Optimization of Drinking Water Distribution Costs Using Vogel's Approximation Method (VAM) and Three Modified Methods of VAM (Case Study: Sikumbang Kampar Spring)

Abstract


INTRODUCTION
Transportation methods manage distribution from sources that provide the same product where it is needed (destination) optimally [1]. Product allocation must be regulated because many differences in the allocation cost from a source to a destination. Moreover, differences in the amount of existing transportation capacity and needs at the destination [2]. Transportation problems can be solved by several methods, namely the Least Cost Method (LCM), North West Corner (NWC), and Vogel's Approximation Method (VAM). VAM usually produces a better initial solution than other methods [2]. VAM is based on the difference between the first most minor transportation cost and the following most negligible cost in a cell in a row or column [3]. VAM also does not require a long time to find the optimal solution [4]. However, in its application, there are still VAM that produce solutions that are not optimal [5]. Several studies have been conducted to improve VAM, including the Improved Vogel's Approximation Method (IVAM), Max-Min Vogel's Approximation Method (MM-VAM), and Modified Vogel's Approximation Method (MVAM). No theory guarantees that the initial solution is the optimal solution; to obtain the optimal solution, one of them is using the stepping stone method [6].
Related research conducted by Shafarda using VAM with Optimal Test stepping stone resulted in an initial solution that was not optimal [7]. Subsequent research conducted by Nahar using IVAM on distribution costs by the Logistics Affairs Agency (BULOG) resulted in a more optimal calculation of transportation costs than the company's calculation [8]. Furthermore, the research conducted by Tampubolon, PT. Perkebunan Nusantara IV produced an initial solution using MM-VAM, which was better than LCM [9]. Research in the application of MVAM to minimize transportation costs at the tofu factory conducted by Afiani resulted in a more optimal transportation cost than the factory calculation with a few iterations [10]. Comparing the revised distribution method and the stepping stone method in the research conducted by Saputri resulted in an optimal solution using the stepping stone method, which is better than the revised distribution method [11]. Based on the research that has been done, the purpose of this research is to minimize the cost of distributing drinking water from Sikumbang Kampar Springs using the VAM, IVAM, MM-VAM, and MVAM methods and using the stepping stone method as an optimization method. VAM usually produces a better initial feasible solution than the NWC and LCM methods [17].

METHOD
The data used in this study is data on the cost of transportation for each means of transportation to the destination for the distribution of drinking water from the Sikumbang Kampar Springs each canister. The source of the data used in the study was obtained from an interview with one of the distributors of drinking water from Sikumbang Kampar Springs, precisely in Pulau Sarak Village, Rumbio, on February 24, 2020.

Transport Model
Problems in distributing products from sources to several destinations can be arranged into a transportation model. The model is part of a linear programming problem that minimizes transportation costs from the source to the destination area [12]. The transportation model can allocate goods effectively from a source to a destination with the lowest possible cost [13]. The transportation model aims Constraints in transportation problems do not go beyond the source and meet the demands of each destination [14]. In solving transportation problems, transportation problems must be balanced. A transportation problem is said to be balanced if the amount of supply is equal to the number of requests [15]. The steps for solving transportation problems begin by translating the problem into tabular form. Then balance the problem by checking that the quantity of supply equals the quantity of demand. Then, determine the initial feasible solution obtained by four methods: VAM, IVAM, MM-VAM, and MVAM. After the initial solution is obtained, the next step is to test the stepping stone method's optimality.

Vogel's Approximation Method (VAM)
VAM usually produces a better initial feasible solution than the NWC and LCM methods, although it is more complicated than the two methods [16]. The steps for solving VAM [6] are to calculate the penalty for each row and column obtained based on the difference between the smallest and the second smallest values of the same row and column. Second, choose the row or column with the most significant penalty value. Third, allocate as many products as possible to the cell that has the most negligible cost. Fourth, repeat the first step and stop the process when all allocations have been met. Fifth, calculate the total cost obtained [6].

Improved Vogel's Approximation Method (IVAM)
IVAM is a method to get an initial feasible solution to transportation problems. IVAM is a modified method of VAM. VAM is modified by using the Total Opportunity Cost (TOC) matrix and alternative allocation costs.
The steps of Improved Vogel's Approximation begin by creating a matrix (TOC) obtained from the sum of the Opportunity Cost (OC) of the row and OC of the column, where for the OC row, it is obtained from the difference of each row with the smallest value of the row [17]. The OC column is obtained from the difference between each column with the smallest value in that column. Next, calculate the penalty for each row and column. The penalty value is obtained from the difference between the smallest value of the row and column with the second smallest value of the same row and column. Then, select the row and column with the three highest penalty costs. After that, the most negligible transportation cost will be chosen from the three highest penalties. These steps are carried out so that all allocations are met. The final step is to calculate the total transportation costs using the initial transportation costs.

