Goal uncertainty and the supermartingale property in an information feedback loop

A method for processing sequentially acquired information is developed in which a decision makers uncertainty concerning the utility of this information is modeled by interval-valued random functions. Interval-valued estimates of the expected utility of each specific course of action chosen from a finite set of possible courses of action are developed. It is assumed that the uncertainty concerning the utility of the various courses of action tends to decrease as information increases, and certain reasonable conditions on the process of updating the estimated utilities are imposed. Under the conditions cited above it is shown that for each possible course of action, the sequential interval estimates of the expected utility forms a set-valued supermartingale with respect to a sequence of expanding σ-fields generated by the history of the information acquisition process     

A process for generating quantitative belief functions

We present a structured methodology for transforming qualitative preference relationships among propositions into appropriate numeric representations. This approach will be useful in the difficult process of knowledge acquisition from experts on the degree of belief in various propositions or the probability of the truthfulness of those propositions. The approach implicitly (through the qualitative assignments) and explicitly (through the vague interval pairwise comparisons) provides for different levels of preference relationships. Among its advantages, it permits the expert to: explore the given problem situation, using linguistic quantifiers; avoid the premature use of numeric measures; and identify input data that are inconsistent with the theory of belief functions.     

Eliciting and mapping qualitative preferences to numeric rankings in group decision making

Group work is becoming the norm in organizations. From strategy planning committees to quality management teams, organizational members are collaborating on problem solving. One area of team support that is often desired is the scoring and ranking of decision alternatives on qualitative/subjective domains, and the aggregation of individual preferences into group preferences. In this paper we present a new conceptual approach to qualitative preference elicitation and aggregation. This approach is based on well established decision analysis techniques. It significantly advances the state of the art of group decision making by addressing four common limitations: (1) the inability to deal with vagueness of human decision makers in articulating preferences; (2) difficulties in mapping qualitative evaluation to numeric estimates; (3) problems in aggregating individual preferences into meaningful group preference; and (4) the lack of simple user friendly techniques for dealing with a large number of decision alternatives. Our approach is easy to implement in stand alone personal computers and groupware. We illustrate this with a real-world problem.     

Markov-type fuzzy decision processes with a discounted reward on a closed interval

We formulate a new multi-stage decision process with Markov-type fuzzy transition, which is termed Markov-type fuzzy decision process. In the general framework of the decision process, both of state and action are assumed to be fuzzy itself. The transition of states is defined using the fuzzy relation with Markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval. To discuss the optimization problem, a partial order of convex fuzzy numbers is introduced. In this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping. Moreover, the optimality equation for the fuzzy decision model is derived under some continuity conditions. Also, an illustrated example is given to explain the theoretical results and the computation in the paper.     

On distributionally robust multiperiod stochastic optimization

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider “ambiguity neighborhoods” around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in Pflug and Pichler (SIAM J Optim 22:1–23, 2012). The ambiguity model is formulated as a minimax problem, where the the optimal decision is to be found, which minimizes the maximal objective function within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We determine the minimax solution and identify the worst trees within the ambiguity set. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.     

The evidential reasoning approach for multi-attribute decision analysis under interval uncertainty

The evidential reasoning (ER) approach is a method for multiple attribute decision analysis (MADA) under uncertainties. It improves the insightfulness and rationality of a decision making process by using a belief decision matrix (BDM) for problem modelling and the Dempster–Shafer (D–S) theory of evidence for attribute aggregation. The D–S theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis, which is not explored in the original ER approach and will be investigated in this paper. Firstly, interval uncertainty will be defined and modelled in the ER framework. Then, an extended ER algorithm, IER, is derived, which enables the ER approach to deal with interval uncertainty in assessing alternatives on an attribute. It is proved that the original ER algorithm is a special case of the IER algorithm. The latter is demonstrated using numerical examples.     

