Rough Interval Research Articles

Solving a Fully Rough Integer Linear Fractional Programming Problem

In this paper, a fully rough integer linear fractional programmingproblem is introduced, in which all coefficients and decision variablesin the objective function and the constraints are rough intervals. Theoptimal value of decision rough variables is rough interval. In order tosolve this problem, we will construct four crisp integer linear fractionalprogramming problems. Via these four crisp problems the roughoptimal integer solution is obtained. An illustrative numerical exampleis given for the developed theory.

Systemic Functions Evaluation based Technological Innovation System for the Sustainability of IoT in the Manufacturing Industry

Technological innovations are regarded as the tools that can stimulate economic growth and the sustainable development of technology. In recent years, as technologies based on the internet of things (IoT) have rapidly developed, a number of applications based on IoT innovations have emerged and have been widely adopted by various public and private sectors. Applications of IoT in the manufacturing industry, such as manufacturing intelligence, not only play a significant role in the enhancement of industrial competitiveness and sustainability, but also influence the diffusion of innovative applications that are based on IoT innovations. It is crucial for policy makers to understand these potential reasons for stimulating IoT industrial sustainability, as they can facilitate industrial competitiveness and technological innovations using supportive means, such as government procurement and financial incentives. Therefore, there is a need to ascertain different factors that may affect IoT industrial sustainability and further explore the relationship between these factors. However, finding a set of factors that affects IoT industrial sustainability is not easy. Recently, the robustness of a theoretical framework, termed the technological innovation system (TIS), has been verified and has been used to explore and analyze technological and industrial development. Thus, it is suitable for this research to use this theoretical model. In order to find out appropriate factors and accurately analyze the causality among factors that influence IoT industrial sustainability, this research presents a Bayesian rough Multiple Criteria Decision Making (MCDM) model based on TIS functions by integrating random forest (RF), decision making trial and evaluation (DEMATEL), Bayesian theory, and rough interval numbers. The proposed analytical framework is validated by an empirical case of defining the causality between TIS functions to enable the industrial sustainability of IoT in the Taiwanese smart manufacturing industry. Based on the empirical study results, the cause group consists of entrepreneurial activities, knowledge development, market formation, and resource mobilization. The effect group is composed of knowledge diffusion through networks’ guidance of the search, and creation of legitimacy. Moreover, the analytical results also provide several policy suggestions promoting IoT industrial sustainability that can serve as the basis for defining innovation policy tools for Taiwan and late coming economies.

Research on Rough Entropy of Interval Set Include Order Rough Set

In view of the characteristics that information incomplete system is difficult to be accurately expressed and described, based on the structure of interval containing ordered rough sets, this paper first gives the concept of granularity and roughness of interval rough sets through the inclusion relation between Boolean algebras. Then the roughness concept of interval set is defined, the structure and property of the measure satisfied are researched.By defining the element measure formula of interval set and the entropy measure of interval set, the rough entropy concept is further introduced in the rough interval set, and the theorem and conclusion satisfied by this measure are researched.For the measurement of roughness and roughness entropy of interval set rough set. For the measurement of roughness and roughness entropy of interval set rough set, the conclusion is that the uncertainty measure monotonically decreases as the granular size of interval set rough set becomes smaller,and finally the verification is carried out by a example.

On solution of constraint matrix games under rough interval approach

The aim of this article is to propose an effective technique for solving constraint matrix games with rough interval payoffs, which are a class of non-cooperative two-person matrix games with rough interval payoffs and the strategies of the players are constrained. Since the payoffs of the rough constraint matrix games are rough intervals, then its game value is also a rough interval. In this technique, we derived four linear programming problems, which are used to obtain the upper–lower bound, lower–lower bound, lower–upper bound and upper–upper bound of the rough interval game values of the players in rough constraint matrix games. Moreover, the expected value operator and trust measure of rough interval have been used to obtain the α-trust equilibrium strategies and the expected equilibrium strategies of the problem under study. In addition, the different advantages of the proposed technique over those existing are discussed. Finally, a numerical experiment of market share game model is given and solved by the three mentioned methods to illustrate the effectiveness and practicality of the proposed method.

