Multi-objective robust optimization using a sensitivity region concept

In multi-objective design optimization, it is quite desirable to obtain solutions that are multi-objectively optimum and insensitive to uncontrollable (noisy) parameter variations. We call such solutions robust Pareto solutions. In this paper we present a method to measure the multi-objective sensitivity of a design alternative, and an approach to use such a measure to obtain multi-objectively robust Pareto optimum solutions. Our sensitivity measure does not require a presumed probability distribution of uncontrollable parameters and does not utilize gradient information; therefore, it is applicable to multi-objective optimization problems that have non-differentiable and/or discontinuous objective functions, and also to problems with large parameter variations. As a demonstration, we apply our robust optimization method to an engineering example, the design of a vibrating platform. We show that the solutions obtained for this example are indeed robust.     

Managing Water Supply and Demand to Achieve Economic and Environmental Objectives: Application of Mathematical Programming and ANFIS Models

The integrated management of water supply and demand has been considered by many policymakers; due to its complexity the decision makers have faced many challenges so far. In this study, we proposed an efficient framework for managing water supply and demand in line with the economic and environmental objectives of the basin. To design this framework, a combination of ANFIS and multi-objective augmented ε-constraint programming models and TOPSIS were used. First, using hydrological data from 2001 to 2017, the rate of water release from the dam reservoir was estimated with the ANFIS model; afterwards, its allocation to agricultural areas was performed by combining multi-objective augmented ε-constraint models and TOPSIS. To prove the reliability of the proposed model, the southern Karkheh basin in Khuzestan province, Iran, was considered as a case study. The results have showed that this model is able to reduce irrigation water consumption and to improve its economic productivity in the basin.

     

Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling

Applications of multi-objective genetic algorithms (MOGAs) in engineering optimization problems often require numerous function calls. One way to reduce the number of function calls is to use an approximation in lieu of function calls. An approximation involves two steps: design of experiments (DOE) and metamodeling. This paper presents a new approach where both DOE and metamodeling are integrated with a MOGA. In particular, the DOE method reduces the number of generations in a MOGA, while the metamodeling reduces the number of function calls in each generation. In the present approach, the DOE locates a subset of design points that is estimated to better sample the design space, while the metamodeling assists in estimating the fitness of design points. Several numerical and engineering examples are used to demonstrate the applicability of this new approach. The results from these examples show that the proposed improved approach requires significantly fewer function calls and obtains similar solutions compared to a conventional MOGA and a recently developed metamodeling-assisted MOGA.     

On improving normal boundary intersection method for generation of Pareto frontier

Gradient-based methods, including Normal Boundary Intersection (NBI), for solving multi-objective optimization problems require solving at least one optimization problem for each solution point. These methods can be computationally expensive with an increase in the number of variables and/or constraints of the optimization problem. This paper provides a modification to the original NBI algorithm so that continuous Pareto frontiers are obtained “in one go,” i.e., by solving only a single optimization problem. Discontinuous Pareto frontiers require solving a significantly fewer number of optimization problems than the original NBI algorithm. In the proposed method, the optimization problem is solved using a quasi-Newton method whose history of iterates is used to obtain points on the Pareto frontier. The proposed and the original NBI methods have been applied to a collection of 16 test problems, including a welded beam design and a heat exchanger design problem. The results show that the proposed approach significantly reduces the number of function calls when compared to the original NBI algorithm.     

Optimizing thermal design of data center cabinets with a new multi-objective genetic algorithm

There is an ever increasing need to use optimization methods for thermal design of data centers and the hardware populating them. Airflow simulations of cabinets and data centers are computationally intensive and this problem is exacerbated when the simulation model is integrated with a design optimization method. Generally speaking, thermal design of data center hardware can be posed as a constrained multi-objective optimization problem. A popular approach for solving this kind of problem is to use Multi-Objective Genetic Algorithms (MOGAs). However, the large number of simulation evaluations needed for MOGAs has been preventing their applications to realistic engineering design problems. In this paper, details of a substantially more efficient MOGA are formulated and demonstrated through a thermal analysis simulation model of a data center cabinet. First, a reduced-order model of the cabinet problem is constructed using the Proper Orthogonal Decomposition (POD). The POD model is then used to form the objective and constraint functions of an optimization model. Next, this optimization model is integrated with the new MOGA. The new MOGA uses a “kriging” guided operation in addition to conventional genetic algorithm operations to search the design space for global optimal design solutions. This approach for optimal design is essential to handle complex multi-objective situations, where the optimal solutions may be non-obvious from simple analyses or intuition. It is shown that in optimizing the data center cabinet problem, the new MOGA outperforms a conventional MOGA by estimating the Pareto front using 50% fewer simulation calls, which makes its use very promising for complex thermal design problems. Recommended by: Monem Beitelmal     

