Single-source capacitated facility location problems are well studied in the operations research literature, yet classic problems often lack practicability by disregarding the customers’ perspective: An authority that assigns customers to open facilities deprives customers from choosing facilities according to their individual preferences. In reality, this can render solutions infeasible, as customers may deviate to their most preferred open facility, thereby breaking capacities of facilities serving more customers than originally planned. Preference constraints aim to prevent this by ensuring that each customer is served at their most preferred open facility. The corresponding problem is called single-source capacitated facility location problem with customer preferences (CFLP-CP) and is known to be strongly NP-hard. In this paper, we provide a deeper understanding of the transition from polynomially solvable cases to strongly NP-hard cases of the CFLP-CP. In particular, we show that the effect of preference constraints on the theoretical complexity can go both ways: Some problems become harder, while others become easier. This is because preference constraints simplify the assignment of customers to facilities, while simultaneously increasing the complexity of locating facilities. We show that the type of customer preferences, e.g., strict preferences or geographically closest assignments, has a vital impact on the complexity. Notably, strict preferences, i.e., customers cannot be indifferent, allow to compute feasible solutions of the CFLP-CP in polynomial time.

In medical appointment assignment, unit jobs representing patients arrive online and are assigned to a time slot within their given feasible time interval. We model this setting as interval-constrained online bipartite matching problem. We consider a variant of this problem where reassignments are allowed and extend it by a notion of time that is decoupled from the job arrival events. As jobs arrive, the current point in time gradually advances, and once the time of a slot is passed, the job assigned to it is fixed and cannot be reassigned anymore. We analyze two algorithms for this problem with respect to the resulting matching size and the number of occurring reassignments. We show that FirstFit with reassignments according to the shortest augmenting path rule is exactly 2/3 -competitive with respect to the matching cardinality. The competitive ratio remains 2/3 if we restrict FirstFit to consider only augmenting paths causing at most a constant number of reassignments, which implies a linear number of reassignments in total. This fills the gap between the known optimal algorithm with no reassignments at all, which is 1/2 -competitive, on the one hand, and an earliest-deadline-first strategy (EDF), which we prove to be 1-competitive in our over-time framework, but which suffers Ω (n^2) reassignments in the worst case, on the other. We further extend the problem setting to the sets of feasible slots per job that are not intervals. In this setting, FirstFit remains 2/3 -competitive, which is optimal with respect to the matching cardinality, while EDF loses its optimality.

Future greenhouse gas neutral energy systems will be dominated by renewable energy technologies providing variable supply subject to uncertain weather conditions. For this setting, we propose an algorithm for capacity expansion planning: We evaluate solutions optimized on a single years’ data under different input weather years, and iteratively modify solutions whenever supply gaps are detected. These modifications lead to solutions with sufficient capacities to overcome periods of cold dark lulls and seasonal demand/supply fluctuations. A computational study on a German energy system model for 40 operating years shows that preventing supply gaps, i.e., finding a robust system, increases the total annual cost by 1.6-2.9%. In comparison, non-robust systems display loss of load close to 50% of total demand during some periods. Results underline the importance of assessing the feasibility of energy system models using atypical time-series, combining dark lull and cold period effects.

Developing algorithms for real-life problems that perform well in practice depends on the availability of realistic data for testing. Obtaining real-life data for optimization problems in health care, however, is often difficult, and such data typically cannot be published, which limits reproducibility by other researchers. This is especially true for patient-related problems because of data privacy policies such as the patient-to-room assignment problem. Therefore, artificially generated instances are commonly used. To improve the generation of realistic instances, we develop a configurable instance generator for the patient-to-room assignment problem and other patient-related problems, featuring an easy-to-use graphical user interface. The design of the generator is based on an extensive empirical analysis of real hospital data, which identifies relevant ward-specific patterns such as patients' age and length-of-stay distributions. Moreover, as randomly generated instances are often infeasible, we address this issue in two ways. We implement a dynamic programming approach in the generator to optionally enforce feasibility and extend existing results from the literature to derive new combinatorial insights into patient-to-room feasibility.

