New results about multi-band uncertainty in Robust Optimization

"The Price of Robustness" by Bertsimas and Sim represented a breakthrough in the development of a tractable robust counterpart of Linear Programming Problems. However, the central modeling assumption that the deviation band of each uncertain parameter is single may be too limitative in practice: experience indeed suggests that the deviations distribute also internally to the single band, so that getting a higher resolution by partitioning the band into multiple sub-bands seems advisable. The critical aim of our work is to close the knowledge gap about the adoption of a multi-band uncertainty set in Robust Optimization: a general definition and intensive theoretical study of a multi-band model are actually still missing. Our new developments have been also strongly inspired and encouraged by our industrial partners, which have been interested in getting a better modeling of arbitrary distributions, built on historical data of the uncertainty affecting the considered real-world problems. In this paper, we study the robust counterpart of a Linear Programming Problem with uncertain coefficient matrix, when a multi-band uncertainty set is considered. We first show that the robust counterpart corresponds to a compact LP formulation. Then we investigate the problem of separating cuts imposing robustness and we show that the separation can be efficiently operated by solving a min-cost flow problem. Finally, we test the performance of our new approach to Robust Optimization on realistic instances of a Wireless Network Design Problem subject to uncertainty.Comment: 15 pages. The present paper is a revised version of the one appeared in the Proceedings of SEA 201

Problem-Based Learning Of Heuristic Methods For Decision Problems In Mathematics, Computer Science And Industrial Engineering

In a digitalized world, most processes can be formalised, measured and described mathematically. The use of analytical methods to optimise such models and decisions constitutes operational research (OR), developing new methods for a specific problem and analysing them are part of discrete optimisation (DO). However, there is limited research on OR and application driven DO in higher education. Furthermore, neither is well integrated into engineering education research. In this work, we present a case study of an interdisciplinary Master’s course on heuristic methods in the context of OR and DO. We discuss to what extent wellestablished approaches from engineering education practice, such as ProblemBased Learning, are applicable. Furthermore, we introduce two practical cases and argue that due to its application-oriented nature, OR and DO specifically stimulate independent student work. Results from evaluations, minute papers and student coursework indicate that the teaching approach successfully contributed to students’ achievement of the intended learning outcomes. To further foster discussion, we not only provide the lecture notes publicly, but also all tutorial and project case data to instructors upon request under a CC BY-NC license

Multithread interval scheduling with flexible machine availabilities: complexity and efficient algorithms

In the known Interval Scheduling problem with Machine Availabilities (ISMA), each machine has a contiguous availability interval, and each job has a specific time interval which has to be scheduled. The objective is to schedule all jobs such that the machines’ availability intervals are respected or to decide that there exists no such schedule. We extend ISMA by introducing machine capacities and flexible machine end times. Using machine capacities we model parallel processing of multiple jobs per machine, which leads to the Multithread Interval Scheduling with Machine Availabilities (MISMA). Limited machine availabilities are usually due to maintenance. Time slots for maintenance at the end of a processing period are often predetermined by staff schedules before the slots are assigned to specific machines. This motivates a variant of MISMA in which the end times of the machines’ availability intervals can be permuted, the Flexible Multithread ISMA (FLEXMISMA). In this paper, we determine a tight classification of conditions that are required for obtaining a polynomial-time algorithm for both MISMA and FLEXMISMA. More specifically, we show that FLEXMISMA is at least as hard as MISMA. For FLEXMISMA, we present polynomial-time algorithms for instances (i) with at most two available machines at a time, and (ii) with constantly many parallel jobs at each point in time, which both also solve MISMA; (iii) with arbitrarily many machines of capacity one each, in which case MISMA is known to be NP-hard; and (iv) with jobs having length one or two, for which the complexity of MISMA remains open Furthermore, we complement result (i) by showing that both problems are NP-hard already for instances with three machines as a special case of the Vertex-Disjoint Paths problem. In contrast to (iii), we prove that increasing the capacity of machines from one to two renders FLEXMISMA NP-hard as well for arbitrarily many machines

Coworking scheduling with network flows

Collaborative usage of resources is becoming increasingly popular in various fields. One common example are coworking spaces — office rooms with work places that can be rented by individuals on hourly basis. We consider the problem of assigning all booking requests for a day to equivalent office rooms with different but fixed opening times and fixed interchangeable closing times. The closing times are flexible due to daily maintenance, e.g. cleaning, which must be done in all rooms in an arbitrary order. This problem is related to the known Interval Scheduling Problem with Machine Availabilities (ISMA), where each machine has a contiguous availability interval, and each job presents a specific time interval which has to be scheduled. According to our coworking scheduling application, we extend ISMA to Flexible Multithread ISMA (FlexMISMA) by introducing machine capacities that model the number of work places per room and by allowing to permute the end times of machines’ availability periods. In this paper, we determine a tight classification of necessary conditions for the existence of a polynomial time algorithm for FlexMISMA, assuming P ≠ NP. More specifically, we develop a network flow model and present polynomial time algorithms for instances (i) with two machines, and (ii) with arbitrarily many machines of capacity one each. In the same time, we prove that increasing the machine capacity to two renders FlexMISMA NP-hard for arbitrarily many machines. Furthermore, we complement result (i) by showing that the problem is NP-hard already for instances with three machines as a special case of the Vertex-Disjoint Paths problem

Robust Algorithms for Sorting Railway Cars

We consider a sorting problem from railway optimization called train classification: incoming trains are split up into their single cars and reassembled to form new outgoing trains. Trains are subject to delay, which may turn a prepared sorting schedule infeasible for the disturbed situation. The classification methods applied today deal with this issue by completely disregarding the input order of cars, which provides robustness against any amount of disturbance but also wastes the potential contained in the a priori knowledge about the input

Robust Minimum Cost Flow Problem Under Consistent Flow Constraints

The robust minimum cost flow problem under consistent flow constraints (RobMCF$\equiv$) is a new extension of the minimum cost flow (MCF) problem. In the RobMCF$\equiv$ problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem on networks with integral capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCF$\equiv$ problem is strongly $\mathcal{NP}$-hard on acyclic digraphs by a reduction from the $(3,B2)$-Sat problem. Further, we demonstrate that the RobMCF$\equiv$ problem is weakly $\mathcal{NP}$-hard on series-parallel digraphs by providing a reduction from Partition and a pseudo-polynomial algorithm based on dynamic programming. Finally, we propose a special case on series-parallel digraphs for which we can solve the RobMCF$\equiv$ problem in polynomial time