the shortest path problem with two objective functions abstract we present methods to find the shortest path in a network where each path is associated with two objectives we describe how to obtain the nondominated paths and the extreme nondominated paths and compare the expected complexity of the methods an improvement in efficiency can be obtained when quasiconcave or quasiconvex utility functions are assumed in the first case we describe how to find the optimal extreme nondominated path and bounds for the optimal path value then the optimal path can be located by calculating the k th shortest path in the second case we suggest a branch and bound method to solve the problem

knotting in stretched polygons the knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon by examining the incidence of cut planes in the polygon we prove a pattern theorem in the stretching regime for large applied forces this theorem can be used to examine the incidence of entanglements such as knotting and writhing in particular we prove that for arbitrarily large positive but finite values of the stretching force the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases in the case of writhing we prove that for stretched polygons of length n and for every function the probability that the absolute value of the mean writhe is less than f n approaches 0 as n for sufficiently large values of the applied stretching force

the shapes of self avoiding polygons with torsion we consider self avoiding polygons on the simple cubic lattice with a torsion fugacity we use monte carlo methods to generate large samples as a function of the torsion fugacity and the number of edges in the polygon using these data we investigate the shapes of the polygons at large torsion fugacity and find evidence that the polygons have substantial helical character in addition we show that these polygons have induced writhe for any non zero torsion fugacity and that torsion and writhe are positively correlated

monte carlo generation of self avoiding walks with fixed endpoints and fixed length we propose a new class of dynamic monte carlo algorithms for generating self avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension and prove that these algorithms are ergodic in all cases we also prove the ergodicity of a variant of the pivot algorithm

a monte carlo algorithm for lattice ribbons a lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions the ribbon can be closed to form an object which is topologically either a cylinder or a mobius band depending on whether its surface is orientable or nonorientable we describe a grand canonical monte carlo algorithm for generating a sample of these ribbons prove that the associated markov chain is ergodic and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons

choice of optimal blocking schemes in two level and three level designs blocking is commonly used in design of experiments to reduce systematic variation and increase precision of effect estimation for 2n k designs in 2p blocks others have studied the confounding pattern between blocking effects and treatment effects and proposed some criteria to choose optimal blocking schemes we propose alternative criteria that allow estimation of more lower order effects extensions to three level fractional factorial design are considered and tables of useful designs are given

path integral calculation of the writhe for circular semiflexible polymers earlier obtained numerical and analytical non path integral results for the writhe of a circular polymer are reproduced exactly employing the available path integral models for the semiflexible chains it is confirmed analytically that the average writhe scales with the root of the chain length we also briefly discuss related topological problems e g knotting of circular polymers which could be formulated and solved with the use of path integrals

optimal two level regular fractional factorial block and split plot designs we propose a general and unified approach to the selection of regular fractional factorial designs which can be applied to experiments that are unblocked blocked or have a split plot structure our criterion is derived as a good surrogate for the model robustness criterion of information capacity in the case of random block effects it takes the ratio of intra and interblock variances into account in most of the cases we have examined there exist designs that are optimal for all values of that ratio examples of optimal designs that depend on the ratio are provided we also demonstrate that our criterion can further discriminate designs that cannot be distinguished by the existing minimum aberration criteria

on anova expansions and strategies for choosing the anchor point the classic lebesgue anova expansion offers an elegant way to represent functions that depend on a high dimensional set of parameters and it often enables a substantial reduction in the evaluation cost of such functions once the anova representation is constructed unfortunately the construction of the expansion itself is expensive due to the need to evaluate high dimensional integrals a way around this is to consider an alternative formulation known as the anchored anova expansion this formulation requires no integrals but has an accuracy that depends sensitively on the choice of a special parameter known as the anchor point we present a comparative study of several strategies for the choice of this anchor point and argue that the optimal choice of this anchor point is the center point of a sparse grid quadrature this choice induces no additional cost and as we shall show results in a natural truncation of the anova expansion the efficiency and accuracy is illustrated through several standard benchmarks and this choice is shown to outperform the alternatives over a range of applications c 2010 elsevier inc all rights reserved

