generalized characteristic polynomials of graph bundles abstract in this paper we find computational formulae for generalized characteristic polynomials of graph bundles we show that the number of spanning trees in a graph is the partial derivative at 0 1 of the generalized characteristic polynomial of the graph since the reciprocal of the bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph consequently the bartholdi zeta function of a graph bundle can be computed by using our computational formulae

weighted coefficients in two phase approach for solving the multiple objective programming problems abstract two phase approach with equal weighted coefficients has been proposed to yield an efficient solution for multiple objective programming problems in this note we will show that the two phase approach as long as the weighted coefficients are positive not necessarily equal will generate an efficient solution a counterexample is given to the case that some weighted coefficients are zero

fuzzy linear programming and applications abstract this paper presents a survey on methods for solving fuzzy linear programs first lp models with soft constraints are discussed then lp problems in which coefficients of constraints and or of the objective function may be fuzzy are outlined pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives in addition to the commonly applied extended addition based on the min operator and used for the aggregation of the left hand sides of fuzzy constraints and fuzzy objectives a more flexible procedure based on yager s parametrized t norm t p is presented finally practical applications of fuzzy linear programs are listed

the spectrum and the signless laplacian spectrum of coronae abstract let g 1 g 2 be two simple connected graphs denote the corona and the edge corona of g 1 g 2 by g 1 g 2 and g 1 g 2 respectively in this paper we first introduce a new invariant the m coronal of a graph matrix m where the matrix m is associated with a graph in a prescribed way then it is used to compute the signless laplacian spectrum of g 1 g 2 and g 1 g 2 in terms of the signless laplacian spectrum of g 1 and g 2 in addition the spectrum of g 1 g 2 are also given in terms of the spectrum of g 1 and g 2 finally as an application of these results we construct many pairs of nonisomorphic signless laplacian cospectral graphs

multiple objective programming problems with possibilistic coefficients abstract we consider the problem of incorporating possibilistic data into a multiple objective linear programming framework a solution concept that is attractive from the standpoints of feasibility and efficiency is specified a necessary and sufficient condition for such a solution is established and ways to single out some satisfying alternative in this context are described a numerical example is also included for the sake of illustration throughout the paper we indicate topics for further research

optimality conditions for linear programming problems with fuzzy coefficients the optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers the solution concepts proposed in this paper will follow from the similar solution concept called the nondominated solution in the multiobjective programming problem under these settings the optimality conditions will be naturally elicited

numerical solution method for general interval quadratic programming abstract recently liu and wang described an interesting numerical method to a special class of interval quadratic programming where the linear term in objective function and constraints involved interval coefficients s t liu r t wang a numerical solution method to interval quadratic programming applied mathematics and computation 189 2007 1274 1281 in this paper we generalize liu and wang s method to general interval quadratic programming where all coefficients in the objective function and constraints are interval numbers although the interval quadratic programming model considered in this paper is more general the proposed method requires less computing compared with liu and wang s method numerical examples are presented to illustrate the whole idea

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

a method for capturing the entire fuzzy non dominated set of a fuzzy multi criteria optimization problem abstract in this study we consider the problem of capturing the complete fuzzy non dominated set of fuzzy multi criteria optimization problem we investigate the problem from a fuzzy geometrical viewpoint the fuzzy feasible regions on the decision space and on the criterion space are formulated using basic fuzzy geometrical ideas the sup min composition of the extension principle and the concept of inverse points in fuzzy geometry are used separately to formulate the fuzzy decision feasible region we show that under a monotone condition on the constraint functions both of the decision feasible regions are identical furthermore we obtain that the inverse points method can greatly reduce the computational cost in evaluating the fuzzy constraint inequalities under the assumption that criteria of the problem are fuzzy number valued functions fuzzy points are obtained in the criterion space that correspond to each point in the fuzzy decision feasible region the complete fuzzy criteria feasible region is the union of all of these fuzzy points a fuzzy non dominated set is defined on the formulated fuzzy criteria feasible region a definition of a proper fuzzy non dominated point is proposed for making the final decision different parts of the proposed methodology are demonstrated with suitable numerical examples and pictorial illustrations

