Rough Intervals Research Articles

تصميم نظام أمثل باستخدام الفاصل التقريبي لبرمجة دي نوڨو متعددة الأهداف

تُعد برمجة دي نوڤو متعددة الأهداف، أداة فعالة تتعامل مع تصميم نظام أمثل، وذلك من خلال تحديد المستوى الأمثل لتخصيص الموارد، وتحسين قيمة دوال الهدف وفقاً لسعر الموارد (هذا فيما لو كانت الظروف طبيعية ولا يوجد تقلب في الاسعار). في هذه البحث اُقتِرحَ نهجاً جديداً لحل مشكلة عدم اليقين لمشكلة برمجة دي نوفو، وذلك باستخدام أنموذج مركب يتضمن نوعين من البرمجة: الأول هو برمجة متعددة الأهداف ذات الفاصل التقريبي (RIMOP)، والثاني: برمجة دي نوڤو(DNP)، علماً بأن معاملات متغيرات القرار لدالة الهدف والقيود عبارة عن فاصل تقريبي RIC. وقد استخدامنا ثلاث طرائق لحل الانموذج المقترح وإيجاد تصميم نظام أمثل، الطريقة الاولى هي طريقة المجموع الموزون WSM والتي تُستخدم قبل اعادة صياغة النموذج المقترح RIMOP (وكانت الموارد معروفة)، وقد وجد ان نتائج طريقة WSM تعطي حلاً واحداً (حل وسط قريب من المثالية) من بين الحلول المقبولة تحت كل حد من حدود المشكلة، وبالتالي يمكن تقديم أربع حلول (بدائل) لصانع القرار (DM). اما منهجية زيلني وطريقة نسب المسار الأمثل فقد استخدمت بعد صياغة الانموذج المقترح (RIMODNP) (وكانت الموارد للطرف الأيمن للقيود ليست معروفة). وجد ان منهجية زيلني تعطي تصميم نظام أمثل واحداً لكل حد من حدود المشكلة، في حين ان طريقة المسار الأمثل طبقت بعد ان تم التأكد من حدود النموذج المقترح وفقاً النظرية شاي فقد استخدمنا ثلاثة أنواع من النسب تحت كل مشكلة فرعية، فقد وجد ان هذهِ الطريقة تعطي ثلاث تصاميم انظمة مثلى تحت كل حد للمشكلة، وقد اتضح من خلال النتائج ان طريقة نسب المسار الامثل هي أكثر كفاءة من غيرها بعد تطبيقها على النموذج المقترح، لأنها اعطت اثنا عشر بديلاً لـ(DM)، لوحظ ان الانموذج المقترح يتوافق مع شروط ونظريات RIC. يُعد الانموذج المقترح مناسباً جداً لظروف عدم اليقين، اخيراً، طبق مثال رقمي على الانموذج المقترح.

Culture influences conscious appraisal of, but not automatic aversion to, acoustically rough musical intervals.

There is debate whether the foundations of consonance and dissonance are rooted in culture or in psychoacoustics. In order to disentangle the contribution of culture and psychoacoustics, we considered automatic responses to the perfect fifth and the major second (flattened by 25 cents) intervals alongside conscious evaluations of the same intervals across two cultures and two levels of musical expertise. Four groups of participants completed the tasks: expert performers of Lithuanian Sutartinės, English speaking musicians in Western diatonic genres, Lithuanian non-musicians and English-speaking non-musicians. Sutartinės singers were chosen as this style of singing is an example of 'beat diaphony' where intervals of parts form predominantly rough sonorities and audible beats. There was no difference in automatic responses to intervals, suggesting that an aversion to acoustically rough intervals is not governed by cultural familiarity but may have a physical basis in how the human auditory system works. However, conscious evaluations resulted in group differences with Sutartinės singers rating both the flattened major as more positive than did other groups. The results are discussed in the context of recent developments in consonance and dissonance research.

Solving the multi-modal transportation problem via the rough interval approach

This research studies a transportation problem to minimize total transportation cost under the rough interval approximation by considering the multi-modal transport framework, referred to here as the rough Multi-Modal Transportation Problem (MMTP). The parameters of MMTP are rough intervals, because the problem is chosen from a real-life scenario. To solve MMTP under a rough environment, we employ rough chance-constrained programming and the expected value operator for the rough interval and then outline the main advantages of our suggested method over those existing methods. Next, we propose an algorithm to optimally solve the problem and present a numerical example to examine the proposed technique. Finally, the solution is analyzed by the proposed method with rough-chance constrained programming and expected value operator.

