inference based sensitivity analysis for mixed integer linear programming a new method of sensitivity analysis for mixed integer linear programming milp is derived from the idea of inference duality the inference dual of an optimization problem asks how the optimal value can be deduced from the constraints in milp a deduction based on the resolution method oftheorem proving can be obtained from the branch and cut tree that solves the primal problem one can then investigate which perturbations ofthe problem leave this proof intact on this basis it is shown that in a minimization problem any perturbation that satisfies a certain system of linear inequalities will reduce the optimal value no more than a prespecified amount one can also give an upper bound on the increase in the optimal value that results from a given perturbation the method is illustrated on two realistic problems

two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

allocation and scheduling of conditional task graphs we propose an original complete and efficient approach to the allocation and scheduling of conditional task graphs ctgs in ctgs nodes represent activities some of them are branches and are labeled with a condition arcs rooted in branch nodes are labeled with condition outcomes and a corresponding probability a task is executed at run time if the condition outcomes that label the arcs in the path to the task hold at schedule execution time this can be captured off line by adopting a stochastic model tasks need for their execution either unary or cumulative resources and some tasks can be executed on alternative resources the solution to the problem is a single assignment of a resource and of a start time to each task so that the allocation and schedule is feasible in each scenario and the expected value of a given objective function is optimized for this problem we need to extend traditional constraint based scheduling techniques in two directions i compute the probability of sets of scenarios in polynomial time in order to get the expected value of the objective function ii define conditional constraints that ensure feasibility in all scenarios we show the application of this framework on problems with objective functions depending either on the allocation of resources to tasks or on the scheduling part also we present the conditional extension to the timetable global constraint experimental results show the effectiveness of the approach on a set of benchmarks taken from the field of embedded system design comparing our solver with a scenario based solver proposed in the literature we show the advantages of our approach both in terms of execution time and solution quality

hierarchical optimization a satisfactory solution abstract hierarchical optimization or multi level programming techniques are extensions of stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization they become more important for contemporary decentralized organizations where each unit or department seeks its own interests traditional approaches include vertex enumeration algorithms and approaches based on kuhn tucker conditions or penalty functions these are not only technically inefficient but also lead to a paradox that the follower s decision power dominates the leader s in this study concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently in the proposed hierarchy decision process the leader first sets memberships of optimalities of his her possible objective values and decisions as well as his her decision power and then asks the follower for his her optima calculated in isolation under given constraints the follower s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached

a comparison of hybrid and quadratic upstream differencing in high reynolds number elliptic flows abstract numerical computations have been performed of various two dimensional elliptic flows at high reynolds number with a view to assessing the relative merits of the widely used hybrid i e upwind central interpolation and the recently proposed quadratic upstream interpolation of leonard known as quick the latter scheme possesses less intrinsic stability than hybrid differencing by selectively evaluating some of the convective elements at a previous iteration level however satisfactorily rapid convergence has been obtained it is concluded from the flows tested that the extra computational effort per node required for quick over the hybrid difference scheme is more than repaid by the greater numerical accuracy that results

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

integer programming duality price functions and sensitivity analysis recently a duality theory for integer programming has been developed here we examine some of the economic implications of this theory in particular the necessity of using price functions in place of prices and the possibility of carrying out sensitivity analysis of optimal solutions in addition we consider the form of price functions that are generated by known algorithms for integer programming

partial orders based on core nilpotent decomposition abstract in this paper we study partial orders in terms of the core nilpotent decomposition we derive some characterizations of the c n partial ordering create a new partial ordering the g drazin partial ordering which is a generalization of c n partial ordering and give some properties and characterizations of the g drazin partial ordering

g outer inverse of banach spaces operators abstract the aim of this paper is to present a new generalized inverse which is called a g outer inverse of operators between two banach spaces as a generalization of g drazin inverses of square matrices and banach space operators several characterizations of this new inverse are given based on the g outer inverse we define and characterize the g outer partial order extending the g drazin partial order to operators between two banach spaces we also relate this new partial order to the minus partial order in the end we study an outer binary relation which is an equivalence relation on the corresponding set

