Nondominated Points Research Articles

A method to identify a representation of the set of non-dominated points for discrete tri-objective optimization problems

A method to identify a representation of the set of non-dominated points for discrete tri-objective optimization problems

Approximating the Pareto frontier for bi-objective preventive maintenance and workshop scheduling: A Lagrangean lower bounding methodology for evaluating contracting forms

AbstractEffective planning of preventive maintenance plays an important role in maximizing the operational readiness of any industrial system. We consider an operating system and a maintenance workshop governed by two stakeholders who collaborate based on a mutual contract: components of the operating system that need maintenance are sent to the maintenance workshop, where necessary maintenance activities are performed and after which the maintained components are returned to the operating systems and ready to be used again. While the maintenance activities must obey the workshop capacity, the components should be returned to the operating system within a contracted time frame. For this problem, we developed in a previous work a mixed-integer linear optimization model incorporating stocks of damaged as well as repaired components, workshop scheduling, and preventive maintenance planning for the operating system. We then investigated an availability contract between the stakeholders and which is in the paper at hand compared with a turn-around time contract type, which is more often used in reality. Since, for real instance sizes, the latter leads to a computationally demanding bi-objective optimization problem, we use Lagrangean relaxation and subgradient optimization to compute local lower bounds on the set of non-dominated points, complemented with math-heuristics to identify good feasible solutions (i.e., local upper bounds). Our suggested method thus provides a bounding of the set of non-dominated points for a turn–around time contract.

Designing sustainable diet plans by solving triobjective integer programs

AbstractWe present an algorithm for triobjective nonlinear integer programs that combines the $$\varepsilon $$ ε -constrained method with available oracles for biobjective integer programs. We prove that our method is able to detect the nondominated set within a finite number of iterations. Specific strategies to avoid the detection of weakly nondominated points are devised. The method is then used to determine the nondominated solutions of triobjective 0–1 models, built to design nutritionally adequate and healthy diet plans, minimizing their environmental impact. The diet plans refer to menus for school cafeterias and we consider the carbon, water and nitrogen footprints as conflicting objectives to be minimized. Energy and nutrient contents are constrained in suitable ranges suggested by the dietary recommendation of health authorities. Results obtained on two models and on real world data are reported and discussed.

Direct MultiSearch optimization of TPMS scaffolds for bone tissue engineering

BackgroundScaffolds designed for tissue engineering must consider multiple parameters, namely the permeability of the design and the wall shear stress experienced by the cells on the scaffold surface. However, these parameters are not independent from each other, with changes that improve wall shear stress, negatively impacting permeability and vice versa. This study introduces a novel multi-objective optimization framework using Direct MultiSearch (DMS) to design triply periodic minimal surface (TPMS) scaffolds for bone tissue engineering. MethodThe optimization algorithm focused on maximizing the permeability of the scaffolds and obtaining a desired value of average wall shear stress (which ranges between the values that promote osteogenic differentiation of 0.1 mPa and 10 mPa). Multiple fluid inlet velocities and target wall shear stress were analyzed. The DMS method successfully generated Pareto fronts for each configuration, enabling the selection of optimized scaffolds based on specific structural requirements. ResultsThe findings reveal that increasing the target wall shear stress results in a greater number of non-dominated points on the Pareto front, highlighting a more robust optimization process. Additionally, it was also demonstrated that the tested Schwartz diamond scaffolds had a better permeability-wall shear stress relation when compared to Schoen gyroid geometries. ConclusionsDirect MultiSearch was proven as an effective tool to aid in the design of tissue engineering scaffolds. This adaptable optimization framework has potential applications beyond bone tissue engineering, including cartilage tissue differentiation.

From Single-Objective to Bi-Objective Maximum Satisfiability Solving

The declarative approach is key to efficiently finding optimal solutions to various types of NP-hard real-world combinatorial optimization problems. Most work on practical declarative solvers—ranging from classical integer programming to finite-domain constraint optimization and maximum satisfiability (MaxSAT)—has focused on optimization under a single objective; fewer advances have been made towards efficient declarative techniques for multi-objective optimization problems. Motivated by significant recent advances in practical solvers for MaxSAT, in this work we develop BiOptSat, an exact declarative approach for finding Pareto-optimal solutions to bi-objective optimization problems, with propositional logic as the underlying constraint language. BiOptSat can be viewed as an instantiation of the lexicographic method. The approach makes use of a single Boolean satisfiability solver that is incrementally employed throughout the entire search procedure, allowing for finding a single Pareto-optimal solution, finding one representative solution for each non-dominated point, and enumerating all Pareto-optimal solutions. We detail several algorithmic instantiations of BiOptSat, each building on recent algorithms proposed for single-objective MaxSAT. We empirically evaluate the instantiations compared to recently-proposed alternative approaches to multi-objective MaxSAT solving on several real-world domains from the literature, showing the practical benefits of our approach.

