improving linguistic pairwise comparison consistency via linguistic discrete regions linguistic pairwise comparison matrices are widely used in decision making procedures however the matrices often give conflicting results when there are multiple criteria under consideration despite intensive research achieving consistency of such matrices remains a daunting task in this paper a novel approach based on linguistic discrete region is proposed to address the challenge unlike existing methods that require a single value for each comparison our approach allows a comparison to be expressed by a discrete region with multiple linguistic terms such front end gives users more freedom to express their opinions in the back end we propose an iterative searching algorithm that is able to achieve approximate optimal consistency for the comparison matrices with discrete region values the final results are single value matrices that not only guarantee approximate optimal consistency but comply with evaluators intentions a well as our approach does not modify any linguistic values like many existing methods we have conducted extensive evaluations and our empirical study confirms that the linguistic discrete region based approach significantly improves the consistency of linguistic pairwise comparison matrices

existence uniqueness and lipschitz dependence for patlak keller segel and navier stokes in mathbb r 2 with measure valued initial data we establish a new local well posedness result in the space of finite borel measures for mild solutions of the parabolic elliptic patlak keller segel pks model of chemotactic aggregation in two dimensions our result only requires that the initial measure satisfy the necessary assumption max x in mathbb r 2 mu x 8 pi this work improves the small data results of biler stud math 114 2 181 192 1995 and the existence results of senba and suzuki j funct anal 191 17 51 2002 our work is based on that of gallagher and gallay math ann 332 287 327 2005 who prove the uniqueness and log lipschitz continuity of the solution map for the 2d navier stokes equations nse with measure valued initial vorticity we refine their techniques and present an alternative version of their proof which yields existence uniqueness and lipschitz continuity of the solution maps of both pks and nse many steps are more difficult for pks than for nse particularly on the level of the linear estimates related to the self similar spreading solutions

a cosine maximization method for the priority vector derivation in ahp abstract the derivation of a priority vector from a pair wise comparison matrix pcm is an important issue in the analytic hierarchy process ahp the existing methods for the priority vector derivation from pcm include eigenvector method ev weighted least squares method wls additive normalization method an logarithmic least squares method lls etc the derived priority vector should be as similar to each column vector of the pcm as possible if a pair wise comparison matrix pcm is not perfectly consistent therefore a cosine maximization method cm based on similarity measure is proposed which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a pcm an optimization model for the cm is proposed to derive the reliable priority vector using three numerical examples the cm is compared with the other prioritization methods based on two performance evaluation criteria euclidean distance and minimum violation the results show that the cm is flexible and efficient

an improved version of the augmented e constraint method augmecon2 for finding the exact pareto set in multi objective integer programming problems generation or a posteriori methods in multi objective mathematical programming momp is the most computationally demanding category among the momp approaches due to the dramatic increase in computational speed and the improvement of mathematical programming algorithms the generation methods become all the more attractive among today s decision makers in the current paper we present the generation method augmecon2 which is an improvement of our development augmecon although augmecon2 is a general purpose method we will demonstrate that augmecon2 is especially suitable for multi objective integer programming moip problems specifically augmecon2 is capable of producing the exact pareto set in moip problems by appropriately tuning its running parameters in this context we compare the previous and the new version in a series of new and old benchmarks found in the literature we also compare augmecon2 s performance in the generation of the exact pareto sets with established methods and algorithms based on specific moip problems knapsack set packing and on published results except from other mathematical programming methods augmecon2 is found to be competitive also with multi objective meta heuristics momh in producing adequate approximations of the pareto set in multi objective combinatorial optimization moco problems

solving multiobjective multiconstraint knapsack problems using mathematical programming and evolutionary algorithms abstract in this paper we solve instances of the multiobjective multiconstraint or multidimensional knapsack problem momckp from the literature with three objective functions and three constraints we use exact as well as approximate algorithms the exact algorithm is a properly modified version of the multicriteria branch and bound mcbb algorithm which is further customized by suitable heuristics three branching heuristics and a more general purpose composite branching and construction heuristic are devised comparison is made to the published results from another exact algorithm the adaptive e constraint method laumanns m thiele l zitzler e 2006 an efficient adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method european journal of operational research 169 932 942 using the same data sets furthermore the same problems are solved using standard multiobjective evolutionary algorithms moea namely the spea2 and the nsgaii the results from the exact case show that the branching heuristics greatly improve the performance of the mcbb algorithm which becomes faster than the adaptive e constraint regarding the performance of the moea algorithms in the specific problems spea2 outperforms nsgaii in the degree of approximation of the pareto front as measured by the coverage metric especially for the largest instance

