Linear Ranking Functions Research Articles

Addressing Fuzzy Linear Programming Problems Through Ranking Functions

In contemporary decision-making, reliance on information is paramount, yet much of it is fraught with uncertainty, making logical decision-making challenging. To tackle this uncertainty, various methods, including the use of fuzzy numbers, have been employed. This paper specifically delves into addressing linear programming (LP) problems characterized by fuzzy coefficients in the objective function, fuzzy values in the right-hand side, and fuzzy coefficients of constraints. The proposed approach involves employing linear ranking functions such as Maleki, Campos, Yager’s F1 and Yager linear ranking functions to solve these fuzzy linear programming (FLP) problems and attain optimal solutions. Furthermore, the paper elucidates the resolution steps through the presentation of numerical examples, in this study, a comprehensive methodology is presented for effectively addressing a wide range of linear programming problems.

A multi-objective berth allocation problem in fuzzy environment

The Berth Allocation Problem (BAP) is an important seaside problem in port logistics, which involves the determination of berthing times and positions of arriving vessels in a wharf. Most studies have considered that vessels arrival and handling times are known fixed quantities available to wharves operators beforehand. However, due to many uncontrollable factors that can affect vessels arrival and handling times, those quantities are highly uncertain and may have a significant impact in the wharf operation. This paper takes fuzzy uncertainty into account and presents a fully fuzzy BAP with two objective functions. Specifically, it is considered the minimisation of the total waiting time of vessels and the makespan of the wharf operation. The problem is solved by using two lexicographic methods for fully fuzzy multi-objective linear programming (FFMOLP). Multiple Pareto optimal fuzzy solutions are obtained using a fuzzy epsilon-constraint method. Results demonstrate the effectiveness of the proposed approaches in handling fuzziness and conflicting objectives in a BAP. Moreover, from a methodological point of view, results show that it is best to use lexicographic ranking criteria instead of linear ranking functions for solving the associated FFMOLP problem.

On the Application of a Lexicographic Method to Fuzzy Linear Programming Problems

The literature on fuzzy variable linear programming (FVLP) and fuzzy number linear programming (FNLP) is prolific in terms of the number of available solution methods. In FVLP problems, only the decision variables and right-hand side values of the constraints are fuzzy numbers. In FNLP problems, except forthe decision variables, all parameters are fuzzy numbers. A widespread approach for solving problemsin FVLP and FNLP is to use linear ranking functions in order to transform the fuzzy problems into conventional ones. Previous studies have shown that linear ranking functions do not guarantee uniqueness of the optimal fuzzy objective values. In this paper, we use a lexicographic method to find unique optimal fuzzy objective values of such problems and compare the results with those obtained via linear ranking function approaches. The paper also discusses applications of the lexicographic method in diet and time–cost trade-off problems in fuzzy environments.

Learning to Resolve Conflicts for Multi-Agent Path Finding with Conflict-Based Search

Conflict-Based Search (CBS) is a state-of-the-art algorithm for multi-agent path finding. On the high level, CBS repeatedly detects conflicts and resolves one of them by splitting the current problem into two subproblems. Previous work chooses the conflict to resolve by categorizing conflicts into three classes and always picking one from the highest-priority class. In this work, we propose an oracle for conflict selection that results in smaller search tree sizes than the one used in previous work. However, the computation of the oracle is slow. Thus, we propose a machine-learning (ML) framework for conflict selection that observes the decisions made by the oracle and learns a conflict-selection strategy represented by a linear ranking function that imitates the oracle's decisions accurately and quickly. Experiments on benchmark maps indicate that our approach, ML-guided CBS, significantly improves the success rates, search tree sizes and runtimes of the current state-of-the-art CBS solver.

