Simulation Optimization Problems Research Articles

Multi-objective decision-making for green infrastructure planning (LID-BMPs) in urban storm water management under uncertainty

Sustainable urban surface runoff management is receiving increased attention due to environmental and ecological consequences related to urbanization. This study presents a useful new framework for green infrastructure (GI) planning for the management of urban runoff quality and quantity. The proposed framework considers physical, technical, economic, and multi-stakeholder aspects related to urban runoff management while simultaneously addressing the uncertainties of the decision-making process and model parameters. The methodology is applied to regionally locating and sizing low impact development-best management practices (LID-BMPs) while considering the hydrologic and hydraulic impacts. A decision-making multi-objective optimization framework is developed by integrating: 1) a Multi-Layer Perceptron neural network founded on a Storm Water Management Model (SWMM-MLP meta-model), 2) NSGA-II multi-objective optimization, 3) fuzzy α-cut technique, and 4) a decision-making support model based on social choice theory to elicit trade-offs among system cost and LID-BMP performance indicators. The decision-making model, based on Fuzzy Social Choice (FSC) theory, is applied to simulate consensus between stakeholders for a partially cooperative group decision-making problem. The proposed methodology is explored in a catchment located in the northeastern part of Tehran, Iran. Results showed that the SWMM could be effectively replaced with a MLP-based meta-model in simulation-optimization problems for urban runoff management. In the application of FSC methods, the optimal scenarios were effective in reducing the volume of urban runoff and contamination loads while maintaining the optimality of the operation costs. Considering the optimal LID scenario, a reduction of more than 99% in runoff volume and biochemical oxygen demand (BOD), and a decrease of more than 92% of total suspended solids (TSS), occurred for the lower bound of uncertainty (lower (left) end of the α-cut = 0.3). For the upper bound of uncertainty (upper (right) end of the α-cut level = 0.3), a maximum reduction of 57% was obtained. In applying FSC methods through the decision-making process, the Borda Counting method considered the preferences of all stakeholders best. Moreover, the proposed framework allows decision-makers to decide on the acceptability and reliability of the optimal management scenarios considering their preferences and uncertainties.

Gaussian process based optimization algorithms with input uncertainty

Metamodels as cheap approximation models for expensive to evaluate functions have been commonly used in simulation optimization problems. Among various types of metamodels, the Gaussian Process (GP) model is popular for both deterministic and stochastic simulation optimization problems. However, input uncertainty is usually ignored in simulation optimization problems, and thus current GP-based optimization algorithms do not incorporate input uncertainty. This article aims to refine the current GP-based optimization algorithms to solve the stochastic simulation optimization problems when input uncertainty is considered. The comprehensive numerical results indicate that our refined algorithms with input uncertainty can find optimal designs more efficiently than the existing algorithms when input uncertainty is present.

Comparative Study of Metamodeling and Sampling Design for Expensive and Semi-Expensive Simulation Models Under Uncertainty

In spite of the wide improvements in computer simulation packages, many complex simulation models, particularly under uncertainty, may be inefficient to run in terms of time, computation, and resources. To address such a challenge, integrating metamodels and robust design optimization has been applied. In the current paper, a systematic comparative study is implemented to evaluate the performance of three common metamodels, namely polynomial regression, kriging, and radial basis function. The required experiments are designed by different space-filling methods including the orthogonal array design and three forms of Latin hypercube sampling such as randomized, maximin, and correlation approaches. Although, the impact of sample size on the performance of metamodels in robust optimization results are investigated. All methods are analyzed using five two-dimensional test problems and one engineering problem while all of them are considered in two forms that are expensive (with a small sample size) and semi-expensive (with a large sample size). Uncertainty is assumed in all problems as a source of variability, so all test problems are conducted in the format of robust optimization in the class of dual response surface in order to estimate robust Pareto frontier. The performances of methods are studied in two terms of accuracy and robustness. Finally, the results of comparison, an applicable guideline is provided to aid the practitioners in selecting the appropriate combination of metamodels and sampling design methods for investigating set of robust optimal points (estimated Pareto frontier) in simulation–optimization problems under uncertainty.

