Classical Assignment Problem Research Articles

Optimized Selective Assembly using Hungarian Algorithm

Assembly of discrete parts guided by hardware design specification constitutes the final phase in product manufacturing. In the course of mass production of components, mating parts with geometric or dimensional deviation from their intended design can be made acceptable by the identification of suitable pairs after analysing design fit and tolerance limits. By transforming this application-specific problem into a unified mathematical model, an optimal solution can be achieved that minimizes the rejection of non-conforming fabricated parts. Regardless of the type and range of a design fit, the problem can be mapped into a matrix using a ranking function defined by the user. The ranking function is modifiable as per the user requirements and may vary based on the selection criteria for an assembly. Based on the type of ranking function used, the tabulated matrix is solved using the Hungarian minimization/maximization algorithm, which is a powerful combinatorial optimization algorithm that solves the classical assignment problem in mathematics. This approach ensures maximum number of suiting pairs as well as nominal suiting of parts with each other resulting in high-quality products and maximum utilization of fabricated resources.

A Feasible Flow-based Algorithm for Solving Classic Assignment Problem with Bottleneck

Classic assignment problem with bottleneck is a fundamental combinatorial optimization problem with numerous practical applications in diverse domains. In order to comprehensively explore the classic assignment problem with bottleneck and to develop efficient algorithm, a feasible flow-based algorithm is proposed for optimizing classic assignment problem with bottleneck. The bottleneck constraints are addressed by updating the bottleneck of the problem, and a feasible-flow approach is leveraged to efficiently identify optimal solutions within the network framework associated with such problems. To assess the effectiveness of our proposed algorithm, a comprehensive set of numerical experiments is carried out across various instances of classic assignment problem with bottleneck. Our numerical results demonstrate the superior performance of our algorithm, particularly when applied to problems with smaller cost scales, making it particularly suitable for addressing medium-sized to smaller-scale problems.

Tabu search and genetic algorithm in rims production process assignment

Abstract The paper discusses the problem of assignment production resources in executing a production order on the example of the car rims manufacturing process. The more resources are involved in implementing the manufacturing process and the more they can be used interchangeably, the more complex and problematic the scheduling process becomes. Special attention is paid to the effective scheduling and assignment of rim machining operations to production stations in the considered manufacturing process. In this case, the use of traditional scheduling methods based on simple calculations, or the know-how of process engineers often turns out to be insufficient to achieve the intended results. Due to the scale of the problems faced in practice, the methods based on approximate approaches (Genetic and Tabu Search) were used to solve them. In this perspective, the problem under consideration involves the extension of the classic assignment problem with the possibility of taking into account: the times of operations, potential changeovers and the capacity of production resources.

Overview of Cyclic Bottleneck Problem

Abstract: In this paper, we are going to overview a newvariant of the classical assignment problem referred to as the cyclic bottleneck assignment problem (CBAP). This problem can arise in the healthcare industry or supply chain management. The problem can arise in surgical room scheduling and can be described as follows. Each of m successive time units can be allocated to any of the m surgeonsavailable. The number of patients to be scheduled for surgery on a given unit and the length of stay of the patient in the recovery unit are dependent on the type of surgery. The problem is to assign one surgeon to each OR time so as to minimize the maximum number of patients that will be expected to occupy the recovery unit in the long run. In this paper we have compared various ways through which the CBAP has been solved , that is Cohort Intelligence[4] , Tabu Search[2] and Iterated Local Search

A measure of the difference between test sets for generating controlled random tests

Objectives . The problem of constructing the characteristics of the difference between test sequences is solved. Its relevance for generating controlled random tests and the complexity of finding measures of difference for symbolic tests are substantiated. The limitations of using the Hamming and Damerau – Levenshtein distances to obtain a measure of the difference between test sets are shown.Methods . Based on the characteristic of the interval used in the theory of the chain of successive events, a new measure of the difference between two symbolic test sets is determined. As a difference measure, the distance AD(Ti, Tk) between the test sets Ti and Tk is calculated using the interval characteristic, which is based on determining independent pairs of same (identical) symbols belonging to two sets and calculating the intervals between them.Results. The combinatorial nature of the calculation of the proposed difference measure for symbolic test sets of an arbitrary alphabet and dimension is shown. An example of calculating this measure for various types of test sets, including such as address test sets, is given. Possible modifications are shown and some properties and limitations are determined. The application of the measure of difference is considered for the case of repeated testing of storage devices based on address sequences pA with even p repetition of addresses. For the case p = 2, mathematical relations are given for calculating the intervals and distances AD(Ti, Tk) for address sequences 2A used for controlled random testing of storage devices. The main attention is paid to binary test sets, when the task of calculating given difference metric is reduced to the classical assignment problem using the Hungarian algorithm. The computational complexity of the Hungarian algorithm is estimated by the relation O(n4). As an alternative to the Hungarian algorithm, an algorithm for calculating the considered difference measure is proposed, the complexity of which is much less and has an estimate equal to O(n2). The experimental studies confirm the effectiveness of the proposed algorithm.Conclusion. The proposed difference measure extends the possibilities of generating test sequences when generating controlled random tests. It is shown that test sets, which are indistinguishable when Hamming distance is used as a measure of difference, have different values of AD(Ti, Tk) that allows to make more accurate classification of randomly generated sets as candidates for test sets.

