Fuzzy Relation Equations (II): The Branch-point-solutions and the Categorized Minimal Solutions

This paper presents some novel theoretical results as well as practical algorithms and computational procedures on fuzzy relation equations (FRE). These results refine and improve what has already been reported in a significant manner. In the previous paper, the authors have already proved that the problem of solving the system of fuzzy relation equations is an NP-hard problem. Therefore, it is practically impossible to determine all minimal solutions for a large system if P ≠ NP. In this paper, an existence theorem is proven: there exists a special branch-point-solution that is greater than all minimal solutions and less than the maximum solution. Such branch-point-solution can be calculated based on the solution-base-matrix. Furthermore, a procedure for determining all branch-point-solutions is designed. We also provide efficient algorithms which is capable of determining as well as searching for certain types of minimal solutions. We have thus obtained: (1) a fast algorithm to determine whether a solution is a minimal solution, (2) the algorithm to search for the minimal solutions that has at least a minimum value at a component in the solution vector, and (3) the procedure of determining if a system of fuzzy relation equations has the unique minimal solution. Other properties are also investigated.     

An Efficient Procedure for Solving a Fuzzy Relational Equation With Max–Archimedean t-Norm Composition

In the literature, a necessary condition for minimal solutions of a fuzzy relational equation with max-product composition shows that each of its components is either zero or the corresponding component's value of the greatest solution. In this paper, we first extend this necessary condition to the situation with max-Archimedean triangular-norm (t-norm) composition. Based on this necessary condition, we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently. Furthermore, rather than work with the actual equations, we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions. Numerical examples with specific cases of the max-Archimedean t-norm composition are provided to illustrate the procedure.     

An algorithm for solving fuzzy relation equations with max-T composition operator

This paper studies the problem of solving a max-T composite finite fuzzy relation equation, where T is a special class of pseudo-t-norms. If the equation is solvable, then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions. Some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value. Under these conditions, some minimal solutions of the system can be obtained easily. Some procedures are also proposed in order to simplify the original system. The simplified system is then decomposed (if possible) into several subsystems with smaller dimensions, which are very easy to solve. Furthermore, a method is presented to solve each subsystem. By combining the method and those procedures, an efficient algorithm is proposed to obtain the set of feasible solutions of the original system. Two examples are also given to illustrate the algorithm.     

Solving equations with sequence variables and sequence functions

Term equations involving individual and sequence variables and sequence function symbols are studied. Function symbols can have either fixed or flexible arity. A sequence variable can be instantiated by any finite sequence of terms. A sequence function abbreviates a finite sequence of functions all having the same argument lists. It is proved that solvability of systems of equations of this form is decidable. A new unification procedure that enumerates a complete almost minimal set of solutions is presented, together with variations for special cases. The procedure terminates if the solution set is finite. Applications in various areas of artificial intelligence, symbolic computation, and programming are discussed.     

Optimization of fuzzy relational equations with max-av composition

Max-min and max-product compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations. Both are members in the class of max-t-norm composition. In this study, the max-av composition is considered for the same optimization model, which does not belong to the max-t-norm composition. However, max-av composition generates some properties of the solution set that are similar to the max-product composition. Thanks to these properties, a simple value matrix with rules can be applied to reduce problem size. Thus, this study proposes an efficient procedure for obtaining optimal solutions without decomposing the problem into two sub-problems or finding all the potential minimal solutions.     

Remarks on minimal solutions of a system of fuzzy relation equations over complete infinitely distributive lattices

In Section 3.6 of Fuzzy relation equations and their applications to knowledge engineering. Kluwer Academic Publishers, Boston (1989), Di Nola et al. presented a procedure to find a minimal solution from a fixed solution of a system of fuzzy relation equations over complete infinitely distributive lattices, and put the question: is the minimal solution found by the procedure unique or not? In this paper, we give a negative answer to the question and make some further remarks. We not only give a necessary and sufficient condition for the uniqueness of such minimal solutions, but also characterize the existence of the least solution and a unique solution of a system of fuzzy relation equations over complete infinitely distributive lattices.     

Application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations

A novel approach to the numerical solution of weakly singular Volterra integral equations is presented using the C^∞ multiquadric (MQ) radial basis function (RBF) expansion rather than the more traditional finite difference, finite element, or polynomial spline schemes. To avoid the collocation procedure that is usually ill-conditioned, we used a global minimization procedure combined with the method of successive approximations that utilized a small, finite set of MQ basis functions. Accurate solutions of weakly singular Volterra integral equations are obtained with the minimal number of MQ basis functions. The expansion and optimization procedure was terminated whenever the global errors were less than 5 · 10⁻⁷.     

