arXiv subjects

Marie-Louise Lackner

Publications and source records attributed to Marie-Louise Lackner.

9 recordsLinked to original sources

arXiv2025

Solving Parallel Machine Scheduling With Precedences and Cumulative Resource Constraints With Calendars

The task of finding efficient production schedules for parallel machines is a challenge that arises in most industrial manufacturing domains. There is a large potential to minimize production costs through automated scheduling techniques, due to the large-scale requirements of modern factories. In the past, solution approaches have been studied for many machine scheduling variations, where even basic variants have been shown to be NP-hard. However, in today's real-life production environments, additional complex precedence constraints and resource restrictions with calendars arise that must be fulfilled. These additional constraints cannot be tackled efficiently by existing solution techniques. Thus, there is a strong need to develop and analyze automated methods that can solve such real-life parallel machine scheduling scenarios. In this work, we introduce a novel variant of parallel machine scheduling with job precedences and calendar-based cumulative resource constraints that arises in real-life industrial use cases. A constraint modeling approach is proposed as an exact solution method for small scheduling scenarios together with state-of-the-art constraint-solving technology. Further, we propose a construction heuristic as well as a tailored metaheuristic using local search to efficiently tackle large-scale problem instances. This metaheuristic approach has been deployed and is currently being used in an industrial setting.

cs.AI↗

arXiv2024

Theoretical Lower Bounds for the Oven Scheduling Problem

The Oven Scheduling Problem (OSP) is an NP-hard real-world parallel batch scheduling problem arising in the semiconductor industry. The objective of the problem is to schedule a set of jobs on ovens while minimizing several factors, namely total oven runtime, job tardiness, and setup costs. At the same time, it must adhere to various constraints such as oven eligibility and availability, job release dates, setup times between batches, and oven capacity limitations. The key to obtaining efficient schedules is to process compatible jobs simultaneously in batches. In this paper, we develop theoretical, problem-specific lower bounds for the OSP that can be computed very quickly. We thoroughly examine these lower bounds, evaluating their quality and exploring their integration into existing solution methods. Specifically, we investigate their contribution to exact methods and a metaheuristic local search approach using simulated annealing. Moreover, these problem-specific lower bounds enable us to assess the solution quality for large instances for which exact methods often fail to provide tight lower bounds.

cs.AI↗

arXiv2022

Exact methods and lower bounds for the Oven Scheduling Problem

The Oven Scheduling Problem (OSP) is a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem via constraint programming (CP) and integer linear programming (ILP) and present corresponding models. For an experimental evaluation, we introduce a multi-parameter random instance generator to provide a diverse set of problem instances. Using state-of-the-art solvers, we evaluate the quality and compare the performance of our CP- and ILP-models. We show that our models can find feasible solutions for instances of realistic size, many of those being provably optimal or nearly optimal solutions. Finally, we derive theoretical lower bounds on the solution cost of feasible solutions to the OSP; these can be computed within a few seconds. We show that these lower bounds are competitive with those derived by state-of-the-art solvers.

cs.AI↗

arXiv2020

A Mathematical Analysis of an Election System Proposed by Gottlob Frege

In 1998 a long-lost proposal for an election law by Gottlob Frege (1848--1925) was rediscovered in the Thüringer Universitäts- und Landesbibliothek in Jena, Germany. The method that Frege proposed for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions in the long run. We find that through a slight modification of Frege's original method even stronger proportionality guarantees can be achieved. This modified version of Frege's method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties.

cs.GT↗

arXiv2020

Runs in labelled trees and mappings

We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional digraph together with a generating functions approach allows us to perform a joint study of ascending and descending runs in labelled trees and mappings, respectively. From the given characterization of the respective generating functions we can deduce bivariate central limit theorems for these quantities. Furthermore, for ascending runs (or descending runs) we gain explicit enumeration formulae showing a connection to Stirling numbers of the second kind. We also give a bijective proof establishing this relation, and further state a bijection between mappings and labelled trees connecting the quantities in both structures.

math.CO↗

arXiv2020

Exact and Metaheuristic Approaches for the Production Leveling Problem

In this paper we introduce a new problem in the field of production planning which we call the Production Leveling Problem. The task is to assign orders to production periods such that the load in each period and on each production resource is balanced, capacity limits are not exceeded and the orders' priorities are taken into account. Production Leveling is an important intermediate step between long-term planning and the final scheduling of orders within a production period, as it is responsible for selecting good subsets of orders to be scheduled within each period. A formal model of the problem is proposed and NP-hardness is shown by reduction from Bin Backing. As an exact method for solving moderately sized instances we introduce a MIP formulation. For solving large problem instances, metaheuristic local search is investigated. A greedy heuristic and two neighborhood structures for local search are proposed, in order to apply them using Variable Neighborhood Descent and Simulated Annealing. Regarding exact techniques, the main question of research is, up to which size instances are solvable within a fixed amount of time. For the metaheuristic approaches the aim is to show that they produce near-optimal solutions for smaller instances, but also scale well to very large instances. A set of realistic problem instances from an industrial partner is contributed to the literature, as well as random instance generators. The experimental evaluation conveys that the proposed MIP model works well for instances with up to 250 orders. Out of the investigated metaheuristic approaches, Simulated Annealing achieves the best results. It is shown to produce solutions with less than 3% average optimality gap on small instances and to scale well up to thousands of orders and dozens of periods and products. The presented metaheuristic methods are already being used in the industry.

cs.AI↗

arXiv2019

On the Likelihood of Single-Peaked Preferences

This paper contains an extensive combinatorial analysis of the single-peaked domain restriction and investigates the likelihood that an election is single-peaked. We provide a very general upper bound result for domain restrictions that can be defined by certain forbidden configurations. This upper bound implies that many domain restrictions (including the single-peaked restriction) are very unlikely to appear in a random election chosen according to the Impartial Culture assumption. For single-peaked elections, this upper bound can be refined and complemented by a lower bound that is asymptotically tight. In addition, we provide exact results for elections with few voters or candidates. Moreover, we consider the Pólya urn model and the Mallows model and obtain lower bounds showing that single-peakedness is considerably more likely to appear for certain parameterizations.

cs.GT↗

arXiv2016

The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations

The Permutation Pattern Matching problem, asking whether a pattern permutation $π$ is contained in a permutation $τ$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $π$ and $τ$ are $321$-avoiding; the second is applicable if $π$ and $τ$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $π$ and $n$ the length of $τ$.

math.CO↗

arXiv2015

Longest increasing subsequences and log concavity

Let $π$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(π)$ the length of a longest increasing subsequence of $π$. Let $\ell_{n,k}$ be the number of permutations $π$ of $[n]$ with $\ell(π)=k$. Chen conjectured that the sequence $\ell_{n,1},\ell_{n,2},\dots,\ell_{n,n}$ is log concave for every fixed positive integer $n$. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Schensted algorithm have certain shapes. In addition, we present a proof of Deift that part of the limiting distribution is log concave. Various other conjectures are discussed.

math.CO↗