three point approximations for continuous random variables this paper compares a number of approximations used to estimate means and variances of continuous random variables and or to serve as substitutes for the probability distributions of such variables with particular emphasis on three point approximations numerical results from estimating means and variances of a set of beta distributions indicate surprisingly large differences in accuracy among approximations in current use with some of the most popular ones such as the pert and triangular density function approximations faring poorly a simple new three point approximation which is a straightforward extension of earlier work by pearson and tukey outperforms the others significantly in these tests and also performs well in related multivariate tests involving the dirichlet family of distributions it offers an attractive alternative to currently used approximations in a variety of applications

technical note an exact algorithm for the time constrained traveling salesman problem the time constrained traveling salesman problem is a variation of the familiar traveling salesman problem that includes time window constraints on the time a particular city or cities may be visited this note presents a concise formulation of the time constrained traveling salesman problem the model assumes that the distances of the problem are symmetrical and that the triangle inequality holds additionally the model allows the salesman to wait at a city if necessary for a time window to open the dual of the formulation is shown to be a disjunctive graph model which is well known from scheduling theory a longest path algorithm is used to obtain bounding information for subproblems in a branch and bound solution procedure computational results are presented for several small to moderate size problems

knot probability of polygons subjected to a force a monte carlo study we use monte carlo methods to study the knot probability of lattice polygons on the cubic lattice in the presence of an external force f the force is coupled to the span of the polygons along a lattice direction say the z direction if the force is negative polygons are squeezed the compressive regime while positive forces tend to stretch the polygons along the z direction the tensile regime for sufficiently large positive forces we verify that the pincus scaling law in the force extension curve holds at a fixed number of edges n the knot probability is a decreasing function of the force for a fixed force the knot probability approaches unity as 1 exp 0 f n o n where 0 f is positive and a decreasing function of f we also examine the average of the absolute value of the writhe and we verify the square root growth law known for f 0 for all values of f

better estimation of pert activity time parameters this paper builds upon earlier work from the decision risk analysis area in presenting simple easy to use approximations for the mean and variance of pert activity times these approximations offer significant advantages over the pert formulas currently being taught and used as well as over recently proposed modifications for instance they are several orders of magnitude more accurate than their pert counterparts in estimating means and variances of beta distributions if the data required for all methods are obtained accurately moreover they utilize probability data that can be assessed more reliably than those required by the pert formulas while still requiring just three points from each activity time probability distribution using the proposed approximations can significantly improve the accuracy of probability statements about project completion time and their use complements ongoing efforts to improve pert analyses of networks involving multiple critical paths

the computational complexity of the criticality problems in a network with interval activity times abstract the paper analyzes the criticality in a network with interval activities duration times a natural generalization of the criticality notion for a path an activity and an event for the case of network with interval activity duration times is given the computation complexity of five problems linked to the introduced criticality notion is presented

determining the k most critical paths in pert networks a fundamental problem in pert networks is to identify a project s critical paths and its critical activities in this paper we define the criticality index of a path as the probability that the duration of the path is greater than or equal to the duration of every other path in the network and define the criticality index of an activity as the sum of the criticality indices of the paths containing that activity the most critical path or k most critical paths in a pert network could be found by enumerating all the paths and calculating the corresponding criticality indices both of which are burdensome tasks this paper uses stochastic dominance to develop a procedure to identify the k most critical paths without using this path enumeration the procedure has been applied to various sized pert networks generated at random and the results are found to be very close to those obtained by extensive monte carlo sampling

blocked nonregular two level factorial designs this article discusses the optimal blocking criteria for nonregular two level designs we extend the optimal blocking criteria of cheng and wu to nonregular designs by adapting the g and g2 minimum aberration criteria discussed by tang and deng to define word length pattern for nonregular designs we extend the notion of word to nonregular designs through a polynomial representation of factorial designs we define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding we propose four new criteria which we use to search for optimal blocking schemes of 12 run 16 run and 20 run two level orthogonal arrays

using blocked fractional factorial designs to construct discrete choice experiments for healthcare studies discrete choice experiments dces are increasingly used for studying and quantifying subjects preferences in a wide variety of healthcare applications they provide a rich source of data to assess real life decision making processes which involve trade offs between desirable characteristics pertaining to health and healthcare and identification of key attributes affecting healthcare the choice of the design for a dce is critical because it determines which attributes effects and their interactions are identifiable we apply blocked fractional factorial designs to construct dces and address some identification issues by utilizing the known structure of blocked fractional factorial designs our design techniques can be applied to several situations including dces where attributes have different number of levels we demonstrate our design methodology using two healthcare studies to evaluate i asthma patients preferences for symptom based outcome measures and ii patient preference for breast screening services copyright 2016 john wiley sons ltd

