Abstract
Recent results of three areas, pickup and delivery, optimal mass transportation, matching under preferences are highlighted. The topics themselves have been selected from the active research fields of Hungarian Operations Research. We also provide a short summary of selected research results from the 34th Hungarian Operations Research Conference, held in Cegléd, Hungary, August 31–September 2, 2021.
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1 Quickest route
The International Conference on Automated Planning and Scheduling (ICAPS) hosted four competitions in 2021. One of these, the Dynamic Pickup and Delivery Problem (DPDP) competition was organized by Huawei Noah’s Ark Lab (Hao et al. 2022). This competition was based on a real-life scenario of Huawei Technologies Co. Ltd. The DPDP can be stated as follows. There is a homogeneous fleet of vehicles to serve pickup-and-delivery requests that arrive over the day. Each request has a pickup and a delivery location, a quantity and a due-date. The vehicles cannot carry more orders than their capacity, and unloading the orders has to respect the last-in-first-out rule. The locations have a limited number of docking ports for loading and unloading the vehicles, which may force the vehicles to wait. The goal is to satisfy all the requests such that a combination of tardiness penalties and traveling costs is minimized. The problem is dynamic in the sense, that the orders are not known in advance, but arrive online, and at certain time points there is an opportunity to make decisions, i.e., to re-plan routes, reflecting on the new information.
The task of the participants was to create an algorithm to assign new orders, and possibly re-assign already allocated orders to vehicles at the decision points. The organizers provided a public benchmark dataset along with a simulator for testing the algorithms. A single run simulated an entire day, i.e., the arrival of new orders, along with the movement of the vehicles and the transportation of the orders. The algorithm was called every ten minutes (in simulation time) to update the route plans of the vehicles.
152 teams joined the competition, including the team ’Quickest Route’, formed by Markó Horváth, Tamás Kis, and Péter Györgyi. The team developed a so-called cost function approximation method for the problem, where they perturbed the objective function in order to obtain vehicle routes that are flexible for possible future changes. Then, the problem at each decision point was solved by a local search-based procedure.
After a two months long preparation and trial submission phase, the three-week competition began, where each team could submit their algorithm for evaluation 5 times per day. The evaluation was performed by the organizers on a hidden dataset, and the teams were ranked based on their best scores. The publicly available leaderboard was updated every three hours, which made the competition very exciting. Six hours before the end of the competition, with their last attempts, the team Quickest Route moved from fifth place to second place. However, another team was able to improve, taking the second place at the penultimate update of the leaderboard. After that, the podium was not changed, so the team Quickest Route took the third place of the competition.
Since then, many articles have addressed the problem (Cai et al. 2022, 2022; Du et al. 2023; Cai et al. 2023). The team also improved their algorithm, with which they obtained significantly better results on the benchmark dataset than the former approaches. Their manuscript is currently under review.
2 Optimal mass transportation
Optimal Mass Transportation (OMT), as its name suggests, provides solutions for transporting a unit measure from one source to another target measure using minimal work/energy with respect to certain cost function, associating a suitable Wasserstein metric. The problem dates back to G. Monge, and its rigorous mathematical approach was developed by L. Kantorovich, who relaxed Monge’s original problem into a linear programming problem. In a breakthrough work by Y. Brenier, the author observed that under very mild assumptions, optimal transport maps between two marginal masses are realized by the gradient of a convex function. In this way, a whole machinery were applied from operation research, nonlinear and convex analysis to study various aspects of OMT. The theory of OMT is still a very active research area with various applications (in geometric optics, isoperimetric inequalities, meteorology, chemistry, etc), culminated in the works of two Fields Medalist, A. Figalli and C. Villani.
By using OMT on sub-riemannian objects, Balogh et al. (2018, 2019) provided a positive answer to the tacitly accepted view that no meaningful geometric inequalities can be proved on singular spaces as Heisenberg and Carnot groups. Their geometric inequalities imply various forms of the sub-riemannian distorted Borell–Brascamp–Lieb, Brunn–Minpkowski and Prékopa–Leindler inequalities.
Balogh and Kristály (2023), by using OMT and Brunn-Minkowski inequality, provided the sharp isoperimetric inequality on Riemannian manifolds with nonnegative Ricci curvature, where the so called asymptotic volume ratio plays a crucial role. As a consequence, one can obtain sharp Sobolev-type inequalities, characterizing also the equality cases. Equality in the geometric inequalities have been also established in Balogh and Kristály (2018), using quantitative OMT theory.
