Optimal computing budget allocation for complete ranking with input uncertainty

In this study, we consider ranking and selection problems where the simulation model is subject to input uncertainty. Under the input uncertainty, we compare system designs based on their worst-case performance, and seek to maximize the probability of selecting the design with the best performance under the worst-case scenario. By approximating the probability of correct selection (PCS), we develop an asymptotically (as the simulation budget goes to infinity) optimal solution of the resulting problem. An efficient selection procedure is designed within the optimal computing budget allocation (OCBA) framework. Numerical tests show the high efficiency of the proposed method.

This chapter reviews the problem of selecting the best of a finite set of alternatives, where best is defined with respect to the highest mean performance, and where the performance is uncertain but may be estimated with simulation. This problem has been explored from several perspectives, including statistical ranking and selection, multiple comparisons, and stochastic optimization. Approaches taken in the literature include frequentist statistics, Bayesian statistics, related heuristics, and asymptotic convergence in probability. This chapter presents algorithms that are derived from Bayesian and related conceptual frameworks to provide empirically effective performance for the ranking and selection problem. In particular, we motivate the optimal computing budget allocation (OCBA) algorithm and expected value of information (EVI) approaches, give example algorithms, and provide pointers to the literature for detailed derivations and extensions of these approaches.KeywordsAllocation RuleCorrect SelectionMultiple AlternativeLinear LossEqual AllocationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Convergence rate analysis for optimal computing budget allocation algorithms

We consider the problem of ranking the topmdesigns out ofkalternatives. Using the optimal computing budget allocation framework, we formulate this problem as that of maximizing the probability of correctly ranking the topmdesigns subject to the constraint of a fixed limited simulation budget. We derive the convergence rate of the false ranking probability based on the large deviation theory. The asymptotically optimal allocation rule is obtained by maximizing this convergence rate function. To implement the simulation budget allocation rule, we suggest a heuristic sequential algorithm. Numerical experiments are conducted to compare the effectiveness of the proposed simulation budget allocation rule. The numerical results indicate that the proposed asymptotically optimal allocation rule performs the best comparing with other allocation rules.

We consider a ranking and selection problem with independent normal observations, and we analyze the asymptotic sampling rates of expected improvement (EI) methods in this setting. Such methods often perform well in practice, but a tractable analysis of their convergence rates is difficult because of the nonlinearity and nonconvexity of the EI calculations. We present new results indicating that, for known sampling noise, variants of EI produce asymptotic simulation allocations that are essentially identical to those chosen by the optimal computing budget allocation (OCBA) methodology, which is known to yield near-optimal asymptotic performance in ranking and selection. This is the first general equivalence result between EI and OCBA, and it provides insight into the good practical performance of EI. We also derive the limiting allocation for EI under unknown sampling variance.

Particle Swarm Optimization (PSO) is a population-based algorithm designed to find good solutions to optimization problems. Its characteristics have encouraged its adoption to tackle a variety of problems in different fields. However, when such problems are subject to noise, the performance of PSO suffers an immediate deterioration which demands the incorporation of noise handling mechanisms. One such mechanism comprises resampling methods, which re-evaluate the solutions multiple times in order to estimate their true objective values. The state-of-the-art integration with which the best results have been obtained utilizes the resampling method named Optimal Computing Budget Allocation (OCBA). This resampling method starts by estimating the objective values of all the solutions via Equal Resampling (ER), and then sequentially allocating further re-evaluations to the estimated best solutions. However, after having a first estimate via ER, we question the importance of the additional efforts to correctly select the true best solutions when a good-enough and accurate one can be selected. In this paper, we propose a new PSO algorithm based on ER in which the additional evaluations are allocated at once to the estimated best solutions, thus skipping the complexity of using OCBA. Experiments on 20 large-scale benchmark functions subject to different levels of noise show that the proposed algorithm produces similar results to PSO with OCBA in most cases.

Markov decision processes (MDPs) are widely used to model stochastic systems to deduce optimal decision-making policies. As the transition probabilities are usually unknown in MDPs, simulation-based policy improvement (SBPI) using a base policy to derive optimal policies when the state transition probabilities are unknown is suggested. However, estimating the Q-value of each action to determine the best action in each state requires many simulations, which results in efficiency problems for SBPI. In this study, we propose a method to improve the overall efficiency of SBPI using optimal computing budget allocation (OCBA) based on accumulated samples. Previous works have mainly focused on improving SBPI efficiency for a single state and without using the previous simulation samples. In contrast, the proposed method improves the overall efficiency until an optimal policy can be found in consideration of the state traversal property of the SBPI. The proposed method accumulates simulation samples across states to estimate the unknown transition probabilities. These probabilities are then used to estimate the mean and variance of the Q-value for each action, which allows the OCBA to allocate the simulation budget efficiently to find the best action in each state. As the SBPI traverses the state, the accumulated samples allow appropriate allocation of OCBA; thus, the optimal policy can be obtained with a lower budget. The experimental results demonstrate the improved efficiency of the proposed method compared to previous works.

