Anytime Lexicographic Enumeration of the Pareto Front in Multi‐Objective Combinatorial Optimisation

This paper proposes a step‐by‐step decision making approach that incorporates preferences of a decision maker into a many‐objective evolutionary algorithm. The algorithm is applied to a challenging many‐objective optimization problem (MaOP) involving integer objective values, tight constraints, heavy computational burden, and limited knowledge about the true Pareto front. The proposed method consists of five steps. (a) The entire Pareto front with high dimensions is roughly approximated by a small set of nondominated solutions. (b) On the basis of the approximated Pareto front, the decision maker specifies a preference. (c) Reference points are distributed through part of the objective space using the preference information. (d) The approximation accuracy of the Pareto front in the region of interest is improved by local search using the reference points. (e) The most preferred solution from a set of the nondominated solutions is chosen. The evolutionary algorithms employed in Steps (a) and (d) have an improved normalization procedure and selection mechanism for the MaOP involving integer objective value(s) and the tight constraints. Steps (b) and (e) use a graphical‐user‐interface‐based interactive tool we developed to efficiently narrow down to the preferred solution(s) among the high‐dimensional, nondominated solutions. This alleviates the cognitive burden on the decision maker and therefore aids the decision maker in handling MaOPs. In the computational experiment, we choose the real‐world application of reconfiguring an electricity distribution network. We demonstrate that the proposed approach can provide satisfying solutions to a decision maker.

We consider a multi-objective optimization problem with objective functions that are expensive to evaluate. The decision maker (DM) has unknown preferences, and so the standard approach is to generate an approximation of the Pareto front and let the DM choose from the generated non-dominated designs. However, especially for expensive to evaluate problems where the number of designs that can be evaluated is very limited, the true best solution according to the DM's unknown preferences is unlikely to be among the small set of non-dominated solutions found, even if these solutions are truly Pareto optimal. We address this issue by using a multi-objective Bayesian optimization (BO) algorithm (see, e.g., [5]) and allowing the DM to select a preferred solution from a predicted continuous Pareto front just once before the end of the algorithm rather than selecting a solution after the end. This allows the algorithm to understand the DM's preferences and make a final attempt to identify a more preferred solution. We demonstrate the idea using ParEGO [2], and show empirically that the found solutions are significantly better in terms of true DM preferences than if the DM would simply pick a solution at the end.

Engineering design reconciles design constraints with decision maker (DM) preferences. The task of eliciting and encoding decision maker preferences is, however, extremely difficult. A Pareto front representing the locus of the nondominated designs is, therefore, often generated to help a decision maker select the best design. In this paper, we show that this method has a shortcoming when there is uncertainty in both the decision problem variables and in the model of decision maker's preferences. In this case, the Pareto front is inconsistent with multi-attribute utility (MAU) theory, unless the decision maker trades off attributes or some functions of them linearly. This is a strong restriction. To account for this, we propose a methodology that enables a decision maker to select the best design on a modified pareto front (MPF) which is acquired using envelopes of a set of certainty equivalent (CE) surfaces. The proposed method does not require separability of the multi-attribute utility function into single-attribute utilities, nor does it require the decision maker to trade the attributes (or any function of them) linearly. We demonstrate our approach on a simple optimization problem and in design of a reduction gear.

In reservoir flood control operation, the safety of upstream and downstream of the dam are the main two optimization goals with conflicts. In addition, the irrigation water demands is also an important issue considered by decision makers. Therefore, the dispatching schemes that meet the final water level constraint are preferred. Considering such preference in decision making, a novel preference-based selection operator is developed and combined with immune inspired optimization technique to form the proposed multi-objective immune algorithm with preference- based selection (MOIA-PS) for reservoir flood control operation. The unique of MOIA-PS is that it intends to obtain a set of preferred Pareto optimal solutions that located within a part of preferred area on the Pareto front rather than to find a good approximation of the entire Pareto front as most existing methods did. Experimental results on four typical floods at the Ankang reservoir have indicated that the preferred non-dominated solutions are distributed within a local area of preferred PF region. And the newly designed preference-based selection operator can guide the search of MOIA-PS towards the preferred PF region. Comparing with the outstanding multi-objective evolutionary algorithm NSGAII and the immune inspired multi-objective optimization algorithm NNIA, the proposed MOIA-PS obtains more non-dominated solutions that densely and evenly scattered within the preferred area of the Pareto front. MOIA-PS can find finding dispatching schemes that not only reduce the flood peak significantly and guarantee the dam safety well but also satisfy the irrigation water demands. It is a more efficient use of the computing efforts.

