MATHEMATICAL MODEL OF RICE COMMODITY SUPPLY CHAIN IN INDONESIA

ABSTRACT


INTRODUCTION
Food is a major need for every individual, which must always be met. Food availability has always been the government's concern in meeting food security. As for Indonesia, food is identified with rice because this particular type of food is a commodity consumed by the majority of Indonesia's population. As such, the rice supply crisis could endanger economic stability and national stability.
National rice consumption from year to year will continue to increase in line with the increase in Indonesia's population. Although Indonesia is the third largest country in the world in producing the most rice, Indonesia itself still depends on rice imports. This is done by the government to sustain national rice stock supplies. Mahbubi's research [1] describes the importance of assessing a sustainable rice supply chain system in efforts to ensure national food security. A supply chain consists of all parties involved directly or indirectly in fulfilling customer demand. The supply chain includes not only manufacturers and suppliers but also transporters, warehouses, retailers, and even the customers themselves. The supply chain includes all functions involved in receiving and filling customer requests [2].
These problems can be studied from a mathematical point of view for expected decision-making. According to Winston [3] a scientific approach to decision-making usually involves the use of one or more mathematical models. A mathematical model is a mathematical method of representation of an actual situation that can be used to produce better decisions or simply to understand the truth. The conceptual model of the decision-making process implies that the final decision is made through three main models, namely the model structure, the impact model, and the evaluation model [4]. Bala et al. [5] present a dynamic model of the rice supply chain system from farmers to consumers for efficient supply chain management in Bangladesh. Mathematical models are in the form of equations to describe the supply and demand for supply at a certain time. The research focuses on ensuring the availability of rice to consumers and the analysis of the sensitivity of the parameters. In the research of Surjasa et al. [6] analyzed the rice supply chain in Indonesia and a supplier selection model to identify and study the important components of the entire supply chain to maintain a stable supply and price of rice. In this study, the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) model was used to drink out the costs and maximize profits.
In this research, the model structure is taken from Badan Pusat Statistik (BPS) data [7] which is processed by involving producers, distributors, and retailers. This research focuses on constructing models related to supply chains in Indonesia. Such a model is also built from producers selling their products through two media, namely those directly to consumers and indirectly through distributors and retailers. The constructed model structure aims to maximize profits.

RESEARCH METHODS
This research is conducted with a literature study method. In this section, we divide it into two subsections, namely the literature review that underlies the research in this paper and the research methodology.

Literature Review
In 2008, Yong's research [8] explained the mismatch between supply, demand and resources faced by the rice supply chain in Malaysia. The study considers the size of domestic production and existing imports from various types of constraints and explains the need for an efficient supply chain improvement. This research is still limited to the basics and challenges of the rice supply chain in the future.
According to Aji [9], supply chain improvement is intended to increase profit competitiveness. In 2013, Mahbubi [1] studied the importance of the rice supply chain for sustainable food security in Indonesia which resulted in a dynamic model. The model formulation is structured by using a flow chart and mathematical formulation. This dynamic model, in order to see the supply chain system and food security, is not included in optimizing the profits of business actors. Research by South et al. [10] shows that the rice supply chain in Indonesia is not yet optimal, as indicated by the relatively small profits of farmers. Farmers as the main actors in the rice supply chain play an important role in meeting consumer demand. Sharma [11] in his research explained that farmers distribute rice directly to consumers to reduce middleman costs and increase income. Indirect distribution is carried out due to low demand or unfulfilled large demand or other reasons. Indirect distribution is carried out through collectors, retailers, and consumers [10].
In 2016 Bala et al. [5] developed a dynamic model of the rice supply chain system from farmers to consumers to address supply chain management scenarios and design policy options for efficient and sustainable supply chain management in Bangladesh. To ensure the availability of rice for the public, the impact of changes in productivity from seasonal rice production, lead time, and average demand variability on supply chain performance is assessed, and different management policies are also analyzed. The stability of rice supply and price can be achieved if rice supplies are always available and the price of rice is affordable to consumers, while the optimum profit is achieved for the actors involved in the rice supply chain [9].
The development of supply chain management models in rice production to improve food endurance and security in the Demak Regency was researched by Kurniawati et.al. [12] in 2020. Collaborative strategy for rice supply chain was researched by Guritno et.al. [13] using a case study on Demak and Sukoharjo regency, Central Java, Indonesia in 2021. In the same year, rice supply management was also researched by Fristin et.al. [14] to ensure self-sufficiency and food security. Furthermore, Wuryantoro et.al. [15] analyzed the supply chain and added value of rice in West Lombok Regency. In 2022, Prasetyo et. al. [16] will review the model and implementation of rice supply chain management. Based on the studies mentioned, this research was conducted.