a simplex based algorithm for the lexicographically extended linear maxmin problem abstract let f x cx c where c is a real m n matrix c a constant m vector and x a variable n vector g x denotes the vector with components f i x in order of ascending magnitude the problem is to maximize g x over x in the lexicographic sense where x is the set of all points such that ax b x 0 a a real r n matrix it is known that this problem can be reduced to a sequence of m maximum problems where the maxmin problems can be solved by the well known simplex method of linear optimization a prerequisite such a procedure is the knowledge of at least one common index per stage it is shown that these common indices can readily be obtained from the respective optimal simplex tableaux some simple tests help to obtain more than just one common index per stage

a new possibilistic programming approach for solving fuzzy multiobjective assignment problem in this paper we propose a new possibilistic programming approach to solve a fuzzy multiobjective assignment problem in which the objective function coefficients are characterized by triangular possibility distributions the proposed solution approach simultaneously minimizes the best scenario the likeliest scenario and the worst scenario for the imprecise objective functions using level sets the level sets are used to define the confidence level of the fuzzy judgments of the decision maker additionally we provide a systematic framework in which the decision maker controls the search direction by updating both the membership values and aspiration levels until a set of satisfactory solutions is obtained numerical examples with dataset from realistic situations are provided to demonstrate the effectiveness of the proposed approach

production planning problem with sequence dependent setups as a bilevel programming problem abstract each of n products is to be processed on two machines in order to satisfy known demands in each of t periods only one product can be processed on each machine at any given time each switch from one item to another requires sequence dependent setup time the objective is to minimize the total setup time and the sum of the costs of production storage and setup we consider the problem as a bilevel mixed 0 1 integer programming problem the objective of the leader is to assign the products to the machines in order to minimize the total sequence dependent setup time while the objective of the follower is to minimize the production storage and setup cost of the machine we develop a heuristics based on tabu search for solving the problem at the end some computational results are presented

solving fuzzy stochastic linear programming problems using superiority and inferiority measures in this paper the author presents a model to measure the superiority and inferiority of fuzzy numbers fuzzy stochastic variables then the new measures are used to convert the fuzzy stochastic linear program into the corresponding deterministic linear program numerical examples are provided to illustrate the effectiveness of the proposed method

the mixed integer linear bilevel programming problem a two person noncooperative game in which the players move in sequence can be modeled as a bilevel optimization problem in this paper we examine the case where each player tries to maximize the individual objective function over a jointly constrained polyhedron the decision variables are variously partitioned into continuous and discrete sets the leader goes first and through his choice may influence but not control the responses available to the follower for two reasons the resultant problem is extremely difficult to solve even by complete enumeration first it is not possible to obtain tight upper bounds from the natural relaxation and second two of the three standard fathoming rules common to branch and bound cannot be applied fully in light of these limitations we develop a basic implicit enumeration scheme that finds good feasible solutions within relatively few iterations a series of heuristics are then proposed in an effort to strike a balance between accuracy and speed the computational results suggest that some compromise is needed when the problem contains more than a modest number of integer variables

multi objective optimization with a max t norm fuzzy relational equation constraint in this paper we consider minimizing multiple linear objective functions under a max t norm fuzzy relational equation constraint since the feasible domain of a max archimedean t norm relational equation constraint is generally nonconvex traditional mathematical programming techniques may have difficulty in yielding efficient solutions for such problems in this paper we apply the two phase approach utilizing the min operator and the average operator to aggregate those objectives to yield an efficient solution a numerical example is provided to illustrate the procedure

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

interactive multiple objective programming subject to flexible constraints abstract an interactive decision support system is introduced which aids in solving multiple objective programming problems subject to strict and flexible constraints integral part is an extension of a well known fuzzy sets approach evaluating possible solutions by their degrees of membership to objectives and constraints this approach is linked to classical multiple objective programming models if the decision maker cannot determine membership functions a priori the system suggests functions dependent on the given information and interactive modifications are allowed

exact solution approaches for bilevel lot sizing abstract in this paper we propose exact solution methods for a bilevel uncapacitated lot sizing problem with backlogs this is an extension of the classical uncapacitated lot sizing problem with backlogs in which two autonomous and self interested decision makers constitute a two echelon supply chain the leader buys items from the follower in order to meet external demand at lowest cost the follower also tries to minimize its costs both parties may backlog we study the leader s problem i e how to determine supply requests over time to minimize its costs in view of the possible actions of the follower we develop two mixed integer linear programming reformulations as well as cutting planes to cut off feasible but suboptimal solutions we compare the reformulations on a series of benchmark instances