Max-Min Vogel's Approximation Method (MM-VAM)
The MM-VAM method is a modification of VAM. MM-VAM modifies several steps in VAM, namely using a matrix (TOC). The steps of MM-VAM [5] are: First, create a matrix (TOC). Next, calculate the penalty for each row and column obtained by subtracting the maximum value from the minimum value for each row and column. Then, pick the two highest penalties. After that, the cell with the most negligible transportation cost and the highest allocation will be selected. The process will stop until all products have been allocated and all requests have been fulfilled. The last step is to calculate the results of the initial feasible solution with the initial transportation costs.

Modified Vogel's Approximation Method (MVAM)
MVAM is one of the modifications of VAM. MVAM produces an initial feasible solution that is more optimal than other methods [18]. The MVAM steps are as follows: First, Subtract the highest transportation cost in each row with each element of each row of the transport table and place it on the top left of the corresponding element. Second, subtract the highest transportation costs in each column with each element of each column of the transportation table and place it on the lower left of the appropriate element. Third, form a reduced matrix whose elements are the sum of the top left and bottom left elements in steps one and two. Then exchange the initial transportation cost with the reduced matrix. Fourth, calculate the penalty by finding the difference between the two most significant numbers in each row and column. Fifth, choose the highest penalty value. If two or more are equal, select the cell with the highest transportation cost (reduced matrix)-sixth, repeating the fourth and fifth steps until all allocations are met. Seventh, calculate the result of the initial feasible solution with the initial transportation cost.

Stepping Stone Method
According to S. Basriati, the stepping stone method is a method that changes the product allocation to get the optimal production allocation using the trial and error method [19]. The optimal solution to the transportation problem can be obtained using the stepping stone method with the following algorithm [7]: First, perform a closed horizontal or vertical jump alternately by resting on an already filled cell. Second, calculate the cost of the empty cells using a closed path basis with signs (+) and (-), which are carried out alternately in subsequent cells until they return to the initial empty cell. Third, if all the calculations in the second step are optimistic, then the transportation matrix is optimal. If there is a negative value of the calculation result, then the transportation matrix is not optimal, and the most significant negative value is chosen for the improvement index. Fourth, select the cell with the smallest unit in the jump that is negative. Add the smallest unit to the positive jump, subtract it from the negative jump. Fifth, revise the table by repeating the first step until there are no negative values in evaluating empty cells.

RESULTS AND DISCUSSION 1. Distribution of Sikumbang Kampar Spring Drinking Water
Sikumbang Kampar Spring is used as a drinking water business in Kampar District, Pulau Sarak Village Rumbio. The distributor of drinking water from Sikumbang Kampar Springs is Mr. Zulfikar. He has started his business as a drinking water distributor for Sikumbang Kampar Springs in 2006. The distribution of drinking water from Mr. Zulfikar's Sikumbang Kampar Springs is distributed to five destinations, namely Pekanbaru, Tapung, Tandun, Ujung Batu and Pulau Payung. Mr. Zulfikar uses four means of transportation to distribute drinking water from the Sikumbang Spring. The total demand in a week with a total supply of transportation equipment is not balanced. So the distribution of drinking water from the Sikumbang Spring is repeated to meet all water demands in a week.

Completion of the First Distributed Transportation Model
Based on transportation problems and data obtained from the drinking water distribution business of Mr. Zulfikar's Sikumbang Kampar Springs, the number of requests is more significant than a total supply of transportation equipment, resulting in an unbalanced transportation model. For the transportation model to be balanced, a dummy supply is added to the supply line. The transportation model can be seen as follows: Purpose Function: Min = 2.741 11 + 2.593 12 + 3.185 13

a. Completion of the First Distribution Using the VAM Method
The first step, calculate the penalty of each row and column. The penalty value is obtained from the difference between the smallest value of the row and column with the second smallest value of the same row and column, then chooses the most significant penalty value, then allocates as many products as possible to the cell with the most negligible cost. Here is a table with all the allocations that have been fulfilled using the VAM method: After all, allocations are met, the initial solution has been obtained with the total cost of the initial solution generated using the VAM method is Z = 880.070.

b. Completion of the First Distribution Using the IVAM Method
The first step in solving transportation problems using IVAM is to create a TOC matrix obtained from the sum of the Opportunity Cost (OC) rows and OC columns, where for row OC, it is obtained from the difference between each row and the smallest value of the row. The OC column is obtained from the difference in the value of each column with the smallest value. The following is the column TOC table for the transportation problem of the Sikumbang Kampar Spring drinking water distribution business: After obtaining the TOC matrix, the next step is to calculate the penalty for each row and column. The penalty value is obtained from the difference between the smallest value of the row and column with the second smallest value of the same row and column, then selecting the row and column with the three highest penalty costs. Then choose the most negligible transportation cost from the three highest penalties. The following is a table with all allocations that have been fulfilled using the IVAM method: After all, allocations are met, the initial solution using the IVAM method has been obtained with a total initial solution cost of Z = 927.525.

