Model of Flexible Periodic Vehicle Routing Problem-Service Choice Considering Inventory Status

The most popular combinatorial optimization problem is the Vehicle Routing Problem (VRP) [1]. VRP was first introduced in 1959 [2]. VRP is considered popular because it can solve companies' problems related to product delivery and pick-up [1]. VRP is a complex problem with various characteristics that allow for development. Some researchers have conducted a taxonomic review of VRP development [3][4]. The development of VRP research includes problems with heterogeneous fleets, time windows, split delivery, and pick-up and delivery problems. Some of the best known VRP developments are the Vehicle Routing Problem with pick-ups and delivery (VRPPD) [5], [6], [7]; Vehicle Routing Problem with split delivery (SDVRP) [8], [9]; Vehicle Routing Problem with time window (VRPTW) [10],[11],[12],[13],[14]. Besides the VRP types above, there are several other types of VRP developments such as the Periodic Vehicle Routing Problem (PVRP), Multi Depot Vehicle Routing Problem (MDVRP), Capacitated Vehicle Routing Problem (CVRP), Green Vehicle Routing Problem (G-VRP). ARTICLE INFO ABSTRACT


Introduction
The most popular combinatorial optimization problem is the Vehicle Routing Problem (VRP) [1]. VRP was first introduced in 1959 [2]. VRP is considered popular because it can solve companies' problems related to product delivery and pick-up [1]. VRP is a complex problem with various characteristics that allow for development. Some Vehicle routing problems and inventory problems need to be integrated in order to improve performance. This research discusses the determination of vehicle routes for product delivery with periodic delivery times that are released at any time depending on the inventory status. A mixed-integer linear programming model in determining periodic flexible visiting vehicles' route considering inventory is proposed to solve this problem. This model also accommodates time window constraints, retailer warehouse capacity. The search for solutions was carried out using the branch-and-bound method with the help of Lingo 18.0. The mathematical model testing result saves shipping costs and inventory costs. In addition, the developing mathematical model offers the flexibility of visiting depending on the inventory status of the consumer. The sensitivity analysis of the model results in the vehicle capacity influence the total cost and routes formed. The model developed in this study is Mixed Integer Linear Programming (MILP) with a branch-and-bound method of solving. Branch-and-bound is a method commonly used to solve complex combination problems [39]. The branch-and-bound method tries to connect each customer in the form of a branch with a certain sequence of visits (route of visits) [40]. The branch-and-bound main idea is to break the solution space in succession into specific subsets (branches) [39]. The branches are removed one by one when a solution is obtained that minimizes the objective function. In general, Lingo 18.0 combines each consumer into a visiting route, taking into account limitations, resulting in a minimum objective function.
The paper is presented as follows: methods are presented in section 2. The results of problem optimization, validation of inventory status, and sensitivity analysis are presented in section 3. The final section presents conclusions and further work.

Methods
In the method section, this study was divided into four sub-sections, namely (1) determining assumptions, notation, and developing mathematical models, (2) data and experiments, (3) model validation using DRP, and (4) sensitivity analysis of the developed model.

Assumptions, Notations, and Mathematical Models
Some of the assumptions used in this problem are: (1) delivery of products to customers in the form of delivery for a single product, (2) the problem of using a single depot for delivery to customers, (3) the depot serves customer requests during the planning horizon, (4) each customer has a certain time window, (5) the vehicle used has a certain capacity, (6) each customer needs service time when the delivery is executed, and (7) each vehicle assigned to each route departs and returns to the warehouse.
Research considers demand over the planning horizon. In addition, this study has a decision variable of the maximum number of shipping lots for each customer. Furthermore, the vehicle visits to customers minimize the total cost, including distribution costs and inventory costs.
The parameters in the mathematical model on the FPVRP-SCI problem are denoted as follows: i  Node i is visited on day t,

Others
As stated earlier, the MILP model was developed from the model proposed by Archetti, et al. [32]. The MILP model is divided into two main parts, namely the objective function and the limiting function. The following is the MILP model proposed to solve the FPVRP-SCI problem.
Objective Function : Subject to : The objective function (1) states the total cost consisting of inventory costs and transportation costs. Transportation costs consist of fixed costs and variable costs of vehicles. Inventory costs consist of the cost of storing products to consumers. Constraint (2) shows the accumulated changes in inventory for each period. Constraint (3) is a barrier to ensure that customer requests are fulfilled. Equation (4) shows the barrier that ensures the warehouse capacity is met. Constraint (5-6) formulates the initial and end period inventory status for each customer. Constraint (7-8) is a barrier to determine the number of vehicles used per day and vehicle capacity. Constraint (9) shows the load on each consumer. Limiter (10) aims to ensure that the consumer is visited on day t. Constraint (11) describes every vehicle that comes to consumer i must leave that customer i. Constraint (12) formulates that each consumer is visited once a day at maximum. Constraint (13) is the accumulated travel time of the vehicle to node j on day t that is the accumulated travel time from node i plus the service time at node i. Constraint (14) is a limitation so that the vehicle travel time does not exceed the maximum vehicle time. Constraint (15) formulates the time window for the consumer. Constructor (16) is a nonnegative limiter for the variable. Constraint (17) represents integer limiter. Constraint (18) is a binary limiter for the decision variable.

