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Teamwork has increasingly become more popular in educational environments. With the also increasing mobility trends in the educational sector, internationalization and other diversity features have gained importance in the structure of teams. In this paper, we discuss an assignment problem arising in the allocation of students to business projects in a master program in Norway. Among other problem features, the students state their preferences on the projects they most want to conduct. There are also requirements from the companies that propose the projects and from the program administration. We develop a bi-objective approach to consider efficiency and fairness criteria in this assignment problem. We test the model using real data of 2017 and 2018, in joint collaboration with the administrative staff of the program. In light of the good results, our proposed solutions have been implemented in practice in 2019 and 2020. The implementation of these solutions have been beneficial for the administration, the students, and the companies.

We consider the patient-to-bed assignment problem that arises in hospitals. Both emergency patients who require hospital admission and elective patients who have had surgery need to be found a bed in the most appropriate ward. The patient-to-bed assignment problem arises when a bed request is made, but a bed in the most appropriate ward is unavailable. In this case, the next-best decision out of a many alternatives has to be made, according to some suitable decision making algorithm. We construct a Markov chain to model this problem in which we consider the effect on the length of stay of a patient whose treatment and recovery consists of several stages, and can be affected by stays in or transfers to less suitable wards. We formulate a dynamic program recursion to optimise an objective function and calculate the optimal decision variables, and discuss simulation techniques that are useful when the size of the problem is too large. We illustrate the theory with some numerical examples.

The top two divisions of the Argentinean professional basketball system have since 2014–2015 used a season schedule format similar to that used by the NBA in which games are played all through the week, replacing the previous setup where all games were scheduled on weekends. This change has confronted the Argentinean league organizers with new scheduling challenges, one of which is the assignment of referees to games. The present article addresses this assignment problem using a tool based on an integer linear programming model. The objective is to minimize the total cost of trips made by the referees while also satisfying a series of other conditions. The problem is broken down into a series of relatively small subproblems representing successive periods of the season, and the solution is obtained using a rolling horizon heuristic. The approach was tested by applying it to the First Division’s 2015–2016 season, the last one before introducing the approach presented in this article, when referees were still assigned using manual methods. The travel costs simulated by the model were 26% lower than the total travel costs actually incurred under the manual assignments, and all of the restrictions that had been requested by league officials were satisfied. The model was used by the First Division for the 2016–2017 and 2017–2018 seasons and in 2017–2018 also by the Second Division.

In this paper, we develop a new bi-objective model for dynamic weapon-target assignment problem. We consider that the initial weapon assignment plan of defense is disrupted during engagement because of a destroyed air target, breakdown of a weapon system or a new incoming air target. The objective functions are defined as the maximization of probability of no-leaker and the maximization of stability in engagement order of weapon systems. Stability is defined as assigning same air target in sequence in engagement order of a weapon system so that reacquisition and re-tracking of air target are not required by sensors. We propose a new solution procedure to generate updated assignment plans by maximizing efficiency of defense while maximizing stability through swapping weapon engagement orders. The proposed solution procedure generates non-dominated solutions from which defense can quickly choose the most-favored course of action. We solve a set of representative problems with different sizes and present computational results to evaluate effectiveness of the proposed approach.

Over the past couple of centuries, with the increase in the significance of the railways to the economy, the complexity of the railway network and consequently the decision making involved and the number of problems faced has only increased. Among the host of problems in railways management, one of the most important is the locomotive assignment problem (LAP). The LAP is as old as the railways itself. The objective of the LAP is to assign a fleet of locomotives at minimum cost to pre-scheduled trains subject to a host of constraints. Most of the models in literature either work on improving the efficiency of solving the LAP or attempt to solve the three levels of LAP separately. Our attempt in this paper is to arrive at a linear mathematical model that integrates the strategic, tactical and operational level aspects of the LAP. We also demonstrate that by the addition of a valid inequality, the lower bounds can be improved substantially resulting in a substantial reduction in solution times for the problem instances. Further, we develop a Dijkstra’s algorithm-based heuristic to test the problem instances and compare the solution times and quality of solutions. Finally, we develop a case based on Southern Zone of the Indian Railways and test the heuristic to solve the LAP for this zone.

