Mahmoud H. Alrefaei

A modification of the stochastic ruler method for discrete stochastic optimization

Abstract We propose a modified stochastic ruler method for finding a global optimal solution to a discrete optimization problem in which the objective function cannot be evaluated analytically but has to be estimated or measured. Our method generates a Markov chain sequence taking values in the feasible set of the underlying discrete optimization problem; it uses the number of visits this sequence makes to the different states to estimate the optimal solution. We show that our method is guaranteed to converge almost surely (a.s.) to the set of global optimal solutions. Then, we show how our method can be used for solving discrete optimization problems where the objective function values are estimated using either transient or steady-state simulation. Finally, we provide some numerical results to check the validity of our method and compare its performance with that of the original stochastic ruler method.

A simulated annealing technique for multi-objective simulation optimization

In this paper, we present a simulated annealing algorithm for solving multi-objective simulation optimization problems. The algorithm is based on the idea of simulated annealing with constant temperature, and uses a rule for accepting a candidate solution that depends on the individual estimated objective function values. The algorithm is shown to converge almost surely to an optimal solution. It is applied to a multi-objective inventory problem; the numerical results show that the algorithm converges rapidly.

Selecting the best stochastic system for large scale problems in DEDS

We consider the problem of selecting the stochastic system with the best expected performance measure, when the number of alternative systems is large. We consider the case of discrete event dynamic systems (DEDS) where the standard clock simulation technique can be used for simulating multiple systems using only one sample path. In this paper, we use a two-phase procedure that uses the standard clock simulation technique. In the first phase, we screen out non-competent alternatives and construct a confidence set that contains the best alternative with a pre-specified large probability. In the second phase, we use the indifference-zone ranking and selection procedure to select the best expected alternative among the survivals of the first phase. We implement this algorithm for solving a practical simulation optimization problem.

Subset selection of best simulated systems

In this paper, we consider the problem of selecting a subset of k systems that is contained in the set of the best s simulated systems when the number of alternative systems is huge. We propose a sequential method that uses the ordinal optimization to select a subset G randomly from the search space that contains the best simulated systems with high probability. To guarantee that this subset contains the best systems it needs to be relatively large. Then methods of ranking and selections will be applied to select a subset of k best systems of the subset G with high probability. The remaining systems of G will be replaced by newly selected alternatives from the search space. This procedure is repeated until the probability of correct selection (a subset of the best k simulated systems is selected) becomes very high. The optimal computing budget allocation is also used to allocate the available computing budget in a way that maximizes the probability of correct selection. Numerical experiments for comparing these algorithms are presented.

Automatic Clustering Using Multi-objective Particle Swarm and Simulated Annealing.

This paper puts forward a new automatic clustering algorithm based on Multi-Objective Particle Swarm Optimization and Simulated Annealing, “MOPSOSA”. The proposed algorithm is capable of automatic clustering which is appropriate for partitioning datasets to a suitable number of clusters. MOPSOSA combines the features of the multi-objective based particle swarm optimization (PSO) and the Multi-Objective Simulated Annealing (MOSA). Three cluster validity indices were optimized simultaneously to establish the suitable number of clusters and the appropriate clustering for a dataset. The first cluster validity index is centred on Euclidean distance, the second on the point symmetry distance, and the last cluster validity index is based on short distance. A number of algorithms have been compared with the MOPSOSA algorithm in resolving clustering problems by determining the actual number of clusters and optimal clustering. Computational experiments were carried out to study fourteen artificial and five real life datasets.

A new search algorithm for discrete stochastic optimization

We present a new method for finding a global optimal solution to a discrete stochastic optimization problem. This method resembles the simulated annealing method for discrete deterministic optimization. However, in our method the annealing schedule (the cooling temperature) is kept fixed, and the mechanism for estimating the optimal solution is different from that used in the original simulated annealing method. We state a convergence result that shows that our method converges almost surely to a global optimal solution under mild conditions. We also present empirical results that illustrate the performance of the proposed approach on a simple example.

Accelerating the convergence of the stochastic ruler method for discrete stochastic optimization

We present two new variants of the stochastic ruler method for solving discrete stochastic optimization problems. These two variants use the same mechanism for moving around the state space as the modified stochastic ruler method we have proposed earlier. However, the new variants use different approaches for estimating the optimal solution. In particular, the modified stochastic ruler method uses the number of visits to each state by the Markov chain generated by the algorithm to estimate the optimal solution. On the other hand, one of our new methods uses the number of visits to each state by the embedded chain of the Markov chain generated by the algorithm to estimate the optimal solution, and our other new method uses the feasible solution with the best average estimated objective function value to estimate the optimal solution. Like our earlier modification of the stochastic ruler method, these two new methods are guaranteed to converge almost surely to the set of global optimal solutions. We present theoretical and numerical results that indicate that our new approaches tend to lead to the set of global optimal solutions faster.

Discrete stochastic optimization via a modification of the stochastic ruler method

In this paper, we present a modification of the stochastic ruler method for solving discrete stochastic optimization problems. Our method generates a stationary Markov chain sequence taking values in the feasible set of the underlying discrete optimization problem. The number of visits to every state by this Markov chain is used to estimate the optimal solution. Unlike the original stochastic ruler method, our method is guaranteed to converge almost surely to a global optimal solution. We present empirical results that illustrate the performance of our method, and we show that these results compare favorably with empirical results obtained using the original stochastic ruler method.

An adaptive Monte Carlo integration algorithm with general division approach

We propose an adaptive Monte Carlo algorithm for estimating multidimensional integrals over a hyper-rectangular region. The algorithm uses iteratively the idea of separating the domain of integration into 2^ssubregions. The proposed algorithm can be applied directly to estimate the integral using an efficient way of storage. We test the algorithm for estimating the value of a 30-dimensional integral using a two-division approach. The numerical results show that the proposed algorithm gives better results than using one-division approach.

Solution quality of random search methods for discrete stochastic optimization

In this paper, we propose a framework for selecting a high quality global optimal solution for discrete stochastic optimization problems with a predetermined confidence level using general random search methods. This procedure is based on performing the random search algorithm several replications to get estimate of the error gap between the estimated optimal value and the actual optimal value. A confidence set that contains the optimal solution is then constructed and methods of the indifference zone approach are used to select the optimal solution with high probability. The proposed procedure is applied on a simulated annealing algorithm for solving a particular discrete stochastic optimization problem involving queuing models. The numerical results indicate that the proposed technique indeed locate a high quality optimal solution.