Multi-item 4D-TPs under budget constraint using rough interval

Multi-item four dimensional transportation problems (MI4D-TPs) are formulated with budget constraint under rough and fuzzy interval environments. Here unit selling prices, unit purchasing costs, unit transportation costs, fixed charges, sources, demands, capacities of conveyances and total budget are represented by rough intervals for rough model and fuzzy numbers for fuzzy model. In this paper, four dimensional transportation problems (4D-TPs) under rough interval environment are transformed to deterministic ones by three different methods – (i) upper and lower approximation intervals, (ii) Expected Value Technique and (iii) rough chance-constraint programming technique. The fuzzy 4D-TP is made deterministic using possibility measures. Reduced problems are solved by generalized reduced gradient (GRG) method using LINGO 12.0 software. The models are illustrated with numerical examples and natures of the solutions i.e. complete and rather satisfactory solutions are presented. As particular cases, the results of solid transportation problems (3D-TPs) and conventional transportation problems (2D-TPs) are obtained. Importance of consideration of routes in MI4D-TPs is pointed out. As a particular case, the results of an earlier investigation are derived. Sensitivity of profit in the rough model through Expected Value Technique is presented against the arithmetic combination parameter. Percentage of increase of profit against the expectation parameter is furnished.

Solving Fully Rough Interval Multi-level Multi-objective linear Fractional Programming Problems via FGP

Solving Fully Rough Interval Multi-level Multi-objective linear Fractional Programming Problems via FGP

Novel approach to group multi-criteria decision making based on interval rough numbers: Hybrid DEMATEL-ANP-MAIRCA model

Interval rough number is introduced to deal with the vagueness in decision-making.A novel DEMATEL-ANP model based oninterval rough numbers.Application of a new multi-criteria technique called MARICA.An interval rough number based on MARICA is proposed to evaluate the alternatives.Multi-criteria techniques were compared based on interval rough and fuzzy approaches. This paper presents a novel approach for treating uncertainty in the multi-criteria decision making process by introducing interval rough numbers (IRN). The IRN approach enables decision making using only the internal knowledge incorporated in the data provided by the decision maker. A hybrid multi-criteria model was developed based on IRN, and demonstrated using the example of the bidder selection process in the state administration public procurement procedure. The first segment of the hybrid model deals with the rough interval DEMATEL-ANP (IR'DANP) model, which enables more objective expert evaluation of criteria in a subjective environment than the traditional/crisp approach. In the second segment, the evaluation is enabled by applying the new rough interval MAIRCA method, which introduces mathematical tools and shows high stability concerning changes in the nature and characteristics of the criteria. The results of the hybrid IR'DANP-MAIRCA model were analyzed using 36 scenarios of sensitivity analysis, which showed high stability of the results. The results of the interval rough method were compared with the fuzzy extensions of the TOPSIS, VIKOR, MABAC, TODIM, ELECTRE I and DEMATEL-ANP models.

A Profit Maximizing Solid Transportation Model Under a Rough Interval Approach

This study attempts to establish an innovative solid transportation problem that intends to maximize profit under the rough interval approximation methodology. Two transportation problems were constructed in this regard with interval coefficients corresponding to the upper approximations and the lower approximations of the rough intervals under study. Furthermore, from the contingent solid transportation problems, four different classical solid transportation problems were derived, which were subsequently solved on the LINGO iteration platform. The concepts of completely satisfactory solution and rather satisfactory solution , surely optimal range , possibly optimal range , and rough optimal range have been discussed with a perspective to its relevance to real-world practical problems. The rough chance-constrained programming and the expected value operator for rough interval have been applied to solve the problem under study. The distinct advantages of the proposed method over those existing have been outlined. Numerical examples have also been provided to illustrate the solution procedure and the methodologies adopted.

