Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 5(4) (2019), Pages 195-202 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: March 14, 2018 Reviewed: March 15, 2018 Accepted: May 31, 2019 DOI: http://dx.doi.org/10.18860/ca.v5i4.4893 Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuri Lutvi Azizah1 1Muhammadiyah University of Sidoarjo, Sidoarjo, Indonesia Email: nurillutviazizah@umsida.ac.id ABSTRACT RASKIN/RASTRA is a rice distribution program for the poor family with the aim of improving food security starting from the household scale. The Aim of this study is to determine whether the mathematics modelling can use in the transportation methods to provide an efficient solution of costs on the rice distribution in the region of Sidoarjo. The method used in this research is the NWC (Northwest Corner) method used to analyze the initial fiscal solution, and refined by MODI (Modified Distribution) to analyze the most optimal cost. From the calculation by transportation method, the optimum cost is lower than the company's calculation of Rp 85.186.040 while the cost of the company's calculation is Rp 87.209.690,750. Thus the use of transportation methods can save RASTRA's distribution cost of Rp 2.023.650,750. Keywords: mathematics modelling, transportation methods, optimum cost INTRODUCTION RASKIN/RASTRA is a staple food subsidy in the form of rice intended for the poor as the government's efforts to increase food security and provide protection to poor families. Bulog has two activities, namely service activities and commercial business activities. But in its implementation almost 90% of the activities of the company is a service activity that is the assignment of the government [1]. β€œPerum BULOG Sub Divre Sidoarjo” as the implementer of RASTRA program for some areas such as Balongbendo, Buduran, Candi, Gedangan, Jabon, Krembung, Krian, Porong, Prambon, Sedati and Sidoarjo spent considerable funds for distribution activities. To minimize the cost of this distribution it is necessary to plan for the distribution of RASTRA so that the cost of distribution issued is as optimal as possible. One method that can be used to optimize distribution costs is by transportation method. The transportation methods require a valid mathematical model before the transporatation’s calculation. Mathematical model is a description of a system using mathematical concepts and language, and the process of developing a mathematical model is termed mathematical modelling. The method of transportation is a method or means used to solve the problem of distribution from sources that provide the same product, to the place where the need optimally so that the cost of distribution issued is minimal [2]. The goal is to minimize the cost of distributing RASTRA from one location to another in the Sidoarjo area, so that the needs of each region are met according to capacity with minimal distribution cost. Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 196 METHODS a. Transportation Method The Mathematical model is used to determine the mathematical equation related to the problems encountered in the problem of distribution cost optimization. The method of transportation is related to the distribution of a single product from multiple sources with limited supply, to some destinations with a particular request to obtain a minimum distribution charge. Because only one kind of product then a destination can meet the demand of one or more sources. Transportation issues have several characteristics that are [3]: a) There are a number of sources and a number of specific goals b) The quantity of goods distributed from each source and requested by any destination is certain c) The quantity or quantity of goods sent from a source of destination in accordance with the request or capacity of the source d) The cost of transportation from a source to a destination is certain Mathematically, transportation problems can be modeled as follows: Objective function π‘€π‘–π‘›π‘–π‘šπ‘’π‘šπ‘˜π‘Žπ‘› 𝑍 = βˆ‘ βˆ‘ 𝐢𝑖𝑗 𝑋𝑖𝑗 𝑛 𝑗=1 π‘š 𝑖=1 (1) Constrain function βˆ‘ 𝐢𝑖𝑗 π‘š 𝑖=1 𝑋𝑖𝑗 = π‘Žπ‘– ; 𝑖 = 1, 2, … . . , π‘š (2) βˆ‘ 𝐢𝑖𝑗 𝑛 𝑗=1 𝑋𝑖𝑗 = 𝑏𝑗 ; 𝑗 = 1, 2, … . . , 𝑛 (3) Operator definition of each variable will be described as Cij = transportation cost per unit of goods from source i to destination