efficient computation of the search region in multi objective optimization multi objective optimization procedures usually proceed by iteratively producing new solutions for this purpose a key issue is to determine and efficiently update the search region which corresponds to the part of the objective space where new nondominated points could lie in this paper we elaborate a specific neighborhood structure among local upper bounds thanks to this structure the update of the search region with respect to a new point can be performed more efficiently compared to existing approaches moreover the neighborhood structure provides new insight into the search region and the location of nondominated points

a reduction dynamic programming algorithm for the bi objective integer knapsack problem abstract this paper presents a backward state reduction dynamic programming algorithm for generating the exact pareto frontier for the bi objective integer knapsack problem the algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept first an approximate core is obtained by eliminating dominated items second the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted sum functions associated with the efficient extreme solutions of the linear relaxation of the multi objective integer knapsack problem third the items are classified according to the values of their upper bounds items with zero upper bounds can be eliminated finally the remaining items are used to form a mixed network with different upper bounds the numerical results obtained from different types of bi objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

algorithms for improving consistency or consensus of reciprocal 0 1 valued preference relations we investigate the consistency and consensus of reciprocal 0 1 valued preference relations also called fuzzy preference relations by many authors based on the multiplicative consistency property which is an important issue in fuzzy set theory an algorithm is first developed to improve the consistency level of a reciprocal 0 1 valued preference relation and the corresponding algorithm for the incomplete reciprocal 0 1 valued preference relation is also developed we further propose the consensus improving algorithms for individual reciprocal 0 1 valued preference relations or incomplete ones the convergence and robustness of the algorithms are proven and some important conclusions are obtained in addition the proposed algorithms can improve the consistency or consensus of reciprocal 0 1 valued preference relations with less interactions with the decision makers which can save a lot of time and obtain the results quickly

on the analysis of the 1 1 evolutionary algorithm many experimental results are reported on all types of evolutionary algorithms but only few results have been proved a step towards a theory on evolutionary algorithms in particular the so called 1 1 evolutionary algorithm is performed linear functions are proved to be optimized in expected time o nlnn but only mutation rates of size 1 n can ensure this behavior for some polynomial of degree 2 the optimization needs exponential time the same is proved for a unimodal function both results were not expected by several other authors finally a hierarchy result is proved moreover methods are presented to analyze the behavior of the 1 1 evolutionary algorithm

assessing priority weights from subsets of pairwise comparisons in multiple criteria optimization problems using a set covering concept we explore the possibility and flexibility of selecting a subset of all possible pairs of comparisons to assess priority weights of the decision criteria different techniques offer different priority weights with different properties they are however interrelated when special subsets such as basic coverings elementary cycles and complete covering are considered this paper provides a framework for unifying existing assessment methods and offers some fundamental theorems which reveal the intricate relationships among the methods over the subsets

an improved version of a core based algorithm for the multi objective multi dimensional knapsack problem in this paper we propose an improved version of a core based algorithm for the multi objective extension of one of the most well known combinatorial optimization problems the multi dimensional knapsack problem the original algorithm was designed only for bi objective problems combining an extension of the core concept and a branch and bound algorithm it is a deterministic algorithm and the core concept exploits the divide and conquer solution strategy which proves very effective in such problems the new version is not limited to bi objective problems it can effectively handle problems with more than two objective functions and it has features that greatly accelerate the solution process namely these features are the use of a heuristic based on the dantzig bound in the fathoming process and the better housekeeping of the incumbent list through filtering of solutions the key parameters of the new algorithm are a the size of the core and b the number of provided cores varying these parameters the user can easily tune the size of the obtained pareto set a comparison with evolutionary algorithms like nsga ii spea2 and moea d has been made according to run time and the most widely used metrics set coverage set convergence etc our new version performs better than the most popular evolutionary algorithms and it is comparable to very recent state of the art metaheuristics like multi objective memetic algorithm based on decomposition momad for multi objective programming

