Factivel
a1Tx = b1
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
How do we represent the disjunction (1) in mixed-integer linearprogramming.
I Let M = maxi=1,2{aTi x − bi : 0 ≤ x ≤ u}.I Fist, we introduce two binary variables, y1 and y2, whose
semantics is explained below:
y1 =
{1 if x satisfies aT1 x ≤ b10 otherwise
y2 =
{1 if x satisfies aT2 x ≤ b20 otherwise
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
How do we represent the disjunction (1) in mixed-integer linearprogramming.
I Let M = maxi=1,2{aTi x − bi : 0 ≤ x ≤ u}.I Fist, we introduce two binary variables, y1 and y2, whose
semantics is explained below:
y1 =
{1 if x satisfies aT1 x ≤ b10 otherwise
y2 =
{1 if x satisfies aT2 x ≤ b20 otherwise
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
Given the above variables, we can introduce the completeformulation:
aT1 x ≤ b1 + M(1− y1)aT2 x ≤ b2 + M(1− y2)
y1 + y2 = 1y1, y2 ∈ {0, 1}
0 ≤ x ≤ u
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
I Disjunctions appear in scheduling problems.
I Tasks 1 and 2 must be processed in a given machine, but notsimultaneously.
I Let pi be the processing time of task i and ti the timeprocessing begins.
I Then, we can express temporal precedence of one task inrelation to the other by a disjunction:
(t1 + p1 ≤ t2) or (t2 + p2 ≤ t1)
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Modeling Strategies
Modeling Disjunctions
Discrete Alternatives and Disjunctions
I Disjunctions appear in scheduling problems.
I Tasks 1 and 2 must be processed in a given machine, but notsimultaneously.
I Let pi be the processing time of task i and ti the timeprocessing begins.
I Then, we can express temporal precedence of one task inrelation to the other by a disjunction:
(t1 + p1 ≤ t2) or (t2 + p2 ≤ t1)
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Modeling Strategies
Modeling Variable Product
Power of Binary Variables
I The power function xp, p ∈ N+, with x ∈ {0, 1} is nonlinear.I Notice that xp = x since:
I xp = 0 if x = 0 andI xp = 1 and x = 1.
I Thus, it is possible to linearize the term xp.
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Modeling Strategies
Modeling Variable Product
Power of Binary Variables
I The power function xp, p ∈ N+, with x ∈ {0, 1} is nonlinear.I Notice that xp = x since:
I xp = 0 if x = 0 andI xp = 1 and x = 1.
I Thus, it is possible to linearize the term xp.
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Modeling Strategies
Modeling Variable Product
Product of Binary Variables
I Consider the term y = x1x2x3, in which xi ∈ {0, 1}.I The nonlinear term can be reformulated as:
y ≤ x1
y ≤ x2
y ≤ x3
y ≥ x1 + x2 + x3 − 2
y ≥ 0
x1, x2, x3 ∈ {0, 1}
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Modeling Strategies
Modeling Variable Product
Product of Binary Variables
I Consider the term y = x1x2x3, in which xi ∈ {0, 1}.I The nonlinear term can be reformulated as:
y ≤ x1
y ≤ x2
y ≤ x3
y ≥ x1 + x2 + x3 − 2
y ≥ 0
x1, x2, x3 ∈ {0, 1}
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Modeling Strategies
Modeling Variable Product
Sign Function: sign(x)
I The function sign(·) can be modeled using a binary variable.
I Assuming that |x | ≤ M, then:
x ≤ Mz ,
x ≥ −M(1− z),
sign(x) = (2z − 1),
z ∈ {0, 1}
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Modeling Strategies
Modeling Variable Product
Some Challenges
Can you model the following functions in mixed-integer linearprogramming?
I y = max{x1, x2}?I y = |x |?
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Modeling Strategies
Modeling Variable Product
Integer Programming
I Thank you for attending this lecture!!!