True Optimal Value Research Articles

A dual‐response surface modeling approach for gamma robust design

AbstractRobust parameter design is a methodology that involves experimental design and strategic modeling for the purpose of simultaneously reducing system variability while achieving a target specification for the system mean. Modeling results obtained under unrealistic assumptions about the underlying data distribution may mislead practitioners when it comes to improving system quality. Using classical normal theory modeling techniques for right‐skewed experimental data in dual‐response modeling has some drawbacks, such as obtaining a misleading fit and estimated optimal operating conditions that are potentially far from the true optimal values. Moreover, using inappropriate modeling methods can create a chain of degradation in the production phase and subsequently result in poor system performance. This paper proposes a new dual‐response surface modeling approach when the quality characteristic of a system follows a gamma distribution. The procedure and its advantages are illustrated using the well‐known printing process experiment.

Integrated multiresponse parameter and tolerance design with model parameter uncertainty

AbstractIntegrated parameter and tolerance design is a cost‐effective method to multiresponse quality improvement. However, previous methods usually ignore model parameter uncertainty, dispersion effect, or correlation among responses. This may lead to the obtained optimal solutions far from the true optimal values of parameters and tolerances. To address the problem, a novel integrated parameter and tolerance design method is proposed to solve correlated multiple response problems under consideration of model parameter uncertainty, the location and dispersion effects of the quality loss, and the tolerance costs simultaneously. As there usually exists uncertainty in the quality loss and tolerance costs, multiobjective optimization is adopted to seek for the robust optimal solutions. The effectiveness and robustness of the proposed method are illustrated with a practical example and a random simulation example. The results show that the proposed method provides more reasonable results in quality improvement and cost reduction than those of the existing methods.

Computational Bounds for Photonic Design

Physical design problems, such as photonic inverse design, are typically solved using local optimization methods. These methods often produce what appear to be good or very good designs when compared to classical design methods, but it is not known how far from optimal such designs really are. We address this issue by developing methods for computing a bound on the true optimal value of a physical design problem; physical designs with objectives smaller than our bound are impossible to achieve. Our bound is based on Lagrange duality and exploits the special mathematical structure of these physical design problems. For a multimode 2D Helmholtz resonator, numerical examples show that the bounds we compute are often close to the objective values obtained using local optimization methods, which reveal that the designs are not only good, but in fact nearly optimal. Our computational bounding method also produces, as a byproduct, a reasonable starting point for local optimization methods.

Outcome-Weighted Learning for Personalized Medicine with Multiple Treatment Options.

To achieve personalized medicine, an individualized treatment strategy assigning treatment based on an individual's characteristics that leads to the largest benefit can be considered. Recently, a machine learning approach, O-learning, has been proposed to estimate an optimal individualized treatment rule (ITR), but it is developed to make binary decisions and thus limited to compare two treatments. When many treatment options are available, existing methods need to be adapted by transforming a multiple treatment selection problem into multiple binary treatment selections, for example, via one-vs-one or one-vs-all comparisons. However, combining multiple binary treatment selection rules into a single decision rule requires careful consideration, because it is known in the multicategory learning literature that some approaches may lead to ambiguous decision rules. In this work, we propose a novel and efficient method to generalize outcome-weighted learning for binary treatment to multi-treatment settings. We solve a multiple treatment selection problem via sequential weighted support vector machines. We prove that the resulting ITR is Fisher consistent and obtain the convergence rate of the estimated value function to the true optimal value, i.e., the estimated treatment rule leads to the maximal benefit when the data size goes to infinity. We conduct simulations to demonstrate that the proposed method has superior performance in terms of lower mis-allocation rates and improved expected values. An application to a three-arm randomized trial of major depressive disorder shows that an ITR tailored to individual patient's expectancy of treatment efficacy, their baseline depression severity and other characteristics reduces depressive symptoms more than non-personalized treatment strategies (e.g., treating all patients with combined pharmacotherapy and psychotherapy).

