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جستجوی مقالات مرتبط با کلیدواژه « fuzzy programming » در نشریات گروه « صنایع »

تکرار جستجوی کلیدواژه «fuzzy programming» در نشریات گروه «فنی و مهندسی»
  • Mohsen Amini Khouzani, Alireza Sadeghi *, Amir Daneshvar, Adel Pourghader Chobar

    The problem of allocation of financial resources in projects is one of the most important problems of mathematical optimization. Incorrect allocation of financial resources can lead to project failure, increased costs, and reduced profitability. The importance of this issue has led to the modeling of a financial resource allocation problem for sustainable projects under uncertainty in this article. A fuzzy programming method was used to control model parameters and GSSA, GA, and SSA algorithms were used to solve the model. In the mathematical model, the goal was to optimize the objective function consisting of predicted return, investment risk, and project sustainability. Mathematical calculation results showed that meta-heuristic algorithms have high efficiency in achieving optimal solutions in a short time. so that the average time to solve them was less than 10 seconds. Also, the calculation results showed that increasing the uncertainty rate leads to increasing the value of the objective function and creating a distance from the optimal point. This is due to increasing costs and decreasing profits in sustainable projects. Finally, usage the TOPSIS method, the ranking of solving algorithms was done, and the GSSA algorithm was the most efficient algorithm among other algorithms with a desirability weight of 0.846.

    Keywords: resource allocation, sustainable projects, fuzzy programming, meta-heuristic algorithms}
  • حسین عموزادخلیلی*

    امروزه به دلیل اهمیت آلاینده های محیط زیستی و افزایش استانداردهای جهانی برای محیط زیست، توجه بیشتری به طراحی شبکه های زنجیره ی تامین حلقه بسته با ملاحظات سبز معطوف شده است. از طرفی شدت کارایی زنجیره های مستقیم و معکوس بر هم اثرگذار است. درنتیجه، عملکرد هر زنجیره بر زنجیره ی دیگر و بر کل زنجیره ی تامین تاثیر خودش را دارد. در این تحقیق، به طراحی یک مدل ریاضی برای شبکه زنجیره ی تامین حلقه بسته ی سبز لاستیک سنگین می پردازیم که با مفهوم قیمت گذاری اقتصادی، محصولات را تحت شرایط عدم قطعیت در نظر می گیرد. قیمت گذاری اقتصادی در این مسیله افزایش سودآوری اقتصادی را به همراه دارد. مدل ریاضی طراحی شده از نوع فازی و دوهدفه است که هدف اول حداقل سازی هزینه ها و هدف دوم حداقل سازی آلاینده های زیست محیطی است. تصمیمات تعیین شده در این مدل، شامل تعیین مکان بهینه هر یک از مراکز بر اساس مکان های بالقوه، میزان بهینه ی تولید، توزیع، جمع آوری، بازیافت و همچنین بازتولید محصولات است. همچنین، از آزمون تی دو نمونه ای مستقل برای اعتبارسنجی نتایج مدل غیر قطعی و قطعی استفاده می گردد. برای حل مدل تابع دوهدفه، روش ε-محدودیت به کار گرفته می شود تا مسیله بتواند جواب های بهینه ی پارتویی قوی را تضمین کند و از جوابهای پارتویی ضعیف جلوگیری کند. در نهایت، برای ارزیابی کارایی روش ارایه شده، یک مطالعه‎ی موردی در زمینه ی لاستیک سنگین به کار گرفته می شود و با حل و پیاده سازی آن نتایج مدیریتی مفیدی ارایه می شود.

    کلید واژگان: برنامه ریزی فازی, زنجیره ی تامین حلقه بسته, هدف زیست محیطی, قیمت گذاری, ε- محدودیت}
    Hossein Amoozadkhalili *

    Today due to the significance of environmental pollutants and the increase of global standards for the environment, more attention has been paid to the design of closed-loop supply chain networks with green considerations. On the other hand, the direct and inverse chains affect each other in terms of the efficiency rate. As a result, the performance of each chain has its own effect on other chains and on the entire supply chain. In this research, we design a mathematical model for the green closed-loop heavy-duty supply chain network, which considers products under conditions of uncertainty with the concept of economic pricing. Economic pricing in this issue increases economic profitability. The designed mathematical model has a fuzzy basis and pursues two objectives. The first goal is to minimize costs and the second goal is to minimize environmental pollutants. The decisions made in this model include determining the optimal location of each center based on potential locations, the optimal amounts of production, distribution, collection and recycling, and also the reproduction of products. Furthermore, a two-sample independent t-test is used to validate the results of the definite and indefinite models. To solve the two-objective function model, the ε-constraint method is used so that the problem can guarantee strong Pareto optimal solutions and prevent weak Pareto solutions. Finally, to evaluate the efficiency of the proposed method, a case in the field of heavy tires is studied, solved and implemented to produce and present valuable management results.