Data and Experiments
This problem was studied from problems in actual conditions. This case study was based on the problems that existed in the franchise businesses in Indonesia. The data collected included the data of vehicle capacity ( ) that was 30 units, the maximum travel time on one tour ( ) was 1,236 minutes, the constant vehicle speed ( ) 1 km/minute, the cost of storing unused products per unit (ℎ) 20,000 IDR, variable costs based on vehicle trips ( ) 2,500 IDR, fixed cost if one vehicle was used( ) 500,000 IDR, and the number of days period used in the ( ) calculation was five days. The number of consumers used in data processing was as many as eight consumers. The detailed data can be seen in Table  1, and the coordinates of the consumer's location can be seen in Table 2. In Table 2, the depot position is depicted at coordinates (0,0) to facilitate modeling and data processing. The data obtained is then processed into two parts. The first is the determination of the route generated by the model, and the second is the analysis of the demand for the results of the model. Data processing uses Lingo 18.0 software which runs on Intel Core i5-8400 @ 2.8GHz and 8 Gb Ram at one depot (single depot) and eight consumers for five days. Lingo 18.0 looks for the optimal solution taking into account several. The optimal solution in this route-finding case is the one that yields the minimum cost.

Result Validation
To ensure consumers' inventory status, validation of the proposed MILP model results is carried out using distribution requirement planning (DRP) [41]. Regarding the actual system used, the validation is done using DRP. DRP can be used as a control parameter in inventory, such as safety stock and when inventory is needed [42]. In addition, DRP is an inventory management system that deals with stock replenishment in a multi-echelon distribution system. DRP helps determine when a layer in a particular echelon places orders in the next echelon. DRP can decide when the order is placed and how much the transaction costs [43].

Sensitivity Analysis
Sensitivity analysis is an essential part of building testing and model development. Sensitivity analysis is performed by checking the model's output when input changes are given [44]. In developing this model, a sensitivity test is carried out by changing the number of vehicle capacities. Vehicle capacity is a simple limit that can affect how much cargo is carried and the number of customers that can be visited. Twelve variations of changes in vehicle capacity are in the sensitivity analysis. The changes of vehicle capacity are tested from a value of 20-600. In this sensitivity analysis, vehicle capacity changes are tested to determine the effect of changes in the number of routes and destination functions.

Route Results
The results of determining the vehicle's route using the lingo software can be seen in Fig. 1, Fig. 2, and Fig. 3. The resulting destination function is 8,713,750 IDR. From data processing, it was obtained the results of sending three times. On the first day. One delivery route is obtained with one vehicle shown in Fig. 1. On the 2nd day, one vehicle route is obtained with one vehicle, which can be seen in Fig. 2. The number of routes generated on the 3rd-day delivery was two routes shown in Fig. 3. The number of vehicles required on the 3rd day is four vehicles. The first vehicle serves consumers 5 and 6 ; second vehicle serves 3 and 8 ; the third vehicle serves 7 and 2 ; and the last vehicle serves 4 and 1 .
The research results in Fig. 1 -Fig. 3 show that every consumer is not visited every day. In a week, some consumers have been visited once ( 3 ), consumers who have been visited twice ( 1 , 4 , 5 , 6 , 7 ), and consumers who have been visited three times ( 2 and 8 ). The consumers are visited with periodic delivery times, but free on any day (flexible). Each customer is visited on a different day, depending on the status of the inventory.

Solution Validation to Inventory Status
Solution validation testing related to the resulting inventory status is carried out with DRP. Validation of the state of each consumer's inventory is tested on consumers who are visited once, twice, and three times a week. Inventory status for 1 , 3, and 8 consumers are described in more detail in Table 3, Table 4, and Table 5.       Validation of each consumer's inventory status is shown in Table 3, Table 4, and Table 5. The table results show that the model developed is valid when viewed from an inventory status perspective. The status of Planned Order Receipts / POR for each consumer is matched with the route formed in Fig. 1 to Fig. 3. In Table 3, the POR status of 1 consumer is in the 3rd and 5th periods while in the results in sub-chapter 3.1, 1 consumers are considered to be included in the visit route on the third and fifth days, as seen in Fig. 2 and Fig. 3. 3 consumer based on their POR status were visited once, which is on the fifth day of the period that is by the results in Fig. 3. While 8 were visited three times. The results of the DRP prove that the model developed is valid when viewed from the inventory status.

Sensitivity Analysis
The developed model's sensitivity test results by making changes to vehicle capacity can be seen in Table 6. The effect analysis by changing the vehicle capacity variable was carried out because this periodic flexible study's initial objective was to optimize vehicle capacity. Therefore, the effect of changes in vehicle capacity should not be too large on total costs.
Based on the results of the sensitivity analysis in Table 6, it can be seen that the effect of vehicle capacity on total costs is not significant. It is because the flexible method has optimized the use of vehicle capacity. Thus, changes in vehicle capacity only affect the number of visiting routes. In Table 6, it is known that there was a change in total costs at ISSN: 1978- the beginning. Changes in total costs occur as the number of routes decreases. Furthermore, after the number of routes is not decreased, it can be seen that the total cost tends to remain unchanged.

Conclusion
The developed MILP model has succeeded in achieving the expected goals. Based on the model trial, it was found that each consumer was not visited every day but was periodically and flexible. From the research results, it was found that there were consumers who were only visited once ( 3 ) and ( 3 ). Some consumers were visited twice ( 1 , 4 , 5 , 6 , 7 ), and some were visited three times ( 2 and 8 ). The flexibility of visiting days that depends on the consumer's inventory status has also been validated by looking at the POR status on the DRP in each consumer. The sensitivity analysis test on the model also shows that vehicle capacity changes result in total costs and the number of routes formed. The bigger the vehicle capacity, the minimum total cost that is generated. This research still has several possible developments that can be done, for example, by developing heuristic solutions to facilitate more consumer nodes and demand from consumers that can be varied by using several distribution patterns to get closer to the actual conditions in real life.