Probabilistic decision graphs for optimization under uncertainty

This paper provides a survey on probabilistic decision graphs for modeling and solving decision problems under uncertainty. We give an introduction to influence diagrams, which is a popular framework for representing and solving sequential decision problems with a single decision maker. As the methods for solving influence diagrams can scale rather badly in the length of the decision sequence, we present a couple of approaches for calculating approximate solutions. The modeling scope of the influence diagram is limited to so-called symmetric decision problems. This limitation has motivated the development of alternative representation languages, which enlarge the class of decision problems that can be modeled efficiently. We present some of these alternative frameworks and demonstrate their expressibility using several examples. Finally, we provide a list of software systems that implement the frameworks described in the paper.     

A comparison of two approaches for solving unconstrained influence diagrams

Influence diagrams and decision trees represent the two most common frameworks for specifying and solving decision problems. As modeling languages, both of these frameworks require that the decision analyst specifies all possible sequences of observations and decisions (in influence diagrams, this requirement corresponds to the constraint that the decisions should be temporarily linearly ordered). Recently, the unconstrained influence diagram was proposed to address this drawback. In this framework, we may have a partial ordering of the decisions, and a solution to the decision problem therefore consists not only of a decision policy for the various decisions, but also of a conditional specification of what to do next. Relative to the complexity of solving an influence diagram, finding a solution to an unconstrained influence diagram may be computationally very demanding w.r.t. both time and space. Hence, there is a need for efficient algorithms that can deal with (and take advantage of) the idiosyncrasies of the language. In this paper we propose two such solution algorithms. One resembles the variable elimination technique from influence diagrams, whereas the other is based on conditioning and supports any-space inference. Finally, we present an empirical comparison of the proposed methods.     

Solving linear-quadratic conditional Gaussian influence diagrams

This paper considers the problem of solving Bayesian decision problems with a mixture of continuous and discrete variables. We focus on exact evaluation of linear-quadratic conditional Gaussian influence diagrams (LQCG influence diagrams) with additively decomposing utility functions. Based on new and existing representations of probability and utility potentials, we derive a method for solving LQCG influence diagrams based on variable elimination. We show how the computations performed during evaluation of a LQCG influence diagram can be organized in message passing schemes based on Shenoy–Shafer and Lazy propagation. The proposed architectures are the first architectures for efficient exact solution of LQCG influence diagrams exploiting an additively decomposing utility function.     

Decision making with hybrid influence diagrams using mixtures of truncated exponentials

Mixtures of truncated exponentials (MTE) potentials are an alternative to discretization for representing continuous chance variables in influence diagrams. Also, MTE potentials can be used to approximate utility functions. This paper introduces MTE influence diagrams, which can represent decision problems without restrictions on the relationships between continuous and discrete chance variables, without limitations on the distributions of continuous chance variables, and without limitations on the nature of the utility functions. In MTE influence diagrams, all probability distributions and the joint utility function (or its multiplicative factors) are represented by MTE potentials and decision nodes are assumed to have discrete state spaces. MTE influence diagrams are solved by variable elimination using a fusion algorithm.     

BDD-based optimization for the quadratic stable set problem

The quadratic stable set problem (QSSP) is a natural extension of the well-known maximum stable set problem. The QSSP is NP-hard and can be formulated as a binary quadratic program, which makes it an interesting case study to be tackled from different optimization paradigms. In this paper, we propose a novel representation for the QSSP through binary decision diagrams (BDDs) and adapt a hybrid optimization approach which integrates BDDs and mixed-integer programming (MIP) for solving the QSSP. The exact framework highlights the modeling flexibility offered through decision diagrams to handle nonlinear problems. In addition, the hybrid approach leverages two different representations by exploring, in a complementary way, the solution space with BDD and MIP technologies. Machine learning then becomes a valuable component within the method to guide the search mechanisms. In the numerical experiments, the hybrid approach shows to be superior, by at least one order of magnitude, than two leading commercial MIP solvers with quadratic programming capabilities and a semidefinite-based branch-and-bound solver.     

