Constraint propagation,relational arithmetic in AI systems and mathematical programs

This paper explores the interrelationships between methods developed in mathematical programming to discover the structure of constraint (feasibility) sets and constraint propagation over networks used by some AI systems to perform inferences about quantities. It is shown that some constraint set problems in mathematical programming are equivalent to inferencing problems for constraint networks with interval labels. This makes the inference and query capabilities associated with AI systems that use logic programming, directly accessible to mathematical programming systems. On the other hand, traditional and newer methods which mathematical programming uses to obtain information about its associated feasibility set can be used to determine the propagation of constraints in a network of nodes of an AI system. When viewed from this point of view, AI problems can access additional mathematical programming analytical tools including new ways to incorporate qualitative data into constraint sets via interval and fuzzy arithmetic.This work was partially supported by the Industrial Consortium to Develop an Intelligent Mathematical Programming System — Amoco Oil Company, General Research Corporation, Ketron Management Science, Shell Oil Company, MathPro, and US West Advanced Technologies.     

A comparison of mathematical programming modeling systems

We compare three mathematical programming modeling languages, GAMS, OMNI and MathPro. To understand the properties of these languages, we formulate four linear programs in each language. The formulations are representative of the kinds of model structures one encounters in practice. Each of the languages focuses on a different view of linear programs. GAMS approximates algebra, OMNI uses the activity view and MathPro uses a block schematic. We summarize our experiences with the languages and suggest areas for further enhancement.     

Faces of mathematical modeling

In this paper I will discuss and exemplify my perspectives on how to teach mathematical modeling, as well as discuss quite different faces of mathematical modeling. The field of mathematical modeling is so enormous and vastly outspread and just not possible to comprehend in one single paper; or in one single book, or even in one single book shelf. Nevertheless, I have found that the more I can illuminate some of the various interpretations and perceptions of mathematical modeling which exists in the world around us when introducing and starting a course in mathematical modeling, the more benefit I will have during the course when discussing the need and purpose of mathematical modeling with the students. The fact that only some models fit within the practical teaching and assessing of a course in mathematical modeling, does not exclude the importance to illustrate that the world of today cannot go on without mathematical modeling. Students are nevertheless much more charmed with some models of reality than others.     

Three modeling paradigms in mathematical programming

Celebrating the sixtieth anniversary since the zeroth International Symposium on Mathematical Programming was held in 1949, this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field. These are: competition, dynamics, and hierarchy. The discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive, equilibrium, and nonlinear programming, and variational inequalities. We will describe the modeling approaches, mathematical formulations, and recent results of these paradigms, and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems. Our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field.     

Analysing production and environmental risks in arable farming systems: A mathematical approach

In response to a growing environmental concern in Dutch society, sustainable production systems in arable farming have been developed. Amongst other things, a reduction of the dependency on chemical inputs is attempted. This paper addresses the role of risk in the adoption by farmers of new systems by means of a model that determines differences in production risks between conventional and sustainable farming systems (CAFS and IAFS).Timing of activities – setting out a management track – is particularly important in sustainable arable farming systems. Resource requirements of crop husbandry activities mainly depend on weather conditions. To assess risks caused by weather conditions, the major aspects of crop husbandry in various crops have been modelled. Using tactics in crop husbandry (decision rules) and weather uncertainty as input, crop husbandry models (HMs) calculate management tracks that require resources. The value distributions of resource requirements of crop husbandry according to different farming systems is calculated in different HMs represented by stochastic dynamic directed networks. Hence, production risks of CAFS and IAFS can be compared.On a farm, all the aspects of crop husbandry in the various crops are to be taken into account. Given the weather conditions, tactics for all the aspects are combined in an LP model of the whole farm where they compete for limited resources. In the LP model, tactics are re-assessed by means of the HMS, using information of the LP solution. This iterative procedure enables production risks of CAFS and IAFS to be compared, considering fixed, allocatable resources for the whole farm firm.     