A study of uncertainty multi-objective nonlinear programming problems for rough intervals

In real conditions, the parameters of multi-objective nonlinear programming (MONLP) problem models can’t be determined exactly. Hence in this paper, we concerned with studying the uncertainty of MONLP problems. We propose algorithms to solve rough and fully-rough-interval multi-objective nonlinear programming (RIMONLP and FRIMONLP) problems, to determine optimal rough solutions value and rough decision variables, where all coefficients and decision variables in the objective functions and constraints are rough intervals (RIs). For the RIMONLP and FRIMONLP problems solving methodology are presented using the weighting method and slice-sum method with Kuhn-Tucker conditions, We will structure two nonlinear programming (NLP) problems. In the first one of this NLP problem, all of its variables and coefficients are the lower approximation (LAI) it’s RIs. The second NLP problems are upper approximation intervals (UAI) of RIs. Subsequently, both NLP problems are sliced into two crisp nonlinear problems. NLP is utilized because numerous real systems are inherently nonlinear. Also, rough intervals are so important for dealing with uncertainty and inaccurate data in decision-making (DM) problems. The suggested algorithms enable us to the optimal solutions in the largest range of possible solution. Finally, Illustrative examples of the results are given.

A Survey on Rough Intervals Programming Formulation and Methodologies

This research states a survey on formulation of linear programming problems with rough intervals coefficients in the objective functions and constraints, basic preliminaries about rough intervals, interval method and trust probability constraints for transforming rough intervals to crisp nature and fully rough intervals problems. Finally, presents different operational research models that contain rough intervals coefficients. This research states a survey on formulation of linear programming problems with rough intervals coefficients in the objective functions and constraints, basic preliminaries about rough intervals, interval method and trust probability constraints for transforming rough intervals to crisp nature and fully rough intervals problems. Finally, presents different operational research models that contain rough intervals coefficients. This research states a survey on formulation of linear programming problems with rough intervals coefficients in the objective functions and constraints, basic preliminaries about rough intervals, interval method and trust probability constraints for transforming rough intervals to crisp nature and fully rough intervals problems. Finally, presents different operational research models that contain rough intervals coefficients.

Solving multi-level multiobjective fractional programming problem with rough interval parameter in neutrosophic environment

In this study, a novel algorithm is developed to solve the multi-level multiobjective fractional programming problems, using the idea of a neutrosophic fuzzy set. The co-efficients in each objective functions is assumed to be rough intervals. Furthermore, the objective functions are transformed into two sub-problems based on lower and upper approximation intervals. The marginal evaluation of pre-determined neutrosophic fuzzy goals for all objective functions at each level is achieved by different membership functions, such as truth, indeterminacy/neutral, and falsity degrees in neutrosophic uncertainty. In addition, the neutrosophic fuzzy goal programming algorithm is proposed to attain the highest degrees of each marginal evaluation goals by reducing their deviational variables and consequently obtain the optimal solution for all the decision-makers at all levels. To verify and validate the proposed neutrosophic fuzzy goal programming techniques, a numerical example is adressed in a hierarchical decision-making environment along with the conclusions.

An Approach for Solving Multi-Objective Linear Fractional Programming Problem with fully Rough Interval Coefficients

In this paper, a multi-objective linear fractional programming (MOLFP) problem is considered where all of its coefficients in the objective function and constraints are rough intervals (RIs). At first, to solve this problem, we will construct two MOLFP problems with interval coefficients. One of these problems is a MOLFP where all of its coefficients are upper approximations of RIs and the other is a MOLFP where all of its coefficients are lower approximations of RIs. Second, the MOLFP problems are transformed into a single objective linear programming (LP) problem using a proposal given by Nuran Guzel. Finally, the single objective LP problem is solved by a regular simplex method which yields an efficient solution of the original MOLFP problem. A numerical example is given to demonstrate the results.

Design optimization and uncertainty analysis of multi-energy complementary system for residential building in isolated area

Due to extreme climatic and geographical factors, conventional power sources, which typically use large power-grids, have difficulties to supply energy to isolated areas. In addition, many isolated areas rely mainly on fossil fuel to ensure adequate energy-supply, which involves transport across long distances and high costs. Thus, finding a multi-energy complementary system (MECS) for residential buildings can help ensure the survival and improve the well-being of about 1.5 billion people that are currently living in isolated areas worldwide. In this study, a rough interval-Copula stochastic planning (RI-CSP) programming model is proposed to address this problem. Its purpose is to optimize the MECS in isolated areas, with the aim to reduce cost and provide high-quality energy. RI-CSP can deal with multiple uncertain parameters that are characterized by rough intervals and probability distributions in the MECS. It can also reveal the combined effect of different random variables. To prove the feasibility of RI-CSP, it was applied to a residential building on Yongxing island in the South China Sea. 24 different scenarios were selected that consider four renewable energy scarcity levels and 2 constraint-violation levels (p). Then, the optimal scenarios with minimum total system costs were selected. This study can help decision makers find better configurations for many alternative power-generation scenarios. The results show that the uncertainties that exist in the MECS have a significant effect on the power-generation configuration and total system cost. The results indicate that the total cost of MECS decreases with the increase in scarcity levels. Our results also revealed that, as the scarcity levels for renewable energy decrease, the fraction of PV power-generation gradually increases.