This work proposes a Bayesian optimisation with Gaussian Process approach to learn decision maker (DM) preferences in the attribute search space of a multi-objective optimisation problem (MOP). The DM is consulted periodically during optimisation of the problem and asked to provide their preference over a series of pairwise comparisons of candidate solutions. After each consultation, the most preferred solution is used as the reference point in an appropriate multiobjective optimisation evolutionary algorithm (MOEA). The rationale for using Bayesian optimisation is to identify the most preferred location in the decision search space with the least number of DM queries, thereby minimising DM cognitive burden and fatigue. This enables non-expert DMs to be involved in the optimisation process and make more informed decisions. We further reduce the number of preference queries required, by progressively redefining the Bayesian search space to reflect the MOEA's decision bounds as it converges toward the Pareto Front. We demonstrate how this approach can locate a reference point close to an unknown preferred location on the Pareto Front, of both benchmark and real-world problems with relatively few pairwise comparisons.

Incorporating decision maker's preference with multi objective optimization problems keeps an active research area. In this paper, we suggest an algorithm to integrate decision maker's preference into multiobjective evolutionary optimization algorithm based on decomposition technique (MOEA/D). Decomposition techniques require a set of evenly distributed weight vectors to generate a diverse set of solutions on the Pareto front. This newly proposed algorithm incorporates preferred weights generated by desirability functions. A set of evenly distributed weights in desirability space are mapped into objective space to represent decision maker's preference. The solutions corresponding to these preferred weights consist preferred population. Further, a second population associated with evenly distributed weights in objective space is utilized to boost the search for promising areas and present to the decision maker a global perspective of view. Experimental results show the algorithm could be able to find a set of trade-offs on the Pareto front.

This paper advances and evaluates a recently proposed progressively interactive evolutionary multi-objective optimization algorithm. The algorithm uses preference information from the decision maker during the intermediate generations of the EMO and produces the most preferred solution on the Pareto-optimal front. The progress towards the Pareto-optimal front is made by approximating decision maker's value function. In this paper, a generalized polynomial value function has been proposed and the procedure to fit the value function to the decision maker's preference information has been described. The generality of the procedure of fitting a value function to the decision maker's preferences has been shown by using other existing value functions from the literature. The proposed generic polynomial value function has been incorporated in the PI-EMO-VF algorithm to efficiently approximate the decision maker's value function. The paper then evaluates the performance of the PI-EMO-VF algorithm on three and five objective test problems with constraints. It also evaluates the efficacy of the procedure in producing the most preferred solution when the decision maker is unable to provide perfect information, i.e., the decision maker finds certain pairs of solutions in the objective space to be incomparable. Results have been presented for three and five objective constrained test problems using the procedure.

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

This paper considers a multicriteria choice model that includes multiple choices, a numerical vector criterion, and an asymmetric preference relationship of the decision maker (DM). The solution of the multicriteria choice problem is a set of selected choices. Certain "rational" axioms are assumed to be true. Provided that the DM's preference relationship is known only partially, one cannot find a set of selected choices; nevertheless, an upper-bound estimate can be obtained from the available information. The information about a DM's preferences is considered to be of the point-multiple type. This means that the DM is ready to take losses in some finite groups of criteria for the sake winning in some other group. The information about the DM's preferences can be used to develop a new vector criterion and build a restriction of the Pareto set that produces a more precise estimate of the unknown set of the selected variants than the original Pareto set.

