Disjunctive Model Research Articles

Disjunctive modeling for optimal control of hybrid systems

In this contribution, a novel approach for the modeling and numerical optimal control of hybrid (discrete–continuous dynamic) systems based on a disjunctive problem formulation is proposed. It is shown that a disjunctive model representation, which constitutes an alternative to mixed-integer model formulations, provides a very flexible, intuitive and effective way to formulate hybrid (discrete–continuous dynamic) optimization problems. The structure and properties of the disjunctive process models can be exploited for an efficient and robust numerical solution by applying generalized disjunctive programming techniques. The proposed modeling and optimization approach will be illustrated by means of optimal control of hybrid systems embedding linear discrete–continuous dynamic models.

Integrated systems for forecasting urban meteorology, air pollution and population exposure

Abstract. Urban air pollution is associated with significant adverse health effects. Model-based abatement strategies are required and developed for the growing urban populations. In the initial development stage, these are focussed on exceedances of air quality standards caused by high short-term pollutant concentrations. Prediction of health effects and implementation of urban air quality information and abatement systems require accurate forecasting of air pollution episodes and population exposure, including modelling of emissions, meteorology, atmospheric dispersion and chemical reaction of pollutants, population mobility, and indoor-outdoor relationship of the pollutants. In the past, these different areas have been treated separately by different models and even institutions. Progress in computer resources and ensuing improvements in numerical weather prediction, air chemistry, and exposure modelling recently allow a unification and integration of the disjunctive models and approaches. The current work presents a novel approach that integrates the latest developments in meteorological, air quality, and population exposure modelling into Urban Air Quality Information and Forecasting Systems (UAQIFS) in the context of the European Union FUMAPEX project. The suggested integrated strategy is demonstrated for examples of the systems in three Nordic cities: Helsinki and Oslo for assessment and forecasting of urban air pollution and Copenhagen for urban emergency preparedness.

The Sensitivity of Probability Assessments to Time Units and Performer Characteristics

This paper examines the sensitivity of probability judgments pertaining to real-world events (e.g., the chance of getting a speeding ticket) to the events’ time horizon (e.g., 1, 2, or 8 weeks) and performer characteristics (e.g., 1, 2, and 8 drivers). Using the participants’ multiple responses to different time horizons and number of performers, we assessed the degree of deviance from the normative prescription and potential explanations for such deviance. In Study 1, two questionnaires were administered to a sample of 120 students. In the first questionnaire, in violation of the normative disjunctive model, most participants did not adjust probabilities sufficiently for various time horizons. This pattern cannot be attributed merely to regression to the mean with a fixed error term (Erev et al. 1994). In the second questionnaire, the same bias was replicated in scenarios featuring different numbers of individuals, indicating no time-specific contributing factors. In Study 2, a sample of 140 students responded to a computerized questionnaire in which the items preceding target questions were controlled. The direction of the deviance did not change as a function of previous items. The results are attributed to the “extension neglect” tendency (Kahneman 2003).

Optimization of Discrete‐Continuous Dynamic Systems Based on Disjunctive Programming

AbstractIn this contribution, a novel approach for the modeling and optimization of discrete‐continuous dynamic systems based on a disjunctive problem formulation is proposed. It will be shown that a disjunctive model representation, which constitutes an alternative to mixed‐integer model formulations, provides a very flexible and intuitive way to formulate discrete‐continuous dynamic optimization problems. Moreover, the structure and properties of the disjunctive process models can be exploited for an efficient and robust numerical solution by applying generalized disjunctive programming techniques. The proposed modeling and optimization approach will be illustrated by means of an optimal control problem that embeds a linear discretecontinuous dynamic system. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simplified Models of the Relationship Between Health and Disease

The concepts of health and disease are crucial in defining the aim and the limits of modern medicine. Accordingly it is important to understand them and their relationship. However, there appears to be a discrepancy between scholars in philosophy of medicine and health care professionals with regard to these concepts. This article investigates health care professionals' concepts of health and disease and the relationship between them. In order to do so, four different models are described and analyzed: the ideal model, the holistic model, the medical model and the disjunctive model. The analysis reveals that each model has its pros and cons, and that health care professionals appear to apply more than one models. Furthermore, the models and the way health care professionals' use them may be helpful for scholars in philosophy of medicine with regard to developing theories and communicating them to health care professionals.

The Implications of Conjunctive and Disjunctive Forgiveness for Sexual Abuse

This article examines the relationship between forgiveness styles (conjunctive and disjunctive models) and sexual abuse. Surveys from 114 university students were analyzed for differences between non-sexually abused and sexually abused subjects on a number of psychological and physical well-being variables. A number of differences were found including higher levels of reported verbal and physical abuse for the sexually abused subjects. Only one sexually abused subject was found that fit the pattern of conjunctive forgiveness (one of reconciliation) towards the offender. The majority of sexually abused subjects preferred to keep their distance from the abuser, regardless of the extent to which the abuser had been forgiven. The results suggest that reconciliation within forgiveness may not be an appropriate goal for all subjects.

