Enhanced forecasting of rice price and production in Malaysia using novel multivariate fuzzy time series models

Abstract

A significant portion of the world’s population relies on rice as a primary source of nutrition. In Malaysia, rice production began in the early 1960s, which led to the cultivation of the country’s most significant food crop up till the present day. Research on various aspects of the price and production of rice has been done by various methods in the past. In this study, we have adopted novel multivariate fuzzy time series models (MFTS) i.e. fuzzy vector autoregressive models (FVAR) alongside conventional vector autoregressive model (VAR) for assessing rice price and production using a dataset from the Malaysian Agricultural Research and Development Institute (MERDI). The proposed method(s) especially with the usage of Trapezoidal Fuzzy Numbers (TrFNs) have commendable accuracy with great future forecasts over the VAR model. The model selection was made by the least MAPE with the corresponding highest Relative Efficiency as criteria. The study fills the gap in applying advanced fuzzy models for rice forecasting, aiming to improve accuracy using fuzzy vector autoregressive (FVAR) models with Triangular Fuzzy Numbers (TFNs) and Trapezoidal Fuzzy Numbers (TrFNs) over traditional VAR models. The study’s findings imply that the enhanced forecasting accuracy of FVAR models with Trapezoidal Fuzzy Numbers (TrFNs) can significantly assist local farmers and stakeholders in making informed decisions about production and pricing. This improved forecasting capability is expected to promote business growth within the Malaysian market and facilitate increased rice exports, ultimately contributing to the country’s economic prosperity.

Introduction

Rice is a cornerstone of Malaysian cuisine and a staple food for its population, making its production and pricing critical issues for the nation’s economy and food security. Climate, agricultural practice, government policy, and the global market also play a role in determining the dynamics of rice production and price in Malaysia. A thorough examination of these considerations will help to comprehend where rice production stands at present and what possibilities lie ahead for Malaysia’s cereal crops. This has not only caused widespread shock waves in the global production of the crop, but this also brought a huge challenge to farmers as well as policymakers. These variances will influence the cost of food and availability, which going to affect living standards for millions of Malaysians. Accordingly, a detailed examination of the main drivers behind these developments is essential to identifying measures needed for market stabilization and maintaining supplies necessary, especially in support of agriculture.

This study aims to assess the behavior of rice production in Malaysia and analyze its pricing by utilizing historical data and trend identification to uncover the root causes of variability. By examining the interplay between local agricultural practices, government interventions, and external market forces, this research endeavors to provide a comprehensive overview of the Malaysian rice industry. The data for this assessment was sourced from several different locations - government reports, market surveys, and agricultural statistics to build up a picture of production patterns and pricing movements. Rice in Malaysia is very much weather-dependent because we are not able to irrigate every single acre of paddy fields with the type of infrastructures that we have implemented technological speed advances and pest control measures also play a vital role in our rice production throughout those years as well. Also, the crop is influenced by government policy concerning subsidies, import controls, and assistance for farmers which all shape what can be produced. Factors such as population increase, diversity of diets, and income levels have a strong influence on the right side of the equation which is the demand side that also affects consumption changes and prices in the market.

The innovations and recent research offer important information to stakeholders (agriculturalists, legislators, scholars) whose decisions can also help ensure survival and sustainability in their choices regarding what improvements make a rice field grow faster. The study provides insight into rice production and pricing which are the major factors in ensuring stakeholders meet their needs, reduce risks, and increase productivity security. Accomplishing this comprehensive review, not only can provide the basis for existing academic databases related to the rice industry in Malaysia but also offers some practical ideas that could be implemented as a means of enhancing the Malaysian paddy sector.

By conducting a comprehensive review, this research not only contributes to the existing academic literature related to the Malaysian rice industry but also presents practical recommendations for enhancing the paddy sector. Additionally, a quantitative examination investigates global market trends i.e. international prices of rice over time, trade policies in major production and consumption areas, and factors related to climate change affecting rice cultivation at the world level Such external factors are important to consider in a rice trade that is becoming more integrated and exposed to global market cycles. Vacuuming local and global perspectives, this assessment can help in gripping the challenges as well as opportunities for Malaysian rice sector development. The determination of rice production and pricing in Malaysia is a complex task that warrants analyzing different factors affecting the industry. The results will help guide collective action towards the development of sound policies and strategies to ensure a stable, sustainable rice supply; which can contribute toward improving Malaysia’s economic health through food security.

