Improving vehicle routing decision for subsidized rice distribution using linear programming considering stochastic travel times

As the highest regency with the absolute poverty rate in Indonesia, Bogor Regency must be able to handle the poverty program appropriately and ensure its effectiveness and efficiency. Subsidized rice is one of the government programs for poor households. This program is important because rice is a food commodity with the largest contribution to the food poverty line. The effectiveness of the distribution is very dependent on the accuracy of the target number of beneficiaries and the accuracy of the amount of rice received at the consumption points. Meanwhile, the distribution efficiency is measured from the distribution route by taking into account the travel time that is directly related to transportation costs. This study discusses the distribution from only 1 (one) warehouse to 57 destinations. This study contributes to the enrichment of a cluster first route second approach to solve the routing problem. This paper applies Ward’s method and K-Means sequentially which generates 4 (four) clusters. Linear programming under the Capacitated Vehicle Routing Problem model is utilized to generate the routing decisions. Clusters 1 and 3 must take 1 and 2 trips, respectively. Meanwhile, clusters 2 and 4 each take 3 trips. As a result, the proposed method ensures all trips consume the shortest time.


Introduction
The poverty line is an indicator to declare the poor population where people below this line are declared poor. The poverty line is stated in two components, namely the food poverty line and the non-food poverty line. The food poverty line has a greater role than the other. The food commodity that contributes the most to the food poverty line is rice, which is 21.81 percent.
The government runs the program to sell low-priced or subsidized rice. This program aims to reduce the burden of poor households' expenditure by meeting some of the basic food needs and preventing the reduction in energy and protein consumption. The effectiveness of this program's distribution is very dependent on the accuracy of the consumption points and the amount of rice. Meanwhile, the distribution efficiency is measured by distribution routes that are directly related to transportation costs.Stakeholders must always be aware of things that can disrupt the smooth running of business processes [1].In this context, it is necessary to exceed the customers' expectations to be classified as excellent [2].
Poverty is not merely a lack of income to meet basic living needs or decent living standards, but it involves the probability of poor people to carry out and develop their businesses and living standards. The absolute poverty rate in Bogor Regency occupies the highest number among all cities/regencies in Indonesia, which is almost 400 thousand individuals. Therefore, this research is conducted in the coverage area of the warehouse which is located in Bogor City. This warehouse not only serves Bogor City but also Bogor Regency and Depok City as Figure 1 shows. This paper assumes the consumption point is at the district level in the 3 (three) cities/regencies. Thus, there are 57 consumption points in total [3]. This study focuses on solutions to the distribution effectiveness problems which is closely related to distribution efficiency by considering stochastic travel times.

Problem formulation
The distribution problem with travel time from one point to another requires different travel times even with the same distance. Travel time is dynamic [4] and varies due to traffic congestion or weather conditions that affect the efficiency and feasibility of shipping routes [5][6]. By minimizing the travel time of the distribution route, operational costs are assumed to be minimal as well to achieve the effectiveness and efficiency of distribution. Any waste should be minimized if not eliminated [7].
A study was carried out related to the effectiveness of the distribution of goods without regard to the distribution efficiency factor. A location-allocation problem with dwell time was discussed to support thedecision-making process [8]. A previous study discussed the transportation mode allocation by considering demand uncertainty using the hill-climbing approach [9].
Previous studies discussed the efficiency of the distribution of goods that had been carried out using exploratory survey methods and the system dynamics approach. The exploration survey method was used to reduce exhaust emissions in the transportation industry in Sweden [10]. Other studies used the system dynamics approach to estimate and minimize costs caused by transportation [11] by considering environmental impacts [12]. These studies did not point out the effectiveness of distribution.
Other studies discussed the effectiveness and efficiency of distribution by various methods, including analytical descriptive methods, simulations, Genetic Algorithms (GA), Fuzzy Genetic Algorithms (FGA), and simulated annealing. A previous study was conducted to measure the effectiveness and efficiency of distribution routes using descriptive-analytical methods [13]. Simulation models were applied to improve the sustainability of the distribution network performance in British logistics companies by taking into account exhaust emissions[14 -15]. Research using GA, FGA, and simulated annealing aims to optimize costs on the distribution of fresh food products [16]. Respectfully, the above research did not use the Capacitated Vehicle Routing Problem (CVRP) as used in this study. A Mixed Integer Non-LinearProgramming (MINLP) was developed model for planning the movement and storage of food grain from surplus states to deficitstates considering the seasonal procurement, silo capacity, demand satisfaction, and vehicle capacity constraints [17]. CVRP model can be used as the powerful model to solve the routing problems considering many characteristics [18]. A previous study performed a CVRP model considering the stochastic travel times to minimize the transportation cost [19]. In performing CVRP, it is necessary to identify 2 (two) types of locationsnamely source point and consumption points. The source point is defined as a single warehouse. There are 1,550 storage warehouses throughout the country responsibleto ensure the distribution of subsidized rice is successfully done.One of them is an important part of this research because it is the only warehouse thatserves Bogor Regency. The distribution activities are carried out at an agreed distribution point such as sub-district offices or places that are closer to the community.
The distribution route is partial shipping from the warehouse to each destination. This study aims to determine and minimize travel time for distribution routes in aparticular area. The intended distribution route is taken from a warehouse to 57 districts in Bogor City, Bogor Regency, and Depok City. By minimizing the length of the distribution route, operational costs are assumed to be minimal and in turn, it can achieve the effectiveness and efficiency of distribution.