The simulation of human behavior is an essential component in the domain of energy demand modeling. However, due to its diverse nature, it is often unclear whether a simulated behavior pattern is fitting to certain contexts. Combined with the poor availability of appropriate activity data, this makes proper validation of behavior models difficult. Existing validation approaches are limited and specialized to the respective use case, and are therefore not reusable or comparable. To address this issue, the new open-source framework ETHOS.ActivityAssure is presented for evaluation of generated activity profiles, supporting the validation of behavior models. For this purpose, an aggregated activity dataset is created from restricted European time use data and published to ensure reusability. Validation is conducted through a set of indicators and comparative plots, taking into account activity duration, frequency, and time. A categorization of person types and activities enables mapping to result categories commonly used in behavior models. The framework's capabilities are demonstrated on the residential demand model LoadProfileGenerator. The developed framework allows for consistent and reproducible validation of synthetic activity profiles targeting Europe, without requiring access to confidential data. This offers the opportunity to enhance both new and existing behavior models by identifying flaws, compare multiple modeling approaches, and thoroughly evaluate model quality for diverse target purposes.

The single-source capacitated facility location problem with customer preferences (SSCFLPCP) is known to be strongly NP-hard. Computational tests imply that state-of-the-art solvers struggle with computing exact solutions. In this paper, we contribute two novel preprocessing methods which reduce the size of the considered integer programming formulation and introduce sets of valid inequalities which decrease the integrality gap. Each of the introduced results utilises structural synergies between capacity constraints and customer preferences in the SSCFLPCP. First, we derive two preprocessing methods where the first method fixes location variables and the second method fixes allocation variables. Afterwards, we study cover-based inequalities. Here, we first strengthen the well-known cover inequalities: when determining covers, we also consider demands of customers not in the cover that must be assigned to the covered facility if a customer in the cover is assigned to it. We further strengthen these inequalities by including information on the assignments of customers in a cover if they are not assigned to the covered facility. Afterwards, we derive a new family of valid inequalities, which expresses the relation of open facilities based on sets of customers covering a facility. We then discuss solution methods for the corresponding separation problems and, finally, test our results for two preference types in a computational study. Our results show a clear positive impact of the preprocessing methods and inequalities, in particular when preferences are defined by assignment costs.

In this article, we motivate and define variants of perfect matching under precedence constraints where a perfect matching is built incrementally and precedence constraints ensure that an edge may only be added to the matching if the edge's predecessor vertices have already been covered. We study the complexity of the problem and particularly consider -canonical precedence constraints where only edges that are "close" to the current matching may be added to the matching. For the & Nscr; & Pscr; -hard perfect matching under 1-canonical precedence constraints, we identify polynomial-time solvable cases.

During a hospital stay, a roommate can significantly influence a patient's overall experience both positivly and negatively. Therefore, hospital staff tries to assign patients together to a room that are likely to be compatible. However, there are more conditions and objectives to be respected by the patient-to-room assignment (PRA), e.g., ensuring gender separated rooms and avoiding transfers. In this paper, we review the literature for reasons why roommate compatibility is important as well as for criteria that can help to increase the probability that two patients are suitable roommates. We further present combinatorial insights about computing patient-to-room assignments with optimal overall roommate compatibility. We then compare different IP-formulations for PRA as well as the influence of different scoring functions for patient compatibility on the runtime of PRA integer programming (IP) optimisation. Using these results and real-world data, we conclude this paper by developing and evaluating a fast IP-based solution approach for the dynamic PRA.

Interval-constrained online bipartite matching problem frequently occurs in medical appointment scheduling: Unit-time jobs representing patients arrive online and are assigned to a time slot within their given feasible time interval. We consider a variant of this problem where reassignments are allowed and extend it by a notion of time that is decoupled from the job arrival events. As jobs appear, the current point in time gradually advances, and once the time of a slot is passed, the job assigned to it is fixed and cannot be reassigned anymore. We analyze two algorithms for the problem with respect to the resulting matching size and the number of reassignments they make. We show that FirstFit with reassignments according to the shortest augmenting path rule is 2/3 -competitive with respect to the matching cardinality, and that the bound is tight. The competitive ratio remains 2/3 if we restrict the algorithm to make only a constant number k ⩾ 1 of reassignments per job arrival. This fills the gap between the known optimal algorithm that makes no reassignments, which is 1/2 -competitive, on the one hand, and an earliest-deadline-first strategy (EDF), which we prove to obtain a maximum matching in this over-time framework, but which suffers Ω (n^2) reassignments in the worst case, on the other. In the full paper, we show that FirstFit remains 2/3 -competitive also if the sets of feasible slots per job that are not intervals, and that this is the best possible deterministic competitive ratio, while EDF loses its optimality.