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

torsion of polygons in the torsion of polygons and self avoiding walks in the cubic lattice is a measure of the self entanglement of these objects we consider several definitions of torsion in polygons and introduce a fugacity conjugate to the torsion in our models we study the thermodynamic behaviour of these models using probabilistic methods and rigorous methods from statistical mechanics in particular we prove that at least one of our models has a non analyticity in its free energy corresponding to a transition between phases with high and low torsion

some results on 2n m designs of resolution iv with weak minimum aberration it is known that all resolution iv regular 2 n m designs of run size n 2 n m where 5n 16 n n 2 must be projections of the maximal even design with n 2 factors and therefore are even designs this paper derives a general and explicit relationship between the wordlength pattern of any even 2 n m design and that of its complement in the maximal even design using these identities we identify some weak minimum aberration 2 n m designs of resolution iv and the structures of their complementary designs based on these results several families of minimum aberration 2 n m designs of resolution iv are constructed

critical path in an activity network with time constraints abstract an acyclic graph with nonnegative weights and with a unique source and destination is called an activity network a project comprised of a set of activities and precedence relationships can be represented by an activity network and the mathematical analysis of the network can provide useful information for managing the project in a traditional activity network it is assumed that an activity can begin any time after all of its preceding activities have been completed this assumption does not adequately describe many practical situations in which some kinds of time constraint are usually associated with an activity in this paper we investigate two types of time constraint commonly encountered in project management the first is the time window constraint which assumes that an activity can begin its execution only in a specified time interval the second is the time schedule constraint which requires that an activity begin only at one of pre specified beginning times an efficient linear time algorithm for finding the longest path critical path and for analyzing the flow time of each arc is developed for activity networks with these time constraints

optimal blocking of two level factorial designs summary blocking of two level factorial designs is considered for block sizes 2 and 4 using the method of fractional partial confounding a d and e optimal designs are obtained for block size 2 within the class of orthogonal designs for which main effects and two factor interactions are all orthogonal to each other before allowing for blocking a d and e optimal designs are obtained for block size 4 within the class of orthogonal designs with main effects orthogonal to blocks the designs obtained also have other favourable properties including orthogonal estimation of effects and orthogonality to superblocks

demonstrating single and multiple currents through the e coli secyeg pore testing for the number of modes of noisy observations we analyze a new dataset from an electrophysiological recording of transmembrane currents through a bacterial membrane channel to demonstrate the existence of single and multiple channel currents protein channels mediate transport through biological membranes knowledge of the channel properties gained from electrophysiological recordings is important for a targeted drug design we investigate the bacterial membrane protein secyeg which is of essential importance for the secretory pathway for sorting of newly synthesized proteins to their place of function in the cell our results strongly indicate that in the secyeg pore the different modes of the density of channel currents are approximately equidistant and correspond to different numbers of open channels in the membrane a current of 12 pa under the present experimental conditions turns out to be characteristic of the presence of a single open secyeg pore a fact that had not been electrophysiologically characterized so far electrophysiological reco

an alternative for robust estimation in project management abstract recently hahn 2008 has proposed the mixture between the uniform and the beta distributions as an alternative to the beta distribution in pert methodology which allows for varying amounts of dispersion and a greater likelihood of more extreme tail area events however this mixture lacks a closed cumulative distribution function expression and its parameters remain a difficult interpretation in addition the kurtosis limit of the beta distribution is 3 due to their higher kurtosis and easier elicitation we consider the two sided power and the generalized biparabolic distributions good candidates for applying hahn s idea 2008 and for building the uniform two sided power u tsp and the uniform generalized biparabolic u gbp distributions using the same example from hahn 2008 we are going to demonstrate that we can obtain more accuracy and a greater likelihood of more extreme tail area events these distributions could be applied in other heavy tailed phenomena which are usual in business contexts

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

estimating the mean completion time of pert networks with exponentially distributed durations of activities abstract in order to compute the exact values of the characteristics for pert networks with exponentially distributed durations of activities one need to explore the state space of a markov chain this method suffers from exponential state explosion an approximate method for estimating the mean completion time of the above networks is proposed in this paper our method is of polynomial computational complexity the accuracy of the method is compared with the best upper bound for the considered kind of networks

