Route optimization model design to support multi-cluster commodity distribution in the Moluccas islands

This study aims to design a route optimization model to support commodity distribution in the Moluccas archipelago zone. The model is designed using an integer programming approach for multi-depot multi-cluster with the aim of minimizing distance. In this case, several points on the island must be serviced. Each service point will only choose one port to serve. Sea transportation routes are formed from selected port nodes with a ring-ring-star hierarchical structure. A model designed to complement transportation routes in meeting the demand for service points by considering local, regional and global route distribution. Each level route will select one or more depots which will represent set to the top route level. The model is applied to several scenarios to see the best distance recommendations from island sample characteristics and point and island sizes.


Introduction
Food is needed to ensure the sustainable development of human society.Obtaining and utilizing food is the most basic and enduring interaction between humans and the environment [1].To maintain the continuity of this relationship, the food system is an essential factor to be strengthened.The food system includes primary agricultural production, food processing, distribution, consumption, and waste disposal.Because of the breadth and complexity of the scope of supply chain activities, there is a need for integrated and collaborative management from the upstream (production) to the downstream (consumption) lines.
The COVID-19 pandemic has added a new perspective to food/commodity supply chain management.Its influence on the food sector impacts inequality in food access and new food vulnerabilities [2].This pandemic shows how humans will accept and support transitioning to a more localized food production system [3].To anticipate the non-occurrence of the same conditions, efforts to integrate the potential of local commodities with an efficient distribution network are needed.This effort at least does not feel big for a country whose territory is mostly land.However, this will be an essential part that needs attention for countries whose territory is primarily islands, such as Indonesia.
Network distribution efficiency is highly dependent on the model used.The reliability of the model is dependent on the factors considered.Clustering, location, and routing integration are core features of network distribution design [4,5].Various clustering methods to improve solution quality in the face of the difficulty of solving integrated location and routing problems have been proposed by Barreto et al. [6].This method was further developed by Lam and Mittenthal [7] in the context of multi-depot site routing.In particular, the clustering of locations by Lam and Mittenthal significantly impacts the overall cost of vehicle routing.Their results motivate the multi-cluster transport network model discussed in this study.
The network systems discussed in this study intersect with ship routing problems or insular vehicle routing problems [8][9][10], hub location routing problems [11], and covering or facility location problems [12].However, it is rare for articles to specifically discuss the efficiency of multi-cluster food distribution networks in the island zone, especially the Moluccas.Research related to transportation in the Maluku archipelago zone has been discussed but only touched on the problem of vehicle routing for product delivery [13].Therefore, in this research, we use a programming approach to develop a quantitative optimization model to analyze the optimal distance of existing food distribution networks and propose efficient network configurations for the archipelago of Maluku Province.
The model in this study considers clustering and routing factors with a ring-ring-star hierarchical structure.This hierarchical structure differs from that discussed by Karimi [14] and Eidy et al. [15].Clustering includes depot selection (local and regional), number of clusters, and cover distances, while routing comprises local, regional, and global transportation routes.Local routes are established between selected local depots and non-depot points.This route is in the shape of a star, different from the shape of regional and global routes, namely rings.Regional routes are formed between selected regional depots and local depots, while global routes include selected regional depots.In this model, routing does not consider vehicle capacity, pickup, and delivery as Hosoda et al. [16].
This model was developed from the facility location problem [17] and the hub location routing problem [11] and uses several other techniques related to multi-cluster and multi-depot.The problem introduced in this study is a new variant of the location, coverage, and routing problem, which is called the multi-depot clustering, location, and routing problem.

Methodology
Two research questions must be met in this study: (1) what is the mathematical formulation of the optimization model?and (2) what is the optimal distance efficiency of the configuration of the distribution network system in Maluku Province, with either two or three regional depots?To answer this question, this research was carried out in three main stages: data collection and processing, optimization model development, and efficiency analysis.
Analysis of the optimal total distance efficiency of the distribution network configuration using equation (1).The optimal total distance is based on the developed optimization model's simulation output.
where, E n = Efficiency of the total distance of the distribution network system with n regional depots (%) Y n = Total optimal distance of the distribution network system from n regional depots (kilometres, km).Determined based on the objective function of the optimization model developed.n = Number of regional depots planned = 3 Villages or islands in Moluccas Province are represented by several sample ports.Samples are taken collectively by considering their role and location on an island.The ports sampled were seaports, ferry ports, or small ports used by a village as transportation facilities.The village port is taken if there is only one village on an island and there are no seaports or ferry ports.
Data collection was carried out through literature studies, interviews, and measurements.Literature studies are carried out through reports, websites, and articles considered scientific and relevant to identify ports and their locations, as well as government policies regarding clustering and sea transportation going forward.Interviews were conducted with related agencies to learn more about marine transportation systems' characteristics and policies.Measurements are carried out on identified port locations to obtain the distance between ports.Distance data between ports is based on table 1.