Analysis of interval programming in different environments and its application to fixed-charge transportation problem

In this paper, we analyze the interval programming using interval and rough interval (RI). Interval programming is one of the tools to tackle uncertainty in mathematical programming. In the real world situations, we often encounter the cases where the data cannot be determined with certainty. So, the value of the data is assessed using an interval. Here, we consider that the parameters of the Fixed-Charge Transportation Problem (FCTP) are imprecise. In our proposed problem, uncertainties are considered in terms of interval and RI. We present a study of the FCTP using the interval programming. An example of the FCTP is included to show the effectiveness and usefulness of our proposed study. Finally, concluding remarks and future scopes of our paper are discussed.

Rough-interval-based multicriteria decision analysis for remediation of 1,1-dichloroethane contaminated groundwater

A rough-interval-based multicriteria decision analysis method (RI-MCDA) is developed for supporting the selection of remediation strategies for 1,1-dichloroethane contaminated sites. The concept of ‘‘rough interval’’ is introduced in the design framework to represent dual-uncertain parameters. Three rough-interval scenarios generated through pair-wise combining the values under three confidence levels (i.e. 68.3%, 95.4% and 99.7%) and one deterministic scenario adopted crisp numbers for parameters are introduced into the framework. The proposed method is then applied to a contaminated site in the Pudong district of Shanghai, China. Fifty remediation alternatives under four duration options (i.e. 5, 10, 15, and 20 years) and ten criteria, including daily total pumping rate, total cost and rough-interval risk information in light of uncertainty parameter (e.g. slope factor), are taken into consideration to compare different alternatives through RI-MCDA. Results indicated that the most desirable remediation strategy lied in A25 for the 5-year, A10 for the 10-year, A15 for the 15-year, and A11 for the 20-year remediation. Compared to the traditional MCDA, the proposed RI-MCDA shows the uniqueness in addressing the interaction between dual intervals of highly uncertain parameters, as well as their joint impact on the decision results, which reduces the subjectivity as much as possible.

A Fuzzy Technique for Solving Rough Interval Multiobjective Transportation Problem

In this paper the rough interval multiobjective transportation problem (RIMOTP) is presented and its solution procedure is introduced. The concept of solving the interval multiobjective transportation problem is applied for solving RIMOTP. So, The rough interval in the objective function and the constrains, is represented by three different models and such models are solved by using fuzzy programming technique based on the right limit, the center and the half-width of each rough interval using possibly region. Numerical examples are provided to illustrate the solution procedure of three possible types of the original problem. General Terms Fuzzy Programming Technique, Transportation Problem, rough interval multiobjective transportation

A proposed method for solving multiobjective linear fractional programming problems with rough coefficients

In this paper, a new method for solving multiobjective linear fractional programming problems with roughcoefficient (MORLFP) is proposed. The MORLFP problem is considered by incorporating rough intervals in thecoefficients of the objective functions. It is provided that a MORLFP problem is converted to an optimization problemwith rough interval valued objective functions which it their bounds are four multiobjective linear fractional functions.The rough efficient solutions are characterized by using a new proposed algorithm. A numerical example is given for thesake of illustration

Solving multiple-server queues in rough environment

In this paper, a multiple-server queue in rough environment is considered in which the arrival rate and service rate are rough interval numbers. A new method namely, multi-level method is proposed to determine the performance measures in the given queueing system as rough interval numbers. The solution procedure for the proposed method is demonstrated with the help of numerical example. The performance measures of the multiple-server queue in rough environment provided by the proposed method are rough interval numbers which are very much useful for decision makers for taking the managerial decision-making.

On multi-objective linear programming problems with inexact rough interval-fuzzy coefficients

This paper deals with a multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOLP. This problem is considered by incorporating an inexact rough interval fuzzy number in both the objective function and constrains. The concept of "Rough interval" is introduced in the modeling framework to represent dualuncertain parameters. A suggested solution procedure is given to obtain rough interval solution for IRFLP(w) problem. Finally,two numerical example is given to clarify the obtained results in this paper.

On the Solution of a Rough Interval three-level Quadratic Programming Problem

In this paper, a three-level quadratic programming (QP) problem is considered where some or all of its coefficients in the objective function are rough intervals. At the first phase of the solution approach and to avoid the complexity of the problem, two QP problems with interval coefficients will be formulated. One of these problems is a QP where all of its coefficients are upper approximation of rough intervals and the other problem is a QP where all of its coefficients are lower approximations of rough intervals. At the second phase, a membership function is constructed to develop a fuzzy model for obtaining the optimal solution of the threelevel quadratic programming problem. Finally, an illustrative numerical example is given to demonstrate the obtained results.