Suppose we like to find non-overlapping periods for a set of events which may have multiple teachers assigned, is this easy or hard in terms of complexity? Or assume that only a single teacher is fixed per event, but we like to allocate rooms and periods simultaneously. What if a single teacher and a room is already given and we look for periods alone? And how do requests of teachers for specific rooms or additional student conflicts change the computational complexities of these questions from university course timetabling (UCT)? We provide a complete hard/easy-list of all UCT subproblems derived from a typical set of hard constraints. We obtain this list with a systematic study of the fine structure of UCT in terms of complexity w.r.t. the order in which rooms, periods and teachers are assigned to events. These kind of subproblems appear in practice when some entities in a timetable are fixed while the assignments of others are (re-)computed, and they also appear as necessary conditions for the existence of feasible timetables. Moreover, we identify which of the seemingly different subproblems are essentially the same computational tasks by reducing them to the same bipartite assignment problem, and we discuss some variations of constraints.

This paper considers the integration of three essential seaport terminal operations: the berth allocation problem, the quay crane assignment problem (QCAP), and the quay crane scheduling problem (QCSP). The paper presents a new mathematical formulation that captures all associated operations and constraints. Different quay crane operational policies are considered, namely permitting versus not permitting bay task preemption in QCSP and static versus dynamic crane allocations in QCAP. Thus, variants of the mathematical formulation are introduced to capture the different combinations of these scenarios. Due to the preemption consideration, the models include disaggregated quay crane (QC) tasks. Specifically, QC tasks are identified by single container movements as opposed to bay or stack task allocations that are commonly used in the literature. A case study based on Abu Dhabi’s container terminal is presented where the use of the proposed mathematical models are compared against the current existing operational approach. Results show that the service times can be significantly decreased by the use of the proposed models. Moreover, the policy choice effect on the total schedule is compared through simulated examples and Abu Dhabi’s container terminal case study. The results show that the policy improvements can depend on the problem’s attributes and thus a better policy cannot be generalized.

The Reformulation Linearization Technique (RLT) of Sherali and Adams (Manag Sci 32(10):1274–1290, 1986; SIAM J Discrete Math 3(3):411–430, 1990), when applied to a pure 0–1 quadratic optimization problem with linear constraints (P), constructs a hierarchy of LP (i.e., continuous and linear) models of increasing sizes. These provide monotonically improving continuous bounds on the optimal value of (P) as the level, i.e., the stage in the process, increases. When the level reaches the dimension of the original solution space, the last model provides an LP bound equal to the IP optimum. In practice, unfortunately, the problem size increases so rapidly that for large instances, even computing bounds for RLT models of level k (called RLTk) for small k may be challenging. Their size and their complexity increase drastically with k. To our knowledge, only results for bounds of levels 1, 2, and 3 have been reported in the literature. We are proposing, for certain quadratic problem types, a way of producing stronger bounds than continuous RLT1 bounds in a fraction of the time it would take to compute continuous RLT2 bounds. The approach consists in applying a specific decomposable Lagrangean relaxation to a specially constructed RLT1-type linear 0–1 model. If the overall Lagrangean problem does not have the integrality property, and if it can be solved as a 0–1 rather than a continuous problem, one may be able to obtain 0–1 RLT1 bounds of roughly the same quality as standard continuous RLT2 bounds, but in a fraction of the time and with much smaller storage requirements. If one actually decomposes the Lagrangean relaxation model, this two-step procedure, reformulation plus decomposed Lagrangean relaxation, will produce linear 0–1 Lagrangean subproblems with a dimension no larger than that of the original model. We first present numerical results for the Crossdock Door Assignment Problem, a special case of the Generalized Quadratic Assignment Problem. These show that just solving one Lagrangean relaxation problem in 0–1 variables produces a bound close to or better than the standard continuous RLT2 bound (when available) but much faster, especially for larger instances, even if one does not actually decompose the Lagrangean problem. We then present numerical results for the 0–1 quadratic knapsack problem, for which no RLT2 bounds are available to our knowledge, but we show that solving an initial Lagrangean relaxation of a specific 0–1 RLT1 decomposable model drastically improves the quality of the bounds. In both cases, solving the fully decomposed rather than the decomposable Lagrangean problem to optimality will make it feasible to compute such bounds for instances much too large for computing the standard continuous RLT2 bounds.

