Initial Basic Feasible Solution Research Articles

A New Technique for Solving Transportation Problems by Using Decomposition-Based Pricing and its Implementation in Real Life

In this paper, we develop a new technique for solving transportation problems (TP) and develop a computer code by using mathematical programming language AMPL. There are many existing techniques for solving TP problems in use. By these techniques one has to determine initial basic feasible solution at first then improve this solution to determine optimal solution by another method. But this process is very lengthy and time consuming. By our technique we can determine optimal solution directly without determining initial basic feasible solution and optimal solution separately and we hope that this technique will provide an easier way than that of the other methods. We use the idea of decomposition based pricing (DBP) method to develop our technique. To our knowledge, there is no other paper which used DBP to solve TP. We demonstrate our technique by solving real life models developed by collecting data from a business organization of Bangladesh.Dhaka Univ. J. Sci. 64(1): 45-50, 2016 (January)

Incessant Allocation Method for Solving Transportation Problems

Industries require planning in transporting their products from production centres to the users end with minimal transporting cost to maximize profit. This process is known as Transportation Problem which is used to analyze and minimize transportation cost. This problem is well discussed in operation research for its wide application in various fields, such as scheduling, personnel assignment, product mix problems and many others, so that this problem is really not confined to transportation or distribution only. In the solution procedure of a transportation problem, finding an initial basic feasible solution is the prerequisite to obtain the optimal solution. Again, development is a continuous and endless process to find the best among the bests. The growing complexity of management calls for development of sound methods and techniques for solution of the problems. Considering these factors, this research aims to propose an algorithm “Incessant Allocation Method” to obtain an initial basic feasible solution for the transportation problems. Several numbers of numerical problems are also solved to justify the method. Obtained results show that the proposed algorithm is effective in solving transportation problems.

A New Approach to Solve Transportation Problems

Finding an initial basic feasible solution is the prime requirement to obtain an optimal solution for the transportation problems. In this article, a new approach is proposed to find an initial basic feasible solution for the transportation problems. The method is also illustrated with numerical examples.

Tuncay Can’s Approximation Method to Obtain Initial Basic Feasible Solution to Transport Problem

Obtaining an initial basic feasible solution to a transport problem – or a corner point in the convex polytope region – is extremely important in terms of reaching the optimal solution to the problem in the shortest time. When a transport problem is basically accepted as a linear programming problem, a degenerated solution is caused by the structure of the simplex method used when modelling with linear programming and located in a corner point sometimes at the optimal solution itself but mostly in close proximity to the optimal solution vector. One of the ways to eliminate this degenerated solution is to employ approximation methods. The main aim of this paper is to introduce Tuncay Can’s approximation method, which was developed as an alternative to the approximation methods in the literature for a balanced transport problem. Tuncay Can’s approximation method usually has less iterations than other approximation methods. In this paper, the Tuncay Can approximation method is introduced as an alternative to The North West Corner Rule, Minimum Cost Method, and the RAM and VAM methods.

On multi-index fixed charge bi-criterion transportation problem

In this paper, we propose an algorithm for solving a multi-index fixed charge bi-criterion transportation problem and present a method for finding the optimum trade-off pair among efficient cost–time trade-off pairs. The proposed algorithm minimizes variable and fixed costs simultaneously to yield an optimal basic feasible solution. The results of the computational experiments suggests that our algorithm is better than the one proposed by Ahuja and Arora (Indian Journal of Pure and Applied Mathematics, 32(5) 739-746, 2001). We also provide an algorithm to obtain an initial basic feasible solution of the multi-index transportation problem that may be used as a better initial solution for other existing procedures to enhance convergence to the optimal solution.