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

weighted pre orders involving the generalized drazin inverse the aim of this paper is to characterize new pre orders defined on the set of all bounded linear operators between two banach spaces thus recent results on pre orders involving the drazin inverse of a complex matrix are extended to a more general setting

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

a branch and check algorithm for minimizing the weighted number of late jobs on a single machine with release dates in this paper we consider the scheduling problem of minimizing the weighted number of late jobs on a single machine 1 rj wjuj a branch and check algorithm is proposed where a relaxed integer programming formulation is solved by branch and bound and infeasible solutions are cut off using infeasibility cuts we suggest two ways to generate cuts first tightened no good cuts are derived using a modification of the algorithm by carlier 1982 ejor v 11 42 47 which was developed for the problem of minimizing maximum lateness on a single machine secondly we show how to create cuts by using constraint propagation the proposed algorithm is implemented in the mosel modelling and optimization language computational experiments on instances with up to 140 jobs are reported a comparison is presented with the exact approach of peridy at al 2003 ejor v 148 591 603

fuzzy goal programming approach to multilevel programming problems the purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach here the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers since the objectives are potentially conflicting in nature a possible relaxation of the higher level decision is considered for avoiding decision deadlock then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values the efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

interactive fuzzy programming for two level linear fractional programming problems this paper presents interactive fuzzy programming for two level linear fractional programming problems with the essentially cooperative decision makers in our interactive method after determining the fuzzy goals of the decision makers at both levels a satisfactory solution is efficiently derived by updating the satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels in an interactive process optimal solutions to the formulated programming problems are obtained by combined use of the bisection method and the phase one of linear programming and the variable transformation by charnes and cooper an illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method

sensitivity analysis for branch and bound integer programming after an integer program is solved we may be interested in the effect of small changes in the right hand side or objective function coefficients without modifying the underlying branch and bound or cutting plane algorithm we show what additional data to collect during the solution of the original integer program to give useful bounds on the solution of the modified problem

interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms abstract in this paper we propose interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms our method is supposed to apply to hierarchical decision problems in which decision making at each level is sequential from upper to lower level and decision makers are essentially cooperative after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers at the upper level with considerations of overall satisfactory balance among all levels an illustrative numerical example for three level 0 1 programming problems is provided to demonstrate the feasibility of the proposed method

determination of relaxation factors for high cell peclet number flow simulation abstract for the numerical simulation of transport problems with high cell peclet number the coefficient matrix of finite difference equations may lose the diagonal dominance if a scheme more accurate than the first order upwind is used to approximate the convection terms hence in many cases it is difficult to choose a suitable relaxation factor a priori for these schemes when the commonly used iterative method is applied to obtain the convergent solution however it is found that after appropriate normalization an easier determined relaxation factor useful to all three schemes studied here i e second order central differencing second order upwinding and quick exists two model problems are used to compare the performances among the three schemes the second order central differencing is found to be less efficient in the cases investigated without the aid of the normalization the second order upwind scheme has the widest range of allowable values for the relaxation factor yet quick may be computationally more efficient after normalization this is mainly because the number of iterations needed for quick is less sensitive to the choice of the relaxation factor when the optimum value isn t known for a flow problem with non constant velocities the present method becomes the iterative algorithm with different relaxation parameters for different points i e a local relaxation method it is demonstrated that in addition to helping choose the relaxation factors this idea may also substantially reduce the computing time compared with the iterative algorithm with the uniform relaxation factor for all the grid points