Solving Fuzzy Bi-Criteria Fixed Charge Transportation Problem Using a New Fuzzy Algorithm

The bi-criteria fixed charge transportation problem is an extension of the classical transportation problem. The bi-criteria fixed charge transportation problem in a crisp environment is, often, not effective in dealing with imprecision or vagueness in the values of the problem parameters. To deal with such situations, it has been proposed that the parameters should be represented as fuzzy numbers. Hence, bi-criteria fixed charge transportation problem in fuzzy environment is considered here. In existing approaches, the programming problems in fuzzy environment are solved by converting them into crisp environment by choosing appropriate membership functions and thus, the solutions are also crisp numbers. However, in this paper, a new algorithm is proposed to solve the above said problem using a linear ranking function, without converting it into crisp environment and the solutions derived are fuzzy numbers. The algorithm is suitably illustrated with a numerical example. Numerical results are compared for both fuzzy and crisp versions of the problem.

A Fuzzy C-means Algorithm for Clustering Fuzzy Data and Its Application in Clustering Incomplete Data

The fuzzy c-means clustering algorithm is a useful tool for clustering; but it is convenient only for crisp complete data. In this article, an enhancement of the algorithm is proposed which is suitable for clustering trapezoidal fuzzy data. A linear ranking function is used to define a distance for trapezoidal fuzzy data. Then, as an application, a method based on the proposed algorithm is presented to cluster incomplete fuzzy data. The method substitutes missing attribute by a trapezoidal fuzzy number to be determined by using the corresponding attribute of q nearest-neighbor. Comparisons and analysis of the experimental results demonstrate the capability of the proposed method.

Solving fully fuzzy multi-objective linear programming problem with fuzzy dominant degrees

In this paper, we investigate all relative relationships between two fuzzy numbers. Then, we introduce new relative measures to compare two fuzzy numbers instead of using absolute value to represent the fuzzy number. These measures address the dominant level that one fuzzy number is better than the other in terms of its position and shape. The so-called absolute fuzzy dominant degree and relative fuzzy dominant degree are developed to measure the differences between two fuzzy numbers applying for different types of constraint. These measures could capture all the shape’s characteristics and relative positions of fuzzy numbers. Finally, the fully fuzzy multi-objective decision making (FFMODM) problem is solved by using these fuzzy dominant degrees. For validation, we compare our approach to the fuzzy ranking method of the linear ranking function. Our obtained results show better performance.

On solving linear programming problem by duality approach in neutrosophic environment.

This paper extends the concept of crisp linear programming problem (lpp) by adopting the coefficients in objective function, technical coefficients, the right hand side coefficients and the decision variables as single valued triangular neutrosophic numbers (Sυtrn number). It is a special type of neutrosophic set. This modified concept is here called neutrosophic linear programming problem (Nlp problem). To develop this notion, a linear ranking function is newly constructed from geometrical concept first. Then the validity of existing crisp results are tested with the help of that ranking function in this new arena. An attempt is also taken to solve an Nlp problem by duality approach. For that, an efficient solution algorithm is developed by obeying the properties of ranking function. Finally, this proposed algorithm is demonstrated to solve a real life problem and some special cases are also illustrated by numerical examples.

On ranking functions for single-path linear-constraint loops

Program termination is a fundamental research topic in program analysis. In this paper, we present a new complete polynomial-time method for the existence problem of linear ranking functions for single-path loops described by a conjunction of linear constraints, when variables range over the reals (or rationals). Unlike existing methods, our method does not depend on Farkas’ Lemma and provides us with counterexamples to existence of linear ranking functions, when no linear ranking function exists. In addition, we extend our results established over the rationals to the setting of the integers. This deduces an alternative approach to deciding whether or not a given SLC loop has a linear ranking function over the integers. Finally, we prove that the termination of bounded single-path linear-constraint loops is decidable over the reals (or rationals).