deriving priorities from pairwise comparison matrices with a novel consistency index abstract it is important to measure the inconsistency level of a pairwise comparison matrix pcm and derive the priority vector in the analytic hierarchy process ahp in the present study a new consistency index is proposed by using the cosine similarity measures of two row column vectors in a pcm it is called as the double cosine similarity consistency index dcsci since the row and column vectors are all considered some interesting properties of dcsci are investigated and the thresholds for inconsistency tolerance level are discussed in detail then following the idea of dcsci we provide a new method for obtaining the priority vector from a pcm through maximizing the sum of the cosine similarity measures between the priority vector and the row column vectors the priority vector is derived by solving the constructed optimization problem by analyzing the proposed double cosine similarity maximization dcsm method it is found that the existing cosine maximization cm method can be retrieved by carrying out numerical examples some comparisons with the existing methods show that the proposed index and method are effective and flexible

c psa constrained pareto simulated annealing for constrained multi objective optimization in recent years evolutionary algorithms eas have been extensively developed and utilized to solve multi objective optimization problems however some previous studies have shown that for certain problems an approach which allows for non greedy or uphill moves unlike eas can be more beneficial one such approach is simulated annealing sa sa is a proven heuristic for solving numerical optimization problems but owing to its point to point nature of search limited efforts has been made to explore its potential for solving multi objective problems the focus of the presented work is to develop a simulated annealing algorithm for constrained multi objective problems the performance of the proposed algorithm is reported on a number of difficult constrained benchmark problems a comparison with other established multi objective optimization algorithms such as infeasibility driven evolutionary algorithm idea non dominated sorting genetic algorithm ii nsga ii and multi objective scatter search ii moss ii has been included to highlight the benefits of the proposed approach

the geometric consistency index approximated thresholds abstract crawford and williams journal of mathematical psychology 29 1985 387 suggested for the row geometric mean method rgmm one of the most extended ahp s priorization procedure a measure of the inconsistency based on stochastic properties of a subjacent model in this paper we formalize this inconsistency measure hereafter called the geometric consistency index gci and provide the thresholds associated with it these thresholds allow us an interpretation of the inconsistency tolerance level analogous to that proposed by saaty multicriteria decision making the analytic hierarchy process new york 1980 for the consistency ratio cr used with the right eigenvector method in conventional ahp

generation of the exact pareto set in multi objective traveling salesman and set covering problems abstract the calculation of the exact set in multi objective combinatorial optimization moco problems is one of the most computationally demanding tasks as most of the problems are np hard in the present work we use augmecon2 a multi objective mathematical programming momp method which is capable of generating the exact pareto set in multi objective integer programming moip problems for producing all the pareto optimal solutions in two popular moco problems the multi objective traveling salesman problem motsp and the multi objective set covering problem moscp the computational experiment is confined to two objective problems that are found in the literature the performance of the algorithm is slightly better to what is already found from previous works and it goes one step further generating the exact pareto set to till now unsolved problems the results are provided in a dedicated site and can be useful for benchmarking with other momp methods or even multi objective meta heuristics momh that can check the performance of their approximate solution against the exact solution in motsp and moscp problems

a geometric averaging procedure for constructing supertransitive approximation to binary comparison matrices abstract the usefulness of encoding the fuzzy evaluations of alternatives and the importance weights of criteria in a multiple objective decision problem through binary comparison matrices or pairwise judgment matrices is receiving considerable attention the methodology for identifying the best alternative in a given decision problem involves the computation of the principal eigenvectors of the binary comparison matrices the eigenvectors transform the fuzzy evaluations of the importance of the criteria and the ratings of the alternatives into a ratio scale a difficulty that is often experienced in using this approach in practice is the inconsistency of the binary evaluations this paper proposes a simple averaging procedure to construct a supertransitive approximation to a binary comparison matrix where inconsistency is a problem it is further suggested that such an adjustment might be necessary to more closely reflect the inherent fuzziness of the evaluations contained in a binary comparison matrix the procedure is illustrated by means of examples