Table 1. Matrix of measuring distances between ports
The optimization model was developed for multi-depot using a linear integer programming approach.The model's mathematical formula is programmed using the Lingo optimization software.The model structure is constrained as follows.First, transportation from the cluster/local depot to the ports it covers occurs directly (direct shipment).Second, transportation between regional depots and between cluster/local depots and regional depots takes place on a consolidated basis.Third, the maximum cover distance and expected number of clusters are deterministic.The model parameters include the distance between ports, the maximum cover distance for depots, the expected number of clusters, and port indicators that classify port types based on readiness (located close to the district/city capital) and future development expansion (located on islands with an area of the land area of more than 365 square kilometres, km2).
The developed model is used as a simulation tool.The simulation is based on the following three scenarios: 1).Ports of Ambon and Tual act as regional depots and double as local depots with a maximum cover distance of no more than 500 kilometres 2).Saumlaki Port as an alternative to the third regional depot besides Ambon and Tual ports, and the three of them have a maximum cover distance of no more than 500 kilometres 3).Port of Moa as an alternative to the third regional depot beside the ports of Ambon and Tual, and the three of them have a maximum cover distance of no more than 500 kilometres The first scenario determines the efficiency of using two regional depots, Ambon and Tual Port.This scenario compares to the other two scenarios to determine the efficiency of the Ports of Saumlaki and Moa as an alternative to the third regional depot.

Port identification
Identification of port locations is based on the type and benefits of the port.Table 2 shows the names and types of ports in Maluku Province that were identified.The first eleven ports are ports that have high readiness as depot alternatives.Of the 73 existing ports, 58 are seaports, while the rest consist of 11 ferry ports and four village ports.The reason for adding village ports is that several countries have an influential position in the route but do not have seaports and ferry ports.It is assumed that the island has a speedboat dock or similar.

Measurement of the distance between ports
The distance between ports is measured using Google Maps in kilometres (km).Measurements are based on the provisions of the shipping route of the ship.The results of these distance measurements can be seen in [18].This data contains parameters of the distance between ports and indicators of port readiness as an alternative to regional or local depots.The distance between ports is measured in kilometres (km).Port readiness is indicated by being located close to the district/city capital, located on an island with a land area of more than 365 square kilometres for future development expansion.= the binary variable indicates if port i is a port that deserves to be selected.

Decision variables. yl i = the binary variable indicates if port i selected as local depot xl ij
= the binary variable indicates if port i is covered by port j wg ij = the binary variable indicates if depot i is covered by depot j xg hij = the binary variable indicates if, in the loop from depot h, the port of depot i is covered by the port of depot j v i = the integer variable indicates the sequence of visits to depot i.

Mathematical Formulation.
The objective of minimizing the total distance of transportation can be seen in equation (2).The components of the left side are the sum of the cover distances from the depot to the ports, while the right side is the total distance of sea transportation between depots.This objective is subject to constraint functions (3) through (28).
subject to: ( ) 3) represents two decision variables: yl is binary, while v is non-negative.Equation ( 4) functions as a controller of depot decisions based on the level of readiness or feasibility for development.Equation ( 5) regulates the selective suitability of the depot based on the desired number of clusters.Equations ( 6) to (9) control the provision for transferring a depot to a port.Equation (10) selects the planned port as a regional depot.Equations ( 11) to (13) set the value of the decision variable wg, which will only be selected if it meets the coverage requirements.Equations ( 14) and (15) set the general requirements for the decision variable xg.Equation ( 16) ensures no cover-up between regional depots, while Equation (17) ensures that ports that are not regional depots cannot cover other depot ports.The amount of coverage is set by Equation ( 18) which must equal the expected number of clusters.Equation ( 19) to (26) governs regional routing conditions.To ensure that sub-tours do not occur on global routes, the MTZ (Miller-Tucker-Zemlin) formula is used, which is represented by equations ( 27) and (28).

Application of model
In the model application, the maximum cover distance parameter setting (cov max ) is determined to be 500 kilometres, while the zk i value for each port can be seen in table 3.

Scenario III.
In this scenario, the Ports of Ambon, Tual, and Moa are assumed as regional centres.The experimental results can be seen in table 6.

Discussion
In addition to analyzing the efficiency of the distance from the two regional depots, efforts to recommend three depots were also carried out through experiments II and III.This was done by considering the large maximum cover of Ambon Port to the south.To analyze this case, two alternative ports were sampled, Saumlaki and Moa.The simulation results of both are illustrated in figures 1 and 2.

Figure 1 .Figure 2 .
Figure 1.The operational areas of the three regional depots, Ambon-Tual-Saumlaki, based on the optimal distance

Table 2 .
Names and Types of Ports in Moluccas

Table 3 .
zk parameters for experiment In this scenario, the Port of Ambon and Port of Tual are assumed as regional centres.The experimental results can be seen in table4.

Table 4 .
The results of scenario I In this scenario, the Ports of Ambon, Tual, and Saumlaki are assumed as regional centres.The experimental results can be seen in table5.

Table 5 .
The results of scenario II

Table 6 .
The results of scenario III