Resource Analysis driven by (Conditional) Termination Proofs

AbstractWhen programs feature a complex control flow, existing techniques for resource analysis produce cost relation systems (CRS) whose cost functions retain the complex flow of the program and, consequently, might not be solvable into closed-form upper bounds. This paper presents a novel approach to resource analysis that is driven by the result of a termination analysis. The fundamental idea is that the termination proof encapsulates the flows of the program which are relevant for the cost computation so that, by driving the generation of the CRS using the termination proof, we produce a linearly-bounded CRS (LB-CRS). A LB-CRS is composed of cost functions that are guaranteed to be locally bounded by linear ranking functions and thus greatly simplify the process of CRS solving. We have built a new resource analysis tool, named MaxCore, that is guided by the VeryMax termination analyzer and uses CoFloCo and PUBS as CRS solvers. Our experimental results on the set of benchmarks from the Complexity and Termination Competition 2019 for C Integer programs show that MaxCore outperforms all other resource analysis tools.

Detecting multiphase linear ranking functions for single-path linear-constraint loops

Single-path linear-constraint loops are important since many analyses in different fields can be reduced to termination proof synthesis for such loops. In 2017, Ben-Amram and Genaim provided a complete polynomial-time solution to the problem of existence and of synthesis of multiphase linear ranking functions ( $$M\varPhi \hbox {RFs}$$ ) of bounded depth to prove the termination of a single-path linear-constraint loop. However, an open question whether one can precompute a bound on the depth of a $$M\varPhi \hbox {RF}$$ for a given loop is raised and the solution has not been accomplished so far. In this paper, we propose an approach to depth bound detection and synthesis of $$M\varPhi \hbox {RFs}$$ for rational single-path linear-constraint loops. We take a step further based on the work of Ben-Amram and Genaim, and detect the depth bound for a certain class of rational single-path linear-constraint loops. Furthermore, the effectiveness of our approach is presented with experimental evidence. The depth bound can be precomputed for loops where existing tools fail to infer a $$M\varPhi \hbox {RF}$$ or prove that none exists.

A branching algorithm to solve binary problem in uncertain environment: an application in machine allocation problem

This paper studies a new algorithm to solve the uncertain generalized assignment problem. The presented technique is based on the concept of branch and bound rather than the usual simplex based techniques. At first, the problem is relaxed to the transportation model which is easy to handle and work with. The model, so obtained is solved by the conventional transportation technique. The obtained solution serves as starting solution for further sub problems. The ambiguity in parameters is represented by triangular fuzzy numbers. We propose a linear ranking function, called the grade function which is based on the centroid method. The grade function is used to rank the triangular fuzzy numbers. The proposed approach is justified numerically by showing its application in generalized machine allocation problem.

Duality in Bipolar Fuzzy Number Linear Programming Problem

We develop a linear ranking function for ordering bipolar fuzzy numbers and study its properties. Using this ranking function, we solve a bipolar fuzzy linear programing problem. Then, we present the dual of the problem and establish several duality results. Also, we presented an application of bipolar fuzzy number in real life problem.

A new procedure for solving linear programming problems in an intuitionistic fuzzy environment

This paper presents a novel approach for solving Linear Programming Problems in an Intuitionistic Fuzzy environment. Here, the cost coefficients in the objective function of an Intuitionistic Fuzzy Number Linear Programming Problem (IFNLPP) are considered to be Symmetric Triangular Intuitionistic Fuzzy Numbers (STriIFNs) and the problem has been solved without converting them to crisp Linear Programming Problem using linear Ranking function. Intuitionistic Fuzzy Simplex algorithm which is a most powerful tool of handling any IFLPP has been used for solution procedure. An example has also been illustrated to reveal the effectiveness of the projected method.

