A lot of researches have been conducted on MOTP with **fuzzy** parameters. Hussein [8] dealt with the complete solutions of MOTP with possibilistic **coefficients**. Das et al. [6] fo- cused on the solution procedure of the MOTP where all the parameters have been expressed as interval values by the decision maker. Ahlatcioglu et al. [1] proposed a model for solving the **transportation** **problem** that supply and demand quantities are given as triangular **fuzzy** numbers bounded from below and above, respectively. Basing on extension principle, Liu and Kao [12] developed a procedure to derive the **fuzzy** objective value of the **fuzzy** transporta- tion **problem** where the **cost** **coefficients**, supply and demand quantities are **fuzzy** numbers. Using signed distance ranking, defuzzification by signed distance, interval-valued **fuzzy** sets and statistical data, Chiang [5] get the **transportation** **problem** in the **fuzzy** sense. Ammar and Youness [3] examined the solution of MOTP which has **fuzzy** **cost**, source and destination parameters. They introduced the concepts of **fuzzy** efficient and parametric efficient solu- tions. Islam and Roy [9] dealt with a multi-objective entropy **transportation** **problem** with an additional delivery time constraint, and its **transportation** costs are generalized trapezoidal **fuzzy** numbers. Chanas and Kuchta [4] proposed a concept of the optimal solution of the **transportation** **problem** with **fuzzy** **cost** **coefficients** and an algorithm determining this solu- tion. Pramanik and Roy [14] showed how the concept of Euclidean distance can be used for modeling MOTP with **fuzzy** parameters and solving them efficiently using priority based **fuzzy** goal programming under a priority structure to arrive at the most satisfactory decision in the decision making environment, on the basis of the needs and desires of the decision making unit.

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and used -cut to obtain a **transportation** **problem** in the **fuzzy** sense expressed in **linear** programming form. Chanas and Kuchta [13] used **fuzzy** numbers of the type L-L to fuzzify **cost** **coefficients** in the objective function and -cut to express the objective function in the form of an interval. Hussien [14] studied the complete set of -possibility efficient solutions of multi objective **transportation** **problem** with possibilistic co-efficients of the objective functions. Li and Lai [15] proposed a **fuzzy** compromise programming **approach** to a multi objective **linear** **transportation** **problem**. Zimmermann [16] showed that solutions obtained by **fuzzy** **linear** programming are always efficient. Subsequently, Zimmermann’s **fuzzy** **linear** programming has developed into several **fuzzy** optimization methods for solving the **transportation** problems. Oheigeartaigh [17] proposed an algorithm for solving **transportation** problems where the capacities and requirements are **fuzzy** sets with **linear** or triangular membership functions. Chanas et al. [18] presented a **fuzzy** **linear** programming model for solving **transportation** problems with crisp **cost** **coefficients** and **fuzzy** supply and demand values. Chanas and Kuchta [19] proposed the concept of the optimal solution for the **transportation** **problem** with **fuzzy** **coefficients** expressed as **fuzzy** numbers and developed an algorithm for obtaining the optimal solution. Saad and Abbas [20] discussed the solution algorithm for solving the **transportation** **problem** in **fuzzy** environment. Liu and Kao [21] described a method for solving **fuzzy** **transportation** problems based on extension principle. Kaur and Kumar [24] proposed a method for solving **fuzzy** **transportation** problems by assuming that a decision maker is uncertain about the precise values of the **transportation** **cost**, availability and demand of the product are represented by generalized trapezoidal **fuzzy** numbers and illustrated with an example. Most of the **fuzzy** **transportation** problems discussed above, the optimal **fuzzy** **transportation** **cost** is negative in nature but in our proposed method we obtain positive optimal **fuzzy** **transportation** **cost** in all most all cases.

