Plausibility relations and multiple expected utilities in decision-making under uncertainty

In this article we develop a new method for dealing with decision-making problems under uncertainty, derived from the one proposed by Fourgeaud, Lenclud and Sentis in 1968. We offer an information gathering procedure based on plausibility relations between ‘possible futures’ which leads to the construction of a polyhedron which contains all the admissible probability measures. Thus it is possible to compare alternatives by associating with each one of them a set of expected utilities calculated from these measures.An application concerning the cost effectiveness of the flexible manufacturing systems figures as an illustration. The utilization facilities of the method and its value as an instrument for supporting decision-making are particularly highlighted here.     

Random volumes under a general matrix-variate model

The convex hull generated by p linearly independent points in Euclidean n-space, n?p will almost surely determine a p-simplex and the corresponding p-parallelotope. The volume of this p-parallelotope is where the rows of the p×n,n?p matrix of rank p represent the p linearly independent points. If the points are random points in some sense then v becomes a random volume. The distribution of this random volume v when the matrix X has a very general real rectangular matrix-variate density is the topic of this paper. The complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on Jacobians of matrix transformations and functions of matrix argument. Apart from the distribution of v under this general model, arbitrary moments of v, connection to the likelihood ratio statistic or λ-criterion for testing hypotheses on the parameters of multivariate normal distributions, connections to Mellin-Barnes integrals and Meijer’s G-function, connection to the concept of generalized variance, various structural decompositions of v and special cases are also examined here.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

On a problem on chamber systems

It is well known that a geometry belonging to a disconnected diagram is the direct sum of geometries corresponding to the connected components of the diagram. On the other hand, chamber systems with a disconnected diagram exist which do not split as direct products of components of smaller rank. Many finite examples of this kind are discussed in Groups of Lie Type and their Geometries (CUP, 1995, pp. 185–214), but none of them is simply connected. In this article, we construct a simply connected finite example.     

The dimension print of most convex surfaces

The dimension print is a concept which contains more detailed information than the usual Hausdorff dimension. So, for example, a sphere and the surface of a cube have same dimension but different dimension prints. Can anything be said about the dimension print of most convex surfaces (in the Baire category sense)?     

The image of a closed convex set under a Fredholm operator

The purpose of this article is two-fold. In the first place, we prove that a set is the image of a non empty closed convex subset of a real Banach space under an onto Fredholm operator of positive index if and only if it can be written as the union of {_Dn:n∈N}$\{D_n:n\in \mathbb{N}\}$, a non-decreasing family of non empty, closed, convex and bounded sets such that _Dn+_Dn+2⊆2_Dn+1$D_n+D_{n+2}\subseteq 2D_{n+1}$ for every n∈N$n\in \mathbb{N}$.     

Social status and long-run effects of monetary policy in a two-sector monetary economy of endogenous growth

We develop a two-sector monetary economy with human capital accumulation and a cash constraint applied to both consumption and investment to examine the ways in which social status affects the impact of monetary policy on the long-run economic growth rate. Our findings suggest that the formation of human capital is an important determinant to the super-neutrality of money in the growth-rate sense. Within an economy with Lucas-type human capital formation, money is super-neutral; however, within an economy where human capital accumulation formation is more generalized, and in which both physical and human capital are used as inputs, the growth rate in money will have a negative effect on the long-run growth rate of the economy. The existence, uniqueness and saddle-path stability of balanced-growth equilibrium are also examined.     

The effects of demand uncertainty on channel structure

Motivated by the business practice whereby some manufacturers open their own retail stores despite the existence of more efficient independent retailers, this paper examines the distribution channel choice of competing manufacturers under demand uncertainty and resale price maintenance. We characterize the conditions for the equilibrium channel structures. We find that (1) manufacturers tend to distribute products with more design attributes through their own retail stores, (2) manufacturers with highly substitutable products are more likely to use independent retailers, and (3) at least one manufacturer has more incentive to open its own retail stores when facing an increase of the market size asymmetry.     

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForXL(E) we denote by the metrical projection ontoX, i.e. the mapping which associates to eachaE the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE2, then for a typicalXL(E) the map is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.     

Direct sum decomposition of geometries based on maximal subgroups

In 1961 J. Tits described a way to define a geometry from a group and a collection of subgroups. Such incidence geometries are now studied by the team of F. Buekenhout in Brussels. Here we present theorems about decomposition of PRI geometries into direct sums and we find the full direct sum decomposition of PRI geometries on solvable groups.     

Note on a general Palais-Smale condition

In this paper, we establish a qualitative deformation theorem for a locally Lipschitz function satisfying a general Palais-Smale type condition. Then we show that, with some assumptions, this compactness condition implies the coercivity of the function.     

A dual strategy for the implementation of the aggregation principle in decision making under uncertainty

A solution method for stochastic programs is proposed based on the aggregation principle, which allows one to find the solution of a stochastic program by aggregating the solutions of individual deterministic scenario problems. The method concentrates on finding good estimates of the dual variables associated with the non-anticipativity constraints.     

