The orienteering problem with stochastic travel and service times

In this paper, we introduce a variant of the orienteering problem in which travel and service times are stochastic. If a delivery commitment is made to a customer and is completed by the end of the day, a reward is received, but if a commitment is made and not completed, a penalty is incurred. This problem reflects the challenges of a company who, on a given day, may have more customers than it can serve. In this paper, we discuss special cases of the problem that we can solve exactly and heuristics for general problem instances. We present computational results for a variety of parameter settings and discuss characteristics of the solution structure.     

A priori orienteering with time windows and stochastic wait times at customers

In the pharmaceutical industry, sales representatives visit doctors to inform them of their products and encourage them to become an active prescriber. On a daily basis, pharmaceutical sales representatives must decide which doctors to visit and the order to visit them. This situation motivates a problem we more generally refer to as a stochastic orienteering problem with time windows (SOPTW), in which a time window is associated with each customer and an uncertain wait time at a customer results from a queue of competing sales representatives. We develop a priori routes with the objective of maximizing expected sales. We operationalize the sales representative’s execution of the a priori route with relevant recourse actions and derive an analytical formula to compute the expected sales from an a priori tour. We tailor a variable neighborhood search heuristic to solve the problem. We demonstrate the value of modeling uncertainty by comparing the solutions to our model to solutions of a deterministic version using expected values of the associated random variables. We also compute an empirical upper bound on our solutions by solving deterministic instances corresponding to perfect information.     

Metaheuristics for the team orienteering problem

The Team Orienteering Problem (TOP) is the generalization to the case of multiple tours of the Orienteering Problem, known also as Selective Traveling Salesman Problem. A set of potential customers is available and a profit is collected from the visit to each customer. A fleet of vehicles is available to visit the customers, within a given time limit. The profit of a customer can be collected by one vehicle at most. The objective is to identify the customers which maximize the total collected profit while satisfying the given time limit for each vehicle. We propose two variants of a generalized tabu search algorithm and a variable neighborhood search algorithm for the solution of the TOP and show that each of these algorithms beats the already known heuristics. Computational experiments are made on standard instances.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

On the reliability of stochastic systems

In this note we consider the problem of computing the probability R(t₀ = P(X(t) > Y(t) for 0 < t ? t₀), where X(t) and Y(t) are stochastic processes. This extends some of the existing results to the case of stochastic processes. Related estimation problems are also considered.     

On the complexity of stochastic integration

We study the complexity of approximating stochastic integrals with error for various classes of functions. For Ito integration, we show that the complexity is of order , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.

     

On the control of stochastic systems

A general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the “iterative” method of the authors or its more recent adaptations for stochastic operator equations. It is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations; the method, therefore, is applicable to the control problem. Application is made to the state space equation $\text{x}\text{?}\text{= Ax + Bu + C,}\text{where}\text{A, B, C}$ are stochastic matrices corresponding to stochastic operators, i.e., involving randomly time varying elements, e.g., $\text{a}{}_{\mathrm{ij}}\text{(t, ω) ? A, t ? T, ω ? (Ω, F,μ)}$, a p.s. It is emphasized that the processes are arbitrary stochatic processes with known statistics. No assumption is made of Wiener or Markov behavior or of smallness of fluctuations and no closure approximations are necessary. The method differs in interesting aspects from Volterra series expansions used by Wiener and others and has advantages over the other methods. Because of recent progress in the solution of the nonlinear case, it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages.     

On stochastic games

In this paper, we consider the stochastic games of Shapley, when the state and action spaces are all infinite. We prove that, under certain conditions, the stochastic game has a value and that both players have optimal strategies.Part of this research was supported by NSF grant. The authors are indebted to L. S. Shapley for the useful discussions on this and related topics. The authors thank the referee for pointing out an ambiguity in the formulation of Lemma 2.4 in an earlier draft of this article.     

On stochastic estimation

We consider a local random searching method to approximate a root of a specified equation. If such roots, which can be regarded as estimators for the Euclidean parameter of a statistical experiment, have some asymptotic optimality properties, the local random searching method leads to asymptotically optimal estimators in such cases. Application to simple first order autoregressive processes and some simulation results for such models are also included.     

On the stochastic porous medium equation

In this paper, we discuss an initial-boundary value problem and a Cauchy problem for the stochastic porous medium equation. Our basic estimates are based on the known results due to (Math. Sbornik 67 (1965) 609-642, Arch. Rational Mech. Anal. 25 (1967) 64-80). By the procedure developed in (Trans. Amer. Math. Soc. 354 (2001) 1117-1135), we obtain solutions over the given probability space rather than martingale solutions. We will also establish the existence of invariant measures when the space domain is bounded.     

On the inverse stochastic reconstruction problem

We consider the reconstruction problem in the class of stochastic differential equations of Itô type on the basis of given motion properties that depend only on part of the variables. We determine the set of controls providing necessary and sufficient conditions for the existence of a given integral manifold.