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

the benders decomposition algorithm a literature review the benders decomposition algorithm has been successfully applied to a wide range of difficult optimization problems this paper presents a state of the art survey of this algorithm emphasizing its use in combinatorial optimization we discuss the classical algorithm the impact of the problem formulation on its convergence and the relationship to other decomposition methods we introduce a taxonomy of algorithmic enhancements and acceleration strategies based on the main components of the algorithm the taxonomy provides the framework to synthesize the literature and to identify shortcomings trends and potential research directions we also discuss the use of the benders decomposition to develop efficient meta heuristics describe the limitations of the classical algorithm and present extensions enabling its application to a broader range of problems

an elementary survey of general duality theory in mathematical programming we survey some recent developments in duality theory with the idea of explaining and unifying certain basic duality results in both nonlinear and integer programming the idea of replacing dual variables prices by price functions suggested by everett and developed by gould is coupled with an appropriate dual problem with the consequence that many of the results resemble those used in linear programming the dual problem adopted has a traditional economic interpretation and dual feasibility then provides a simple alternative to concepts such as conjugate functions or subdifferentials used in the study of optimality in addition we attempt to make precise the relationship between primal dual and saddlepoint results in both the traditional lagrangean and the more general duality theories and to see the implications of passing from prices to price functions finally and perhaps surprisingly it appears that all the standard algorithms terminate by constructing primal and dual feasible solutions of equal value i e by satisfying generalised optimality conditions

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

weighted g drazin inverses and a new pre order on rectangular matrices abstract this paper deals with weighted g drazin inverses which is a new class of matrices introduced to extend to the rectangular case g drazin inverses recently considered by wang and liu for square matrices first we define and characterize weighted g drazin inverses next we consider a new pre order defined on complex rectangular matrices based on weighted g drazin inverses finally we characterize this pre order and relate it to the minus partial order and to the weighted drazin pre order

joint optimization of product family configuration and scaling design by stackelberg game abstract product family design is generally characterized by two types of approaches module based and scale based while the former aims to enable product variety based on module configuration the latter is to variegate product design by scaling up or down certain design parameters the prevailing practice is to treat module configuration and scaling design as separate decisions or aggregate two design problems as a single level all in one optimization problem in practice optimization of scaling variables is always enacted within a specific modular platform and meanwhile an optimal module configuration depends on how design parameters are to be scaled the key challenge is how to deal with explicitly the coupling of these two design optimization problems this paper formulates a stackelberg game theoretic model for joint optimization of product family configuration and scaling design in which a bilevel decision structure reveals coupled decision making between module configuration and parameter scaling a bilevel mixed 0 1 non linear programming model is developed and solved comprising an upper level optimization problem and a lower level optimization problem the upper level seeks for an optimal configuration of modules and module attributes by maximizing the shared surplus of an entire product family the lower level entails parametric optimization of attribute values for optimal technical performance of each individual module a case study of electric motors demonstrates that the bilevel joint optimization model excels in leveraging optimal scaling in conjunction with optimal module configuration which is advantageous over the existing paradigm of product family scaling design that cannot change the product family configuration

performance comparison of the nwf and dc methods for implementing high resolution schemes in a fully coupled incompressible flow solver this paper reports on the use of the normalized weighting factor nwf method and the deferred correction dc approach for the implementation of high resolution hr convective schemes in an implicit fully coupled pressure based flow solver four hr schemes are realized within the framework of the nwf and dc methods and employed to solve the following three laminar flow problems i lid driven flow in a square cavity ii sudden expansion in a square cavity and iii flow in a planar t junction over three grid systems with sizes of 104 5i 104 and 3i 105 control volumes the merit of both approaches is demonstrated by comparing the computational costs required to solve these problems using the various hr schemes on the different grid systems whereas previous attempts to use the nwf method in a segregated flow solver failed to produce converged solutions current results clearly demonstrate that both methods are suitable for utilization in a coupled flow solver in terms of cpu efficiency there is no global and consistent superiority of any method over another even though the dc method outperformed the nwf method in two of the three test problems solved

ray pattern of drazin inverses of some matrices abstract let q a be the set of complex matrices with the same ray pattern as a for any a q a if the drazin inverses of a and a have the same ray pattern then a is said to have ray drazin inverse in this paper we give some properties on matrices whose drazin inverses or m p inverses are ray unique and use these properties to obtain necessary and sufficient conditions for some block matrices to have ray drazin inverses