Modified VOGEL Method to Find Initial Basic Feasible Solution (IBFS) Introducing a New Methodology to Find Best IBFS

This research presents transportation modeling approach to solve transportation problems that flow of items from a certain number of suppliers with various production capacities to a certain points of destination. The aim of the paper is to achieve minimum cost of transportation flow for an initial basic feasible solution (IBFS) which consequently will be closer to the optimal solution of the model. Classical transportation methods for distribution are discussed in details and their IBFS are calculated and compared to the results of newly introduced methodology. The results show that the new method has an advantage over the classical transportation methods through achieving a competitive cost for the IBFS.

LU Decomposition in DEA with an Application to Hospitals

A fundamental problem that usually appears in linear systems is to find a vector $$\mathbf{x}$$x satisfying $$\mathbf{Bx}=\mathbf{b}$$Bx=b. This linear system is encountered in many research applications and more importantly, it is required to be solved in many contexts in applied mathematics. LU decomposition method, based on the Gaussian elimination, is particularly well suited for spars and large-scale problems. Linear programming (LP) is a mathematical method to obtain optimal solutions for a linear system that is more being considered in various fields of study in recent decades. The simplex algorithm is one of the mostly used mathematical techniques for solving LP problems. Data envelopment analysis (DEA) is a non-parametric approach based on linear programming to evaluate relative efficiency of decision making units (DMUs). The number of LP models that has to be solved in DEA is at least the same as the number of DMUs. Toloo et al. (Comput Econ 45(2):323---326, 2015) proposed an initial basic feasible solution for DEA models which practically reduces at least 50 % of the whole computations. The main contribution of this paper is in utlizing this solution to implement LU decomposition technique on the basic DEA models which is more accurate and numerically stable. It is shown that the number of computations in applying the Gaussian elimination method will be fairly reduced due to the special structure of basic DEA models. Potential uses are illustrated with applications to hospital data set.

On Intuitionistic Fuzzy Transportation Problem Using Hexagonal Intuitionistic Fuzzy Numbers

In this paper we introduce Hexagonal intuitionistic fuzzy number with its membership and non membership functions. The main objective of this paper is to introduce an Intuitionistic Fuzzy Transportation problem with hexagonal intuitionistic fuzzy number. The arithmetic operations on hexagonal intuitionistic fuzzy numbers are performed. Based on this new intuitionistic fuzzy number, we obtain a initial basic feasible solution and optimal solution of intuitionistic fuzzy transportation problem. The solutions are illustrated with suitable example.

New Procedure of Finding an Initial Basic Feasible Solution of the Time Minimizing Transportation Problems

Minimization of transportation time is a great concern of the transportation problems like the cost minimizing transportation problems. In this writing, a transportation algorithm is developed and applied to obtain an Initial Basic Feasible Solution (IBFS) of transportation problems in minimizing transportation time. The developed method has also been illustrated numerically to test the efficiency of the method where it is observed that the proposed method yields a better result.

Comparison of Transportation Algorithms Using Data from Katsina State Transport Authority, Katsina State, Nigeria

Transportation problem has been one of the most important applications of Linear programming. Transportation problems have become vastly applied in industrial organizations with multiple manufacturing units, warehouses and distribution centers. In this stugy, the methods of finding initial basic feasible solution of balanced transportation problem are studied and compared to find among them the best in terms of efficiency. The initial basic feasible solution tableaus of all the methods are constructed using data collected from Katsina State Transport Authority. The costing of the allocated cells associated with the initial basic feasible solutions of the five methods are computed and compared with that of optimal solution which was found to be N1,098,000:00. It was observed that Vogel’s approximation method, Least-Cost method and Column minimum method yielded better starting solutions. The North-West Corner method and Row minimum methods though simple to compute yielded starting solutions far from the optimal solution. In addition, Vogel’s approximation method is more difficult and requires more iteration. The best transportation network for Katsina State Transport Authority was obtained.Keywords: Comparison, Transportation algorithms, Initial Basic Feasible Solution, Optimal solution, Linear programming solver

Transportation Algorithm with Volume Discount on Distribution Cost (A Case Study of the Nigerian Bottling Company Plc Owerri Plant)

