Compromise Solutions for Robust Combinatorial Optimization with Variable-Sized Uncertainty

In classic robust optimization, it is assumed that a set of possible parameter realizations, the uncertainty set, is modeled in a previous step and part of the input. As recent work has shown, finding the most suitable uncertainty set is in itself already a difficult task. We consider robust problems where the uncertainty set is not completely defined. Only the shape is known, but not its size. Such a setting is known as variable-sized uncertainty. In this work we present an approach how to find a single robust solution, that performs well on average over all possible uncertainty set sizes. We demonstrate that this approach can be solved efficiently for min-max robust optimization, but is more involved in the case of min-max regret, where positive and negative complexity results for the selection problem, the minimum spanning tree problem, and the shortest path problem are provided. We introduce an iterative solution procedure, and evaluate its performance in an experimental comparison

Multiobjective optimization for multiproduct batch plant design under economic and environmental considerations

This work deals with the multicriteria cost–environment design of multiproduct batch plants, where the design variables are the size of the equipment items as well as the operating conditions. The case study is a multiproduct batch plant for the production of four recombinant proteins. Given the important combinatorial aspect of the problem, the approach used consists in coupling a stochastic algorithm, indeed a genetic algorithm (GA) with a discrete-event simulator (DES). Another incentive to use this kind of optimization method is that, there is no easy way of calculating derivatives of the objective functions, which then discards gradient optimization methods. To take into account the conflicting situations that may be encountered at the earliest stage of batch plant design, i.e. compromise situations between cost and environmental consideration, a multiobjective genetic algorithm (MOGA) was developed with a Pareto optimal ranking method. The results show how the methodology can be used to find a range of trade-off solutions for optimizing batch plant design

Variable-sized uncertainty and inverse problems in robust optimization

In robust optimization, the general aim is to find a solution that performs well over a set of possible parameter outcomes, the so-called uncertainty set. In this paper, we assume that the uncertainty size is not fixed, and instead aim at finding a set of robust solutions that covers all possible uncertainty set outcomes. We refer to these problems as robust optimization with variable-sized uncertainty. We discuss how to construct smallest possible sets of min–max robust solutions and give bounds on their size. A special case of this perspective is to analyze for which uncertainty sets a nominal solution ceases to be a robust solution, which amounts to an inverse robust optimization problem. We consider this problem with a min–max regret objective and present mixed-integer linear programming formulations that can be applied to construct suitable uncertainty sets. Results on both variable-sized uncertainty and inverse problems are further supported with experimental data

A Combinatorial Optimization Approach to the Selection of Statistical Units

In the case of some large statistical surveys, the set of units that will constitute the scope of the survey must be selected. We focus on the real case of a Census of Agriculture, where the units are farms. Surveying each unit has a cost and brings a different portion of the whole information. In this case, one wants to determine a subset of units producing the minimum total cost for being surveyed and representing at least a certain portion of the total information. Uncertainty aspects also occur, because the portion of information corresponding to each unit is not perfectly known before surveying it. The proposed approach is based on combinatorial optimization, and the arising decision problems are modeled as multidimensional binary knapsack problems. Experimental results show the effectiveness of the proposed approach

Mathematical Optimization of the Tactical Allocation of Machining Resources in Aerospace Industry

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers (which includes, measures of quality and production lead-times) and to maintain control of the tied-up working capital. We introduce a new multi-item, multi-level capacitated planning model with a medium-to-long term planning horizon. The model can be used by most companies having functional workshops where costly and/or time- and resource demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimizes the maximum excess resource loading above a given loading threshold, while incurring as low qualification costs as possible. In Paper I (Bi-objective optimization of the tactical allocation of jobtypes to machines), we propose a new bi-objective mathematical optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. Another contribution is a modified version of the bi-directional $\epsilon$ -constraint method especially tailored for our problem. We perform numerical tests on industrial test cases generated for our class of problem which indicates computational superiority of our method over conventional solution approaches. In Paper II (Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs), we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. We also suggest a solution approach for identifying all the relevant robust efficient (RE) solutions. Our proposed approach is significantly faster than an existing approach for robust bi-objective optimization problems

Trust-Based Mechanisms for Robust and Efficient Task Allocation in the Presence of Execution Uncertainty

Vickrey-Clarke-Groves (VCG) mechanisms are often used to allocate tasks to selfish and rational agents. VCG mechanisms are incentive-compatible, direct mechanisms that are efficient (i.e. maximise social utility) and individually rational (i.e. agents prefer to join rather than opt out). However, an important assumption of these mechanisms is that the agents will always successfully complete their allocated tasks. Clearly, this assumption is unrealistic in many real-world applications where agents can, and often do, fail in their endeavours. Moreover, whether an agent is deemed to have failed may be perceived differently by different agents. Such subjective perceptions about an agent’s probability of succeeding at a given task are often captured and reasoned about using the notion of trust. Given this background, in this paper, we investigate the design of novel mechanisms that take into account the trust between agents when allocating tasks. Specifically, we develop a new class of mechanisms, called trust-based mechanisms, that can take into account multiple subjective measures of the probability of an agent succeeding at a given task and produce allocations that maximise social utility, whilst ensuring that no agent obtains a negative utility. We then show that such mechanisms pose a challenging new combinatorial optimisation problem (that is NP-complete), devise a novel representation for solving the problem, and develop an effective integer programming solution (that can solve instances with about 2×105 possible allocations in 40 seconds).

Stability Analysis in Multicriteria Discrete Portfolio Optimization.