The berth allocation and quay crane assignment problem (BACAP) is a complex port operation planning problem susceptible to uncertainties, such as vessel arrival time fluctuation to its estimated time of arrival and maritime markets. For promoting reliability and sustainability of container terminals, this paper addresses the optimization of BACAP under the uncertain vessels’ arrival times and fluctuation of loading and unloading volumes. We propose a proactive BACAP strategy considering minimum recovery cost under uncertainty using a reactive strategy. A stochastic programming model is formulated to minimize the basic cost in the baseline schedule, and the recovery cost in real uncertain scenarios. A two-stage meta-heuristic framework based on GA is developed for solving this problem. Numerical experiments and scenario analysis are conducted to validate the effectiveness of the proposed model and the proposed solution approaches.

In this article, we investigate the vehicle assignment problem with demand uncertainty in a hybrid hub-and-spoke network with a single hub. The problem is deciding both the transportation routes and the number and types of vehicles to be deployed to minimize the sum of costs to transport all quantities in a hybrid hub-and-spoke network which allows direct transportation between spokes. Daily changes in quantities are reflected with a finite number of scenarios. Regularly scheduled vehicles and temporarily scheduled vehicles are considered to meet the demand variation. We propose a Dantzig–Wolfe decomposition approach which yields a strong LP relaxation bound by introducing a set of feasible direct route patterns. We develop an algorithm which incorporates a column generation procedure at the root node and repeats iteratively a variable fixing and column generation procedure at the non-root nodes until an integral solution is found. Finally, we present computational results using the well-known CAB data sets and real-life data from the Korea Post. The results show that our algorithm can find near optimal solutions very efficiently.

Falls represent a considerable public health problem, especially in older population. We describe and evaluate data-driven operations research models for detection and situational assessment of falls and near falls with a system of wearable sensors. The models are formulated as instances of the multidimensional assignment problem. Our computational studies provide some initial empirical evidence of the potential usefulness of this new application of the multidimensional assignment problem.

A two-phase approach was adopted to solve the problem given during the ROADEF/EURO Challenge 2014 competition. The problem focuses on rolling stock management at railway sites, as defined by French Railways (SNCF). In the first phase, a train assignment problem is solved by combining a greedy heuristic procedure with integer programming. The objective is to maximize the number of assigned departures while meeting technical constraints. The second phase consists of scheduling train movements inside the station while minimizing the number of cancelled (uncovered) arrivals and departures. This schedule has to comply with resource constraints such as capacity, length, order of trains, etc. A constructive heuristic is used to build a feasible schedule, which is subject to improvement by an iterative procedure based on a local search. Experiment results, that demonstrate the effectiveness of our approach on the large scale instances provided by SNCF, are presented hereafter.

We consider a bilevel extension of the classical linear assignment problem motivated by network interdiction applications. Specifically, given a bipartite graph with two different (namely, the leader’s and the follower’s) edge costs, the follower solves a linear assignment problem maximizing his/her own profit, whereas the leader is allowed to affect the follower’s decisions by eliminating some of the vertices from the graph. The leader’s objective is to minimize the total cost given by the cost of the interdiction actions plus the cost of the assignments made by the follower. The considered problem is strongly $${ NP}$$ -hard. First, we formulate this problem as a linear mixed integer program (MIP), which can be solved by commercial MIP solvers. More importantly, we also describe a greedy-based construction heuristic, which provides (under some mild conditions) an optimal solution for the case, where the leader’s and the follower’s edge costs are equal to one. Finally, we present the results of our computational experiments comparing the proposed heuristic against an MIP solver.

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

Population growth and the massive production of automotive vehicles have lead to the increase of traffic congestion problems. Traffic congestion today is not limited to large metropolitan areas, but is observed even in medium-sized cities and highways. Traffic engineering can contribute to lessen these problems. One possibility, explored in this paper, is to assign tolls to streets and roads, with the objective of inducing drivers to take alternative routes, and thus better distribute traffic across the road network. This assignment problem is often referred to as the tollbooth problem and it is NP-hard. In this paper, we propose mathematical formulations for two versions of the tollbooth problem that use piecewise-linear functions to approximate congestion cost. We also apply a biased random-key genetic algorithm on a set of real-world instances, analyzing solutions when computing shortest paths according to two different weight functions. Experimental results show that the proposed piecewise-linear functions approximate the original convex function quite well and that the biased random-key genetic algorithm produces high-quality solutions.