Simulation Optimization for Complex Multi-Domain Physical Systems Based on Partial Resolving

The iterative process of simulation optimization is a time-consuming task, as it involves executing the main simulation program in order to evaluate the optimal constraints and objective functions repeatedly according to the values of tuner parameters. Parameter optimization for a model of a multi-domain physical system based on Modelica is a typical simulation optimization problem. Traditionally, each simulation during each iterative step needs resolve all the variables in all the mass differential-algebraic equations (DAE) generated from the simulation model through constructing and traversing the solving dependency graph of the model. In order to improve the efficiency of the simulation optimization process, a new method named partial simulation resolving algorithm based on the set of input parameters and output variables for complex simulation model was proposed. By using this algorithm, a minimum solving graph (MSG) of the simulation model was built according to the set of parameters, constraints, and objective functions of the optimization model. The simulation during the optimization iterative process needs only to resolve the variables on the MSG, and therefore this method could decrease the simulation time greatly during every iterative step of the optimization process. As an example, the parameter optimization on economy of fuel for a heavy truck was realized to demonstrate the efficiency of this solving strategy. This method has been implemented in MWorks—a Modelica-based simulation platform.

Comparative study of metamodeling and sampling design for expensive and semi-expensive simulation models under uncertainty

In spite of the wide improvements in computer simulation packages, many complex simulation models, particularly under uncertainty, may be inefficient to run in terms of time, computation, and resources. To address such a challenge, integrating metamodels and robust design optimization has been applied. In the current paper, a systematic comparative study is implemented to evaluate the performance of three common metamodels, namely polynomial regression, kriging, and radial basis function. The required experiments are designed by different space-filling methods including the orthogonal array design and three forms of Latin hypercube sampling such as randomized, maximin, and correlation approaches. Although, the impact of sample size on the performance of metamodels in robust optimization results are investigated. All methods are analyzed using five two-dimensional test problems and one engineering problem while all of them are considered in two forms that are expensive (with a small sample size) and semi-expensive (with a large sample size). Uncertainty is assumed in all problems as a source of variability, so all test problems are conducted in the format of robust optimization in the class of dual response surface in order to estimate robust Pareto frontier. The performances of methods are studied in two terms of accuracy and robustness. Finally, the results of comparison, an applicable guideline is provided to aid the practitioners in selecting the appropriate combination of metamodels and sampling design methods for investigating set of robust optimal points (estimated Pareto frontier) in simulation–optimization problems under uncertainty.

An Introduction to Multiobjective Simulation Optimization

The multiobjective simulation optimization (MOSO) problem is a nonlinear multiobjective optimization problem in which multiple simultaneous and conflicting objective functions can only be observed with stochastic error. We provide an introduction to MOSO at the advanced tutorial level, aimed at researchers and practitioners who wish to begin working in this emerging area. Our focus is exclusively on MOSO methods that characterize the entire efficient or Pareto-optimal set as the solution to the MOSO problem; later, this set may be used as input to the broader multicriteria decision-making process. Our introduction to MOSO includes an overview of existing theory, methods, and provably convergent algorithms that explicitly control sampling error for (1) MOSO on finite sets, called multiobjective ranking and selection; (2) MOSO with integer-ordered decision variables; and (3) MOSO with continuous decision variables. In the context of integer-ordered and continuous decision variables, we focus on methods that provably converge to a local efficient set under the natural ordering. We also discuss key open questions that remain in this emerging field.

Productivity Improvement Through Multi-Objective Simulation Optimization—A Case Study

To maximize profitability and meet customers’ expectations, assembly plants strive to enhance operations and minimize costs. Using discrete-event system simulation and optimization, this paper investigates different scenarios to enhance operations efficiency and presents a decision support tool to aid operational decision making in an air conditioner assembly plant. Specifically, simulation is used to evaluate the system-wide impact of an optimal sequencing rule at a bottleneck station; in addition, a multi-objective simulation optimization problem is solved to enable manufacturers to evaluate their time/cost tradeoffs. Using actual data from a previous month, the simulation study illustrates possible savings throughout the evaluated alternatives. To sustain the study outcomes, we combine mathematical programming and simulation into an integrated decision support tool. Overall, this study illustrates the benefits of the solution approach to a real-life system.

Stochastic Simulation Optimization for Route Selection Strategy Based on Flight Delay Cost

The continuous and strong growth of the civil aviation in the world combined with the severe adverse weather problem have made necessary the collaboration between the different civil aviation agents to improve the management of the capacity-demand imbalances in the airspace. In this paper, we consider a stochastic simulation optimization problem for air route selection strategy based on flight delay cost. The problem takes consideration of airspace capacity and demand uncertainty, three strategies, including collaborative reroute strategy (CRS), full information reroute strategy (FIRS) and hybrid stated route preference strategy (HSR), are employed to mitigate the flight delay cost. To find the best strategy, a discrete event simulation model is built by Arena Software, and the Monte Carlo method combined with the OCBA simulation optimization technique is employed for assessing a common severe convective weather scenario in the Central and Southern China airspace. Simulation results imply that HSR schemes show better system-wide performance than CRS and FIRS, these benefits are supposed to come from the batch allocations method. Although the airline can receive full information in advance, FIRS does not show obvious advantage in reducing the total airborne waiting time than CRS. For the system-wide performance FIRS is better than CRS, but not as good as HSR.