fuzzy multiple level programming the use of fuzzy set concept to overcome the computational difficulties of multi level optimization for large systems appears to be a powerful approach and can solve practical large problems in a very simplified manner furthermore unlike the duo ploy or multi ploy approaches the proposed interactive concept provides flexibility which is essential due to the different styles of management in different organizations the approach is formulated based on exploring the fuzziness or indefiniteness which is almost always present in a decentralized large hierarchy organization the advantage of this approach is that not only the problem is much more simplified for solution purposes the original problem is also much more realistically represented

logic based benders decomposition benders decomposition uses a strategy of learning from one s mistakes the aim of this paper is to extend this strategy to a much larger class of problems the key is to generalize the linear programming dual used in the classical method to an inference dual solution of the inference dual takes the form of a logical deduction that yields benders cuts the dual is therefore very different from other generalized duals that have been proposed the approach is illustrated by working out the details for propositional satisfiability and 0 1 programming problems computational tests are carried out for the latter but the most promising contribution of logic based benders may be to provide a framework for combining optimization and constraint programming methods

a computational study for bilevel quadratic programs using semidefinite relaxations abstract in this paper we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem for this purpose we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent karush kuhn tucker kkt conditions then we use the single level formulations to obtain mixed integer linear programming milp models and semidefinite programming sdp relaxations thus we compute optimal solutions and upper bounds using linear programming lp and sdp relaxations our numerical results indicate that the sdp relaxations are considerably tighter than the lp ones consequently the sdp relaxations allow finding tight feasible solutions for the problem especially when the number of variables in the leader problem is larger than in the follower problem moreover they are solved at a significantly lower computational cost for large scale instances

a study of finite difference approximations to steady state convection dominated flow problems abstract five different finite difference schemes first order upwind skew upwind second order upwind second order central differencing and quick approximating the convection terms in the transport equation with fluid motion have been studied three simple test problems are used to compare the performances by the five schemes for high cell peclet number flows they are also used to demonstrate the restraints on the accuracy of the numerical approximations set by the types of the boundary conditions by the presence of the source term in the flow region and by the skewness of the numerical grid lines the basic reasons behind the spurious oscillations in a numerical solution are studied among all five schemes studied the second order upwind is found to be in general the most satisfactory

minimal inequalities and subadditive duality in this note we use a duality theorem for mixed integer programs first explicitly stated by johnson 1973 for the one row group problem to characterize minimal inequalities this characterization extends earlier results which assumed either a rational or a bounded constraint set blair 1978 jeroslow 1979 johnson 1976 by relaxing these assumptions either entirely or almost so it also extends results given first by gomory and johnson 1969 for the group problem

a fuzzy approach for bi level integer non linear programming problem abstract bi level programming a tool for modeling decentralized decisions consists of the objective of the leader at its first level and that is of the follower at the second level integer programming deals with the mathematical programming problems in which some or all the variables are to be integer this paper studies a bi level integer non linear programming problem with linear or non linear constraints and in which the non linear objective function at each level are to maximized the bi level integer non linear programming bli nlp problem can be thought as a static version of the stackelberg strategy which is used leader follower game in which a stackelberg strategy is used by the leader or the higher level decision maker hldm given the rational reaction of the follower or the lower level decision maker lldm this paper proposes a two planner integer model and a solution method for solving this problem this method uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy max min decision model for generating pareto optimal solution for this problem an illustrative numerical example is given to demonstrate the obtained results

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

the polynomial hierarchy and a simple model for competitive analysis the multi level linear programs of candler norton and townsley are a simple class of sequenced move games in which players are restricted in their moves only by common linear constraints and each seeks to optimize a fixed linear criterion function in his her own continuous variables and those of other players all data of the game and earlier moves are known to a player when he she is to move the one player case is just linear programming

a fuzzy approach to hierarchical multiobjective programming problems and its application to an industrial pollution control problem in this paper we focus on hierarchical multiobjective programming problems where multiple decision makers in a hierarchical organization have their own multiple objective functions and propose an interactive algorithm based on the dual decomposition method to obtain the satisfactory solution which reflects not only the hierarchical relationships between multiple decision makers but also their own preferences for their objective functions in the proposed algorithm assuming that each of the decision makers has fuzzy goals for his her objective functions corresponding membership functions are elicited from the decision makers in their subjective manner in order to deal with hierarchical multiobjective programming problems a new kind of pareto optimality concept in membership spaces is defined and the concept of decision powers of multiple decision makers in a hierarchical decision structure is introduced the proposed algorithm is applied to the industrial pollution control problem in osaka city in japan in order to show the efficiency of the proposed method