j Xij = number of goods distributed from source i to destination j ai = quantity of goods offered or capacity from source i bj = quantity of goods requested or ordered by destination j m = number of sources n = number of goals b. Transportation Models A transport problem is said to be balanced (balance program) if the amount of supply at source i is equal to the amount of demand on the destination [4].Can be written: βˆ‘ π‘Žπ‘– = βˆ‘ 𝑏𝑗 𝑛 𝑗=1 π‘š 𝑖=1 (4) Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 197 The first step to solve the problem of transportation is to determine the initial feasible solution. There are three methods for determining the initial feasible solution [5] : 1. North West Corner (NWC) Method 2. The Least Cost Method 3. Vogel Approximation Method (VAM) After obtaining the initial feasible solution then sought the optimal solution. There are two methods to determine the optimal solution [5]: 1. The stepping stone method (Stepping Stone) 2. Modified Method Of Distribution (MODI) c. Step of NWC Methods The method discussed in this research is Northwest Corner method for initial solution and MODI for optimal solution. The step of the NWC methods are [6]: 1. Step 1: Allocate the maximum amount available to the selected cell and adjust the associated supply and demand quantities by subtracting the allocated quantity. 2. Step 2: Exit the row or the column when the supply or demand reaches zero and cross it out, to show that it cannot make any more allocations that row or column. If a row or column simultaneously reach zero, only cross out one (the row or the column) and leave a zero supply (demand) in the row (column) that is not crosses out. 3. Step 3: if exactly one row or column is left that is not crosses out, stop. Otherwise, advance to the cell to the right if a column has just been crossed out, or to the cell below if a row was crossed out. Continue with Step 1. d. Step of MODI Methods While the MODI method steps are [6]: 1. Step 1: Determine the π‘ˆπ‘– values for each row and 𝑉𝑗 values for each column by using the relation 𝐢𝑖𝑗 = π‘ˆπ‘– + 𝑉𝑗 for all base variables and set that the value of π‘ˆπ‘– is zero. 2. Step 2 : Calculate the cost charges for each non based variable using 𝑋𝑖𝑗 = 𝐢𝑖𝑗 βˆ’ π‘ˆπ‘– βˆ’ 𝑉𝑗 3. Step 3: If there is a negative 𝑋𝑖𝑗 value, then the solution is not optimal. Select the 𝑋𝑖𝑗 variable with the largest negative value as the entering variable. 4. Step 4 : Allocated goods entering variable 𝑋𝑖𝑗 according stepping stone or NWC process 5. Step 5: Repeat Step 1 through Step 4 until all 𝑋𝑖𝑗 values are zero or positive. e. Data Collection Technique The main problem faced in this research is not optimal the fund spent for rice distribution cost in β€œPerum Bulog Sub Divre Sidoarjo”, so after mathematical modelling used in this research, the transportation method will be applied to get the most optimal cost solution. Here is a Data Collection Technique used are: a. Study of Literature b. Secondary Data Retrieval This study uses secondary data obtained from β€œPerum Bulog Sub Divre Sidoarjo”, and can be done using assumptions to be used for similar problems. c. Observation This activity is done to synchronize the secondary data obtained by doing direct observation of the spaciousness to ensure the data obtained is valid. d. System Planning Using Mathematics model e. Implementation of transportation methods Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 198 RESULTS AND DISCUSSION a. Rice Distribution Data for RASKIN/RASTRA The RASTRA distribution data consist of the amount of RASTRA rice distributed per month in kilograms (kg), for each area sub-div described in several districts/cities. In each sub div of the districts, there are number of distribution points (TD), and at the village level with the number of Target Household (RTS) receiving RASTRA [7]. In this research, RASTRA distribution in point emphasizes on the distribution from warehouse to the sub- district level to find out how much the cost from the warehouse to the distribution place in the area to the sub district where rice distribution area. For the value to be measured for completion in this research is the amount of distribution or distribution of RASTRA rice per month in kilogram (kg) then be converted in tons/year as needed. The amount of