Embedding Ordinal Optimization into Tree–Seed Algorithm for Solving the Probabilistic Constrained Simulation Optimization Problems

Probabilistic constrained simulation optimization problems (PCSOP) are concerned with allocating limited resources to achieve a stochastic objective function subject to a probabilistic inequality constraint. The PCSOP are NP-hard problems whose goal is to find optimal solutions using simulation in a large search space. An efficient “Ordinal Optimization (OO)” theory has been utilized to solve NP-hard problems for determining an outstanding solution in a reasonable amount of time. OO theory to solve NP-hard problems is an effective method, but the probabilistic inequality constraint will greatly decrease the effectiveness and efficiency. In this work, a method that embeds ordinal optimization (OO) into tree–seed algorithm (TSA) (OOTSA) is firstly proposed for solving the PCSOP. The OOTSA method consists of three modules: surrogate model, exploration and exploitation. Then, the proposed OOTSA approach is applied to minimize the expected lead time of semi-finished products in a pull-type production system, which is formulated as a PCSOP that comprises a well-defined search space. Test results obtained by the OOTSA are compared with the results obtained by three heuristic approaches. Simulation results demonstrate that the OOTSA method yields an outstanding solution of much higher computing efficiency with much higher quality than three heuristic approaches.

Surrogate-Based Promising Area Search for Lipschitz Continuous Simulation Optimization

We propose an adaptive search algorithm for solving simulation optimization problems with Lipschitz continuous objective functions. The method combines the strength of several popular strategies in simulation optimization. It employs the shrinking ball method to estimate the performance of sampled solutions and uses the performance estimates to fit a surrogate model that iteratively approximates the response surface of the objective function. The search for improved solutions at each iteration is then based on sampling from a promising region (a subset of the decision space) adaptively constructed to contain the point that optimizes the surrogate model. Under appropriate conditions, we show that the algorithm converges to the set of local optimal solutions with probability one. A computational study is also carried out to illustrate the algorithm and to compare its performance with some of the existing procedures.

Minimizing Expected Cycle Time of Stochastic Customer Orders Through Bounded Multi-Fidelity Simulations

This paper considers the scheduling of stochastic customer orders to minimize expected cycle time. Customer orders dynamically arrive at a machine station, and each order consists of multiple product types. Random workloads are required by each product type, and the workloads are assigned to a set of unrelated parallel machines in the station to be processed. The objective is to obtain the minimal long-run expected order cycle time through proper workload assignments. In view of the difficulty in evaluating the objective function, this paper models the targeted problem as a simulation optimization problem and proposes to solve it under the multifidelity model framework. To improve the efficiency in evaluating candidate solutions, a low-fidelity model is constructed to select solutions with better performances for high-fidelity simulations. The effectiveness of this low-fidelity model is demonstrated through a series of theoretical evidences. A simulation optimization algorithm, named Bound-Multi-fidelity Optimization with Ordinal Transformation and Optimal Sampling Bound-(MO2TOS), is developed by taking advantage of the properties of the low-fidelity model. Numerical experiments are conducted to evaluate the performance of the proposed algorithm against three other well-known simulation optimization algorithms in the literature. Results indicate that the Bound-MO2TOS outperforms all the other tested algorithms, and its performance is robust against changes in problem scale. Note to Practitioners —Customer order scheduling models have wide applications in industries where every single order may contain multiple product types and the entire order requires one shipment delivery. In many applications, such as pigments dyes manufacturing, auto repair, and grocery, consideration of customer orders rather than individual jobs is often preferred. Given the stochastic nature and the synchronization constraints of the problem, a multifidelity simulation algorithm named Bound-Multi-fidelity Optimization with Ordinal Transformation and Optimal Sampling Bound-(MO2TOS) is proposed. Experimental results suggest that the proposed algorithm outperforms three other popular simulation optimization algorithms under a variety of scenarios. It is admitted that further improvement can be made in refining the computation resource distribution process of Bound-MO2TOS. In the future research, such an improvement will be more thoroughly elaborated so that practitioners can utilize it to coordinate their productions better.