Rank pooling dynamic network: Learning end-to-end dynamic characteristic for action recognition

Abstract In video recognition, rank-pooling operators are a type of models for sorting video sequences, which act on either the raw inputs or the intermediate feature maps of convolutional neural network (CNN). However, such models are currently restricted in the optimization of the linear ranking function by Rank-SVM and Rank-SVR. In this paper, we first propose a CNN architecture called RGB Rank Pooling Dynamic Network (RGB-RPDN), mapping a video to multiple frame-level dynamic spaces with the same size as the input. Importantly, a classical classification (e.g. FC, CNN) advanced in 2D image can be jointly positioned behind the generated representation for action classification, thus the joint architecture can be trained in an end-to-end manner. Second, we analyze how the flow-level evolution can be modeled by the hand-crafted rank-pooling machine, and extend the dynamic space of frame-level to that of flow-level by the Flow Rank Pooling Dynamic Network (Flow-RPDN). Third, equivalence relations between hand-crafted rank-pooling and RPDN are formulated, further the comparison of computing cost are qualitatively analyzed. Finally, the frame-level and flow-level pipelines are combined to achieve the final prediction by the late fusion. Specifically, with the models pre-trained on the large-scale Kinetics dataset, we train the two-stream RPDN on the UCF101 and HMDB51, where the parameters are initialized by the pre-trained models above. Experimental results demonstrate that the RPDN significantly improves the hand-crafted rank-pooling machines by a large margin of promotion, and achieves the correct rate of more excellent classification in action recognition.

Solution of Matrix Games with Generalised Trapezoidal Fuzzy Payoffs

ABSTRACTIn this paper, it is shown that once linear ranking function is chosen, then the fuzzy matrix game (FMG) is converted into crisp one and then it is solved easily by the proposed method. The main aim of this paper is to reduce the computational complexity of the FMG using the ranking function. The applicability and feasibility of the FMG is illustrated with a numerical example. This paper finalises with the conclusion and to include an outlook for future study in this direction.

A Compromise Solution for the Fully Fuzzy Multiobjective Linear Programming Problems

A new approach is undertaken to solve the fully fuzzy multiobjective linear programming (FFMLP) problem. The coefficients of the objective functions, constraints, right-hand-side parameters, and variables are of the triangular fuzzy number (T r FN)s. A solution strategy, called compromise solution algorithm (CSA), is presented using a three-step procedure. First, a revised simplex method together with Gaussian elimination in the environment of the linear ranking function is used to convert the FFMLP problems partially into semi fully fuzzy multiobjective linear programming (SFFMLP) problems. Then, the obtained SFFMLP problems are gathered together as a single problem. Finally, the gathered problem is solved by one of four different methods to find a fuzzy compromise solution for the FFMLP problems. The CSA is then numerically applied to a FFMLP problem to illustrate the practicability of the proposed procedure.

Bit-Precise Procedure-Modular Termination Analysis

Non-termination is the root cause of a variety of program bugs, such as hanging programs and denial-of-service vulnerabilities. This makes an automated analysis that can prove the absence of such bugs highly desirable. To scale termination checks to large systems, an interprocedural termination analysis seems essential. This is a largely unexplored area of research in termination analysis, where most effort has focussed on small but difficult single-procedure problems. We present a modular termination analysis for C programs using template-based interprocedural summarisation. Our analysis combines a context-sensitive, over-approximating forward analysis with the inference of under-approximating preconditions for termination. Bit-precise termination arguments are synthesised over lexicographic linear ranking function templates. Our experimental results show the advantage of interprocedural reasoning over monolithic analysis in terms of efficiency, while retaining comparable precision.

A Novel Technique for Solving Multiobjective Fuzzy Linear Programming Problems

This study considers multiobjective fuzzy linear programming (MFLP) problems in which the coefficients in the objective functions are triangular fuzzy numbers. The study proposing a new technique to transform MFLP problems into the equivalent single fuzzy linear programming problem and then solving it via linear ranking function using the simplex method, supported by numerical example.

SOME RESULTS OF DUALITY IN FUZZY LINEAR PROGRAMMING

In this paper, based on linear ranking functions we investigate some properties for the fuzzy number linear programming problems and its dual with the relationships between them. AMS Subject Classification: 90C05, 90C46, 90C70