An interactive procedure for linguistic multiple attribute decision making with incomplete weight information

In this paper, we consider the multiple attribute decision making (MADM) problems, in which the information about attribute weights is partly known and the attribute values are expressed in linguistic labels. We first define the concepts of linguistic positive ideal point, linguistic negative ideal point, and satisfactory degree of alternative. Based on these concepts, we then establish some linear programming models, through which the decision maker interacts with the analyst. Furthermore, we establish a practical interactive procedure for solving the MADM problems considered in this paper. The interactive process can be realized by giving and revising the satisfactory degrees of alternatives till an optimum satisfactory solution is achieved. Finally, a practical example is given to illustrate the developed procedure.     

Sequential valuation networks for asymmetric decision problems

This paper deals with representation and solution of asymmetric decision problems. We describe a new representation called sequential valuation networks that is a hybrid of Covaliu and Oliver’s sequential decision diagrams and Shenoy’s valuation networks. The solution algorithm is based on the idea of decomposing a large asymmetric problem into smaller sub-problems and then using the fusion algorithm of valuation networks to solve the sub-problems. Sequential valuation networks inherit many of the strengths of sequential decision diagrams and valuation networks while overcoming many of their shortcomings. We illustrate our technique by representing and solving a modified version of Covaliu and Oliver’s [Manage. Sci. 41(12) (1995) 1860] Reactor problem in complete detail.     

Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization,Part 2: Application

In this paper, we describe an application of the planar conjugate gradient method introduced in Part 1 (Ref. 1) and aimed at solving indefinite nonsingular sets of linear equations. We prove that it can be used fruitfully within optimization frameworks; in particular, we present a globally convergent truncated Newton scheme, which uses the above planar method for solving the Newton equation. Finally, our approach is tested over several problems from the CUTE collection (Ref. 2).This work was supported by MIUR, FIRB Research Program on Large-Scale Nonlinear Optimization, Rome, Italy.The author acknowledges Luigi Grippo and Stefano Lucidi, who contributed considerably to the elaboration of this paper. The exchange of experiences with Massimo Roma was a constant help in the investigation. The author expresses his gratitude to the Associate Editor and the referees for suggestions and corrections.     

Influence diagrams for Bayesian decision analysis

Influence diagrams for representing Bayesian decision problems are redefined in a formal way using conditional independence. This makes the graphs somewhat more helpful for exploring the consequences of a clients state beliefs. Some important results about the manipulation of influence diagrams are extended and reviewed as is an algorithm for computing an optimal policy. Two new results about the manipulation of influence diagrams are derived. A novel influence diagram representing a practical decision problem is used to illustrate the methodologies presented in this paper.     

Time-critical interactive dynamic influence diagram

Multiagent time-critical dynamic decision making is a challenging task in many real-world applications where a trade-off between solution quality and computational tractability is required. In this paper, we present a formal representation for modelling time-critical multiagent dynamic decision problems based on interactive dynamic influence diagrams (I-DIDs). The new representation called time-critical I-DIDs (TC-IDIDs) represents space-temporal abstraction by providing time-index to nodes and the model is defined in terms of the condensed and deployed forms. The condensed form is a static model of TC-IDIDs and can be expanded into its dynamic version. To facilitate the conversion between the two forms, we exploit the notion of object-orientation design to develop flexible and reusable TC-IDIDs. The difficulty on expanding TC-IDIDs is to select a proper time sequence to index nodes in the condensed form so that the expanded TC-IDIDs can be solved efficiently without compromising the quality of the policy. For this purpose, we propose two methods to build the condensed form of TC-IDIDs. We evaluate the solution quality and time complexity in three well-studied problems and provide results in support.     

