Chance constrained bottleneck spanning tree problem

In this paper we consider a stochastic version of the bottleneck spanning tree in which edge costs are independent random variables. Our problem is to find an optimal spanning tree and optimal satisficing level of the chance constraint with respect to the bottleneck (maximum cost) edge of the spanning tree. The problem is first transformed into a deterministic equivalent problem. Then a subproblem in order to solve the problem is introduced and the close relation between these problems is clarified. Finally, based on the relation, polynomial time solution procedures to solve the problem are proposed under two special functions of satisficing level which are given as examples to be solved easily.     

Stochastic spanning tree problem

The minimal spanning tree problem has been well studied and until now many efficient algorithms such as [5,6] have been proposed. This paper generalizes it toward a stochastic version, i.e., considers a stochastic spanning tree problem in which edge costs are not constant but random variables and its objective is to find an optimal spanning tree satisfying a certain chance constraint. This problem may be considered as a discrete version of P-model first introduced by Kataoka [4].First it is transformed into its deterministic equivalent problem P. Then, an auxiliary problem P(R) with a positive parameter R is defined. After clarifying close relations between P and P(R), this paper proposes a polynomial order algorithm fully utilizing P(R). Finally, more improvement of the algorithm and applicability of this type algorithm to other discrete stochastic programming problems are discussed.     

Planning logistics operations in the oil industry

In this paper we apply stochastic programming modelling and solution techniques to planning problems for a consortium of oil companies. A multiperiod supply, transformation and distribution scheduling problem—the Depot and Refinery Optimization Problem (DROP)—is formulated for strategic or tactical level planning of the consortium's activities. This deterministic model is used as a basis for implementing a stochastic programming formulation with uncertainty in the product demands and spot supply costs (DROPS), whose solution process utilizes the deterministic equivalent linear programming problem. We employ our STOCHGEN general purpose stochastic problem generator to ‘recreate’ the decision (scenario) tree for the unfolding future as this deterministic equivalent. To project random demands for oil products at different spatial locations into the future and to generate random fluctuations in their future prices/costs a stochastic input data simulator is developed and calibrated to historical industry data. The models are written in the modelling language XPRESS-MP and solved by the XPRESS suite of linear programming solvers. From the viewpoint of implementation of large-scale stochastic programming models this study involves decisions in both space and time and careful revision of the original deterministic formulation. The first part of the paper treats the specification, generation and solution of the deterministic DROP model. The stochastic version of the model (DROPS) and its implementation are studied in detail in the second part and a number of related research questions and implications discussed.     

Stochastic uncapacitated hub location

We study stochastic uncapacitated hub location problems in which uncertainty is associated to demands and transportation costs. We show that the stochastic problems with uncertain demands or dependent transportation costs are equivalent to their associated deterministic expected value problem (EVP), in which random variables are replaced by their expectations. In the case of uncertain independent transportation costs, the corresponding stochastic problem is not equivalent to its EVP and specific solution methods need to be developed. We describe a Monte-Carlo simulation-based algorithm that integrates a sample average approximation scheme with a Benders decomposition algorithm to solve problems having stochastic independent transportation costs. Numerical results on a set of instances with up to 50 nodes are reported.     

A branch and bound algorithm for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.In this paper a branch and bound algorithm for the robust spanning tree problem is proposed. The method embeds the extension of some results previously presented in the literature and some new elements, such as a new lower bound and some new reduction rules, all based on the exploitation of some peculiarities of the branching strategy adopted.Computational results obtained by the algorithm are presented. The technique we propose is up to 210 faster than methods recently appeared in the literature.     

On the Stochastic Non-sequential Production-planning Problem

This paper examines a stochastic non-sequential capacitated production-planning problem where the demand of each period is a continuous random variable. The stochastic non-sequential production-planning problem is examined with sequence-independent and then with sequence-dependent set-up costs, and the worst-case error determined when an approximate solution is obtained by solving the deterministic equivalent. We prove in general that the worst-case error is not dependent on the nature of the set-up cost, and identify a family of approximations for the stochastic non-sequential production-planning problem.     

A Benders decomposition approach for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data. Computational results highlight the efficiency of the new method, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.     

A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.     