Least order,stable solution of the exact model matching problem

The set of all solutions of the minimal design problem (MDP) is presented in parametric form. This result is obtained by first deriving a parametric representation of all minimal bases of a particular vector space. From this the set of all solutions of the MDP are obtained. Then conditions are placed on the parameters, which conditions define the set of stable solutions of the MDP.Solutions to the nonminimal model matching problem are also presented in parametric form, and it is shown how in principle solutions of successively higher degree can be searched for a stable solution, should the MDP have no stable solution, so that a solution to the model matching problem can be found which has the lowest order consistent with a stability constraint.     

Facility location problem on a network under multiple criteria—fuzzy set theoretic approach

The investigation deals with the optimal location of a number p of facilities at the nodes of a given network under multiple criteria. The 0–1 algorithm is used to generate the set of non-dominated solutions. A fuzzy set theoretic methodology is used for specifying the preferred solution from the set of non-dominated solutions. The solution procedure is illustrated with the help of a numerical example.     

Infinite fuzzy relation equations with continuous t-norms

This study develops a concept of infinite fuzzy relation equations with a continuous t-norm. It extends the work by Xiong and Wang [Q.Q. Xiong, X.P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets and Systems 151 (2005) 393-402]. We describe attainable (respectively, unattainable) solutions, which are closely related to minimal solutions to the equations. It is shown that a solution set comprises both attainable and unattainable solutions. The study offers a characterization of these solutions. Under some assumptions, the solution set is presented and discussed. Two applications are also given.     

Performance bounds on the splitting algorithm for binary testing

Summary In machine fault-location, medical diagnosis, species identification, and computer decisionmaking, one is often required to identify some unknown object or condition, belonging to a known set of M possibilities, by applying a sequence of binary-valued tests, which are selected from a given set of available tests. One would usually prefer such a testing procedure which minimizes or nearly minimizes the expected testing cost for identification. Existing methods for determining a minimal expected cost testing procedure, however, require a number of operations which increases exponentially with M and become infeasible for solving problems of even moderate size. Thus, in practice, one instead uses fast, heuristic methods which hopefully obtain low cost testing procedures, but which do not guarantee a minimal cost solution. Examining the important case in which all M possibilities are equally likely, we derive a number of cost-bounding results for the most common heuristic procedure, which always applies next that test yielding maximum information gain per unit cost. In particular, we show that solutions obtained using this method can have expected cost greater than an arbitrary multiple of the optimal expected cost.The authors are indebted to R. L. Rivest for simplying the original proof of Theorem 1 and to P. J. Burke for his many valuable comments.     

An algorithm for constructing the basis of the solution set for systems of linear Diophantine equations over the ring of integers

A polynomial algorithm is proposed to construct the minimal generating set of solutions and the basis of the solution set for systems of linear Diophantine equations over the ring of integer. The algorithm is based on a modified TSS method.     

Efficient heuristic algorithms for the weighted set covering problem

In this paper two heuristic algorithms are presented for the weighted set covering problem. The first algorithm uses a simple, polynomial procedure to construct feasible covering solutions. The procedure is shown to possess a worst case performance bound that is a function of the size of the problem. The second algorithm is a solution improvement procedure that attempts to form reduced cost composite solutions from available feasible covering solutions. Computational results are presented for both algorithms on several large set covering problems generated from airline crew scheduling data.     

Path relinking for the vertex separator problem

This paper presents the first population-based path relinking algorithm for solving the NP-hard vertex separator problem in graphs. The proposed algorithm employs a dedicated relinking procedure to generate intermediate solutions between an initiating solution and a guiding solution taken from a reference set of elite solutions (population) and uses a fast tabu search procedure to improve some selected intermediate solutions. Special care is taken to ensure the diversity of the reference set. Dedicated data structures based on bucket sorting are employed to ensure a high computational efficiency. The proposed algorithm is assessed on four sets of 365 benchmark instances with up to 20,000 vertices, and shows highly comparative results compared to the state-of-the-art methods in the literature. Specifically, we report improved best solutions (new upper bounds) for 67 instances which can serve as reference values for assessment of other algorithms for the problem.     

