Finding the critical path in an activity network with time-switch constraints

An activity network is an acyclic graph with non-negative weights and with a unique source and destination. A project consisting of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network provides useful information for managing the project. In a traditional activity network, it is assumed that an activity always begins after all of its preceding activities have been completed. This assumption may not be adequate enough to describe some practical applications where some forms of time constraints are attached to an activity. In this paper, we investigate one type of time constraint called time-switch constraint which assumes that an activity begins only in a specified time interval of a cycle with some pairs of exclusive components. Polynomial time algorithms are developed to find the critical path (or longest path) and analyze the float of each arc in this time-constrained activity network. The analysis shows that the critical path and float in this context differ from those of a traditional activity network in some management perspectives and thus, consideration of the time-switch constraint leads to enhanced project management through more effective use of budgets and resource allocation.     

Responsibility for proving and defining in abstract algebra class

There is considerable variety in inquiry-oriented instruction, but what is common is that students assume roles in mathematical activity that in a traditional, lecture-based class are either assumed by the teacher (or text) or are not visible at all in traditional math classrooms. This paper is a case study of the teaching of an inquiry-based undergraduate abstract algebra course. In particular, gives a theoretical account of the defining and proving processes. The study examines the intellectual responsibility for the processes of defining and proving that the professor devolved to the students. While the professor wanted the students to engage in all aspects of defining and proving, he was only successful at devolving responsibility for certain aspects and much more successful at devolving responsibility for proving than conjecturing or defining. This study suggests that even a well-intentioned instructor may not be able to devolve responsibility to students for some aspects of mathematical practice without using a research-based curriculum or further professional development.     

Intermediate buffer analysis for a production system

In this paper, a production system consisting of two serial machines and an intermediate buffer is studied. A shortage cost is incurred when the upstream machine is down and the buffer is exhausted. The practical example for this type of system can be an automated work center or an automobile general assembly.Researches on a similar two-machine system have been done in some articles where maintenance and an intermediate buffer are considered, but the spare parts are not involved. Nevertheless, spare parts are essential for maintenance implementation, and there is interaction between the buffer inventory and the spare parts due to maintenance activity. This paper is aimed to investigate three types of cost related to the intermediate buffer inventory, and obtain their expectations as functions of several decision parameters on maintenance, buffer, and spare parts during a renewal cycle, by using mathematical analysis. The proposed method can be an important basis for further study of system cost calculation and decision making optimization.     

Meaning and Formalism in Mathematics

This essay is an exploration of possible sources (psychological, not mathematical) of mathematical ideas. After a short discussion of plationism and constructivism, there is a brief review of some suggestions for these sources that have been put forward by various researcher (including this author). These include: mental representations, deductive reasoning, metaphors, natural language, and writing computer programs.The problem is then recast in terms of the relation between meaning and formalism. On one hand, formalism can be seen as a tool for expressing meaning that is already present in an individual's mind. On the other hand, and the discussion of this point is the main contribution of this paper, it is not only possible, but a standard activity of mathematicians, to use formalism to construct meaning and this can also be a source of mathematical ideas.Although using formalism to construct meaning is a very difficult method for students to learn, it may be that this is the only route to learning large portions of mathematics at the upper high school and tertiary levels. The essay ends with an outline of a pedagogical strategy for helping students travel this route.This revised version was published online in September 2005 with corrections to the Cover Date.     

IS FISHING COMPATIBLE WITH ENVIRONMENTAL CONSERVATION: A STOCHASTIC MODEL WITH AN ELEMENT OF SELF‐PROTECTION

Abstract The purpose of this paper is to introduce the impact of fishing activity on a marine ecosystem. The fishing activity is considered not only through annual harvest but also through a second component, called the degree of protection of the fishery environment. This characterizes the environmental impact of fishing. A stochastic dynamic programming problem is presented in infinite horizon, where a sole owner seeks to maximize a discounted expected profit. The main hypothesis states that the stock–recruitment relationship is stochastic and that both components of the fishing activity have an impact on the probability law of the state of the fishery environment. The optimal fishing policy is obtained and compared with standard models. This optimal policy has the following properties: is not a constant escapement policy and indicates an element of self‐protection by the fishery manager. The paper ends with a discussion on the existence of degrees of protection of the fishery environment that take into account the environmental conservation and preservation of economic activity.     

