Dynamic programming and minimum risk paths

This paper addresses the problem of computing minimum risk paths by taking as objective the expected accident cost. The computation is based on a dynamic programming formulation which can be considered an extension of usual dynamic programming models: path costs are recursively computed via functions which are assumed to be monotonic. A large part of the paper is devoted to analyze in detail this formulation and provide some new results. Based on the dynamic programming model a linear programming model is also presented to compute minimum risk paths. This formulation turns out to be useful in solving a biobjective version of the problem, in which also expected travel length is taken into consideration. This leads to define nondominated mixed strategies. Finally it is shown how to extend the basic updating device of dynamic programming in order to enumerate all nondominated paths.     

An Optimal Chance Constraint Multivariate Stratified Sampling Design Using Auxiliary Information

When we are dealing with multivariate problem then we need an allocation which is optimal for all the characteristics in some sense because the individual optimum allocations usually differ widely unless the characteristics are highly correlated. So an allocation called “Compromise allocation” is to be worked out suggested by Cochran. When auxiliary information is also available, it is customary to use it to increase the precision of the estimates. Moreover, for practical implementation of an allocation, we need integer values of the sample sizes. In the present paper the problem is to determine the integer optimum compromise allocation when the population means of various characteristics are of interest and auxiliary information is available for the separate and combined ratio and regression estimates. This paper considers the optimum compromise allocation in multivariate stratified sampling with non-linear objective function and probabilistic non-linear cost constraint. The probabilistic non-linear cost constraint is converted into equivalent deterministic one by using Chance Constrained programming. The formulated multi-objective nonlinear programming problem is solved by Fuzzy Goal programming approach and Chebyshev approximation. Numerical illustration is also given to show the practical utility of the approaches.     

Two Probabilistic Models for Solving an Oxygen-bottle Inventory Problem

In this paper, an inventory oxygen-bottles problem is formulated as (i) an (M/M/C):(GD/N/∞) queueing model with different service rates; (ii) a probabilistic goal programming (PGP) model. By using the first model, the optimum required rate for oxygen bottles to be stored is determined. By using the second model, the decision-maker can determine his policy, which minimizes the total cost. Finally, a case study of the storage of oxygen bottles for the Egyptian Air Force (1980-1984) is presented.     

Two basic problems in reliability-based structural optimization

Optimization of structures with respect to performance, weight or cost is a well-known application of mathematical optimization theory. However optimization of structures with respect to weight or cost under probabilistic reliability constraints or optimization with respect to reliability under cost/weight constraints has been subject of only very few studies. The difficulty in using probabilistic constraints or reliability targets lies in the fact that modern reliability methods themselves are formulated as a problem of optimization. In this paper two special formulations based on the so-called first-order reliability method (FORM) are presented. It is demonstrated that both problems can be solved by a one-level optimization problem, at least for problems in which structural failure is characterized by a single failure criterion. Three examples demonstrate the algorithm indicating that the proposed formulations are comparable in numerical effort with an approach based on semi-infinite programming but are definitely superior to a two-level formulation.     

Scheduling ambulance crews for maximum coverage

This paper addresses the problem of scheduling ambulance crews in order to maximize the coverage throughout a planning horizon. The problem includes the subproblem of locating ambulances to maximize expected coverage with probabilistic response times, for which a tabu search algorithm is developed. The proposed tabu search algorithm is empirically shown to outperform previous approaches for this subproblem. Two integer programming models that use the output of the tabu search algorithm are constructed for the main problem. Computational experiments with real data are conducted. A comparison of the results of the models is presented.     

Markov Decision Processes with a Constraint on the Asymptotic Failure Rate

In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a stationary policy which minimizes the infinite horizon expected average cost, given that the system never fails. Using Perron-Frobenius theory of non-negative matrices and spectral analysis, we show that the problem can be reduced to a linear programming problem. Finally, we apply this method to a real problem for an aeronautical system.     