an interactive possibilistic programming approach for multiple objective supply chain master planning providing an efficient production plan that integrates the procurement and distribution plans into a unified framework is critical to achieving competitive advantage in this paper we consider a supply chain master planning model consisting of multiple suppliers one manufacturer and multiple distribution centers we first propose a new multi objective possibilistic mixed integer linear programming model mopmilp for integrating procurement production and distribution planning considering various conflicting objectives simultaneously as well as the imprecise nature of some critical parameters such as market demands cost time coefficients and capacity levels then after applying appropriate strategies for converting this possibilistic model into an auxiliary crisp multi objective linear model molp we propose a novel interactive fuzzy approach to solve this molp and finding a preferred compromise solution the proposed model and solution method are validated through numerical tests computational results indicate that the proposed fuzzy method outperforms other fuzzy approaches and is very promising not only for solving our problem but also for any molp model especially multi objective mixed integer models

computing efficient solutions to fuzzy multiple objective linear programming problems in this paper we improve the fuzzy compromise approach of guu and wu by automatically computing proper membership thresholds instead of choosing them indeed in practice choosing membership thresholds arbitrarily may result in an infeasible optimization problem although we can adjust minimum satisfaction degree to get fuzzy efficient solution it sometimes makes the process of interaction more complicated in order to overcome this drawback a theoretically and practically more efficient two phase max min fuzzy compromise approach is proposed in this paper moreover the efficiency of the two phase approach is verified by an example

on the construction of graphs determined by their generalized characteristic polynomials abstract given a graph g with adjacency matrix a g the generalized characteristic polynomial of g is defined as g g t det i a g t d g d g is the degree diagonal matrix g is said to be determined by generalized characteristic polynomial ds for short if for any graph h g h implies that h is isomorphic to g in this paper we show that a connected graph g is ds if and only if the subdivision graph g s is ds this gives a simple method to construct large ds graph from smaller ones some further generalization of this result as wells as some non trivial examples of constructing ds graphs are also provided

fuzzy programming in multiobjective linear programming a parametric approach abstract zimmermann s fuzzy approach to the compromise solution concept in the multi objective linear programming problem is considered it is shown that given a class of membership functions of fuzzy goals assigned to objective functions in the problem wider than primarily proposed by zimmermann the use of classical linear programming methods to solve and analyse the problem is also possible as a result of application of the method based on the parametric programming technique one can obtain a fuzzy solution of the problem this solution is a certain fuzzy subset of the set of weakly efficient solutions of the problem the solution which belongs to this set to the highest degree is a maximizing one in bellman and zadeh s terminology it may also be obtained by use of the usual non parametric simplex method

pareto optimal solutions in fuzzy multi objective linear programming the problem of solving multi objective linear programming problems by assuming that the decision maker has fuzzy goals for each of the objective functions is addressed several methods have been proposed in the literature in order to obtain fuzzy efficient solutions to fuzzy multi objective programming problems in this paper we show that in the case that one of our goals is fully achieved a fuzzy efficient solution may not be pareto optimal and therefore we propose a general procedure to obtain a non dominated solution which is also fuzzy efficient two numerical examples illustrate our procedure

linear programming with multiple fuzzy goals abstract this paper describes the use of fuzzy set theory in goal programming gp problems in particular it is demonstrated how fuzzy or imprecise aspirations of the decision maker dm can be quantified through the use of piecewise linear and continuous functions models are then presented for the use of fuzzy goal programming with preemptive priorities with archimedean weights and with the maximization of the membership function corresponding to the minimum goal examples are also provided

multiple objective linear fractional programming this paper presents a simplex based solution procedure for the multiple objective linear fractional programming problem by 1 departing slightly from the traditional notion of efficiency and 2 augmenting the feasible region as in goal programming the solution procedure solves for all weakly efficient vertices of the augmented feasible region the article discusses the difficulties that must be addressed in multiple objective linear fractional programming and motivates the solution algorithm that is developed

fuzzy multiple objective programming and compromise programming with pareto optimum abstract a fuzzy multiple objective decision making approach is proposed based on the desirable features of compromise programming and the fuzzy set theory the proposed two phase approach guarantees both nondominated and balanced solutions for solving both the crisp and the fuzzy multiple objective decision making problems furthermore it is shown that the original compromise programming does not guarantee nondominated solutions if the distance parameter is assumed to be infinite and the resulting programming problem s solution is not unique

solving linear programming problems under fuzziness and randomness environment using attainment values in this paper the author presents a model to measure attainment values of fuzzy numbers fuzzy stochastic variables these new measures are then used to convert the fuzzy linear programming problem or the fuzzy stochastic linear programming problem into the corresponding deterministic linear programming problem numerical comparisons are provided to illustrate the effectiveness of the proposed method