Allocation of Subjects in an Educational Institution by Pascal Triangular Approach

The classical assignment problems cannot be successfully used for the real-world situation it is because the work efficiency varies to some extent from person to person hence, the use of fuzzy assignment problems is more appropriate. Based on the performance of the previous record in this study, on taking the scores obtained by the Chairman, by Course Director, by faculty members, and the student’s feedback, we applied Pascal triangular approach in fuzzy assignment problem for allocating the subjects in a department of coming semester. A study of the technique for 4 jobs and 4 persons has been made.

A Generalized Primal-Dual Algorithm with Improved Convergence Condition for Saddle Point Problems

We generalize the well-known primal-dual algorithm proposed by Chambolle and Pock for saddle point problems, and improve the condition for ensuring its convergence. The improved convergence-guaranteeing condition is effective for the generic setting, and it is shown to be optimal. It also allows us to discern larger step sizes for the resulting subproblems, and thus provides a simple and universal way to improve numerical performance of the original primal-dual algorithm. In addition, we present a structure-exploring heuristic to further relax the convergence-guaranteeing condition for some specific saddle point problems, which could yield much larger step sizes and hence significantly better performance. Effectiveness of this heuristic is numerically illustrated by the classic assignment problem.

Verification of multi-layered assignment problems

The class of assignment problems is a fundamental and well-studied class in the intersection of Social Choice, Computational Economics and Discrete Allocation. In a general assignment problem, a group of agents expresses preferences over a set of items, and the task is to allocate items to agents in an “optimal” way. A verification variant of this problem includes an allocation as part of the input, and the question becomes whether this allocation is “optimal”. In this paper, we generalize the verification variant to the setting where each agent is equipped with multiple incomplete preference lists: Each list (called a layer) is a ranking of items in a possibly different way according to a different criterion. In particular, we introduce three multi-layer verification problems, each corresponds to an optimality notion that weakens the notion of global optimality (that is, pareto optimality in multiple layers) in a different way. Informally, the first notion requires that, for each group of agents whose size is exactly some input parameter k, the agents in the group will not be able to trade their assigned items among themselves and benefit in at least \(\alpha \) layers; the second notion is similar, but it concerns all groups of size at most k rather than exactly k; the third notion strengthens these notions by requiring that groups of k agents will not be part of possibly larger groups that benefit in at least \(\alpha \) layers. We study the three problems from the perspective of parameterized complexity under several natural parameterizations such as the number of layers, the number of agents, the number of items, the number of allocated items, the maximum length of a preference list, and more. We present an almost comprehensive picture of the parameterized complexity of the problems with respect to these parameters.

Problems on Shortest k-Node Cycles and Paths

The paper is devoted to the construction of mathematical models for problems on the shortest cycles and paths, that pass through a given number of nodes of a directed graph. Such cycles and paths are called k-node, where 1<k <n, n is the number of nodes in the graph. Section 1 formulates two problems for finding the shortest k-node cycle – a mixed Boolean and linear programming problem and a discrete programming problem. Both problems include constraints from the classical assignment problem, describing a one-time entry into a node and a one-time exit from a node for those nodes through which the cycle passes. The cycle connectivity in the first problem is ensured by modeling the flow problem, and in the second problem, it is ensured by using the A. Tucker constraints for the travelling salesman problem. Section 2 establishes a connection between the formulations of both problems from Section 1 and the travelling salesman problem and investigates the efficiency of their solution using modern versions of gurobi and cplex programs and the AMPL modeling language. Section 3 contains the formulation of the shortest k-node path problem, which is represented by a mixed Boolean and linear programming problem. With its help, the optimal routes were found for visiting the wine-making points of the Malopolskie Wine Route in the direction Lviv-Wroclaw-Lviv (Section 4). Here a map for the 20 most visited wine-making points of the Malopolskie Wine Route and a table of the distances between them and the distances from them to Lviv and Wroclaw, calculated using the Google Maps web service, are presented. The developed mathematical models of the problems of finding the shortest k-node paths and cycles and the developed software in the AMPL modeling language can be used for the design and arrangement of technical objects, optimization of the transportation of products, analysis and forecasting of economic processes, determination of optimal routes when planning passenger and freight traffic, optimal organization of the process of managing a set of transactions and queries during their implementation in network databases and other classes of applied optimization problems. Keywords: digraph, shortest path, Boolean variable, linear programming, Hamiltonian cycle, Hamiltonian path, travelling salesman problem, AMPL, gurobi, cplex.