Relaxations and approximations of chance constraints under finite distributions

Optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications. Such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity. In the setting of a finite distribution of the stochastic parameters, an optimization problem with linear chance constraints can be formulated as a mixed integer linear program (MILP). The natural MILP formulation has a weak relaxation bound and is quite difficult to solve. In this paper, we review some recent results on improving the relaxation bounds and constructing approximate solutions for MILP formulations of chance constraints. We also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems. This algorithm uses a relaxation to construct a solution whose (constraint violation) risk level may be larger than the pre-specified threshold, but is within a constant factor of it, and whose objective value is also within a constant factor of the true optimal value. Finally, we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios.

Optimization of multistage artificial aging parameters on Al-Cu alloy mechanical properties

Purpose: This research is aimed to describe heat treatment process by using multistage artificial aging for Al-Cu alloy with Taguchi method in Minitab 16 to optimize the heat treatment parameters. This research conducted due to the applied of aluminium alloy in automotive industrial and aircraft industrial that has good properties for fabrication. Design/methodology/approach: Methodology that use in this paper is experimental design with statistical approach. Three controllable parameters were selected, they were temperature aging, holding time of aging, and the number of stages. The hardness value and impact value after multistage artificial aging were chosen as quality characteristics. The experiment was performed using orthogonal arrays of L9 (33). Findings: The finding that resulted in this research are the most significant parameters that affected hardness and toughness value of Al-Cu alloy against multistage artificial aging. The optimal hardness and toughness for Al-Cu alloy were obtained with heat treatment at temperature 200ºC, holding time for 6 hours, with two stages artificial aging. Research limitations/implications: The limitation that found in this research is even optimal level had been determined, it is unable to determine the true optimal value of each design parameters. Practical implications: This optimization process can be applied in manufacture process in industrial without spend expensive cost and time. Originality/value: According to research result, can be understood that by conducting these experiments, the impact value and the hardness value of Al-Cu alloy increase with multistage artificial aging treatment.

Measurement and optimization of atmospheric plasma sprayed CoMoCrSi coatings parameters on Ti-6Al-4V substrates affecting microstructural and properties using hybrid abductor induction mechanism

Abstract A hybrid artificial intelligence method (AIM) is used to optimize the atmospheric plasma spraying (APS) processing parameters for CoMoCrSi coatings deposited on Ti-6Al-4V substrates. In the proposed approach, the Taguchi design method is used to obtain an initial solution for the optimal set of APS parameters. The true optimal values of the spray distance, chamber pressure, current, argon gas flow rate and hydrogen gas flow rate are then obtained using an artificial neural network (ANN) and genetic algorithm (GA). The coatings deposited using the processing parameters determined using the Taguchi design method alone have a porosity of 8.5%. However, following the ANN/GA optimization procedure, the porosity reduces to just 5.6%. The XRD analysis results show that the as-sprayed coatings contain Cr 3 Si phase as a result of the high-temperature plasma flame. In addition, it is shown that the structure of the Ti phase in the coating transforms from a α-Ti crystal structure (i.e., HCP) to a β-Ti crystal structure (i.e., BCC) during thermal treatment at temperatures to 1200 °C. Finally, it is shown that the hardness of the annealed coatings increases with both an increasing annealing time and an increasing annealing temperature. The higher hardness under higher temperature conditions is attributed to the precipitation of Cr 3 Si phase. Overall, the coatings obtained in the present study using the optimal spray-coating parameters have a low porosity, a high hardness and a good thermal stability at high temperatures.

Truncated Approximate Dynamic Programming with Task-Dependent Terminal Value

We propose a new class of computationally fast algorithms to find close to optimal policy for Markov Decision Processes (MDP) with large finite horizon T.The main idea is that instead of planning until the time horizon T, we plan only up to a truncated horizon H << T and use an estimate of the true optimal value function as the terminal value. Our approach of finding the terminal value function is to learn a mapping from an MDP to its value function by solving many similar MDPs during a training phase and fit a regression estimator. We analyze the method by providing an error propagation theorem that shows the effect of various sources of errors to the quality of the solution. We also empirically validate this approach in a real-world application of designing an energy management system for Hybrid Electric Vehicles with promising results.

Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff---Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.