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From such a point of view, in this paper, assuming that the decision maker adopts the **fuzzy** decision to integrate membership functions, two types of **fuzzy** approaches are proposed for both **multiobjective** **fuzzy** **linear** programming **problem** with random variable **coefficients** and **fuzzy** random variable **coefficients**. In section 2, a **fuzzy** **approach** is pro- posed for **multiobjective** **fuzzy** **linear** programming **problem** with random variable **coefficients**. In section 3, a **fuzzy** **approach** is proposed for **multiobjective** **fuzzy** **linear** pro- gramming **problem** with **fuzzy** random variable **coefficients**. Section 4 provides a numerical example to demonstrate the proposed **fuzzy** **approach** for **multiobjective** **fuzzy** **linear** pro- gramming **problem** with **fuzzy** random variable **coefficients**. Finally, in section 5, we conclude this paper.

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Amid et al. (2009) formulated a weighted additive **fuzzy** multi-objective model for the supplier selection **problem** under all-unit price discounts. They developed the **problem** in such a way as to concurrently consider the imprecision of information and specify the order quantities to each supplier based on price breaks. The **problem** contains the three objective functions: minimizing the **cost**, minimizing the rejected goods and delayed deliveries, while satisfying capacity and demand requirement limitations. They argued in actfor supplier selection problems. Most of the input information is not known accurately, so they employ the **fuzzy** optimization theory to deal with this vagueness. Moreover, a **fuzzy** weighted additive and mixed integer **linear** programming is extended.

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Abstract—This paper represents a non **linear** bi-criterion generalized multi-index **transportation** **problem** (BGMTP) is considered. The generalized **transportation** **problem** (GTP) arises in many real-life applications. It has the form of a classical **transportation** **problem**, with the additional assumption that the quantities of goods change during the **transportation** process. Here the **fuzzy** constraints are used in the demand and in the budget. An efficient new solution procedure is developed k eeping the budget as the first priority. All efficient time-**cost** trade-off pairs are obtained. D 1 -distance is calculated to each trade-off pair from the

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where t m = t n i j i=1 j=1 (Balanced condition) Such a matrix A, with all of its nonzero elements found in blocks centered roughly on and just below the main diagonal, is called a staircase matrix because of its resemblance to a set of steps. The sub matrices At, t = 1, ... , T, are called diagonal blocks and are of dimensions m t x n t where m = m t t=1 and t=1 n = n t For any nonzero column of an off diagonal block, the associated column of A is called a linking column, with the corresponding linking variable being the appropriate component of the vector x. On the other hand, an all zero column in B t is associated with a variable that is said to be local to period t (since it has no effect on period t + 1 through the matrix B t ). With [c Lt Rt ,c ] is an interval representing the uncertain **cost** for the **transportation** **problem**. the **problem** may be restated as Minimize Z= c x t t t=1 m = m t t=1 n = n t t=1 t 0 m = t n for all t M and t N t t=1 t=1 (7)

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ABSTRACT: The assignment **problem** is a special type of **linear** programming **problem** in which our objectives is to a minimum **cost** (time). The mathematical formulation of the **problem** suggests that this is a programming **problem** and is highly degenerate. All the algorithms developed to find optimal solution of **transportation** problems are applicable to assignment **problem**. However, due to its highly degeneracy nature, a specially designed algorithm widely known as Hungarian method proposed by khun is used for its solution. Examples of these types of problems are assigning men to offices, crew (drivers and conductors) to buses, trucks to delivery routes etc.

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The **transportation** **problem** is one of the earliest applications of **linear** programming problems. In the literature, several methods are proposed for solving **transportation** problems in **fuzzy** environment but in all the proposed methods, the parameters are normal **fuzzy** numbers. In this paper, a general **fuzzy** **transportation** **problem** is discussed. In the proposed method, **transportation** **cost**, availability and demand of the product are represented by symmetric triangular **fuzzy** numbers. We develop **fuzzy** version of Vogel’s algorithms for finding **fuzzy** optimal solution of **fuzzy** **transportation** **problem**. A numerical example is given to show the efficiency of the method.

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3) Now there will be at least one 0 in each row and each column in the reduced **cost** matrix. Suppose first zero (row wise) occurring in (I j) th cell, find total sum of all the costs in ith cell and jth column. Repeat the process for all the zeros. Allocate the maximum possible amount to the cell at the of zero where sum is maximum. Delete the row or column for further calculation where supply from a given source is depleted or demand for a given destination is satisfied.

