Bi-objective Combinatorial Optimization Problems Research Articles

Multiple financial analyst opinions aggregation based on uncertainty-aware quality evaluation

Financial analysts’ opinions are pivotal in investment decision-making, as they provide valuable expert knowledge. Aggregating these opinions offers a promising way to unlock their collective wisdom. However, existing opinion aggregation methods are hindered by their inability to effectively assess differences in opinion quality, resulting in suboptimal outcomes. This Study introduces a novel model called SmartMOA, which addresses this limitation by automatically evaluating the quality of each opinion and integrating this evaluation into the aggregation process. Our model begins with a novel Bayesian neural network that leverages the implicit knowledge embedded in the interactions between analysts and stock characteristics. This methodology produces an assessment of individual opinions that accounts for uncertainties. We then formulate a bi-objective combinatorial optimization problem to determine optimal weights for combining multiple analysts’ opinions, simultaneously minimizing the error and uncertainty of the aggregated outcome. Therefore, SmartMOA systematically highlights high-quality opinions during the aggregation process. Using a real dataset spanning eight years, we present comprehensive empirical evidence that demonstrates the superior performance of SmartMOA in heterogeneous analyst opinion aggregation.

Computing representations using hypervolume scalarizations

In this article, we use the hypervolume indicator as a scalarizing function for biobjective combinatorial optimization problems. In particular, we describe a generic solution approach that determines the nondominated set of a biobjective optimization problem by solving a sequence of hypervolume scalarizations with appropriate choices of the reference point. Moreover, this solution technique can also provide a compact representation of the efficient set that is a (1−1/e)-approximation to the optimal representation in terms of the hypervolume in an a priori manner. We illustrate these concepts for a particular variant of the biobjective knapsack problem and for a biobjective shortest path problem. Numerical results are presented for the former problem.

Effective Selection of a Compact and High-Quality Review Set with Information Preservation

Consumers increasingly make informed buying decisions based on reading online reviews for products and services. Due to the large volume of available online reviews, consumers hardly have the time and patience to read them all. This article aims to select a compact set of high-quality reviews that can cover a specific set of product features and related consumer sentiments. Selecting such a subset of reviews can significantly save the time spent on reading reviews while preserving the information needed. A unique review selection problem is defined and modeled as a bi-objective combinatorial optimization problem, which is then transformed into a minimum-cost set cover problem that is NP-complete. Several approximation algorithms are then designed, which can sustain performance guarantees in polynomial time. Our effective selection algorithms can also be upgraded to handle dynamic situations. Comprehensive experiments conducted on twelve real datasets demonstrate that the proposed algorithms significantly outperform benchmark methods by generating a more compact review set with much lower computational cost. The number of reviews selected is much smaller compared to the quantity of all available reviews, and the selection efficiency is deeply increased by accelerating strategies, making it very practical to adopt the methods in real-world online applications.

Bi-objective Branch-and-Cut Algorithms Based on LP Relaxation and Bound Sets

Most real-world optimization problems are multi-objective by nature, with conflicting and incomparable objectives. Solving a multi-objective optimization problem requires a method that can generate all rational compromises between the objectives. This paper proposes two distinct bound set-based branch-and-cut algorithms for general bi-objective combinatorial optimization problems based on implicit and explicit lower-bound sets. The algorithm based on explicit lower-bound sets computes, for each branching node, a lower-bound set and compares it with an upper-bound set. The other fathoms branching nodes by generating a single point on the lower-bound set for each local nadir point. We outline several approaches for fathoming branching nodes, and we propose an updating scheme for the lower-bound sets that prevents us from solving the bi-objective linear programming relaxation of each branching node. To strengthen the lower-bound sets, we propose a bi-objective cutting-plane algorithm that adjusts the weights of the objective functions such that different parts of the feasible set are strengthened by cutting planes. In addition, we suggest an extension of the branching strategy “Pareto branching.” We prove the effectiveness of the algorithms through extensive computational results.

A Discrete Multiobjective Particle Swarm Optimizer for Automated Assembly of Parallel Cognitive Diagnosis Tests.