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

stochastic bounds on distributions of optimal value functions with applications to pert network flows and reliability in many classical combinatorial optimization problems including critical and shortest paths maximum flow and network reliability the introduction of uncertainty considerably complicates the calculation of system performance in fact in these contexts computing system performance exactly can often be an impossible task therefore obtaining stochastic bounds on the system s performance becomes an attractive and useful alternative this paper studies several stochastic bounds that are applicable to these contexts and to a broader set of problems that can be described by the general combinatorial concepts of clutters and blocking clutters we begin our discussion by defining these unifying concepts and illustrating their specialization in several problem contexts

a new structural mechanism for reducibility of stochastic pert networks abstract in recent years many techniques were developed to compute completion time distribution function of stochastic pert networks none of the existing techniques clearly consider the subject from structural viewpoint in this paper a structural mechanism based on graph theory is presented this mechanism changes structure of network to series parallel network with contraction of some arcs in contraction operation one arc is removed and its two nodes are changed to individual node in this operation probably dependency between activities and number of existing paths in the network would be different from the original network an algorithm is presented in correspondence with the developed structural mechanism through this algorithm the mechanism acts as an effective approach to estimate f t the paper contains a comparative analysis this comparison is made between the computational results of developed algorithm and those of dodin s and kleindorfer s approaches the comparative analysis demonstrates that the mechanism has capability to be used for the networks with actual sizes also to show that the method works for practical problems in the real world a real case from a car factory of iran is considered again for this case of the developed method dodin s and kleindorfer s approaches are applied whose results approve the acceptability of proposed method completely

approximating the criticality indices of the activities in pert networks a stochastic pert network is a directed acyclic network in which the arc lengths are independent random variables with known distributions a fundamental problem in pert networks is to identify the activities which are critical to the achievement of the project objectives in an activity network if the duration of each activity is not a random variable then it is easy to identify the criticality of each activity represented by its float time however when the duration of any activity is a random variable it is not easy to identify the criticality of each activity in this case the criticality of an activity is known as the criticality index which is defined as the sum of the criticality indices of the paths containing it the criticality index of a path is the probability that the duration of the path is greater than or equal to the duration of every other path in the network clearly the criticality index of an activity can be obtained by determining the criticality indices of the paths which requires identifying all the paths determining their criticality indices then identifying the paths containing the activity in this paper we develop a theory which leads to a procedure to approximate the criticality indices of all the activities without going through the above three steps the procedure has been applied to large size pert networks generated at random and the results are found to be very close to those obtained by extensive monte carlo sampling

optimal blocking schemes for 2n and 2n p designs systematic sources of variations in factorial experiments can be effectively reduced without biasing the estimates of the treatment effects by grouping the runs into blocks for full factorial designs optimal blocking schemes are obtained by applying the minimum aberration criterion to the block defining contrast subgroup a related concept of order of estimability is proposed for fractional factorial designs because of the intrinsic difference between treatment factors and block variables the minimum aberration approach has to be modified a concept of admissible blocking schemes is proposed for selecting block designs based on multiple criteria the resulting 2 n and 2 n p designs are shown to have better overall properties for practical experiments than those in the literature

reply an answer to sasieni s question on pert times in a recent note sasieni 1986 questioned the classical pert formula for finding the average time t a 4m b 6 where a is the least possible time b the greatest possible time m the modal time and t the estimate of the mean time the original source of this formula navy 1958 has the answer this answer is summarized here since the original document is now difficult to obtain as sasieni points out the beta distribution is the usual assumption for the times in pert analysis the standardized beta density with parameters a 1 and 3 1 for the random variable x is given by

2n l designs with weak minimum aberration since not all 2 n l fractional factorial designs with maximum resolution are equally good fries and hunter introduced the minimum aberration criterion for selecting good 2 n l fractional factorial designs with the same resolution we modify the concept of minimum aberration and define weak minimum aberration and show the usefulness of this new design concept using some techniques from finite geometry we construct 2 n l fractional factorial designs of resolution iii with weak minimum aberration further several families of 2 n l fractional factorial designs of resolution iii and iv with minimum aberration are obtained

branch and price and cut for the split delivery vehicle routing problem with time windows this paper addresses the split delivery vehicle routing problem with time windows sdvrptw that consists of determining least cost vehicle routes to service a set of customer demands while respect

mathematical formulations for a 1 full truckload pickup and delivery problem abstract we address a generalization of the asymmetric traveling salesman problem where routes have to be constructed to satisfy customer requests which either involve the pickup or delivery of a single commodity a vehicle is to be routed such that the demand and the supply of the customers is satisfied under the objective to minimize the total distance traveled the commodities which are collected from the pickup customers can be used to accommodate the demand of the delivery customers in this paper we present three mathematical formulations for this problem class and apply branch and cut algorithms to optimally solve the model formulations for two of the models we derive benders cuts based on the classical and the generalized benders decomposition finally we analyze the different mathematical formulations and associated solution approaches on well known data sets from the literature

improved moment estimation formulas using more than three subjective fractiles pert type subjective estimations are used in many stochastic decision models to estimate the random variables mean and standard deviation s d the approach is based on the beta distribution assumption also most pert type formulas use only three estimated fractiles we point out that i it is desirable to consider a substantially richer set of distributions than the beta in developing pert type formulas ii it may be beneficial to use more than three fractile estimates in pert type formulas we then develop formulas for estimating the mean and s d that are based on a substantially richer set of distributions than the beta and that use more than three estimated fractiles these formulas perform better than the best currently vailable formulas when the subjective distribution is not restricted to be beta

the most critical path in a pert network a heuristic approach abstract the classical pert approach uses a deterministic critical path to estimate the probability of completing a project by a given deadline in general such a path is not the most critical path and does not provide the lowest probability of completion time this paper studies the most critical path problem and formulates it as an optimal path problem through a deterministic network with a two attribute fractional objective function a heuristic approach is developed for the optimal path problem which gives a solution to the most critical path problem the illustrative examples and computational results demonstrate that the proposed procedure provides critical activities and estimates for the probabilities of completion time that are more accurate than those resulting from the classical pert approach further our computational experiments show that the proposed heuristic performs well in identifying the most critical paths

a note on the definition of resolution for blocked 2 k p designs when two level fractional factorial designs are blocked the application of the standard definition of resolution requires careful consideration sometimes linear contrasts that superficially appear to be estimates of higher order interaction effects are in reality estimates of first order effects experimenters may therefore inadvertently choose designs that are of lower resolution than intended or unknowingly confound important effects in this note i discuss this subtle problem and propose an additional rule to the usual definition of resolution that provides a conservative but more realistic assessment of the resolution with this more realistic characterization experimenters are provided with a warning about possible confounding i also show that my amendment to the definition of resolution may be useful when characterizing designs in which several two level contrasts are combined to accommodate factors with four or more levels

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

the writhe of polygons on the face centred cubic lattice for polygons on the simple cubic lattice there is an important theorem due to lacher and sumners which shows that the writhe of a polygon is the average of the linking numbers of the polygon and its pushoffs in four particular directions this implies that the writhe of a polygon on the simple cubic lattice is always rational we prove a related theorem for the face centred cubic fcc lattice but show that on this lattice the probability that a polygon of length n has irrational writhe tends to unity as n tends to infinity in addition we show that the expectation of the absolute value of the writhe increases at least as fast as n 1 2 for large n

a linear bayesian stochastic approximation to update project duration estimates by relaxing the unrealistic assumption of probabilistic independence on activity durations in a project this paper develops a hierarchical linear bayesian estimation model statistical dependence is established between activity duration and the amount of resource as well as between the amount of resource and the risk factor upon observation or assessment of the amount of resource required for an activity in near completion the posterior expectation and variance of the risk factor can be directly obtained in the bayesian scheme then the expected amount of resources required for and the expected duration of upcoming activities can be predicted we simulate an application project in which the proposed model tracks the varying critical path activities on a real time basis and updates the expected project duration throughout the entire project in the analysis the proposed model improves the prediction accuracy by 38 36 compared to the basic pert approach

risk taking in stochastic pert networks abstract this paper studies a pert problem which incorporates project managers risk taking behavior in stochastic environments in this stochastic pert network which we refer to as riskpert activity durations are nonnegative random variables and a project manager uses a disutility function to evaluate each path and chooses that path which maximizes his expected disutility as an optimal or critical path this path as opposed to the classical critical path is used as a basis to analyze the pert network and to estimate the probability of completion time the general riskpert is difficult to solve however special cases when activity durations are statistically independent and disutility functions are linear exponential quadratic or linear exponential are solvable exactly the illustrative examples demonstrate that riskpert captures managers risk taking behavior and provides results that are more realistic than those of the classical pert models furthermore this paper extends some of the riskpert models to include multi dimensional pert networks

existence and construction of randomization defining contrast subspaces for regular factorial designs regular factorial designs with randomization restrictions are widely used in practice this paper provides a unified approach to the construction of such designs using randomization defining contrast subspaces for the representation of randomization restrictions we use finite projective geometry to determine the existence of designs with the required structure and develop a systematic approach for their construction an attractive feature is that commonly used factorial designs with randomization restrictions are special cases of this general representation issues related to the use of these designs for particular factorial experiments are also addressed

chance constrained programming in activity networks a critical evaluation abstract we review the chance constrained programming ccp model for event realization in activity networks ans and the more recent contribution to it by kress from the points of view of validity and accuracy we present a classification scheme of stochastic programming and confirm that the ccp does not solve the problem of finding a point estimate that satisfies a required confidence level and is of little help in determining the cumulative distribution function c d f of the project completion time we show that the ccp leads to an extremely weak lower bound l b on the exact c d f a backtracking approach is proposed that improves this l b via the ccp which is still too weak to be of any practical value finally we utilize the ccp approach to derive a result due to kress in a simpler and more direct way we also demonstrate that the result itself unfortunately is of questionable validity

bounding the project completion time distribution in pert networks we consider the pert model of a project composed of activities whose durations are random variables with known distributions for the situations in which the activity durations are completely independent we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below the bounding distribution can be used to obtain an upper bound on the mean completion time of the project we also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

a repeated matching heuristic for the single source capacitated facility location problem facility location models form an important class of integer programming problems with application in many areas such as the distribution and transportation industries an important class of solution methods for these problems are so called lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust the general facility location problem can be divided into a number of special problems depending on the properties assumed in the capacitated location problem each facility has a specific capacity on the service it provides we describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility the approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied the method generates feasible solutions in each iteration in contrast to lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution numerical results show that the approach produces solutions which are of similar and often better than those produced using the best lagrangean heuristics

minimizing task completion time with the execution set method we consider a partial ordered set poset of assembling operations with known execution durations that must be accomplished the assembling operations can be executed on an acyclic network with an identical set of robots on each conveyer arc the number of depots nodes is a known integer between each pair of depots we can locate only one conveyer we seek an arrangement of the network and a plan that divides the task operations among the conveyers minimizing the overall task completion time we use linear programming optimization subject to reasonably general rules for distributing the operation fragments among the conveyers

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 diaer 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 eaective use of budgets and resource allocation 2000 elsevier science b v all rights reserved

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

blocked regular fractional factorial designs with maximum estimation capacity in this paper the problem of constructing optimal blocked regular fractional factorial designs is considered the concept of minimum aberration due to fries and hunter is a well accepted criterion for selecting good unblocked fractional factorial designs cheng steinberg and sun showed that a minimum aberration design of resolution three or higher maximizes the number of two factor interactions which are not aliases of main effects and also tends to distribute these interactions over the alias sets very uniformly we extend this to construct block designs in which i no main effect is aliased with any other main effect not confounded with blocks ii the number of two factor interactions that are neither aliased with main effects nor confounded with blocks is as large as possible and iii these interactions are distributed over the alias sets as uniformly as possible such designs perform well under the criterion of maximum estimation capacity a criterion of model robustness which has a direct statistical meaning some general results on the construction of blocked regular fractional factorial designs with maximum estimation capacity are obtained by using a finite projective geometric approach

blocked regular fractional factorial designs with minimum aberration this paper considers the construction of minimum aberration ma blocked factorial designs based on coding theory the concept of minimum moment aberration due to xu 2003 for unblocked designs is extended to blocked designs the coding theory approach studies designs in a row wise fashion and therefore links blocked designs with nonregular and supersaturated designs a lower bound on blocked wordlength pattern is established it is shown that a blocked design has ma if it originates from an unblocked ma design and achieves the lower bound it is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros sufficient conditions are given for constructing ma blocked designs from unblocked ma designs the theory is then applied to construct ma blocked designs for all 32 runs 64 runs up to 32 factors and all 81 runs with respect to four combined wordlength patterns

induced writhe in linked polygons we consider a pair of polygons on the simple cubic lattice forming a link of specified type we investigate the mean writhe of the polygons as a function of the number of edges in each polygon and of the link type although two unknotted polygons which are linked as a hopf link have zero mean writhe we show that other torus links induce writhe in the polygons we also consider non torus links including a link in which one polygon is knotted

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

a new warmstarting strategy for the primal dual column generation method this paper presents a new warmstarting technique in the context of a primal dual column generation method applied to solve a particular class of combinatorial optimization problems the technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after new columns arrive conditions on the maximum size of the cuts and on a suitable initial point are discussed additionally the strategy ensures that the duality gap of the warmstart is bounded by the old duality gap multiplied with a small constant which depends on the relation between the old and modified problems computational experiments demonstrate the gains achieved when compared to a coldstart approach

estimating network characteristics in stochastic activity networks this paper describes a monte carlo method based on the theory of quasirandom points for estimating the distribution functions and means of network completion time and shortest path time in a stochastic activity network in particular the method leads to estimators whose absolute errors converge as log kn k where k denotes the number of replications collected in the experiment and n is the number of dimensions for sampling this rate compares favorably with the standard error of estimate o1 k1 2 which obtains for experiments that use random sampling moreover since quasirandom points are nonrandom the upper bound log kn k is deterministic in contrast to the random sampling rate o1 k1 2 which is probabilistic the paper demonstrates how the use of a cutset of the network reduces n in the bound when estimating the distribution functions two examples illustrate the benefits and costs of using quasirandom points

characterization of minimum aberration 2 sp n k designs in terms of their complementary designs a general result is obtained that relates the word length pattern of a 2 n k design to that of its complementary design by applying this result and using group isomorphism we are able to characterize minimum aberration 2 n k designs in terms of properties of their complementary designs the approach is quite powerful for small values of 2 n k n 1 in particular we obtain minimum aberration 2 n k designs with 2 n k n 1 1 to 11 for any n and k

a comparison of procedures for estimating the parent probability distribution from a given set of fractiles abstract on one hand eliciting subjective probabilities fractiles is a well established procedure on the other hand knowledge of a subjective variable s central moments or distribution function is often assumed however the problem of converting elicited fractiles into the required moments or distribution function has been largely ignored we show that this conversion problem is far from trivial and that the most commonly used conversion procedures often produce huge errors alternative procedures are proposed the tocher s curve and linear function of fractiles methods are shown to be much more accurate than the commonly used procedures

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

algorithms for the vehicle routing and scheduling problems with time window constraints this paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints given the intrinsic difficulty of this problem class approximation methods seem to offer the most promise for practical size problems after describing a variety of heuristics we conduct an extensive computational study of their performance the problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems the percentage of customers with time windows their tightness and positioning and the scheduling horizon we found that several heuristics performed well in different problem environments in particular an insertion type heuristic consistently gave very good results

a complementary design theory for doubling a complementary design theory for doubling by hongquan xu 1 and ching shui cheng 2 university of california los angeles and university of california berkeley august 11 2006 chen and cheng 2006a discussed the method of doubling for con structing two level fractional factorial designs they showed that for 9n 32 n 5n 16 all minimum aberration designs with n runs and n factors are projections of the maximal design with 5n 16 factors which is constructed by repeatedly doubling the 2 5 1 design de ned by i abcde this paper develops a general complementary design the ory for doubling for any design obtained by repeated doubling general identities are established to link the wordlength patterns of each pair of complementary projection designs a rule is developed for choosing minimum aberration projection designs from the maximal design with 5n 16 factors it is further shown that for 17n 64 n 5n 16 all minimum aberration designs with n runs and n factors are projections of the maximal design with n runs and 5n 16 factors ams 2000 subject classi cations primary 62k15 key words and phrases maximal design minimum aberration pless power moment identity wordlength pattern running title doubling and complementary designs supported in part by nsf grant dms 0505728 supported in part by nsf grant dms 0505556

computing the writhe on lattices given a polygonal closed curve on a lattice or space group we describe a method for computing the writhe of the curve as the average of weighted projected writhing numbers of the polygon in a few directions these directions are determined by the lattice geometry the weights are determined by areas of regions on the unit 2 sphere and the regions are formed by the tangent indicatrix to the polygonal curve we give a new formula for the writhe of polygons on the face centred cubic lattice and prove that the writhe of polygons on the body centred cubic lattice the hexagonal simple lattice and the diamond space group is always a rational number and discuss applications to ring polymers

an upper bound on the expected completion time of pert networks abstract in this paper the problem of determining an upper bound on the expected project completion time described by the pert network is considered it is assumed that activity durations are independent random variables with given means the exact forms of probability distributions do not have to be known however their cumulative distribution functions are expected to belong to the so called nbue class very simple algorithms for deriving this bound are presented the computations can even be performed manually for more involved networks our approach producing a pessimistic evaluation of the expected value of the project duration extends considerably the information obtained through the use of the classical pert that always underestimates this value the results are illustrated by a simple example and errors of approximations are discussed

blocked two level regular factorial designs with weak minimum aberration this paper considers the construction of blocked two level regular designs with weak minimum aberration we first obtain the minimum value of the number of two factor interactions which are aliased with the block effects based on this result two methods are then proposed in two different scenarios to construct weak minimum aberration blocked two level designs with respect to some existing combined wordlength patterns copyright 2013 oxford university press

theory of optimal blocking of 2 n m designs in this paper we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment defining words and block defining words the concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design by blending techniques of coding theory and finite projective geometry we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked 2 n m design and the split wordlength pattern of its blocked residual design based on these identities we establish general rules for identifying minimum aberration blocked 2 n m designs in terms of their blocked residual designs using these rules we study the structures of some blocked 2 n m designs with minimum aberration

some identities on q sp n m designs with application to minimum aberration designs chen and hedayat and tang and wu studied and characterized minimum aberration 2 n m designs in terms of their complemetary designs based on a new and more powerful approach we extend the study to identify minimum aberration q n m designs through their complementary designs by using macwilliams identities and krawtchouk polynomials in coding theory we obtain some general and explicit relationships between the wordlength pattern of a q n m design and that of its complementary design these identities provide a powerful tool for characterizing minimum aberration q n m designs the case of q 3 is studied in more details

combinatorial optimization with rational objective functions a communication megiddo megiddo n 1979 combinatorial optimization with rational objective functions math oper res 4 414 424 has proposed an algorithm to solve a class of combinatorial optimization problems with rational objective functions in polynomial time the minimum ratio path problem is claimed to be one such problem this claim is not correct as indicated below

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

fractional resolution and minimum aberration in blocked 2 n k designs systematic sources of variation can be reduced in fractional factorial experiments by grouping the runs into blocks this is accomplished through the use of blocking factors the concepts of resolution and minimum aberration design optimization criteria ordinarily used to rank unblocked fractional factorial designs are extended to such blocked fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup some limited theoretical results are derived and tables of minimumaberration blocked two level fractional factorial designs are presented and considered the relationship between clear effects effects that are estimable when higher order effects are assumed negligible and minimum aberration in the presence of blocking is discussed

the design of blocked fractional factorial split plot experiments in industrial screening experiments if some factors are hard to vary and others are easy to vary randomization restrictions may lead to the use of fractional factorial split plot ffsp designs blocked ffsp bffsp designs arise when all runs cannot be performed under homogeneous conditions this article introduces three approaches for blocking ffsp designs a catalog of designs is presented in which bffsp designs are ranked according to the minimum aberration criterion in addition additional optimality criteria are used to assist in the ranking procedure the case study that motivated this research is also discussed in detail

the mean squared writhe of alternating random knot diagrams the writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram it plays an important role not only in knot theory itself but also in various applications of knot theory to fields such as molecular biology and polymer physics the mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one half however such a diagram is usually not minimal if we restrict ourselves to a minimal knot diagram then the choice of which strand is the over or under strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams this paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced alternating knot diagrams we demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates in particular this is true when the sample space contains only diagrams of a special structure when the sample space is large enough to contain not only diagrams of a special type then the mean squared writhe for n crossing diagrams tends to grow linearly with n but at a faster rate than n indicating an intrinsic property of alternating knot diagrams studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used which is an important area of study in the applications of random knot theory