A Hyper-Transformer model for Controllable Pareto Front Learning with Split Feasibility Constraints

Controllable Pareto front learning (CPFL) approximates the Pareto optimal solution set and then locates a non-dominated point with respect to a given reference vector. However, decision-maker objectives were limited to a constraint region in practice, so instead of training on the entire decision space, we only trained on the constraint region. Controllable Pareto front learning with Split Feasibility Constraints (SFC) is a way to find the best Pareto solutions to a split multi-objective optimization problem that meets certain constraints. In the previous study, CPFL used a Hypernetwork model comprising multi-layer perceptron (Hyper-MLP) blocks. Transformer can be more effective than previous architectures on numerous modern deep learning tasks in certain situations due to their distinctive advantages. Therefore, we have developed a hyper-transformer (Hyper-Trans) model for CPFL with SFC. We use the theory of universal approximation for the sequence-to-sequence function to show that the Hyper-Trans model makes MED errors smaller in computational experiments than the Hyper-MLP model.

Numerical algorithms for generating an almost even approximation of the Pareto front in nonlinear multi-objective optimization problems

A multiobjective optimization problem (MOP) returns a set of non-dominated points, the so-called Pareto front. Since this set is usually infinite, it is impossible to generate it completely in practice. Therefore, a discrete approximation of the Pareto front is created. One of the most important features of this approximation is a uniform distribution of points on the full Pareto front in order to present a wide variety of solutions to the decision maker who chooses a final solution. While a few algorithms consider this property, two algorithms based on the Pascoletti-Serafini (PS) scalarization approach are proposed. In addition, six well-known test problems with convex and non-convex Pareto fronts are considered to show the effectiveness of the proposed algorithms. Their results are compared with some algorithms including Normal Constraint (NC), Benson type, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), S-Metric Selection Evolutionary Multiobjective Algorithm (SMS-EMOA), Differential Evolution (DE) with Binomial Crossover and MOEA/D-DE. The computational results on CPU time and reasonable distribution of points obtained on the Pareto front show that the presented algorithms perform better than other algorithms on these criteria. In addition, although the proposed algorithms compete closely with some algorithms in terms of CPU time, they have more non-dominated solutions and more appropriate distribution than they do in most problems.

The Bus Rapid Transit investment problem

Bus Rapid Transit (BRT) systems can provide a fast and reliable service to passengers at low investment costs compared to tram, metro and train systems. Therefore, they can be of great value to attract more passengers to use public transport. This paper thus focuses on the BRT investment problem: Which segments of a single bus line should be upgraded such that the number of newly attracted passengers is maximized? Motivated by the construction of a new BRT line around Copenhagen, we consider a setting in which multiple parties are responsible for the financing of different segments of the line. As each party has a limited willingness to invest, we solve a bi-objective problem to quantify the trade-off between the number of attracted passengers and the investment budget. We model different problem variants: First, we consider two potential passenger responses to upgrades on the line. Second, to prevent scattered upgrades along the line, we consider different restrictions on the number of upgraded connected components on the line. We propose an epsilon-constraint-based algorithm to enumerate the complete set of non-dominated points and investigate the complexity of this problem. Moreover, we perform extensive numerical experiments on artificial instances and a case study based on the BRT line around Copenhagen. Our results show that we can generate the full Pareto front for real-life instances and that the resulting trade-off between investment budget and attracted passengers depends both on the origin–destination demand and on the passenger response to upgrades. Moreover, we illustrate how the generated Pareto plots can assist decision makers in selecting from a set of geographical route alternatives in our case study.

Inexact Exponential Penalty Function with the Augmented Lagrangian for Multiobjective Optimization Algorithms

This paper uses an augmented Lagrangian method based on an inexact exponential penalty function to solve constrained multiobjective optimization problems. Two algorithms have been proposed in this study. The first algorithm uses a projected gradient, while the second uses the steepest descent method. By these algorithms, we have been able to generate a set of nondominated points that approximate the Pareto optimal solutions of the initial problem. Some proofs of theoretical convergence are also proposed for two different criteria for the set of generated stationary Pareto points. In addition, we compared our method with the NSGA-II and augmented the Lagrangian cone method on some test problems from the literature. A numerical analysis of the obtained solutions indicates that our method is competitive with regard to the test problems used for the comparison.