A generalized stochastic goal programming model

In this paper we show how one can get stochastic solutions of Stochastic Multi-objective Problem (SMOP) using goal programming models. In literature it is well known that one can reduce a SMOP to deterministic equivalent problems and reduce the analysis of a stochastic problem to a collection of deterministic problems. The first sections of this paper will be devoted to the introduction of deterministic equivalent problems when the feasible set is a random set and we show how to solve them using goal programming technique. In the second part we try to go more in depth on notion of SMOP solution and we suppose that it has to be a random variable. We will present stochastic goal programming model for finding stochastic solutions of SMOP. Our approach requires more computational time than the one based on deterministic equivalent problems due to the fact that several optimization programs (which depend on the number of experiments to be run) needed to be solved. On the other hand, since in our approach we suppose that a SMOP solution is a random variable, according to the Central Limit Theorem the larger will be the sample size and the more precise will be the estimation of the statistical moments of a SMOP solution. The developed model will be illustrated through numerical examples.     

The Multi-Handler Knapsack Problem under Uncertainty

The Multi-Handler Knapsack Problem under Uncertainty is a new stochastic knapsack problem where, given a set of items, characterized by volume and random profit, and a set of potential handlers, we want to find a subset of items which maximizes the expected total profit. The item profit is given by the sum of a deterministic profit plus a stochastic profit due to the random handling costs of the handlers. On the contrary of other stochastic problems in the literature, the probability distribution of the stochastic profit is unknown. By using the asymptotic theory of extreme values, a deterministic approximation for the stochastic problem is derived. The accuracy of such a deterministic approximation is tested against the two-stage with fixed recourse formulation of the problem. Very promising results are obtained on a large set of instances in negligible computing time.     

Computing a Stationary Base-Stock Policy for a Finite Horizon Stochastic Inventory Problem with Non-linear Shortage Costs

Abstract

We consider a periodic-review stochastic inventory problem in which demands for a single product in each of a finite number of periods are independent and identically distributed random variables. We analyze the case where shortages (stockouts) are penalized via fixed and proportional costs simultaneously. For this problem, due to the finiteness of the planning horizon and non-linearity of the shortage costs, computing the optimal inventory policy requires a substantial effort as noted in the previous literature. Hence, our paper is aimed at reducing this computational burden. As a resolution, we propose to compute “the best stationary policy.” To this end, we restrict our attention to the class of stationary base-stock policies, and show that the multi-period, stochastic, dynamic problem at hand can be reduced to a deterministic, static equivalent. Using this important result, we introduce a model for computing an optimal stationary base-stock policy for the finite horizon problem under consideration. Fundamental analytic conclusions, some numerical examples, and related research findings are also discussed.     

带有边集限制的最均匀支撑树问题

本在无向网络中，建立了带有边集限制的最均匀支撑树问题的网络模型．中首先解决最均匀支撑树问题，并给出求无向网络中最均匀支撑树的多项式时间算法；然后，给出了求无向网络中带有边集限制的最小树多项式时间算法；最后，在已解决的两个问题的基础上解决了带有边集限制的最均匀支撑树问题．     

Warm-Start Heuristic for Stochastic Portfolio Optimization with Fixed and Proportional Transaction Costs

We consider a probabilistic portfolio optimization model including fixed and proportional transaction costs. We derive a deterministic equivalent of the probabilistic model for fat-tailed portfolio returns. We develop a method which finds provably near-optimal solutions in minimal amount of time for industry-sized (up to 2000 assets) problems. To solve the mixed-integer nonlinear programming (MINLP) deterministic formulation equivalent to the stochastic problem, we design a mathematical programming-based warm-start heuristic. The tests show the computational efficiency of the heuristic which is more than an order of magnitude faster than Cplex in finding high-quality solutions.     

A warm-start approach for large-scale stochastic linear programs

We describe a way of generating a warm-start point for interior point methods in the context of stochastic programming. Our approach exploits the structural information of the stochastic problem so that it can be seen as a structure-exploiting initial point generator. We solve a small-scale version of the problem corresponding to a reduced event tree and use the solution to generate an advanced starting point for the complete problem. The way we produce a reduced tree tries to capture the important information in the scenario space while keeping the dimension of the corresponding (reduced) deterministic equivalent small. We derive conditions which should be satisfied by the reduced tree to guarantee a successful warm-start of the complete problem. The implementation within the HOPDM and OOPS interior point solvers shows remarkable advantages.     

