MODEL OF VEHICLE ROUTING PROBLEM WITH SPLIT DELIVERY MULTI TRIPS, MULTI PRODUCTS AND COMPARTMENTS FOR DETERMINING FUEL DISTRIBUTION ROUTES

The industrial development in Indonesia encourages companies to have high sensitivity in competing to meet consumer demands promptly by considering minimum distribution costs. One of the factors that can affect distribution costs is route determination. Determining the distribution route is the Vehicle Routing Problem (VRP). The purpose of VRP is to arrange the order of distribution routes to produce a minimum total distance. This study aims to determine the fuel distribution route at TBBM Rewulu in one delivery period to obtain the optimal distribution route and minimize the vehicle mileage. Delivery is carried out using three types of tanker trucks with heterogeneous capacities. This study uses a mathematical model of Mixed Integer Linear Programming (MILP) by considering split delivery, multi trips, multi-products, and compartments. The branch and bound method in the LINGO solver has been used to solve this problem. This model was tested on a simple case using data of 8 customers with different distances and demand shipped by truck. The results obtained indicate that no boundaries are violated, and all consumers are served. The mathematical model built is still general, so it can solve similar cases. A model can be developed for further research by adding VRP variants such as time windows and adding the product types to represent the entire existing system.


Introduction
The increase in Indonesia's population impacts all aspects of people's lives, especially on the consumption of fuel oil (BBM) in the transportation sector. Companies are required to have high competitiveness against changes that occur to survive and compete with other companies. The projected energy demand in the transportation sector has increased from 362 million BOE (barrel oil equivalent) in 2017 to 1,252 million BOE (barrel oil equivalent) in 2050 or with an average growth rate of 3.8% per year (BPPT, 2019).
The distribution system is an important thing that must be considered because it becomes the main supporting factor after the production process. Distribution activities ensure that the product arrives in the hands of consumers on time, following the conditions ordered, with minimal shipping costs (Febriandini et al., 2020). In distribution problems, there is often a delay in receiving goods to consumers, which is the cause of not paying attention to the problem of finding the fastest or shortest path and setting the order of customers to be visited by starting and ending at the central depot.
One of the efforts made by the company to avoid delays in delivery to consumers is to make deliveries quickly, do not cost a lot, and have a minimum distance. This effort can be carried out by considering the configuration of the right distribution route and following the company's characteristics. Given the company has a limited number of vehicles and vehicle capacity. The absence of control over the distribution of products can cause losses for the company (Rofiq et al., 2016).
The problem of determining the route is generally known as the Vehicle Routing Problem (Karim et al., 2017). The Vehicle Routing Problem (VRP) is a problem in finding the optimal route for delivery or collection of goods or services from one or more depots to a number of cities or customers by meeting a number of constraints (Baradaran et al., 2019).
In general, the purpose of VRP is to minimize distribution costs and mileage as well as minimize the number of vehicles used, balance travel routes, and minimize consumer dissatisfaction such as delays in delivery times (Toth and Vigo, 2002).
The vehicle routing problem (VRP) is one of the most important and studied combinatorial optimization problems in distribution, logistics, and transportation (Feng et al., 2017). There are much literature that discusses how to solve this problem. Many studies have developed various variants of the classic VRP model, such as the VRP model with particular characteristics of multiple routes or multiple trips (VRPMT) conducted by Prins (2002), Olivera, and Viera (2007), Mingozzi et al. (2013).
This characteristic illustrates that one vehicle can serve more than one route in one planning horizon. Prins (2002) discusses a real case of a French furniture manufacturer to minimize the total travel time and the number of vehicles needed. Heuristic and Tabu Search methods were applied to this study to provide solutions in real cases with 775 stores for distribution purposes. Olivera and Viera (2007) proposed the Adaptive Memory Programming (AMP) algorithm, while Mingozzi et al. (2013) used an exact method with two-setpartitioning in formulating existing problems. The researcher proposes four Column and Cut Generations and then uses an integer programming solver to reduce the existing problems. Pramuditha (2014) conducted a VRP study on drug distribution in pharmaceutical companies with the characteristics of multiple product distribution systems from single distribution centers to multiple buyers using multivehicles. Shipments are made to 56 customers for each delivery period. The model is built with the characteristics of integer linear programming, and the solution is searched using LINGO software.