Parallel test assembly has long been an important yet challenging topic in educational assessment. Cognitive diagnosis models (CDMs) are a new class of assessment models and have drawn increasing attention for being able to measure examinees' ability in detail. However, few studies have been devoted to the parallel test assembly problem in CDMs (CDM-PTA). To fill the gap, this paper models CDM-PTA as a subset-based bi-objective combinatorial optimization problem. Given an item bank, it aims to find a required number of tests that achieve optimal but balanced diagnostic performance, while satisfying important practical requests in the aspects of test length, item type distribution, and overlapping proportion. A set-based multiobjective particle swarm optimizer based on decomposition (S-MOPSO/D) is proposed to solve the problem. To coordinate with the property of CDM-PTA, S-MOPSO/D utilizes an assignment-based representation scheme and a constructive learning strategy. Through this, promising solutions can be built efficiently based on useful assignment patterns learned from personal and collective search experience on neighboring scalar problems. A heuristic constraint handling strategy is also developed to further enhance the search efficiency. Experimental results in comparison with three representative approaches validate that the proposed algorithm is effective and efficient.

Scoring of Side-Chain Packings: An Analysis of Weight Factors and Molecular Dynamics Structures.

The protein side-chain packing problem (PSCPP) is a central task in computational protein design. The problem is usually modeled as a combinatorial optimization problem, which consists of searching for a set of rotamers, from a given rotamer library, that minimizes a scoring function (SF). The SF is a weighted sum of terms, that can be decomposed in physics-based and knowledge-based terms. Although there are many methods to obtain approximate solutions for this problem, all of them have similar performances and there has not been a significant improvement in recent years. Studies on protein structure prediction and protein design revealed the limitations of current SFs to achieve further improvements for these two problems. In the same line, a recent work reported a similar result for the PSCPP. In this work, we ask whether or not this negative result regarding further improvements in performance is due to (i) an incorrect weighting of the SFs terms or (ii) the constrained conformation resulting from the protein crystallization process. To analyze these questions, we (i) model the PSCPP as a bi-objective combinatorial optimization problem, optimizing, at the same time, the two most important terms of two SFs of state-of-the-art algorithms and (ii) performed a preprocessing relaxation of the crystal structure through molecular dynamics to simulate the protein in the solvent and evaluated the performance of these two state-of-the-art SFs under these conditions. Our results indicate that (i) no matter what combination of weight factors we use the current SFs will not lead to better performances and (ii) the evaluated SFs will not be able to improve performance on relaxed structures. Furthermore, the experiments revealed that the SFs and the methods are biased toward crystallized structures.

Column generation algorithms for bi-objective combinatorial optimization problems with a min–max objective

Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon $$ -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon $$ -constraint presented is also better than a standard $$\varepsilon $$ -constraint method in terms of the quality of the bound sets.

Selecting Efficient Service-providers in Electric Power Distribution Industry Using Combinatorial Reverse Auction

In this paper, a combinatorial reverse auction mechanism is proposed for selecting the most efficient service-providers for resolving sustained power interruptions in multiple regions of an electric power distribution company’s responsibility area. Through this mechanism, supplying the required service in each region is assigned to only one potential service-provider considering two criteria including cost and service time. So, the corresponding winner determination problem of the proposed auction mechanism is formulated as a bi-objective combinatorial optimization problem. However, finding a feasible solution for the formulated problem as well as its solving is NP-complete. Since exact optimization algorithms are failed in solving this kind of problems in a reasonable time, a problem specific metaheuristic called NSGA-II that is an evolutionary algorithm for solving multi-objective optimization problems is developed to estimate the set of Pareto optimal solutions of the formulated bi-objective winner determination problem. In our developed NSGA-II, two problem-specific operators are proposed for creating initial feasible solutions and converting infeasible solutions to feasible ones. Furthermore, a new method for determining the population size based on the size of problem instance is proposed. We conduct a computational experiment in which several randomly generated problem instances are solved using the proposed NSGA-II in different settings. Computational results of proposed algorithm in different settings are compared using a quality measure and statistical hypothesis tests. The results of performance comparison show that the proposed NSGA-II with a population size determined by proposed method and a different form of binary tournament method performs better in finding non-dominated solutions for different instances of formulated bi-objective optimization problem.