    Keywords: Fuzzy Programming, Closed-loop Supply Chain, Environmental purpose, Pricing, e-constraint}
  • Ali Sabbaghnia *, Jafar Heydari, Jafar Razmi
    This study investigates the joint production planning and warehouse layout under uncertainty. Today’s competitive business world needs to be investigated by models which are capable of considering uncertain nature of the problems, especially when the historical data is not available or the level of uncertainty is high. Joint production planning and warehouse layout problems is almost a novel and new area in both academics and practice. For warehousing problem, the eventually of rental warehouses and new allocations is enabled in each planning horizon period. A bi-objective MILP model is proposed and fuzzy distributed parameters and chance constraints are taken into considerations. One of the objective functions deals with the cost associated parameters and variables while the second one minimizes the fluctuations of the work labor in each planning period. A simple test problem along with a case study is investigated by the proposed model. The obtained results prove the applicability of the proposed model in real-world scale problems.
    Keywords: warehouse layout, Production Planning, robust possibilistic programming, fuzzy programming}
  • Hamiden Khalifa*, E. E. Ammar

    This paper deals with a multi- objective linear fractional programming problem involving probabilistic parameters in the right- hand side of the constraints. These probabilistic parameters are randomly distributed with known means and variances through the use of Uniform and Exponential Distributions. After converting the probabilistic problem into an equivalent deterministic problem, a fuzzy programming approach is applied by defining a membership function. A linear membership function is being used for obtaining an optimal compromise solution. The stability set of the first kind without differentiability corresponding to the obtained optimal compromise solution is determined. A solution procedure for obtaining an optimal compromise solution and the stability set of the first kind is presented. Finally, a numerical example is given to clarify the practically and the efficiency of the study.

    Keywords: Multi-objective linear fractional programming, Uniform distribution, Exponential distribution, Linear membership function, Fuzzy programming, optimal compromise solution, parametric study}
  • Shiva Ghaffari, Hassan khademi Zareh*, Ahmad Sadeghieh, Ali Mostafaeipour

    This paper proposes a mathematical model for ride-sharing vehicles with a common destination. A number of cars should assign to individuals by a company to pick up other participants in their way to the common destination. Traveling time as an important parameter is considered an uncertain parameter to enhance the applicability of the model which is formulated using fuzzy programming and necessity concept. Moreover, to have a better solution with better productivity, maximizing the earliest departure time of the individuals is considered beside of minimizing total traveling time. This helps to make justice among individuals for departure time. Goal programming is employed to work with objective functions and solve the model. Furthermore, a numerical example is implemented on the model to evaluate the applicability of the model which indicates the efficiency of employing fuzzy programming and considering both of the objective functions using goal programming. Results of the numerical example indicate the importance of considering both of the objective functions together in which ignoring each of them leads to inefficient solutions.

    Keywords: Ride-sharing vehicles, mathematical modelling, fuzzy programming, goal programming}
  • Hadis Derikvand, Seyed Mohammad Hajimolana *, Armin Jabarzadeh, Esmaeil Najafi
    Emergency blood distribution seeks to employ different means in order to optimize the amount of blood transported while timely provision. This paper addresses the concept of blood distribution management in disastrous conditions and develops a fuzzy scenario-based bi-objective model whereas blood compatibility concept is incorporated in the model, and the aim is to minimize the level of unsatisfied demand of affected areas (AAs) while minimizing the cost of the supply chain. The blood supply chain network under investigation consists of blood suppliers (hospitals or blood centers), blood distribution centers (BDCs), and AAs. Demand and capacity, as well as cost, are the sources of uncertainty and in accordance with the nature of the problem, the fuzzy-stochastic programming method is applied to deal with these uncertainties. After removing nonlinear terms, Ɛ-constraint solves the bi-objective model as a single objective one. Finally, we apply a case from Iran to show the applicability of the model, results prove the role of blood distribution management in decreasing the unsatisfied demand about 38%.
    Keywords: blood supply chain, disaster, fuzzy programming, Stochastic programming, Ɛ-constraint}
  • Tanmay Kundu *, Sahidul Islam