local stability intervals in the analytic hierarchy process abstract this paper provides a sensitivity analysis of the judgements used in the analytic hierarchy process ahp in relation to the rank reversal produced in two different situations the selection of the best alternative p problem and the ranking of all the alternatives p problem in both cases under the supposition that we employ the row geometric mean method rgmm to determine the local priorities we obtain in the case of a single criterion a local stability interval for each judgement for each alternative and for the paired comparisons matrix which allow us to guarantee the best alternative and the ranking of all of them the p and p problems respectively with respect to these three situations judgement alternative and matrix and two problems p and p we also calculate a local stability index to detect the critical values of the resolution process both the local stability intervals and indexes are used as management tools in the final stage of the decision making process that is to say the exploitation phase especially in the negotiation process and the search for consensus between the actors

logarithmic least squares prioritization and completion methods for interval fuzzy preference relations based on geometric transitivity abstract this article introduces the notion of geometric transitivity which may be used to define consistent interval fuzzy preference relations some useful properties are presented for consistent interval fuzzy preference relations a close investigation reveals that existing methods for determining the analogous property of consistency are not robust to permutations of the pairwise judgments and not able to reflect the hesitancy in the decision maker s preference a parametric transformation formula is put forward to convert a normalized interval weight vector into a consistent interval fuzzy preference relation by minimizing the squared difference between the logarithm of the ratio of the original judgment and the logarithm of the ratio of the converted consistent one a logarithmic least squares model is developed to derive interval weights from any interval fuzzy preference relation based on geometric transitivity a logarithmic least squares model is established to rectify inconsistency for complete and inconsistent interval fuzzy preference relations and a logarithmic least squares completion approach is further developed to estimate missing values for incomplete interval fuzzy preference relations numerical examples are furnished to show the validity of the proposed models and compare with other existing methods

consistency and consensus improving methods for pairwise comparison matrices based on abelian linearly ordered group the aim of a valued pairwise comparison matrix is to derive the priority structure over a set of criteria or alternatives in decision making the consistency and consensus of a pairwise comparison matrix should be measured and improved to avoid a misleading priority structure the basic entries of a pairwise comparison matrix can be described in different forms hence different consistency and consensus methods should be developed for different types of matrices to provide a general framework the pairwise comparison matrix based on abelian linearly ordered group is first introduced a consistency index is defined by constructing the nearest consistent pairwise comparison matrix from an inconsistent one and two consistency improving methods are introduced a group pairwise comparison matrix is derived a consensus index of individual pairwise comparison matrices is defined and two consensus improving methods are developed by introducing a general aggregation operator based on abelian linearly ordered group the proposed consistency and consensus methods are convergent and can provide a general framework for existing methods

bi objective multi mode project scheduling under risk aversion abstract the paper proposes a model for stochastic multi mode resource constrained project scheduling under risk aversion with the two objectives makespan and cost activity durations and costs are assumed as uncertain and modeled as random variables for the scheduling part of the decision problem the class of early start policies is considered in addition to the schedule the assignment of execution modes to activities has to be selected to take risk aversion into account the approach of optimization under multivariate stochastic dominance constraints recently developed in other fields is adopted for the resulting bi objective stochastic integer programming problem the pareto frontier is determined by means of an exact solution method incorporating a branch and bound technique based on the forbidden set branching scheme from stochastic project scheduling randomly generated test instances partially derived from a test case from the psplib are used to show the computational feasibility of the approach

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

on the representation of the search region in multi objective optimization given a finite set n of feasible points of a multi objective optimization moo problem the search region corresponds to the part of the objective space containing all the points that are not dominated by any point of n i e the part of the objective space which may contain further nondominated points in this paper we consider a representation of the search region by a set of tight local upper bounds in the minimization case that can be derived from the points of n local upper bounds play an important role in methods for generating or approximating the nondominated set of an moo problem yet few works in the field of moo address their efficient incremental determination we relate this issue to the state of the art in computational geometry and provide several equivalent definitions of local upper bounds that are meaningful in moo we discuss the complexity of this representation in arbitrary dimension which yields an improved upper bound on the number of solver calls in epsilon constraint like methods to generate the nondominated set of a discrete moo problem we analyze and enhance a first incremental approach which operates by eliminating redundancies among local upper bounds we also study some properties of local upper bounds especially concerning the issue of redundant local upper bounds that give rise to a new incremental approach which avoids such redundancies finally the complexities of the incremental approaches are compared from the theoretical and empirical points of view