Estimating Response Ratios from Continuous Outcome Data

Several methods for imputing the number of responders from summary continuous outcome data in randomized controlled trials exist. A method by Furukawa and others was used in the quite common case that only such summary continuous outcome measures, but not the actual numbers of responders, are reported in order to estimate response rates (probabilities) for different treatments and response ratios between treatments in such trials. The authors give some empirical justification, but encourage search for theoretical support and further empirical exploration. In particular, a problem that needs to be addressed is that randomness in baseline score is not taken into consideration. This will be done in the present paper. Assuming a binormal model for the data, we compare theoretically the true response rate for a single treatment arm to the theoretical response rate underlying two versions of the suggested imputation method. We also assess the performance of the method numerically for some choices of model parameters. We show that the method works satisfactorily in some cases, but can be seriously biased in others. Moreover, assessing the uncertainty of the estimates is problematic. We suggest an alternative Bayesian estimation procedure, based directly on the normal model, which avoids these problems and provides more precise estimates when applied to simulated data sets.     

Intelligibility and intensionality

A common argumentative strategy employed by anti-reductionists involves claiming that one kind of entity cannot be identified with or reduced to a second because what can intelligibly be predicated of one cannot be predicated intelligibly of the other. For instance, it might be argued that mind and brain are not identical because it makes sense to say that minds are rational but it does not make sense to say that brains are rational. The scope and power of this kind of argument — if valid — are obvious; but if it turns out that ‘It makes sense to say that...’ creates an opaque context, such arguments will fail. I analyze a possible counterexample to validity and show that it is not conclusive, as it depends on what syntactical construction is given to the premises. This leads to the general observation that the argument form under consideration works for some constructions but not others, and thus to the conclusion that further analysis of intelligibility is called for before it can be known whether the argumentative strategy is open to the anti-reductionist or not.     

Economics of collective monitoring: a study of environmentally constrained electricity generators

This paper investigates the costs of monitoring of a distributed multi-agent economic activity in the presence of constraints on the agents’ joint outputs. If the regulator monitors agents individually she calculates each agent’s optimal contribution to the constrains by solving a constrained welfare-maximisation problem. This will maximise welfare but may be expensive because monitoring cost rises with the number of agents. Alternatively, the regulator could monitor agents collectively, using a detector, or detectors, to observe if each constraint is jointly satisfied. This will ease implementation cost, but lower welfare. We define the welfare difference between each regime of monitoring for a fairly inclusive electricity generation model and formulate some predictions. The behaviour of two generators in a coupled-constrained, three-node case study reproduces these predictions. We find that the welfare loss from collective monitoring can be small if the constraints are tight. We also learn that, under either regime, the imposition of transmission and environmental restrictions may benefit the less efficient generator and shift surplus share towards the emitters, decreasing consumer surplus.     

Setting gates for activities in the stochastic project scheduling problem through the cross entropy methodology

This paper addresses the problem of scheduling activities in projects with stochastic activity durations. The aim is to determine for each activity a gate—a time before it the activity cannot begin. Setting these gates is analogous to setting inventory levels in the news vendor problem. The resources required for each activity are scheduled to arrive according to its gate. Since activities’ durations are stochastic, the start and finish time of each activity is uncertain. This fact may lead to one of two outcomes: (1) an activity is ready to start its processing as all its predecessors have finished, but it cannot start because the resources required for it were scheduled to arrive at a later time. (2) The resources required for the activity have arrived and are ready to be used but the activity is not ready to start because of precedence constraints. In the first case we will incur a “holding” cost while in the second case, we will incur a “shortage” cost. Our objective is to set gates so as to minimize the sum of the expected holding and shortage costs. We employ the Cross-Entropy method to solve the problem. The paper describes the implementation of the method, compares its results to various heuristic methods and provides some insights towards actual applications.     

Setting gates for activities in the stochastic project scheduling problem through the cross entropy methodology

This paper addresses the problem of scheduling activities in projects with stochastic activity durations. The aim is to determine for each activity a gate—a time before it the activity cannot begin. Setting these gates is analogous to setting inventory levels in the news vendor problem. The resources required for each activity are scheduled to arrive according to its gate. Since activities’ durations are stochastic, the start and finish time of each activity is uncertain. This fact may lead to one of two outcomes: (1) an activity is ready to start its processing as all its predecessors have finished, but it cannot start because the resources required for it were scheduled to arrive at a later time. (2) The resources required for the activity have arrived and are ready to be used but the activity is not ready to start because of precedence constraints. In the first case we will incur a “holding” cost while in the second case, we will incur a “shortage” cost. Our objective is to set gates so as to minimize the sum of the expected holding and shortage costs. We employ the Cross-Entropy method to solve the problem. The paper describes the implementation of the method, compares its results to various heuristic methods and provides some insights towards actual applications.     

