Bogdana Stanojević

Enhanced savings calculation and its applications for solving capacitated vehicle routing problem

The vehicle routing problem (VRP) is one of the most explored combinatorial problems in operations research. A very fast and simple algorithm that solves the VRP is the well known Clarke-Wright savings algorithm. In this paper we introduce a new way of merging routes and the corresponding formula for calculating savings. We also apply the enhanced merging to develop a new heuristic - Extended Savings Algorithm (ESA) that dynamically recalculates savings during iterations. Computational results show that, on average ESA gives better solutions than the original savings algorithm. Implementing randomization of some steps of our heuristic we obtained even better results which competes with more complex and well known heuristics. The ESA is further used to generate good routes as part of a set-covering-based algorithm for the Capacitated VRP (CVRP). The numerical results of our experiments are reported.

On the cardinality of the nondominated set of multi-objective combinatorial optimization problems

Abstract In this paper we formulate and prove two upper bounds for the nondominated set of multi-objective combinatorial optimization problems with integer coefficients. We also show that under certain assumptions, reasonable and applicable in the majority of practical problems, the number of nondominated points grows following a polynomial function.

A note on ‘Taylor series approach to fuzzy multiple objective linear fractional programming’

Abstract Toksari, in his paper ‘Taylor series approach to fuzzy multiobjective linear fractional programming’, published in Information Sciences 178 (2008), proposed a new method to solve multiple objective linear fractional programming problems and declared that it is more effective when compare to previous methods. He constructed fuzzy goals and then used first order Taylor-polynomials to approximate the corresponding linear fractional membership functions by linear functions. Aggregating the linear functions Toksari obtained a crisp linear programming problem and claimed that it is ‘equal’ to the fuzzy fractional one. In this paper, we indicate the fallacy that arises by using Taylor approximation and propose an improved method that guarantees the efficiency of the solution it provides. Both practical applications from Toksari’s paper are recalled to show that the improvements we suggest are effective.

On the ratio of fuzzy numbers – exact membership function computation and applications to decision making

AbstractIn the present paper, we propose a new approach to solving the full fuzzy linear fractional programming problem. By this approach, we provide a tool for making good decisions in certain problems in which the goals may be modelled by linear fractional functions under linear constraints; and when only vague data are available. In order to evaluate the membership function of the fractional objective, we use the α-cut interval of a special class of fuzzy numbers, namely the fuzzy numbers obtained as sums of products of triangular fuzzy numbers with positive support. We derive the α-cut interval of the ratio of such fuzzy numbers, compute the exact membership function of the ratio, and introduce a way to evaluate the error that arises when the result is approximated by a triangular fuzzy number. We analyse the effect of this approximation on solving a full fuzzy linear fractional programming problem. We illustrate our approach by solving a special example – a decision-making problem in production planning.

Comment on “Fuzzy mathematical programming for multi objective linear fractional programming problem”

Abstract Chakraborty and Gupta, in their paper “Fuzzy mathematical programming for multi objective linear fractional programming problem”, published in Fuzzy Sets and Systems 125 (2002), claimed that their methodology proposed for solving multi objective linear fractional programming problem always yields an efficient solution. This paper indicates that their claim is generally wrong and this is due to the non-equivalence of the original problem and the associated linear problem.

Extended procedure for computing the values of the membership function of a fuzzy solution to a class of fuzzy linear optimization problems

In 1994, Chanas and Kuchta suggested one approach to determine the membership function of a fuzzy solution to the fuzzy linear optimization problem with fuzzy coefficients in the objective function, based on computing the sum of lengths of certain intervals. In 2012, Dempe and Ruziyeva introduced a methodology for realizing Chanas and Kuchtas idea, and derived explicit formulas for computing the endpoints of the suggested intervals in the particular case of triangular fuzzy numbers. The purpose of this paper is to extend Dempe and Ruziyevas approach by handling the possible degeneracy of basic feasible solutions, and derive new formulas for computing the values of the membership function. The special example considered by Dempe and Ruziyeva is again used to illustrate the relevance of the extended solving procedure and new formulas.

Optimization of thresholds in serial multimodal biometric systems

Multimodal biometric verification systems use information from several biometric modalities to verify an identity of a person. The false acceptance rate (FAR) and false rejection rate (FRR) are metrics generally used to measure the performance of such systems. In this paper we propose a novel approach to determine the upper and lower acceptance thresholds in sequential multimodal biometric matching, in such a way that the expected values of FAR and FRR for the entire system are minimized. We linearize locally the score distributions of both genuine users and impostors using the least squares method, and derive formulas for the approximated FAR and FRR for each matcher. Further, we aim to minimize both probabilities for entire processing chain. In order to find the best compromise between them, we analyze the efficient solutions to the associated bi-objective programming problem. The results of our experiments are also reported in the paper. They showed a good performance of the sequential multiple biometric matching system based on optimized thresholds comparing with the widely adopted parallel fusion multimodal biometric systems.

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.

Set-Covering-Based Approximate Algorithm Using Enhanced Savings for Solving Vehicle Routing Problem

Transportation is widely present in human activities and supports many economic activities. Using phones, reading mails, traveling, and flying involve the routing of messages, people, and goods. One of the present aims of research is to fill the gap between academic research and practical applications. Our aim is to present a simple and flexible heuristic for solving the capacitated vehicle routing problem and heterogeneous fleet vehicle routing problems, and discuss its advantages compared to other well-known heuristics. The flexibility of our approach comes from the simplicity of the solution procedure and is especially important when the algorithm is going to be applied to solving real-life problems.