multi objective robust pid controller tuning using two lbests multi objective particle swarm optimization in this paper two lbests multi objective particle swarm optimization 2lb mopso is applied to design multi objective robust proportional integral derivative pid controllers for two mimo systems namely distillation column plant and longitudinal control system of the super maneuverable f18 harv fighter aircraft multi objective robust pid controller design problem is formulated by minimizing integral squared error ise and balanced robust performance criteria during the search 2lb mopso can focus on small regions in the parameter space in the vicinity of the best existing fronts as the lbests are chosen from the top fronts in a non domination sorted external archive of reasonably large size the offspring obtained can be more diverse with good fitness the performance of various optimal pid controllers is compared in terms of the sum of ise and balanced robust performance criteria for the purpose of comparison 2lb mopso nsga ii as well as earlier reported riccati iga and osa methods are considered the performance of pid controllers obtained using 2lb mopso is better than that of others in addition hypervolume based comparisons are carried out to show the superior performance of 2lb mopso over nsga ii the results reveal that 2lb mopso yields better robustness and consistency in terms of the sum of ise and balanced robust performance criteria than various optimal pid controllers

a note on ahp group consistency for the row geometric mean priorization procedure abstract this work analyses the consistency in group decision making for the analytic hierarchy process ahp when using the weighted geometric mean method wgmm as the aggregation procedure the row geometric mean method rgmm as the priorization procedure and the geometric consistency index as the inconsistency measure the paper proves that the inconsistency of the group is smaller than the largest individual inconsistency this result complements that obtained by xu eur j oper res 126 2000 683 for the eigenvector priorization method em and its associated consistency index saaty multicriteria decision making the analytic hierarchy process mcgraw hill 1980 moreover our result guarantees that by using the rgmm priorization procedure the group priorities obtained through the aggregation of the individual priorities verify the requirement of consistency proposed in ahp methodology if the individual priorities also verify this requirement

on the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices abstract several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons however some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them in this paper we first introduce the consistency approximated thresholds for the salo hamalainen index by deriving an inconsistency tolerance level which is analogous to that proposed for the consistency ratio by saaty for the geometric consistency index by aguaron moreno jimenez and for the consistency measure by koczkodaj then via a simulation we propose a transitivity threshold for n n matrices which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements on varying n we derive the transitivity thresholds for salo hamalainen and for the other consistency indices proposed in literature

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

inconsistency in the ordinal pairwise comparisons method with and without ties abstract comparing alternatives in pairs is a well known method used to create ranking experts are asked to perform a series of binary comparisons and then using mathematical methods the final ranking is constructed experts conduct a series of single assessments however they may not always be consistent the level of inconsistency among individual assessments is widely accepted as a measure of the ranking quality the higher the ranking quality the higher its credibility one of the earliest and most widespread inconsistency indices is the consistency coefficient defined by kendall and babington smith in their work the authors consider binary pairwise comparisons i e those where the result of an individual comparison can only be better or worse in the presented work the maximal number of inconsistent triads in the set of ordinal pairwise comparisons with ties of arbitrary size is determined formula 14 this in turn opens the possibility of effectively extending the kendall and babington smith index to pairwise comparisons where the result of an individual comparison can be better worse or equal hence this effectively extends the use of this index to the analytic hierarchy process and other quantitative methods based on comparing alternatives in pairs the work also introduces the notions of a generalized tournament and a double tournament as graphs that model ordinal pairwise comparisons with ties and the maximally inconsistent set of pairwise comparisons with ties respectively the relationship between the most inconsistent set of pairwise comparisons with ties and the set cover problem is also shown

acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices this paper investigates consistency and uncertainty measurements as well as acceptability checking for triangular fuzzy multiplicative preference relations tfmprs a geometric consistency index is proposed to measure the consistency degree of a tfmpr and thresholds of acceptable consistency are discussed for tfmprs uncertainty indices are introduced to measure indeterminacy degrees of triangular fuzzy judgments and of tfmprs and a notion of acceptable tfmprs are put forward by considering both acceptable consistency and acceptable uncertainty in tfmprs we also explore the elicitation of triangular fuzzy weights from acceptable tfmprs and the aggregation of local triangular fuzzy weights for solving hierarchical multi criteria decision making problems a geometric mean based equation and a logarithmic least square model are developed to elicit normalized acceptable triangular fuzzy multiplicative weights from an acceptable tfmpr an equation and a linear program are established to transform triangular fuzzy multiplicative weights into triangular fuzzy additive weights a geometric weighting based equation and a pair of linear programs are devised to generate global priority fuzzy weights from local triangular fuzzy multiplicative weights three numerical examples including a hierarchical multi criteria decision making problem are used to illustrate how to apply the developed framework and comparative studies are provided to show its availability and advantages

relative measurement and the power function abstract relative measurement is considered in the context of nomology a theory that defines and relates categories of activity that arise in qualitative decision making a nomological variable is defined to correspond to the amount weight or importance of any naturally forming cluster or category of qualitative information this article shows that in general the mind measures nomological variables relatively and changes in a nomological variable relative to its current value the way the mind assesses the relationship between two related nomological variables generally should be modelled as a power function and it is appropriate to use a geometric mean to average multiple observations empirical evidence from psychophysics is shown to support these results when synthesising direct and indirect relative comparisons of nomological variables it is proven that the indirect comparisons should be given a weight that is inversely proportional to the number of relative measurements made in the indirect comparison this result is shown to be consistent with but not identical with the use of the row geometric means approach for synthesising pairwise relative comparison matrices such as in the analytical hierarchy process in the case of more than three variables other terms should be included in the synthesis the article reports on empirical research that endorses the use of nomological structuring and relative weighting of criteria trees it shows that decision makers generally prefer scoring within intervals when comparing alternatives but can also use relative measurement where there are difficulties with identifying intervals the former leads to modelling by an additive value function the latter to a power function

eigenvector and logarithmic least squares abstract the eigenvector method deals with two questions simultaneously closeness and order together they belong to the field topology of order the metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations there is usually a condition which relates order in the derived scale to order in the data this is precisely the kind of condition captured by the eigenvector but not by the logarithmic least squares estimate

an intrinsic consistency threshold for reciprocal matrices abstract in this paper we present a new intrinsic and scale independent consistency threshold for reciprocal matrices in the analytic hierarchy process ahp its derivation is based upon the relation between the consistency measure of the ahp and the reduction of weight of a new alternative we compare this standard with an existing one

solving the bi objective multi dimensional knapsack problem exploiting the concept of core this paper deals with the bi objective multi dimensional knapsack problem we propose the adaptation of the core concept that is effectively used in single objective multi dimensional knapsack problems the main idea of the core concept is based on the divide and conquer principle namely instead of solving one problem with n variables we solve several sub problems with a fraction of n variables core variables the quality of the obtained solution can be adjusted according to the size of the core and there is always a trade off between the solution time and the quality of solution in the specific study we define the core problem for the multi objective multi dimensional knapsack problem after defining the core we solve the bi objective integer programming that comprises only the core variables using the multicriteria branch and bound algorithm that can generate the complete pareto set in small and medium size multi objective integer programming problems a small example is used to illustrate the method while computational and economy issues are also discussed computational experiments are also presented using available or appropriately modified benchmarks in order to examine the quality of pareto set approximation with respect to the solution time extensions to the general multi objective case as well as to the computation of the exact solution are also mentioned

a new algorithm for generating all nondominated solutions of multiobjective discrete optimization problems abstract most real life decision making activities require more than one objective to be considered therefore several studies have been presented in the literature that use multiple objectives in decision models in a mathematical programming context the majority of these studies deal with two objective functions known as bicriteria optimization while few of them consider more than two objective functions in this study a new algorithm is proposed to generate all nondominated solutions for multiobjective discrete optimization problems with any number of objective functions in this algorithm the search is managed over p 1 dimensional rectangles where p represents the number of objectives in the problem and for each rectangle two stage optimization problems are solved the algorithm is motivated by the well known e constraint scalarization and its contribution lies in the way rectangles are defined and tracked the algorithm is compared with former studies on multiobjective knapsack and multiobjective assignment problem instances the method is highly competitive in terms of solution time and the number of optimization models solved