Scheduling non-critical activities using multicriteria approach

In many projects the problem of selecting the start time of a non-critical activity arises. Usually it is possible to use the “as soon as possible” or “as late as possible” rules. In some situations, however, the result of such a decision depends on external factors such as exchange rate. This leads to an approach in which the problem of scheduling non-critical activities is solved using an expanded Cox–Ross–Rubinstein (CRR) binomial tree method. In the paper a bi-criteria problem of determining the start time of a non-critical activity is considered. We assume that the early start and the late start of the activity have been identified using Critical Path Method, but the project manager is free to select the time when the activity will actually be started. This decision cannot, however, be changed later, as it is associated with the allocation of key resources. Two main criteria are considered: cost and risk. While cost depends on exchange rate, risk increases with the delay of the start of the activity. The problem can be described as a dynamic process. We propose a new interactive technique for solving such a bi-criteria decision making problem under risk. The procedure uses trade-offs to identify a candidate solution. The CRR binomial method is applied to evaluate the cost of the activity.

     

A Blind-Spot Argument Against Dispositionalist Accounts of Belief

Dispositionalist accounts of belief define beliefs in terms of specific sets of dispositions. In this article, I provide a blind-spot argument against these accounts. The core idea of the argument is that beliefs having the form [p and it is not manifestly believed that p] cannot be manifestly believed. This means that one cannot manifest such beliefs in one’s assertions, conscious thoughts, actions, behaviours, or any other type of activity. However, if beliefs are sets of dispositions, they must be manifestable in some way. Therefore, according to my argument, beliefs are not sets of dispositions. The argument is defended against some possible objections.     

A 3/2 Algorithm for Two-Machine Open Shop with Route-Dependent Processing Times

This paper considers the problem of minimizing the schedule length of a two-machine shop in which not only can a job be assigned any of the two possible routes, but also the processing times depend on the chosen route. This problem is known to be NP-hard. We describe a simple approximation algorithm that guarantees a worst-case performance ratio of 2. We also present some modifications to this algorithm that improve its performance and guarantee a worst-case performance ratio of 3/2.     

Structured construction and simulation of nondeterministic stochastic activity networks

In this work we deal with nondeterministic stochastic activity networks (NDSANs). Their stochastic character results from activity durations, which are given by nonnegative continuous random variables. The nondeterministic behavior of an NDSAN is a consequence of its variable topology, based on two additional features. First, by associating choice probabilities with the immediate successors of an activity, some branches of execution are not always taken. Second, by allowing iterated executions of a group of activities according to predetermined probabilities, the number of times an activity is to be executed is not determined a priori. These properties lead to a wide variety of activity networks, capable of modelling many real situations in process engineering and project management. We describe a simple, recursively structured construction of NDSANs, which both provides a coherent syntactic mechanism to incorporate the two abovementioned nondeterminism features and allows the analytic formulation of completion time. This construction also directly gives rise to a recursive simulation algorithm for NDSANs, whose repeated execution produces an estimate of the probability distribution of the completion time of the network. We also report on real-world case studies, using the Komolgorov–Smirnov statistic for validation.     

Modified Kuhn-tucker conditions when a minimum is not attained

The Kuhn-Tucker conditions for constrained minimization assume that the minimum is attained. When there is a finite infimum, but a minimum is not attained, an asymptotic version of Kuhn-Tucker conditions is obtained for linear problems, in general in infinite dimensions, with some restriction on the feasible set. This result is extended to some nonlinear problems, not necessarily convex, with some further restriction on differentiability.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

Should proof be minimal? Ms T's evaluation of secondary school students’ proofs

Calls for reform in mathematics education around the world state that proofs should be part of school mathematics at all levels. Turning these calls into a reality falls on teachers’ shoulders. This paper focuses on one secondary school teacher's reactions to students’ suggested proofs and justifications in elementary number theory. To determine whether the justifications are acceptable, the teacher used not only her SMK regarding mathematical aspects, but also her PCK about what a student giving this justification might know or not know. A discussion of the findings is followed by some questions that arise.     

A new model for Boolean algebra

This new model for set theory is a graph. It is similar in many ways to a Venn diagram or Karnaugh map, but it does not pose as a rival, merely as an alternative model which may be useful in some contexts. Defined with reference to the duality of lines and points, the graph is a fitting framework in which to display the rich duality of Boolean algebra.

In the first four sections the graph is developed as a natural embodiment of Boolean theory and it is hoped that it will be seen, not as a more computational device but as helpful for demonstrating Boolean theory. The second half of the article is devoted to practical applications. The graph can be applied (and has been applied in school teaching) extensively in set theory, in logic, in probability, in genetics and in switching circuits, but space does not allow the elaboration of all these in detail. So this article concentrates mainly on one of these applications, switching circuits. The graph is used to simplify and minimize logic circuits with techniques different from Karnaugh's and in some instances more comprehensive.     

Central limit theorems under weak dependence

This article is motivated by a central limit theorem of Ibragimov for strictly stationary random sequences satisfying a mixing condition based on maximal correlations. Here we show that the mixing condition can be weakened slightly, and construct a class of stationary random sequences covered by the new version of the theorem but not Ibragimov's original version. Ibragimov's theorem is also extended to triangular arrays of random variables, and this is applied to some kernel-type estimates of probability density.