comment to interactive fuzzy goal programming approach for bilevel programming problem by s r arora and r gupta the fuzzy bilevel programming problem is solved in the paper 1 using an approach of fuzzy goal programming using a simple example we show that this will not lead to a satisfactory solution especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example this constraint is used to model the hierarchy in the approach in 1 at the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute

a benders approach for the constrained minimum break problem this paper presents a hybrid ip cp algorithm for designing a double round robin schedule with a minimal number of breaks both mirrored and non mirrored schedules with and without place constraints are considered the algorithm uses benders cuts to obtain feasible home away pattern sets in few iterations and this approach leads to significant reductions in computation time for hard instances furthermore the algorithm is capable of solving a number of previously unsolved benchmark problems for the traveling tournament problem with constant distances

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

second order upwind and central difference schemes for recirculating flow computation two dimensional driven cavity flows with the reynolds number ranging from 102 to 3 2 x 10 3 are used to assess the performance of second order upwind and central difference schemes for the convection terms three different implementations of the second order upwind scheme are designed and tested in the context of the simple algorithm with the grid size varying from 21 x 21 to 161 x 161 uniformly spaced nodes converged solutions are obtained for all reynolds numbers although these different implementations of the second order upwind scheme have the same formal order of accuracy significant differences in numerical accuracy are observed it is demonstrated that better performance can be obtained for the second order upwind scheme if the discretization is cast in accordance with the finite volume formulation although both the second order upwind and central difference schemes exhibit no oscillations in the solution the upwind scheme is more accurate in assessing and comparing the performance of these schemes the distribution of cell reynolds number is discussed and its impact on numerical accuracy illustrated

on the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates abstract the issues of adopting the velocity components as dependent velocity variables including the cartesian and culvilinear ones for the navier stokes flow computations are investigated the viewpoint advocated is that a numerical algorithm should preferably honor both the physical conservation law in differential form and the geometric conservation law in discrete form it is demonstrated that with the curvilinear velocity vectors the curvatures of the grid lines introduce extra source terms into the governing equations with the cartesian velocity vector on the other hand the governing equations in curvilinear coordinates can retain the full conservation law form and honor the physical conservation laws the nonlinear combinations of the metric terms also cause the algorithms based on curvilinear velocity components to be more difficult to satisfy the geometric conservation law and hence more sensitive to grid skewness effect for the combined utilization of the cartesian velocity vector and the staggered grid arrangement the implications of spurious pressure oscillation arising from the 90 turning are discussed it is demonstrated that these spurious oscillations can possibly appear only under a very specific circumstance namely the meshes in the region with 90 turning must be parallel to the cartesian coordinates and of uniform spacing along coordinates otherwise no spurious oscillations can appear several flow solutions for domain with 90 and 360 turnings are presented to demonstrate that satisfactory results can be obtained by using the cartesian velocity components and the staggered grid arrangement

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

assessment of finite difference approximations for the advection terms in the simulation of practical flow problems abstract an assessment of seven discretization schemes for the advection terms of the transport equation to reduce numerical diffusion in practical flow problems has been established the schemes have been evaluated using three test cases for laminar flow problems the test cases consist of the transport of a scalar step in a uniform velocity field two interacting parallel streams and a slot jet the performance of the schemes is evaluated for advection dominated flows in transient and steady state solutions the considered schemes include four that have not been evaluated before for practical flow problems in general schemes which produced less numerical diffusion suffered from more numerical dispersion or oscillations two bounding techniques considered in the study were effective in significantly elliminating numerical dispersion