    This paper presents an application of interactive fuzzy goal programming to the nonlinear multi-objective reliability optimization problem considering system reliability and cost of the system as objective functions. As the decision maker always have an intention to produce highly reliable system with minimum cost, therefore, we introduce the interactive method to design a high productivity system here. This method plays an important role to maximize the worst lower bound to obtain the preferred compromise solution which is close to the best upper bound of each objective functions. Until the preferred compromise solution is reached, new lower bounds corresponding to each objective functions will be determined based on the present solution to develop the updated membership functions as well as aspiration levels to resolve the proposed problem. Considering judgmental vagueness of decision maker, here we consider the resources as trapezoidal fuzzy numbers and apply total integral value of fuzzy number to transform into crisp one. To illustrate the methodology and performance of this approach, numerical examples are presented and evaluated by comparing with the other method at the end of this paper.

    Keywords: Reliability, Fuzzy programming, Multi, objective programming, Interactive methods, Goal programming}
  • alireza hamidieh, alireza arshadi khamseh, Bahman Naderi
    Nowadays, the design of a strategic supply chain network under the incidence of disruption is regarded as one of the important priorities of governments. Supplying sustainable petrochemical products is considered as a strategic goal by managers who require reliable infrastructure design. Crisis conditions such as natural disasters and sanctions have a destructive effect on the raw materials and product flows. On the other hand, the uncertainty of input parameters affects the business environment and intensifies the condition of disruption. In the present research, a new model of resilient supply chain network is introduced in a critical condition, which consists in a combination of reactive and preventive resilient strategies. In order to deal with the parametric uncertainties caused by changes in the business environment and inadequate knowledge, an effective hybrid possibilistic-flexible robust programming method was presented. The proposed model was capable of controlling the adverse effects of uncertainties and risk-aversion level of output decisions. The extended model was analyzed in the national project of polyethylene strategic supply chain network using real data, which included the flexibility of demand, capacity, and lead time components. The results indicated that optimality and feasibility robustness were guaranteed by presenting efficiency solutions.
    Keywords: Fuzzy programming, reliability, Resilience, Robustness, Supply Chain}
  • Mohammad Ali Shafia *, Sayyede Ashraf Moousavi Loghman, Aghdas Badiee, Kamran Shahanaghi
    Production is a key economic activity with potential long-term social benefits that can be thoroughly realised only if governments comply with their duties towards domestic production. Governments are responsible for the production of sustainable agricultural products via appropriate allocation of subsidies and regulation of price policies that would help take advantage of the potentials underlying agricultural production. In this paper, a model is developed to investigate the interaction between two decision makers in the stackelberg game, government as leader and agriculture as follower, with the ultimate aim of providing benefits to all sectors in the society in the sustainable agriculture paradigm. The proposed model is validated and its efficiency demonstrated via a case study of cotton production as a strategic agricultural production. The model is first solved using a combination of fuzzy mathematical and grey quadratic programming methods to account for the inherent uncertainty in a number of problem parameters. The model is then analyzed against various government-producer interaction scenarios and finally, the analysis results are compared.
    Keywords: government, Sustainable agriculture, Stackelberg game, Social benefit, Grey quadratic programming, Fuzzy programming}
  • Farzaneh Ferdowsi *, Milad Nasiri
