Discrete Optimization Introduction & applications

Transcription

1 Discrete Optimization /21 Discrete Optimization Introduction & applications Bertrand Cornélusse ULg - Institut Montefiore 2013

2 Discrete Optimization /21 Outline Introduction Some applications Communication Electrical power systems Bioinformatics Scheduling References

3 Discrete Optimization /21 A popular discrete problem: the traveling salesman problem (TSP) Given n cities, in which order should one visit them in order to minimize the total distance? Example: Visit 23 EU cities with minimal traveling distance

4 Discrete Optimization /21 A popular discrete problem: the traveling salesman problem (TSP) Given n cities, in which order should one visit them in order to minimize the total distance? Example: Visit 23 EU cities with minimal traveling distance Optimal solution

5 Discrete Optimization /21 Why is that so complicated? After all, there is a finite number of solutions In particular, n! possible permutations Imagine we can check possibilities per second That is already a pretty amazing machine! 10! 0 sec 20! 28 days 30! 8400 billion years 40! 5 quadrillions times the age of the Earth... Let us not dare to continue...

6 Discrete Optimization /21 Applications of the TSP Truck routing Arranging school buses routes (the very first application) Scheduling of a machine to drill holes in a circuit board Delivery of meals to homebound people Genome sequencing Cities are local strings, and the cost is the measure of likelihood that a sequence follows another Link points through fiber optic connections in order to minimize the total distance and ensure that any failure leaves the whole system operational Interactive example Thanks to Pierre Schaus, UCL.

7 Discrete Optimization /21 Milestones in the solution of TSP instances Year Research team Number of cities 1954 Dantzig, Fulkerson, Johnson 49 cities 1971 Held, Karp 64 cities 1977 Grötschel 120 cities 1980 Crowder, Padeberg 318 cities 1987 Padberg, Rinaldi 2392 cities 1994 Applegate, Bixby, Chvatal, Cook 7397 cities 2006 Applegate, Bixby, Chvatal, Cook, Helsgaun cities

8 Discrete Optimization /21 Milestones in the solution of TSP instances Year Research team Number of cities 1954 Dantzig, Fulkerson, Johnson 49 cities 1971 Held, Karp 64 cities 1977 Grötschel 120 cities 1980 Crowder, Padeberg 318 cities 1987 Padberg, Rinaldi 2392 cities 1994 Applegate, Bixby, Chvatal, Cook 7397 cities 2006 Applegate, Bixby, Chvatal, Cook, Helsgaun cities Techniques: formulation as an integer programming problem, branch-and-bound, cutting planes. topics covered in the class

9 Discrete Optimization /21 Modeling a discrete problem This course focuses mainly on mathematical programming formulations. We consider c, f, g linear min c(x) s.t. f (x) b g(x) = 0 x X. X = Z n + This is called Integer (Linear) Programming (IP). Remarks: Mixed Integer Programming (MIP) when some variables have a continuous domain. If c is nonlinear, very little has been done (except the quadratic case) If f or g is nonlinear: even less

10 Discrete Optimization /21 Modeling a discrete problem (...) We will also consider alternative approaches. They are important because sometimes better for some applications. Constraint Programming (CP): uses reasoning on the constraints to discard infeasible variable values. Mostly targetted towards feasibility problems. Local Search: when starting from a feasible solution, try improving the solution by applying small perturbations that do not violate constraints. Dynamic Programming: many applications, when problem has an optimal substructure property. In many practical solutions, they are used in conjunction: CP with IP. Local search to refine solutions provided by a global approach (CP or MIP).

11 Discrete Optimization /21 Outline Introduction Some applications Communication Electrical power systems Bioinformatics Scheduling References

12 Discrete Optimization /21 Location of GSM transmitters A mobile phone operator decides to equip a currently uncovered geographical zone. The management allocates a budget of 10 million e to equip this region. 7 locations are possible for the construction of the transmitters. Every transmitter only covers a certain number of communities. Where should the transmitters be built to cover the largest population with the given budget?

13 Discrete Optimization /21 Electricity production An electricity producer wants to plan the level of production of his main power plants in order to fulfill the demand and minimize his costs. The demand for 6 time periods in the day are listed in the following table. Time 0-4h 4-8h 8-12h 12-16h 16-20h 20-0h Demand (GWh)

14 Discrete Optimization /21 Electricity production An electricity producer wants to plan the level of production of his main power plants in order to fulfill the demand and minimize his costs. The demand for 6 time periods in the day are listed in the following table. Time 0-4h 4-8h 8-12h 12-16h 16-20h 20-0h Demand (GWh) The producer has 6 coal power plants. 3 of them have a power of 1GW while the remaining 3 have a power of 2GW. The cost of operating a 1GW plant is 100 e/gwh while the cost of operating a 2GW plant is 200 e/gwh.

15 Discrete Optimization /21 Electricity production An electricity producer wants to plan the level of production of his main power plants in order to fulfill the demand and minimize his costs. The demand for 6 time periods in the day are listed in the following table. Time 0-4h 4-8h 8-12h 12-16h 16-20h 20-0h Demand (GWh) The producer has 6 coal power plants. 3 of them have a power of 1GW while the remaining 3 have a power of 2GW. The cost of operating a 1GW plant is 100 e/gwh while the cost of operating a 2GW plant is 200 e/gwh. There is a fixed cost for using a plant of 100 eper hour of use. Finally, starting up a plant costs 400 e. Type Number Power Varying Fixed Startup Cost Cost Cost (GW) e/gwh e/h e

16 Discrete Optimization /21 Electricity production An electricity producer wants to plan the level of production of his main power plants in order to fulfill the demand and minimize his costs. The demand for 6 time periods in the day are listed in the following table. Time 0-4h 4-8h 8-12h 12-16h 16-20h 20-0h Demand (GWh) The producer has 6 coal power plants. 3 of them have a power of 1GW while the remaining 3 have a power of 2GW. The cost of operating a 1GW plant is 100 e/gwh while the cost of operating a 2GW plant is 200 e/gwh. There is a fixed cost for using a plant of 100 eper hour of use. Finally, starting up a plant costs 400 e. Type Number Power Varying Fixed Startup Cost Cost Cost (GW) e/gwh e/h e What is the optimal production planning to minimize the costs while satisfying the demand?