on the computational efficiency of certain upwinding schemes abstract the computational efficiency of certain upwinding schemes for the discretization of the one dimensional advection dominated transport equation are examined in particular the first order upwind the second order upwind the weighted scheme and the quick scheme are considered the total computational cost for a prescribed accuracy of these schemes was compared using a general purpose algorithm for the integration of time dependent partial differential equations using the method of lines the method of lines approach enables us to utilize the well developed methodology for the assessment of computational efficiency of ode integrators moreover in the time integration the resulting system of odes is integrated by an efficient variable order stepsize numerical integration scheme the results indicate that for small cell peclet number central differences are superior for moderate values of the cell peclet number the quick scheme is the most attractive under high advective flow conditions however the second order upwinding is the most reliable

the value function of an integer program we consider integer programs in which the objective function and constraint matrix are fixed while the right hand side varies the value function gives for each feasible right hand side the criterion value of the optimal solution we provide a precise characterization of the closed form expression for any value function

interactive random fuzzy two level programming through possibility based probability model this paper considers interactive decision making methods for random fuzzy two level linear programming problems assuming that the decision makers concern about the probabilities that their own objective function values are smaller than or equal to certain target values fuzzy goals of the decision makers for the probabilities are introduced then the possibility based probability model to maximize the degrees of possibility with respect to the attained probability is considered interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented an illustrative numerical example demonstrates the feasibility and efficiency of the proposed method

compensatory fuzzy multiple level decision making abstract fuzzy set theory has been shown to be an effective tool to overcome the computational difficulties encountered in solving large multiple level programming problems shih et al 1996 in this paper compensatory operators are introduced for adjusting the decision making process between the different levels and also between the decision makers of the same level after a brief consideration of the bi level and three level systems the large decentralized organizations with both equal and unequal goals are investigated various numerical examples are given to compare the influences of compensation and to illustrate the approaches

on two level optimization decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

combinatorial benders cuts for the strip packing problem we study the strip packing problem in which a set of two dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height with the aim of minimizing the height used the problem is important because it models a large number of real world applications including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste scheduling problems in which tasks require a contiguous subset of identical resources and container loading problems arising in the transportation of items that cannot be stacked one over the other the strip packing problem has been attacked in the literature with several heuristic and exact algorithms nevertheless benchmark instances of small size remain unsolved to proven optimality in this paper we propose a new exact method that solves a large number of the open benchmark instances within a limited computational effort our method is based on a benders decomposition in whi

wavenumber extended high order upwind biased finite difference schemes for convective scalar transport this paper proposes some new wavenumber extended high order upwind biased schemes the dispersion and dissipation errors of upwind biased finite difference schemes are assessed and compared by means of a fourier analysis of the difference schemes up to 11th order upwind biased schemes are analyzed it is shown that both the upwind biased scheme of order 2n 1 and the corresponding centered differencing scheme of order 2nhave the same dispersion characteristics thus the former can be considered to be the latter plus a correction that reduces the numerical dissipation the new second order wavenumber extended scheme is tested and compared with some well known schemes the range of wavenumbers that are accurately treated by the upwind biased schemes is improved by using additional constraints from the fourier analysis to construct the new schemes the anisotropic behavior of the dispersion and dissipation errors is also analyzed for both the conventional and the new wavenumber extended upwind biased finite difference schemes

the χ schemes a new consistent high resolution formulation based on the normalized variable methodology this paper deals with the formulation and testing of a new class of consistent high resolution schemes denoted as the schemes these schemes combine consistency accuracy and boundedness across systems of equations and are suitable for use in the simulation of multi phase and multi component flows the consistency feature refers to the capability of these schemes to implicitly satisfy the additional algebraic constraint representing a global conservation relation governing certain sets of equations e g species mass fraction volume fraction etc four schemes are implemented within an unstructured grid finite volume framework tested by solving four multi component pure advection test problems and shown to be consistent