Approximation of multiresponse deterministic engineering simulations: a dependent metamodeling approach

A new approach to metamodeling is introduced whereby a sequential technique is used to construct and simultaneously update mutually dependent metamodels for multiresponse, high-fidelity deterministic simulations. Unlike conventional approaches which produce a single metamodel for each scalar response independently, the present method uses the correlation among different simulation responses in the construction of the metamodel. These dependent metamodels are solved as a system of equations to estimate all individual responses simultaneously. Since several responses contribute to the construction of each individual metamodel, more information from the computed responses is used, thus improving the accuracy of the obtained metamodels. Examples are used to explore the relative performance of the proposed approach and show that the new approach outperforms conventional metamodeling approaches in terms of approximation accuracy. The new method should be particularly useful in problems that require very computationally intensive simulations.     

A modified Benders decomposition method for efficient robust optimization under interval uncertainty

The goal of robust optimization problems is to find an optimal solution that is minimally sensitive to uncertain factors. Uncertain factors can include inputs to the problem such as parameters, decision variables, or both. Given any combination of possible uncertain factors, a solution is said to be robust if it is feasible and the variation in its objective function value is acceptable within a given user-specified range. Previous approaches for general nonlinear robust optimization problems under interval uncertainty involve nested optimization and are not computationally tractable. The overall objective in this paper is to develop an efficient robust optimization method that is scalable and does not contain nested optimization. The proposed method is applied to a variety of numerical and engineering examples to test its applicability. Current results show that the approach is able to numerically obtain a locally optimal robust solution to problems with quasi-convex constraints (≤ type) and an approximate locally optimal robust solution to general nonlinear optimization problems.     

New Approximation Assisted Multi-objective collaborative Robust Optimization (new AA-McRO) under interval uncertainty

Existing collaborative optimization techniques with multiple coupled subsystems are predominantly focused on single-objective deterministic optimization. However, many engineering optimization problems have system and subsystems that can each be multi-objective, constrained and with uncertainty. The literature reports on a few deterministic Multi-objective Multi-Disciplinary Optimization (MMDO) techniques. However, these techniques in general require a large number of function calls and their computational cost can be exacerbated when uncertainty is present. In this paper, a new Approximation-Assisted Multi-objective collaborative Robust Optimization (New AA-McRO) under interval uncertainty is presented. This new AA-McRO approach uses a single-objective optimization problem to coordinate all system and subsystem multi-objective optimization problems in a Collaborative Optimization (CO) framework. The approach converts the consistency constraints of CO into penalty terms which are integrated into the system and subsystem objective functions. The new AA-McRO is able to explore the design space better and obtain optimum design solutions more efficiently. Also, the new AA-McRO obtains an estimate of Pareto optimum solutions for MMDO problems whose system-level objective and constraint functions are relatively insensitive (or robust) to input uncertainties. Another characteristic of the new AA-McRO is the use of online approximation for objective and constraint functions to perform system robustness evaluation and subsystem-level optimization. Based on the results obtained from a numerical and an engineering example, it is concluded that the new AA-McRO performs better than previously reported MMDO methods.     

A prescriptive production-distribution approach for decision makingin new product design

The production problem in product design consists of determining technically feasible options and implementing collective decision making to decide which alternatives to produce. A problem with considering collective decision making in the production problem is that it may not be possible to define, a priori, a group utility function, due to the difficulty of making interpersonal comparisons of the members' preferences. Constructing a group utility function and solving the production problem may lead to unacceptable alternatives for some of the members. The distribution problem involves consideration of individual preferences in satisfying group decision-making situations. If production and distribution of the products are considered simultaneously, the preferences of the members are taken into account, explicitly leading to acceptable solutions. This approach also allows for explicit consideration of a company business strategy in determining which products to develop. A production-distribution approach is outlined and demonstrated for the design of a power electronics module product with a group of decision makers consisting of three customers and a manufacturer. In addition to explicit consideration of the customers' preference, some business strategies are also considered, leading to product alternatives acceptable to the customers and manufacturer     

A minimax reduction method for multi-objective decomposition-based design optimization

In this paper, a method for the design optimization of multi-objective engineering problems (or systems) which are decomposedhierarchically ornonhierarchically into several subproblems is presented. The method is based on a minimax formulation of the overall problem and the application of some reduction measures to reduce the number of variables in this problem. Two well-known examples are selected from the literature and included to demonstrate the proposed solution steps.