an exact e constraint method for bi objective combinatorial optimization problems application to the traveling salesman problem with profits abstract this paper describes an exact constraint method for bi objective combinatorial optimization problems with integer objective values this method tackles multi objective optimization problems by solving a series of single objective subproblems where all but one objectives are transformed into constraints we show in this paper that the pareto front of bi objective problems can be efficiently generated with the constraint method furthermore we describe heuristics based on information gathered from previous subproblems that significantly speed up the method this approach is used to find the exact pareto front of the traveling salesman problem with profits a variant of the traveling salesman problem in which a profit or prize value is associated with each vertex the goal here is to visit a subset of vertices while addressing two conflicting objectives maximize the collected prize and minimize the travel costs we report the first exact results for this problem on instances derived from classical vehicle routing and traveling salesman problem instances with up to 150 vertices results on approximations of the pareto front obtained from a variant of our exact algorithm are also reported

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

multi objective integer programming a general approach for generating all non dominated solutions in this paper we develop a general approach to generate all non dominated solutions of the multi objective integer programming moip problem our approach which is based on the identification of objective efficiency ranges is an improvement over classical e constraint method objective efficiency ranges are identified by solving simpler moip problems with fewer objectives we first provide the classical e constraint method on the bi objective integer programming problem for the sake of completeness and comment on its efficiency then present our method on tri objective integer programming problem and then extend it to the general moip problem with k objectives a numerical example considering tri objective assignment problem is also provided

well posedness of a model of point dynamics for a limit of the keller segel system abstract the purpose of this paper is to prove well posedness for a problem that describes the dynamics of a set of points by means of a system of parabolic equations it has been seen in velazquez point dynamics in a singular limit of the keller segel model 1 motion of the concentration regions siam j appl math to appear that the considered model is the limit of a singular perturbation problem for a system of the keller segel type

contribution of individual judgments toward inconsistency in pairwise comparisons pairwise comparison pc is a well established method to assist decision makers in estimating their preferences in pcs the acquired judgments are used to construct a pc matrix pcm that is used to check whether the inconsistency in judgments is acceptable or requires revision the use of consistency ratio cr a widely used measure for inconsistency has been widely debated and the literature survey has identified a need for a more appropriate measure considering this need a new measure termed congruence is proposed in this paper the measure is shown to be useful in finding the contribution of individual judgments toward overall inconsistency of a pcm and therefore can be used to detect and correct cardinally inconsistent judgments the proposed measure is applicable to incomplete sets of pc judgments without modification unlike cr which requires a complete set of pc judgments to address ordinal inconsistency another measure termed dissonance is proposed as a supplement to the congruence measure the two measures appear useful in detecting both outliers and the phenomenon of consistency deadlock where all judgments equally contribute toward the overall inconsistency

an efficient adaptive parameter variation scheme for metaheuristics based on the epsilon constraint method abstract this paper discusses methods for generating or approximating the pareto set of multiobjective optimization problems by solving a sequence of constrained single objective problems the necessity of determining the constraint value a priori is shown to be a serious drawback of the original epsilon constraint method we therefore propose a new adaptive scheme to generate appropriate constraint values during the run a simple example problem is presented where the running time measured by the number of constrained single objective sub problems to be solved of the original epsilon constraint method is exponential in the problem size number of decision variables although the size of the pareto set grows only linearly we prove that independent of the problem or the problem size the time complexity of the new scheme is o km 1 where k is the number of pareto optimal solutions to be found and m the number of objectives simulation results for the example problem as well as for different instances of the multiobjective knapsack problem demonstrate the behavior of the method and links to reference implementations are provided

scaling binary comparison matrices a comment on narasimhan s proposal and other methods abstract recently narasimhan has proposed an adjustment to saaty s solution for scaling square judgment matrices consisting of pairwise ratio comparisons he recommends that prior to the application of saaty s solution the judgment matrix be adjusted to satisfy the supertransitivity requirement the present note shows that the proposed adjustment is equivalent to an alternative solution to this problem based upon calculation of the geometric means of the matrix rows thus the proposed adjustment should be regarded as an alternative to saaty s solution rather than a preliminary stage finally four solutions to this scaling problem are compared and the need for further research in this area is reiterated