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

fuzzy stochastic linear programming survey and future research directions in many real life problems one has to base decision on information which is both fuzzily imprecise and probabilistically uncertain although consistency indexes providing a union nexus between possibilistic and probabilistic representation of uncertainty exist there are no reliable transformations between them this calls for new paradigms for incorporating the two kinds of uncertainty into mathematical models fuzzy stochastic linear programming is an attempt to fulfill this need it deals with modelling and problem solving issues related to situations where randomness and fuzziness co occur in a linear programming framework in this paper we provide a survey of the essential elements methods and algorithms for this class of linear programming problems along with promising research directions being a survey the paper includes many references to both give due credit to results in the field and to help readers obtain more detailed information on issues of interest

interactive decision making in fuzzy linear optimization problems abstract this paper presents an interactive method for solving a multi criteria linear program where coefficients of the objective functions and or of the constraints are known exactly but imprecisely assuming fuzzy aspiration levels for each of the goals and basing on a comparison of flat fuzzy numbers the original fuzzy problem is transformed into a crisp satisfactory model to determine a suitable compromise solution we will maximize the minimum of the surpluses over the aspiration levels this induces a problem with a nonlinear objective function which is equivalent to a linear program

optimisation of gravity fed water distribution network design a critical review abstract this paper surveys the literature on the optimisation of water distribution network design the water distribution network design wdnd optimisation problem entails finding the material and diameter of each pipe in the network so that the total cost of the network is minimised without violating any hydraulic constraints this is a difficult combinatorial optimisation problem in which decision variables are discrete and both cost function and constraints are non linear over the past 30 years a large number of methods especially in the field of meta heuristics have been developed to solve this problem most of which obtain good results on the available benchmark networks in addition to outlining the basic features of each method a detailed computational comparison is presented based on this comparison some issues with the current state of the art in this domain are discussed and some future research directions are suggested additionally the need for an adequate set of benchmark instances is motivated and the minimal requirements for an instance set generator are discussed

a nonlinear interval number programming method for uncertain optimization problems in this paper a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints based on an order relation of interval number the uncertain objective function is transformed into two deterministic objective functions in which the robustness of design is considered through a modified possibility degree the uncertain inequality and equality constraints are changed to deterministic inequality constraints the two objective functions are converted into a single objective problem through the linear combination method and the deterministic inequality constraints are treated with the penalty function method the intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non constraint optimization problem two numerical examples are investigated to demonstrate the effectiveness of the present method

a triangular type 2 multi objective linear programming model and a solution strategy abstract we consider a multi objective linear programming model with type 2 fuzzy objectives the considered model has the flexibility for the user to specify the more general membership functions for objectives to reflect the inherent fuzziness while being simple and practical we develop two solution strategies with reasonable computing costs the additional cost as compared to the type 1 fuzzy model is indeed insignificant these two algorithms compute pareto optimal solutions of the type 2 problems one being based on a maxmin approach and the other on aggregating the objectives finally applying the proposed algorithms we work out two illustrative examples

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

graphs determined by their generalized characteristic polynomials abstract for a given graph g with 0 1 adjacency matrix a g the generalized characteristic polynomial of g is defined to be g g t det i a g td g where i is the identity matrix and d g is the diagonal degree matrix of g in this paper we are mainly concerned with the problem of characterizing a given graph g by its generalized characteristic polynomial g we show that graphs with the same generalized characteristic polynomials have the same degree sequence based on which a unified approach is proposed to show that some families of graphs are characterized by g we also provide a method for constructing graphs with the same generalized characteristic polynomial by using gm switching

a two phase possibilistic linear programming methodology for multi objective supplier evaluation and order allocation problems in this study an integration of the analytic hierarchy process ahp and a multi objective possibilistic linear programming moplp technique is developed to account for all tangible intangible quantitative and qualitative factors which are used to evaluate and select suppliers and to define the optimum order quantities assigned to each a multi objective linear programming technique is first employed to solve the problem to model the uncertainties encountered in the integrated supplier evaluation and order allocation methodology fuzzy theory is adopted hence possibilistic linear programming plp is proposed for solving the problem as it is believed to be the best approach for absorbing the imprecise nature of the real world in the supplier evaluation phase environmental criteria are also considered

