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Multinomial selection is concerned with selecting the most probable (best) multinomial alternative. The alternatives compete in a number of independent trials. In each trial, each alternative wins with an unknown probability specific to that alternative. A long-standing research goal has been to find a procedure that minimizes the expected number of trials subject to a lower bound on the probability of correct selection ( (CS)). Numerous procedures have been proposed over the past 55 years, all of them suboptimal, for the version where the number of trials is bounded. We achieve the goal in the following sense: For a given multinomial probability vector, lower bound on (CS), and upper bound on trials, we use linear programming (LP) to construct a procedure that is guaranteed to minimize the expected number of trials. This optimal procedure may necessarily be randomized. We also present a mixed-integer linear program (MIP) that produces an optimal deterministic procedure. In our computational stud...

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.

We consider a very general online scheduling problem with an objective to minimize the maximum level of resource allocated. We find a simple characterization of an optimal deterministic online algorithm. We develop further results for the two, more specific problems of single resource scheduling and hierarchical line balancing. We determine how to compute optimal online algorithms for both problems using linear programming and integer programming, respectively. We show that randomized algorithms can outperform deterministic algorithms, but only if the amount of work done is a nonconcave function of resource allocation.

In practical applications of stochastic programming the involved probability distributions are never known exactly. One can try to hedge against the worst expected value resulting from a considered set of permissible distributions. This leads to a min-max formulation of the corresponding stochastic programming problem. We show that, under mild regularity conditions, such a min-max problem generates a probability distribution on the set of permissible distributions with the min-max problem being equivalent to the expected value problem with respect to the corresponding weighted distribution. We consider examples of the news vendor problem, the problem of moments and problems involving unimodal distributions. Finally, we discuss the Monte Carlo sample average approach to solving such min-max problems.

In this paper we study a Monte Carlo simulation--based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates, stopping rules, and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem.

The Dynamic and Stochastic Knapsack Problem (DSKP) is defined as follows. Items arrive according to a Poisson process in time. Each item has a demand (size) for a limited resource (the knapsack) and an associated reward. The resource requirements and rewards are jointly distributed according to a known probability distribution and become known at the time of the item's arrival. Items can be either accepted or rejected. If an item is accepted, the item's reward is received; and if an item is rejected, a penalty is paid. The problem can be stopped at any time, at which time a terminal value is received, which may depend on the amount of resource remaining. Given the waiting cost and the time horizon of the problem, the objective is to determine the optimal policy that maximizes the expected value (rewards minus costs) accumulated. Assuming that all items have equal sizes but random rewards, optimal solutions are derived for a variety of cost structures and time horizons, and recursive algorithms for computing them are developed. Optimal closed-form solutions are obtained for special cases. The DSKP has applications in freight transportation, in scheduling of batch processors, in selling of assets, and in selection of investment projects.

In this paper a dynamic and stochastic model of the well-known knapsack problem is developed and analyzed. The problem is motivated by a wide variety of real-world applications. Objects of random weight and reward arrive according to a stochastic process in time. The weights and rewards associated with the objects are distributed according to a known probability distribution. Each object can either be accepted to be loaded into the knapsack, of known weight capacity, or be rejected. The objective is to determine the optimal policy for loading the knapsack within a fixed time horizon so as to maximize the expected accumulated reward. The optimal decision rules are derived and are shown to exhibit surprising behavior in some cases. It is also shown that if the distribution of the weights is concave, then the decision rules behave according to intuition.