Maxima and near-maxima of a Gaussian random assignment field

The assumption that the elements of the cost matrix in the classical assignment problem are drawn independently from a standard Gaussian distribution motivates the study of a particular Gaussian field indexed by the symmetric permutation group. The correlation structure of the field is determined by the Hamming distance between two permutations. The expectation of the maximum of the field is shown to go to infinity in the same way as if all variables of the field were independent. However, the variance of the maximum is shown to converge to zero at a rate which is slower than under independence, as the variance cannot be smaller than the one of the cost of the average assignment. Still, the convergence to zero of the variance means that the maximum possesses a property known as superconcentration. Finally, the dimension of the set of near-optimal assignments is shown to converge to zero.

School bus routing problem considering affinity among children

School bus routing problem is widely studied, however, social elements such as the interaction between children traveling on the same route have not been considered so far. In this way, this article has as its main objective to propose a methodology to solve the school bus routing problem, including affinity as a strategy to increase positive interrelationships between children, and with this, support in bullying situations during school trips. The methodology includes two stages, assigning children to vehicles considering affinities and defining vehicle routes. The main contribution is the consideration of affinity in the process of forming the groups of children that will be taken on the bus, evidencing a balance in the affinity of the groups. Additionally, from the methodological point of view, the integration of a modified group technology algorithm and a new assignment model are proposed that simplify the classic quadratic assignment problem. Consideration of affinity in school bus routing generates benefits from a social point of view.

Аналіз оперативних дій рятувальних формувань за допомогою методу мережевого планування

A network model of operational actions of emergency rescue teams in the elimination of the consequences of emergency situations in a graphical and mathematical representation has been developed. The use of the model makes it possible to plan or analyze the process of organizing operational actions of civil protection units, to manage the course of its implementation. This is relevant for the planned period of emergency response. The task of forming performers of individual works in the elimination of the consequences of emergency situations has been formulated. The essence of the task is reduced to the ability to choose from a variety of civil protection units the necessary performers and assign them to work. Moreover, upon the appointment, the entire complex of works was completed within a given deadline and with minimal costs. Formalization of the corresponding problem made it possible to bring it to the classical assignment problem, which is solved by Kuhn's method. The use of a dynamic pro-gramming algorithm made it possible to obtain an initial approximation of the solution of the problem, at which the cost of performing a complex of emergency rescue operations will be minimal. To optimize the network graph of operational actions by reducing the length of the critical path, a dynamic programming method is proposed. The research results are synthesized into an algorithm. The implementation of the algorithm is to consistently clarify the assignments of performers to work. This makes it possible to determine the minimum costs for the implementation of the rescue plan within a given time frame (if such a solution exists), as well as to estimate the minimum time for carrying out emergency rescue operations for a given set of possible performers.

Decomposition Method for Solving a Three-Index Planar Assignment Problem

The classical assignment problem is considered. A third index is introduced, which can characterize, for example, the location of the work carried out. An iterative decomposition algorithm is proposed. At each step, a problem is solved with three constraints from different groups of conditions and with one connecting variable. The triples of problems are also solved with one restriction, and according to certain rules, the coefficients of the objective functions change. The iterative process that is monotonic in the objective function either arrives at the exact optimum solution of the original problem or indicates the nonuniqueness of the solution. In the latter case, a simple procedure finds the optima.

Joseph Schumpeter Lecture: Sharing a Government

Abstract This paper develops a simple theoretical framework to study a set of regions, each with its own regional government, who share a union or central government. These governments must decide whether to implement or discard a large number of projects that produce local benefits for the region that implements them, and externalities for the rest of the regions. Conflict or disagreement arises since different regions value projects differently. The classic assignment problem consists of deciding who decides these projects, either the union or the regional governments. It is well known that regional governments are insensitive to externalities. The key observation here is that the union government is insensitive to local benefits. Thus, each government maximizes only a piece of the value of projects, and disregards the other one. This observation leads to simple and clear rules for solving the assignment problem.