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

group decision making based on multiplicative consistent reciprocal preference relations the main aim of this paper is to investigate the consistency and consensus of multiplicative consistent reciprocal preference relations based on the multiplicative consistency property a straightforward method to obtain the priority vector for multiplicative consistent reciprocal preference relation is introduced together with some of its desired properties in addition some interesting properties on consistency and consensus for reciprocal preference relations are further investigated a consensus optimization model for group decision making with reciprocal preference relations is proposed which enables the decision makers to reach consensus with the highest overall consensus level numerical examples are provided to make a discussion and comparison with some existing similar methods

stochastic decision making using multiplicative ahp abstract the analytic hierarchy process ahp has found a number of applications in decision making problems its multiplicative version called the multiplicative ahp mahp has been proposed to overcome some of the criticisms of the conventional version both these methodologies operate by obtaining expert judgements on the ratios of perceived importance of objects under consideration the literature on mahp in dealing with these judgements when they are specified without uncertainty is well developed however stochastic aspects of these judgements have not received much consideration in the literature so far stochastic judgements are considered in this paper for use in mahp the fact that weight derivation in mahp can be handled using mathematical programming is exploited and the literature on stochastic programming is adapted to the mahp context

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

a technical note on two inconsistency indices for preference relations a case of functional relation abstract different representations of valued preferences have been studied and used over the years however one common factor has been the prominence of the concept of consistency inconsistency indices have been introduced to estimate the deviation of preferences from a fully consistent form in this note we shall recall two types of preference relations reciprocal relations and multiplicative preference relations and show that two inconsistency indices introduced in these two different frameworks are functionally related besides this main result some reflections on the consequences of being functionally related are presented

the multi objective generalized consistent vehicle routing problem abstract more and more companies in the routing industry are providing consistent service to gain competitive advantage however improved service consistency comes at the price of higher routing cost i e routing cost and service consistency are conflicting objectives in this paper we extend the generalized consistent vehicle routing problem genconvrp by considering several objective functions improving driver consistency and arrival time consistency and minimizing routing cost are independent objectives of the problem we refer to the problem as the multi objective generalized consistent vehicle routing problem mogenconvrp a multi objective optimization approach enables a thorough trade off analysis between the conflicting objective functions the results of this paper should help companies in finding adequate consistency goals to aim for results are generated for several test instances by two exact solution approaches and one heuristic the exact approaches are based on the constraint framework and are used to solve small test instances to optimality large instances with up to 199 customers and a planning horizon of 5 days are solved by multi directional large neighborhood search mdlns that combines the multi directional local search framework and the lns for the genconvrp the solution quality of the heuristic is evaluated by examining five multi objective quality indicators we find that mdlns is an eligible solution approach for performing a meaningful trade off analysis our analysis shows that a 70 percent better arrival time consistency is achieved by increasing travel cost by not more than 3 84 percent on average visiting each customer by the same driver each time is significantly more expensive than allowing at least two different drivers per customer in many cases arrival time consistency and driver consistency can be improved simultaneously

on a parabolic elliptic system with chemotaxis and logistic type growth we consider a nonlinear pdes system of two equations of parabolic elliptic type with chemotactic terms the system models the movement of a biological population u towards a higher concentration of a chemical agent w in a bounded and regular domain rn for arbitrary n n after normalization the system is as follows ut u div um w u 1 u in t 0 t w w u in t for some positive constants m and with positive initial datum u0 and neumann boundary conditions we study the range of parameters and constrains for which the solution exists globally in time if either m 1 or m 1 and n 22 m 1 n the solution exists globally in time moreover if m 1 and 2 and there exist positive constants u 0 and u 0 such that 0 u 0 u0 u 0 we have that u 1 l w 1 l 0 as t

dynamic programming algorithms for the bi objective integer knapsack problem abstract this paper presents two new dynamic programming dp algorithms to find the exact pareto frontier for the bi objective integer knapsack problem first a property of the traditional dp algorithm for the multi objective integer knapsack problem is identified the first algorithm is developed by directly using the property the second algorithm is a hybrid dp approach using the concept of the bound sets the property is used in conjunction with the bound sets next the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set it means that the upper bound set is underestimated then an extended upper bound set is proposed on the basis of the set of linear relaxation solutions the efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set the numerical results obtained from different types of bi objective instances show the effectiveness of the proposed approach