Almost every problem of design, planning and management in the technical and organizational systems has several conflicting goals or interests. Nowadays, multicriteria decision models represent a rapidly developing area of operation research. While solving practical optimization problems, it is necessary to take into account various kinds of uncertainty due to lack of data, inadequacy of mathematical models to real-time processes, calculation errors, etc. In practice, this uncertainty usually leads to undesirable outcomes where the solutions are very sensitive to any changes in the input parameters. An example is the investment managing. Stability analysis of multicriteria discrete optimization problems investigates how the found solutions behave in response to changes in the initial data (input parameters). This thesis is devoted to the stability analysis in the problem of selecting investment project portfolios, which are optimized by considering different types of risk and efficiency of the investment projects. The stability analysis is carried out in two approaches: qualitative and quantitative. The qualitative approach describes the behavior of solutions in conditions with small perturbations in the initial data. The stability of solutions is defined in terms of existence a neighborhood in the initial data space. Any perturbed problem from this neighborhood has stability with respect to the set of efficient solutions of the initial problem. The other approach in the stability analysis studies quantitative measures such as stability radius. This approach gives information about the limits of perturbations in the input parameters, which do not lead to changes in the set of efficient solutions. In present thesis several results were obtained including attainable bounds for the stability radii of Pareto optimal and lexicographically optimal portfolios of the investment problem with Savage's, Wald's criteria and criteria of extreme optimism. In addition, special classes of the problem when the stability radii are expressed by the formulae were indicated. Investigations were completed using different combinations of Chebyshev's, Manhattan and Hölder's metrics, which allowed monitoring input parameters perturbations differently.Siirretty Doriast

Mixed uncertainty sets for robust combinatorial optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a “true” model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-of-sample performance, the combination of multiple sets can give better results than each set on its own

Matheuristics for robust optimization: application to real-world problems

In the field of optimization, the perspective that the problem data are subject to uncertainty is gaining more and more interest. The uncertainty in an optimization problem represents the measurement errors during the phase of collecting data, or unforeseen changes in the environment while implementing the optimal solution in practice. When the uncertainty is ignored, an optimal solution according to the mathematical model can turn out to be far from optimal, or even infeasible in reality. Robust optimization is an umbrella term for mathematical modelling methodologies focused on finding solutions that are reliable against the data perturbations caused by the uncertainty. Among the relatively more recent robust optimization methodologies, an important concept studied is the degree of conservativeness, which can be explained as the amount of targeted reliability against the uncertainty while looking for a solution. Because the reliability and solution cost usually end up being conflicting objectives, it is important for the decision maker to be able to configure the conservativeness degree, so that the desired balance between the cost and reliability can be obtained, and the most practical solution can be found for the problem at hand. The robust optimization methodologies are typically proposed within the framework of mathematical programming (i.e. linear programming, integer programming). Thanks to the nature of mathematical programming, these methodologies can find the exact optimum, according to the various solution evaluation perspectives they have. However, dependence on mathematical programming might also mean that such methodologies will require too much memory from the computer, and also too much execution time, when large-scale optimization problems are considered. A common strategy to avoid the big memory and execution time requirements of mathematical programming is to use metaheuristic optimization algorithms for solving large problem instances.In this research, we propose an approach for solving medium-to-large-sized robust optimization problem instances. The methodology we propose is a matheuristic (i.e. a hybridization of mathematical programming and metaheuristic). In the matheuristic approach we propose, the mathematical programming part handles the uncertainty, and the metaheuristic part handles the exploration of the solution space. Since the exploration of the solution space is entrusted onto the metaheuristic search, we can obtain practical near-optimal solutions while avoiding the big memory and time requirements that might be brought by pure mathematical programming methods. The mathematical programming part is used for making the metaheuristic favor the solutions which have more protections against the uncertainty. Another important characteristic of the methodology we propose is concurrency with information exchange: we concurrently execute multiple processes of the matheuristic algorithm, each process taking the uncertainty into account with a different degree of conservativeness. During the execution, these processes exchange their best solutions. So, if a process is stuck on a bad solution, it can realize that there is a better solution available thanks to the information exchange, and it can get unstuck. In the end, the solutions of these processes are collected into a solution pool. This solution pool provides the decision maker with alternative solutions with different costs and conservativeness degrees. Having a solution pool available at the end, the decision maker can make the most practical choice according to the problem at hand. In this thesis, we first discuss our studies in the field of robust optimization: a heuristic approach for solving a minimum power multicasting problem in wireless actuator networks under actuator distance uncertainty, and a linear programming approach for solving an aggregate blending problem in the construction industry, where the amounts of components found in aggregates are subject to uncertainty. These studies demonstrate the usage of mathematical programming for handling the uncertainty. We then discuss our studies in the field of matheuristics: a matheuristic approach for solving a large-scale energy management problem, and then a matheuristic approach for solving large instances of minimum power multicasting problem. In these studies, the usage of metaheuristics for handling the large problem instances is emphasized. In our study of solving minimum power multicasting problem, we also incorporate the mechanism of information exchange between different solvers. Later, we discuss the main matheuristic approach that we propose in this thesis. We first apply our matheuristic approach on a well-known combinatorial optimization problem: capacitated vehicle routing problem, by using an ant colony optimization as the metaheuristic part. Finally, we discuss the generality of the methodology that we propose: we suggest that it can be used as a general framework on various combinatorial optimization problems, by choosing the most appropriate metaheuristic algorithm according to the nature of the problem