An outer approximation algorithm for generating the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems

In this paper, we present an outer approximation algorithm for computing the Edgeworth–Pareto hull of multi-objective mixed-integer linear programming problems (MOMILPs). It produces the extreme points (i.e., the vertices) as well as the facets of the Edgeworth–Pareto hull. We note that these extreme points are the extreme supported non-dominated points of a MOMILP. We also show how to extend the concept of geometric duality for multi-objective linear programming problems to the Edgeworth–Pareto hull of MOMILPs and use this extension to develop the algorithm. The algorithm relies on a novel oracle that solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the extreme supported non-dominated points in the case of MOMILPs. Thus, for MOMILPs for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay—a result that was formerly only known for the computation of extreme supported non-dominated points. Our algorithm can be an attractive option to compute lower bound sets within multi-objective branch-and-bound algorithms for solving MOMILPs. This is for several reasons as (i) the algorithm starts from a trivial valid lower bound set then iteratively improves it, thus at any iteration of the algorithm a lower bound set is available; (ii) the algorithm also produces efficient solutions (i.e., solutions in the decision space); (iii) in any iteration of the algorithm, a relaxation of the MOMILP can be solved, and the obtained points and facets still provide a valid lower bound set. Moreover, for the special case of multi-objective linear programming problems, the algorithm solves the problem to global optimality. A computational study on a set of benchmark instances from the literature is provided.

Supported nondominated points as a representation of the nondominated set: An empirical analysis

AbstractThe nondominated set of a multiple objective discrete optimization problem is known to contain unsupported nondominated points, which outnumber the supported ones and are more difficult to obtain. We treat supported nondominated points as a representation and analyse their quality using different metrics beyond their sheer numbers. Under different data generation schemes on multiobjective knapsack and assignment problems, we observe that supported nondominated points almost always provide a good representation of the entire nondominated set.

A modified version of a Benson-type algorithm proposed for obtaining solutions with better dispersion on the non-dominated set of a non-convex multi-objective programming problem

This paper proposes a new method to approximate the non-dominated set of a general multi-objective programming (MOP) problem with continuous variables (that is, without any convexity assumption). The algorithm is based on a recent modification of Benson's method [14], and like Benson's idea, generates an inner approximation and an outer approximation of the (weakly) non-dominated set of an MOP. It is proved that this method provides a set of (weakly) $ \varepsilon $-non-dominated points. To evaluate the performance of the proposed algorithm, some different test problems are presented. The results show that the algorithm provides a reliable and practical approach to the approximation of non-dominated sets.

Density Function-Based Trust Region Algorithm for Approximating Pareto Front of Black-Box Multiobjective Optimization Problems

In this paper, we consider a black-box multiobjective optimization problem, whose objective functions are computationally expensive. We propose a density function-based trust region algorithm for approximating the Pareto front of this problem. At every iteration, we determine a trust region and then in this trust region, select several sample points, at which are evaluated objective function values. In order to obtain non-dominated solutions in the trust region, we convert given objective functions into one function: scalarization. Then, we construct quadratic models of this function and the objective functions. In current trust region, we find optimal solutions of all single-objective optimization problems with these models as objectives. After that, we remove dominated points from the set of obtained solutions. In order to estimate the distribution of non-dominated solutions, we introduce a density function. By using this density function, we obtain the most “isolated” point among the non-dominated points. Then, we construct a new trust region around this point and repeat the algorithm. We prove convergence of proposed algorithm under the several assumptions. Numerical results show that even in case of tri-objective optimization problems, the points generated by proposed algorithm are uniformly distributed over the Pareto front.

A test instance generator for multiobjective mixed-integer optimization

Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments.

A criterion space decomposition approach to generalized tri-objective tactical resource allocation

We present a tri-objective mixed-integer linear programming model of the tactical resource allocation problem with inventories, called the generalized tactical resource allocation problem (GTRAP). We propose a specialized criterion space decomposition strategy, in which the projected two-dimensional criterion space is partitioned and the corresponding sub-problems are solved in parallel by application of the quadrant shrinking method (QSM) (Boland in Eur J Oper Res 260(3):873–885, 2017) for identifying non-dominated points. To obtain an efficient implementation of the parallel variant of the QSM we suggest some modifications to reduce redundancies. Our approach is tailored for the GTRAP and is shown to have superior computational performance as compared to using the QSM without parallelization when applied to industrial instances.