    Development of the infrastructure of alternative fuel stations is one of the best ways to extend the use of alternative fuel vehicles. Hence, constructing refueling stations with minimum cost is an important issue. On the other hand, considering the exact value of cost is not match with real cases. In this regard, the cost of building station is considered as a trapezoidal fuzzy value and a mathematical fuzzy programming model is presented in this paper. In order to solve the fuzzy model, first the model is converted to an interval programming model, then the equivalent bi-objective crisp model of the interval programming problem is written. Finally, two interactive fuzzy solution approaches are used to solve the respective bi-objective crisp model. The results show that the performance of the solution approaches is the same.
    Keywords: Refueling station‎, ‎Facility location‎, ‎Fuzzy programming}
  • Mohsen Sadegh Amal Nik *, Javad Ansarifar, Faezeh Akhavizadegan
    Management and scheduling of flights and assignment of gates to aircraft play a significant role to improve the performance of the airport, due to the growing number of flights and decreasing the flight times. This research addresses the assignement and scheduling problem of runways and gates simultaneously. Moreover, this research is the first study that considers the constraint of unavailability of runway’s and the uncertain parameters relating to both areas of runway and gate assignment. One of the distinguishing contributions of the proposed model is that the problem is formulated as a bi-level bi-objective one. The leader objective function minimizes the total waiting time for runways and gates for all aircrafts based on their importance coefficient. Meanwhile, the total distance traveled by all passengers in the airport terminal is minimized by a follower objective function. To solve the proposed model, Benders’ decomposition method is applied. Empirical data are used to show the validation and application of the proposed model. A comparison shows the effectiveness of the model and its significant impact on decreasing the costs.
    Keywords: Aircraft scheduling, gate assignment, Multi, Objective, bi, level, fuzzy programming, Bender's decomposition algorithm}
  • Masoud Rabbani *, Farzad Mehrpour, Amir Farshbaf-Geranmayeh
    Lean manufacturing is a strategic concern for companies which conduct mass production and it has become even more significant for those producing in a project-oriented way by modularization. In this paper, a bi-objective optimization model is proposed to design and plan a supply chain up to the final assembly centre. The delivery time and the quality in the procurement and low fluctuation of the production are the most important lean production principles that are considered. Because of the long-horizon planning and the subjective data gathered, it is necessary to handle uncertainty. Therefore, a robust credibility-based fuzzy programming (RCFP) approach is proposed to perform the robust optimization and to obtain the crisp equivalent of an MILP model using the chance constraint programming method in terms of simultaneous credibility measurement. A real industrial case study is provided to present the usefulness and applicability of the proposed model and programming approach.
    Keywords: Lean manufacturing, Supply chain network design, robust optimization, credibility measure, fuzzy programming}
  • M. Bashiri_H. R Rezaei_A. Farshbaf Geranmayeh_F. Ghobadi
    Most of existing researches for multi response optimization are based on regression analysis. However, the artificial neural network can be applied for the problem. In this paper, two approaches are proposed by consideration of both methods. In the first approach, regression model of the controllable factors and S/N ratio of each response has been achieved, then a fuzzy programming has been applied to find the optimal factor's levels. In the second approach, a tuned Artificial Neural Network (ANN) is used to relate controllable factors and overall exponential desirability function then Genetic Algorithm(GA) is used to find factors optimum value. Mentioned approaches have been discussed in a real case study of oil refining industry. Experimental results for the suggested levels confirm efficiency of the both proposed methods; however, the Neural Network based approach shows more suitability in our case study.
    Keywords: Multi, response optimization, Taguchi Method, Artificial Neural Network, Genetic algorithm, Fuzzy programming}
  • اسماعیل مهدی زاده*، سینا کشاوری