Research Methodology
The rice supply chain model was obtained from the BPS [7], which was then simplified, developed, then analyzed. The steps taken in this study are studying previous research on the rice commodity supply chain model; determining the model assumptions to be developed; constructing a supply chain profit model by determining the demand function, total revenue, and total costs for each actor; proving that the model has the optimal solution for the centralized system; determining the optimal solution for the centralized system; finding suitable parameters for numerical simulations; finally, analyze the sensitivity of parameters to the advantages of the centralized system.

RESULTS AND DISCUSSION
Writing the results and discussion can be separated into different subs or can also be combined into one sub. The summary of results can be presented in the form of graphs and figures. The results and discussion sections must be free from multiple interpretations. The discussion must answer research problems, support and defend answers with results, compare with relevant research results, state the study's limitations, and find novelty.

The Model Framework
The structure of a simplified rice supply chain in Indonesia is presented in Figure 1. Based on Figure 1, the supply chain model for rice commodities in Indonesia is presented below, involving producers, distributors, and retailers. It is assumed that all actors in the supply chain sell standard rice. Producers sell domestic rice through direct and indirect media and sell rice abroad through export activities.
Distributors and retailers receive rice supplies from producers and imported rice. The decision to determine the optimum price uses a centralized system. The notation used in the model is presented in Table 1.  The following assumptions are used in the study 1. The amount of available rice produced is equal to the amount of consumer demand, so there is no shortage of rice. 2. There is only one producer, one distributor and one retailer. 3. The rice production costs by producer are equal to Rp5.200,00/kg based on the average Indonesia rice production costs. 4. Price of rice 1 , 2 , dan 3 always greater than rice productions costs . 5. The values of 2 < < and < < .

Demand Function
The number of initial units of demand made by producers of direct media for the country is . The initial unit of demand for producers for domestic consumers decreases as the selling price of rice changes according to the elasticity parameter of the selling price in domestic direct media, namely 1 1 . Furthermore, changes in domestic consumer preferences in buying rice from producers through the media are directly influenced by the price of rice from retail traders, so that the unit demand for producer increases in line with the price sensitivity parameters of local rice and imported rice, respectively 2 and 3 . The longer the consumers get imported rice because there is a lead time that is done by the distributor in the indirect media, which results in an increase in producer demand along with the sensitivity of the waiting time, namely 1 . Thus, the function of producer demand for domestic direct media is Producers also sell rice to overseas consumers through export activities. Initial demand unit from producers for export activities is . The unit of demand will decrease along with the increase in the selling price of rice, which changes according to the parameter of the selling price elasticity in direct media for foreign consumers, namely 2 3 . The existence of lead time for service and delivery of producer for export activities also reduces the unit demand and 2 . The existence of the rupiah against the US dollar affects the reduction of demand for units of 1 . Thus, the demand function for direct media producers for foreign consumers is The demand function for direct media is the sum of Equation (1) and Equation (2), namely: The initial unit of demand on the indirect medium is (1 − ) . In indirect media, the initial proportion for local rice is . The unit of initial demand for local rice in the media indirectly decreases in line with the increase in the selling price of rice which changes according to the parameter of selling price elasticity at retail traders for local rice by 4 . The existence of consumer preference in buying local rice at retailers results in additional demand units due to rising producer prices and imported rice prices by 1 1 and 3 and 3 . In addition, there is a lead time for delivery due to import activities that have an effect on the addition of local rice demand units of . Thus, the demand function on the indirect media for local rice is The proportion of units of initial demand for imported rice is 1 − . The unit of demand will decrease in line with the increase in the selling price of rice which changes according to the elasticity parameter of the selling price of rice from importers and the selling price of imported rice for domestic consumers, which are respectively 5 and 3 , . In addition, the unit of demand will also increase. Demand for imported rice will increase by 1 1 and 2 in line with rising producer prices and local rice prices. The existence of lead time due to export activities will also reduce the demand for imported rice in the indirect media by . . The rupiah exchange rate against the US dollar also has an effect on reducing the demand unit by 2 . For that, the function of imported rice demand in indirect media is Thus, the function of the media indirect demand is sum of Equation (4) and Equation (5): = − − 4 − 3 − 5 + 2 1 1 + 2 + 3 + − − 2 . (6) The total demand function for the rice supply chain is sum of Equation (3) and Equation (6) as follows:

Profit Function
Producers sell rice through direct and indirect media so the producer profit function is obtained from the sum of profits in the direct and indirect media. The costs incurred by producers are the production costs of multiplied by consumer demand for direct and indirect media for local rice. Revenues earned distributors are derived from the sale of imported rice at a price multiplied by the number of requests for imported rice media indirect and sale of local rice at a price multiplied by request of indirect-media local rice. Distributor incur costs by purchasing rice from producer at a price of 2 . Apart from that, distributor also pay a fee of to obtain imported rice. Then, retailers earn income from selling imported rice at a price of which is multiplied by the demand for imported rice in indirect media. In addition, retail traders also earn income by selling local rice at the price of which is multiplied by the local demand for rice in direct media. Retail traders will pay the same amount as the purchase of rice from the distributor. So that, the optimal profit function is obtained

Supply Chain Centralized
The centralized model in the rice supply chain is a profit function model without the constraints obtained by adding up the profits of producer, distributor, and retailer with price decision variables 1 , 3 , and . In this model, suppose 2 = 1 1 , = 2 , dan = 3 . So, the profit function of the centralized model is sum of Equation ( From the profit function (Π C ) of the system, a Hessian matrix can be constructed to prove that the system profit function is a straightly concave or has a single maximum solution.
The principal minor determinants of the Hessian matrix in Equation ( Based on the 6 th and 7 th assumption is obtained |H 11 | < 0, |H 22 | > 0, |H 33 | < 0, and |H 44 | > 0. So, it can be concluded that Π ( 1 , 3 , , ) is straightly concave function. It's proof that the system profit function is a straightly concave so that it is guaranteed that the function has a single maximum solution. The

Numerical Experiment
The solution obtained is then carried out with a numerical simulation approach with a centralized system. The parameter values used in this study are described in Table 2. Based on the numerical simulations, the optimum solutions of the rice commodity supply chain model are obtained as follows The sensitivity analysis in this study uses the effect of changing proportions on direct and indirect (θ) media. The parameter value = (0.0.1] is taken because when > 0.1 causes the price of 2 > > so that it does not meet the 5th assumption. The effect of changes in parameter value at the interval (0.0.1] on the profits of each supply chain actor is presented in Figure 2.

CONCLUSIONS
This research examines the rice supply chain in Indonesia obtained from BPS, which is then simplified by involving the main actors of producers, distributors, and retailers. Producers sell rice to domestic consumers through direct and indirect media and sell rice abroad through export activities. Distributors and retailers receive rice supplies from producers and imported rice. The mathematical model is constructed to obtain the maximum total profits by using a centralized system. Based on the numerical simulation results, changes in the parameters of proportions on the direct and indirect media greatly affect the optimal benefits obtained.

ACKNOWLEDGMENT
We are very grateful to University of Sebelas Maret, Surakarta.