comparison of methodologies for fuzzy and stochastic multi objective programming abstract methods for solving a multicriteria linear program with coefficients of the objective functions and the constraints being flat fuzzy numbers and those dealing with a stochastic multicriteria linear program where imprecision of some data is modelled by probability distributions are surveyed the methodologies are compared and evaluated some ideas for future research are emphasized

distinguishing graphs with zeta functions and generalized spectra abstract conjecturally almost all graphs are determined by their spectra this problem has also been studied for variants such as the spectra of the laplacian and signless laplacian here we consider the problem of determining graphs with ihara and bartholdi zeta functions which are also computable in polynomial time these zeta functions are geometrically motivated but can be viewed as certain generalizations of characteristic polynomials after discussing some graph properties determined by zeta functions we show that large classes of cospectral graphs can be distinguished with zeta functions and enumerate graphs distinguished by zeta functions on 11 vertices this leads us to conjecture that almost all graphs which are not determined by their spectrum are determined by zeta functions along the way we make some observations about the usual types of spectra and disprove a conjecture of setyadi and storm about ihara zeta functions determining degree sequences

siting recycling drop off stations in urban area by genetic algorithm based fuzzy mulitobjective nonlinear integer programming modeling due to the rapid depletion of landfill space and the time consuming process for siting and building new municipal incinerators solid waste management strategies have to be reorganized in light of the success of recycling recovery and reuse of secondary materials effective planning of solid waste recycling programs however is currently a substantial challenge in many solid waste management systems one of such efforts is how to effectively allocate the recycling drop off stations with appropriate size in the solid waste collection network to maximize the recycling achievement with minimum expense this paper illustrates a new approach with a view to optimizing siting and routing aspects using a fuzzy multiobjective nonlinear integer programming model as a means that is particularly solved by a genetic algorithm the case study based on one of the administrative districts in the city of kaohsiung in taiwan presents the application potential of such a planning methodology

on bi regular graphs determined by their generalized characteristic polynomials abstract it was shown that whether a graph g is determined by its generalized characteristic polynomial is equivalent to the same problem for a well defined bi regular graph g also a unified approach is proposed to show that some new families of bi regular graphs are characterized by their generalized characteristic polynomials

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

studies on optimum path ratios in multicriteria de novo programming problems abstract this paper proposes several optimum path ratios for enforcing different budget levels of resources so as to find alternative optimal system designs for solving multicriteria de novo programming problems then the study of optimum path ratios is applied to solve an optimal pattern matching problem which is formulated by the de novo programming with a given initial budget level and an optimal pattern preferred by the decision maker an interactive algorithm is developed to continuously reshape the problem for matching the optimal pattern a numerical example is also used to illustrate the algorithm

computational complexity of the capacitated lot size problem in this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings for the single item case new properties are introduced classes of problems solvable by polynomial time algorithms are identified and efficient solution procedures are given we show that special classes are np hard and that the problem with two items and independent setups is np hard under conditions similar to those where the single item problem is easy topics for further research are discussed in the last section

a new possibilistic optimization model for multiple criteria assignment problem this paper presents a new multiple criteria optimization model of an assignment problem with imprecise coefficients besides minimizing the total cost total time of finishing jobs and maximization of the overall achieved quality we introduce a new criterion that minimizes the number of workers employed to finish all jobs it contributes significantly in multi job assignment to adjust the number of workers assigned to at least one job for balancing work allocation among the workers furthermore we employ new diversification constraints to obtain a reasonable tradeoff between the number of workers employed and number of jobs assigned a new interactive possibilistic programming approach is developed for trapezoidal possibility distributions which uses alpha level sets to incorporate confidence levels of the decision maker in his fuzzy judgments leading to alpha efficient solutions numerical experiments are conducted using data coming from a manpower planning problem to demonstrate working of the proposed multiple criteria assignment model and effectiveness of the fuzzy interactive approach

on finding compromise solutions in multicriteria problems using the fuzzy min operator abstract this paper presents an application of fuzzy approaches to the linear vectormaximum problem it shows that using the fuzzy min operator together with linear as well as special nonlinear membership functions the obtained solutions are always compromise solutions of the original multicriteria problem

an interactive fuzzy programming system abstract an interactive system is introduced which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals one part of the system is the determination of membership functions representing goals to this purpose fuzzy extreme solutions are computed and are presented to the decision maker these and each of the proposed compromise solutions are fuzzy efficient