Precise Time System Efficiency of a Frame Slotted Aloha based Anti-Collision Algorithm in a RFID System

The tag identification efficiency of a reader in a RFID system with Frame Slotted Aloha (FSA) based anti-collision Algorithm (ACA) can be maximized by selecting the optimal frame length with respect to the number of interrogating tags. Conventional analytical models that have been used widely to derive such an optimal frame length are inaccurate because they lack either precise characterization of the timing details of the underlying ACA or do not consider the physical layer capture effect. In this study, one of the most popular conventional analytical models has been extended not only to deliberate the exact timing details of the underlying ACA but also to consider the physical layer capture effect. Rigorous numerical analysis shows that the optimal frame length derived from the new extended model is precise, whereas that of from the conventional model deviates significantly from the true optimal value, particularly when the number of tags is high or the capture probability is low.

Energy-Efficient Design for Downlink OFDMA with Delay-Sensitive Traffic

The tremendous popularity of smart phones and electronic tablets has spurred the explosive growth of high-rate multimedia services and promptly boomed energy consumption in wireless networks. Therefore, energy-efficient design in wireless networks is very important and is attracting more and more attention, just like the conventional spectral-efficient design. In this paper, we study energy-efficient design in downlink orthogonal frequency division multiple access (OFDMA) networks with effective capacity-based delay provisioning for delay-sensitive traffic. By integrating information theory with the concept of effective capacity, we formulate an energy efficiency (EE) optimization problem with statistical delay provisioning, which is a complicated nonconvex combinatorial fractional programming problem. To solve the problem, we first relax it with an upper bound on the original one and then prove and exploit the quasiconcave property of the EE-versus-transmit power curve, which facilitates the optimal algorithm development. Then, we demonstrate that the resultant solution is quite close to the true optimal value when the number of subcarriers is larger than that of the users. We also analyze the tradeoff between EE and delay, the relationship between spectral-efficient and energy-efficient designs, and the impact of system parameters, including circuit power and delay exponents, on the overall performance. Numerical results show that the proposed energy-efficient design scheme greatly improves EE while maintaining the delay requirement.

Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

We observe that in a simple one-dimensional polynomial optimization problem (POP), the `optimal' values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviors of the SDP solvers are given. This result gives a warning to users of the SDP relaxation method for POPs to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly.

Solving the probabilistic reserve selection problem

The Reserve Selection Problem consists in selecting certain sites among a set of potential sites for biodiversity protection. In many models of the literature, the species present and able to survive in each site are supposed to be known. Here, for every potential site and for every species considered, only the probability that the species survives in the site is supposed to be known. The problem to select, under a budgetary constraint, a set of sites which maximizes the expected number of species is known in the literature under the name of probabilistic reserve selection problem. In this article, this problem is studied with species weighting to deal differently with common species and rare species. A spatial constraint is also considered preventing to obtain too fragmented reserve networks. As in Polasky et al. (2000), the problem is formulated by a nonlinear mathematical program in Boolean variables. Camm et al. (2002) developed a mixed-integer linear programming approximation that may be solved with standard integer programming software. The method gives tight approximate solutions but does not allow to tell how far these solutions are from the optimum. In this paper, a slightly different approach is proposed to approximate the problem. The interesting aspect of the approach, which also uses only standard mixed-integer programming software, is that it leads, not only to an approximate solution, but also to an upper limit on the true optimal value. In other words, the method gives an approximate solution with a guarantee on its accuracy. The linear reformulation is based on an upper approximation of the logarithmic function by a piecewise-linear function. The approach is very effective on artificial instances that include up to 400 sites and 300 species. Within an average CPU time of about 12 min, near-optimal solutions are obtained with an average relative error, in comparison to the optimum, of less than 0.2%.

Sample average approximation of stochastic dominance constrained programs

In this paper we study optimization problems with second-order stochastic dominance constraints. This class of problems allows for the modeling of optimization problems where a risk-averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks. Here we deal with the case of multi-variate stochastic dominance under general distributions and nonlinear functions. We introduce the concept of $${\mathcal{C}}$$ -dominance, which generalizes some notions of multi-variate dominance found in the literature. We apply the Sample Average Approximation (SAA) method to this problem, which results in a semi-infinite program, and study asymptotic convergence of optimal values and optimal solutions, as well as the rate of convergence of the feasibility set of the resulting semi-infinite program as the sample size goes to infinity. We develop a finitely convergent method to find an $${\epsilon}$$ -optimal solution of the SAA problem. An important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value. We also show that the bounds are asymptotically tight as the sample size goes to infinity.