c. Completion of the First Distribution using the MM-VAM Method
First, create a TOC matrix. After the TOC matrix is obtained, then calculate the penalty for each row and column. The penalty value is obtained from subtracting the maximum value with the minimum value for each row and column. Then, pick the two highest penalties. Next, select the cell that has the most negligible transportation cost. Here is a table with all the allocations that have been fulfilled using the MM-VAM method:

d. Completion of the First Distribution Using the MM-VAM Metode Method
The first step, subtract the highest transportation costs in each row with each element of each row of the transportation table and place it on the top left of the corresponding element. Then subtract the highest transportation cost in each column by each element of each column of the transportation table and place it on the lower left of the appropriate element. Next, form a reduced matrix whose elements are the sum of the top left and bottom left elements in steps one and two. Then exchange the initial transportation cost with the reduced matrix. After the reduced matrix is obtained, calculate the penalty by finding the difference between the most significant numbers in each row and column. After choosing the highest penalty value, after all, allocations are fulfilled, enter all cell allocations from the reduced matrix to the transport table, then calculate the result of the initial feasible solution.

e. The First Distribution Optimal Solution Using the Stepping Stone Method
The first distribution optimal solution for the initial solution using the VAM, IVAM, MM-VAM, and MVAM methods produces the exact total optimal solution cost and allocation. However, to get the optimal solution, the VAM method obtains the optimal solution with two iterations, the IVAM method, MM-VAM, and MVAM with four iterations using the stepping stone method. The optimal solution for the first distribution can be seen in the following table: methods obtain the optimal solution with three iterations, the IVAM and MM-VAM methods with one iteration using the stepping stone method. The optimal solution for the second distribution can be seen in the following table: The optimal solution for the distribution of drinking water from the Sikumbang Kampar Springs uses the optimal stepping stone test method to distribute the two drinking water from the Sikumbang Kampar Springs resulting in a total of Rp 1,130,075.00. Engkel is distributing 135 canisters to Pekanbaru. Grandmax is distributing 80 canisters to Pekanbaru, L300 is distributing 90 canisters to Pekanbaru and Carry is distributing 65 canisters to Pekanbaru.

Completion of the Third Distribution Transport Model
After obtaining the optimal solution in the second distribution, not all requests were met. It was carried out again looking for the optimal solution for the distribution of the three drinking water, based on the optimal table obtained from the distribution of the two remaining unfulfilled requests for Pekanbaru 90 canisters, Tandun 540 canisters, and Ujung Batu as many as 110 canisters and with the same supply of means of transportation. Based on the remaining demand, the initial solution for the third distribution using the VAM method is Rp 1,228,895, IVAM Rp.1,235.005, MM-VAM Rp 1,285,005, and MVAM Rp 1,232,195. After the initial solution is obtained, the optimization test is carried out using the stepping stone method. The results of the third distribution optimization for the initial solution using the VAM, IVAM, MM-VAM, and MVAM methods produce the exact total initial solution cost and allocation but to get the optimal solution. The VAM method obtains the optimal solution with one iteration, the IVAM and MM-VAM methods with two iterations, and the MM-VAM method obtains the optimal solution with three iterations of a stepping stone. The optimal solution for the first distribution can be seen in the following table: The optimal solution for the distribution of drinking water from the Sikumbang Kampar Springs using the optimal stepping stone test method for distributing the four-drinking water from the Sikumbang Kampar Springs resulted in a total of Rp 1,346,625.00 Engkel is distributing 135 canisters to Tandun. Grandmax distributes 80 canisters to Ujung Batu, L300 distributes 60 canisters to Tandun and 30 canisters to Ujung Batu, and Carry distributes 65 canisters to Tandun. After obtaining the optimal solution for the fourth distribution, all requests have been fulfilled, then the search for the optimal solution for the distribution of drinking water from Sikumbang Kampar Springs is stopped.

CONCLUSIONS AND SUGGESTIONS
The optimal transportation cost for drinking water distribution from the Sikumbang Spring in one week using the transportation model is Rp 4,580,485.00. This result is more effective than the distributor's calculation before using the transportation model, Rp 5,050,000.00. Based on the research results, the VAM method is more efficient than the IVAM, MM-VAM, and MVAM methods because it has the fewest iterations to obtain the optimal solution using the Stepping Stone method.
Based on the research results that have been done, the authors hope that distributors can consider the use of Vogel's Approximation Method in optimizing the distribution costs of drinking water from Sikumbang Kampar Springs. Optimization of transportation costs using this method is more efficient and can result in greater profits for distributors. It is recommended that readers use other transportation methods to solve transportation problems to obtain a more efficient method.