second order upwind differencing in a recirculating flow in this study we have investigated the accuracy and stability of the second order upwind differencing scheme originally suggested by price et al and recently used by several others the solution algorithm employed is based on a coupled solution of the nonlinear finite difference equations by the multigrid technique calculations have been made of the driven cavity flow for several reynolds numbers and finite difference grids we observe that in comparison with the hybrid differencing the second order upwind differencing is somewhat more accurate but that it is not monotonically accurate with mesh refinement also the convergence of the solution algorithm deteriorates with the use of the second order upwind differencing nomenclature a finite difference coefficient c convective flux d diffusive flux d cavity depth f right hand side of the equation l f i restriction prolongation operator p pressure r residual in the solution re reynolds number uw d v s source term u ar direction velocity uw top wall u velocity d direction velocity x y coordinate directions e error tolerance t smoothing factor v fluid kinematic viscosity p fluid density general transport variable

planning and scheduling by logic based benders decomposition we combine mixed integer linear programming milp and constraint programming cp to solve an important class of planning and scheduling problems tasks are allocated to facilities using milp and scheduled using cp and the two are linked via logic based benders decomposition tasks assigned to a facility may run in parallel subject to resource constraints cumulative scheduling we solve problems in which the objective is to minimize cost makespan or total tardiness we obtain significant computational speedups of several orders of magnitude for the first two objectives relative to the state of the art in both milp and cp we also obtain better solutions and bounds for problems than cannot be solved to optimality

equilibrium programming the path following approach and dynamics a general formulation of equilibrium is introduced which subsumes a large range of equilibrium models in economics mathematical programming game theory networks organizations etc the existence of a solution is proved by path following and without invocation of a fixed point theorem the path following procedure provides new means for establishing existence of equilibria for constructing them and for interpreting them a dynamic equilibrium model is also presented and the existence stability and computation of dynamic equilibrium paths are discussed moreover these paths can often be interpreted as adjustment processes

a bm lambda cut and goal programming based algorithm for fuzzy linear multiple objective bilevel optimization bilevel programming techniques are developed to handle decentralized problems with two level decision makers which are leaders and followers who may have more than one objective to achieve this paper proposes a cut and goal programming based algorithm to solve fuzzy linear multiple objective bilevel flmob decision problems first based on the definition of a distance measure between two fuzzy vectors using cut a fuzzy linear bilevel goal flbg model is formatted and related theorems are proved then using a cut for fuzzy coefficients and a goal programming strategy for multiple objectives a cut and goal programming based algorithm to solve flmob decision problems is presented a case study for a newsboy problem is adopted to illustrate the application and executing procedure of this algorithm finally experiments are carried out to discuss and analyze the performance of this algorithm

an adaptive grid method for navier stokes flow computation a study has been conducted to investigate the adaptive grid method based on the concept of equidistribution applied to navier stokes flow computations it is demonstrated that due to the fundamental differences between the structure of the pressure field and the velocity field an adaptive grid strategy that employs multiple one dimensional grid adjustment procedures using different weighting functions for different directions is useful a consistency check is devised to evaluate the a posteriori error of a given numerical solution two different flow configurations with reynolds number 500 have been calculated on several grid systems with different nodal points as well as different distributions it is shown that as a result of this adaptive grid method the numerical accuracy obtained with given number of nodal points can be improved it is demonstrated that without attempting simultaneously to extremize several measures in the whole flow domain the present multiple one dimensional adaptive grid method enables one to concentrate the grids more easily in the regions needed this method should prove useful for many navier stokes flow calculations

needed an empirical science of algorithms deductive algorithmic science has reached a high level of sophistication but its worst case and average case results seldom tell us how well an algorithm is actually going to work in practice i argue that an empirical science of algorithms is a viable alternative i respond to misgivings about an empirical approach including the prevalent notion that only a deductive treatment can be theoretical or sophisticated np completeness theory for instance is interesting partly because it has significant if unacknowledged empirical content an empirical approach requires not only rigorous experimental design and analysis but also the invention of empirically based explanatory theories i give some examples of recent work that partially achieves this aim

a solution method for the linear static stackelberg problem using penalty functions a new solution technique is presented for the linear constrained static stackelberg problem for a given value of x the leader s decision vector the follower is at its rational reaction set when the duality gap of the second level problem becomes zero the outer problem is solved by appending to the leader s objective a function that minimizes the duality gap of the follower s problem this structure leads to the decomposition of the composite problem into a series of linear programs leading to an efficient algorithm it is proved that optimality is reached for an exact penalty function and the method is illustrated with some examples