Finding the Nondominated Set and Efficient Integer Vectors for a Class of Three-Objective Mixed-Integer Linear Programs

We consider a class of three-objective mixed-integer linear programs (MILPs) where at least one of the objective functions takes only discrete values. These problems commonly occur in MILPs where one or more of the three objective functions contain only integer decision variables. In such problems, the nondominated set consists of the union of nondominated edges and individual nondominated points. The nondominated edges can provide valuable insights on the trade-offs between the two continuous objectives at different levels of the discrete-valued objective. We develop an objective-space search algorithm that keeps partitioning the search space by progressively creating cones in the two-dimensional feasible space of the two continuous objectives for relevant values of the discrete-valued objective. The algorithm generates the nondominated points or edges in the nonincreasing order of the feasible values of the selected discrete-valued objective. Additionally, the algorithm uncovers all efficient integer variable vectors, including different vectors that lead to the same nondominated points or edges. We apply the algorithm to the day-ahead electricity market clearing problem. This paper was accepted by Chung Piaw Teo, optimization. Supplemental Material: The data files and e-companion are available at https://doi.org/10.1287/mnsc.2023.4712 .

Application of the Plant Propagation Algorithm and NSGA-II to Multiple Objective Linear Programming

Multiple Objective Linear Programming (MOLP) problems are usually solved by exact methods. However, nature-inspired population based stochastic algorithms such as the plant propagation algorithm are becoming more and more prominent. This paper applies the multiple objective plant propagation algorithm (MOPPA) and nondominated sorting genetic algorithm II (NSGA-II) for the first time to MOLP and compares their outcomes with those of prominent exact methods. Computational results from a collection of 51 existing MOLP instances suggests that MOPPA compares favourably with four of the most prominent exact methods namely extended multiple objective simplex algorithm (EMSA), affine scaling interior MOLP algorithm (ASIMOLP), Benson’s outer-approximation algorithm (BOA) and parametric simplex algorithm (PSA), and returns best nondominated points which are of higher quality than those returned by NSGA-II. However, the nondominated points approximated by NSGA-II are evenly distributed across the nondominated front. The methods compare well with the four exact methods especially on the large instances which the exact methods failed to solve even when given generous amounts of computation times.

An exact criterion space search method for a bi-objective nursing home location and allocation problem

A health care system will be survived if it is economically affordable and elderly people can receive its services in case of availability. Hence, a network should be designed for home nursing of elderly people considering both economic and well-fare objectives. Finding optimal solutions to both mentioned objectives in health care systems, especially for elderly people, is challenging nowadays. Hence, it is necessary to find near-optimal solutions in action. This paper presents a new criterion space search algorithm for achieving all non-dominated points in a bi-objective nursing home location–allocation problem. Concepts of Box and Balanced Box methods and modified e-constraint are used in designing the proposed algorithm. The main focus of the research is to decrease potential areas that could involve non-dominated points by defining Minimum Feasible Rectangles during the search procedure. The performance of the proposed algorithm is analyzed by using two real data sets taken from the literature in the field of nursing home location–allocation problems. The proposed algorithm can outperform previous recently developed exact methods based on obtained results, especially from a computational time viewpoint.

Space-Partitioned ND-Trees for the Dynamic Nondominance Problem

We present techniques for improving the efficiency of ND-trees in the solution of the dynamic nondominance problem, i.e., for building and maintaining a Pareto archive. We propose algorithms for (re)building a tree from a set of nondominated points, either as a space-partitioned ND-tree or by a clustering approach. Numerical experiments confirm that rebuilding the tree at intervals, combined with modified strategies for updating the lower and upper bounds in each node and for computing the distances of points to subtrees in between the rebuilds, can lead to significant speedups over using plain ND-tree-based updates throughout, in particular for higher dimensions.

Multi-Objective Optimal Control of Autocatalytic Esterification Process Using Control Vector Parameterization (CVP) and Hybrid Strategy (HS)

The semi-batch esterification of propionic anhydride (PA) with 2-butanol (BT) in the presence of catalyst can be optimised using an optimal control strategy, which utilises the reactor temperature (TR) and feed (FR) flowrate. However, the opposing objective functions, which are maximum conversion (XM) and minimum process time (tf) in the autocatalytic esterification process, could complicate the optimisation strategy. Simultaneous optimisation of various objectives results in a multi-objective optimal control (MOOC) problem with numerous solutions known as non-dominated (ND) points. In this paper, control vector parameterisation (CVP) and hybrid strategy (HS) are utilised to form Pareto Front (PF) for two opposite targets, which are first to increase XM and secondly to reduce tf. Each ND point comprises variant optimal dynamic tracks of TR and FR, which results in various targets of XM and tf. These solutions provide numerous options for evaluating trade-offs and deciding on the most efficient operating strategy. It is found that the ND point in zone II can be selected as the trade-off of the optimal TR and FR in this study.