Scheduling and Timetabling Integer linear programming models Marjan - - PowerPoint PPT Presentation
Scheduling and Timetabling Integer linear programming models Marjan van den Akker 1 I ntro. Marjan van den Akker Lecturer/research Algorithms and Complexity group: Coordination Software- and Gameproject Master courses :
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Scheduling and Timetabling
Integer linear programming models Marjan van den Akker
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I ntro…….
Marjan van den Akker Lecturer/research Algorithms and Complexity group:
Coordination Software- and Gameproject Master courses :
Algorithms for decision support (COSC), Advanced Linear Programming (Mastermath)
Research on planning/scheduling algorithms:
Public transportation Sustainable energy systems integer linear programming and local search simulation
4/24/2018
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Contents
Integer linear programming is well-known modelling and
solution technique
First, linear programming Modelling integer linear programming problems This is material from the course Algorithms for Decision
Support
NB: we will work a lot on the blackboard.
You are strongly advised to take notes.
A beautiful reader is available at
http://www.cs.uu.nl/docs/vakken/mads/LectureNotesILP.pdf
ADS, ILP 1
4 SCM, chapter 3
4 Exam ple: Ajax
Three types of computers: Alpha, Beta, and Gamma. Net profit: $350,- per Alpha, $470,- per Beta, and
$610,- per Gamma.
Every computer can be sold at the given profit. Testing: Alpha and Beta computers on the A-line,
Gamma computers on the C-line.
Testing takes 1 hour per computer. Capacity A-line: 120 hours; capacity C-line: 80
hours.
Required labor: 10 hours per Alpha, 15 hours per
Beta, and 20 hours per Gamma.
Total amount of labor available: 2000 hours.
5 SCM, chapter 3
5 Exam ple: Ajax
) labor ( 2000 ) line C ( 48 line) (A 120 MC , MB , MA MC 20 MB 15 MA 10 MC MB MA subject to MC 610 MB 470 MA 350 Z max
Decision variables: MA number of alpha’s produced, etc Objective function Constraints
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Min cTx s.t. Ax ≤ b x ≥ 0 With ∈ , ∈ ℚ, A ∈ ℚ, and ∈ ℚ
ADS, ILP 1
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LP:Geom etric
5 x2
1
2 1
x x
x2
2 1
max x x 2
(4,4)
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2 1
x x
2 1 3 1 2 3
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The sim plex m ethod
Example dictionaries
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Solution options
A linear programming problem can
be infeasible
Example:
max 6 4 3, 2 2 8, , 0
be unbounded
Example:
max 6 4 3, 2 2 8, , 0 for any 0 we have that , is feasible
have a bounded optimum
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Solution m ethod for linear program m ing
Simplex method
Slower than polynomial Practical
Ellipsoid method
Polynomial (Khachian, 1979) Not practical
Interior points methods
Polynomial (Karmakar, 1984) Outperforms Simplex for very large instances
ADS, ILP 1
Seems not too hard to
problems you run into numerical problems. Use a standard solver (Gurobi, CPLEX, GLPK)
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Knapsack problem Item 1:paper 2:book 3:bread 4:smart
5:water Utility 8 12 7 15 12 Volume 4 8 5 2 6
Knapsack with volume 15 What should you take with you to maximize utility?
ADS, ILP 1
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Knapsack problem ( 2 )
max z= 8 x1 + 12 x2 + 7 x3 + 15 x4 + 12 x5 subject to 4 x1 + 8 x2 + 5 x3 + 2 x4 + 6 x5 ≤ 15 x1, x2, x3 , x4 , x5 Є {0,1}
x1 = 1 if item 1 is selected, 0 otherwise, x2, ……
ADS, ILP 1
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Min cTx + dT y s.t. Ax + By ≤ b x,y ≥ 0 x integral (or binary) Extension of LP:
Good news: more possibilities for modelling Bad news: larger solution times
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( Mixed) I nteger linear program Min cTx s.t. Ax + By ≤ b x,y ≥ 0 x integral (or binary) LP-relaxation Min cTx s.t. Ax + By ≤ b x,y ≥ 0 Lower bound (or upper bound in case of maximization)
ADS, ILP 1
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Modeling
Decision variables Objective function Constraints
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Assignm ent problem
n persons, n jobs. Each person can do at most one job Each job has to be executed Cij cost if person i performs job j We want to minimize cost
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Maxim um independent set
Given a graph G=(V,E) V: nodes E: edges An independent set I is a set of nodes, such that every
edge has at most one nodes in I, i.e., no pair of nodes in I is connected.
What is the maximum number of nodes in an independent
set?
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Facility location Possible locations: n Customers:m
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Capacitated facility location
Data:
m customers, Customer demand: Di n possible locations of depots (facilities) cij unit cost of serving customer i by depot j Capacity depot: Cj Fixed cost for opening depot DC: Fj
Which depots are opened and which customer is served by which
depot?
ADS, ILP 1
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Uncapacitated facility location
Data:
m customers, n possible locations of depot Each customer is assigned to one depot dij cost of serving customer i by depot j Fixed cost for opening depot DC: Fj
Which depots are opened and which customer is served by which
depot?
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Uncapacitated facility location
Two models FL and AFL Same optimal value ZIP for ILP Set of feasible solutions for LP-relaxation satisfy:
⊆
The LP-relaxation of FL gives a stronger bound
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Single m achine scheduling exam ple
n jobs, 1 machine Job j requires uninterrupted processing time Pj Job j has release date rj Machine can process at most one job at a time We want to minimize the weighted sum of the completion
times
This is going to be denoted by
ADS, ILP 1