fuzzy mathematical programming based on some new inequality relations abstract this paper deals with inequality relations in fuzzy mathematical programming problem fmp not necessarily linear moreover fuzzy parameters may have nonlinear membership functions a new approach for comparing fuzzy sets is proposed which is more general than the well known proposals in the literature

fuzzy approaches to multicriteria de novo programs abstract multicriteria de novo programming is a promising tool for optimal system design however there exist no algorithms for solving a general multicriteria de novo program only special cases have been discussed this paper proposes a two step fuzzy approach based on the ideal and negative ideal solutions it is shown that this approach is very efficient and is applicable to the general multicriteria de novo programming the fuzzy version of this problem is also formulated and analyzed numerical examples are given to illustrate the approaches

solving multiobjective linear programming problems a generic approach abstract we propose a fuzzy approach which induces some methodologies as special cases for solving the multiple objective linear programming problem a numerical example is then provided for demonstration

designing robust coverage networks to hedge against worst case facility losses in order to design a coverage type service network that is robust to the worst instances of long term facility loss we develop a facility location interdiction model that maximizes a combination of initial coverage by p facilities and the minimum coverage level following the loss of the most critical r facilities the problem is formulated both as a mixed integer program and as a bilevel mixed integer program to solve the bilevel program optimally a decomposition algorithm is presented whereby the original bilevel program is decoupled into an upper level master problem and a lower level subproblem after sequentially solving these problems supervalid inequalities can be generated and appended to the upper level master in an attempt to force it away from clearly dominated solutions computational results show that when solved to optimality the bilevel decomposition algorithm is up to several orders of magnitude faster than performing branch and bound on the mixed integer program

the characteristic polynomial of a generalized join graph abstract for a graph g with adjacency matrix a g and degree diagonal matrix d g cvetkovic et al introduced a bivariate polynomial g x t d e t x i a g t d g where i is the identity matrix the polynomial g x t not only generalizes the characteristic polynomials of some well known matrices related to g such as the adjacency the laplacian matrices but also has an elegant combinatorial interpretation as being equivalent to the bartholdi zeta function let g h g 1 g 2 g k be the generalized join graph of g 1 g 2 g k determined by graph h in this paper we first give a decomposition formula for g x t the decomposition formula provides us a new method to construct infinitely many pairs of non regular cospectral graphs then as applications explicit expressions for g x t of some special kinds of graphs are given

interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems abstract this paper presents interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems by adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level our aim is to obtain a preferred compensatory compromise pareto optimal solution for dmllp for this the weights of objectives at each level are assigned by the next upper level decision maker dm by using analytic hierarchy process ahp or any other weighting methods the weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker dm 0 using these weights equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights werners compensatory fuzzy and operator is offered to solve dmllp problem the most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights thanks to this equivalence dmllp problem has been transformed to the multi objective linear programming molp problem at level 0 the equivalence is reflected to the compensatory model within the constraints and the equivalence also enables all dms to obtain proportional satisfactions with their weights as much as possible so in our compensatory model a reduction on equivalent satisfactory level of one dm can be compensated for by an increase in the equivalent satisfactory level of another dm furthermore being developed a finite interactive iterative procedure with maximum interaction step a set of compensatory solutions which are also pareto optimal is obtained depending on compensation parameter giving a theorem we will show that the solutions generated by werners compensatory fuzzy and operator do guarantee pareto optimality for our dmllp problem and comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for dmllp illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for dmllp

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

a simple tabu search method to solve the mixed integer linear bilevel programming problem abstract multilevel programming is characterized as mathematical programming to solve decentralized planning problems the models partition control over decision variables among ordered levels within a hierarchical planning structure of which the linear bilevel form is a special case of a multilevel programming problem in a system with such a hierarchical structure the high level decision making situations generally require inclusion of zero one variables representing yes no decisions we provide a mixed integer linear bilevel programming formulation in which zero one decision variables are controlled by a high level decision maker and real value decision variables are controlled by a low level decision maker an algorithm based on the short term memory component of tabu search called simple tabu search is developed to solve the problem and two supplementary procedures are proposed that provide variations of the algorithm computational results disclose that our approach is effective in terms of both solution quality and efficiency