Vector Autoregressive model

Vector Autoregressive (VAR) models are the most popular and widely used in time series research. Their primary purpose is to investigate the dynamic interactions that exist between variables that interact with one another. Furthermore, they are considered to be significant forecasting tools that are used by the majority of organizations that are involved in macroeconomics or policymaking.

General form of the VAR $\:p$ model

The general form of a Vector Autoregressive (VAR) model of order $\:p$ can be expressed as follows in Eq. (1):

$${y_t}=c+\sum\limits_{{i=1}}^{p} {{A_i}{y_{t - i}}+{u_t}}$$

(1)

Where

${y_t}$is an n-dimensional vector of endogenous variables at a time t. c is an n-dimensional vector of intercept terms (constants). ${A_i}$, $\forall$$i=1,2,3,\ldots,p$are $n \times n$ coefficients matrices. ${u_t}$is an n-dimensional vector of error terms (innovations or shocks) at a time t, usually assumed to be white noise with mean zero and covariance matrix $\sum\nolimits_{u} {}$.

For specific bivariate VAR with 12-lags, the reduced form is under in Eq. (2):

$${y_t}=c+\sum\limits_{{i=1}}^{{12}} {{A_i}{y_{t - i}}+{u_t}}$$

(2)

where ${y_t}=\left( \begin{gathered} {x_t} \hfill \\ {y_t} \hfill \\ \end{gathered} \right)$, $c=\left( \begin{gathered} {c_x} \hfill \\ {c_y} \hfill \\ \end{gathered} \right)$, ${A_i}=\left( {\begin{array}{*{20}{c}} {{a_{xx,i}}}&{{a_{xy,i}}} \\ {{a_{yx,i}}}&{{a_{yy,i}}} \end{array}} \right)$, and${u_t}=\left( \begin{gathered} {u_{xt}} \hfill \\ {u_{yt}} \hfill \\ \end{gathered} \right)$

Fuzzy numbers

One of the most essential concepts in fuzzy set theory, which is a mathematical framework for dealing with uncertainty and imprecision, is the idea of fuzzy numbers. Unlike classical numbers, which have precise and crisp values, fuzzy numbers represent values with degrees of membership, capturing the idea that some values are more likely or more representative than others within a certain range.

Definition

One instance of a fuzzy set on real numbers${\mathbb{R}}$is referred to as a fuzzy number. One of its distinguishing features is a membership function $\mu :{\mathbb{R}} \to \left[ {0,1} \right]$ that assigns to each real number a degree of membership. This membership function describes how much the number “” belongs to the fuzzy set, $\mu \left( x \right)=1$ indicating full membership and indicating no membership.

Triangular fuzzy numbers

The triangle fuzzy number, often known as the TFN, is the most widely used of all the available forms of fuzzy numbers. A fuzzy number is represented by three points, and the following is how it is expressed in Eq. (3):

$$A=\left( {{a_{j - 1}},{a_j},{a_{j+1}}} \right)$$

(3)

having membership functions as follows in Eq. (4) and represented by Fig. 1.

$$t=\left\{ {\begin{array}{*{20}{c}} {\frac{{1+{\gamma _1}}}{{\frac{1}{{{a_1}}}+\frac{{{\gamma _1}}}{{{a_2}}}}}}&{j=1} \\ {\frac{{{\delta _1}+1+{\delta _2}}}{{\frac{{{\delta _1}}}{{{a_{j - 1}}}}+\frac{1}{{{a_j}}}+\frac{{{\delta _2}}}{{{a_{j+1}}}}}}}&{2 \leqslant j \leqslant m - 1} \\ {\frac{{{\gamma _2}+1}}{{\frac{{{\gamma _2}}}{{{a_{j - 1}}}}+\frac{1}{{{a_j}}}}}}&{j=m} \end{array}} \right.$$

(4)

Where ${a_{j - 1}}$,${a_j}$, and ${a_{j+1}}$ are the midpoints of the interval of fuzzy sets ${X_{k - 1}}$, ${X_k}$ and ${X_{k+1}}$ carriers respectively, while ${\gamma _1}$, ${\gamma _2}$, ${\delta _1}$ and ${\delta _2}$ are called parameters.

Fig. 1

Triangular fuzzy numbers.