Selecting a Compact and High-Quality Representative Review Set with Information Preservation

Consumers increasingly make informed buying decisions based on reading online reviews for products and services. Due to the large volume of available online reviews, consumers hardly have the time and patience to read them all. The goal of this paper is to select a compact set of high-quality reviews that can cover a specific set of product features and related consumer sentiments. The set of features covered by the selected reviews can be all product features discussed in available reviews or a subset of features tailored to the specific interests of individual consumers if such information is available. Given the features, the selected reviews will cover all the information related to them in terms of sentiment polarity to preserve information completeness. Selecting such a subset of reviews can significantly save the time spent on reading reviews while preserving the information needed. A unique review selection problem is defined and modeled as a bi-objective combinatorial optimization problem, which is then transformed into a minimum-cost set cover problem that is NP-complete. Several approximation algorithms are then designed, which can sustain performance guarantees in polynomial time. Our effective selection algorithms can also be upgraded to handle dynamic situations. Comprehensive experiments conducted on six real-world datasets demonstrate that the proposed algorithms significantly outperform benchmark methods by generating a more compact review set with much lower computational cost. The number of reviews selected is much smaller compared with the quantity of all available reviews, and the selection efficiency is deeply increased by accelerating strategies, making it very practical to adopt the methods in real-world online applications.

Finding representations for an unconstrained bi-objective combinatorial optimization problem

Typically, multi-objective optimization problems give rise to a large number of optimal solutions. However, this information can be overwhelming to a decision maker. This article introduces a technique to find a representative subset of optimal solutions, of a given bounded cardinality for an unconstrained bi-objective combinatorial optimization problem in terms of $$\epsilon $$ -indicator. This technique extends the Nemhauser–Ullman algorithm for the knapsack problem and allows to find a representative subset in a single run. We present a discussion on the representation quality achieved by this technique, both from a theoretical and numerical perspective, with respect to an optimal representation.

Introduction to the special issue on search‐based software engineering (NasBASE 2015)

It is our pleasure to introduce to the reader this special issue based on extended papers from the 1st North American Search Based Software Engineering Symposium (NasBASE 2015). The NasBASE symposium was conceived as a vehicle to bring together the rapidly growing North American Search Based Software Engineering (SBSE) community in a welcoming forum for discussion and dissemination, and to sustain the recent flourishing of interest in this field of research and practice. A total of 16 papers were submitted to the symposium. Each paper was subjected to at least three reviews, and finally nine were accepted for publication and presentation at the symposium. From these nine papers, four were chosen and the authors invited to extend their paper for submission to this special issue. These four invited papers were extended by their authors and then subjected to the rigorous JSEP reviewing process, undergoing several further rounds of review and revision. Finally, the three papers described below successfully completed the review process and are contained in this special issue. In “An Empirical Investigation of Single-Objective and Multi-objective Evolutionary Algorithms for Developer's Assignment to Bugs” by Md. Mainur Rahman, Muhammad Rezaul Karim, Guenther Ruhe, Vahid Garousi, and Thomas Zimmermann, the authors model the problem of developer assignment to bugs as a single objective (to minimize bug fix time) and as a bi-objective (minimize bug fix time and cost) combinatorial optimization problem. Two models of developer assignment are considered where in the first model a single developer is assigned per bug (single developer model), whereas in the second model, a single developer is assigned for each competency area of a bug (individual competency model). The performance of the proposed approaches was evaluated for 2040 bugs in 19 open-source milestone projects from the Eclipse platform. The results and analysis demonstrate that the bi-objective model is far better than the single developer model, with an average bug fix time reduction of 39.7% across all projects. In “Error Leakage and Wasted Time: Sensitivity and Effort Analysis of a Requirements Consistency Checking Process” by Jane Hayes, Bram Adams, Yann-Gaël Guéhéneuc, Giuliano Antoniol, Wenbin Li, and Mirek Truszczynski, the authors model and study a process for checking the consistency of temporal requirements and assess error leakage and wasted time. They perform an analysis of the input factors of the proposed model to determine the effect that sources of uncertainty may have on the final accuracy of the consistency checking process. The results show that an analyst's effort varies depending on the precision and recall of the subprocesses, and that the number and capability of analysts affect their effort. In “Search-Based Model Transformations” by Martin Fleck, Javier Troya, and Manuel Wimmer, the authors address the problem of model optimization through transformation. Current approaches based on rule orchestration suffer from the limitations that the optimization objectives are hidden in the transformation rules, and that manually finding a good sequence of rules to apply in a particular scenario is a very challenging task. In this paper, the transformation orchestration problem is formulated as an optimization problem, which enables it to be addressed by search-based exploration of the transformation space driven by explicitly stated transformation objectives. The authors embody these ideas in a framework and test it on a number of typical model-based software engineering problems, thus demonstrating it to be usable by non-SBSE experts as well as capable of producing feasible solutions that are easy to comprehend. We hope that readers will enjoy reading these papers and find that they make a useful contribution to their knowledge and understanding. There are many people we would like to thank: the authors who submitted papers to this special issue and the reviewers who did an excellent job of helping to bring these papers to such a high level, the editorial board of the Journal of Software: Evolution and Process and especially its editor, Gerardo Canfora, who provided his expertise in a very timely fashion throughout the process.