On the bicriterion – minimal cost/minimal label – spanning tree problem

We address a bicriterion spanning tree problem relevant in some application fields such as telecommunication networks or transportation networks. Each edge is assigned with a cost value and a label (such as a color). The first criterion intends to minimize the total cost of the spanning tree (the summation of its edge costs), while the second intends to get the solution with a minimal number of different labels. Since these criteria, in general, are conflicting criteria we developed an algorithm to generate the set of non-dominated spanning trees. Computational experiments are presented and results discussed.     

Stochastic chaos in a Duffing oscillator and its control

Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations. A system with random parameters is usually called a stochastic system. The modifier ‘stochastic’ here implies dependent on some random parameter. As the system itself is stochastic, so is the response, even under harmonic excitations alone. In this paper stochastic chaos and its control are verified by the top Lyapunov exponent of the system. A non-feedback control strategy is adopted here by adding an adjustable noisy phase to the harmonic excitation, so that the control can be realized by adjusting the noise level. It is found that by this control strategy stochastic chaos can be tamed down to the small neighborhood of a periodic trajectory or an equilibrium state. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation, so that the problem of controlling stochastic chaos can be reduced into the problem of controlling deterministic chaos in the equivalent system. Then the top Lyapunov exponent of the equivalent system is obtained by Wolf’s method to examine the chaotic behavior of the response. Numerical simulations show that the random phase control strategy is an effective way to control stochastic chaos.     

Noise-induced Transitions for an SIV Epidemic Model with Medical-resource Constraints

We consider a stochastically forced epidemic model with medical-resource constraints. In the deterministic case, the model can exhibit two type bistability phenomena, i.e., bistability between an endemic equilibrium or an interior limit cycle and the disease-free equilibrium, which means that whether the disease can persist in the population is sensitive to the initial values of the model. In the stochastic case, the phenomena of noise-induced state transitions between two stochastic attractors occur. Namely, under the random disturbances, the stochastic trajectory near the endemic equilibrium or the interior limit cycle will approach to the disease-free equilibrium. Besides, based on the stochastic sensitivity function method, we analyze the dispersion of random states in stochastic attractors and construct the confidence domains (confidence ellipse or confidence band) to estimate the threshold value of the intensity for noise caused transition from the endemic to disease eradication.     

Interactive decision making for uncertain minimum spanning tree problems with total importance based on a risk-management approach

This paper deals with a minimum spanning tree problem where each edge cost includes uncertainty and importance measure. In risk management to avoid adverse impacts derived from uncertainty, a d-confidence interval for the total cost derived from robustness is introduced. Then, by maximizing the considerable region as well as minimizing the cost-importance ratio, a biobjective minimum spanning tree problem is proposed. Furthermore, in order to satisfy the objects of the decision maker and to solve the proposed model in mathematical programming, fuzzy goals for the objects are introduced as satisfaction functions, and an exact solution algorithm is developed using interactive decision making and deterministic equivalent transformations. Numerical examples are provided to compare our proposed model with some previous models.     

A stochastic programming model to find optimal sample sizes to estimate unknown parameters in an LP

An LP is considered where the technology coefficients are unknown and random samples are taken to estimate them. A stochastic programming problem is formulated to find the optimal sample sizes where it is required that a confidence interval should cover the unknown deterministic optimum value by a given probability and the cost of sampling be minimum.     

New solution approaches for the maximum-reliability stochastic network interdiction problem

We investigate methods to solve the maximum-reliability stochastic network interdiction problem (SNIP). In this problem, a defender interdicts arcs on a directed graph to minimize an attacker’s probability of undetected traversal through the network. The attacker’s origin and destination are unknown to the defender and assumed to be random. SNIP can be formulated as a stochastic mixed-integer program via a deterministic equivalent formulation (DEF). As the size of this DEF makes it impractical for solving large instances, current approaches to solving SNIP rely on modifications of Benders decomposition. We present two new approaches to solve SNIP. First, we introduce a new DEF that is significantly more compact than the standard DEF. Second, we propose a new path-based formulation of SNIP. The number of constraints required to define this formulation grows exponentially with the size of the network, but the model can be solved via delayed constraint generation. We present valid inequalities for this path-based formulation which are dependent on the structure of the interdicted arc probabilities. We propose a branch-and-cut (BC) algorithm to solve this new SNIP formulation. Computational results demonstrate that directly solving the more compact SNIP formulation and this BC algorithm both provide an improvement over a state-of-the-art implementation of Benders decomposition for this problem.