Tabu search and iterated local search for the cyclic bottleneck assignment problem

In this paper, we study a new variant of the classical assignment problem referred to as the cyclic bottleneck assignment problem (CBAP). This problem arises in surgical and operating room (OR) scheduling and can be described as follows. Each of n successive block times can be allocated to anyone of n surgeons. The number of patients to be scheduled for surgery on a given block and the length of stay in the recovery unit of each operated patient are dependent on the type of surgery. The problem is to assign one surgeon to each OR time so as to minimize the maximum number of patients that will be expected to occupy the recovery unit in the long run.This problem has been shown to be NP-hard. Consequently, we present two algorithms that can be used to solve the problem efficiently. One algorithm is based on the Tabu search method, and the other follows an iterated local search scheme. We evaluate our algorithms on 470 benchmark instances and 470 newly generated instances, and compare them with available metaheuristics and commercial solver CPLEX. The computational results for most of our problem instances show that our proposed approaches achieve excellent performance. Our algorithms outperform existing approaches and CPLEX. The tabu search approach works slightly better than the iterated local search approach.

Choosing a winning team for mixed medley events

The swimming medley event consists of teams of four swimmers (all men or all women) competing to complete a medley of four strokes in the quickest time. A new idea has emerged where swimming teams of two women and two men may now compete against each other in the same medley event. This new problem may be formulated as a classical assignment problem with an additional side constraint. We propose a simple back-of-the-envelope algorithm for selecting the fastest mixed team. The mixed-medley model is also generalised to other possible sports events (e.g., in gymnastics and track and field). Reformulating the problem by standard Lagrangian relaxation allows us to solve it in an iterative fashion with the well-known Hungarian method. Thus, the problem is suitable in a classroom environment for students and practitioners of OR.

A hybrid heuristic algorithm for the sequencing generalized assignment problem in an assembly line

The generalized assignment problem (GAP) is an enlarged version of the classical assignment problem in which the assignment of several tasks to an agent is possible. The only restriction that can limit the assignment of the tasks to the agents is the capacity of the agent. In this study, a manpower planning is modeled as a GAP in an assembly line. The considered assembly line consists of a number of workstations in series where each workstation contains several jobs in parallel. The present research seeks to assign the best operator to the jobs in order to minimize the processing time in each workstation and in the whole of the system. The generalized assignment formulation is adapted to this case. A mixed-integer mathematical model is proposed for this problem and proved that the considered problem is NP-complete. The model is solved in various sizes by employing Gurobi mixed-integer solver. This paper presents a greedy heuristic combined with a local search for the generalized assignment problem. The slight deviation from the optimal solutions and the computational times which is extremely reduced, validate the efficiency of the proposed heuristic approach for solving such problems.

Querying Similar Process Models Based on the Hungarian Algorithm

The structural similarity between two process models is usually considered as the main measurement for ranking the process models for a given query model. Current process query methods are inefficient since too many expensive computations of the graph edit distance are involved. To address this issue, using Petri-net as the modeling method, this paper presents the Hungarian algorithm based similarity query method. Unlike previous work where the non-task nodes (i.e., place nodes in the Petri-net) were lightly studied or even ignored, we think these non-task nodes also play an essential role in measuring the structural similarity between process models. First, we extract the context for each place and define the similarity for a pair of place nodes that are from different process models from two perspectives: commonality and the graph edit distance. Then, the place mapping is transformed to classical assignment problem that can be solved by Hungarian algorithm efficiently. Furthermore, we propose a new process similarity measurement on the basis of the place similarity. The extensive experimental evaluation shows that our Hungarian based methods outperform the baseline algorithm in both retrieval quality and query response time.

Suitable assignment of individuals to positions based on consensus formation

Abstract This paper investigates how to assign individuals to suitable positions effectively and fairly. Firstly, some related standard mathematical evaluation methods such as DEA, total ranking method based on DEA and consensus formation method are reviewed. Secondly, using these methods and combining them under multi-criteria and scenarios, in Ishii (2014) we proposed two allocation models to assign suitable individuals to suitable positions, i.e., a ranking data model with respect to each position and a scenario model. Modifying a previous paper and the relative distance method by Cook and Kress (1984), a new consensus method is then proposed and applied to these two models. These models are firstly transformed into a transportation problem with a special structure since our model assumes that the number of positions does not exceed that of individuals, that is, it includes the case in which the number of positions is insufficient to cover all individuals. The transportation problem is then transformed into the classic assignment problem. From an optimal solution of the assignment problem, an optimal allocation of individuals to positions is found for each model. Finally the paper concludes by showing the results and discussing applicability of the model to other allocation problems and to other evaluation methods that use linguistic terms denoting the suitability of individuals to positions.

A Hungarian penalty-based construction algorithm to minimize makespan and total flow time in no-wait flow shops

This paper presents a penalty-based construction algorithm for the no-wait flow shop scheduling problem with the objective of minimizing makespan and total flow time of jobs. The proposed method, derived from Hungarian penalty method originally used for the classic assignment problem is employed to generate an initial schedule of jobs, which is further improved by an insertion technique to obtain an optimal or near-optimal schedule. The results of computational experiments on a large number of test problems show that the proposed method performs significantly better than the state-of-the-art procedures while requiring comparable computational effort.