Nowadays another side of the OMT has become very active, the so-called non-commutative / quantum optimal transport theory. In this setting, the quantum analogue replaces the usual measures by quantum states (which are positive-semidefinite trace-class operators of a unit trace) acting on a separable Hilbert space. A very successful event “Workshop: Optimal Transport on Quantum Structures” was organized by T. Titkos and D. Virosztek at the Alfréd Rényi Institute of Mathematics, 26–30 September 2022, attracting leading experts in OMT.
Gehér et al. (2023a) considered two quadratic quantum cost operators on a qubit state space (i.e., on a two-dimensional Hilbert space) by classifying the corresponding isometries.
A series of papers has been published by see e.g. Gehér et al. (2023b), Gehér et al. (2022), Gehér et al. (2019), where aspects of the isometry group of Wasserstein spaces have been investigated in various aspects (Hilbert case, metric case, discrete case, etc).
3 Matching under preferences
In this section we will present the most important recent research results by the Hungarian OR community on the topic of matching under preferences. We will cover papers on classical two-sided matching markets (such as the college admission problem), allocation problems (e.g., vaccine allocation), pairing at tournaments, exchange problems (with application of kidney exchanges), assignment games and auctions for power and gas markets, and stable flow and network problems.
The seminal paper of Gale and Shapley (1962) on college admission problem initiated the scientific research on the interdisciplinary field of matching under preferences. They provided a model and efficient algorithm for solving the college admission problem, where both the students and the colleges have preferences and the matching is computed by a centralised algorithm. Hungarian college admissions has four special features, ties, lower quotas, common quotas and paired applications, and each of the last three features makes the problem NP-hard to solve. The Hungarian application uses a Gale-Shapley type heuristic algorithm, however, Ágoston et al. (2016) showed that lower quotas can be solved for optimality by IP-techniques, and in a recent paper Ágoston et al. (2022) covered also the cases of common quotas and ties. The CEMS Business project allocation is a task every year at Corvinus University of Budapest, that has been solved also at partner universities in Germany and Poland by also IP-techniques for this special setting of matching under distributional constraints by Ágoston et al. (2018). General cutoff-score based solutions have been studied by Aziz et al. (2022) also with IP-technique. A recent survey on the combinatorics of two-sided stable matching problems has been written by Fleiner (2023).
In object allocation problems only one side has preferences. A new solution concept of constrained efficient allocations was proposed by Biró and Gudmundsson (2021) motivated by school choice and refugee allocation applications. The general problem is NP-hard, but IP-techniques were used for conducting simulations on an Estonian kindergarten allocation problem. Vaccine allocation was an interesting application for preference based allocation problem during the COVID-pandemic, especially in Hungary where besides the EU-approved vaccines Russian and Chinese vaccines were also available in large quantities. The vaccine acceptance choices made by the Hungarian population were studied in Kutasi et al. (2022). Computer simulations for vaccine allocation were conducted by Reguly et al. (2022) and Angelov et al. (2023). Another application developed during the pandemic was for allocating voluntary mentors to students in Hungary by using optimisation techniques by Biró and Gyetvai (2023).
Pairing teams or individuals in the rounds of a competition is also a matching problem. Various tournament designs were compared by simulations by Sziklai et al. (2022), where the Swiss tournament was found to be the best in approximating the real power rankings. The Swiss systems are also the most popular design in chess tournaments, these were analysed by Führlich et al. (2022). A fair scheduling problem for The International Young Physicists’ Tournament was solved with optimisation techniques by Cechlárová et al. (2023).
Paired kidney exchanges match incompatible donor-recipient pairs together in order for swapping their donors, so that everyone can receive immunologically compatible pairs. Larger cycles are also possible, but due to the simultaneity of the transplants, only 3-way exchanges are allowed in most applications. In Europe there are at least ten large countries operating national or international kidney exchange programs with various optimisation criteria, surveyed by Biró et al. (2021). The ENCKEP COST Action during 2016–2021 provided a network for studying and coordinating these applications. As part of that project the ENCKEP-simulator was also developed that can be used to evaluate the performance national and international kidney exchange policies. This simulator was extended by Matyasi and Biró (2023) for analysing failure-aware polices. Individual fairness of the solution can be potentially improved if the goal of the kidney exchange program is not simply maximising the number of transplants, but rather finding stable/core solutions. This approach was studied with IP-techniques by Klimentova et al. (2023), and also by Biró et al. (2023), where the latter paper also showed that core solutions provide good incentives to the recipients to bring better or more donors to the pool. More general circulation problems were studied by Andersson et al. (2021) and Biró et al. (2022), motivated by time banks and portfolio compression, here agents are exchanging their time or liabilities with each other.
