# A Compensatory Approach to Multiobjective Linear Transportation Problem with Fuzzy Cost Coefficients

## Top PDF A Compensatory Approach to Multiobjective Linear Transportation Problem with Fuzzy Cost Coefficients: ### A Compensatory Fuzzy Approach to Multi-Objective Linear Transportation Problem with Fuzzy Parameters

A lot of researches have been conducted on MOTP with fuzzy parameters. Hussein  dealt with the complete solutions of MOTP with possibilistic coefficients. Das et al.  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.  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  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  get the transportation problem in the fuzzy sense. Ammar and Youness  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  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  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  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. ### Fuzzy Transportation Linear Programming Models based on L R Fuzzy Numbers

and used  -cut to obtain a transportation problem in the fuzzy sense expressed in linear programming form. Chanas and Kuchta  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  studied the complete set of  -possibility efficient solutions of multi objective transportation problem with possibilistic co-efficients of the objective functions. Li and Lai  proposed a fuzzy compromise programming approach to a multi objective linear transportation problem. Zimmermann  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  proposed an algorithm for solving transportation problems where the capacities and requirements are fuzzy sets with linear or triangular membership functions. Chanas et al.  presented a fuzzy linear programming model for solving transportation problems with crisp cost coefficients and fuzzy supply and demand values. Chanas and Kuchta  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  discussed the solution algorithm for solving the transportation problem in fuzzy environment. Liu and Kao  described a method for solving fuzzy transportation problems based on extension principle. Kaur and Kumar  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. ### Fuzzy Approaches for Multiobjective Fuzzy Random Linear Programming Problems Through a Probability Maximization Model

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. ### A Compensatory Fuzzy Approach to Multi-Objective Supplier Selection Problem in All- Unit Discount Environments With Multiple-Item

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. ### Multi-Index Bi-Criterion Transportation Problem: A Fuzzy Approach

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 ### Fuzzy Approach; Stair-case linear programming Problem with cost coefficient

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) ### Objective Linear Programming Approach for Travelling Salesman Problem Fuzzy Ranking Method

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. ### Solving fuzzy transportation problem using symmetric triangular fuzzy number

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. ### A New approach for Solving Transportation Problem

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. ### A New And Efficient Proposed Approach For Optimizing The Initial Basic Feasible Solution Of A Linear Transportation Problem

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. ### A New Dimension Algorithmic Approach to Solve Fuzzy Linear Complementarity Problem with Interval Numbers Approach

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. ### A Possibility Linear Programming Approach to Solve a Fuzzy Single Machine Scheduling Problem

(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. ### Determination of cost coefficients of a priority based water allocation linear programming model – a network flow approach

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. ### Fuzzy Sumudu Transform for Solving System of Linear Fuzzy Differential Equations with Fuzzy Constant Coefficients

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 . The previous works done did not demonstrated both cases efficiently, for example, the work in . 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. ### Interactive Fuzzy Decision Making for Hierarchical Multiobjective Stochastic Linear Programming Problems

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. ### On approximation of the ‎f‎ully fuzzy fixed charge transportation ‎problem

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 ### Solving Intuitionistic Fuzzy Linear Programming Problem

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 ### Revised Distribution Method for Intuitionistic Fuzzy Transportation Problem

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. ### Fuzzy Goal Programming Procedure to Bilevel Multiobjective Linear Fractional Programming Problems

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. 