The p-median problem under uncertainty

Consider the need to currently locate p facilities but it is possible that up to q additional facilities will have to be located in the future. There are known probabilities that 0 ? r ? q facilities will need to be located. The p-median problem under uncertainty is to find the location of p facilities such that the expected value of the objective function in the future is minimized. The problem is formulated on a graph, properties of it are proven, an integer programming formulation is constructed, and heuristic algorithms are suggested for its solution. The heuristic algorithms are modified to reduce the run time by about two orders of magnitude with minimal effect on the quality of the solution. Optimal solutions for many problems are found effectively by CPLEX. Computational results using the heuristic algorithms are presented.     

The optimal harvesting problem under price uncertainty

In this paper we study the exploitation of a one species forest plantation when timber price is governed by a stochastic process. The work focuses on providing closed expressions for the optimal harvesting policy in terms of the parameters of the price process and the discount factor, with finite and infinite time horizon. We assume that harvest is restricted to mature trees older than a certain age and that growth and natural mortality after maturity are neglected. We use stochastic dynamic programming techniques to characterize the optimal policy and we model price using a geometric Brownian motion and an Ornstein–Uhlenbeck process. In the first case we completely characterize the optimal policy for all possible choices of the parameters. In the second case we provide sufficient conditions, based on explicit expressions for reservation prices, assuring that harvesting everything available is optimal. In addition, for the Ornstein–Uhlenbeck case we propose a policy based on a reservation price that performs well in numerical simulations. In both cases we solve the problem for every initial condition and the best policy is obtained endogenously, that is, without imposing any ad hoc restrictions such as maximum sustained yield or convergence to a predefined final state.     

Programming under uncertainty: The complete problem

We define the complete problem of a two-stage linear programming under uncertainty, to be:

$$\begin{gathered} Minimize z(x) = E_\xi \{ cx + q^ + y^ + + q^ - y^ - \} \hfill \\ subject to Ax = b \hfill \\ Tx + Iy^ + + Iy^ - = \xi \hfill \\ x \geqq 0,y^ + \geqq 0,y^ - \geqq 0 \hfill \\ \end{gathered} $$     

The consequences of irreversibility on optimal intertemporal emission policies under uncertainty

This paper investigates how irreversibility affects optimal intertemporal emission policies when negative stock externalities exist. In particular it discusses the effect of irreversible emission, i.e., it concerns the physical issue whether it is possible to recollect pollutants that have been emitted or not. We depict our analysis with the greenhouse effect as a topical example and model the uncertainty with respect to the future evolution of the world’s temperature (i.e., the uncertain factor that determines the costs) as Itô-process with the drift provided by current carbon-dioxide emissions. We show analytically that irreversibility affects the optimal emission policy only if the future impact of today’s emissions is uncertain. Under uncertainty, irreversibility leads to a conservationist policy such that emissions are reduced at any level of environmental concentration of the pollutant. The level where stopping emissions is optimal decreases in the presence of irreversibility. Furthermore, the expected duration of fossil fuel use is derived. A numerical example which is calibrated to roughly reflect the global CO₂ problem illustrates the analytical findings.     

Algebraic radicals and incidence algebras

In this note we prove the uniqueness of U in a group G with a spherical split-BN-pair of rank ,i.e., if G has such a BN-pair with a nilpotent normal subgroup of B, and , then and is a normal subgroup of G. Here is the corresponding group of Lie-type and the subgroup of generated by all root-subgroups corresponding to positive roots. Received: 19 May 2003     

Dynamic journeying under uncertainty

We introduce a journey planning problem in multi-modal transportation networks under uncertainty. The goal is to find a journey, possibly involving transfers between different transport services, from a given origin to a given destination within a specified time horizon. Due to uncertainty in travel times, the arrival times of transport services at public transport stops are modeled as random variables. If a transfer between two services is rendered unsuccessful, the commuter has to reconsider the remaining path to the destination. The problem is modeled as a Markov decision process in which states are defined as paths in the transport network. The main contribution is a backward induction method that generates an optimal policy for traversing the public transport network in terms of maximizing the probability of reaching the destination in time. By assuming history independence and independence of successful transfers between services we obtain approximate methods for the same problem. Analysis and numerical experiments suggest that while solving the path dependent model requires the enumeration of all paths from the origin to the destination, the proposed approximations may be useful for practical purposes due to their computational simplicity. In addition to on-time arrival probability, we show how travel and overdue costs can be taken into account, making the model applicable to freight transportation problems.     

The mixed capacitated general routing problem under uncertainty

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known. Since each vehicle has a limited capacity, the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence, become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrained integer programming formulation of the problem for which a deterministic equivalent is derived. The introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristic solution approach exploring promising parts of the search space. The effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem.