In Computer Science, Optimal Computing Budget Allocation (OCBA) is a simulation optimization method designed to maximize the Probability of Correct Selection (PCS) while minimizing computational costs. First introduced by Dr. Chun-Hung Chen in the mid-1990s, OCBA determines how many simulation runs (or how much computational time) or the number of replications each design alternative needs to identify the best option while using as few resources as possible.
OCBA has also been shown to enhance partition-based random search algorithms for solving deterministic global optimization problems. Over the years, OCBA has been applied in manufacturing systems design, healthcare planning, and financial modeling. It has also been extended to handle more complex scenarios, such as balancing multiple objectives, feasibility determination, and constrained optimization.
The goal of OCBA is to provide a systematic approach to efficiently run a large number of simulations by focusing only on the critical alternatives, in order to select the best alternative.
In other words, OCBA prioritizes only the most critical alternatives, minimizing computation time and reducing the variances of these critical estimators. The expected outcome is maintaining the required level of accuracy while requiring fewer computational resources.
The problem is mathematically formulated as:
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Subject to:
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where:
OCBA optimizes the allocation of simulation replications by focusing on alternatives with higher variances or smaller performance gaps relative to the best alternative. The ratio of replications between two alternatives, such as and , is determined by the following formula:
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Here:
This formula ensures that alternatives with smaller performance gaps () or higher variances () receive more simulation replications. This maximizes computational efficiency while maintaining a high Probability of Correct Selection (PCS), ensuring computational efficiency by reducing replications for non-critical alternatives and increasing them for critical ones. Numerical results show that OCBA can achieve the same simulation quality with only one-tenth of the computational effort compared to traditional methods.
According to Szechtman and Yücesan (2008), OCBA is also helpful in feasibility determination problems. This is where the decisions makers are only interested in differentiating feasible alternatives from the infeasible ones. Further, choosing an alternative that is simpler, yet similar in performance is crucial for other decision makers. In this case, the best choice is among top-r simplest alternatives, whose performance rank above desired levels.
In addition, Trailovic and Pao (2004) demonstrate an OCBA approach, where we find alternatives with minimum variance, instead of with best mean. Here, we assume unknown variances, voiding the OCBA rule (assuming that the variances are known). During 2010 research was done on an OCBA algorithm that is based on a t distribution. The results show no significant differences between those from t-distribution and normal distribution. The above presented extensions of OCBA is not a complete list and is yet to be fully explored and compiled.
Multi-Objective Optimal Computing Budget Allocation (MOCBA) is the OCBA concept that applies to multi-objective problems. In a typical MOCBA, the PCS is defined as
in which
We notice that, the Type I error and Type II error for identifying a correct Pareto set are respectively
and .
Besides, it can be proven that
and
where is the number of objectives, and follows posterior distribution Noted that and are the average and standard deviation of the observed performance measures for objective of design , and is the number of observations.
Thus, instead of maximizing , we can maximize its lower bound, i.e., Assuming , the Lagrange method can be applied to conclude the following rules:
in which
and
The primary performance measure can be called the main objective while the secondary performance measures are referred as the constraint measures. This falls into the problem of constrained optimization. When the number of alternatives is fixed, the problem is called constrained ranking and selection where the goal is to select the best feasible design given that both the main objective and the constraint measures need to be estimated via stochastic simulation. The OCBA method for constrained optimization (called OCBA-CO) can be found in Pujowidianto et al. (2009) and Lee et al. (2012).
Define
Suppose all the constraints are provided in form of , . The probability of correctly selecting all the feasible designs is
and the budget allocation problem for feasibility determination is given by Gao and Chen (2017)
Let and . The asymptotic optimal budget allocation rule is
Intuitively speaking, the above allocation rule says that (1) for a feasible design, the dominant constraint is the most difficult one to be correctly detected among all the constraints; and (2) for an infeasible design, the dominant constraint is the easiest one to be correctly detected among all constraints.
Specifically, the expected opportunity cost is
where,
The budget allocation problem with the EOC objective measure is given by Gao et al. (2017)
where is the proportion of the total simulation budget allocated to design . If we assume for all , the asymptotic optimal budget allocation rule for this problem is
where is the variance of the simulation samples of design . This allocation rule is the same as the asymptotic optimal solution of problem (1). That is, asymptotically speaking, maximizing PCS and minimizing EOC are the same thing.
Assuming that the uncertainty set contains a finite number of scenarios for the underlying input distributions and parameters, Gao et al. (2017) introduces a new OCBA approach by maximizing the probability of correctly selecting the best design under a fixed simulation budget, where the performance of a design is measured by its worst-case performance among all the possible scenarios in the uncertainty set.
Predictive Multi-Fidelity Models: Gaussian mixture models (GMMs) predict relationships between low- and high-fidelity simulations, enabling OCBA to focus on the most promising alternatives. Multi-fidelity models combine insights from low-fidelity simulations, which are computationally inexpensive but less accurate, and high-fidelity simulations, which are more accurate but computationally intensive. The integration of GMMs into this process allows OCBA to strategically allocate computational resources across fidelity levels, significantly reducing simulation costs while maintaining decision accuracy.
Dynamic Resource Allocation in Healthcare: A Bayesian OCBA framework has been applied to allocate resources in hospital emergency departments, balancing service quality with operational efficiency. By minimizing expected opportunity costs, this approach supports real-time decision-making in high-stakes environments. Additionally, the integration of OCBA with real-time digital twin-based optimization has further advanced its application in predictive simulation learning, enabling dynamic adjustments to resource allocation in healthcare settings. Furthermore, a contextual ranking and selection method for personalized medicine leverages OCBA to optimize resource allocation in treatments tailored to individual patient profiles, demonstrating its potential in personalized healthcare.
Sequential Allocation using Machine-learning Predictions as Light-weight Estimates (SAMPLE): SAMPLE is an extension of OCBA that presents a new opportunity for the integration of machine learning with digital twins for real-time simulation optimization and decision-making. Current methods for applying machine learning on simulation data may not produce the optimal solution due to errors encountered during the predictive learning phase since training data can be limited. SAMPLE overcomes this issue by leveraging lightweight machine learning models, which are easy to train and interpret, then running additional simulations once the real-world context is captured through the digital twin.