Juan F. Monge

On stochastic dynamic programming for solving large-scale planning problems under uncertainty

The stochastic dynamic programming approach outlined here, makes use of the scenario tree in a back-to-front scheme. The multi-period stochastic problems, related to the subtrees whose root nodes are the starting nodes (i.e., scenario groups), are solved at each given stage along the time horizon. Each subproblem considers the effect of the stochasticity of the uncertain parameters from the periods of the given stage, by using curves that estimate the expected future value (EFV) of the objective function. Each subproblem is solved for a set of reference levels of the variables that also have nonzero elements in any of the previous stages besides the given stage. An appropriate sensitivity analysis of the objective function for each reference level of the linking variables allows us to estimate the EFV curves applicable to the scenario groups from the previous stages, until the curves for the first stage have been computed. An application of the scheme to the problem of production planning with logical constraints is presented. The aim of the problem consists of obtaining the planning of tactical production over the scenarios along the time horizon. The expected total cost is minimized to satisfy the product demand. Some computational experience is reported. The proposed approach compares favorably with a state-of-the-art optimization engine in instances on a very large scale. Scope and purpose: For quite some time, we have known that traditional methods of deterministic optimization are not suitable to capture the truly dynamic nature of most real-life problems, in view of the fact that the parameters which represent information concerning the future are uncertain. Many of the problems in planning under uncertainty, have logical constraints that require 0-1 variables in their formulation and can be solved via stochastic integer programming using scenario tree analysis. Given the dimensions of the deterministic equivalent model (DEM) of the stochastic problem, certain decomposition approaches can be considered by exploiting the structure of the models. Traditional decomposition schemes, such as the Benders and Lagrangean approaches, do not appear to provide the solution for large-scale problems (mainly in the cardinality of the scenario tree) in affordable computing effort. In this work, a stochastic dynamic programming approach is suggested, which we feel is particularly suited to exploit the scenario tree structure and, therefore, very amenable to finding solutions to very large-scale DEMs. The pilot case used involves a classical tactical production planning problem, where the structure is not exploited by the proposed approach so that it is generally applicable.

Design and analysis of hybrid metaheuristics for the Reliability p-Median Problem

In the p-Median Problem, it is assumed that, once the facilities are opened, they may not fail. In practice some of the facilities may become unavailable due to several factors. In the Reliability p-Median Problem some of the facilities may not be operative during certain periods. The objective now is to find facility locations that are both inexpensive and also reliable. We present different configurations of two hybrid metaheuristics to solve the problem, a genetic algorithm and a scatter search approach. We have carried out an extensive computational experiment to study the performance of the algorithms and compare its efficiency solving well-known benchmark instances.

Expected Future Value Decomposition Based Bid Price Generation for Large-Scale Network Revenue Management

This paper studies a multistage stochastic programming SP model for large-scale network revenue management. We solve the model by means of the so-called expected future value EFV decomposition via scenario analysis, estimating the impact of the decisions made at a given stage on the objective function value related to the future stages. The EFV curves are used to define bid prices on bundles of resources directly, as opposed to the traditional additive approach. We compare our revenues to those obtained by additive bid prices, such as the bid prices derived from the deterministic equivalent model DEM of the compact representation of the SP model. Our computational experience shows that the revenues obtained by our approach are better for middle-range values of the load factor of demand, whereas the differences among all the approaches we have tested are insignificant for extreme values. Moreover, our approach requires significantly less computation time than does the optimization of DEM by plain use of optimization engines. Problem instances with 72 pairs of bundle-fare classes have been solved in less than one minute, with 800 pairs in less than five minutes, and with 4,000 pairs in less than one hour. The time taken by DEM was, in general, of one order of magnitude higher. Finally, for the three largest problem instances, and after two hours, the expected revenue returned by DEM was below that obtained by EFV by 13.47%, 17.14%, and 38.94%, respectively.

On the time-consistent stochastic dominance risk averse measure for tactical supply chain planning under uncertainty

Abstract In this work a modeling framework and a solution approach have been presented for a multi-period stochastic mixed 0–1 problem arising in tactical supply chain planning (TSCP). A multistage scenario tree based scheme is used to represent the parameters’ uncertainty and develop the related Deterministic Equivalent Model. A cost risk reduction is performed by using a new time-consistent risk averse measure. Given the dimensions of this problem in real-life applications, a decomposition approach is proposed. It is based on stochastic dynamic programming (SDP). The computational experience is twofold, a comparison is performed between the plain use of a current state-of-the-art mixed integer optimization solver and the proposed SDP decomposition approach considering the risk neutral version of the model as the subject for the benchmarking. The add-value of the new risk averse strategy is confirmed by the computational results that are obtained using SDP for both versions of the TSCP model, namely, risk neutral and risk averse.