Using column generation to compute lower bound sets for bi-objective combinatorial optimization problems

We discuss the use of column generation in a bi-objective setting. Just as in single objective combinatorial optimization, the role of column generation in the bi-objective setting is to compute dual bounds (i.e. lower bounds for minimization problems and upper bounds for maximization problems) which can be used to guide the search for efficient solutions or to evaluate the quality of approximate solutions. The general idea used in this paper is to first transform the bi-objective problem into single objective by a scalarization method and then solve the transformed problem several times by varying the necessary parameters. We show that irrespective of the scalarization method used, similar subproblems are solved when applying column generation. For this reason, we investigate possible ways of intelligently searching for columns for these subproblems in order to accelerate the column generation method.

A Support-Based Algorithm for the Bi-Objective Pareto Constraint

Bi-Objective Combinatorial Optimization problems are ubiquitous in real-world applications and designing approaches to solve them efficiently is an important research area of Artificial Intelligence. In Constraint Programming, the recently introduced bi-objective Pareto constraint allows one to solve bi-objective combinatorial optimization problems exactly. Using this constraint, every non-dominated solution is collected in a single tree-search while pruning sub-trees that cannot lead to a non-dominated solution. This paper introduces a simpler and more efficient filtering algorithm for the bi-objective Pareto constraint. The efficiency of this algorithm is experimentally confirmed on classical bi-objective benchmarks.

Choquet optimal set in biobjective combinatorial optimization

We study in this paper the generation of the Choquet optimal solutions of biobjective combinatorial optimization problems. Choquet optimal solutions are solutions that optimize a Choquet integral. The Choquet integral is used as an aggregation function, presenting different parameters, and allowing to take into account the interactions between the objectives. We develop a new property that characterizes the Choquet optimal solutions. From this property, a general method to easily generate these solutions in the case of two objectives is defined. We apply the method to two classical biobjective optimization combinatorial optimization problems: the biobjective knapsack problem and the biobjective minimum spanning tree problem. We show that Choquet optimal solutions that are not weighted sum optimal solutions represent only a small proportion of the Choquet optimal solutions and are located in a specific area of the objective space, but are much harder to compute than weighted sum optimal solutions.