    در این مقاله مسئله مکان یابی مسیریابی همراه با زمان های سفر و زمان های تحویل فازی مورد مطالعه قرار گرفته و یک مدل برنامه ریزی ریاضی دو هدفه پیشنهاد می شود. هدف های در نظر گرفته شده شامل کمینه کردن هزینه های شبکه توزیع و جمع موزون دیرکردها می باشد. هزینه های شبکه توزیع شامل هزینه های نصب دپوها و هزینه های حمل ونقل است و برای هر یک از مشتری ها یک موعد تحویل فازی در نظر گرفته می شود. از آنجا که مسئله مورد نظر در زمره مسائل NP-hard قرار دارد، از دو الگوریتم ژنتیک با مرتب سازی نامغلوب3 و شبیه سازی تبرید چند هدفه برای حل بهره گرفته می شوند. برای تنظیم پارامترهای الگوریتم ها از روش تاگوچی استفاده می شود و برای مقایسه الگوریتم های پیشنهادی تعدادی مسئله در سه مقیاس کوچک، متوسط و بزرگ تولید و حل شده استو نتایج محاسباتی نشان می دهند که الگوریتم شبیه سازی چند هدفه از کارایی بالاتری برخوردار است.
    کلید واژگان: مسئله مکان یابی مسیریابی, مکان یابی تسهیلات, برنامه ریزی فازی, تصمیم گیری چند معیاره}
    Esmaeil Mehdizadeh *, Sina Keshavari
    In this study the location routing problem with fuzzy parameters is taken into account, this problem involves determining the location of the depots and routing of the vehicles in order to serve the customers. In this study a location routing problem with fuzzy travel times and due dates is considered and two objective models are proposed. The considered objectives are minimizing the total costs of the network and minimizing the total weighted tardiness. The costs of the network include the fixed installation costs and the transportation costs. In order to solve this problem a mathematical model is proposed. However since this problem is categorized into NP-hard problem; the mathematical model cannot be solved efficiently. Therefore meta-heuristic algorithms are proposed to efficiently solve this problem.
    Keywords: Location Routing Problem, Facility Location, Fuzzy Programming, Multi, Criteria Decision Making}
  • Saeed Khalili, Mohammad Mehdi Lotfi
    Among the various existing models for the warehousing management, the simultaneous use of private and public warehouses is as the most well-known one. The purpose of this article is to develop a queuing theory-based model for determining the optimal capacity of private warehouse in order to minimize the total corresponding costs. In the proposed model, the available space and budget to create a private warehouse are limited. Due to the ambiguity, some parameters are naturally simulated by expert-based triangular fuzzy numbers and two well-known methods are applied to solve the queuing-based fuzzy programming model and optimize the private warehouse capacity. The numerical results for three cases confirm that unlike the previous approaches, the proposed one may easily and efficiently be matched with various lines of manufacturing environments and conditions.
    Keywords: Optimal warehouse capacity, Queuing theory, Fuzzy programming, Multi, objective}
  • A. Mohajeri *, M. Fallah, F Hosseinzadeh Lotfi
    Recovery of used products is receiving much attention recently due to growing environmental concern. In this paper, we address the carbon footprint based problem arising in closed-loop supply chain where returned products are collected from customers. These returned products can either be disposed or be remanufactured to be sold as new ones again. Given this environment, an optimization model for a closed-loop supply chain in which the carbon emission is expressed in terms of environmental constraints, namely carbon emission constraints, is developed. These constraints aim at limiting the carbon emission per unit of product supplied with different transportation mode. Here, we design a closed-loop network where capacity limits, single-item management and uncertainty on product demands and returns are considered. First, the fuzzy mathematical programming is introduced for uncertain modeling. Therefore, the statistical approach towards possibility to synthesize fuzzy information is utilized. So, using defined possibilistic mean and variance, we transform the proposed fuzzy mathematical model into a crisp form to facilitate efficient computation and analysis. The model is applied to an illustrative example of an uncertain green supply chain (GSC).
    Keywords: Closed-loop supply chain, fuzzy programming, interval programming, carbon footprints, transport mode, preference model, Emission-constraint model}
  • مهدی بشیری، امیر فرشباف گرانمایه
    تنظیم مناسب پارامترها، مهمترین نقش را در عملکرد شبکه عصبی ایفا می کند. لیکن بدلیل اینکه هنوز هیچ روش مشخصی برای انجام این کار وجود ندارد، تنظیم پارامترها همچنان مهمترین مشکل در استفاده از شبکه عصبی مصنوعی محسوب میشود. در این مطالعه، سه معیار به عنوان معیارهای عملکرد شبکه عصبی مصنوعی و سه عامل کنترلی که بیشترین تاثیر را روی معیارهای عملکرد دارند، معرفی شده اند. علاوه بر آن طرح باکس- بنکن برای طراحی آزمایشات مربوط به تنظیم پارامترها بکار برده شده است. بررسی یک فرآیند واقعی برای چنین موضوعاتی از اهمیت ویژه ای برخوردار می باشد، لیکن به منظور بررسی و پیش بینی عملکرد روش پیشنهادی، فرض شده است که متغیر خروجی فرآیند شبیه سازی شده دارای یک تابع نسبتا پیچیده اما معین با متغیرهای ورودی می باشد. بعد از استخراج روابط عوامل کنترلی و معیارهای عملکرد تعیین شده مرتبط با شبکه عصبی طراحی شده، روش برنامه ریزی فازی برای یافتن بهینه ترین ترکیب عوامل کنترلی جهت بهترین عملکرد شبکه عصبی مصنوعی بکار گرفته شده است. نتایج حاصل از مثال عددی بررسی شده، کارایی روش پیشنهادی را نشان میدهد. این رویکرد می تواند برای تنظیم پارامترها در حل انواع مختلف مسائل با استفاده از شبکه عصبی مصنوعی استفاده شود.