Methodology
This paper aims to determine the distribution route started withclustering analysis and followed by the CVRP model as shown in Figure 2. The scope of this study is the distribution route Bogor City, Bogor Regency, and Depok City. A collection of materials, data, and information is derived from books, field studies, and relevant scientific articles to strengthen the understanding of the distribution route. At the same time, some data are collected such as travel time between distribution points, the average consumption of rice per person in a certain time along with the type, and capacity of transportation modes. After the data is collected, the analysis is carried out and then formulate the recommendations.
A previous study used K-Means to determine the location of distribution centers as the representative of each region [20]. This paper uses the same logical proportion of demand [20]since the data is obtained at the regency/city level.
The travel time data acts as an input for clustering analysis and CVRP model respectively. The transportation cost is approximated by the travel time between the distribution points. Travel time should follow the normal distribution, so the average value is taken as transportation cost.
To obtain the travel time between distribution points, the coordinates of each destination are needed. After obtaining the coordinates, the travel time can be found using the Google Maps application by repeating at least 30 (thirty) times in working time and workdays. Working time is defined as the active time during the hours of 8:00 to 16:00, while workdays are defined as Monday to Friday. The warehouse capacity is assumed to be ableto meet the demand.

Results and discussion
The data on the poorhouseholds were obtained only at the city/regency level, so this paperperforms a proportional calculation of the poorhouseholds to the total population at the district level. Each person is assumed to consume 114.6 kg per year [21] or 0.3140 kg per day. This assumption also indicates that the distribution is carried out every day to the consumption point. Meanwhile, the transportation mode is assumed to have a capacity of 25,000 kg or 25 tons. The consumption point with the lowest and highest demand is in the Limo District coded DE7 (784 kg) and Gunung Putri District coded KA18 (10.392 kg) respectively.

Clustering analysis
Because CVRP is Non-Polynomial hard (NP-hard), clustering analysis is performed beforehand so that the computing time would not be too long. In the clustering analysis stage, we conduct two stages, namely hierarchical clustering analysis using Ward's method and non-hierarchical using K-Means.

Ward's method.
The results of Ward's method are shown in Table 1. This table explains the number of clusters formed as seen from the percentage increase in the coefficient value after a merger between the two clusters. Coefficient values describe the distance that occurs between clusters formed at a particular stage. The greater the increase in the coefficient value between one stage with another stage indicates that the two clusters are increasingly heterogeneous. For example, at stage 2 there is a merger between points 4 and 7 with a coefficient of 0,000 while at stage 3 there is a merger between points 51 and 57 with coefficients 0.001. The increase in the coefficient value for both stages is equal to 1.056893. Eventually, each point will merge into one large cluster. The average increase in the value of the coefficient for each stage is 0.183805 or 18.2805%. The value of the increase in the coefficient that is greater than the average value is a candidate for the number of clusters to be formed. The biggest increase occurred during stage 2, which amounted to 1.056893, but this value could not be taken because the number of clusters was too large, namely 56 clusters. The second-largest increasing coefficient occurs at stage 54 and indicates 4 (four) clusters are formed, with a value of 0.613912. Figure 3 also states the same thing, namely an increase in the coefficient seen from the elbow graphic shape that increases significantly at the 54th stage. The value of 4 (four) clusters will be input (k value) on K-Means.   Figure 4 where clusters 1, 2, 3, and 4 have 10, 19, 7, and 21 members respectively. Cluster 1, 2, 3, and 4 are shown in red, yellow, blue, and green icons respectively. Each cluster proceeds to the CVRP calculation stage separately.

Capacitated vehicle routing problem (CVRP) computation
To calculate CVRP, it takes travel time between points in each cluster. It requires different computational time to reach the optimal global solution for each cluster due to the different number of points. In clusters 1 and 3, the computation time is very short (finished instantaneously), while in cluster 2 it must spend 7 hours 13 minutes and 50 seconds. The longer time needed by cluster 4 which is 8 hours 43 minutes 17 seconds to solve 22 points. Table 2 explains the proposed route to be taken by each cluster. Clusters 1 and 3 must travel 1 trip and 2 trips, respectively. Meanwhile, clusters 2 and 4 each took 3 trips. The time required for all trips is the shortest and longest taken by cluster 3 (447.4 minutes) and cluster 4 (646.8 minutes) respectively.
Furthermore, the extension analysis may include uncertainty demand [22]. The commodity can be transported using heterogeneous fleet [23]. When it comes to fleet management, both corrective and preventive maintenance policies need to be taken into account for every transportation mode [24]. Future research may also associate the workers' motivation in delivering goods [25] with their performances [26].The performance measure can consider the last mile distribution until the smallmedium enterprise stage [27].