Stochastic MOLP with Incomplete Information: An Interactive Approach with Recourse

Numerous multiobjective linear programming (MOLP) methods have been proposed in the last two decades, but almost all for contexts where the parameters of problems are deterministic. However, in many real situations, parameters of a stochastic nature arise. In this paper, we suppose that the decision-maker is confronted with a situation of partial uncertainty where he possesses incomplete information about the stochastic parameters of the problem, this information allowing him to specify only the limits of variation of these parameters and eventually their central values. For such situations, we propose a multiobjective stochastic linear programming methodology; it implies the transformation of the stochastic objective functions and constraints in order to obtain an equivalent deterministic MOLP problem and the solving of this last problem by an interactive approach derived from the STEM method. Our methodology is illustrated by a didactical example.     

A generalized DEA model for inputs/outputs estimation

In this paper we discuss the question: among a group of decision making units (DMUs), if a DMU changes some of its input (output) levels, to what extent should the unit change outputs (inputs) such that its efficiency index remains unchanged? In order to solve this question we propose a solving method based on Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP). In our suggested method, the increase of some inputs (outputs) and the decrease due to some of the other inputs (outputs) are taken into account at the same time, while the other offered methods do not consider the increase and the decrease of the various inputs (outputs) simultaneously. Furthermore, existing models employ a MOLP for the inefficient DMUs and a linear programming for weakly efficient DMUs, while we propose a MOLP which estimates input/output levels, regardless of the efficiency or inefficiency of the DMU. On the other hand, we show that the current models may fail in a special case, whereas our model overcomes this flaw. Our method is immediately applicable to solve practical problems.     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Progressive hedging as a meta-heuristic applied to stochastic lot-sizing

In a great many situations, the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved. This gives rise to a need for stochastic programming (SP) methods that create solutions that are hedged against future uncertainty. The progressive hedging algorithm (PHA) of Rockafellar and Wets is a general method for SP. We cast the PHA in a meta-heuristic framework with the sub-problems generated for each scenario solved heuristically. Rather than using an approximate search algorithm for the exact problem as is typically the case in the meta-heuristic literature, we use an algorithm for sub-problems that is exact in its usual context but serves as a heuristic for our meta-heuristic. Computational results reported for stochastic lot-sizing problems demonstrate that the method is effective.     

Range-based Multi-Actor Multi-Criteria Analysis: A combined method of Multi-Actor Multi-Criteria Analysis and Monte Carlo simulation to support participatory decision making under uncertainty

Concerns about environmental and social effects have made Multi-Criteria Decision Making (MCDM) increasingly popular. Decision making in complex contexts often – possibly always – requires addressing an aggregation of multiple issues to meet social, economic, legal, technical, and environmental objectives. These values at stake may affect different stakeholders through distributional effects characterized by a high and heterogeneous uncertainty that no social actors can completely control or understand. On this basis, we present a new process framework that aims to support participatory decision making under uncertainty: the range-based Multi-Actor Multi-Criteria Analysis (range-based MAMCA). On the one hand, the process framework explicitly considers stakeholders’ objectives at an output level of aggregation. On the other hand, by means of a Monte Carlo analysis, the method also provides an exploratory scenario approach that enables the capture of the uncertainty, which stems from the complex context evolution. Range-based MAMCA offers a unique participatory process framework that enables us (1) to identify the alternatives pros and cons for each stakeholder group; (2) to provide probabilities about the risk of supporting mistaken, or at least ill-suited, decisions because of the uncertainty regarding to the decision-making context; (3) to take the decision-makers’ limited control of the actual policy effects over the implementation of one or several options into account. The range-based MAMCA framework is illustrated by means of our first case study that aimed to assess French stakeholders’ support for different biofuel options by 2030.     

Target setting in the general combined-oriented CCR model using an interactive MOLP method

The traditional data envelopment analysis (DEA) model does not include a decision maker’s (DM) preference structure while measuring relative efficiency, with no or minimal input from the DM. To incorporate DM’s preference information in DEA, various techniques have been proposed. An interesting method to incorporate preference information, without necessary prior judgment, is the use of an interactive decision making technique that encompasses both DEA and multi-objective linear programming (MOLP). In this paper, we will use Zionts-Wallenius (Z-W) method to reflecting the DM’s preferences in the process of assessing efficiency in the general combined-oriented CCR model. A case study will conducted to illustrate how combined-oriented efficiency analysis can be conducted using the MOLP method.     