Approximation schemes for bi-objective combinatorial optimization and their application to the TSP with profits

We describe two approximation schemes for bi-objective combinatorial optimization problems with nonnegative, integer valued objectives. The procedures compute a subset of the efficient points such that any other efficient point is within an arbitrary factor from a computed one with respect to both objectives. Both schemes are simple modifications of classical algorithms for the construction of the whole efficient set.In both procedures, a properly defined single objective subproblem has to be solved at every iteration. We show that, in both cases, the number of subproblems to be solved and the number of returned efficient points are polynomial in the input size and the reciprocal of the allowed error. We also show that a fast post-processing guarantees that the number of returned efficient points is at most three times the minimum possible number needed to approximate the efficient set within the specified tolerance.We test the procedures on the Traveling Salesman Problem with Profits, where profits and costs are treated as conflicting objectives. Results are taken on randomly generated instances and on TSPLIB instances. We show that both algorithms return a guaranteed approximation with significant time sparing with respect to exact procedures. We also give empirical evidence that in the specific application the number of points returned by the approximation schemes is close to the minimum.

Single Machine Scheduling with Rejection

In this paper, the authors consider a single machine scheduling problem with rejection. In traditional research, it is assumed all jobs must be processed. However, in the real-world situation, certain jobs can be rejected. In this study, the jobs can be either accepted and scheduled or be rejected at the cost of a penalty. Two objective functions are considered simultaneously: (1) minimization of the sum of weighted completion times for the accepted jobs, and (2) minimization of the sum of penalties for the rejected jobs. The authors apply two-phase method (TPM), which is a general technique to solve bi-objective combinatorial optimization problems, to find all supported and non-supported solutions for small-sized problems. The authors present a mathematical model for implementing both phases. On the other hand, three different bi-objective simulated annealing algorithms have also been developed to find a good estimation of Pareto-optimal solutions for large-sized problems. Finally the authors discuss the results obtained from each of these algorithms.

An exact -constraint method for bi-objective combinatorial optimization problems: Application to the Traveling Salesman Problem with Profits

Abstract This paper describes an exact ϵ -constraint method for bi-objective combinatorial optimization problems with integer objective values. This method tackles multi-objective optimization problems by solving a series of single objective subproblems, where all but one objectives are transformed into constraints. We show in this paper that the Pareto front of bi-objective problems can be efficiently generated with the ϵ -constraint method. Furthermore, we describe heuristics based on information gathered from previous subproblems that significantly speed up the method. This approach is used to find the exact Pareto front of the Traveling Salesman Problem with Profits, a variant of the Traveling Salesman Problem in which a profit or prize value is associated with each vertex. The goal here is to visit a subset of vertices while addressing two conflicting objectives: maximize the collected prize and minimize the travel costs. We report the first exact results for this problem on instances derived from classical Vehicle Routing and Traveling Salesman Problem instances with up to 150 vertices. Results on approximations of the Pareto front obtained from a variant of our exact algorithm are also reported.

A bi-objective coordination setup problem in a two-stage production system

This paper addresses a problem arising in the coordination between two consecutive departments of a production system, where parts are processed in batches, and each batch is characterized by two distinct attributes. Due to the lack of interstage buffering between the two stages, these departments have to follow the same batch sequence. In the first department, a setup occurs every time the first attribute of a new batch is different from the one of the previous batch. In the downstream department, there is a setup when the second attribute changes in two consecutive batches. The problem consists in finding a batch sequence optimizing the number of setups paid by each department. This case results in a particular bi-objective combinatorial optimization problem. We present a geometrical characterization for the feasible solution set of the problem, and we propose three effective heuristics, as shown by an extensive experimental campaign. The proposed approach can be also used to solve a class of single-objective problems, in which setup costs in the two departments are general increasing functions of the number of setups.

An exact parallel method for a bi-objective permutation flowshop problem

In this paper, we propose a parallel exact method to solve bi-objective combinatorial optimization problems. This method has been inspired by the two-phase method which is a very general scheme to optimally solve bi-objective combinatorial optimization problems. Here, we first show that applying such a method to a particular problem allows improvements. Secondly, we propose a parallel model to speed up the search. Experiments have been carried out on a bi-objective permutation flowshop problem for which we also propose a new lower bound.

Bound sets for biobjective combinatorial optimization problems

In this paper we introduce the concept of bound sets for multiobjective discrete optimization. We prove general results on lower and upper bound sets for combinatorial optimization problems with multiple objectives. We present general algorithms for constructing lower and upper bound sets for biobjective problems and provide numerical results on five different problem types.