Solving binary programming problems using homotopy theory ideas

PurposeThe aim of the study is to show that merging two areas of mathematics – topology and discrete optimization – could result in a viable option to solve classical or specialized integer problems.Design/methodology/approachIn the paper, discrete topology concepts are applied to propose a metaheuristic algorithm that is capable to solve binary programming problems. Particularly, some of the homotopy for paths principles are used to explore the solution space associated with four well-known NP-hard problems herein considered as follows: knapsack, set covering, bi-level single plant location with order and one-max.FindingsComputational experimentation confirms that the proposed algorithm performs in an effective manner, and it is able to efficiently solve the sets of instances used for the benchmark. Moreover, the performance of the proposed algorithm is compared with a standard genetic algorithm (GA), a scatter search (SS) method and a memetic algorithm (MA). Acceptable results are obtained for all four implemented metaheuristics, but the path homotopy algorithm stands out.Originality/valueA novel metaheuristic is proposed for the first time. It uses topology concepts to design an algorithmic framework to solve binary programming problems in an effective and efficient manner.

A New Binary Programming Formulation and Social Choice Property for Kemeny Rank Aggregation

Rank aggregation is widely used in group decision making and many other applications, where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem—whose ranking inputs may be complete and incomplete, with and without ties. Moreover, it leverages the equivalence of two ranking aggregation problems, namely, that of minimizing the Kemeny-Snell distance and of maximizing the Kendall-τ correlation, to compare the newly introduced binary programming formulation to a modified version of an existing integer programming formulation associated with the Kendall-τ distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the nonstrict extended Condorcet criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Condorcet criterion. Unlike its parent properties, the new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and with benchmark instances from PrefLib, a library of preference data.

The Assignment Problem in Human Resource Project Management under Uncertainty

The assignment problem (AP) is a discrete and combinatorial problem where agents are assigned to perform tasks for efficiency maximization or cost (time) minimization. AP is a part of human resource project management (HRPM). The AP optimization model, with deterministic parameters describing agent–task performance, can be easily solved, but it is characteristic of standard, well-known projects realized in a quiet environment. When considering new (innovation or innovative) projects or projects performed in very turbulent times, the parameter estimation becomes more complex (in extreme cases, even the use of the probability calculus is not recommended). Therefore, we suggest an algorithm combining binary programming with scenario planning and applying the optimism coefficient, which describes the manager’s nature (attitude towards risk). The procedure is designed for one-shot decisions (i.e., for situations where the selected alternative is performed only once) and pure strategies (the execution of a weighted combination of several decision variants is not possible).

Binary Programming Model for Rostering Ambulance Crew-Relevance for the Management and Business

The nature of health care services is very complex and specific, thus delays and organizational imperfections can cause serious and irreversible consequences, especially when dealing with emergency medical services. Therefore, constant improvements in various aspects of managing and organizing provision of emergency medical services are vital and unavoidable. The main goal of this paper is the development and application of a binary programming model to support decision making process, especially addressing scheduling workforce in organizations with stochastic demand. The necessary staffing levels and human resources allocation in health care organizations are often defined ad hoc, without empirical analysis and synchronization with the demand for emergency medical services. Thus, irrational allocation of resources can result in various negative impacts on the financial result, quality of medical services and satisfaction of both patients and employees. We start from the desired staffing levels determined in advance and try to find the optimal scheduling plan that satisfies all significant professional and regulatory constraints. In this paper a binary programming model has been developed and implemented in order to minimize costs, presented as the sum of required number of ambulance crews. The results were implemented for staff rostering process in the Ambulance Service Station in Subotica, Serbia. Compared to earlier scheduling done ad hoc at the station, the solution of the formulated model provides a better and equable engagement of crews. The developed model can be easily modified and applied to other organizations with the same, stochastic, nature of the demand.

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

Puzzles and games enhance the quality of teaching by creating an enjoyable, interactive, and playful atmosphere. The knight exchange is a famous, very old, and amusing game on the chessboard. This puzzle was used by the author to teach modeling in a mathematical programming course designed for graduate students. The aim was to teach the students the efficiency of the models. Accordingly, first, a binary programming formulation was developed. This formulation was, however, found to be inefficient, and tremendous time (i.e., more than four hours) and a large amount of processing memory were needed to solve the puzzle. The puzzle was subsequently formulated as a minimum cost network flow problem. The latter formulation outperformed the general binary formulation by solving the puzzle in less than a minute. The network formulation could also save the required processing memory. The results could help students to learn the value of modeling combinatorial optimization problems as network flows.

Multi-Objective Human Resource Allocation Approach for Sustainable Traffic Management

Efficient human resource deployment is one of the key aspects of road traffic management for maintaining the lifelines of any metropolitan city. The problem becomes relevant when collaboration between human resources with different skills in day-to-day operations is necessary to maintain public and commercial transport, manage various social events and emergency situations, and hence reduce congestion, injuries, emissions, etc. This study proposes a two-phase fuzzy multi-objective binary programming model for optimal allocation of five different categories of human resources to minimize the overall operational cost, maximize the allocation to accident-prone road segments, minimize the number of volunteer personnel and maximize the direct contact to reduce emissions and road traffic violations, simultaneously. A binary programming model is formulated to provide an efficient individual manpower allocation schedule for multiple road segments at different shifts. A case study is proposed for model evaluation and to derive managerial implications. The proposed model can be used to draw insights into human resource allocation planning in traffic management to reduce road traffic congestion, injuries and vehicular emissions.