A production-repairing inventory model considering demand and the proportion of defective items as rough intervals

This paper explores a production inventory model in an imperfect production system under a rough interval environment. The occurrences of shortages are allowed and are fully backlogged. In this paper, two models are presented. In the first model, the demand of the product is considered as a rough interval. In the second model, both the demand and the defective rate are considered as rough intervals. In reality, these two factors cannot be assessed exactly due to a lack of available data. Moreover, there are few scenarios where the demand of the product is assessed in two-layer information. The inner layer demand is obvious, and it may extend up to the outer layer depending on the situation. Also, the same thing may happen for the defective rate of the product. In those cases, it is more reasonable to express the demand and the defective rate as rough intervals instead of fuzzy or interval numbers. Based on the expected value of rough intervals, the necessary and sufficient conditions are determined analytically to obtain the optimal inventory policies. The main advantage of this method is that the optimal profit and the optimal lot size are determined as crisp numbers rather than rough intervals. Obtaining a precise value for these two quantities is desirable for any production manager. Numerical examples for both models are provided to illustrate the solution methodology. Sensitivity analysis of the optimal solutions concerning the key parameters is conducted for identifying several managerial implications.

A Novel Approach for Solving a Fully Rough Multi-Level Quadratic Programming Problem and Its Application

The most widely used actions and decisions of the real-world tasks frequently appear as hierarchical systems. To deal with these systems, the multi-level programming problem presents the most flourished technique. However, practical situations involve some the impreciseness regarding some decisions and performances; RST provides a vital role by considering the lower and upper bounds of any aspect of uncertain decision. By preserving the advantages of it, in the present study, solving fully rough multi-level quadratic programming problems over the variables, parameters of the objective functions, and the constraints such as rough intervals are focused on. The proposed approach incorporates the interval method, slice-sum method, Frank and Wolfe algorithm, and the decomposition algorithm to reach optimal values as rough intervals. The proposed is validated by an illustrative example, and also environmental-economic power dispatch is investigated as a real application. Finally, the proposed approach is capable of handling the fully rough multi-level quadratic programming models.

On Solving Fully Rough Multi-Objective Integer Linear Programming Problems

In this paper a suggested algorithm to solve fully rough multi-objectiveinteger linear programming problem [FRMOILP] is described. In orderto solve this problem and find rough value efficient solutions anddecision rough integer variables by the slice-sum method with thebranch and bound technique, we will use two methods, the first one isthe method of weights and the second is e- Constraint method. The basicidea of the computational phase of the algorithm is based onconstructing two LP problems with interval coefficients, and then to fourcrisp LPs. In addition to determining the weights and the values of e-constraint. Also, we reviewed some of the advantages and disadvantagesfor them. We used integer programming because many linearprogramming problems require that the decision variables are integers.Also, rough intervals (RIs) are very important to tackle the uncertaintyand imprecise data in decision making problems. In addition, theproposed algorithm enables us to search for the efficient solution in thelargest range of possible solutions range. Also, we obtain N suggestedsolutions and which enables the decision maker to choose the bestdecisions. Finally, two numerical examples are given to clarify theobtained results in the paper.

A mathematical model for solving fuzzy integer linear programming problems with fully rough intervals

A suggested algorithm to solve triangular fuzzy rough integer linear programming (TFRILP) problems with α-level is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are triangular fuzzy rough numbers. In real-life situations, the parameters of a linear programming problem model may not be defined precisely, because of the current market globalization and some other uncontrollable factors. In order to solve this problem, a proper methodology is adopted to solve the TFRILP problems by the slice-sum method with the branch-and-bound technique, through which two fuzzy integers linear programming (FILP) problems with triangular fuzzy interval coefficients and variables were constructed. One of these problems is an FILP problem, where all of its coefficients are the upper approximation interval and represent rather satisfactory solutions; the other is an FILP problem, where all of its coefficients are the lower approximation interval and represent completely satisfactory solutions. Moreover, α-level at α = 0.5 is adopted to find some other rough value optimal solutions and decision rough integer variables. Integer programming is used, since a lot of the linear programming problems require that the decision variables be integers. In addition, the motivation behind this study is to enable the decision makers to make the right decision considering the proposed solutions, while dealing with the uncertain and imprecise data. A flowchart is also provided to illustrate the problem-solving steps. Finally, two numerical examples are given to clarify the obtained results.