In this research, a new **approach** (Loop Product Difference) for optimizing the initial basic feasible solution of a balanced **transportation** **problem** is proposed. The proposed technique has been tested and proven efficient by solving several number of **cost** minimizing **transportation** problems and it was discovered that the method gives the same result as that of optimal solution obtained by using MODI/Stepping stone methods. Conclusively, it can be said that proposed technique is easy to adopt and close to optimality if employed with the Inverse Coefficient of Variation Method as an improved technique of obtaining Initial Basic Feasible Solution.

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The above line is a k-**problem** in that a point on the line would solve the k – **problem** if not for ̃ < 0. The intent of the algorithm in following the line is to reach an endpoint where ̃ is zero. An endpoint is reached whenever a variable ̃ = ̃ or ̃ , ̃ ≤ ̃ , becomes nonbasic, that is , equal to zero. The nondegeneracy assumption (3.I) insures that only one variable becomes zero at an endpoint. Distiquising three cases, we now show that an endpoint solves the LCP or uniquely leads to another variable dimension line, which the algorithm follows next. We characterize the new variable dimension line by identifying the variable which is zero at the endpoint and which becomes nonzero along the line.

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(1999) proposes a branch-and-bound **approach** where machine idle time is not allowed. Chang (1999) deals with a no-weighted **problem** without preemption. Rabadia et al. (2004) investigate the **problem** in which due dates of all jobs are identical and setup times depend on jobs’ sequencing. Mazzini and Armentano (2001) have developed a heuristic for minimizing total earliness and tardiness **cost** in a single machine scheduling **problem** with distinct ready times and due dates. Mondal and Sen (2001) suggest an algorithm to solve the **problem** with a common due date. This algorithm uses a graph search space. Wan and Yen (2002) propose an **approach** to combine a Tabu search (TS) procedure and an optimal timing algorithm for solving the **problem** with distinct due windows. Feldmann and Biskup (2003) address the restrictive common due date **problem** by using three meta-heuristic algorithms (evolutionary search (ES), simulated annealing (SA) and threshold accepting (TA)).Ventura and Radhakrishnan (2003) use a Lagrangian relaxation procedure that utilizes the subgradient algorithm to tackle the **problem**. Tavakkoli-Moghaddam et al. (2005) consider the common due date **problem** with the objective of minimizing the sum of maximum earliness and tardiness costs. They propose an algorithm, named idle insert algorithm, to solve this **problem**. Lin et al. (2006) use a sequential exchange **approach** to solve the common due date **problem**.

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reservoir evaporation, and channel routing effects. In pure NFP-based models, these features have been handled through the use of successive iterations (Ilich, 2008, 2009). These it- erative processes are external to the algorithmic solving pro- cedure. Usually the lower or upper limits of links are itera- tively adjusted to meet non-NFP constraints; thus, the prior- ities specified by link costs are unchanged during iterations. By contrast, an LP solver can directly incorporate non-NFP features into the formulation and the algorithmic solving pro- cedure. However, this flexibility may impair the character- istic of priority-based water allocation of NFP. One simple example is that water may be allocated to a junior-priority demand with less flow loss, rather than a senior demand with greater flow loss, if the objective function is not appropri- ately set up in the LP formulation. Another example is the effect of channel flow routing, which may be easily mod- eled by the Muskingum method and incorporated into an LP formulation. Suppose that there are two demands located at the upstream and downstream ends of a river channel, re- spectively, with junior and senior priorities. The travel time required for water to flow through the channel from the loca- tion of upstream (junior) demand to downstream (senior) de- mand exceeds the unit time period of an LP-based simulation model. The portion of water that does not reach the point of downstream demand cannot explicitly contribute to the ob- jective function in the current unit time period. The solution to this issue, similar to that for the flow loss case, consists of allocating water to the junior demand first instead of maxi- mizing satisfaction of the senior downstream demand, if the discrepancy between their assigned **cost** **coefficients** is not large enough to compensate for the retained and ineffective portion of water.

