Linear programming with rough interval 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

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.

Many problems appear to be arising in the present as a result of variations in transportation networks. The stochastic fixed-charge transportation problem (SFCTP) is one such problem. The SFCTP is transformed into a chance-constrained programming (CCP) problem where supply and demand are stochastic and objective functions are in a rough interval. In this model, to analyze the multi-objective rough interval stochastic fixed-charge transportation problem (MORISFCTP), where the objective function coefficients are represented by rough intervals and the supply and destination factors are probabilistic constraints. This model operates an expected value operator to deal with uncertainty, in which the coefficient of the objective functions in the fuzzy is changed to a crisp form, and the probabilistic constraints are converted to a deterministic form by the Weibull distribution. To produce the optimal compromise solutions of the proposed model, three distinct methods are used: the fuzzy programming approach, the method of a linear weighted sum, and the €-constraint method. Lastly, the paper delivers a practical illustration of a MORISFCTP to demonstrate the usefulness and feasibility of the suggested methodology.

In this paper, a linear fractional programming problem with interval coefficients in both objective function and constraints is solved. The uncertain objective function is transformed into two deterministic objective functions. The two objective functions are converted into a single objective problem through the linear combination method. Through a modified possibility degree, the uncertain inequality and equality constraints are transformed into deterministic inequality constraints. The existing results are concerning the qualitative and quantitative analysis of basic notions in parametric linear fractional programming problem. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. A numerical example is investigated to demonstrate the effectiveness of the present approach.

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.

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ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

The multi-objective linear fractional programming is an interesting topic with many applications in different fields. Until now, various algorithms have been proposed in order to solve the multi-objective linear fractional programming (MOLFP) problem. An important point in most of them is the use of non-linear programming with a high computational complexity or the use of linear programming with preferences of the objective functions which are assigned by the decision maker. The current paper, through combining goal programming and data envelopment analysis (DEA), proposes an iterative method to solve MOLFP problems using only linear programming. Moreover, the proposed method provides an efficient solution which fairly optimizes each objective function when the decision maker has no information about the preferences of the objective functions. In fact, along with normalization of the objective functions, their relative preferences are fairly determined using the DEA. The implementation of the proposed method is demonstrated using numerical examples.

Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

With the rapid increase in computing, storage and networking resources, data is not only collected and stored, but also analyzed. This creates a serious privacy problem which often inhibits the use of this data. In this chapter, we investigate and resolve the privacy issues in a fundamental optimization problem -- linear programming (LP) which is formulated by data collected from different parties. We first consider the case where the objective function and constraints of the linear programming problem are not partitioned between two parties where one party privately holds the objective function while the other party privately holds the constraints. Second, we present a privacy preserving technique for the case that objective function and constraints are arbitrarily partitioned between two parties where each party privately holds a share of objective function and constraints. Finally, we extend the technique for securely solving two-party arbitrarily partitioned linear programming problems to a multi-party scenario. In summary, we propose a set of efficient and secure transformation based techniques that create significant value-added benefits of being independent of the specific algorithms used for solving the linear programming problem.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

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.