On the linear combinations of stochastic variables

Summary In this article we give a general distribution of the linear combinations of independent non-negative stochastic variables. This result generalizes a class of distributions of the linear combinations of independent non-negative variates and further a number of other particular cases can be obtained from the general distribution discussed in this article. In deriving the general distribution we used the properties of the most generalized special function, namely, the H-function ofFox [1961] andBraaksma [1964].     

On a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time

This paper seeks to highlight two approaches to the solution of stochastic control and optimal stopping problems in continuous time. Each approach transforms the stochastic problem into a deterministic problem. Dynamic programming is a well-established technique that obtains a partial/ordinary differential equation, variational or quasi-variational inequality depending on the type of problem; the solution provides the value of the problem as a function of the initial position (the value function). The other method recasts the problems as linear programs over a space of feasible measures. Both approaches use Dynkin’s formula in essential but different ways. The aim of this paper is to present the main ideas underlying these approaches with only passing attention paid to the important and necessary technical details.     

A multi-stage stochastic programming approach for pre-positioning of relief supplies considering returns

A dynamic pre-positioning problem is proposed to efficiently respond to victims’ need for relief supplies under uncertain and dynamic demand in humanitarian relief. The problem is formulated as a multi-stage stochastic programming model that considers pre-positioning with the dynamic procurement and return decisions about relief supplies over a time horizon. To validate the advantages of dynamic pre-positioning, three additional pre-positioning strategies are presented: pre-positioning with one-time procurement and without returns, pre-positioning with one-time procurement and returns, and pre-positioning with dynamic procurement and without returns. Using data from real-world disasters in the United States in the Emergency Events Database, we present a numerical analysis to study the applicability of the proposed models. We develop a sample average approximation approach to solving the proposed model in large-scale cases. Our main contribution is that we integrate dynamic procurement and return strategies into pre-positioning to decrease both costs and shortage risks in uncertain and dynamic contexts. The results illustrate that dynamic pre-positioning outperforms the other three strategies in cost savings. It also indicates that a higher return price is particularly helpful for decreasing unmet demand. The proposed models can help relief agencies evaluate and choose the solutions that will have the greatest overall effectiveness in the context of different relief practices.     

The complexity of linear programming

The simplex method for linear programming has always been very successful from a practical point of view. In the worst case, however, the method may require a computational effort that increases exponentially with problem size. Recently L.G. K hachian proposed an entirely different solution method whose running time is bounded by a polynomial function of problem size, thereby settling a major open problem in computational complexity theory. We review the developments preceding K hachian 's discovery, describe the algorithm and discuss its implications.     

Dynamic programming for non-additive stochastic objectives

We derive the existence of an optimum and the techniques of dynamic programming for non-additive stochastic objectives. Our key assumption for non-negative objectives is that asymptotic impatience exceeds asymptotic ‘mean’ growth, where ‘mean’ growth is derived not only from intertemporal inelasticity and the random return on investment but also from the curvature of the non-additive stochastic aggregator (i.e. the ‘certainty equivalent’). We provide broad families of new, interesting, and tractable examples. They illustrate that ‘mean’ growth can exist even when the distribution of returns has unbounded support, that power discounting often implies infinite asymptotic impatience, and that non-positive objectives are easily handled with few restrictions on growth.     

General parametric linear fractional programming

This paper deals with the behaviour of solutions to a linear fractional programming problem when the coefficients of the objective function are allowed to vary.     

Two-stage stochastic integer programming: a survey

Stochastic integer programming is more complicated than stochastic linear programming, as will be explained for the case of the two-stage stochastic programming model. A survey of the results accomplished in this recent field of research is given.     

A linear programming solution to the faculty assignment problem

This paper investigates the problem of assigning faculty to courses at a university. A program is developed which is both efficient in that integer programming is not required, and effective, in that it facilitates interaction by administration in determining the optimal solution. The results of some empirical tests are also reported.     

Analyses of the solution to a linear fractional functionals programming

Summary In this paper the effect of changing one constant of the Linear Fractional Functionals Programming problem has been discussed under the condition that the optimal basis for the original problem remains unaffected. In the changed form the optimal solution and new value of the objective functions are obtained.     

Multi-objective linear programming with partial ranking of objectives

This paper is concerned with multi-objective linear programming problems in which the objective functions can be partially ranked. We represent the set of admissible weight vectors by a system of linear constraints and solve for the policy most likely to be optimum. If each admissible weight vector has the same probability of being correct, the optimum policy maximizes the hypervolume of the polytope of weight vectors having this policy as a solution. The proposed algorithm requires the enumeration of the subset of admissible efficient solutions of a multi-objective linear program. For each admissible solution, we estimate the ratio of the volumes of the corresponding polytope of weight vectors and the polytope of all admissible weight vectors. An algorithm is outlined for numerical integration using the Monte Carlo method. The model is extended to the case where several objectives are expressed as linear constraints with multiple parameter vectors and there is uncertainty about the weighting of these parameters. A numerical example is provided.     

A linear programming model of housing market equilibrium

This paper reviews Wheaton's assertion that the Herbert-Stevens Linear Programming Model produces solutions which do not meet Alonso's criteria for market equilibrium. It demonstrates that, although Wheaton's criticism is valid in general, under certain conditions an alternative specification of the objective function coefficients and the dual variables in the linear program produces a model whose solutions satisfy Alonso's criteria. Finally, this paper shows that the NBER model is an application in which a linear programming problem computes market equilibria.