Trapezoidal fuzzy numbers

The Trapezoidal Fuzzy Numbers are generalizations of the triangular fuzzy numbers. A trapezoidal fuzzy number ${{\tilde {n}}}$ can be defined as $\left( {{n_1},{n_2},{n_3},{n_4}} \right)$, which has the membership function ${t_{\tilde {n}}}\left( x \right)$ as follows in Eq. (5) and represented by Fig. 2.

$${t_{\tilde {n}}}\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {0~~~~~~~~~~~~~~~~~~~x~<{n_1}} \\ {\frac{{x - {n_1}}}{{{n_2} - {n_1}}}~~~~~~~{n_1} \leqslant x \leqslant {n_2}} \\ {1~~~~~~~~~~~~{n_2} \leqslant x \leqslant {n_3}} \\ {\frac{{x - {n_4}}}{{{n_3} - {n_4}}}~~~~~~~~~~~~~~~x \geqslant {n_4}} \end{array}} \right.$$

(5)

Fig. 2

Trapezoidal fuzzy numbers.

We will see how Eqs. (2),  (4), and (5) form our proposed models in the next sections.

The novelty of this study lies in the application of advanced multivariate fuzzy time series models (MFTS), specifically fuzzy vector autoregressive models (FVAR), for forecasting rice production and pricing in Malaysia. By incorporating Triangular Fuzzy Numbers (TFNs) and Trapezoidal Fuzzy Numbers (TrFNs), the study addresses the uncertainties inherent in agricultural data, resulting in improved predictive accuracy compared to traditional vector autoregressive (VAR) models. This research not only fills a significant gap in the literature regarding fuzzy modeling in agricultural forecasting but also provides actionable insights for stakeholders, ultimately enhancing decision-making and supporting food security and economic growth in Malaysia.

Literature review

Rice is hardly an economic concern in most places, but Malaysia relies heavily upon rice as a staple foodstuff required for all three daily meals, and thus so does Thai. There have been many studies investigating different dimensions of rice production and pricing considering factors such as climatic, and technological advances or government policies, and occasional global market influence. This literature review aims to provide a holistic understanding of the determinants that influence rice production and pricing in Malaysia through synthesizing prior research.

Case studies on the effect of climate change on rice production exist in Malaysia. For instance¹, has shown rice to be most at risk from changing temperatures and precipitation patterns. The report revealed that floods and droughts, extreme weather events, had caused severe disruption to rice farming practices with rice production levels varying accordingly. Similarly, in² the implications of climate change for rice productivity were addressed, and emphasized the necessity to introduce adaptive farming practices, which could reduce negative consequences. A study by³ showed that the management of water resources is critical for getting high rice yields under fluctuating climatic conditions and their results were in line with such viewpoint.

The use of new technologies is one of the keys to increasing rice yields research e.g⁴. has also highlighted the positive implications of new agricultural practices such as precision farming, upgraded irrigation systems, or high-yield rice seeds. These innovations have led to more productive and stress-responsive crops. The researchers^5,6 highlight that rice costs are considerably influenced, but modernization that affects labor and harvest efficiency from such automation advances seems to be something that would alter the total cost structure of pricing for a product like rice over time. Further⁷, imposed has been given towards alleviation of losses and improving production also brought the role of biotechnology in designing pest-resistant genotypes.

Government policies largely determine the rice production terrain in Malaysia. Moreover, a study by⁸ argues for subsidies and price controls as potential stabilizing instruments in the rice market. After identifying such behavior in their model, which allowed the movement of supply from one region to another with a different set price control levels on markets there for output and inputs like fertilizer or wage goods (like corn), they found that subsidies are good policy because lowering costs is effective at helping farmers. The study⁹ examined the impact of import restrictions and trade policies on domestic rice prices, arguing that these interventions are necessary to shield local producers from fluctuating international market forces. The researchers¹⁰ assessed the role of public investment in agriculture infrastructure on crop productivity and market stability.

Rice production and pricing are also influenced by socioeconomic variables, such as population growth, income levels, or dietary preferences. They¹¹ explained the impact of prices from the population pressure growth and income level increase will raise people’s consumption more rice. The potential impact of changes in dietary habits driven by urbanization and lifestyle shifts on consumption patterns, and therefore the supply-demand balance targeted to be achieved in the rice market¹², and investigated the effects of urbanization on labor supply available from communes, demonstrating that migration movements can leave rice production villages with scarce workers.

This situation is explained in part because of the relationship among global markets, meaning that international trends and policies do somewhat affect Malaysia’s rice industry. For example¹³, shows the influence of global rice prices and trade agreements at the local market level. Changes in world rice prices exogenously affect domestic retail prices through global supply shortages or surpluses. At least partly, this is because the competitiveness of rice produced in Malaysia for export also depends on trade policies and agreements (e.g. under ASEAN) that affect it when comes to global markets. This perception has been also perpetuated by a recent paper discussing the potential impact of international trade disputes and tariffs on rice, calling for more strategic trade policies.