A Sample Approximation Approach for Optimization with Probabilistic Constraints

We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with a risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.

Opportunistic scheduling for multiclass services with different QoS constraints in wireless data networks

In this letter, we focus on the problem with the objective to maximize the system performance, while guaranteeing specified QoS constraints for multiple user classes in wireless data networks. First, we propose two opportunistic scheduling algorithms that exploit time-varying channel conditions for the special two-constraint case, and then propose an opportunistic scheduling algorithm for the general case. Simulation results illustrate that the proposed scheduling algorithms guarantee the different constraints, and achieve high-system performance that is close to the true optimal value using a known general-purpose optimization package, lingo.

Recursive Learning Automata Approach to Markov Decision Processes

In this note, we present a sampling algorithm, called recursive automata sampling algorithm (RASA), for control of finite-horizon Markov decision processes (MDPs). By extending in a recursive manner Sastry's learning automata pursuit algorithm designed for solving nonsequential stochastic optimization problems, RASA returns an estimate of both the optimal action from a given state and the corresponding optimal value. Based on the finite-time analysis of the pursuit algorithm by Rajaraman and Sastry, we provide an analysis for the finite-time behavior of RASA. Specifically, for a given initial state, we derive the following probability bounds as a function of the number of samples: 1) a lower bound on the probability that RASA will sample the optimal action and 2) an upper bound on the probability that the deviation between the true optimal value and the RASA estimate exceeds a given error.

An application of Taguchi method of experimental design for new product design and development process

PurposeThis paper aims to illustrate an application of Taguchi method of experimental design (TMED) for the development of a new ignition coil for an automotive vehicle.Design/methodology/approachThe application of TMED for optimisation of manufacturing processes has been widely published in the existing literature. However, the applications of TMED in the design and development of new products are not yet widely reported. This case study presents the results of a designed experiment which utilises a 16‐trial experiment to study 14 design parameters and one interaction. The case study strictly follows a systematic and disciplined methodology outlined in the paper.FindingsThe optimal settings of the critical design parameters are determined. The optimal settings have resulted in increased customer satisfaction, improved market share and low defect rate in the hands of customers.Research limitations/implicationsAlthough the optimal levels are determined from one large experiment, it was unable to determine the true optimal values of each design parameter.Practical implicationsManufacturers may use TMED to optimise processes (either design or manufacturing) without expensive and time‐consuming experimentation. This case study demonstrates the true power of a well planned and designed experiment over the traditional varying one‐factor‐at‐a‐time approach to experimentation which is rather unreliable, not cost‐effective and may lead to false optimal conditions.Originality/valueThe paper provides an excellent resource for those people who are involved in the design optimisation of a new product.

Robust Solution of Nonconvex Global Optimization Problems

The concept of ?-approximate optimal solution as widely used in nonconvex global optimization is not quite adequate, because such a point may correspond to an objective function value far from the true optimal value, while being infeasible. We introduce a concept of essential ?-optimal solution, which gives a more appropriate approximate optimal solution, while being stable under small perturbations of the constraints. A general method for finding an essential ?-optimal solution in finitely many steps is proposed which can be applied to d.c. programming and monotonic optimization.

Por que otimização multivariada?

Normalmente, estudos que objetivam a otimização de variáveis experimentais são realizados através de procedimentos que avaliam o efeito de uma variável por vez, o que, de maneira geral, impede o estabelecimento de ótimos verdadeiros, em razão da freqüência com que as variáveis se apresentam altamente correlacionadas. Neste trabalho, algumas aplicações práticas são apresentadas, visando demonstrar a conveniência dos procedimentos multivariados de otimização, principalmente sistemas de planejamento fatorial de experimentos. A partir de três exemplos práticos relacionados com a otimização de processos de degradação, discutem-se aspectos relacionados com o desenho experimental e a interpretação de resultados.