We study the parameterized complexity of scheduling unit-time jobs on parallel, identical machines under generalized precedence constraints for minimization of the makespan and the sum of completion times ( P|gen-prec, p(j) = 1|y, y is an element of {C-max, Sigma(j) C-j}). In our setting, each job is equipped with a Boolean formula (precedence constraint) over the set of jobs. A schedule satisfies a job's precedence constraint if setting earlier jobs to true satisfies the formula. Our definition generalizes several common types of precedence constraints: classical and-constraints if every formula is a conjunction, or-constraints if every formula is a disjunction, and and/or-constraints if every formula is in conjunctive normal form. We prove fixed-parameter tractability when parameterizing by the number of predecessors. For parameterization by the number of successors, however, the complexity depends on the structure of the precedence constraints. If every constraint is a conjunction or a disjunction, we prove the problem to be fixed-parameter tractable. For constraints in disjunctive normal form, we prove W[1]-hardness. We show that the and/or-constrained problem is NP-hard, even for a single successor. Moreover, we prove NP-hardness on two machines if every constraint is a conjunction or a disjunction. This result not only proves para-NP-hardness for parameterization by the number of machines but also complements the polynomial-time solvability on two machines if every constraint is a conjunction ([4]) or if every constraint is a disjunction ([11]). 2012 ACM Subject Classification Mathematics of computing -> Combinatorial optimization

Patient-to-room assignment (PRA) is a scheduling problem in decision support for hospitals. It consists of assigning patients to rooms during their stay at a hospital according to certain conditions and objectives, e.g., ensuring gender separated rooms, avoiding transfers and respecting single-room requests. This work presents combinatorial insights about the feasibility of PRA and about how (many) single-room requests can be respected. We further compare different integer programming (IP) formulations for PRA as well as the influence of different objectives on the runtime using real-world data. Based on these results, we develop a fast IP-based solution approach, which obtains high quality solutions. In contrast to previous IP-formulations, the results of our computational study indicate that large, real-world instances can be solved to a high degree of optimality within (fractions of) seconds. We support this result by a computational study using a large set of realistic but randomly generated instances with 50% to 95% capacity utilisation.

Algorithms for computing fractional solutions to the quickest transshipment problem have been significantly improved since Hoppe and Tardos first solved the problem in strongly polynomial time. For integral solutions, however, no structural improvements on their algorithm itself have yet been proposed. Runtime improvements are limited to general progress on submodular function minimization (SFM), which is an integral part of Hoppe and Tardos' algorithm. In fact, SFM constitutes the main computational load of the algorithm, as the runtime is blown up by using it within Megiddo's parametric search algorithm. We replace this part of Hoppe and Tardos' algorithm with a more efficient routine that solves only a linear number of SFM and, in contrast to previous techniques, exclusively uses minimum cost flow algorithms within Megiddo's parametric search. Our \,approach improves the state-of-the-art runtime from (O) over tilde (m(4)k(15)) down to (O) over tilde (m(2)k(5) + m(4)k(2))(2,) where k is the number of terminals and m is the number of arcs.

We propose a variant of the shortest path problem where the order in which vertices occur in the path is subject to precedence constraints. Precedence constraints are defined in terms of vertex pairs (a,b)$$ \left(a,b\right) $$ which indicate that a vertex a$$ a $$ is the predecessor of a vertex b$$ b $$. A feasible (not necessarily simple) path may visit a vertex only upon having covered all its predecessors. The problem generalizes the graphic TSP Path, which makes it APX-hard. We propose a dynamic program and identify input classes for which the dynamic program yields an optimal solution in polynomial time. We also explore the limits of efficient solvability by proving that the problem remains hard even when significantly restricting the structure of the graph or the structure of the precedence constraints: Surprisingly, the problem remains hard even when restricted to spiders.