Abstract
The industrial development in Indonesia encourages companies to have high sensitivity in competing to meet consumer demands promptly by considering minimum distribution costs. One of the factors that can affect distribution costs is route determination. Determining the distribution route is the Vehicle Routing Problem (VRP). The purpose of VRP is to arrange the order of distribution routes to produce a minimum total distance. This study aims to determine the fuel distribution route at TBBM Rewulu in one delivery period to obtain the optimal distribution route and minimize the vehicle mileage. Delivery is carried out using three types of tanker trucks with heterogeneous capacities. This study uses a mathematical model of Mixed Integer Linear Programming (MILP) by considering split delivery, multi trips, multi-products, and compartments.
The branch and bound method in the LINGO solver has been used to solve this problem. This model was tested on a simple case using data of 8 customers with different distances and demand shipped by truck. The results obtained indicate that no boundaries are violated, and all consumers are served. The mathematical model built is still general, so it can solve similar cases. A model can be developed for further research by adding VRP variants such as time windows and adding the product types to represent the entire existing system. The optimization can reduce distribution costs by 22.97% and save time by 233.88 minutes. Setiawan (2016) developed a variant of the Heterogenous-VRP model by adding multi-trips and multi-products variants. A variation of this VRP is where the company has a heterogeneous fleet in terms of capacity, fixed and variable costs, and speed. The artificial data that has been built is then solved using the exact branch, and bound method and the results are compared with the genetic algorithm.
This research tries to consider other variants of VRP, namely in the conditions of split delivery, multiple trips, multiple products, and compartments. This research was applied to a real case at Pertamina TBBM located in Rewulu, Yogyakarta. So far, the delivery of BBM products has only been based on the driver's experience. This method cannot guarantee that route of each vehicle is optimal route. Therefore, this study will determine the fuel delivery route in one delivery period to obtain the optimal distribution route and reach the vehicle mileage.

A. System Characteristics
PT. Pertamina TBBM Rewulu is one of the Pertamina companies that acts as a distribution center (DC) for fuel oil. TBBM Rewulu delivered petrol products to several gas stations located in Yogyakarta and parts of Central Java, namely Purworejo, Magelang, Temanggung, and Klaten. The fuel supply is obtained from the Pertamina RU IV Cilacap refinery through a pipeline network and then distributed by TBBM Rewulu to any customers of gas stations. Fuel delivery from the depot uses a tank truck which consists of three types of capacities, namely 16 kiloliters, 24 kiloliters, and 32 kiloliters.
Each tank is divided into compartments with a homogeneous compartment capacity of 8 kiloliters. In general, there are 140 customer gas stations served by TBBM Rewulu. However, in the case of fuel distribution, each gas station is not required to make a request every day, so the number of gas stations served by the depots are various because it depends on whether or not there is a request from the gas station. The products that will be discussed in this research are biodiesel and pertalite with deterministic demand characteristics. This research will use the exact branch and bound method in the LINGO 18.0 software. The advantage of the branch and bound method lies in its effectiveness in solving problems with accurate results (Lubis and Herliansyah, 2017). The model that will be developed in the fuel distribution has several limitations, namely: 1. Split Delivery Products delivered to one customer can be split into one or more vehicles.

Multi Trips
One vehicle can serve customers or routes in one tour products delivery.

Multi Products and Compartments
The vehicle has more than one compartment, and the products being delivered are more than one kind of product.
But, one compartment can only be filled with one kind of product.