a two phase approach for multi objective programming problems with fuzzy coefficients in this study a two phase procedure is introduced to solve multi objective fuzzy linear programming problems the procedure provides a practical solution approach which is an integration of fuzzy parametric programming fpp and fuzzy linear programming flp for solving real life multiple objective programming problems with all fuzzy coefficients the interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision maker dm the procedure can be also performed to obtain lexicographic optimal and or additive solutions if it is needed in the first phase of the procedure a family of vector optimization models is constructed by using fpp then in the second phase each model is solved by flp the solutions are optimal and each one is an alternative decision plan for the dm

fuzzy multiple choice knapsack problem abstract this article is intended to present a multiple choice knapsack problem with fuzzy coefficients we propose an approximate algorithm for solving the problem and obtaining some non dominated approximate solutions in the interpretation of the constraints the degree satisfying the inequality between fuzzy numbers is defined on the basis of the agreement index finally a numerical example is shown

a parametric approach to fuzzy linear programming in this paper we discuss how to deal with decision problems that are described with lp models and formulated with elements of imprecision and uncertainty more precisely we will study lp models in which the parameters are not fully known but only with some degree of precision even with incomplete information the model builder or model user is normally able to give a realistic interval for the parameters of an lp model for the constraint vector this is combined with some wishes or some leeway on the constraints even with ambiquity in the objective function there is normally some preference ordering to be found among alternative ways of action we will demonstrate that these modelling complications can be handled with the help of some results developed in the theory of fuzzy sets after an overview of some central contributions to fuzzy linear programming we will develop an lp model in which the parameters are not fully known only with some degree of precision and show that the model can be parametrised in such a way that the optimal solution becomes a function of the degree of precision the fuzzy lp model derived in this way appears to be fairly easy to handle computationally which is demonstrated with a numerical example

a multicriteria fuzzy linear programming method for water supply system development planning abstract a method is presented for solving a multicriteria linear program where the coefficients of the objective functions and the constraints are fuzzy numbers of the l r type assuming the aspiration levels for particular criteria to be fuzzy and basing on comparison of fuzzy numbers the original problem is transformed into a multicriteria linear fractional program the latter is solved using an interactive technique involving a linear programming procedure in the calculation phase this method has been elaborated with a view to the application to the development of a water supply system

aggregate production planning with multiple objectives in a fuzzy environment abstract this work presents a novel fuzzy linear programming flp method for solving the aggregate production planning app problem with multiple objectives where the product price unit cost to subcontract work force level production capacity and market demands are fuzzy in nature also discussed herein are limitations of applying the conventional mathematical programming technique to medium term production planning in addition the specific flp model is proposed moreover an interactive solution procedure is developed to provide a compromise solution finally a numerical example is presented to demonstrate the effectiveness of the proposed procedure the proposed procedure allows a decision maker to model a problem according to the current information the proposed model is more appropriate than the unfuzzy problem formulation in terms of reflecting a realistic situation consequently the information costs are generally decreased

linear programming under randomness and fuzziness this paper is concerned with the use of fuzzy sets theory well suited for nuanced human behavior to extend the applicability of linear programming in a way to let both randomness and fuzziness uncertainties be taken into account a fuzzy stochastic linear programming problem is considered and ways to handle it suggested the basic idea behind our approaches is to translate the original problem in deterministic terms via tools provided by fuzzy sets theory to get a classical one some conditions for the resulting problems to be solved without extra computational effort are stressed and illustrative examples given

fuzzy programming and the pareto optimal set it is shown how fuzzy programming using either the min or product operator may be used to generate the whole pareto optimal set for nonlinear concave or convex multiobjective programming problems also a new solution procedure for a fuzzy program when the min operator is employed is presented

a goal programming procedure for solving problems with multiple fuzzy goals using dynamic programming abstract this paper describes how the preemptive priority based goal programming gp can be used to solve a class of fuzzy programming fp problems with the characteristics of dynamic programming dp in the proposed approach the membership functions of the objective goals of a problem with fuzzy aspiration levels are defined first then under the framework of preemptive priority based gp a multi stage dp model of the problem for achievement of the highest degree unity of each of the membership functions is developed in the decision process the goal satisficing philosophy of gp is used recursively to arrive at the most satisfactory solution two numerical examples are provided to illustrate the approach

two phase approach for solving the fuzzy linear programming problems we propose a two phase approach to solve the fuzzy linear programming problem although several methods in the literature have been proposed to treat this problem the two phase approach has the merit stated as if the decision maker is seeking an efficient solution which can improve the max min operators solution so that each membership degree should be improved then the two phase method automatically attains this desire if there is room to improve an interpretation of this result is that the two phase method not just pursue the highest membership degree in the objective but also pursue a better utilization of each constrained resource