RASTRA rice distribution per month in North Surabaya region can be seen in Table 1. Table 1. RASTRA Rice Distribution per Month NO SUBDIVRE Distribution per Month KAB/KOTA RTS-PM Kg I SURABAYA UTARA 221.845 3.327.675 1. Kota Surabaya 65.991 989.865 2. Kab. Sidoarjo 78.103 1.171.545 3. Kab. Gresik 77.751 1.166.265 Sidoarjo District consist of 18 Sub-District formed five clusters for distribution of rice RASTRA, details for five clusters with members called Distribution Point (TD) that include within each cluster can be seen in Table 2. Table 2. Capacity Data for Each Region in RASTRA Distribution No Warehouse Location Capacity (per kg) 1 Warehouse 1 Gedangan 265045 2 Warehouse 2 Gedangan 257500 3 Warehouse 3 Gedangan 230000 4 Warehouse 4 Porong 201000 5 Warehouse 5 Porong 218000 Total 1171545 b. Description of Distance Data Distance data is retrieved through Google Maps data source to know the distance of warehouse with the sub-district or rice distribution/distribution area for RASTRA [7]. The verification of the distance data that has been collected can be seen in Table 3, it contains the relative distance data collection from the warehouse to the sub-districts where the distribution to the RTS. Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 199 Table 3. Relative Distance Data from Warehouse to the Sub-District No. Location Warehouse/Sub District Distance(km) 1 Balongbendo 28,02 2 Buduran 0,5 3 Candi 9 4 Gedangan 3,5 5 Jabon 19,3 6 Krembung 24,9 7 Krian 18,3 8 Porong 17,6 9 Prambon 27,7 10 Sedati 7,8 11 Sidoarjo 6,1 12 Sukodono 7,5 13 Taman 12,2 14 Tanggulangin 12,5 15 Tarik 32,8 16 Tulangan 18,8 17 Waru 10,8 18 Wonoayu 15,1 The following data on the amount of RASTRA rice required depend on the Distance Data by Google Maps Application and can be seen on Table 4. Table 4. RASKIN Rice Required Amount in each Region in kg of rice NO WAREHOUSE SUB DISTRICT RICE CAPACITY (KG)/DEMAND TOTAL 1 Warehouse 1 Buduran 69000 280425 Sedati 71035 Gedangan 71985 Sidoarjo 68405 2 Warehouse 2 Candi 67600 246160 Tanggulangin 53500 Wonoayu 60035 Tulangan 65025 3 Warehouse 3 Sukodono 50025 234635 Taman 52000 Krian 75545 Waru 57065 4 Warehouse 4 Balongbendo 68050 198080 Tarik 61030 Prambon 69000 5 Warehouse 5 Krembung 67000 212245 Porong 77245 Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 200 Jabon 68000 TOTAL 1171545 1171545 C. Mathematical Modelling Preparation of data is done by translating the problems and descriptions into the form of mathematical models. Formation of the mathematical model of the data obtained need to pay attention to things that affect the calculation process, such as: 1. Decision Variable 2. Objective Function 3. Constrains The steps to be taken in the transport model are the determination of decision variables, objective functions, and constraints. 1. Decision Variable Here are the variables for delivery allocation to each warehouse Alocation Distribusion from GBB Banjar Kemantren 1 (Warehouse 1) a. Allocation from warehouse 1 to several point distribution π‘ΏπŸπŸ = Allocation from Warehouse 1 to Point Distribution 1, π‘ΏπŸπŸ = Allocation from Warehouse 1 to Point Distribution 2, π‘ΏπŸπŸ‘ = Allocation from Warehouse 1 to Point Distribution 3, π‘ΏπŸπŸ’ = Allocation from Warehouse 1 to Point Distribution 4, π‘ΏπŸπŸ“ = Allocation from Warehouse 1 to Point Distribution 5 b. Allocation from warehouse 2 to several point distribution π‘ΏπŸπŸ = Allocation from Warehouse 2 to Point Distribution 1 π‘ΏπŸπŸ = Allocation from Warehouse 2 to Point Distribution 2 π‘ΏπŸπŸ‘ = Allocation from Warehouse 2 to Point Distribution 3 π‘ΏπŸπŸ’ = Allocation from Warehouse 2 to Point Distribution 4 π‘ΏπŸπŸ“ = Allocation from Warehouse 2 to Point Distribution 5 c. Allocation from warehouse 3 to several point distribution π‘ΏπŸ‘πŸ = Allocation from Warehouse 3 to Point Distribution 1 π‘ΏπŸ‘πŸ = Allocation from Warehouse 3 to Point Distribution 2 π‘ΏπŸ‘πŸ‘ = Allocation from Warehouse 3 to Point Distribution 3 π‘ΏπŸ‘πŸ’ = Allocation from Warehouse 3 to Point Distribution 4 π‘ΏπŸ‘πŸ“ = Allocation from Warehouse 3 to Point Distribution 5 d. Allocation from warehouse 4 to several point distribution π‘ΏπŸ’πŸ = Allocation from Warehouse 4 to Point Distribution 1 π‘ΏπŸ’πŸ = Allocation from Warehouse 4 to Point Distribution 2 π‘ΏπŸ’πŸ‘ = Allocation from Warehouse 4 