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 :  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 2 4/24/2018

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 3 ADS, ILP 1

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. 4 4 SCM, chapter 3

Exam ple: Ajax Decision variables: MA number of alpha’s produced, etc Objective function    max Z 350 MA 470 MB 610 MC    subject to MA MB 120 (A line) Constraints   MC 48 ( C line )    10 MA 15 MB 20 MC 2000 ( labor )  MA , MB , MC 0 5 5 SCM, chapter 3

Linear program m ing Min c T x s.t. Ax ≤ b x ≥ 0 With � ∈ � � , � ∈ ℚ � , A ∈ ℚ ��� , and � ∈ ℚ � 6 ADS, ILP 1

LP:Geom etric x 2  5 3 (4,4)  x  3 x 1 2  x  1 x 2 1 2 1 x 1  0 2 x  max x 1 2 x 2  0 1 2 3 ADS, ILP 1 7

The sim plex m ethod  Example dictionaries 8 ADS, ILP 1

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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 16 ADS, ILP 1

Solution m ethod for linear program m ing Seems not too hard to implement. But, for larger  Simplex method problems you run into  Slower than polynomial numerical problems. Use a  Practical standard solver (Gurobi,  Ellipsoid method CPLEX, GLPK)  Polynomial (Khachian, 1979)  Not practical  Interior points methods  Polynomial (Karmakar, 1984)  Outperforms Simplex for very large instances LP  P 17 ADS, ILP 1

Knapsack problem Knapsack with volume 15 What should you take with you to maximize utility? Item 1:paper 2:book 3:bread 4:smart 5:water -phone Utility 8 12 7 15 12 Volume 4 8 5 2 6 ADS, ILP 1 18 18

Knapsack problem ( 2 ) x 1 = 1 if item 1 is selected, 0 otherwise, x 2 , …… max z= 8 x 1 + 12 x 2 + 7 x 3 + 15 x 4 + 12 x 5 subject to 4 x 1 + 8 x 2 + 5 x 3 + 2 x 4 + 6 x 5 ≤ 15 x 1 , x 2 , x 3 , x 4 , x 5 Є {0,1} 19 ADS, ILP 1

( Mixed) I nteger linear program m ing Min c T x + d T 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 20 ADS, ILP 1

( Mixed) I nteger linear program LP-relaxation Min c T x Min c T x s.t. Ax + By ≤ b s.t. Ax + By ≤ b x,y ≥ 0 x,y ≥ 0 x integral (or binary) Lower bound (or upper bound in case of maximization) 21 ADS, ILP 1

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Modeling  Decision variables  Objective function  Constraints 23 ADS, ILP 1

Assignm ent problem  n persons, n jobs.  Each person can do at most one job  Each job has to be executed  C ij cost if person i performs job j  We want to minimize cost 24 ADS, ILP 1

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? 25 ADS, ILP 1

Facility location Possible locations: n Customers: m 26 ADS, ILP 1

Capacitated facility location  Data:  m customers,  Customer demand: D i  n possible locations of depots (facilities)  c ij unit cost of serving customer i by depot j  Capacity depot: C j  Fixed cost for opening depot DC: F j  Which depots are opened and which customer is served by which depot? 27 ADS, ILP 1

Uncapacitated facility location  Data:  m customers, n possible locations of depot  Each customer is assigned to one depot  d ij cost of serving customer i by depot j  Fixed cost for opening depot DC: F j  Which depots are opened and which customer is served by which depot? 28 ADS, ILP 1

Uncapacitated facility location  Two models FL and AFL  Same optimal value Z IP for ILP  Set of feasible solutions for LP-relaxation satisfy: � �� ⊆ � ���  Set of optimal values satisfy: � ��� � � �� � � ��  The LP-relaxation of FL gives a stronger bound 29 ADS, ILP 1

Single m achine scheduling exam ple  n jobs, 1 machine  Job j requires uninterrupted processing time P j  Job j has release date r j  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 30 ADS, ILP 1