17 1 In practive, for most of them, through retailers. Discrete Optimization /21 Day-ahead Electricity markets The day is divided in T equal time periods. Suppliers and Consumers 1 submit orders to the market, defined by a price p o, this is the limit price at which suppliers (resp. consumers) are willing to sell (resp. buy); a power level q o. This quantity must be accepted totally or rejected. Let us assume q o is negative for demand, positive for supply; start and end periods (to start, to end ). The market operator must decide which orders are accepted and rejected. Deals are remunarated at a (period-dependent) uniform price MCP t. Suppliers (resp. consumers) receive (resp. pay) money according to the power they actually sell (resp. buy) and the market prices. Total supply must equal total demand at any period. The market is designed to maximize the social welfare One cannot improve the surplus of one participant without degrading the surplus of another participant of a greater amount. No participant can have a negative surplus (although in some cases that would increase the total surplus). Illustration:

18 Discrete Optimization /21 And more... There are many more applications in power systems: hydro-electric generation scheduling network planning with contingencies topological optimization of the network (e.g. to minimize losses) investements planning

19 Discrete Optimization /21 Multiple Sequence Alignment Compare proteins described by a sequence of letters (amino acids). Aligning sequences means inserting symbols in the sequences such that similar letters appear in the same column (or index in the sequence). Many objective functions can be defined, depending on the properties of the proteins that are compared. Example Input: sequence A: AKZRTJVBTMRJV sequence B: HKZWTXVDTMR Objective: Maximize number of letters matched perfectly, i.e. do not allow mismatches. Output: sequence A: -AKZ-RTJ-V-BTMRJV sequence B: H-KZW-T-XVD-TMR-- Solution methods: Heuristics and approximation algorithms, Dynamic Programming, Branch-and-Cut.

20 Discrete Optimization /21 Probe Design for Microarray Experiments Probes (i.e. experiments) can tell us if some targets (viruses or bacteria) are present in a biological sample. Assume only one target is present in each sample, what is the smallest subset of probes necessary to identify it? Example Targets: t i, i = 1, 2,..., M, probes: p i, i = 1, 2,..., N. Input: p 1 p 2 p 3 p 4 p 5 p 6 p 7 t t t t Means p 1 can tell us if any of targets t 2 and t 3 is present in the sample, p 2 can tell us if t 4 is present,...

21 Discrete Optimization /21 Probe Design for Microarray Experiments Probes (i.e. experiments) can tell us if some targets (viruses or bacteria) are present in a biological sample. Assume only one target is present in each sample, what is the smallest subset of probes necessary to identify it? Example Targets: t i, i = 1, 2,..., M, probes: p i, i = 1, 2,..., N. Input: p 1 p 2 p 3 p 4 p 5 p 6 p 7 t t t t Means p 1 can tell us if any of targets t 2 and t 3 is present in the sample, p 2 can tell us if t 4 is present,... Solution: p 4 p 6 t t t t 4 1 0

22 Discrete Optimization /21 Other applications Sequence-Structure alignment for RNA: compared to DNA and proteins, there is additional structure to the sequence that must be accounted for (folding). Protein 3D structure prediction from sequence. See [Althaus et al.(2009)althaus, Klau, Kohlbacher, Lenhof, and Reinert].

23 Discrete Optimization /21 Sequencing jobs on a bottleneck machine In workshops, it may happen that a single machine determines the throughput of the production critical machine. A set of tasks is to be processed. The execution is non-preemptive. For every task i, a release date and duration are given. Different possible objectives: total processing time (makespan), average processing time, total tardiness (if due dates are given)

24 Discrete Optimization /21 Scheduling of telecommunications via satellite A digital telecommunications system via satellite consists of a satellite and a set of stations on earth. We consider for example 4 stations in the US that communicate with 4 stations in Europe through the satellite. The total traffic is given through a matrix TRAF tr. The satellite has a switch that allows any permutation between the 4 transmitters and the 4 receivers. The cost of a switch is the duration of its longest packet. The objective is to find a schedule with minimal total duration.

25 Discrete Optimization /21 Scheduling nurses A working day in a hospital is subdivided in 12 periods of two hours. The personnel requirements change from period to period. A nurse works 8 hours a day and is entitled to a break after having worked for 4 hours. Determine the minimum number of nurses required to cover all requirements? If only 80 nurses are avalable, and assuming that it is not enough, we may allow for a certain number of nurses to work for a fifth period right after the last one. Determine a schedule the minimum number of nurses working overtime.

26 Discrete Optimization /21 Outline Introduction Some applications Communication Electrical power systems Bioinformatics Scheduling References

27 Discrete Optimization /21 References Ernst Althaus, Gunnar W Klau, Oliver Kohlbacher, Hans-Peter Lenhof, and Knut Reinert. Integer linear programming in computational biology. In Efficient Algorithms, pages Springer, 2009.