In several studies done in the Malaysian rice sector, considerable challenges and opportunities have been identified. Climate uncertainty, resource limitation, and fluctuating market environment are the main thorny sides. For instance¹⁴, addressed the sustainability challenges associated with water management in rice production. At the same time, there are opportunities for sustainable agriculture and technology adoption as well as policy frameworks. Research by¹⁵ highlighted the opportunity of moving towards sustainable development through integrated farming systems and organic rice production Additionally¹⁶, also emphasized the key role of farmer education and training programs for sustainable farming practices upliftment in crop productivity. The authors¹⁷ discussed rice (Oryza sativa L.) as a staple food for over half the world. Rice milling produces bran, rich in bioactive compounds like phenolics and γ-oryzanol, which may help prevent diseases. Although often discarded or used as animal feed, rice bran’s functional ingredients have potential applications in food and health industries for preventing metabolic disorders.

The general modeling concepts for this study are enhanced by the following studies: the study¹⁸ models mammalian prey-predator dynamics, factoring in prey growth, Allee effects, and environmental influences like moonlight and water. It analyzes stability and bifurcation, showing how these factors affect interactions, with simulations confirming the results. Her studies¹⁹ model nutrient effects on organism growth with two competing predators, analyzing stability and bifurcation, supported by simulations. The researchers²⁰ analyzed prey-predator dynamics affected by global warming and wind, focusing on stability and equilibrium points essential for ecosystem persistence. It identifies bifurcation criteria and demonstrates that high warming causes periodic behavior, aiding ecological understanding. The authors²¹ developed a model for prey and generalist predator interactions using a Crowley–Martin response, finding that excessive predator growth and hunting cooperation can lead to prey extinction, while external food and seasonal variations influence stability. The study²² explores inventory control related to carbon emissions and global warming, emphasizing fuzzy logic. It applies Fractional Calculus to model memory effects with triangular fuzzy numbers, finding optimal costs and higher profits with strong memory effects while highlighting the importance of fuzzy logic in reducing carbon emissions. The studies²² offer an omnivore-predator-prey model with II-Holling and nonlinear responses, focusing on stability and bifurcation, validated by simulations.

The full variety of the factors influencing national rice production and pricing in Malaysia was sufficiently covered by prior literature. Factors such as climatic conditions, technological evolution, government intervention, and policies to stimulate domestic productivity for selling goods abroad participate actively in the melting pot along with socio-economic factors and global market dynamics. A policy that ties all these elements together is mandatory to achieve the stability and growth of the rice industry. In the future, studies should explore the development of adaptive strategies that can mitigate current and emerging challenges to enhance the resilience and sustainability rice sector in Malaysia.

Model formulation

Let ${y_i}=i=2,3, \ldots ,n$ be a time series. For chain growth rate in Eq. (6):

$${T_i}=\left( {\frac{{{y_i}}}{{{y_{i - 1}}}} - 1} \right) \times 100\%$$

(6)

Hereafter, Eq. (1) has the following steps:

• Step 1. UoD as the Set, where, and divide into equal “m” intervals through a frequency distribution via class boundaries.

• Step 2. The larger classes are divided into smaller partitions for accuracy.

• Step 3. Fuzzification of variables through fuzzy sets, on each partition interval as the triangular fuzzy numbers (TFN).

• Step 4. Defuzzification of the fuzzy set to crisp values is done by Eq. (7):

  $$t=\left\{ {\begin{array}{*{20}{c}} {\frac{{1+{\gamma _1}}}{{\frac{1}{{{a_1}}}+\frac{{{\gamma _1}}}{{{a_2}}}}}}&{j=1} \\ {\frac{{{\delta _1}+1+{\delta _2}}}{{\frac{{{\delta _1}}}{{{a_{j - 1}}}}+\frac{1}{{{a_j}}}+\frac{{{\delta _2}}}{{{a_{j+1}}}}}}}&{2 \leqslant j \leqslant m - 1} \\ {\frac{{{\gamma _2}+1}}{{\frac{{{\gamma _2}}}{{{a_{j - 1}}}}+\frac{1}{{{a_j}}}}}}&{j=m} \end{array}} \right.$$

  (7)

Where ${a_{j - 1}},{a_j},\;\;{\text{and}}\;{a_{j+1}}$ are the midpoints of the interval of fuzzy sets ${X_{k - 1}},{X_k},\;\;{\text{and}}\;{X_{k+1}}$, carriers respectively, while ${\gamma _1}$, ${\gamma _2}$, ${\delta _1}$ and ${\delta _2}$ are called parameters.

