An Improved Algorithm for Optimal Solution of Unbalanced Transportation Problems

Unbalanced transportation problems are particular kind of transportation problems, but an optimal solution is hard to find for unbalanced transportation problems. Still there is a need for minimizing the transportation cost. Unbalanced–TP deals with two different cases, (i) Excess of accessibility ∑ 𝐀 𝐢 𝐢 (cid:3408) ∑ 𝐁 𝐣 , 𝐣 (ii) Deficiency in accessibility ∑ 𝐀 𝐢 𝐢 (cid:3407) ∑ 𝐁 𝐣 , 𝐣 here in this paper both the cases for getting better optimal solution are discussed. Proposed algorithm is based on dummy rows and dummy columns, by taking the absolute differences (penalty) of Initial & Last cost cells of each row / column in transportation cost-matrix, where the objective function is to find an optimal solution. This method is easy to understand and apply than the other existing methods using Initial Basic Feasible Solution–IBFS. Therefore, the proposed method is very helpful to get optimal solution for unbalanced transportation problems.


INTRODUCTION
In Operation Research-OR, Transportation Problem-TP is specific kind of sub-division of linear programming-LP problems. It is commonly significant in the scheme of decision making [1]. Wherein purpose is to lessen the cost of transportation for businesses with number of destination, while sustaining supply limit and demand prerequisite [2]. To get this aim of cost, we have the quantity and locality of already accessible supplies and the amount demanded on top of the participation united accompanied by each 'sending'. The term 'Transportation Model' is some time deceive because it can be used for plant location, machine assignment, product mix problems and many more as well. Whereas the model is really not limited to transportation/distribution only [5]. Linear programming is a mathematical technique which is widely used in fields of social sciences and business domain. For example: Transportation Problems, Allocation Problems, Diet Problems, Agriculture Problems, Network Flow Problems, Marketing Engineering and several others. L.F. Hitchcock (1941) initially established basic transportation problems. The motive of transportation problem controls optimal distribution arrangements between origins/sources and destinations, which are basically accumulated at many origins to diverse destinations in such a way that the cost of total transportation is minimized [4,7,10]. There are two forms of transportation problems.

Balanced and Unbalanced Transportation Problem
A transportation problem is known as balanced when the summation of all supply bases are equal to the summation of all demand purposes, i.e. ∑ ∑ otherwise it is known as unbalanced transportation problem, i.e. ∑ ∑ or ∑ ∑ , to satisfy both the conditions of unbalanced transportation problems, they are presented with a dummy row or dummy column. If the sum of sources is greater than the sum of requirements, then the dummy destinations are presented with requirements, to solve the difference between sum of sources and sum of requirements cost, the dummy rows and dummy columns are set equal to zero transportation cost [6,8].
In previous, studies many researchers have discussed different methods to solve the transportation problems of Initial Basic Feasible Solution-IBFS and optimal methods. The most advantageous and useful ways for initial  [9] He proposed a different approach which can help the decision makers in handling unbalanced transportation assignment problems to minimize the costs of total transportation and maximize the total profits. A. Sridhar & N. Girmay, et al [11,12] suggests a Heuristic Approach to get improved results of unbalanced transportation problems, and improved the existing VAM. This study has minimized the transportation costs in an easy and effective manner. A. Quddoos, et al [13] has proposed method named ASM-Method to find optimal solution of transportation problems, directly. This method requires easy arithmetical and logical calculation with less time period, and results are comparable to existing MODI-Method and Stepping Stone Method. Rajendra B. Patel [14] the presented method is a little modification of ASM-Method. M. Palanivel, et al [15] analyzed an innovative procedure for solving problems of transportation by using Harmonic Mean Approach. Ravi Kumar R, et al [16] they proposed a new technique named Direct Sum Method using VAM, to discover the I.B.F.S of transportation problems. A. Seethalakshmy & N. Srinivasan [17] proposed a different technique to resolve of unbalanced transportation assignment problems, based on finding the position of '1' by using systematic procedure. This article gives same result as that of the Hungarian Method with less consumption of time. M. A. Metlo, et al [18] Modified North-West Corner Method-MNWCM is developed to discover I.B.F.S based on NWCM. The main objective of this study is to reduce the size of iterations and is used to achieve an optimal solution with minimum time estimation, also the results are compared with existing NWCM.
This paper is a motivation for finding better optimal solution for unbalanced transportation problems. This leads to a different approach than the previous methods to minimize the total cost and to maximize the profit in transporting commodities from sources to destinations. The proposed method is introduced below:

METHODOLOGY: 2.1 Mathematical Formulation of Unbalanced Transportation Model:
Mathematically, the general transportation model is given as follows: Minimize: (Total transportation cost) Z = ∑ ∑ (1) Subject to: (Supply from sources) The unbalanced transportation problems are of two cases (i) ∑ ∑ and ∑ ∑ . There are two feasible cases: i. Step#02:-Choose the smallest cost cell C ij in row/column and allocate with selected maximum penalty (P i /P j ).
Step#03:-"In case of ties", If two or more of the smallest cost cells C ij in row/column are same then select the top entry of them to allocate. If two or more of maximum penalty (P i /P j ) are same then select the smallest cost cell C ij in row/column. Step#04:-Repeat steps (1 to 3) for remaining transportation cost-matrix, continue this procedure until all the supply (A i ) and demand (B j ) are become zero. Note: The dummy row/column should be allocated in last.

CONCLUSION
In this paper, we have promoted an improved algorithm for obtaining better optimal solution of unbalanced transportation problems. According to work computation, the proposed method is easy and less arithmetical calculation required to get optimal solution compared to existing MODI-Method. Also a comparison of proposed algorithm is made with NWC-Method, LC-Method and MODI-Method. Various numerical examples of different -sizes of both the cases are tested and critically observed that the proposed algorithm provide optimal solution directly.