Antoine Gautier

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.

A parametric analysis of a nonlinear asset allocation management model

We show in this note how an asset allocation model can be formulated as a nonlinear parametric network flow problem with one additional linear constraint. We then recommend a method by which a decision-maker could understand qualitatively how to respond to changes in the environment such as variations in interest rates, taxes or asset prices without any additional computation.

Heuristics for determining economic processing rates in a flexible manufacturing system

Abstract In this paper, models are presented for determining economic processing speeds and tool loading to minimize the makespan required to produce a given set of parts in a flexible manufacturing system. Using Taylors tool life equation, models for determining the optimal processing speeds and the tools to be loaded into finite capacity machine magazines are formulated to minimize the maximum processing time in the system. These problems are evaluated for computational complexity, and several heuristics for obtaining good feasible solutions to the problem are discussed. The quality of the solutions obtained using these heuristics is evaluated by computational experiments against lower bounds established by either relaxations or optimal solutions when possible.

Ripples, complements, and substitutes in singly constrained monotropic parametric network flow problems

We extend the qualitative theory of sensitivity analysis for minimum-cost flow problems developed by Granot and Veinott to minimum-cost flow problems with one additional linear constraint. Two natural extensions of the “less dependent on” partial ordering of the arcs are presented. One is decidable in linear time, whereas the other yields more information but is NP-complete in general. The Ripple Theorem gives upper bounds on the absolute value of optimal-flow variations as a function of variations in the problem parameters. The theory of substitutes and complements presents necessary and sufficient conditions for optimal-flow changes to consistently have the same (or the opposite) sign in two given arcs. The Monotonicity Theory links the changes in the value of the parameters to the change in the optimal arc-flows, and bounds on the rates of changes are discussed. The departure from the pure network structure is shown to have a profound effect on computational issues. Indeed, the complexity of determining substitutes and complements, although linear for the unconstrained (no additional constraint) case, is shown to be NP-complete in general for the constrained case. However, for all intractable problems, families of cases arise from easily recognizable graph structures which can be computed in linear time.

The Québec Ministry of Natural Resources Uses Linear Programming to Understand the Wood-Fiber Market

In spring 1996, Qu�©becs Ministry of Natural Resources began using a descriptive mathematical programming model to support various negotiations in the wood-fiber markets. The model, which uses linear programming to solve an economic-equilibrium program, allows the representative of the ministry to come to industry roundtables with accurate scenario analyses for the wood-fiber market. The tool we developed and implemented uses the large amounts of data available to government agencies to foresee and explain the general economic trends facing both lumber and paper producers. During its development, our team of operations-research experts, economists, engineers, and civil servants developed an unprecedented understanding of the wood-fiber market. The ministry incorporated these insights in subsequent government policy aimed at improving sawmill yield and stabilizing market behavior.

Ripples, complements, and substitutes in generalized networks

We extend the qualitative theory of sensitivity analysis for minimum-cost pure network flows of Granot and Veinott [17] to generalized network flow problems, that is, network flow problems where the amount of flow picked up by an arc is multiplied by a (positive) gain while traversing the arc. Three main results are presented. The ripple theorem gives upper bounds on the absolute value of optimal-flow variations as a function of variations in the problem parameter(s). The theory of substitutes and complements provides necessary and sufficient conditions for optimal-flow changes to consistently have the same (or the opposite) sign(s) in two given arcs, whereas the monotonicity theorem links changes in the value of the parameters to changes in optimal arc flows. Bounds on the rates of changes are also discussed. Compared with pure networks, the presence of gains makes qualitative sensitivity analysis here a much harder task. We show the profound effect on computational issues caused by the departure from the pure network structure.

Qualitative Sensitivity Analysis in Monotropic Programming

Optimal selections are parameter-dependent optimal solutions of parametric optimization problems whose properties can be used in sensitivity analysis. Here we present a qualitative theory of sensitivity analysis for linearly-constrained convex separable (i.e., monotropic) parametric optimization problems. Three qualitative sensitivity analysis results previously derived for network flows are extended to monotropic problems: The Ripple and Smoothing Theorems give upper bounds on the magnitude of optimal-variable variations as a function of variations in the problem parameter(s), the theory of substitutes and complements provides necessary and sufficient conditions for optimal-variable changes to consistently have the same (or the opposite) sign(s) in two given variables, and the Monotonicity Theorem links changes in the value of the parameters to changes in optimal decision variables. We introduce a class of optimal selections for which these results hold, thereby answering a long-standing question due to Granot and Veinott (1985) with a simple and elegant method. Although a number of results are known to depend on the resolution of NP-complete problems, easily computable nonnetwork classes of monotropic problems such as unimodular systems of constraints emerge in the light of the present approach.

On the equivalence of constrained and unconstrained flows

A linear-time algorithm that reduces the set of flows on a directed graph with an additional linear equality constraint to that of lower dimensional subgraphs is presented. These subgraphs form a partition of the original graph, and at most one is singly constrained while the others are unconstrained. If none of the subgraphs is constrained, the algorithm provides a linear-time recognition and transformation of constrained to unconstrained network flows. Applications to constrained network flow optimization are given.

Alternative foreign exchange management protocols: an application of sensitivity analysis

Abstract This paper considers the choice between two foreign exchange management protocols, namely whether to hold currencies received until required, or whether to convert foreign currencies into the home reporting currency and back as needed. The alternative protocols involve a tradeoff between saving transaction costs versus stability of home currency values and economies of scale in interest earned on working capital balances. Qualitative sensitivity analysis is applied to the currency management problem in this form to investigate sensitivities and interdependencies in sources of and needs for currencies. The analysis reveals several implications which are not apparent without viewing the problem in such a context.

Qualitative Sensitivity Analysis

In (post-optimal) sensitivity analysis, one usually draws information from both the objective function and the feasible set. In this chapter, we propose an approach based mainly on the latter, and whose predictions are valid as long as the objective function satisfies some regularity conditions such as additivity, convexity and submodularity. The resulting “pre-optimal” analysis is qualitative and provides sensitivity analysis results prior to solving the problem, which is an attractive feature for large-scale problems where the use of conventional parametric programming methods has a very high computational cost. Among the results discussed, the Ripple Theorem provides an upper bound on the variation in the optimal value of a variable resulting from changes in problem parameters. The theory of substitutes and complements leads to necessary and sufficient conditions for changes in optimal values of two arbitrary decision variables to consistently be of the same (or opposite) sign. The Monotonicity and Smoothing Theorems link changes in problem parameters to changes in optimal variable values, and provide bounds on the absolute change in an optimal solution. Computational issues are discussed, and a qualitative analysis of a production/inventory problem and an asset allocation problem are presented as tutorial examples.