Pascal Lang

Distributional efficiency in multiobjective stochastic linear programming

Several concepts of distributional efficiency are proposed for the Multiobjective Stochastic Linear Programming (MSLP) problem, in contexts where the probability distribution of random parameters is known and the decision maker (DM) has an unknown multi-attribute utility function belonging to a given glass U. We present a general efficient set, the U-admissible solutions, and two subsets, the U-unanimous and U-advocated solutions, the latter being particularly relevant to the case of a single DM. We show how advocated solutions can be generated and/or tested when U is the class of non-decreasing additive concave functions.

Piecewise affine approximations for the control of a one-reservoir hydroelectric system

Abstract We analyze the computation of optimal and approximately optimal policies for a discrete-time model of a single reservoir whose discharges generate hydroelectric power. Inflows in successive periods are random variables. Revenue from hydroelectric production is represented by a piecewise linear function. We use the special structure of optimal policies, together with piecewise affine approximations of the optimal return functions at each stage of dynamic programming, to decrease the computational effort by an order of magnitude compared with ordinary value iteration. The method is then used to obtain easily computable lower and upper bounds on the value function of an optimal policy, and a policy whose value function is between the bounds.

Lower bounding aggregation and direct computation for an infinite horizon one-reservoir model

Abstract We present a specialized policy iteration method for the computation of optimal and approximately optimal policies for a discrete-time model of a single reservoir whose discharges generate hydroelectric power. The model is described in (Lamond et al., 1995) and (Drouin et al., 1996), where the special structure of optimal policies is given and an approximate value iteration method is presented, using piecewise affine approximations of the optimal return functions. Here, we present a finite method for computing an optimal policy in O(n3) arithmetic operations, where n is the number of states in the associated Markov decision process, and a finite method for computing a lower bound on the optimal value function in O(m2n) where m is the number of nodes of the piecewise affine approximation.

Approximate stochastic dynamic programming for hydroelectric production planning

This paper presents a novel approach for approximate stochastic dynamic programming (ASDP) over a continuous state space when the optimization phase has a near-convex structure. The approach entails a simplicial partitioning of the state space. Bounds on the true value function are used to refine the partition. We also provide analytic formulae for the computation of the expectation of the value function in the “uni-basin” case where natural inflows are strongly correlated. The approach is experimented on several configurations of hydro-energy systems. It is also tested against actual industrial data.

Social Performance and Financial Performance: A Controversial Relationship

Different factors explaining divergent results on the relationship between corporate Social Performance (SP) and Financial Performance (FP) can be found in the academic literature. The main objective of this chapter is to test the impact of these factors on these divergent results. It also aims to assess the intensity of the sensitivity of this relationship to these factors considered individually or in combination. The results of our experimental research show that the estimated relationship depends on the methodological choice. More specifically, the relationship varies according to the measurement of the SP, the measurement of FP and the chosen sample. This relationship is neither stable nor necessarily linear, as many relevant academic works in the literature assume. This work concentrates on the knowledge gained from this literature and suggests lines of reflection to better understand the studied relationship in a field which is still evolving.

Controlled approximation of the value function in stochastic dynamic programming for multi-reservoir systems

We present a new approach for adaptive approximation of the value function in stochastic dynamic programming. Under convexity assumptions, our method is based on a simplicial partition of the state space. Bounds on the value function provide guidance as to where refinement should be done, if at all. Thus, the method allows for a trade-off between solution time and accuracy. The proposed scheme is experimented in the particular context of hydroelectric production across multiple reservoirs.

Distributional Unanimity in Multiobjective Stochastic Linear Programming

Several notions of efficiency are conceivable for the multiobjective stochastic linear programming problem. In this paper, assuming that the problem’s randomness can be described by discrete scenarios with known probabilities and that decision makers’ preferences, although unknown, can be represented by a class of utility functions, we examine a set of strongly efficient solutions, the unanimous solutions. We state inclusion relations between this and other classes of efficient solutions (admissible and advocated solutions) previously studied. Under plausible assumptions about decision makers’ risk attitudes, we examine how candidates for unanimity can be generated and then tested.

Controlled Approximation of the Stochastic Dynamic Programming Value Function for Multi-Reservoir Systems

We present an approximation of the Stochastic Dynamic Programming (SDP) value function based on a partition of the state space into simplices. The vertices of such simplices form an irregular grid over which the value function is computed. Under convexity assumptions, lower and upper bounds are developed over the state space continuum. The partition is then refined where the gap between these bounds is largest. This process readily provides a controllable trade-off between accuracy and solution time.

Stochastic optimization of multi-reservoir systems with power plants and spillways.

We examine a stochastic optimization model of a multiple reservoir water resource system in which the spilled outflows may have a different routing than the turbined outflows. We extend some results about the monotonicity of optimal decision rules, which were known for particular routings, and we show their validity for arbitrary routings of spilled outflows, provided they satisfy an intuitive monotonicity condition. Special cases are when the spilled outflows are expelled from the system, or when the spilled outflows are routed to the next reservoir downstream. The monotonicity of optimal policies and of the corresponding future value function can be exploited to develop efficient computational algorithms based on a dynamic programming methodology, especially when the rewards are given by a concave, piecewise linear function of electricity generation.

Dominance and Efficiency in Multiobjective Stochastic Linear Programming

In contrast to deterministic multiobjective problems, where the notion of Pareto-efficiency is well accepted, several notions of efficiency are conceivable for the Multiobjective Stochastic Linear Programming problem. Their relevance depends on the available state of information about the decision situation, regarding particularly the Decision Maker’s preference structure and probabilistic anticipations. We investigate efficient sets arising naturally from some extreme reference cases of states of information. We study these sets from the point of view of their relative inclusions and provide some indications as to their computability.