English

A suggested algorithm to solve fully rough integer linear programming (FRILP) problems is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are rough intervals (RIs). In real-life situations, the parameters of linear programming problem model may not be defined precisely, because of globalization of the market, uncontrollable factors, etc., hence for that the FRILP problem solving methodology is presented using the slice-sum method with the branch and bound technique, where we will construct two integers linear programming (ILP) problems with interval coefficients and variables. One of these problems is an ILP problem, where all of its coefficients are upper approximations interval (UAI) of rough intervals and represents rather satisfactory solutions, the other is an ILP problem where all of its coefficients are lower approximations interval (LAI) of rough intervals and represents complete solutions. Thereafter, the two ILP problems are sliced into four crisp problems. Integer programming is used because many linear programming (LP) problems require that the decision variables should be integers. In addition, rough intervals are very important to tackle the uncertainty and imprecise data in decision making problems. Furthermore, the proposed algorithm enables us to search for the optimal solution in the largest range of possible solutions. A flowchart is also provided to illustrate the problem-solving steps. Finally, some examples are given to demonstrate the results. Key words: Integer linear programming, rough set theory, full rough interval coefficients and variables, upper approximation, lower approximation, optimal solution, crisp coefficients. &nbsp

Multi-objective fixed-charge transportation problem using rough programming

This paper analyses the multi-objective fixed-charge transportation problem (MOFCTP) using rough programming. Due to globalisation of market, the parameters of MOFCTP may not be defined precisely, so the parameters of the MOFCTP are treated as rough intervals. Expected value operator is used to convert rough MOFCTP to deterministic MOFCTP. Fuzzy programming method and linear weighted-sum method are used to obtain Pareto-optimal solution from deterministic MOFCTP. A comparative study is made between the obtained solutions extracted from the methods; and thereafter we perform a procedure to analyse the sensitive analysis of the parameters in MOFCTP. Finally, in order to show the applicability of our proposed study, an example on MOFCTP is included in this paper.

Rough set approach to non-cooperative continuous differential games

In this article, we analyze the non-cooperative continuous differential games under rough interval environment. The combination of rough set and continuous differential games expresses a new class defined as rough non-cooperative continuous differential games. We develop an effective and simple technique for solving the problem under study; in this methodology, the rough non-cooperative continuous differential games are transformed into two problems with interval environment corresponding to the upper and lower approximation of rough intervals. Furthermore, four crisp problems of non-cooperative continuous differential games are derived as follows: upper lower problem (ULP), lower lower problem (LLP), lower upper problem (LUP) and upper upper problem (UUP). Moreover, the trust measure and the expected value operator of rough interval are used to find the α-trust equilibrium strategies and the expected equilibrium strategies. Sufficient and necessary conditions for equilibrium strategies of rough non-cooperative continuous differential games are also derived. Finally, applicability and validity of the models and technique proposed in this article are illustrated with a practical example.

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.

Constrained FC 4D MITPs for Damageable Substitutable and Complementary Items in Rough Environments

Very often items that are substitutable and complementary to each other are sent from suppliers to retailers for business. In this paper, for these types of items, fixed charge (FC) four-dimensional (4D) multi-item transportation problems (MITPs) are formulated with both space and budget constraints under crisp and rough environments. These items are damageable/breakable. The rates of damageability of the items depend on the quantity transported and the distance of travel i.e., path. A fixed charge is applied to each of the routes (independent of items). There are some depots/warehouses (origins) from which the items are transported to the sales counters (destinations) through different conveyances and routes. In proposed FC 4D-MITP models, per unit selling prices, per unit purchasing prices, per unit transportation expenditures, fixed charges, availabilities at the sources, demands at the destinations, conveyance capacities, total available space and budget are expressed by rough intervals, where the transported items are substitutable and complementary in nature. In this business, the demands for the items at the destinations are directly related to their substitutability and complementary natures and prices. The suggested rough model is converted into a deterministic one using lower and upper approximation intervals following Hamzehee et al. as well as Expected Value Techniques. The converted model is optimized through the Generalized Reduced Gradient (GRG) techniques using LINGO 14 software . Finally, numerical examples are presented to illustrate the preciseness of the proposed model. As particular cases, different models such as 2D, 3D FCMITPs for two substitute items, one item with its complement and two non substitute non complementary items are derived and results are presented.

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.

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.