A genetic algorithm approach for multi-objective optimization of supply chain networks

Supply chain network (SCN) design is to provide an optimal platform for efficient and effective supply chain management. It is an important and strategic operations management problem in supply chain management, and usually involves multiple and conflicting objectives such as cost, service level, resource utilization, etc. This paper proposes a new solution procedure based on genetic algorithms to find the set of Pareto-optimal solutions for multi-objective SCN design problem. To deal with multi-objective and enable the decision maker for evaluating a greater number of alternative solutions, two different weight approaches are implemented in the proposed solution procedure. An experimental study using actual data from a company, which is a producer of plastic products in Turkey, is carried out into two stages. While the effects of weight approaches on the performance of proposed solution procedure are investigated in the first stage, the proposed solution procedure and simulated annealing are compared according to quality of Pareto-optimal solutions in the second stage.     

Semidefinite programming for uncertain linear equations in static analysis of structures

This paper presents a method for computing a minimal bound that contains the solution set to the uncertain linear equations. Particularly, we aim at finding a bounding ellipsoid for static response of structures, where both external forces and elastic moduli of members are assumed to be imprecisely known and bounded. By using the S-lemma, we formulate a semidefinite programming (SDP) problem which provides an outer approximation of the minimal bounding ellipsoid. Bounding ellipsoids are computed for nodal displacements of uncertain braced frames as the solutions of the presented SDP problems by using the primal-dual interior-point method.     

An iterative approach to robust and integrated aircraft routing and crew scheduling

In airline scheduling a variety of planning and operational decision problems have to be solved. We consider the problems aircraft routing and crew pairing: aircraft and crew must be allocated to flights in a schedule in a minimal cost way. Although these problems are not independent, they are usually formulated as independent mathematical optimisation models and solved sequentially. This approach might lead to a suboptimal allocation of aircraft and crew, since a solution of one of the problems may restrict the set of feasible solutions of the problem solved later. Also, when minimal cost solutions are used in operations, a short delay of one flight can cause very severe disruptions of the schedule later in the day. We generate solutions that incur small costs and are also robust to typical stochastic variability in airline operations. We solve the two original problems iteratively. Starting from a minimal cost solution, we produce a series of solutions which are increasingly robust. Using data from domestic airline schedules we evaluate the benefits of the approach as well as the trade-off between cost and robustness. We extend our approach considering the aircraft routing problem together with two crew pairing problems, one for technical crew and one for flight attendants.     

Local branching-based algorithms for the disjunctively constrained knapsack problem

The disjunctively constrained knapsack problem (DCKP) is a variant of the well-known single constrained knapsack problem with special disjunctive constraints. This paper investigates the use of the local branching techniques for solving large-scale DCKP. Three versions of the algorithm are considered. The first version is based on the standard local branching which uses a starting solution provided by a specialized rounding solution procedure. The second version applies a two-phase solution procedure embedded in the local branching. For each subtree, the procedure serves to construct the set of objects containing the assigned variables and a second set including the free variables. The first set provides a partial local solution to the DCKP, whereas, for the second set, a truncated exact tree-search is applied for completing the partial local feasible solution. Finally, a diversification strategy is considered constituting the third version of the algorithm. All versions of the proposed algorithm are computationally analyzed on a set of benchmark instances of the literature and the obtained solutions are compared to those provided by existing algorithms. Encouraging results have been obtained.     

Error estimates for discretized differential inclusions

We present an estimate for the Hausdorff distance between the set of solutions of a differential inclusion and the set of solutions of its Euler discrete approximation, using an averaged modulus of continuity for multifunctions. A computational procedure to obtain a certain solution of the discretized inclusion is proposed.     

A path cost-based GRASP for minimum independent dominating set problem

The minimum independent dominating set problem (MIDS) is an extension of the classical dominating set problem with wide applications. In this paper, we describe a greedy randomized adaptive search procedure (GRASP) with path cost heuristic for MIDS, as well as the classical tabu mechanism. Our novel GRASP algorithm makes better use of the vertex neighborhood information provided by path cost and thus is able to discover better and more solutions and to escape from local optimal solutions when the original GRASP fails to find new improved solutions. Moreover, to further overcome the serious cycling problem, the tabu mechanism is employed to forbid some just-removed vertices back to the candidate solution. Computational experiments carried out on standard benchmarks, namely DIMACS instances, show that our algorithm consistently outperforms two MIDS solvers as well as the original GRASP.