    کلید واژگان: شبکه عصبی مصنوعی, تنظیم پارامترها, طرح باکس - بنکن, برنامه ریزی فازی}
    M. Bashiri, A. Farshbaf, Geranmayeh
    Tuning parameters has the most important effect in performance of Neural Network. Main problem in using of ANN is parameter tuning because there is no definite and explicit method to select optimal parameters for the ANN parameters. In this study, three artificial neural network performance criteria and also three important factors which affect the selected criteria have been introduced. Moreover, Box-Behnken design has been applied to analyze the ANN structure parameters and its performance. The proposed approach has been implemented for a simulated process according to a complex mathematical function. After extracting relation between controllable factors and performance criteria, fuzzy programming is used for finding the optimal combination of controllable factors in order the best performance of Neural Network. Results attained from the numerical example show efficiency of the proposed method. Generally, proposed approach can be used for tuning of neural network’s parameters in other problems.
    Keywords: Artificial Neural Network, parameter tuning, Box, Behnken design, Fuzzy programming}
  • حسن حسینی نسب*، محسن شاه محمدی، علی اکبر قاسمی گجوان
    مسئله چیدمان پویای تسهیلات، شاخه ای از مسائل چیدمان تسهیلات است که در آن مقدار جریان مواد در دوره های مختلف زمانی در طول افق برنامه ریزی، ثابت نبوده و با تغییرات همراه است. غالب مطالعات صورت گرفته در این زمینه، با فرض معلوم و قطعی بودن اطلاعات جریان در دوره های مختلف زمانی، همراه بوده اند. حال آنکه در محیط های کسب و کار امروزی که پویایی و عدم قطعیت، دو جزء لاینفک آنهاست، این فرض چندان منطقی به نظر نمی رسد. در این مقاله مسئله چیدمان پویای تسهیلات، در شرایط وجود ابهام در اطلاعات جریان، مورد مطالعه قرار گرفته است. در این راستا، حجم تقاضای محصولات (و به تبع آن حجم جریان مواد) در قالب اعداد فازی، با توابع عضویت متفاوت، تعریف شده اند. مدل سازی مسئله در چارچوب برنامه ریزی فازی و در قالب سه مدل ارزش انتظاری، برنامه ریزی با قید احتمالی و برنامه ریزی بر مبنای شانس، انجام شده و برای حل مدل ها، یک الگوریتم هوشمند ترکیبی ارائه شده است. در پایان با ذکر یک مثال عددی، کارایی الگوریتم در حل مدل های فازی مسئله چیدمان پویای تسهیلات، مورد بررسی قرار گرفته است.
    کلید واژگان: مسئله چیدمان پویای تسهیلات, برنامه ریزی فازی, الگوریتم هوشمند ترکیبی, شبیه سازی فازی, الگوریتم ژنتیک}
    Hasan Hosseini Nasab*, Mohsen Shahmohammadi, Ali Akbar Ghasemi
    The dynamic facility layout problem involves the design of facility layouts in which the flows of materials between activities can change during a multi-period planning horizon. In the static layout problem, it is assumed that all the activities are constant. However, in today’s volatile markets, the business conditions are changing. So the similar changes are imposed on the facility projects and the flows of material between activities. In this paper, a new model of the dynamic facility layout problem is studied that the flows of materials are considered as fuzzy numbers with different membership functions. For modeling the problem, the expected value, the chance programming and the probabilistic programming methods are utilized using fuzzy programming. A new method is proposed for solving the model and a numerical example is given to describe the method.
    Keywords: Dynamic facility layout problem, Fuzzy programming, Hybrid intelligent algorithm, Fuzzy simulation Genetic algorithm}
  • S.J .Sadjadi *, M.B. Aryanezhad, A .Sarfaraz

    In this paper, the researchers present a multi-objective linear fractional inventory model using fuzzy pro-gramming. The proposed method in this paper is applied to a problem with two objective functions. The re-sulted fuzzy model is transformed into an ordinary linear programming. The implementation of the develop-ing model presented in this paper is demonstrated through the use of some numerical examples.

    Keywords: Multi-objective linear fractional programming, Inventory model, Fuzzy programming, Min operator}
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