Uncertainty in reliability evaluation processes and a simulation approach to treating it

This paper considers the problem of uncertainty in reliability data that are used in many decision making processes and describes a simulation approach to dealing explicitly with this uncertainty. The causes of uncertainty are discussed for three different situations: (a) in the process of designing new systems, when failure data are not yet available, (b) after performing reliability test and gathering failure data, and (c) in mission reliability prediction. It is concluded that, in performing reliability prediction or failure rate prediction, one should use interval estimates rather than point estimates. These intervals can be used to perform component classification and then, employing simulation, to obtain tables or scattergrams for the mean time between failures of a system or for system reliability.     

Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker’s risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme.

     

Estimating Mixture of Dirichlet Process Models

Abstract

Current Gibbs sampling schemes in mixture of Dirichlet process (MDP) models are restricted to using “conjugate” base measures that allow analytic evaluation of the transition probabilities when resampling configurations, or alternatively need to rely on approximate numeric evaluations of some transition probabilities. Implementation of Gibbs sampling in more general MDP models is an open and important problem because most applications call for the use of nonconjugate base measures. In this article we propose a conceptual framework for computational strategies. This framework provides a perspective on current methods, facilitates comparisons between them, and leads to several new methods that expand the scope of MDP models to nonconjugate situations. We discuss one in detail. The basic strategy is based on expanding the parameter vector, and is applicable for MDP models with arbitrary base measure and likelihood. Strategies are also presented for the important class of normal-normal MDP models and for problems with fixed or few hyperparameters. The proposed algorithms are easily implemented and illustrated with an application.     

On the incorporation of adversarial reasoning in course of action analysis and resource allocation

Course of Action analysis and Resource Management are concerned with the allocation of resources over time to effect desired actions as a result of the perceived situation awareness. Decision Support Systems provide automated recommended courses of action to decision makers, considering relevant resource capabilities and constraints. Incorporating potential adversary actions and reactions to the current course of action decision (and the resources effecting the actions) in the decision making process will make the decision support system more robust and increase confidence that the recommended decisions are appropriate responses to the unfolding situations. We discuss research results from the inclusion of possible adversary actions and reactions into the course of action/resource allocation decision making framework. The overall decision problem is formulated as a multi-stage mathematical program. As the problem is NP-hard, an heuristic is developed through a natural problem decomposition. Simulated results show the effectiveness of the heuristic in producing good-quality solutions in an efficient manner.     

Evidential reasoning based preference programming for multiple attribute decision analysis under uncertainty

Multiple attribute decision analysis (MADA) problems having both quantitative and qualitative attributes under uncertainty can be modelled and analysed using the evidential reasoning (ER) approach. Several types of uncertainty such as ignorance and fuzziness can be consistently modelled in the ER framework. In this paper, both interval weight assignments and interval belief degrees are considered, which could be incurred in many decision situations such as group decision making. Based on the existing ER algorithm, several pairs of preference programming models are constructed to support global sensitivity analysis based on the interval values and to generate the upper and lower bounds of the combined belief degrees for distributed assessment and also the expected values for ranking of alternatives. A post-optimisation procedure is developed to identify non-dominated solutions, examine the robustness of the partial ranking orders generated, and provide guidance for the elicitation of additional information for generating more desirable assessment results. A car evaluation problem is examined to show the implementation process of the proposed approach.     

Optimal decision under ambiguity for diffusion processes

In this paper we consider stochastic optimization problems for an ambiguity averse decision maker who is uncertain about the parameters of the underlying process. In a first part we consider problems of optimal stopping under drift ambiguity for one-dimensional diffusion processes. Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in a two-parameter function space. In a second part we solve optimal stopping problems when the underlying process may crash down. These problems are reduced to one optimal stopping problem and one Dynkin game. Examples are discussed.     