to Point Distribution 3 π‘ΏπŸ’πŸ’ = Allocation from Warehouse 4 to Point Distribution 4 π‘ΏπŸ’πŸ“ = Allocation from Warehouse 4 to Point Distribution 5 e. Allocation from warehouse 5 to several point distribution π‘ΏπŸ“πŸ = Allocation from Warehouse 5 to Point Distribution 1 π‘ΏπŸ“πŸ = Allocation from Warehouse 5 to Point Distribution 2 π‘ΏπŸ“πŸ‘ = Allocation from Warehouse 5 to Point Distribution 3 π‘ΏπŸ“πŸ’ = Allocation from Warehouse 5 to Point Distribution 4 π‘ΏπŸ“πŸ“ = Allocation from Warehouse 5 to Point Distribution 5 Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 201 2. Objective Function Optimization The function 𝑍 𝑍 = βˆ‘ βˆ‘ 𝐢𝑖𝑗 𝑋𝑖𝑗 𝑛 𝑗=1 π‘š 𝑖=1 (5) So that the formulation based on the data became a linear programming formulation and can be formulated by matrix calculation. 𝑍 = 71,25𝑋11 + 75𝑋12 + 80𝑋13 + 110,25𝑋14 + 102,675𝑋15 + 73𝑋21 + 77,5𝑋22 + 76,15𝑋23 + 117,15𝑋24 + 81,765𝑋25 + 70,5𝑋31 + 78,25𝑋32 + 82,75𝑋33 + 105,25𝑋34 + 88,85𝑋35 + 80𝑋41 + 85,75𝑋42 + 76,75𝑋43 + 71,05𝑋44 + 71,75𝑋45 + 101,75𝑋51 + 90,25𝑋52 + 80,75𝑋53 + 71,85𝑋54 + 70,25𝑋55. (6) 3. Constrains Constrains in this research can be divided into two categories. First constrain is for demanding problem and the second is for bargaining problem. Here is the same about demanding constrain and bargaining constrains. 𝑋11 + 𝑋21 + 𝑋31 + 𝑋41 + 𝑋51 = 280425 (7) 𝑋12 + 𝑋22 + 𝑋32 + 𝑋42 + 𝑋52 = 246160 (8) 𝑋13 + 𝑋23 + 𝑋33 + 𝑋43 + 𝑋53 = 234635 (9) 𝑋14 + 𝑋24 + 𝑋34 + 𝑋44 + 𝑋54 = 198080 (10) 𝑋15 + 𝑋25 + 𝑋35 + 𝑋45 + 𝑋55 = 21224 (11) D. Program Formulation The compilation from the formulation and equations is using the computational program as a QM for Windows script. The problem solving for RASTRA rice distribution cost optimization can be seen on the Table 5 by Northwest Corner Methods to determine initial fiscal point. Table 5. Solution Report for Northwest Corner (NWC) Methods SOLUTION VALUE DISTRICT 1 DISTRICT 2 DISTRICT 3 DISTRICT 4 DISTRICT 5 $87440020 WAREHOUSE 1 265045 (-0.75) (0.25) (35.7) (-29.725) WAREHOUSE 2 15380 242120 (-5.85) (40.85) (-7.065) WAREHOUSE 3 (-3.25) 4040 225969 (28.2) (13.4) WAREHOUSE 4 (12.25) (13.5) 8675 192325 (2.3) WAREHOUSE 5 (33.2) (17.2) (3.2) 5755 212245 After the initial value or point obtained from NWC method, then the calculation and the program compilation is continued with MODI method to get the optimal Solution, it can be seen on the Table 6. Table 6. Solution Report for MODI by using initials NWC Method SOLUTION VALUE DISTRICT 1 DISTRICT 2 DISTRICT 3 DISTRICT 4 DISTRICT 5 $85186040 WAREHOUSE 1 18885 246160 (5.6) (41.55) (35.575) WAREHOUSE 2 31540 (0.75) 225960 (46.7) (12.915) WAREHOUSE 3 230000 (4) (9.1) (37.3) (22.5) Mathematical Modelling On Transportation Method Apllication For Rice Distribution Cost Optimization Nuril Lutvi Azizah 202 WAREHOUSE 4 (6.4) (8.4) 8675 192325 (2.3) WAREHOUSE 5 (27.35) (12.1) (3.2) 5755 212245 The solution report from QM for windows compilation obtained that the NWC method has the solution value about $87440020. The MODI method by using the initial fiscal point NWC method has the solution value about $85186040. CONCLUSION Mathematical modelling on the transportation methods will produce a mathematical equations useful for optimization problem solving if the model used is a valid model. Based on the result of QM for Windows compilations is known that transportation with the initial NWC method to determine the initial value has the solution value about Rp 87.440.020. The MODI method is used to determine the optimal solution by using initial fiscal point in NWC method has the solution value about Rp 85.186.040. The MODI method obtain the optimal solution and it can save cost about Rp 2.253.980 from the NWC method. The secondary data that given from the calculation β€œPerum Bulog Sub-Divre Sidoarjo” has minimal cost about Rp 87.209.690,750. So that, with the MODI method can save RASTRA rice distribution cost about Rp 2.023.650,750. REFERENCES [1] H, A. Taha. (1997). Operation Research An Introduction. University of Arkansas, Fayetteville: Prentice Hall, vol. 6, pp. 202-230. [2] Clark, S. D. (2002). Handbook of Transportation Science. Journal of the Operational Research Society (Vol. 53, pp. 470–471). https://doi.org/10.1017/CBO9781107415324.004 [3] T, D, A, Dimyati. 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