Gradient-Based Myopic Allocation Policy: An Efficient Sampling Procedure in a Low-Confidence Scenario

In this note, we study a simulation optimization problem of selecting the alternative with the best performance from a finite set, or a so-called ranking and selection problem, in a special low-confidence scenario. The most popular sampling allocation procedures in ranking and selection do not perform well in this scenario, because they all ignore certain induced correlations that significantly affect the probability of correct selection in this scenario. We propose a gradient-based myopic allocation policy that takes the induced correlations into account, reflecting a tradeoff between the induced correlation and the two factors (mean-variance) found in the optimal computing budget allocation formula. Numerical experiments substantiate the efficiency of the new procedure in the low-confidence scenario.

Constrained optimization of black-box stochastic systems using a novel feasibility enhanced Kriging-based method

Stochastically constrained simulation optimization problems are challenging because the inherent noise terms to a black-box system lead to the need of considering the uncertainty at both the objective and the constraint function. To address this problem, we propose a Kriging-based optimization framework, which uses stochastic Kriging models to approximate the objective and the constraint functions and an adaptive sampling approach to sequentially search for the next point that is promising to have a better objective. The main contribution of this work is to incorporate a feasibility-enhanced term to the infill sampling criterion, which improves the Kriging-based algorithm's capabilities of returning a truly feasible near-optimal solution for stochastic systems. The efficacy of the Kriging-based algorithm is demonstrated with eight benchmark problems and a case study from the pharmaceutical manufacturing domain.

A novel optimal replication allocation strategy for particle swarm optimization algorithms applied to simulation optimization problem

This study proposes a simulation optimization algorithm, called optimal replication allocation strategy (ORAS), for particle swarm optimization (PSO) algorithms in simulation models with high computing cost in large design spaces. In this study, the PSO is employed to explore and exploit near-optimal or optimal solution in the design space. Given the uncertainties in real-word applications, a simulation model is constructed to evaluate the performance of each design alternative. Optimal computing budget allocation (OCBA), a state-of-the-art resampling method, is combined with metaheuristic principles to improve the accuracy of estimating best solutions and enhancing efficiency by intelligently allocating the number of replications. However, the solution space is nonlinear or multimodal; that is, various local or global solutions exist. In OCBA, the probability of correct selection (P(CS)) in the currect best solution serves as an important measurement. P(CS) refers to the probability that the “best” of k populations is selected according to some specified criteria of best. OCBA can halt allocation when P(CS) reaches higher than the desired value, i.e., P(CS)*. The multimodal solution space features a high probability of reaching a low P(CS) as many solutions perform closely to the current best of each generation. This situation indicates that P(CS) cannot achieve P(CS)*; OCBA cannot stop allocation, and additional computational cost may be wasted. In this study, we redefine and modify P(CS). The new version is P(CSE), which considers calculation of global best, called the super individual, instead of current best solution within a confidence level. The proposed ORAS can provide an asymptotically optimal allocation rule for combining with population metaheuristics based on P(CSE). We apply ORAS using an original particle swarm optimization (PSO) and two variants of PSO to address stochastic buffer allocation problem and stochastic function optimization problem compared with several state-of-the-art technologies from literature. The resulting ORAS with three PSOs is an effective procedure as it intelligently utilizes limited computing resources. Numerical tests indicate that ORAS increases P(CSE) in each generation and subsequently enhances the efficiency of PSO algorithms.

A simulation-based optimization framework for integrating scheduling and model predictive control, and its application to air separation units

The integration of dynamic process models in scheduling calculations has recently received significant attention as a mean to improve operational performance in increasingly dynamic markets. In this work, we propose a novel framework for the integration of scheduling and model predictive control (MPC), which is applicable to industrial size problems involving fast changing market conditions. The framework consists on identifying scheduling-relevant process variables, building low-order dynamic models to capture their evolution, and integrating scheduling and MPC by, (i) solving a simulation-optimization problem to define the optimal schedule and, (ii) tracking the schedule in closed-loop using the MPC controller. The efficacy of the framework is demonstrated via a case study that considers an air separation unit operating under real-time electricity pricing. The study shows that significant cost reductions can be achieved with reasonable computational times.

SCORE Allocations for Bi-objective Ranking and Selection

The bi-objective ranking and selection (R8S) problem is a special case of the multi-objective simulation optimization problem in which two conflicting objectives are known only through dependent Monte Carlo estimators, the decision space or number of systems is finite, and each system can be sampled to some extent. The solution to the bi-objective R8S problem is a set of systems with non-dominated objective vectors, called the set of Pareto systems. We exploit the special structure of the bi-objective problem to characterize the asymptotically optimal simulation budget allocation, which accounts for dependence between the objectives and balances the probabilities associated with two types of misclassification error. Like much of the R8S literature, our focus is on the case in which the simulation observations are bivariate normal. Assuming normality, we then use a certain asymptotic limit to derive an easily-implementable Sampling Criteria for Optimization using Rate Estimators (SCORE) sampling framework that approximates the optimal allocation and accounts for correlation between the objectives. Perhaps surprisingly, the limiting SCORE allocation exclusively controls for misclassification-by-inclusion events, in which non-Pareto systems are falsely estimated as Pareto. We also provide an iterative algorithm for implementation. Our numerical experience with the resulting SCORE framework indicates that it is fast and accurate for problems having up to ten thousand systems.