Sequential and non-sequential acceptance sampling plans for autocorrelated processes using ARMA(p,q) models

This paper presents variable acceptance sampling plans based on the assumption that consecutive observations on a quality characteristic(X) are autocorrelated and are governed by a stationary autoregressive moving average (ARMA) process. The sampling plans are obtained under the assumption that an adequate ARMA model can be identified based on historical data from the process. Two types of acceptance sampling plans are presented: (1) Non-sequential acceptance sampling: In this case historical data is available based on which an ARMA model is identified. Parameter estimates are used to determine the action limit (k) and the sample size(n). A decision regarding acceptance of a process is made after a complete sample of size n is selected. (2) Sequential acceptance sampling: Here too historical data is available based on which an ARMA model is identified. A decision regarding whether or not to accept a process is made after each individual sample observation becomes available. The concept of Sequential Probability Ratio Test (SPRT) is used to derive the sampling plans. Simulation studies are used to assess the effect of uncertainties in parameter estimates and the effect of model misidentification (based on historical data) on sample size for the sampling plans. Macros for computing the required sample size using both methods based on several ARMA models can be found on the author’s web page .     

A policy gradient method for semi-Markov decision processes with application to call admission control

Solving a semi-Markov decision process (SMDP) using value or policy iteration requires precise knowledge of the probabilistic model and suffers from the curse of dimensionality. To overcome these limitations, we present a reinforcement learning approach where one optimizes the SMDP performance criterion with respect to a family of parameterised policies. We propose an online algorithm that simultaneously estimates the gradient of the performance criterion and optimises it using stochastic approximation. We apply our algorithm to call admission control. Our proposed policy gradient SMDP algorithm and its application to admission control is novel.     

Detecting a hidden obstacle via the time domain enclosure method. A scalar wave case

The characterization problem of the existence of an unknown obstacle behind a known obstacle is considered by using a singe observed wave at a place where the wave is generated. The unknown obstacle is invisible from the place by using a visible ray. A mathematical formulation of the problem using the classical wave equation is given. The main result consists of two parts: (a) one can make a decision whether the unknown obstacle exists or not behind a known impenetrable obstacle by using a single wave over a finite time interval under some a‐priori information on the position of the unknown obstacle; (b) one can obtain a lower bound on the Euclidean distance of the unknown obstacle to the center point of the support of the initial data of the wave. The proof is based on the idea of the time domain enclosure method and employs some previous results on the Gaussian lower/upper estimates for the heat kernels and domination of semigroups.     

Strategy,performance and operational research

This paper explores the nature of operational research and its interactions with performance measurement and strategy. It is argued that operational research (OR) is well fitted to handle strategic issues as the modelling approach of OR facilitates understanding and learning, and the evaluation of strategies prior to action. The development of problem structuring methods is also a key aid to strategy and policy formulation. OR is also beginning to play a role in performance measurement and there is an opportunity for OR to lead in the improvement of performance measurement systems.     

Concepts of probabilistic sets

In the field of pattern recognition or decision making theory, the important subjects are as follows: (1) ambiguity of property of objects, (2) variety of character of objects, (3) subjectivity of observers, (4) evolution of knowledge of observers (i.e. learning). Considering these points, the concept of probabilistic set is proposed. It is based on both probability theory and fuzzy concepts. A probabilistic set on a total space is defined by a point wise measurable function from a parameter space (which is a probability space) to a characteristic space (which is a measurable space). It is shown that the family of all probabilistic sets constitutes a complete pseudo-Boolean algebra. Moment analysis is possible by using a probability measure of the parameter space. Other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers.     

Ambiguity aversion and ambiguity content in decision making under uncertainty

The Subjectively Weighted Linear Utility (SWLU) model for decision making under uncertainty can accommodate non-neutral attitudes toward ambiguity. We first characterize ambiguity aversion in terms of the SWLU model parameters. In addition, we show that ambiguity content may reasonably be regarded as residing in the decision maker's subjective probability distribution of induced utility. In particular, (a) a special kind of mean preserving spread of the induced utility distribution will always increase ambiguity content, and (b) utility distributions which are more shiftable by new information have higher ambiguity content.     

Examining alternatives in the interval analytic hierarchy process using complete enumeration

The analytic hierarchy process as an aid to decision makers has gained in popularity over the years. Recently the procedure has been generalised into the interval analytic hierarchy process. Rather than select a precise ratio for the comparisons between the alternatives offered to the decision maker an interval is proposed. One popular strategy in examining this problem is to adopt a simulation technique. Here two simulation methods are considered and it is demonstrated that total enumeration is a viable alternative.