Variable neighbourhood search heuristics for the probabilistic multi-source Weber problem

The Multi-source Weber Problem (MWP) is concerned with locating m facilities in the Euclidean plane, and allocating these facilities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation for the probabilistic MWP called the PMWP. For its solution, we propose two heuristics based on variable neighbourhood search (VNS). Computational results obtained on a number of test instances show that the VNS heuristics improve the performance of a probabilistic alternate location-allocation heuristic referred to as PALA. In its original form, the applicability of the new heuristics depends on the existence of a closed-form expression for the expected distances between facilities and customers. Unfortunately, such an expression exists only for a few distance function and probability distribution combinations. We therefore use two approximation methods for the expected distances, which make the VNS heuristics applicable for any distance function and bivariate distribution of customer locations.     

EXPECTED NUMBER OF ITERATIONS OF INTERIOR-POINT ALGORITHMS FOR LINEAR PROGRAMMING

We study the behavior of some polynomial interior-point algorithms for solving random linear programming (LP) problems. We show that the expected and anticipated number of iterations of theseTodd‘s probabilisticalgorithms is bounded above by O(n^1.5). The random LP problem is model with the Cauchy distribution.     

Stochastic chance constrained mixed-integer nonlinear programming models and the solution approaches for refinery short-term crude oil scheduling problem

Stochastic chance constrained mixed-integer nonlinear programming (SCC-MINLP) models are developed in this paper to solve the refinery short-term crude oil scheduling problem which concerns crude oil unloading, mixing, transferring and multilevel inventory control under demands uncertainty of distillation units. The objective of these models is the minimum expected value of total operation cost. It is the first time that the uncertain demands of Crude oil Distillation Units (CDUs) in these problems are set as random variables which have discrete and continuous joint probability distributions. This situation is close to the real world industry use. To reduce the computation complexity, these SCC-MINLP models are transformed into their equivalent stochastic chance constrained mixed-integer linear programming models (SCC-MILP). Stochastic simulation and stochastic sampling technologies are introduced in detail to solve these complex SCC-MILP models. Finally, case studies are effectively solved with the proposed approaches.     

An exact decomposition algorithm for a chance-constrained new product risk model

We use the generalized two-sided Chebyshev inequality to reformulate a certain nonlinear, chance-constrained new product risk model. The problem has a linear cost objective and a constraint set featuring a probabilistic lower bound on an event which depends on a collection of mutually-independent, uniform random parameters. Our reformulation permits a reduction of the problem to a sequence of second-order cone programs. We, therefore, identify a new family of non-convex programs whose members are amenable to convex programming solution techniques.     

A probabilistic model applied to emergency service vehicle location

This paper is concerned with the formulation and the solution of a probabilistic model for determining the optimal location of facilities in congested emergency systems. The inherent uncertainty which characterizes the decision process is handled by a new stochastic programming paradigm which embeds the probabilistic constraints within the traditional two-stage framework. The resulting model drops simplifying assumptions on servers independence allowing at the same time to handle the spatial dependence of demand calls. An exact solution method and different tailored heuristics are presented to efficiently solve the problem. Computational experience is reported with application to various networks.     

Probabilistic Networks and R & D Portfolio Selection

Mathematical programming methods have been suggested and used as an aid to R & D project portfolio selection. One of the main criticisms of the use of such models is that the stochastic nature of the problem has been largely ignored. This paper presents a method which takes into account the stochastic nature of resource requirements and project benefits, using a combination of probabilistic networks, simulation and mathematical programming. A case study based on data from an industrial R & D laboratory is presented and compared with the use of expected value methods. The results of the study indicate that in this particular case the deterministic linear programming solution is robust.     

Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm

In this article, a capacitated location allocation problem is considered in which the demands and the locations of the customers are uncertain. The demands are assumed fuzzy, the locations follow a normal probability distribution, and the distances between the locations and the customers are taken Euclidean and squared Euclidean. The fuzzy expected cost programming, the fuzzy β-cost minimization model, and the credibility maximization model are three types of fuzzy programming that are developed to model the problem. Moreover, two closed-form Euclidean and squared Euclidean expressions are used to evaluate the expected distance between customers and facilities. In order to solve the problem at hand, a hybrid intelligent algorithm is applied in which the simplex algorithm, fuzzy simulation, and a modified genetic algorithm are integrated. Finally, in order to illustrate the efficiency of the proposed hybrid algorithm, some numerical examples are presented.     