Surprisingly, the problem of balancing the benefits of international kidney exchange programmes by compensation schemes, see e.g., Benedek et al. (2021), is closely related to partitioned assignment games. A comprehensive survey on generalised assignment games was written by Benedek et al. (2023). The classical assignment game of Shapley and Shubik (1971) was studied by Solymosi (2023) with respect of the sensitivity of the solution by the preferences and also for changing market participation by Solymosi (2023). The problem setting with the presence of middlemen in assignment games was studied by Atay et al. (2023). Another, related TU-game, the liability games were studied by Csóka et al. (2022).
The matching with payment model is also the base of auctions. Auctions in power markets have been studied by Csercsik (2023) in a complex setting with coordinated balancing capacity procurement, and by Csercsik (2021) for minimum income conditions. Predicting electricity consumption by AI-techniques can help reaching desirable solutions in power markets, as shown in Čegovnik et al. (2023). Auctions have been also used in European gas markets for pipe network capacities, analysed by Csercsik (2022). Game theory models were used to predict the effect of gas network changes in Europe in Sziklai et al. (2020). Finally, a combinatorial auction model was proposed for efficient and fair channel assignments by Csercsik and Jorswieck (2023).
Stability in network was studied by Fleiner et al. (2019), where they proved the existence of a so-called trail-stable solution in a very general environment with choice functions over possible contracts. The computational complexity of computing stable solutions was studied by Fleiner et al. (2023). Stable flows were also studied by using the stable fractional solutions by the Scarf-algorithm by Csáji (2022). Finally, Cseh et al. (2022) analysed the special case of three dimensional stable matchings with cyclic preferences using constraints programming.
4 The 34th Hungarian Operations Research Conference
The 34th Hungarian Operations Research Conference, organized by the Hungarian Society for Economic Modelling, co-organized by the János Bolyai Mathematical Society and by the Hungarian Operations Research Society was held in Cegléd, August 31-September 2, 2021.
Ágoston and E-Nagy (2024) apply mixed integer linear programming for the minimum sum-of-squares clustering problem in order to find global optimum, instead of a local one, usually found by the KMEANS algorithm. The MILP formulations are flexible: additional constraints, regarding e.g. balanced cluster sizes, can also be included.
Bánhidi and Dobos (2024) investigate the sensitivity of the ranking calculated by TOPSIS, a multi-criteria decision model (see, e.g. in Dzemydienė et al. (2022)). Data of Digital Economy and Social Index (DESI) with respect to five criteria (connectivity, human capital, use of internet, integration of digital technology, digital public services) are used.
Borgulya (2024) develops a hybrid estimation of distribution algorithm for the offline two-dimensional variable-sized bin packing problem, where the goal is to pack the rectangles into the bins without overlap, parallel to the sides, so that the total area of the used bins (or total cost) is minimized. The proposed algorithm uses a probabilistic model or selection and mutation operators to generate descendants. Local search procedures improve the quality of the solutions.
Dobos and Vörösmarty (2024) focus on data transformation in several models of Data Envelopment Analysis (DEA, (see, e.g. in Hosseinzadeh et al. (2023); Wu et al. (2023); Zýková (2022))) and draw the attention to the possibility of biases. Supplier selection is used as an illustrative example.
Domokos and Regős (2024) examine geological crack patterns and develop a planar, discrete time evolution model. They define two local, random evolutionary events, corresponding to secondary fracture and rearrangement of cracks. The existence of limit points is proved for several types of trajectories.
Hajba, Horváth, Heitz and Psenák (2024) study the replenishment problem of a European petrol company. Station replenishment problem is a special vehicle routing problem, where different types of fuels are transported from the depots to the customers. A mixed integer linear programming model is developed. Customers are clustered in order to reduce the size of the problem.
Orbán-Mihálykó, Mihálykó and Gyarmati (2024) apply paired comparison methods to aggregate rankings of different groups. Results are applied to the data from the European Handball Federation Women’s Champions League (2020/2021, two groups).
Varga (2024) provides a guide to determine optimised child and spousal support payment. The aim is to minimize the risk of poverty after divorce for the child and the custodial parent. A dynamic, globally applicable model is proposed based on simulations based on German child support data.
Vörös (2024) proves that the maximal loss on a loan portfolio is located at the frontier of the given Mahalanobis distance, furthermore explicit formulas are given. In case of two macro state variables, efficient initial solutions are generated.
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Acknowledgements
The contribution of Tamás Kis, Péter Györgyi and Markó Horváth to Sect. 1 is highly appreciated.
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Biró, P., Bozóki, S., Király, T. et al. Optimization methods and algorithms. Cent Eur J Oper Res 32, 1–9 (2024). https://doi.org/10.1007/s10100-023-00898-6
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DOI: https://doi.org/10.1007/s10100-023-00898-6