Purpose We present an approach to multiple criteria mechanical design problems, for cases where problem complexity precludes derivation of the whole Pareto front. For such problems we propose to limit search, and hence also derivation, of the Pareto front exclusively to regions of the direct designer’s interest, thus saving on computing efforts and gaining on tractable problem sizes. Design/methodology/approach To achieve the purpose, we frame the decision making process (design) into a combination of three specific concepts, namely decision maker's preference capture, local Pareto front search and approximate multiobjective optimization with assessments of the Pareto optimality gap. We illustrate the approach with two small design problems, namely Pareto optimal round tube beam and Pareto optimal pneumatic high speed machine drive selection. We solve these problems in a setting which can be regarded as representative for problem solving in real environment. Findings On the decision making side, the proposed approach has turned out to be a versatile tool for selecting designs from the Pareto suboptimal ones, where each such a Pareto suboptimal design has an explicit assessment of the Pareto optimality gap. On the technical (optimization) side, it has been demonstrated that the approach seamlessly works with evolutionary computations, structured to the specific needs of the approach. Research limitations/implications It has been shown that the navigation over the Pareto front can be achieved with limited effort, both on the cognitive and the computing side. Moreover, navigation over the Pareto front can be focused from the very beginning of the design selection process on the regions of the Pareto front which are of the direct designer’s interest. This eliminates the need to derive (or only approximate) the whole Pareto front, a tangible asset as the derivation of that set is the main factor precluding scalability of design selection problems to higher dimensions (to higher problem sizes). Practical implications Because of the general formulation of the Pareto optimal design selection problem considered in the paper, the absence of any assumptions on its form and easiness of implementation of the underlying procedure of the proposed approach, the paper offers a clear option to approaches based on classical optimization computations. Originality/value The approach offers derivation of Pareto suboptimal designs with assessments of the Pareto optimality gap, whereas currently available multiobjective evolutionary optimization algorithms which derive Pareto suboptimal designs as well, offer no such assessments. Thus, the approach provides a firm ground to valuate designs resulting from approximate multiobjective optimization computations.

This is the author accepted manuscript. The final version is available from IEEE via the DOI in this record

In multiobjective optimization, objectives are typically conflicting, i.e., they cannot reach their individual optima simultaneously. Identifying the best one among Pareto-optimal solutions is not a simple task for the decision maker (DM), since the Pareto-optimal set can potentially contain a very large number of solutions. To ease this task, it is possible to resort to aggregate objective functions that should take into consideration the DM's preferences and objectives; however, accurately specifying meaningful weights can be a challenge for many practitioners. Moreover, for the same DM's preferences, different criteria give different results. A new post-Pareto analysis methodology, based on sum of ranking differences, is proposed to rank and detect the possible groupings of similar solutions of the Pareto front that match the DM's preferences. This way the proposed technique provides the DM a smaller set of optimal solutions. The proposed method was tested in two practical benchmark problems, in the design of a brushless dc motor and in the optimization of a die press model (TEAM Workshop Problem 25).

Engineering design reconciles design constraints with decision maker preferences. The task of eliciting and encoding decision maker preferences is, however, extremely difficult. A Pareto front representing the locus of the non-dominated designs is therefore, often generated to help a decision maker select the best design. In this paper, we show that this method has a shortcoming. We show that when there is uncertainty in both the decision problem variables and in the decision maker’s preferences, this methodology is inconsistent with multi-attribute utility theory, unless the decision maker trades off attributes or some functions of them linearly. This is a strong restriction. To account for this, we propose a methodology that enables a decision maker to select the best design on a modified Pareto front which is acquired using envelopes of a set of certainty equivalent surfaces. This methodology does not require separability of the multi-attribute utility function into single attribute utilities, nor does it require the decision maker to trade the attributes (or any function of them) linearly. We demonstrate this methodology on a simple optimization problem and in design of a reduction gear.

We present a new two-step approach for automatized a posteriori decision making in multi-objective optimization problems, i.e., selecting a solution from the Pareto front. In the first step, a knee region is determined based on the normalized Euclidean distance from a hyperplane defined by the furthest Pareto solution and the negative unit vector. The size of the knee region depends on the Pareto front’s shape and a design parameter. In the second step, preferences for all objectives formulated by the decision maker, e.g., 50–20–30 for a 3D problem, are translated into a hyperplane which is then used to choose a final solution from the knee region. This way, the decision maker’s preference can be incorporated, while its influence depends on the Pareto front’s shape and a design parameter, at the same time favorizing knee points if they exist. The proposed approach is applied in simulation for the multi-objective model predictive control (MPC) of the two-dimensional rocket car example and the energy management system of a building.

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.