An SDP approach for multiperiod mixed 0-1 linear programming models with stochastic dominance constraints for risk management

In this paper we consider multiperiod mixed 0-1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both the first-order and second-order constraints. We propose a stochastic dynamic programming (SDP) solution approach, where one has to overcome the negative impact of the cross-scenario constraints on the decomposability of the model. In our computational experience we compare our SDP approach against a commercial optimization package, in terms of solution accuracy and elapsed time. We use supply chain planning instances, where procurement, production, inventory, and distribution decisions need to be made under demand uncertainty. We confirm the hardness of the testbed, where the benchmark cannot find a feasible solution for half of the test instances while we always find one, and show the appealing tradeoff of SDP, in terms of solution accuracy and elapsed time, when solving medium-to-large instances.

Strengthening the reliability fixed-charge location model using clique constraints

The Reliability Fixed-Charge Location Problem is an extension of the Simple Plant Location Problem that considers that some facilities have a probability of failure. In this paper we reformulate the original mathematical programming model of the Reliability Fixed-Charge Location Problem as a set packing problem. We study certain aspects of its polyhedral properties, identifying all the clique facets. We also discuss how to obtain facets of the Reliability Fixed-Charge Location Problem from facets of the Simple Plant Location Problem. Subsequently, we study some conditions for optimal solutions. Finally, we propose an improved compact formulation for the problem and we check its performance by means of an extensive computational study.

An effective heuristic for multistage linear programming with a stochastic right-hand side

The multistage Stochastic Linear Programming (SLP) problem may become numerically intractable for huge instances, in which case one can solve an approximation for example the well known multistage Expected Value (EV) problem. We introduce a new approximation to the SLP problem that we call the multistage Event Linear Programming (ELP) problem. To obtain this approximation the SLP constraints are aggregated by means of the conditional expectation operator. Based on this new problem we derive the ELP heuristic that produces a lower and an upper bound for the SLP problem. We have assessed the validity of the ELP heuristic by solving large scale instances of the network revenue management problem, where the new approach has clearly outperformed the EV approach. One limitation of this paper is that it only considers randomness on the right-hand side, which is assumed to be discrete and stagewise independent.

A model for risk minimisation on water resource usage failure

We present a framework for solving the strategic problem of assigning transboundary water resources to demand centres under uncertainty in the water exogenous inflow in the reservoirs and other segments of the basin system along the time horizon. The functional to maximise is the probability of satisfying different targets on the stored water and different demands over a set of scenarios. A scenario tree-based scheme is used to represent the Deterministic Equivalent Model (DEM) of the stochastic mixed 0-1 programme with complete recourse. The constraints are modelled by a splitting variable representation via scenarios and, so, a Stochastic Integer Programming (SIP) scheme can be used to exploit the excess probability functional structure as well as the non-anticipativity constraints for the water assignment.

On relaxing the integrality of the allocation variables of the reliability fixed-charge location problem

The aim of the reliability fixed-charge location problem is to find robust solutions to the fixed-charge location problem when some facilities might fail with probability q. In this paper we analyze for which allocation variables in the reliability fixed-charge location problem formulation the integrality constraint can be relaxed so that the optimal value matches the optimal value of the binary problem. We prove that we can relax the integrality of all the allocation variables associated to non-failable facilities or of all the allocation variables associated to failable facilities but not of both simultaneously. We also demonstrate that we can relax the integrality of all the allocation variables whenever a family of valid inequalities is added to the set of constraints or whenever the parameters of the problem satisfy certain conditions. Finally, when solving the instances in a data set we discuss which relaxation or which modification of the problem works better in terms of resolution time and we illustrate that relaxing the integrality of the allocation variables inappropriately can alter the objective value considerably.

Alternative formulations for the Set Packing Problem and their application to the Winner Determination Problem

An alternative formulation for the set packing problem in a higher dimension is presented. The addition of a new family of binary variables allows us to find new valid inequalities, some of which are shown to be facets of the polytope in the higher dimension. We also consider the Winner Determination Problem, which is equivalent to the set packing problem and whose special structure allows us to easily implement these valid inequalities in a very easy way. The computational experiments illustrate the performance of the valid inequalities and obtain good results.