A Neural Network Application in Personnel Scheduling

In this paper, we present and evaluate a neural network model for solving a typical personnel-scheduling problem, i.e. an airport ground staff rostering problem. Personnel scheduling problems are widely found in servicing and manufacturing industries. The inherent complexity of personnel scheduling problems has normally resulted in the development of integer programming-based models and various heuristic solution procedures. The neural network approach has been admitted as a promising alternative to solving a variety of combinatorial optimization problems. While few works relate neural network to applications of personnel scheduling problems, there is great theoretical and practical value in exploring the potential of this area. In this paper, we introduce a neural network model following a relatively new modeling approach to solve a real rostering case. We show how to convert a mixed integer programming formulation to a neural network model. We also provide the experiment results comparing the neural network method with three popular heuristics, i.e. simulated annealing, Tabu search and genetic algorithm. The computational study reveals some potential of neural networks in solving personnel scheduling problems.     

Decision Networks

Decision Networks is a technique for solving problems which involve a sequence of decisions. It is similar in style to critical path analysis in that it consists of arrow diagrams which give a visual representation of the problem and are used as a basis for a simple calculation procedure. The technique can deal with deterministic and stochastic problems and in the latter case is more general than decision trees. The decision network approach meets the need for a method of solution for multi-stage decision problems which is easily understood, helps the user to visualize the nature of the problem and is routine in application.     

Influence diagrams with super value nodes involving imprecise information

The concept of super value nodes was established to allow dynamic programming to be performed within the theory of influence diagrams and to reduce the computational complexity in solving problems by means of influence diagrams. This paper is focused on how influence diagrams with super value nodes are affected by the presence of imprecise information. We analyze how to reduce the complexity when evaluating an influence diagram in this framework by modelling these kinds of nodes and random magnitudes in terms of fuzzy random variables. Finally, an applied example of the theoretical results is developed.     

Fuzzy number linear programming: A probabilistic approach (3)

In the real world there are many linear programming problems where all decision parameters are fuzzy numbers. Several approaches exist which use different ranking functions for solving these problems. Unfortunately when there exist alternative optimal solutions, usually with different fuzzy value of the objective function for these solutions, these methods can not specify a clear approach for choosing a solution. In this paper we propose a method to remove the above shortcoming in solving fuzzy number linear programming problems using the concept of expectation and variance as ranking functions.     

Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities

This paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory. The so-called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes. In fact, generally, it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences. In such a case, the possibility theory offers a suitable modeling framework. Different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams. This paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks. The proposed approach is implemented via a possibilistic influence diagram toolbox (PIDT).     

On the use of binary decision diagrams for solving problems on simple games

Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.     

Graph coloring with decision diagrams

Mathematical Programming - We introduce an iterative framework for solving graph coloring problems using decision diagrams. The decision diagram compactly represents all possible color classes,...     

Beyond realistic considerations: modeling conceptions and controls in task examples with simple word problems

Using simple word problems, we analyze possible teacher conceptions on the process of problem solving, its goals and the choices that a problem solver can make in problem mathematization. We identify several possible teacher conceptions that would be responsible for the different didactical contracts that teachers create in the mathematics class. Using especially chosen and designed task examples, we demonstrate the diagnosis of teacher own controls in solving problems and in evaluating problem solutions. We also discuss characteristics of task examples that might promote a shift from a problem solving perspective to a modeling perspective that goes beyond merely accepting alternative solutions due to realistic considerations. This shift in perspective would be exhibited through a new understanding of the process of fitting mathematical models in problem situations.     

A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs

Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing. Supported in part by the National Science Foundation under Grants DDM-9215921 and SES-9211937.     

基于改进的证据推理方法的多指标决策模型

对于一个多指标决策问题，证据理论可以通过构造辨识框架和基本概率分配函数、采取递归的证据合成方法。计算出原始数据在反映多个指标联合作用的情况下对不同判别结果的支持程度，并可以在信息复杂或数据不完整的条件下做出评估决策。本文首先建立基于证据推理的多指标评估问题的基本模型，然后引入了模糊数据方法以处理具有模糊概念或推理关系的复杂问题，同时还考虑了实际问题中可能出现加权证据或者相关证据的情况，其目的是为了建立一套具有实用性的、准确有效的多指标评估模型。文章最后设计一个风险评估的算例，分析了该方法的优点以及需要进一步完善之处。