The SS Method to Obtain an Optimal Solution of Transportation Problem

Abstract


Introduction
Transportation is used to regulate distribution from sources that provide the same product, to places that need it optimally. Transportation is used to solve business problems, capital expenditure, allocation of funds for investment, location analysis, assembly line balancing and production planning and scheduling (Candra, 2016). The case of transportation arises when someone tries to determine the method of delivery (distribution) of a type of goods (item) from several sources (locations of supply) to several destinations (locations of demand) that minimize costs. The goal in this transportation problem is to allocate goods at the source in such a way that all needs at the destination (demand location) are met. According to Adi Nugroho et.al (2015), transportation is the delivery an item and people from the place of origin to the destination. Whereas according to Harvanda et. Al (2023), transportation is basically a means of moving people or items from one place to another. So that it can be concluded that the transportation problem is distributing an item from several sources to several destinations so that it can be minimize transportation costs.
To obtain a feasible solution in solving transportation problems, there are several methods, namely the North West Corner Method, Least Cost Method, and Vogel's Approximation Method. After obtaining the feasible solution, the next step is to perform an optimality test to obtain an optimal solution with using the Stepping Stone Method or the Modified Distribution Method (Utami & Dewi, 2019). Along with the development of the times, several new methods have emerged that immediately get optimal solutions including the Zero Suffix Method (Ngastiti & Surasoro, 2018), Zero Point Method (Pratiwi, 2016), Zero Neighbouring Method (Aneja & Bhatia, 2018), Exponential Approach Method (Hidayat, 2016) and others.
Zero Suffix Method is one of the methods of optimizing transportation problems that directly tests the optimumancy of the transportation problem table without having to determine the initial solution (Karnila et. al, 2019). Improved Zero Point Method is very useful method to solve all kinds of transportation problems, this method provides an optimal solution without help of any other modification method (Utami & Dewi, 2019). Some of these methods focus on the cost of a reduction result that is worth zero. Most of those direct methods manage to provide the optimal solution on the issue of balanced transportation, while on the issue of unbalanced transportation does not necessarily result in an optimal solution.
The SS method (Sheethalakshmy & Srinivasan, 2016) is a new method in solving transportation problems so it is necessary to study whether the method it provides optimal solutions to transportation problems both balanced and unbalanced. In this research discusses about steps to find the optimum solution on the transportation problem with using the SS Method (Sheethalakshmy & Srinivasan, 2016) so as to determine the optimal quantity of items that must be distributed from several sources to several destinations for a minimum cost transportation. Transportation problem first raised by F. L. Hitchcook (1941) and is known for the Hitchcook distribution problem which is a problem arrangement of the distribution of similar items from a number of sources to a number of places which requires optimally. The model of transportation problem is as follows (Karnila, et. Al, 2019): ≥ 0, = 1,2, … , ; = 1,2, … , In the transportation model, the ability of sources to provide an items (∑ =1 ) is not necessarily equal to the level of demand from a number of objectives (∑ =1 ) so there are three possibilities that will occur namely (Karnila, et. Al, 2019): The first possibility is balanced transportation, while the second and third possibilities are unbalanced transportation. A transportation model is said to be balanced if the amount of supply must be equal to the number of demand. While transportation problems said to be unbalanced if the amount of supply is not equal the amount of demand. But every transportation problems can be made balanced by inserting artificial variable.
If the amount of demand exceeds the amount of supply, then a dummy source that will supply the shortage, i.e. as much as ∑ =1 − ∑ =1 . Conversely, if the amount of supply exceeds the amount of demand, then a dummy goal that will absorb the excess is a much as ∑ =1 − ∑ =1 . Transportation cost per unit ( ) from dummy source and dummy destination is zero. It causes in reality from the source of dummy does not occur delivery. In order to understand transportation problems properly and precisely, transportation problems can be described in the form of Table 1. as follows (Dimyati, A. & Dimyati, A., 2010): Table 1. Transportation Table   Sources Destination (

Method
The method used by the author in this study is:

Results
Transportation problems are problems with the distribution of an item or product from multiple sources to multiple destination with a view to minimizing transportation costs or maximizing profits.
There are steps to get the optimal solution to the transportation problem by searching the feasible solution then performs an optimization test. Along with the development of the times, several new simply methods appeared, one of which was the SS Method proposed by A. Seethalakshmy and Dr. N. Srinivasan (Sheethalakshmy & Srinivasan, 2016). The SS Method can be provided quick steps to get the optimal solution of the transportation problem.
The following is a definition of a feasible solution and optimal solution to the transportation problem.
Here is a theorem to guarantee that transportation problems have a feasible solution:

The transportation problem is said to be unbalanced if the amount of inventory is not the same with the number of requests is
. Transportation problems not balanced can occur in Theorem 2. (Pandian & Natarajan, 2010) For any optimal solution to transportation problems (P1) is the optimal solution of the transportation problem (P). The SS Method is a direct method to obtain an optimal solution of transportation problem without finding initial feasible solution. SS Method steps to determine the optimum solution of transportation problems (minimum cases) are follows:

Form a transportation table
For transportation issues case minimum cost ( ). For unbalanced transportation problem needs to be balanced by adding rows or a dummy column.

Row Reduction
Reducing the row by specifying the smallest element of the on each row, then subtract each row element by the smallest element on that row, which is ( − ).

Column Reduction
Reducing the column by determining the smallest element on each column, then subtract each column element by the smallest element in that column, namely ( − − ).

Allocate
In a transportation table there will be at least one zero cell on each rows and columns, which is ( − − ) = 0, then calculate the cost reduction, i.e. is the sum of each element on the ith rows and the j-th column of the cell that is zero. Next allocate the value demand or supply on selected cells that have a cost value the largest reduction. If there is the largest summation result equal to or more than one, then zero cells are selected that have a minimum supply or demand.

Improvements to the transportation table
Create a new transportation table by ignoring or marking rows or a column in which supply or demand has been fulfilled. Checks whether the new transportation table has at least one zero in every row and column. If not, go back to steps 2 and 3, but if already have at least one zero on each row and column, go to step 6.
6. Repeat step 4 and 5 until all rows and columns the demand is fulfilled.
As for the theorem that guarantees that any solution of the transportation problem the minimum case International Journal of Mathematics and Mathematics Education (IJMME) with the use of the SS Method will be obtained optimal solution, they are as follows:

Theorem 4. Solution that obtained by the SS Method for any problem minimum cases transportation both balanced and unbalanced constitutes optimal solution.
Proof.
Given any balanced transportation problem with the function of the destination minimize. Analogous to the unbalanced transportation problem of minimum cases. For the unbalanced transportation problem, it needs to be balanced with adding a dummy row or column. Drawing up a preliminary transportation table of minimum cases transportation issue given.
Provided that ( ) is the smallest value of the i-th row of the table ( ). Then subtract any i-th row element with ( ) so that a table is obtained ( − ).
Provided that ( ) is the smallest value of the j-th column of the table ( − ). Then subtract each element of the j-th column with ( ) so that a table is obtained ( − − ).
Then calculate the cost of reducing , which is the sum of each element on the i-th row and j-th column on the zero valued ij cell and allocates a demand or supply value on the selected cell that has the largest reduction cost value. If there is the largest summation result which is equal to or more than one, then a zero cell is selected that has an inventory or minimum demand. Based on Theorem 4, using the SS Method will be obtained optimal solution to any well balanced and unbalanced transportation problem with minimum cases. Step 2: After forming the initial transportation table, the next step is to perform row reduction by specifying the smallest element on each row, then subtracts each row element by the smallest element on that row. At this step, taken the smallest element on the first row which is 6, then subtract element on the first row by 6 and do the same thing on the other row so obtained the table as follows: Step 3: Reducing columns by specifying the smallest element in each column, then subtract each column element by the smallest element on the column. The smallest element in the third column is 4, then subtract each element in the third column by 4. Similarly, column fourth, the smallest element in the fourth column is 1, then subtracts any element in the fourth column with 1 so that a table is obtained: Step 4: Cause there is already at least one zero cell in each row and column, then calculates the cost of reduction, is the sum of each element on i-th row and jth column of the zero valued ij cells. Then allocate demand or supply value on selected cells that have a cost value the largest reduction. If there is the largest summation result equal to or more from one, a zero cell is selected that has a supply or demand that minimum cells with zero value include cell (1,2), (1,3), (2,2), (3,1), (3,4), (4,1) with reduction cost, are:
Step 6 for Example of Balanced Transportation Table   Source Destination (in ten of thousands of rupiah) Supply Warehouse 1 Warehouse 2 Warehouse 3 Warehouse 4 Step 7: Cause it still has at least one zero cell in each row and column, it can skip to the next step which is to repeat the fourth and fifth steps until the demand row and supply column are fulfilled.
The cell with zero value include cells (2,2), (3,1), (3,4), and (4,1) then calculated the cost of reduction by summing each of the elements that has not been marked on the cell.