On Sampling Rates in Simulation-Based Recursions

We consider the context of “simulation-based recursions,” that is, recursions that involve quantities needing to be estimated using a stochastic simulation. Examples include stochastic adaptations of fixed-point and gradient descent recursions obtained by replacing function and derivative values appearing within the recursion by their Monte Carlo counterparts. The primary motivating settings are simulation optimization and stochastic root finding problems, where the low point and the zero of a function are sought, respectively, with only Monte Carlo estimates of the functions appearing within the problem. We ask how much Monte Carlo sampling needs to be performed within simulation-based recursions in order that the resulting iterates remain consistent and, more importantly, efficient, where “efficient” implies convergence at the fastest possible rate. Answering these questions involves trading off two types of error inherent in the iterates: the deterministic error due to recursion and the “stochastic” error due to sampling. As we demonstrate through a characterization of the relationship between sample sizing and convergence rates, efficiency and consistency are intimately coupled with the speed of the underlying recursion, with faster recursions yielding a wider regime of “optimal” sampling rates. The implications of our results for practical implementation are immediate since they provide specific guidance on optimal simulation expenditure within a variety of stochastic recursions.

Gradient-Based Adaptive Stochastic Search for Simulation Optimization Over Continuous Space

We extend the idea of model-based algorithms for deterministic optimization to simulation optimization over continuous space. Model-based algorithms iteratively generate a population of candidate solutions from a sampling distribution and use the performance of the candidate solutions to update the sampling distribution. By viewing the original simulation optimization problem as another optimization problem over the parameter space of the sampling distribution, we propose to use a direct gradient search on the parameter space to update the sampling distribution. To improve the computational efficiency, we further develop a two-timescale updating scheme that updates the parameter on a slow timescale and estimates the quantities involved in the parameter updating on the fast timescale. We analyze the convergence properties of our algorithms through techniques from stochastic approximation, and demonstrate the good empirical performance by comparing with two state-of-the-art model-based simulation optimization methods. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0771 .

A simulation–optimization strategy to deal simultaneously with tens of decision variables and multiple performance measures in manufacturing

This work addresses the multiple criteria simulation–optimization problem. This problem entails using an optimization strategy to manipulate the parameters of a simulation model to arrive at the best possible configurations in the presence of several performance measures in conflict. Pareto Efficiency conditions are used in an iterative framework based on experimental design and pairwise comparison. In particular, this work improves upon and replaces the use of Data Envelopment Analysis to determine the efficient frontier, and replaces the use of a single-pass algorithm previously proposed by our research group. The results show a rapid convergence to a more precise characterization of the Pareto-efficient frontier. In addition, the capability of the method to deal with fifty decision variables simultaneously is demonstrated through a study regarding the fine-tuning of a manufacturing line.

Embedding Advanced Harmony Search in Ordinal Optimization to Maximize Throughput Rate of Flow Line

Flow line systems are production systems in which successive operations are performed on a product in a manner so that it moves through the factory in a certain direction. This work firstly formulates a flow line system as an integer-ordered inequality-constrained simulation–optimization problem and present a stochastic simulation procedure to estimate the throughput rate. The mathematical formulation and simulated procedure can be used for any distribution of processing rate and can be applied to high-dimensional problems. An approach that embeds advanced harmony search (AHS) in ordinal optimization (OO), abbreviated as AHSOO, is developed to find a near-optimal design of the flow line system to maximize the throughput rate. The proposed approach comprises three levels, which are meta-modeling, diversification and intensification. A radial basis function network is a meta-model to approximate the performance of a design. The proposed approach integrates the AHS approach for diversification with improved optimal computing budget allocation (IOCBA) for intensification. AHS favorably explores the solution space initially and moves toward exploiting good solutions close to the end. The IOCBA maximized the overall simulation efficiency for finding an optimal solution. The proposed AHSOO is tested on three examples. In the moderately sized example, simulation results reveal that the average best-so-far performances that were determined using PSO, GA and ES were 6.12, 9.65 and 8.53% less than that obtained using AHSOO—even after the former took more than 50 times the CPU time that was consumed by AHSOO upon completion. Analytical results reveal that the proposed method yields designs of much higher quality with a much higher computing efficiency than the seven competing methods.