This study is focused on the Application of Transportation Algorithm with volume Discount on distribution cost using Nigerian Bottling Company Plc Owerri Plant. This paper is intended to determine the quantity of Fanta (in crates), Coke (in crates) and Sprite (also in crates) that the Company should distribute in a month in order to minimize transportation cost and maximize profit. A problem of this nature was identified as a Nonlinear Transportation Problem (NTP), formulated in mathematical terms and solved by the Karush-Kuhn-Tucker (KKT) optimality condition for the NTP. A statistical software package was used to obtain the initial basic feasible solution using the Least Cost Method. Thus, analysis revealed that the optimal solution that gave minimum achievable cost of supply was the supply of 5000 crates of Fanta and 6000 crates of the same product to Umuahia market zone and Afikpo respectively. 7000 crates of Coke, 9000 crates and 1000 crates of the same product should be supplied to Orlu, Mbaise, and Afikpo market zones respectively. 6000, and 5000 crates of Sprite should be allocated to Mbaise and Umuahia market zones respectively, at a total cost of N377, 000.

Finding an Initial Basic Feasible Solution for DEA Models with an Application on Bank Industry

Nowadays, algorithms and computer programs, which are going to speed up, short time to run and less memory to occupy have special importance. Toward these ends, researchers have always regarded suitable strategies and algorithms with the least computations. Since linear programming (LP) has been introduced, interest in it spreads rapidly among scientists. To solve an LP, the simplex method has been developed and since then many researchers have contributed to the extension and progression of LP and obviously simplex method. A vast literature has been grown out of this original method in mathematical theory, new algorithms, and applied nature. Solving an LP via simplex method needs an initial basic feasible solution (IBFS), but in many situations such a solution is not readily available so artificial variables will be resorted. These artificial variables must be dropped to zero, if possible. There are two main methods that can be used to eliminate the artificial variables: two-phase method and Big-M method. Data envelopment analysis (DEA) applies individual LP for evaluating performance of decision making units, consequently, to solve these LPs an IBFS must be on hand. The main contribution of this paper is to introduce a closed form of IBFS for conventional DEA models, which helps us not to deal with artificial variables directly. We apply the proposed form to a real-data set to illustrate the applicability of the new approach. The results of this study indicate that using the closed form of IBFS can reduce at least 50 % of the whole computations.

A computational study on the number of iterations to solve the transportation problem

The Transportation Problem is one of the classical problems in Operations Research. The development of more powerful personal computers in the past years made it possible to solve larger problems, creating huge opportunities of applications. The MODI method is the most cited method in the literature to solve the Transportation Problem. This method requires an initial basic feasible solution and for this purpose the literature often presents the Northwest Corner Rule, the Lest Cost Method and the Vogel Method. This paper aims to study these three different methods and analyze the number of iterations necessary to reach the optimal solution when they are used.

Paradox Algorithm in Application of a Linear Transportation Problem

Paradox seldom occurs in a linear transportation problem, but it is related to the classical transportation problem. For specific reasons of this problem, an increase in the quantity of goods or number of passengers (as used in this paper) to be transported may lead to a decrease in the optimal total transportation cost. Two numerical examples were used for the study. In this paper, an efficient algorithm for solving a linear programming problem was explicitly discussed, and it was concluded that paradox does not exist in the first set of data, while paradox exists in the second set of data. The Vogel’s Approximation Method (VAM) was used to obtain the initial basic feasible solution via the Statistical Software Package known as TORA. The first set of data revealed that paradox does not exist, while the second set of data showed that paradox exists. The method however gives a step by step development of the solution procedure for finding all the paradoxical pair in the second set of data.