B. Distribution Process
To distribute fuel from the depot to the gas station, PT. Pertamina is supported by its subsidiary, PT Patra Niaga, responsible for managing the delivery of fuel using a fleet of tank trucks, most of which are rental cars. Figure 1 describes the distribution process from depot to customers. The distribution flow begins with the gas station reporting the stock demand for fuel via SMS one day before petrol products are sent to the gas station. Then the SMS will be received by the server. The depot sales service will confirm the request whether the quantity and type of BBM ordered is available, and the depot can send BBM on demand. If the depot can make deliveries, the gas station will get a payment code to pay according to the amount ordered. The bank confirms the purchase and then issues an SO (Sales Order).
The list of buyers will be collected by the sales service in the form of LO (Loading Order) then a distribution plan will be made. The fuel delivery plan that the depot sales service has prepared is submitted to Patra Niaga as the tank truck's steward for further delivery to the gas station. Patra Niaga usually makes deliveries based on previous experience because there is no optimization of shipping routes. Generally, gas stations located close to the depot will be supplied first and then to gas station locations that are far away.
Fuel delivery using tank trucks with different capacities. These tank trucks are compartmentalized and do not have flow meter. This implies that the contents of a compartment cannot be used to replenish more than one underground reservoir. Consequently, each compartment of the delivery vehicle must be filled with one of the products to be delivered on its route. Each compartment has a capacity of 8 kiloliters. So, consumers who want to order petrol products must order with multiple capacities of 8 kiloliters.

A. Mathematic Formulation
The mathematical model used in this study refers to Normasari and Warangga (2019) research with modifications model according to the specified system characteristics. The mathematical model consists of several sets, index, parameters, and variable.
Subject to : The objective function in formula (1) is to minimize the total distance of vehicle. Constraint (2) ensures that the vehicle can be used more than once. Constraint (3) ensures the continuity of the routing process, ensuring that vehicles leaving the depot will return to the depot. Constraint (4) ensures a vehicle of various origins to h must exit the h to wherever the next destination, except h = depot. Constraint (5) states that there are no vehicles that return to the previous point that has been visited. Constraint (6) ensures that each customer can be visited by several vehicles. Constraint (7) ensures that dan Synchronize. The maximum amount of fuel a vehicle can deliver during a single trip is bounded by the number of compartments dedicated to this type of fuel, multiplied by the compartment capacity (Constraint 8). Constraint (9) and (10) define a binary variable.

B. Model Verification
The model that has been built is then verified with a simple case using the exact branch and bound method through the LINGO 18.0. Model verification is carried out to ensure that the model created is running according to the expected logic, namely following the behavior of the system. The problems for the following simple case are using one depot and 8 customers with two types of products sent in biodiesel and pertalite. The vehicles used are 3 tank trucks with a capacity of 16 kL, 24 kL dan 32 kL. Each customer has their code to describe the petrol station. There are: The following table is consumer demand data, distance matrix from depot to the costumer and between costumers can be seen in the Table 2.   Table 3 is a matrix of the distance from the depot to the consumer and the distance between consumers in kilometers. Distance data is obtained from the real distance by google maps. The mathematical model was successfully tested using the branch and bound method on LINGO 18.0 with the simple case above, so the results of delivery routes are: vehicle 1 route 1 serving consumers 1 -3 -5 -1 vehicle 2 route 1 serving consumers 1 -4 -1 vehicle 2 route 2 serving consumers 1 -4 -1 vehicle 3 route 1 serving consumers 1 -8 -9 -1 vehicle 4 route 1 serving consumers 1 -2 -7 -6 -1 After knowing the results of the LINGO solver, then in table 4 it will be shown verification of the output obtained to find out if there are any violated in the mathematical model. Based on the table above, it can be seen that the output of the optimization results meets the limits specified in the model, namely the output of the vehicle can make more than one trip in a distribution period, no load that exceeds the capacity of the vehicle in one route, and all consumers are served. From the optimization results above, it can be said that the model output does not violate the limits, so the LINGO output has been verified.

Conclusion
This research was conducted by building a mathematical model on the VRP split delivery, multi-trips, and multi-product problems using the exact branch and bound method to minimize vehicle mileage. The results indicate that no boundaries are violated, and all consumers are served. The mathematical model built is still general to solve similar cases. For further research, a model can be developed by adding VRP variants such as time windows and adding product types to represent the entire existing system.