Patient transportation systems are instrumental in lowering access barriers in primary care by taking patients to their GPs. As part of this setting, each transportation request of a chronic or walk-in patient consists of an outbound trip to the GP and an inbound trip back home. The economic sustainability of patient transportation systems mainly depends on their utilization and how well transportation requests can be bundled through ride sharing. To ease the latter, we consider a flexible scheduling of chronic patients in which only a certain range for an appointment is fixed a priori while the exact time is determined by the scheduling of the outbound trip. This leads to a novel extension of the dial-a-ride problem that we call the dial-a-ride problem with combined requests and flexible scheduling (DARPCF). In this paper, we introduce two heuristics for the DARPCF that exploit this increased flexibility. Both approaches initially compute socalled mini-clusters of outbound requests. Then, the mini-clusters are linked by (i) solving a traveling salesman problem and creating routes of outbound rides with a splitting procedure or by (ii) using a rolling horizon approach and solving bipartite matching problems for the vehicle assignment. Our computational study shows that by using the presented algorithms with the flexible scheduling of chronic appointments, the average number of served requests can be increased by 16% compared to a non-flexible setting.

When planning transportation whose operation requires non-consumable resources, the peak demand for allocated resources is often of higher interest than the duration of resource usage. For instance, it is more cost-effective to deliver parcels with a single truck over eight hours than to use two trucks for four hours, as long as the time suffices. To model such scenarios, we introduce the novel minimum peak cost flow over time problem, whose objective is to minimise the maximum cost at all points in time rather than minimising the integral of costs. We focus on minimising peak costs of temporally repeated flows. These are desirable for practical applications due to their simple structure. This yields the minimum-peak-cost Temporally Repeated flow problem (MPC-TRF). We show that the simple structure of temporally repeated flows comes with the drawback of arbitrarily bad approximation ratios compared to general flows over time. Furthermore, our complexity analysis shows the integral version of MPC-TRF is strongly NP-hard, even under strong restrictions. On the positive side, we identify two benign special cases: unit-cost series-parallel networks and networks with time horizon at least twice as long as the longest path in the network (with respect to the transit time). In both cases, we show that integral optimal flows if the desired flow value equals the maximum flow value and fractional optimal flows for arbitrary flow values can be found in polynomial time. For each of these cases, we provide an explicit algorithm that constructs an optimal solution.

Assigning patients to rooms is a fundamental task in hospitals and, especially, within wards. For this so-called patient-to-room assignment problem (PRA) many heuristics have been proposed with a large variety of different practical constraints. However, a thorough investigation of the problem’s structure itself has been neglected so far. In this paper, we present insights about the basic, underlying combinatorial problem of PRA with a focus on minimizing the number of patient transfers which occur if patients have to change rooms during their stay. Particularly, we prove that in the case of double bedrooms, although minimizing the total number of transfers is NP-complete, each patient has to be transferred at most once, and provide upper bounds on the minimum total number of transfers.

In this paper, we study a robust two-stage concept for combinatorial optimization problems under discrete demand uncertainty. Combinatorial optimization problems are based on a finite set of elements for which we decide whether they are part of a solution. We divide the elements into two types, the so-called fixed and free elements. In a first stage, we irrecoverably decide whether some fixed elements are part of a solution despite uncertain demand. In a second stage, we decide whether some free elements complete a solution after a demand scenario is realized. The objective is to find a robust solution that minimizes the worst-case cost over a finite scenario set. We show the NP-hardness of this robust two-stage version of several specific combinatorial optimization problems under discrete demand uncertainty but provide a polynomial-time solvable special case. In particular, we apply this concept to three combinatorial optimization problems: the representative multi-selection, the shortest path, and the minimum weight perfect b-matching problem. We prove the NP-hardness and present special cases solvable in pseudo-polynomial and polynomial time.

Because of the long planning periods and their long life cycle, railway infrastructure has to be outlined long ahead. At the present, the infrastructure is designed while only little about the intended operation is known. Hence, the timetable and the operation are adjusted to the infrastructure. Since space, time and money for extension measures of railway infrastructure are limited, each modification has to be done carefully and long lasting and should be appropriate for the future unknown demand. To take this into account, we present the robust network design problem for railway infrastructure under capacity constraints and uncertain timetables. Here, we plan the required expansion measures for an uncertain long-term timetable. We show that this problem is NP-hard even when restricted to bipartite graphs and very simple timetables and present easier solvable special cases. This problem corresponds to the fixed-charge network design problem where the expansion costs are minimized such that the timetable is conductible. We model this problem by an integer linear program using time expanded networks. To incorporate the uncertainty of the future timetable, we use a scenario-based approach. We define scenarios with individual departure and arrival times and optional trains. The network is then optimized such that a given percentage of the scenarios can be operated while minimizing the expansion costs and potential penalty costs for not scheduled optional trains.