Mixed‐logic dynamic optimization applied to batch distillation process design

AbstractAn approach for the optimal configuration and sequencing of complex batch‐distillation processes is presented. The proposed method is based on formulating a mixed‐logic dynamic optimization problem in which a performance index is minimized subject to a disjunctive dynamic model and constraints. The disjunctive model comprises a given set of equipment modules, such as stacks of trays of different size or feed and product vessels, available for the distillation process under consideration along with logical relationships provided by the design context. The optimization based on this model may yield unconventional design and operational strategies, and thus improve the economic efficiency to a significant extent. The logic‐based solution technique proposed in this work offers some favorable properties when compared to established mixed‐integer dynamic optimization problem solution techniques reported in the literature. The design of a multistage batch‐distillation process serves as an example to illustrate the proposed modeling and solution method.

Retrofit of multiproduct batch plants through generalized disjunctive programming

The retrofit problem for multiproduct batch plants deals with the modification of the original structure of the plant to meet new production conditions such as the introduction of new products, a new supply pattern, etc. For this problem, a disjunctive model is presented, which takes into consideration every usual alternative about the configuration of units and storage tanks. A disjunction is generated for that purpose. The selection of one of those alternatives sets up all the corresponding constraints, which are: operation time, unit size and cost, etc. Finally, the performance of this approach is analyzed through the resolution of a set of examples.

Robustness optimisation of the minimum makespan schedules in a job shop

In general, many minimum makespan schedules exist in job shops. Therefore, it is important to select the best one based on secondary criterion. In this paper, we adopt a measure called robustness of the schedule as the selection criterion. The robustness of the schedule is the maximum value of the sensitivity, which is a measure for the delay of completion times by one time unit, for all of the operations. As there are generally many minimum makespan schedules, it is difficult to enumerate all makespan-minimum active schedules and then to find the most robust schedule. A branch and bound method is introduced to the minimisation of makespan while optimising robustness based on a disjunctive graph model and the propositions proposed by Carlier and Pinson. A lower bound of robustness for each partial schedule is calculated using the corresponding disjunctive graph. The effectiveness of the proposed approach is clarified by solving test problems.

Test systems and mathematical models for transmission network expansion planning

The data of four networks that can be used in carrying out comparative studies with methods for transmission network expansion planning are given. These networks are of various types and different levels of complexity. The main mathematical formulations used in transmission expansion studies-transportation models, hybrid models, DC power flow models, and disjunctive models are also summarised and compared. The main algorithm families are reviewed-both analytical, combinatorial and heuristic approaches. Optimal solutions are not yet known for some of the four networks when more accurate models (e.g. the DC model) are used to represent the power flow equations-the state of the art with regard to this is also summarised. This should serve as a challenge to authors searching for new, more efficient methods.

A Mixed Integer Disjunctive Model for Transmission Network Expansion

The classical nonlinear mixed integer formulation of the transmission network expansion problem cannot guarantee finding the optimal solution due to its nonconvex nature. We propose an alternative mixed integer linear disjunctive formulation, which has better conditioning properties than the standard disjunctive model. The mixed integer program is solved by a commercial Branch and Bound code, where an upper bound provided by a heuristic solution is used to reduce the tree search. The heuristic solution is obtained using a GRASP metaheuristic, capable of finding suboptimal solutions with an affordable computing effort. Combining the upper bound given by the heuristic and the mixed integer disjunctive model, optimality can be proven for several hard problem instances.

A mixed integer disjunctive model for transmission network expansion

The classical nonlinear mixed integer formulation of the transmission network expansion problem cannot guarantee finding the optimal solution due to its nonconvex nature. We propose an alternative mixed integer linear disjunctive formulation, which has better conditioning properties than the standard disjunctive model. The mixed integer program is solved by a commercial branch and bound code, where an upper bound provided by a heuristic solution is used to reduce the tree search. The heuristic solution is obtained using a GRASP metaheuristic, capable of finding sub-optimal solutions with an affordable computing effort. Combining the upper bound given by the heuristic and the mixed integer disjunctive model, optimality can be proven for several hard problem instances.

Integrating complex economic objectives with the design and planning of offshore oilfield infrastructures

The design and planning of offshore oilfield infrastructures includes several discrete decisions over a long time horizon, as well as complexities arising from nonlinear reservoir behavior and business considerations. This leads to very large mixed integer nonlinear programming (MINLP) models that are very difficult and expensive to solve. It is for this reason that simplified economic objective functions have been used in the past. We show that when complex business rules, such as tariff, tax and royalty calculations are added to the model, substantial increases in the net present value (NPV) of the project are obtained. However, including these calculations leads to more than an order of magnitude increase in the computational effort required. We address this issue by proposing a disjunctive modeling framework for complex economics, and we show that in a particular instance it solves twice as fast as an intuitive big-M modeling framework.