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

strange an interactive method for multi objective linear programming under uncertainty abstract in the field of investment planning within a time horizon problems typically involve multiple objectives and basic data are uncertain in a large number of cases these decision problems can be written as linear programming problems in which time dependent uncertainties affect the coefficients and the right hand side of constraints given the possibility of defining plausible scenarios on basic data discrete sets of such coefficients are given each with its subjective probability of occurrence the corresponding structure is then characteristic for multi objective stochastic linear programming moslp in the paper an interactive procedure is described to obtain a best compromise for such a moslp problem this algorithm called strange extends the stem method to take the random aspects into account it involves in particular the concepts of stochastic programming with recourse in its interactive steps the efficiency projection techniques are used to provide the decision maker with detailed graphical information on efficient solution families as an illustration of the successive steps a didactic example is solved in some detail and the results of a case study in energy planning are given

application of fuzzy sets to aggregate production planning with multiproducts and multitime periods the objective of this study is to develop a fuzzy mathematical programming method to solve aggregate production planning app decision problems that involve multiproducts and multitime periods in a fuzzy environment the fuzzy app model that is developed here attempts to minimize total cost with respect to inventory carrying levels available labor levels machine capacity and warehouse space and the constraint of available budget the proposed app method evaluates monetary interest of related operating cost categories and provides greater computational efficiency and flexibility by adopting triangular fuzzy numbers and piecewise linear membership functions to represent both imprecise data and fuzzy goals the actual performance of an industrial company was used to demonstrate the feasibility of applying the proposed method to real world app decisions the proposed method yields an efficient solution and presents overall decision maker satisfaction with the given goal values this paper also presents several significant management implications that are related to the practical application of the proposed method

stability on multiobjective linear programming problems with fuzzy parameters this paper deals with multiobjective linear programming problems with fuzzy parameters fmolp for such problems some stability notions as the solvability set and the stability set of the first kind ssk1 are defined and characterized two algorithms for the determination of the set ssk1 are proposed one of these algorithms is applied when the fuzzy multiobjective problem remains linear after introducing the so called level set of the fuzzy numbers while the other algorithm is applied when the fuzzy multiobjective problem becomes nonlinear finally a numerical example is given to illustrate the theory developed in the paper

fuzzy stochastic based violation analysis method for planning water resources management systems with uncertain information in this study a fuzzy stochastic based violation analysis fsva approach is developed for the planning of water resources management systems with uncertain information based on a multistage fuzzy stochastic integer programming fsip model in fsva a number of violation variables for the objective and constraints are allowed such that in depth analyses of tradeoffs among economic objective satisfaction degree and constraint violation risk can be facilitated besides the developed method can deal with uncertainties expressed as probability distributions and fuzzy sets it can also reflect the dynamics in terms of decisions for water allocation and surplus flow diversion through transactions at discrete points of a complete scenario set over a multistage context the developed fsva method is applied to a case study of water resources management within a multi stream multi reservoir and multi period context the results indicate that the satisfaction degrees and system benefits would be different under varied violation levels moreover different violation levels can also lead to changed water allocation and surplus flow diversion plans violation analyses are also conducted to demonstrate that violating different constraints have different effects on system benefit and satisfaction degree

fuzzy chance constrained linear programming model for optimizing the scrap charge in steel production optimizing the charge in secondary steel production is challenging because the chemical composition of the scrap is highly uncertain the uncertainty can cause a considerable risk of the scrap mix failing to satisfy the composition requirements for the final product in this paper we represent the uncertainty based on fuzzy set theory and constrain the failure risk based on a possibility measure consequently the scrap charge optimization problem is modeled as a fuzzy chance constrained linear programming problem since the constraints of the model mainly address the specification of the product the crisp equivalent of the fuzzy constraints should be less relaxed than that purely based on the concept of soft constraints based on the application context we adopt a strengthened version of soft constraints to interpret fuzzy constraints and form a crisp model with consistent and compact constraints for solution simulation results based on realistic data show that the failure risk can be managed by proper combination of aspiration levels and confidence factors for defining fuzzy numbers there is a tradeoff between failure risk and material cost the presented approach applies also for other scrap based production processes