Solving Investment Problem with Inexact Rough Interval Data through Dynamic Programming Approach

DOI: 10.9734/BJMCS/2015/15849 Editor(s): (1) Dragos Patru Covei, Department of Mathematics, University “Constantin Brâncusi” of Târgu-Jiu, România. Reviewers: (1) Nurulizwa bte Abdul Rashid, Faculty of Technology and Technopreneurship, UTeM & Malaysia, Malaysia. (2) Anonymous, China. (3) Sergey A. Surkov, International Institute of Management, Russia. Complete Peer review History: http://www.sciencedomain.org/review-history.php?iid=1032&id=6&aid=8696

A multi-objective optimization model for hub network design under uncertainty: An inexact rough-interval fuzzy approach

This article proposes a multi-objective mixed-integer model to optimize the location of hubs within a hub network design problem under uncertainty. The considered objectives include minimizing the maximum accumulated travel time, minimizing the total costs including transportation, fuel consumption and greenhouse emissions costs, and finally maximizing the minimum service reliability. In the proposed model, it is assumed that for connecting two nodes, there are several types of arc in which their capacity, transportation mode, travel time, and transportation and construction costs are different. Moreover, in this model, determining the capacity of the hubs is part of the decision-making procedure and balancing requirements are imposed on the network. To solve the model, a hybrid solution approach is utilized based on inexact programming, interval-valued fuzzy programming and rough interval programming. Furthermore, a hybrid multi-objective metaheuristic algorithm, namely multi-objective invasive weed optimization (MOIWO), is developed for the given problem. Finally, various computational experiments are carried out to assess the proposed model and solution approaches.

A grey rough set model for evaluation and selection of software cost estimation methods

Purpose – Due to the limitation of acknowledgment, the complexity of software system and the interference of noises, this paper aims to solve the traditional problem: traditional software cost estimation methods face the challenge of poor and uncertain inputs. Design/methodology/approach – Under such circumstances, different cost estimation methods vary greatly on estimation accuracy and effectiveness. Therefore, it is crucial to perform evaluation and selection on estimation methods against a poor information database. This paper presents a grey rough set model by introducing grey system theory into rough set based analysis, aiming for a better choice of software cost estimation method on accuracy and effectiveness. Findings – The results are very encouraging in the sense of comparison among four machine learning techniques and thus indicate it an effective approach to evaluate software cost estimation method where insufficient information is provided. Practical implications – Based on the grey rough set model, the decision targets can be classified approximately. Furthermore, the grey of information and the limitation of cognition can be overcome during the use of the grey rough interval correlation cluster method. Originality/value – This paper proposed the grey rough set model combining grey system theory with rough set for software cost estimation method evaluation and selection.

Linear programming with rough interval coefficients

In this paper, a linear programming LP problem is considered where some or all of its coefficients in the objective function and/or constraints are rough intervals. In order to solve this problem, we will construct two LP problems with interval coefficients. One of these problems is an LP where all of its coefficients are upper approximations of rough intervals and the other is an LP where all of its coefficients are lower approximations of rough intervals. Via these two LPs, two newly solutions completely and rather satisfactory are defined. Some examples are given to demonstrate the results.

A mathematical model for design problem of hydraulic valve product

In the production of the assembly product with specific performance requirements, it will be of great significance to study how to determine the range of geometrical parameters of the product components in the product design. Due to the effect of various factors in the processing, characteristics of assembly product and geometric parameters affect characteristics typically have a range of permitted tolerances, namely, with a certain uncertainty, so the key is to study the interval prediction model between the characteristics and geometric parameters of the assembly product. In this study, an interval prediction model was proposed based on rough set theory and support vector interval regression networks. To illustrate the applicability and capability of the developed model, a specific hydraulic valve production was selected as a case study. The experimental results showed that the model can perfectly fit the interval corresponding relationship between the product characteristics and geometric parameters.