An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem

Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generation of efficient frontiers in multi-objective optimization problems by generalized data envelopment analysis

In many practical problems such as engineering design problems, criteria functions cannot be given explicitly in terms of design variables. Under this circumstance, values of criteria functions for given values of design variables are usually obtained by some analyses such as structural analysis, thermodynamical analysis or fluid mechanical analysis. These analyses require considerably much computation time. Therefore, it is not unrealistic to apply existing interactive optimization methods to those problems. On the other hand, there have been many trials using genetic algorithms (GA) for generating efficient frontiers in multi-objective optimization problems. This approach is effective in problems with two or three objective functions. However, these methods cannot usually provide a good approximation to the exact efficient frontiers within a small number of generations in spite of our time limitation. The present paper proposes a method combining generalized data envelopment analysis (GDEA) and GA for generating efficient frontiers in multi-objective optimization problems. GDEA removes dominated design alternatives faster than methods based on only GA. The proposed method can yield desirable efficient frontiers even in non-convex problems as well as convex problems. The effectiveness of the proposed method will be shown through several numerical examples.     

未决赔款准备金评估的随机性Munich链梯法

精算实务界通常采用链梯法等确定性方法评估未决赔款准备金,这些评估方法存在一定缺陷,一方面不能有效考虑保险公司历史数据中所包含的已决赔款和已报案赔款数据信息,另一方面只能得到未决赔款准备金的均值估计,不能度量不确定性。为了克服这些缺陷,本文结合Mack模型假设和非参数Bootstrap重抽样方法,提出了未决赔款准备金评估的随机性Munich链梯法,并应用R软件对精算实务中的实例给出了数值分析。     

Revising the OWA operator for multi criteria decision making problems under uncertainty

Multi criteria decision making (MCDM) problems are usually under uncertainty. One of these uncertain parameters is the decision maker (DM)’s degree of optimism, which has an important effect on the results. Fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from DM and then, because of its uncertainty it is assumed to have stochastic nature. A new approach, entitled FSROWA, is introduced to combine the Fuzzy and Stochastic features into a Revised OWA operator.     

A Semidefinite Programming approach for solving Multiobjective Linear Programming

Several algorithms are available in the literature for finding the entire set of Pareto-optimal solutions of Multiobjective Linear Programmes (MOLPs). However, all of them are based on active-set methods (simplex-like approaches). We present a different method, based on a transformation of any MOLP into a unique lifted Semidefinite Program (SDP), the solutions of which encode the entire set of Pareto-optimal extreme point solutions of any MOLP. This SDP problem can be solved, among other algorithms, by interior point methods; thus unlike an active set-method, our method provides a new approach to find the set of Pareto-optimal solutions of MOLP.     

Multivariate Gaussian criteria in SMAA

We consider stochastic multicriteria decision-making problems with multiple decision makers. In such problems, the uncertainty or inaccuracy of the criteria measurements and the partial or missing preference information can be represented through probability distributions. In many real-life problems the uncertainties of criteria measurements may be dependent. However, it is often difficult to quantify these dependencies. Also, most of the existing methods are unable to handle such dependency information.In this paper, we develop a method for handling dependent uncertainties in stochastic multicriteria group decision-making problems. We measure the criteria, their uncertainties and dependencies using a stochastic simulation model. The model is based on decision variables and stochastic parameters with given distributions. Based on the simulation results, we determine for the criteria measurements a joint probability distribution that quantifies the uncertainties and their dependencies. We then use the SMAA-2 stochastic multicriteria acceptability analysis method for comparing the alternatives based on the criteria distributions. We demonstrate the use of the method in the context of a strategic decision support model for a retailer operating in the liberated European electricity market.     

Large-scale optimization with the primal-dual column generation method

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation process. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.     

On Analyses of the Solution to a Linear Programming Problem

Many complex problem situations in various contexts have been represented in recent years by the linear programming model. The simplex method can then be used to give the optimal values of the variables corresponding to a given set of values of the parameters. However, in many situations it is useful to have the solution to many other related problems which differ from the original problem only in the values of some of the parameters. This paper presents procedures by which the solutions to the changed problems can be derived from the simplex solution tableau corresponding to the original problem. The method will be illustrated by means of an example problem, and it will be shown how quantitative information obtained from such analyses can aid management in decision making.     

Fuzzy stochastic linear programming: Survey and future research directions

In many real-life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain. Although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist, there are no reliable transformations between them. This calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models. Fuzzy stochastic linear programming is an attempt to fulfill this need. It deals with modelling and problem solving issues related to situations where randomness and fuzziness co-occur in a linear programming framework. In this paper we provide a survey of the essential elements, methods and algorithms for this class of linear programming problems along with promising research directions. Being a survey, the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest.     

Stochastic Approaches to Uncertainty Quantification in CFD Simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

Stochastic approaches to uncertainty quantification in CFD simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.