gauss jordan elimination method for computing outer inverses this paper deals with the algorithm for computing outer inverse with prescribed range and null space based on the choice of an appropriate matrix g and gauss jordan elimination of the augmented matrix g i the advantage of such algorithms is the fact that one can compute various generalized inverses using the same procedure for different input matrices in particular we derive representations of the moore penrose inverse the weighted moore penrose inverse the drazin inverse as well as 2 4 and 2 3 inverses numerical examples on different test matrices are presented as well as the comparison with well known methods for generalized inverses computation

comparison of passive scalar transport models coupled with the lattice boltzmann method abstract a complex fluid flow often involves passive scalars that are transported with the flow field but can be neglected regarding the fluid motion such as concentration distributions markers or temperature fields using the lattice boltzmann lb method for fluid dynamic simulations different approaches for coupling advection solvers to the fluid motion exist the aim of this work is to give an overview of four different methods used for solving the advection equation two of these solvers are based on finite differences namely the lax wendroff method lax and wendroff 1960 and an extension from succi et al 1999 that artificially reduces numerical diffusion the other two methods onishi et al 2005 and osmanlic and korner 2016 take advantage of the already available lb distribution functions instead of depending on macroscopic quantities only in the ansatz of onishi et al 2005 the local concentration flux is calculated from distribution functions directly while in a similar approach used by osmanlic and korner 2016 an additional case distinction is made between incoming and outgoing flux besides investigating the order of accuracy and grid dependency numerical diffusion is examined closely with the objective to simulate fluid flow that is solely determined by advection and contains no physical diffusion

fuzzy goal programming algorithm for solving decentralized bi level multi objective programming problems this paper presents fuzzy goal programming fgp algorithm for solving decentralized bi level multi objective programming dbl mop problems with a single decision maker at the upper level and multiple decision makers at the lower level a fgp algorithm for dbl multi objective linear programming dbl molp problems is proposed this algorithm is extended to solve bi level multi objective linear fractional programming dbl molfp problems in the proposed algorithm the membership functions for the defined fuzzy goals of all objective functions at the two levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by uldm are developed in the model formulation of the problem then fgp approach is used to achieve highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers illustrative numerical examples are given to demonstrate the proposed algorithm

weighting method for bi level linear fractional programming problems in this paper we consider the solution of a bi level linear fractional programming problem bllfpp by weighting method a non dominated solution set is obtained by this method in this article decision makers dms provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control we convert the hierarchical system into scalar optimization problem sop by finding proper weights using the analytic hierarchy process ahp so that objective functions of both levels can be combined into one objective function here the relative weights represent the relative importance of the objective functions

generalized benders decomposition for one class of minlps with vector conic constraint in this paper we mainly study one class of mixed integer nonlinear programming problems minlps with vector conic constraint in banach spaces duality theory of convex vector optimization problems applied to this class of minlps is deeply investigated with the help of duality we use the generalized benders decomposition method to establish an algorithm for solving this minlp several convergence theorems on the algorithm are also presented the convergence theorems generalize and extend the existing results on minlps in finite dimensional spaces

on the suppression of numerical oscillations using a non linear filter the idea of using a non linear filtering algorithm to eliminate numerically generated oscillations is investigated a detailed study is conducted to follow the development of numerical oscillations and their interaction with the filter a relaxation procedure is also proposed to enhance the effectiveness of the filter three model problems a 2d steady state scalar convection diffusion equation a 1d unsteady gas dynamics flow with shock and a 1d linear wave equation have been designed to test the performance of the filtering algorithm the effectiveness of the filter is assessed for convection schemes of different dispersive and diffusive characteristics demonstrating that it is effective in eliminating oscillations with short wavelength but oscillations of longer wavelengths are virtually unaffected it is concluded that a proper combination of non linear filter and dispersive numerical scheme is a viable alternative to dissipative schemes in resolving flows with sharp gradients and discontinuities