Exact And Representative Algorithms For Multi Objective Optimization

In most real-life problems, the decision alternatives are evaluated with multiple conflicting criteria. The entire set of non-dominated solutions for practical problems is impossible to obtain with reasonable computational effort. Decision maker generally needs only a representative set of solutions from the actual Pareto front. First algorithm we present is for efficiently generating a well dispersed non-dominated solution set representative of the Pareto front which can be used for general multi objective optimization problem. The algorithm first partitions the criteria space into grids to generate reference points and then searches for non-dominated solutions in each grid. This grid-based search utilizes achievement scalarization function and guarantees Pareto optimality. The results of our experimental results demonstrate that the proposed method is very competitive with other algorithms in literature when representativeness quality is considered; and advantageous from the computational efficiency point of view. Although generating the whole Pareto front does not seem very practical for many real life cases, sometimes it is required for verification purposes or where DM wants to run his decision making structures on the full set of Pareto solutions. For this purpose we present another novel algorithm. This algorithm attempts to adapt the standard branch and bound approach to the multi objective context by proposing to branch on solution points on objective space. This algorithm is proposed for multi objective integer optimization type of problems. Various properties of branch and bound concept has been investigated and explained within the multi objective optimization context such as fathoming, node selection, heuristics, as well as some multi objective optimization specific concepts like filtering, non-domination probability, running in parallel. Potential of this approach for being used both as a full Pareto generation or an approximation approach has been shown with experimental studies

Biobjective Optimization over the Efficient Set Methodology for Pareto Set Reduction in Multiobjective Decision Making: Theory and Application

A large number of available solutions to choose from poses a significant challenge for multiple criteria decision making. This research develops a methodology that reduces the set of efficient solutions under consideration. This dissertation is composed of three major parts: (i) the formalization of a theoretical framework; (ii) the development of a solution approach; and (iii) a case study application of the methodology. In the first part, the problem is posed as a multiobjective optimization over the efficient set and considers secondary robustness criteria when the exact values of decision variables are subjected to uncertainties during implementation. The contributions are centered at the modeling of uncertainty directly affecting decision variables, the use of robustness to provide additional trade-off analysis, the study of theoretical bounds on the measures of robustness, and properties to ensure that fewer solutions are identified. In the second part, the problem is reformulated as a biobjective mixed binary program and the secondary criteria are generalized to any convenient linear functions. A solution approach is devised in which an auxiliary mixed binary program searches for unsupported Pareto outcomes and a novel linear programming filtering excludes any dominated solutions in the space of the secondary criteria. Experiments show that the algorithm tends to run faster than existing approaches for mixed binary programs. The algorithm enables dealing with continuous Pareto sets, avoiding discretization procedures common to the related literature. In the last part, the methodology is applied in a case study regarding the electricity generation capacity expansion problem in Texas. While water and energy are interconnected issues, to the best of our knowledge, this is the first study to consider both water and cost objectives. Experiments illustrate how the methodology can facilitate decision making and be used to answer strategic questions pertaining to the trade-off among different generation technologies, power plant locations, and the effect of uncertainty. A simulation shows that robust solutions tend to maintain feasibility and stability of objective values when power plant design capacity values are perturbed

LEO: Learning Efficient Orderings for Multiobjective Binary Decision Diagrams

Approaches based on Binary decision diagrams (BDDs) have recently achieved state-of-the-art results for multiobjective integer programming problems. The variable ordering used in constructing BDDs can have a significant impact on their size and on the quality of bounds derived from relaxed or restricted BDDs for single-objective optimization problems. We first showcase a similar impact of variable ordering on the Pareto frontier (PF) enumeration time for the multiobjective knapsack problem, suggesting the need for deriving variable ordering methods that improve the scalability of the multiobjective BDD approach. To that end, we derive a novel parameter configuration space based on variable scoring functions which are linear in a small set of interpretable and easy-to-compute variable features. We show how the configuration space can be efficiently explored using black-box optimization, circumventing the curse of dimensionality (in the number of variables and objectives), and finding good orderings that reduce the PF enumeration time. However, black-box optimization approaches incur a computational overhead that outweighs the reduction in time due to good variable ordering. To alleviate this issue, we propose LEO, a supervised learning approach for finding efficient variable orderings that reduce the enumeration time. Experiments on benchmark sets from the knapsack problem with 3-7 objectives and up to 80 variables show that LEO is ~30-300% and ~10-200% faster at PF enumeration than common ordering strategies and algorithm configuration. Our code and instances are available at https://github.com/khalil-research/leo

Optimization Models for Sustainable Design and Management of Biopower Supply Chains

This dissertation presents optimization models to aid with the sustainable design and management of biopower (biomass cofiring) supply chains. We address three main challenges associated with today’s biopower projects: i) high cost of biomass collection, storage and delivery, ii) inefficiency of the mechanisms used to incentivize biomass usage for generating electricity, and iii) lack of clear understanding about the trade-offs between economic and environmental impacts of biopower supply chains. In order to address the high cost of delivering biomass, we present a novel mixed integer nonlinear program that integrates production and transportation decisions at power plants. Proposed model captures the loss in process efficiencies from using biomass, in-vestment and operational costs associated with cofiring, and savings due to production tax credit (PTC), a major governmental incentive to support biopower. We develop a La-grangian relaxation approach to provide upper bounds, and two linear approximations to provide lower bounds for the problem. An important finding is that the one-size-fits-all approach of PTC is not effective in motivating plants to utilize biomass and there is a need for sophisticated incentive schemes. In order to address the second issue, we propose alter-natives for the existing PTC incentive. The proposed flexible alternatives are functions of plant capacity and biomass cofiring ratio. We use a resource allocation framework to model and analyze the profit-earning potentials and fairness of the proposed incentive schemes. Finally, in order to address the last challenge, we propose a stochastic biobjective optimiza-tion model to analyze the economic and environmental impacts of biopower supply chains. The economic objective function maximizes the potential profits in the supply chain and the environmental objective function minimizes the life cycle greenhouse gasses (GHG). We use a life cycle assessment (LCA) approach to derive the emission factors for this objective function. We capture uncertainties of biomass quality and supply via the use of chance constraints. The results of this dissertation work are useful for electric utility companies and policy makers. Utility companies can use the proposed models to identify ways to improve biopower production, have better environmental performance, and make use of the existing incentives. Policy makers would gain insights on designing incentive schemes for a more efficient utilization of biomass and a fairer distribution of tax-payers money

Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance