robust and adaptive network flows we study network flow problems in an uncertain environment from the viewpoint of robust optimization in contrast to previous work we consider the case that the network parameters e g capacities are known and deterministic but the network structure e g nodes and arcs is subject to uncertainty in this paper we study the robust and adaptive versions of the maximum flow problem and minimum cut problems in networks with node and arc failures and establish structural and computational results the adaptive two stage model adjusts the solution after the realization of the failures in the network this leads to a more flexible model and yields less conservative solutions compared to the robust model we show that the robust maximum flow problem can be solved in polynomial time but the robust minimum cut problem is np hard we also prove that the adaptive versions are np hard we further characterize the adaptive model as a two person zero sum game and prove the existence of an equilibrium in such g

nonlinear control of constrained linear systems via predictive reference management a method based on conceptual tools of predictive control is described for solving set point tracking problems wherein pointwise in time input and or state inequality constraints are present it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time a virtual sequence among a family of linearly parameterized command sequences by solving a convex constrained quadratic optimization problem and feeds the primal system according to a receding horizon control philosophy the overall system is proved to fulfill the constraints be asymptotically stable and exhibit an offset free tracking behavior provided that an admissibility condition on the initial state is satisfied though the cg can be tailored for the application at hand by appropriately choosing the available design knobs the required online computational load for the usual case of affine constraints is well tempered by the related relatively simple convex quadratic programming problem

the continuous ordered weighted geometric operator and its application to decision making the aim of this paper is to develop a continuous ordered weighted geometric c owg operator which is based on the continuous ordered weighted averaging c owa operator recently introduced by the author and the geometric mean we study some desirable properties of the c owg operator and present its application to decision making with interval multiplicative preference relation and finally an illustrative example is pointed out

aggregation approach for the minimum binary cost tension problem the aggregation technique dedicated to two terminal series parallel graphs or ttsp graphs and introduced lately to solve the minimum piecewise linear cost tension problem is adapted here to solve the minimum binary cost tension problem or bct problem even on ttsp graphs the bct problem has been proved to be np complete as far as we know the aggregation is the only algorithm with mixed integer programming proposed to solve exactly the bct problem on ttsp graphs a comparison of the efficiency of both methods is presented here

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

pairwise comparison based interval analysis for group decision aiding with multiple criteria interval ahp analytic hierarchy process was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria in this paper interval ahp is applied to group decision problems interval ahp is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker under individual interval weight vectors obtained from individual pairwise comparison matrices three approaches to obtaining a consensus interval weight vector are proposed one is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors by this approach we can count out indubitably inferior alternatives the second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when all individual opinions are similar the third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors by this approach we can find agreeable group preference between alternatives when individual opinions are not similar the usefulness of the proposed three approaches is demonstrated by simple numerical examples

distance based vertex identification in graphs the outer multiset dimension abstract the characterisation of vertices in a network in relation to other peers has been used as a primitive in many computational procedures such as node localisation and de anonymisation this article focuses on a characterisation type known as the multiset metric representation formally given a graph g and a subset of vertices s w 1 w t v g the multiset representationof a vertex u v g with respect to s is the multiset m u s d g u w 1 d g u w t a subset of vertices s such that m u s m v s u v for every u v v g s is said to be a multiset resolving set and the cardinality of the smallest such set is the outer multiset dimension we study the general behaviour of the outer multiset dimension and determine its exact value for several graph families we also show that computing the outer multiset dimension of arbitrary graphs is np hard and provide methods for efficiently handling particular cases

computational complexity certification for real time mpc with input constraints based on the fast gradient method this paper proposes to use nesterov s fast gradient method for the solution of linear quadratic model predictive control mpc problems with input constraints the main focus is on the method s a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre specified suboptimality is obtained for any possible state of the system we investigate cold and warm starting strategies and provide an easily computable lower iteration bound for cold starting and an asymptotic characterization of the bounds for warm starting moreover we characterize the set of mpc problems for which small iteration bounds and thus short solution times are expected the theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal dual interior point method

fulfilling hard constraints in uncertain linear systems by reference managing a method based on conceptual tools of predictive control is described for tackling tracking problems of uncertain linear systems wherein pointwise in time input and or state inequality constraints are present the method consists of adding to a primal compensated system a nonlinear device called predictive reference filter which manipulates the desired reference in order to fulfill the prescribed constraints provided that an admissibility condition on the initial state is satisfied the control scheme is proved to fulfill the constraints as well as stability and set point tracking requirements for all systems whose impulse step responses lie within given uncertainty ranges

a new method of obtaining the priority weights from an interval fuzzy preference relation in analyzing a multiple criteria decision making problem the decision maker may express her his opinions as an interval fuzzy or multiplicative preference relation then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights in this paper a new consistent interval fuzzy preference relation is defined and the corresponding properties are derived the transformation formulae between interval fuzzy and multiplicative preference relations are further given which show that two preference relations consistent interval fuzzy and multiplicative preference relations can be transformed into each other based on the transformation formula the definition of acceptably consistent interval fuzzy preference relation is given furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented finally three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures the numerical results show that the given procedure is feasible effective and not requisite to solve any mathematical programing

can a linear value function explain choices an experimental study abstract we investigate in a simple bi criteria experimental study whether subjects are consistent with a linear value function while making binary choices many inconsistencies appeared in our experiment however the impact of inconsistencies on the linearity vs non linearity of the value function was minor moreover a linear value function seems to predict choices for bi criteria problems quite well this ability to predict is independent of whether the value function is diagnosed linear or not inconsistencies in responses did not necessarily change the original diagnosis of the form of the value function our findings have implications for the design and development of decision support tools for multiple criteria decision making problems

on the metric dimension of corona product graphs given a set of vertices s v 1 v 2 v k of a connected graph g the metric representation of a vertex v of g with respect to s is the vector r v s d v v 1 d v v 2 d v v k where d v v i i 1 k denotes the distance between v and v i s is a resolving set for g if for every pair of distinct vertices u v of g r u s r v s the metric dimension of g dim g is the minimum cardinality of any resolving set for g let g and h be two graphs of order n 1 and n 2 respectively the corona product g h is defined as the graph obtained from g and h by taking one copy of g and n 1 copies of h and joining by an edge each vertex from the ith copy of h with the ith vertex of g for any integer k 2 we define the graph g kh recursively from g h as g kh g k 1h h we give several results on the metric dimension of g kh for instance we show that given two connected graphs g and h of order n 1 2 and n 2 2 respectively if the diameter of h is at most two then dim g kh n 1 n 2 1 k 1dim h moreover if n 2 7 and the diameter of h is greater than five or h is a cycle graph then dim g kh n 1 n 2 1 k 1dim k 1 h

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

numerical experience with lower bounds for miqp branch and bound the solution of convex mixed integer quadratic programming miqp problems with a general branch and bound framework is considered it is shown how lower bounds can be computed efficiently during the branch and bound process improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming qp problems that have to be solved the branch and bound approach is also shown to be superior to other approaches in solving miqp problems numerical experience is presented which supports these conclusions

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

exact complexity certification of active set methods for quadratic programming active set methods are recognized to often outperform other methods in terms of speed and solution accuracy when solving small size quadratic programming qp problems making them very attractive in embedded linear model predictive control mpc applications a drawback of active set methods is the lack of tight bounds on the worst case number of iterations a fundamental requirement for their implementation in a real time system extensive simulation campaigns provide an indication of the expected worst case computation load but not a complete guarantee this paper solves such a certification problem by proposing an algorithm to compute the exact bound on the maximum number of iterations and floating point operations required by a state of the art dual active set qp solver the algorithm is applicable to a given qp problem whose linear term of the cost function and right hand side of the constraints depend linearly on a vector of parameters as in the case of linear mpc in addition a new solver is presented that combines explicit and implicit mpc ideas guaranteeing improvements of the worst case computation time the ability of the approach to exactly quantify memory and worst case computation requirements is tested on a few mpc examples also highlighting when online optimization should be preferred to explicit mpc

computational complexity certification for dual gradient method application to embedded mpc abstract in this paper we analyze the computational complexity of the dual gradient method for solving linearly constrained convex problems when it is difficult to project on the primal feasible set described by linear constraints we use the lagrangian relaxation to handle the complicated constraints and then we apply the dual gradient algorithm for solving the corresponding dual we give a unified convergence rate analysis for the dual gradient algorithm we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions our analysis relies on the lipschitz property of the dual function or an error bound property furthermore the iteration complexity analysis is based on two types of approximate primal solutions an average primal sequence or the last primal iterate sequence we also discuss complexity certifications and implementation aspects of the dual gradient algorithm on constrained mpc problems for embedded linear systems

multiple attribute decision making based on fuzzy preference information on alternatives ranking and weighting this paper investigates the multiple attribute decision making madm problem with fuzzy preference information on alternatives and proposes an eigenvector method to rank them three optimization models are introduced to assess the relative importance weights of attributes in a madm problem which integrate subjective fuzzy preference relations and objective information in different ways since the decisions these models lead to are based on both subjective and objective information they are more balanced and credible than those that consider only one type of information an example is provided to illustrate the application of the proposed models

neighbor locating colorings in graphs abstract a k coloring of a graph g is a k partition s 1 s k of v g into independent sets called colors a k coloring is called neighbor locating if for every pair of vertices u v belonging to the same color s i the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v the neighbor locating chromatic number n l g is the minimum cardinality of a neighbor locating coloring of g we establish some tight bounds for the neighbor locating chromatic number of a graph in terms of its order maximum degree and independence number we determine all connected graphs of order n 5 with neighbor locating chromatic number n or n 1 we examine the neighbor locating chromatic number for two graph operations join and disjoint union and also for two graph families split graphs and mycielski graphs

an eigenvector method for generating normalized interval and fuzzy weights crisp comparison matrices produce crisp weight estimates it is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate in this paper an eigenvector method em is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix which differs from csutora and buckley s lambda max method in several aspects first the proposed em produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model while the lambda max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process next the em directly solves the principal right eigenvector of an interval or fuzzy comparison matrix while the lambda max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems finally the lambda max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights while the em has no such requirement it is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights situations where the em is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed three numerical examples including a hierarchical structure decision making problem are examined using the em to demonstrate its applications

preference programming and inconsistent interval judgments the problem of derivation of the weights of altematives from pairwise comparison matrices is long standing in this paper lexicographic goal programming lgp has been used to find out weights from pairwise inconsistent interval judgment matrices a number of properties and advantages of lgp as a weight determination technique have been explored an algorithm for identification and modification of inconsistent bounds is also provided the proposed technique has been illustrated by means of numerical examples

a linear optimization problem to derive relative weights using an interval judgement matrix abstract in this paper we propose a new decision making model enabling to assess a finite number of alternatives according to a set of bounds on the preference ratios for the pairwise comparisons between alternatives that is an interval judgement matrix in the case in which these bounds cannot be achieved by any assessment vector we analyze the problem of determining of an efficient or pareto optimal solution from a multi objective optimization problem this multi objective formulation seeks for assessment vectors that are near to simultaneously fulfil all the bound requirements imposed by the interval judgement matrix our new model introduces a linear optimization problem in order to define a consistency index for the interval matrix by solving this optimization problem it can be associated a weakly efficient assessment vector to the consistency index in those cases in which the bound requirements are infeasible otherwise this assessment vector fulfils all the bound requirements and has geometrical properties that make it appropriate as a representative assessment vector of the set of feasible weights

a goal programming method for obtaining interval weights from an interval comparison matrix crisp comparison matrices lead to crisp weight vectors being generated accordingly an interval comparison matrix should give an interval weight estimate in this paper a goal programming gp method is proposed to obtain interval weights from an interval comparison matrix which can be either consistent or inconsistent the interval weights are assumed to be normalized and can be derived from a gp model at a time the proposed gp method is also applicable to crisp comparison matrices comparisons with an interval regression analysis method are also made three numerical examples including a multiple criteria decision making mcdm problem with a hierarchical structure are examined to show the potential applications of the proposed gp method

interval articulation of superiority and precise elicitation of priorities this paper develops an approach to obtaining precise point estimates of priorities of objects in particular coefficients of importance of criteria it is assumed that the judgements regarding the preference superiority of one object over another are given in the forms of intervals that is by indicating the lower and upper bounds the suggested approach is suitable both in the case of consistent and inconsistent preferences in the former case the available intervals are processed by contracting them to the maximum degree in the latter by extending them to the minimum degree the degrees of contraction or extension are regarded as the indexes of indeterminacy or inconsistency of preferences and treated as multiple objective functions a method of transformation of such a multicriterial problem to a sequence of several linear programming problems is suggested

a note on a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making abstract in a recently published paper by liu et al liu f zhang w g wang z x 2012 a goal programming model for incomplete interval multiplicative preference relations and its application in group decision making european journal of operational research 218 747 754 two equations are introduced to define consistency of incomplete interval multiplicative preference relations imprs and employed to develop a goal programming model for estimating missing values this note illustrates that such consistency definition and estimation model are technically incorrect new transitivity conditions are proposed to define consistent imprs and a two stage goal programming approach is devised to estimate missing values for incomplete imprs

evidential reasoning approach for multiattribute decision analysis under both fuzzy and interval uncertainty many multiple attribute decision analysis mada problems are characterized by both quantitative and qualitative attributes with various types of uncertainties incompleteness or ignorance and vagueness or fuzziness are among the most common uncertainties in decision analysis the evidential reasoning er and the interval grade er ier approaches have been developed in recent years to support the solution of mada problems with interval uncertainties and local ignorance in decision analysis in this paper the er approach is enhanced to deal with both interval uncertainty and fuzzy beliefs in assessing alternatives on an attribute in this newly developed fuzzy ier fier approach local ignorance and grade fuzziness are modeled under the integrated framework of a distributed fuzzy belief structure leading to a fuzzy belief decision matrix a numerical example is provided to illustrate the detailed implementation process of the fier approach and its validity and applicability

scheduling series parallel task graphs to minimize peak memory abstract we consider a variant of the well known np complete problem of minimum cut linear arrangement for directed acyclic graphs in this variant we are given a directed acyclic graph and we are asked to find a topological ordering such that the maximum number of cut edges at any point in this ordering is minimum in our variant the vertices and edges have weights and the aim is to minimize the maximum weight of cut edges in addition to the weight of the last vertex before the cut there is a known polynomial time algorithm liu 1987 17 for the cases where the input graph is a rooted tree we focus on the instances where the input graph is a directed series parallel graph and propose a polynomial time algorithm thus expanding the class of graphs for which a polynomial time algorithm is known directed acyclic graphs are used to model scientific applications where the vertices correspond to the tasks of a given application and the edges represent the dependencies between the tasks in such models the problem we address reads as minimizing the peak memory requirement in an execution of the application our work combined with liu s work on rooted trees addresses this practical problem in two important classes of applications

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

efficiency analysis to incorporate interval scale data abstract we develop an approach to efficiency analysis to enable us to incorporate interval scale data in addition to ratio scale data our approach introduces a measure of inefficiency and identifies efficient units as is done in data envelopment analysis the basic idea in our approach is to find the best hyperplane separating the units that are better and worse than each unit best is defined in such a way that the number of not better units is maximal the efficiency measure is defined as a proportion of not better units to all units the results are invariant under a strictly increasing linear re scaling of any input or output variables thus zeroes or negative values do not cause problems for the analysis the approach is used to analyze the data of the research evaluation exercise recently carried out at the university of joensuu finland

interval weight generation approaches based on consistency test and interval comparison matrices a simple yet pragmatic method of consistency test is developed to check whether an interval comparison matrix is consistent or not linear programming method is used to derive consistent interval weights from consistent interval comparison matrices and to aggregate local interval weights to generate global interval weights in cases where an interval comparison matrix is inconsistent an eigenvector method based nonlinear programming nlp approach is developed to generate interval weights that can meet pre determined consistency requirements a simple and effective preference ranking method is utilized to compare the interval weights of criteria or rank alternatives three numerical examples including a hierarchical ahp decision problem are provided to illustrate the validity and practicality of the proposed methods

planar orientations with low out degree and compaction of adjacency matrices abstract we consider the problem of orienting the edges of a planar graph in such a way that the out degree of each vertex is minimized if for each vertex v the out degree is at most d then we say that such an orientation is d bounded we prove the following results each planar graph has a 5 bounded acyclic orientation which can be constructed in linear time each planar graph has a 3 bounded orientation which can be constructed in linear time a 6 bounded acyclic orientation and a 3 bounded orientation of each planar graph can each be constructed in parallel time o log n log n on an erew pram using o n log n log n processors as an application of these results we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time the data structure uses linear storage and can be constructed in linear time

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

resolvability and the upper dimension of graphs abstract for an ordered set w w 1 w 2 w k of vertices and a vertex v in a connected graph g the metric representation of v with respect to w is the k vector r v w d v w 1 d v w 2 d v w k where d x y represents the distance between the vertices x and y the set w is a resolving set for g if distinct vertices of g have distinct representations a new sharp lower bound for the dimension of a graph g in terms of its maximum degree is presented a resolving set of minimum cardinality is a basis for g and the number of vertices in a basis is its metric dimension dim g a resolving set s of g is a minimal resolving set if no proper subset of s is a resolving set the maximum cardinality of a minimal resolving set is the upper dimension dim g the resolving number res g of a connected graph g is the minimum k such that every k set w of vertices of g is also a resolving set of g then 1 dim g dim g res g n 1 for every nontrivial connected graph g of order n it is shown that dim g res g n 1 if and only if g k n while dim g res g 2 if and only if g is a path of order at least 4 or an odd cycle the resolving numbers and upper dimensions of some well known graphs are determined it is shown that for every pair a b of integers with 2 a b there exists a connected graph g with dim g dim g a and res g b also for every positive integer n there exists a connected graph g with res g dim g n and dim g dim g n

a new fuzzy programming method to derive the priority vector from an interval reciprocal comparison matrix in this paper we discuss the interval reciprocal comparison matrices and suggest a new fuzzy programming method nfpm to derive the priority vector from an interval reciprocal comparison matrix based on mikhailov s membership function we first give a membership function which is used to measure the decision maker s satisfaction degree for the priority vector derived from each interval constraint then by using the membership function we propose a new fuzzy programming method to get the most satisfaction degree of each priority vector the nfpm has many prominent characteristics especially it is capable of deriving the priority vector from a consistent or inconsistent interval reciprocal comparison matrix moreover the nfpm can overcome some drawbacks of mikhailov s fuzzy programming method fpm for inconsistent interval reciprocal comparison matrices finally several examples are given to illustrate the applicability and advantage of the proposed method

multicriteria decision analysis with fuzzy pairwise comparisons abstract we present a modification of the fuzzy multi criteria method proposed by van laarhoven and pedrycz 1983 first the weights of the decision criteria are calculated by the minimization of a logarithmic regression function next the weights of the decision alternatives are calculated for each criterion separately lastly the fuzzy final scores of the alternatives are determined by an appropriate aggregationm of the calculated weights we will show that our modified method in contrast with the original one yields weights which are optimal with respect to the logarithmic regression function we also present a version of our method with geometric ratio scales rather than saaty s equi distant scale to quantify the gradations of human comparative judgement this approach also enables us to develop an efficient procedure for investigating the scale sensitivity of the calculated weights and the final scores of the alternatives

the evidential reasoning approach for multi attribute decision analysis under interval uncertainty abstract the evidential reasoning er approach is a method for multiple attribute decision analysis mada under uncertainties it improves the insightfulness and rationality of a decision making process by using a belief decision matrix bdm for problem modelling and the dempster shafer d s theory of evidence for attribute aggregation the d s theory provides scope and flexibility to deal with interval uncertainties or local ignorance in decision analysis which is not explored in the original er approach and will be investigated in this paper firstly interval uncertainty will be defined and modelled in the er framework then an extended er algorithm ier is derived which enables the er approach to deal with interval uncertainty in assessing alternatives on an attribute it is proved that the original er algorithm is a special case of the ier algorithm the latter is demonstrated using numerical examples

metric selection in fast dual forward backward splitting the performance of fast forward backward splitting or equivalently fast proximal gradient methods depends on the conditioning of the optimization problem data this conditioning is related to a metric that is defined by the space on which the optimization problem is stated selecting a space on which the optimization data is better conditioned improves the performance of the algorithm in this paper we propose several methods with different computational complexity to find a space on which the algorithm performs well we evaluate the proposed metric selection procedures by comparing the performance to the case when the euclidean space is used for the most ill conditioned problem we consider the computational complexity is improved by two to three orders of magnitude we also report comparable to superior performance compared to state of the art optimization software

transitive calibration of the ahp verbal scale abstract one of the strengths of the analytic hierarchy process ahp is that it allows decision makers to specify their preferences using a verbal scale yet as is well known a strict reliance on the corresponding saaty 1 9 numeric scale can induce some inconsistency hence we argue that in certain situations it may be appropriate to calibrate the verbal scale the result of this calibration is a geometric scale based on a single parameter we present some limited evidence that this geometric scale marginally outperforms the saaty 1 9 scale moreover we suggest that this calibration can be used to do a simple sensitivity analysis in cases where judgements are uncertain

an interactive approach for placing alternatives in preference classes abstract in this paper we consider the multiple criteria decision making problem of partitioning alternatives into preference classes we develop an interactive procedure based on the assumption that the decision maker dm has a linear utility function the approach requires the dm to place an alternative in a preference class from time to time based on the preference information derived from the dm s placement as well as from dominance and linearity we try to place other alternatives we present results on the performance of the algorithm applied to four different sets of alternatives

parallel recognition of series parallel graphs recently he and yesha gave an algorithm for recognizing directed series parallel graphs in time o log2 n with linearly many erew processors we give a new algorithm for this problem based on a structural characterization of series parallel graphs in terms of their ear decompositions our algorithm can recognize undirected as well as directed series parallel graphs it can be implemented in the crcw model of parallel computation to take time o log n in the erew model the time is o log2 n but the number of processors required improves the bounds of the previous algorithm

on the normalization of interval and fuzzy weights the normalization of interval and fuzzy weights is often necessary in multiple criteria decision analysis mcda under uncertainty especially in analytic hierarchy process ahp with interval or fuzzy judgements the existing normalization methods based on interval arithmetic and fuzzy arithmetic are found flawed and need to be revised this paper presents the correct normalization methods for interval and fuzzy weights and offers relevant theorems in support of them numerical examples are examined to show the correctness of the proposed normalization methods and their differences from those existing normalization methods

k metric antidimension the study and analysis of social graphs impacts on a wide range of applications such as community decision making support and recommender systems with the boom of online social networks such analyses are benefiting from a massive collection and publication of social graphs at large scale unfortunately individuals privacy right might be inadvertently violated when publishing this type of data in this article we introduce k anonymity a novel privacy measure aimed at evaluating the resistance of social graphs to active attacks k anonymity is based on a new problem in graph theory the k metric antidimension defined as follows let g v e be a simple connected graph and s w 1 w t v an ordered subset of vertices the metric representation of a vertex u v with respect to s is the t vector r u s d g u w 1 d g u w t where dg u v represents the length of a shortest u v path in g we call s a k antiresolving set if k is the largest positive integer such that for every vertex v v s there exist other k 1 different vertices v 1 v k 1 v s with r v s r v 1 s r v k 1 s the k metric antidimension of g is the minimum cardinality among all the k antiresolving sets for g we address the k metric antidimension problem by proposing a true biased algorithm with success rate above 80 when considering random graphs of size at most 100 the proposed algorithm is used to determine the privacy guarantees offered by two real life social graphs with respect to k anonymity we also investigate theoretical properties of the k metric antidimension of graphs in particular we focus on paths cycles complete bipartite graphs and trees

a flexible approach to ranking with an application to mba programs we develop a model for flexibly ranking multi dimensional alternatives units into preference classes via mixed integer programming we consider a linear aggregation model but allow the criterion weights to vary within pre specified ranges this allows the individual alternatives units to play to their strengths we illustrate the use of the model by considering the financial times global mba program rankings and discuss the implications we argue that in many applications neither the data nor the weights or the aggregation model itself is precise enough to warrant a complete ranking providing an argument for sorting or what we call flexible ranking

distance based and ad hoc consensus models in ordinal preference ranking abstract this paper examines the problem of aggregating ordinal preferences on a set of alternatives into a consensus this problem has been the subject of study for more than two centuries and many procedures have been developed to create a compromise or consensus we examine a variety of structures for preference specification and in each case review the related models for deriving a consensus two classes of consensus models are discussed namely ad hoc methods evolving primarily from parliamentary settings over the past 200 years and distance or axiomatic based methods we demonstrate the levels of complexity of the various distance based models by presenting the related mathematical programming formulations for them we also present conditions for equivalence that is for yielding the same consensus ranking for some of the methods finally we discuss various extensions of the basic ordinal ranking structures paying specific attention to partial ranking voting member weighted consensus ranking with intensity of preference and rank correlation methods as alternative approaches to deriving a consensus suggestions for future research directions are given

online constraint removal we show how to use a lyapunov function to accelerate mpc for linear discrete time systems with linear constraints and quadratic cost our method predicts in the current time step which constraints will be inactive in the next time step these constraints can be removed from the online optimization problem of the next time step the criterion for the detection of inactive constraints is based on the decrease of the lyapunov function along the trajectory of the controlled system the criterion is simple easy to implement in existing mpc algorithms and its computational cost is small

comparison of weighting judgments in multiattribute utility measurement this paper compares four weighting methods in multiattribute utility measurement the ratio method the swing weighting method the tradeoff method and the pricing out method 200 subjects used these methods to weight attributes for evaluating nuclear waste repository sites in the united states the weighting methods were compared with respect to their internal consistency convergent validity and external validity internal consistency was measured by the degree to which ordinal and cardinal or ratio responses agreed within the same weighting method convergent validity was measured by the degree of agreement between the weights elicited with different methods external validity was determined by the degree to which weights elicited in this experiment agreed with weights that were elicited with managers of the department of energy in terms of internal consistency the tradeoff method fared worst in terms of convergent validity the pricing out method turned out to be an outlier in terms of external validity the pricing out method showed the best results while the ratio and swing methods are quite consistent and show a fair amount of convergent validity their external validity problems cast doubt on their usefulness the main recommendation for applications is to improve the internal consistency of the tradeoff method by careful interactive elicitation and to use it in conjunction with the pricing out method to enhance its external validity

interval priorities in ahp by interval regression analysis abstract this paper proposes an interval approach for obtaining interval weights of priorities in the analytic hierarchy process ahp reflecting inconsistency of pairwise comparison ratios given by a decision maker even if pairwise comparison values with respect to preference are given as crisp values the priority weights should be estimated as intervals because of a decision maker s uncertainty of judgments we deal with two types of comparison matrices represented as crisp and interval elements this allows some uncertainty of estimations made by a decision maker the proposed approach can obtain the estimated interval priorities since interval regression analysis is based on the concept of possibility the interval weights obtained by the proposed method can be interpreted as the possible ranges estimated from the given data the sum of widths of the obtained intervals can be regarded as the degree of inconsistency included in the given data it is emphasized that interval data of preference can obtain the lower and upper models for the estimated weight intervals

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

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

interval estimation of priorities in the ahp this paper extends and modifies the analytic hierarchy process ahp and the synthetic hierarchy method shm of priority estimation to accommodate random data in the pairwise comparison matrices it employs a cauchy distribution to describe the pairwise comparison of alternatives in saaty matrices and shows how to modify these matrices in order to handle random data the use of random data yields saaty matrices that are not reciprocally symmetrical several variants of the ahp are then modified i to accommodate reciprocally asymmetric matrices and ii to allow each priority estimate to be expressed on an interval of possible values rather than as a single discrete point the merits of interval estimation are illustrated by an example

post optimality analysis of priority vectors derived from interval comparison matrices by lexicographic goal programming abstract in a recent paper by wang y m wang on lexicographic goal programming method for generating weights from inconsistent interval pair wise comparison matrices applied mathematics and computation 173 2006 985 991 it is shown that lexicographic goal programming method results in different rankings for upper and lower triangular interval judgments arranged in a positive reciprocal pair wise comparison matrix this was unexpected from a prioritization point of view because upper and lower triangular entries in reciprocal interval pair wise comparison matrices essentially provide the same judgment information a research question was posed at the same article which ranking is true or more reliable motivated by this question in this paper we provide a post optimality analysis to detect the most reliable priority vector among m vectors derived by lexicographical goal programming for an inconsistent interval pair wise comparison matrix of order n the method is based on violations in the order of preferences among the resultant priorities and runs in o mn 2 time when there exists a tie between some vectors we also illustrate a tie breaking algorithm running in o mn 4 time based on violations in the order of the preference intensities among the resultant priorities we re visited the examples discussed in the afore mentioned article and validate the practical usability of this method

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

partition dimension of certain classes of series parallel graphs abstract the partition dimension problem is to decompose the vertex set of a graph into minimum number of vertex disjoint sets such that the ordered distances between every vertex of the graph and the disjoint sets of that decomposition have different values in the representation this problem has emerging applications in areas such as structure activity issues in drug design navigation of robots pattern recognition and image processing in the present study we provide an upper bound for the partition dimension of parallel composition of any graphs and derive an exact result for parallel composition of paths with different lengths on the other side we find the partition dimension of the series composition of cycles and complete graphs with different sizes and an upper bound is derived for series composition of any graphs further this problem is applied to a graph which has been generated by parallel and series composition together on paths

brief paper fast large scale model predictive control by partial enumeration partial enumeration pe is presented as a method for treating large linear model predictive control applications that are out of reach with available mpc methods pe uses both a table storage method and online optimization to achieve this goal versions of pe are shown to be closed loop stable pe is applied to an industrial example with more than 250 states 32 inputs and a 25 sample control horizon the performance is less than 0 01 suboptimal with average speedup factors in the range of 80 220 and worst case speedups in the range of 4 9 39 2 compared to an existing mpc method small tables with only 25 200 entries were used to obtain this performance while full enumeration is intractable for this example

a two stage logarithmic goal programming method for generating weights from interval comparison matrices a two stage logarithmic goal programming tlgp method is proposed to generate weights from interval comparison matrices which can be either consistent or inconsistent the first stage is devised to minimize the inconsistency of interval comparison matrices and the second stage is developed to generate priorities under the condition of minimal inconsistency the weights are assumed to be multiplicative rather than additive in the case of hierarchical structures a nonlinear programming method is used to aggregate local interval weights into global interval weights a simple yet practical preference ranking method is investigated to compare the interval weights of criteria or rank alternatives in a multiplicative aggregation process the proposed tlgp is also applicable to fuzzy comparison matrices when they are transformed into interval comparison matrices using a level sets and the extension principle six numerical examples including a group decision analysis problem with a group of comparison matrices a hierarchical decision problem and a fuzzy decision problem using fuzzy comparison matrix are examined to show the applications of the proposed methods comparisons with other existing procedures are made whenever possible

mixed metric dimension of graphs abstract let g v e be a connected graph a vertex w v distinguishes two elements vertices or edges x y e v if dg w x dg w y a set s of vertices in a connected graph g is a mixed metric generator for g if every two distinct elements vertices or edges of g are distinguished by some vertex of s the smallest cardinality of a mixed metric generator for g is called the mixed metric dimension and is denoted by dim m g in this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds we also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph finally we prove that the problem of determining the mixed metric dimension of a graph is np hard in the general case

a multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares model predictive control mpc is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables to circumvent the main drawback of mpc i e the need to solve a quadratic program qp on line to compute the control action explicit mpc was proposed in the past to precompute the control law off line using multiparametric qp mpqp the resulting form of the mpc law is piecewise affine which is extremely easy to code can be computed online by simple arithmetic operations and requires a maximum number of iterations that can be exactly determined a priori on the other hand the offline computations to solve the mpqp problem require detecting emptiness full dimensionality and minimal hyperplane representations of polyhedra and other computational geometric operations while most of the existing methods solve such operations via linear programming the approach proposed in this paper relies on a nonnegative least squares nnls solver that is very simple to code and fast to execute in addition the new approach exploits qp duality to identify and construct critical regions and to handle degeneracy issues

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

a weighted gram schmidt method for convex quadratic programming range space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases in this paper we describe a range space method based upon updating a weighted gram schmidt factorization of the constraints in the active set the updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active set strategy

on lexicographic goal programming method for generating weights from inconsistent interval comparison matrices in a recent paper by european journal of operational research 97 1997 53 62 the lexicographic goal programming lgp method was developed to generate weights from inconsistent interval comparison matrices and a number of its properties and advantages as a weight determination technique were explored it is shown in this paper that lgp is in fact defective in theory although the upper and lower triangular judgments of an inconsistent interval comparison matrix provide exactly the same information on the preferences of weights in general they lead to distinctive priorities and rankings if lgp is used

efficient on line computation of constrained optimal control we consider constrained finite time optimal control problems for discrete time linear time invariant systems with constraints on inputs and outputs based on linear and quadratic performance indices the solution to such problems is a time varying piecewise affine pwa state feedback law and can be computed by means of multiparametric programming by exploiting the properties of the value function and the piecewise affine optimal control law of the constrained finite time optimal control cftoc we propose two new algorithms that avoid storing the polyhedral regions the new algorithms significantly reduce the on line storage demands and computational complexity during evaluation of the pwa feedback control law resulting from the cftoc

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

computing the metric dimension of wheel related graphs abstract an ordered set w w 1 w k v g of vertices of g is called a resolving set or locating set for g if every vertex is uniquely determined by its vector of distances to the vertices in w a resolving set of minimum cardinality is called a basis for g and this cardinality is the metric dimension or location number of g denoted by g in this paper we study the metric dimension of certain wheel related graphs namely m level wheels an infinite class of convex polytopes and antiweb gear graphs denoted by w n m q n and awj 2 n respectively we prove that these infinite classes of wheel related graphs have unbounded metric dimension the study of an infinite class of convex polytopes generated by wheel denoted by q n also gives a negative answer to an open problem proposed by imran et al 2012 in 8 open problem is it the case that the graph of every convex polytope has constant metric dimension it is natural to ask for characterization of graphs with unbounded metric dimension

a quadratic programming algorithm based on nonnegative least squares with applications to embedded model predictive control this technical note proposes an active set method based on nonnegative least squares nnls to solve strictly convex quadratic programming qp problems such as those that arise in model predictive control mpc the main idea is to rephrase the qp problem as a least distance problem ldp that is solved via a nnls reformulation while the method is rather general for solving strictly convex qp s subject to linear inequality constraints it is particularly useful for embedded mpc because i is very fast compared to other existing state of the art qp algorithms ii is very simple to code requiring only basic arithmetic operations for computing ldl t decompositions recursively to solve linear systems of equations iii contrary to iterative methods provides the solution or recognizes infeasibility in a finite number of steps

primal and dual active set methods for convex quadratic programming computational methods are proposed for solving a convex quadratic program qp active set methods are defined for a particular primal and dual formulation of a qp with general equality constraints and simple lower bounds on the variables in the first part of the paper two methods are proposed one primal and one dual these methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal dual form the primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero the dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities in each of these methods the search directions satisfy a kkt system of equations formed from hessian and constraint components associated with an appropriate column basis the composition of the basis is specified by an active set strategy that guarantees the nonsingularity of each set of kkt equations each of the proposed methods is a conventional active set method in the sense that an initial primal or dual feasible point is required in the second part of the paper it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple bound constraints and adding a penalty term to the objective function any conventional column basis may be made optimal for such a primal dual pair of shifted penalized problems the shifts are then updated using the solution of either the primal or the dual shifted problem an obvious application of this approach is to solve a shifted dual qp to define an initial feasible point for the primal or vice versa the computational performance of each of the proposed methods is evaluated on a set of convex problems from the cutest test collection

on minimum average stretch spanning trees in polygonal 2 trees a spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges for a polygonal 2 tree on n vertices we present an algorithm to compute a minimum average stretch spanning tree in o n log n time this algorithm also finds a minimum fundamental cycle basis in polygonal 2 trees we show that there is a unique minimum cycle basis in a polygonal 2 tree and it can be computed in linear time

acceptable consistency analysis of interval reciprocal comparison matrices when a decision maker expresses his her opinions by means of an interval reciprocal comparison matrix the study of consistency becomes a very important aspect in decision making in order to avoid a misleading solution in the present paper an acceptably consistent interval reciprocal comparison matrix is defined which can be reduced to an acceptably consistent crisp reciprocal comparison matrix when the intervals become exact numbers an interval reciprocal comparison matrix with unacceptable consistency can be easily adjusted such that the revised matrix possesses acceptable consistency utilizing a convex combination method a family of crisp reciprocal comparison matrices with acceptable consistency can be obtained whose weights are further found to exhibit a style of convex combination and aggregated to obtain interval weights from an acceptably consistent interval reciprocal comparison matrix a novel simple yet effective formula of possibility degree is presented to rank interval weights numerical results are calculated to show the quality and quantity of the proposed approaches and compare with other existing procedures

joins ears and castelnuovo mumford regularity abstract we introduce a new class of polynomial ideals associated to a simple graph g let k e g be the polynomial ring on the edges of g and k v g the polynomial ring on the vertices of g we associate to g an ideal i x g defined as the preimage of x i 2 x j 2 i j v g k v g by the map k e g k v g which sends a variable t e associated to an edge e i j to the product x i x j of the variables associated to its vertices we show that k e g i x g is a one dimensional cohen macaulay graded ring and we relate its degree with the number of vertices and connected components of the graph we show that i x g is a binomial ideal and that with respect to a fixed monomial order its initial ideal has a generating set independent of the field k we focus on the castelnuovo mumford regularity of i x g providing the following sharp upper and lower bounds g reg i x g v g b 0 g 1 where g is the maximum vertex join number of the graph and b 0 g is the number of its connected components we show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition

a note on the partition dimension of cartesian product graphs abstract let g v e be a connected graph the distance between two vertices u v v denoted by d u v is the length of a shortest u v path in g the distance between a vertex v v and a subset p v is defined as min d v x x p and it is denoted by d v p an ordered partition p 1 p 2 p t of vertices of a graph g is a resolving partition of g if all the distance vectors d v p 1 d v p 2 d v p t are different the partition dimension of g denoted by pd g is the minimum number of sets in any resolving partition of g in this article we study the partition dimension of cartesian product graphs more precisely we show that for all pairs of connected graphs g h pd g h pd g pd h and pd g h pd g dim h where dim h denotes the metric dimension of h consequently we show that pd g h dim g dim h 1