On the optimality of a full-service policy for a queueing system with discounted costs

We provide weak sufficient conditions for a full-service policy to be optimal in a queueing control problem in which the service rate is a dynamic decision variable. In our model there are service costs and holding costs and the objective is to minimize the expected total discounted cost over an infinite horizon. We begin with a semi-Markov decision model for a single-server queue with exponentially distributed inter-arrival and service times. Then we present a general model with weak probabilistic assumptions and demonstrate that the full-service policy minimizes both finite-horizon and infinite-horizon total discounted cost on each sample path.     

概率约束随机规划的一种近似方法及其它的有效解模式

根据最小风险的投资最优问题，我们给出了一个统一的概率约束随机规划模型。随后我们提出了求解这类概率约束随机规划的一种近似算法，并在一定的条件下证明了算法的收敛性。此外，提出了这种具有概率约束多目标随机规划问题的一种有效解模型。     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

Re-solving stochastic programming models for airline revenue management

We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi-stage stochastic programming formulation to the seat allocation problem. Our theoretical results show that the proposed approximation is robust, in the sense that solving more successive two-stage programs can never worsen the expected revenue obtained with the corresponding allocation policy. Although intuitive, such a property is known not to hold for the traditional deterministic linear programming model found in the literature. We also show that this property does not hold for some bid-price policies. In addition, we propose a heuristic method to choose the re-solving points, rather than re-solving at equally-spaced times as customary. Numerical results are presented to illustrate the effectiveness of the proposed approach.     

Probabilistic programming models for traffic incident management operations planning

This paper proposes mathematical programming models with probabilistic constraints in order to address incident response and resource allocation problems for the planning of traffic incident management operations. For the incident response planning, we use the concept of quality of service during a potential incident to give the decision-maker the flexibility to determine the optimal policy in response to various possible situations. An integer programming model with probabilistic constraints is also proposed to address the incident response problem with stochastic resource requirements at the sites of incidents. For the resource allocation planning, we introduce a mathematical model to determine the number of service vehicles allocated to each depot to meet the resource requirements of the incidents by taking into account the stochastic nature of the resource requirement and incident occurrence probabilities. A detailed case study for the incident resource allocation problem is included to demonstrate the use of proposed model in a real-world context. The paper concludes with a summary of results and recommendations for future research.     

Production planning and warehouse management in supply networks with inter-facility mold transfers

This paper, considers with the problem of production capacity and warehouse management in a supply network in which inter-plant mold transfers are enabled. The supply network has a limited number of very expensive molds which can be transferred from a plant to another making it possible for each plant to produce the entire product gamut. It is assumed that warehouses in this supply network can be activated and deactivated as required, and that material transfers from a warehouse to another are also possible. The objective is to develop a capacity and warehouse management plan that satisfies the expected market demands with the lowest possible cost. A mixed integer programming model for the problem is suggested and its properties are discussed. A linear programming-based heuristic that combines Lagrangian relaxation and linear programming duality to generate lower and upper bounds for the problem is proposed. Finally, based on a designed experiment the performance of the heuristic on a set of generated test problems is reported and discussed.     

How big is too big? Trading off the economies of scale of larger telecommunications network elements against the risk of larger outages

This paper builds a probabilistic model to analyze the risk–reward tradeoffs that larger telecommunications network elements present. Larger machines offer rewards in the form of cost savings due to economies of scale. But large machines are riskier because they affect more customers when they fail. Our model translates the risk of outages into dollar costs, which are random variables. This step enables us to combine the deployment cost and outage cost into a total cost. Once we express the decision makers’ preferences via a utility function, we can find the machine size that minimizes the total cost’s expected utility, thereby achieving an optimal tradeoff between reward and risk. The expected utility answers the question “how big is too big?”.