Given examples of balanced transportation problem solved by SS Method.
A company engaged in the field of beverage production located in the Rungkut Industry area, Surabaya.
The company has several depots, namely Sier, Tandes, and Gempol with successive production capacities of 3600 bottles, 2100 bottles, and 1500 bottles. This items are distributed from the depot to several supermarkets A by 2500 bottles, B 850 bottles, C 1800 bottles, and D for 2000 bottles.
As for the distribution fee (in ten rupiah) from a depot to several supermarkets varies according to the distance traveled. By data obtained, problems that occur in this beverage company are unbalanced transportation problem. The amount of demand for beverage is smaller than the number of beverafe that this company provides. From the problem the minimum cost of distributing goods to this company will be determined. Data on the problem contained in this beverage company as follow:  Table 4 is unbalanced transportation problem ∑ =1 > ∑ =1 where the amount of inventory to be distributed exceeds the amount of demand so it needs to be balanced by adding a dummy column of ∑ =1 − ∑ =1 in order to solve transportation problems so that the table becomes: Step 2: After adding a dummy column, the next step is to do reduction rows by specifying the smallest element on each row, then substracts each line element by the smallest element on that row. The smallest element in each row is 0 so that when a row reduction is performed, the table will not change and go directly to the next step.
Step 3: Reducing column by specifying the smallest element in each column, then subtract each column element by the smallest element. The smallest element in the first column is 3, then subtracts each element in the first column with 3 and do the same thing on the other column so that the following table is obtained: Step 4: Calculating the cost of reducing is the sum of each element on the i-th row and the j-th column of the zero value ij cell. Then allocate demand or supply value in zero cells that have reduction cost value biggest. If there is the largest summation result equal to or more than one, then a zero cell is selected that has a minimum supply or demand.
The cell with zero value are cells (1,1), (1,2), (1,3), (1,5), (2,4), (2,5), (3,5) with reduction costs are: The cell that has the largest summation result is (3,5) cell and obtained table as follow: Step 5: Check whether the table already has at least one zero cell in the row and column. If so, you can go directly to the next step. But if no, it is necessary to carry out the reduction of rows and columns.
So that, the minimum distribution cost is Rp341.000,00.

Conclusion
Based on the discussion, conclusions can be drawn that SS Method is an alternative method that can be used to solving transportation problems. This method provides steps that simple in determining the optimal solution without having to look for a feasible solution on transportation problem so that it is easier to understand and faster calculations determine its optimal solution. Completion of this method begins with perform row and column reduction then calculate the cost of reducing by summing each element in the i-th row and the j-th column on ij cells that are worth zero. Next allocate the value of the inventory or query on the selected zero cell that has the largest summation value. If there is the same largest summation result, then a zero cell is selected which have a minimum supply value or demand value. The results obtained using the SS Method on cases transportation issues minimum both balanced and unbalance is the optimal solution.

Recommendations
This research is limited to the problem of minimum case transportation, whether balanced or not balanced. further research can continue to be developed in other cases that occur in a company more specifically