Moving toward shared realities through empathy in mathematical modeling: An ecological systems theory approach

The mathematics education community has routinely called for mathematics tasks to be connected to the real world. However, accomplishing this in ways that are relevant to students’ lived experiences can be challenging. Meanwhile, mathematical modeling has gained traction as a way for students to learn mathematics through real-world connections. In an open problem to the mathematics education community, this paper explores connections between the mathematical modeling and the nature of what is considered relevant to students. The role of empathy is discussed as a proposed component for consideration within mathematical modeling so that students can further relate to real-world contexts as examined through the lens of Ecological Systems Theory. This is contextualized through a classroom-tested example entitled “Tiny Homes as a Solution to Homelessness” followed by implications and conclusions as they relate to mathematics education.     

Hierarchical sets in mathematical programming modeling languages

The mathematical notation commonly applied for the formulation of mathematical programming models is extended to include hierarchical structures. The proposed notation is related to hierarchical set concepts in the languages UIMP, AMPL, and LPL. With the proposed notation it is possible to aggregate and disaggregate over hierarchical structures. In addition, views are introduced to permit the use of hierarchical substructures and to create new hierarchies out of existing ones. The proposed notation for hierarchical sets and views is illustrated by applying it to the representation and estimation of social accounting matrices (SAMs).     

Representation of time in mathematical programming modeling languages

Time-staged mathematical programming models have a planning horizon that is divided into a sequence of consecutive time periods. For the modeling of this sequence of time periods the use of calendars is proposed as an additional set concept for mathematical programming modeling languages. The definition of calendars is based on familiar notions such as set, ordering, interval length and functions. A calendar is an interval set and can be used to verify automatically the proper time referencing in stock balances. When a calendar is also a difference set, then backward and forward time referencing can be stated with the explicit use of time units. For models with a rolling horizon, concise and flexible ways to specify the structure of calendars are presented. The aggregation of raw data into model parameter values is supported by linking calendars that represent different time scales. The influence of the proposed calendar concept on the human ability to understand, maintain and verify models is analyzed throughout the paper on the basis of selected examples.     

Recent results in mathematical modeling of infectious diseases

Some results connected with a simple mathematical model of infectious disease are discussed in order to demonstrate the approach to the modelling of such real processes. A more complicated model of antiviral immune response is presented. A new modification of this model in which targets for the viruses are immunocompetent cells is suggested.     

Applications of mathematical systems theory in population biology

This paper is a review of recent developments of a research line proposed on the turn of the decades, 1980s to 1990s. The main results concern basic qualitative properties of nonlinear models of population biology, such as controllability and observability. The methods applied are different for the density-dependent models of population ecology and for the frequency-dependent models of population genetics and evolutionary theory. While in the first case the classical theorems of nonlinear systems theory can be used, in the second one an extension of classical results to systems with invariant manifold is necessary. Supported by the Hungarian NFSR (OTKA K 62000, K 68187).     

Symmetry methods in mathematical modeling of Aedes aegypti dispersal dynamics

A model of Aedes aegypti, the main vector of the yellow fever, is considered. The Lie point symmetries are found and some classes of exact solutions are shown.     

A mixed-integer mathematical modeling approach to exam timetabling

This paper explores mathematical programming models for an exam timetabling problem related to Kuwait University (KU). In particular, we consider two subproblems: (a) the ExamTimetabling Problem (ETP), which is concerned with assigning exams to designated exam-periods and classrooms, and (b) the Proctor Assignment Problem (PAP), which deals with the assignment of proctors to exams. While this exam timetabling problem is ubiquitous in many academic institutions worldwide, idiosyncrasies of the problem related to gender-based policies and having multiple exam centers at KU require novel models. A mixed-integer exam timetabling programming model (ETM) is developed for ETP, which takes into account restrictions related to exam-period conflicts, facility and human resources, and commuting and traffic considerations. Assuming that exam-periods are specified for all exams as determined by ETM, another mixed-integer programming model is formulated for PAP, denoted by PAM, which incorporates the proctors’ preferences for specific days and exam-periods. Computational results are reported and analyzed for solving ETM and PAM directly using the CPLEX Optimization Software (version 9.0), and for implementing a specialized sequential LP-based heuristic for solving PAM. The results obtained significantly improve upon those derived via the existing manual approach implemented at KU, in terms of eliminating conflicts as well as from the overall efficiency and equity points of view.