A Method for Solving Fuzzy Transportation Problem (FTP) using Fuzzy Russell's Method

The basic transportation problem was originally developed by Hitchcock. In the literature several methods are proposed for solving Fuzzy transportation problem. In this paper, we propose a new algorithm called Fuzzy Russell's method for the initial basic feasible solution to a Fuzzy transportation problem. To examine the proposed method a numerical example is solved. Fuzzy numbers may be normal or abnormal, triangular or trapezoidal or any LR fuzzy number. We can use this proposed method for any kind of Fuzzy numbers.

Solving Linear Programming Problem With Fuzzy Right Hand Sides A Penalty Method

Linear programming problems with trapezoidal fuzzy variables (FVLP) have recently attracted some interest. Some methods have been developed for solving these problems. Fuzzy primal and dual simplex algorithms have been recently proposed to solve these problems. These methods have been developed with the assumption that an initial Basic Feasible Solution (BFS) is at hand. In many cases, finding such a BFS is not straightforward and some works may be needed to get the simplex algorithm started. In this paper, we propose a penalty method to solve FVLP problems in which the BFS is not readily available.

A novel approach for obtaining initial basic solution of transportation problem

The importance of finding efficient solutions for large-sized transportation problems (TPs) are borne out by many practical problems in industries, military affairs, etc. In this way, each optimal solution algorithm needs an initial basic feasible solution that could lead to a better start of the heuristic solution and less works to reach an optimal solution. With this motivation, we develop a high-impact heuristic approach entitled maximum demand to obtain the initial solution, which leads to fast, accurate and efficient optimal solutions of TPs. For the evaluation of the proposed heuristic, computational experiments were generated to compare it with other well-known heuristics in the literature. The results demonstrate the high performance of our heuristic in terms of different criteria.

A Primal-Dual Simplex Algorithm for Solving Linear Programming Problems with Symmetric Trapezoidal Fuzzy Numbers

Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri [2]. The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.

On a Paradox in Linear Plus Linear Fractional Transportation Problem

A paradoxical situation arises in a linear plus linear fractional transportation problem (LPLFTP), when value of the objective function falls below the optimal value and this lower value is attainable by transporting larger amount of quantity. In this paper, a new heuristic is proposed for finding initial basic feasible solution for LPLFTP and a sufficient condition for the existence of a paradoxical solution is established in LPLFTP. Two numerical examples are proposed for explanation of the algorithms. Keywords: Transportation problem; linear plus linear fractional function; more-forless paradox.

Hybrid-LP: Finding advanced starting points for simplex, and pivoting LP methods

The simplex method has proven its efficiency in practice for linear programming (LP) problems of various types and sizes. However, its theoretical worst-case complexity in addition to its poor performance for very large-scale LP problems has driven researchers to develop alternative methods for LP problems. In this paper, we develop the hybrid-LP; a two-phase approach for solving LP problems. Rather than following a path of extreme points on the boundary of the feasible region as in the simplex method, the first phase of the hybrid-LP moves through the interior of the feasible region to obtain an improved and advanced initial basic feasible solution (BFS). Then, in the second phase simplex or other LP methods can be used to find the optimal solution.Since the introduction of polynomial-time methods for LP, a considerable amount of research has focused on interior-point methods for solving large-scale LP problems. Although fewer iterations are needed for interior-point methods to converge to a solution, the iterations are computationally intensive. Our approach is a hybrid method that uses a computationally efficient pivot to move in the interior of the feasible region in its first phase. This single iteration is able to bypassing several extreme points to an improved BFS, which can then be used as a starting point in any LP method in the second phase of the method. Our approach can also be modified to perform a number of interior pivots in the first phase based on the trade-off between the number of iterations and the running time.The hybrid-LP uses an efficient pivoting iteration which is computationally comparable to the standard simplex iteration. Another feature is adaptability in finding the advanced starting point by avoiding the boundaries of the feasible region. In addition, the hybrid-LP has the ability to start from a feasible point which may not be a BFS. Our computational experiments demonstrate that the hybrid-LP reduces both the number of iterations and the running time compared to the simplex method on a wide range of test problems.