A widely acknowledged challenge in ranking and selection is how to allocate the simulation budget such that the probability of correction selection (PCS) is maximized. However, there is yet another challenge: when the input distributions are estimated using finite real-world data, simulation output is subject to input uncertainty and we may fail to identify the best system even using infinite simulation budget. We propose a new formulation that captures the tradeoff between collecting input data and running simulations. To solve the formulation, we develop an algorithm for two-stage allocation of finite budget. We use numerical experiment to demonstrate the performance of our algorithm.

Stochastic point location (SPL) is to search for a target point on the line in stochastic environment. An SPL solver can be described as a Learning Machine (LM) attempting to locate a target point on a line. By using the prompts from stochastic environment, possibly erroneous, the LM moves along the line yielding updated estimates to approximate the target point. This paper proposes an SPL algorithm based on Optimal Computing Budget Allocation (OCBA), named as SPL-OCBA, which employs OCBA and the historical sample information to guide to the location of a target point in stationary and stochastic environment. The proposed algorithm partitions or combines the subintervals of the target line adaptively. Then, OCBA considers such subintervals as its designs and allocates the sample budget for them based on the historical data, thereby resulting in a new method. Extensive experiments show that the newly proposed algorithm is significantly more efficient than the existing ones.

We consider the optimal computing budget allocation (OCBA) problem where the simulated designs are correlated. The exact optimal allocation is presented for two designs, and an approximation proposed for more than two designs. Numerical experiments for several examples compare the approximation with some other allocation procedures. Results on convergence rates and the non-Gaussian setting are also discussed.

In this paper, a simulation-based optimization approach, named NHOCBA, for a typical resource allocation problem is presented. The hybrid algorithm based on neighborhood algorithm is applied to explore toward optimal direction in design space. For increasing efficiency, an optimal computing budget allocation (OCBA) is adopted to compute the optimal number of replications and to provide reliable evaluation of variance. In addition, we deal with the resource allocation problem which exist multiple global optima in design space. Therefore, a trim procedure which prevents to allocate extra numbers of replication to local optima has been proposed to enhance efficiency. Then, we use confidence interval at the end of algorithm procedure to find an optimal set instead of an optimal solution from design space. Finally, we compare the NHOCBA with different algorithms by experimentation study which shows that NHOCBA approach can perform better than the other algorithms under certain conditions.

Evolutionary algorithm assisted by surrogate model in the framework of ordinal optimization and optimal computing budget allocation

In a Ranking and Selection problem, the objective of allocation is vital in deriving the rule. However, most of these objectives do not have a closed form. Due to the high cost of a direct approximation, several cheap but biased substitutes were applied to simplify the problem. These simplifications however could potentially affect the optimality of efficiency and therefore influence its finite performance. Fortunately, due to the increasing accessibility of parallel hardware (e.g. GPU), a direct approximation is becoming more tractable. Thus, we want to test the performance of an allocation rule based on an unbiased and direct approximation, expecting an acceleration on the performance. In this paper, we target on one of the famous objectives, the Probability of Correct Selection (PCS). Numerical experiments were done, showing a considerable improvement in finite performance of our algorithm comparing to a traditional one.

Real-time planning and scheduling in a shop floor are not easy to accomplish due to the concurrent flow of various parts as well as sharing of different types of resources. Multi-pass scheduling is a well-known method for solving the aforementioned problem. Its success depends largely on selecting the best decision-making rule fast and effectively. Although many efforts have been made in the past, a way to minimize the computational load of rule evaluation and selection has yet to appear. The objective of the paper is to apply a nested partitioning (NP) method and an optimal computing budget allocation (OCBA) method to reduce the computational load without the loss of the performance of multi-pass scheduling. The experimental design and analysis was performed to validate that NP and OCBA can be successfully applied to multi-pass scheduling in order to enhance the performance of multi-pass scheduling.

A hybrid differential evolution(DE) algorithm,namely OHTDE,is proposed to solve the stochastic flow shop scheduling problem(FSSP) with limited buffers between consecutive machines.Two different criteria are considered.The first is to minimize a general earliness/tardiness cost function and the second is to minimize the total completion time.The proposed algorithm is a hybrid of DE and two techniques:the optimal computing budget allocation(OCBA) technique and hypothesis test(HT) .DE is used to execute both global and local search.OCBA is utilized to reasonably allocate a limited sampling budget for every solution,assigning more for better individuals to improve the confidence level of getting good solutions under noise environment.HT is adopted to perform a statistical comparison on solutions performances so that the repeated search on the similar region of the solution space can be avoided to some extent.Moreover,a special crossover,which is performed on some good solutions identified by OCBA and HT,is designed to enhance the capability of local search.Furthermore,the stochastic convergence property of OHTDE is analyzed by using the theory of finite Markov chain.Experimental results and comparison show the effectiveness and robustness of the proposed OHTDE.