A Disjunctive Model for Scheduling in a Manufacturing and Packing Facility with Intermediate Storage

The problem considered in this paper deals with the short term scheduling of a two stage continuous process with intermediate storage tanks. The major scheduling decisions in this problem are: a) the assignment of orders to various storage tanks; b) the sequence of orders in each unit; c) the timing of various operations in different stages. The problem is highly combinatorial in nature. The major challenge is to develop strong integer programming formulations and to devise efficient solution techniques. An initial model is presented in the form of a disjunctive program which is later transformed to a Mixed Integer Linear Programming (MILP) problem. A number of example problems are solved which highlight the limitations of this model as the number of orders increases. A heuristic based on partial preordering is considered which solves industrial sized problems very quickly. The objective function values for the heuristic solutions are within 7% of the optimal values.

Processing Deductive Databases under the Disjunctive Stable Model Semantics

Cyclic covers are shown to characterise disjunctive stable models of unstratified deductive databases, and to facilitate top-down query processing, query compilation and view updating under the disjunctive stable model semantics. Such processing is shown to be more complex than comparable processing of stratified databases.

Credulous argumentation with the disjunctive stable semantics

The relationship between the disjunctive stable semantics and argumentation is rarely explored. In particular, the problem of how to perform argumentation with disjunctive logic programs by the disjunctive stable semantics is still open. This paper attempts to address this problem and a satisfied solution is provided, in which a disjunctive logic program is naturally transformed into an argument framework and the credulous argumentation is characterized as the maximal members of all acceptable hypotheses. In this semantic framework, some interesting results are obtained. In particular, it is shown that the formalism of credulous argumentation can be implemented through the disjunctive stable models. As a result, the work provides not only a new way of performing argumentation (abduction) in disjunctive deductive databases, but also a natural and complete extension for the disjunctive stable semantics.

The least fixpoint transformation for disjunctive logic programs

The paradigm of disjunctive logic programming (DLP) enhances greatly the expressive power of normal logic programming (NLP) and many (declarative) semantics have been defined for DLP to cope with various problems of knowledge representation in artificial intelligence. However, the expressive ability of the semantics and the soundness of program transformations for DLP have been rarely explored. This paper defines an immediate consequence operatorT P G for each disjunctive program and shows thatT P G has the least and computable fixpointLft(P). Lft is, in fact, a program transformation for DLP, which transforms all disjunctive programs into negative programs. It is shown thatLft preserves many key semantics, including the disjunctive stable models, well-founded model, disjunctive argument semantics DAS, three-valued models, etc. This means that every disjunctive programP has a unique canonical formLft(P) with respect to these semantics. As a result, the work in this paper provides a unifying framework for studying the expressive ability of various semantics for DLP. On the other hand, the computing of the above semantics for negative programs is just a trivial task, therefore,Lft(P) is also an optimization method for DLP. Another application ofLft is to derive some interesting semantic results for DLP.

Argumentation in disjunctive logic programming

Argumentation (abduction) is widely applied in artificial intelligence (AI) and law reasoning. However, the problem of how to perform argumentation in disjunctive logic programming (DLP) is still open. In addition, a unifying semantic framework is required for incorporating various semantics for DLP. An argumentation-theoretic framework for DLP by taking the disjuncts of negative literals as abducibles is presented. This semantics not only is a simple and intuitive framework for performing argumentation and abduction in DLP, but also provides a unifying framework for many key semantics of disjunctive logic programs. In particular, it is shown that the EGCWA, well-founded model and disjunctive stable models can all be embedded into this semantics.

An alternative explanation of disjunctive formulations

The purpose of this paper is to explain the property of Disjunctive Formulations for Mixed Integer Programmes in a simpler way, relying on the Linear Programme (LP) definition of a dual. Disjunctive Formulations are contrasted with standard formulations. They have proved a very powerful way of modelling Integer Programmes. If a ‘complete’ disjunctive formulation is carried out we have a model whose LP Relaxation yields an integer solution. It is shown that a Disjunctive Formulation has a natural dual which is an LP. Therefore the dual of the dual is an LP representation of the original Disjunctive Model. It is in fact a Disjunctive Formulation.

Analyzing the characteristics of entrepreneurial ventures with behavioral decision models

The descriptive data on venture investments collected by MacMillan et al. is analyzed here with noncompensatory conjunctive and disjunctive models plus the familiar compensatory linear model. The results show that risk handling ability and attention to detail are pre-eminent precursors of venture performance. On the other hand, product and market variables appear to have diminished effectiveness when all the diverse venture data are pooled and considered together, as distinct from when the ventures are clustered into separate familial groups.