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Scientists had continue utilizing FDEs to construct a more complex model which consists of several FDEs. The stepping stone for this effort is the development of system of **linear** first order **fuzzy** differential equations (SLFDEs). There has been several works regarding SLFDEs with **fuzzy** constant **coefficients** (FCCs) in the literature [19–24]. These include the implementation of variational iteration method and homotopy analysis method. When dealing with FDEs interpreted under strongly generalized differentiability concept, there are two cases of differentiability to be considered [11]. The previous works done did not demonstrated both cases efficiently, for example, the work in [19]. Particularly, only the first case is demonstrated by the authors. Because of this, we intend to use FST for solving SLFDEs with FCCs, and both cases of differentiability interpreted under the mentioned concept will be fully demonstrated. Plus, to the best of our knowledge, this is the first time FST is used to handle such system.

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In this paper, we focus on hierarchical **multiobjective** stochastic **linear** programming problems, and propose an interactive algorithm to obtain a satisfactory solution from among a Pareto optimal solution set. In the proposed method, by considering the conflict between permissible objective levels and and permissible probability levels, the corre- sponding membership functions are integrated through the **fuzzy** decision. In the integrated membership space, Pareto optimal concept is introduced. In section II, hierarchical **multiobjective** programming problems through a probability maximization model is formulated. In section III, hierarchi- cal **multiobjective** programming problems through a fractile optimization model is formulated. It is shown that the two kinds of formulations to obtain Pareto optimal solutions are same. In section IV, an interactive algorithm based on **linear** programming technique is proposed to obtain a satisfactory solution.

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With regard to solving the **fuzzy** fixed-charge **transportation** **problem** (FFCTP), a research has hardly been conducted. Therefore, any method, which provides a good solution for it, will be dis- tinguished. To this end, a new method is pro- posed to find an approximation solution close to the optimal solution to the FFCTP such that all of the parameters (**transportation** **cost**, fixed **cost**, demand and supply) are triangular **fuzzy** numbers (TFNs). The present paper, first, tries to convert the FFCTP into the **fuzzy** **transportation** prob- lem (FTP) by using the development of Balin- ski’s formula. This becomes a **linear** version of the FFCTP for the next stage, and then, tries to obtain the **fuzzy** optimal solution the **linear** ver- sion of the FFCTP. The proposed method obtains a lower and upper bounds both on the **fuzzy** op- timal value of the FFCTP can be easily obtained by using the approximation solution. Since the

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In fact, there are some cases where due to insufficiency in the available infor- mation, the evaluation of the membership and non-membership functions to- gether gives better and/or satisfactory result than considering either the mem- bership value or the non-membership value. Accordingly, there remains a part indeterministic on which hesitation survives. Certainly **fuzzy** optimization is unable to deal such hesitation since in this case here membership and non-membership functions are complement to each other. Here, we extend Zimmermann’s optimization technique for solving FLPP. In our proposed tech- nique, sum of membership degree and non-membership degree always taken as strictly less than one and hence hesitation arises. Consequently, to achieve the aspiration level Z 0 of the objective function, our proposed method for solving

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In section 4, we explain the method of ranking for TIFNs with example. In section 5, we define a intuitionistic **fuzzy** **transportation** **problem**. In section 6, we give the revised distribution algorithm to solve the intuitionistic **fuzzy** **transportation** **problem**. In section 7, a numerical example is given.

This paper presents a **fuzzy** goal programming FGP procedure for solving bilevel **multiobjective** **linear** fractional programming BL-MOLFP problems. It makes an extension work of Moitra and Pal 2002 and Pal et al. 2003. In the proposed procedure, the membership functions for the defined **fuzzy** goals of the decision makers DMs objective functions at both levels as well as the membership functions for vector of **fuzzy** goals of the decision variables controlled by first-level decision maker are developed first in the model formulation of the **problem**. Then a **fuzzy** goal programming model to minimize the group regret of degree of satisfactions of both the decision makers is developed to achieve the highest degree unity of each of the defined membership function goals to the extent possible by minimizing their deviational variables and thereby obtaining the most satisfactory solution for both decision makers. The method of variable change on the under- and over-deviational variables of the membership goals associated with the **fuzzy** goals of the model is introduced to solve the **problem** eﬃciently by using **linear** goal programming LGP methodology. Illustrative numerical example is given to demonstrate the procedure.

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