• Step 5. The proposed FVAR model by TFN is as under in Eq. (8):

$${y_F}={c_F}+\sum\limits_{{i=1}}^{{12}} {{A_i}{y_{F - i}}+{u_F}}$$

(8)

Where ${y_F}=\left( \begin{gathered} {y_{Pi}} \hfill \\ {y_{PDi}} \hfill \\ \end{gathered} \right)$, ${c_F}=\left( \begin{gathered} {c_{{y_{Pi}}}} \hfill \\ {c_{{y_{PDi}}}} \hfill \\ \end{gathered} \right)$, ${A_i}=\left( {\begin{array}{*{20}{c}} {{a_{{y_{Pi}}{y_{Pi}},i}}}&{{a_{{y_{Pi}}{y_{PDi}},i}}} \\ {{a_{{y_{PDi}}{y_{Pi}},i}}}&{{a_{{y_{PDi}}{y_{PDi}},i}}} \end{array}} \right)$, ${u_F}=\left( \begin{gathered} {u_{{y_{Pi}}t}} \hfill \\ {u_{{y_{PDi}}t}} \hfill \\ \end{gathered} \right)$ and

${y_{Pi}}={y_{Pi - 1}}\left( {\frac{t}{{100}} - 1} \right)$, $i=2,3,4,\ldots,n$, ${y_{Pi}}$is the new price variable. The same steps 1–4 will be followed for the new production variable ${y_{PDi}}$.

For utilization of trapezoidal fuzzy numbers (TrFN), the defuzzification of fuzzy sets ${X_k},\;\;k=1,2,3, \ldots m$ is done by Eq. (9):

$${t_{\tilde {n}}}\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {0~~~~~~~~~~~~~~~~~~~x~<{n_1}} \\ {\frac{{x - {n_1}}}{{{n_2} - {n_1}}}~~~~~~~{n_1} \leqslant x \leqslant {n_2}} \\ {1~~~~~~~~~~~~{n_2} \leqslant x \leqslant {n_3}} \\ {\frac{{x - {n_4}}}{{{n_3} - {n_4}}}~~~~~~~~~~~~~~~x \geqslant {n_4}} \end{array}} \right.$$

(9)

Where $\left( {{n_1},{n_2},{n_3},{n_4}} \right)$are the parameters while $\:{t}_{\stackrel{\sim}{n}}\left(x\right)$ is the membership function.

The new price and production variable by using TrFN are ${y_{TPi}}={y_{TPi - 1}}+\frac{{{t_n}\left( x \right)}}{{100}}$, $i=2,3,4,\ldots,n$ and ${y_{TPDi}}={y_{TPDi - 1}}+\frac{{{t_n}\left( x \right)}}{{100}}$, Eq. (8) is the proposed FVAR model for these updated variables.

Initially, all parameters are specified to 0.5 value while the optimized parameters will be utilized by Maple in the last.

Data description and results

We have the price and production monthly data from 2018 to 2022 from the Malaysian Agricultural Research and Development Institute (MERDI) for assessment. After diagnostic checks of the data, the results for Eq. (2) as under in Tables 1, 2 and 3 for Vector Autoregressive (VAR) Model:

Table 1 Production modelling by VAR.

By Table 1, R-square value of 0.505956 indicates that approximately 50.6% of the variability in production is explained by the model. The adjusted R-square value of 0.579569, which adjusts for the number of predictors, suggests an improved model fit. However, the overall F-statistic is not significant (p-value = 0.519089), indicating that the model may not be a good fit for predicting production. While none of the F-tests for zero restrictions are significant, indicating that neither the lags of production nor the lags of price significantly contribute to the model.

In Table 2, with an adjusted R-square value of 0.606620, the model appears to have a pretty good fit, as indicated by the R-square value of 0.807495, which shows that the model explains roughly 80.7% of the variability in price. As a result of the fact that the F-statistic is significant (p-value = 0.000691), it may be concluded that the model is an excellent contender for price forecasting. To add insult to injury, the F-tests demonstrate that the lags of production and the lags of pricing both make significant contributions to the model, which further substantiates the validity of the model.

Table 2 Price modelling by VAR.

Table 3; Fig. 3 present the rice price forecasts for 2023-24 indicating an overall upward trend with increasing uncertainty as the year progresses. The predictions and confidence intervals provide valuable insights for stakeholders to understand the potential price movements and associated risks. However, the high level of uncertainty, particularly in certain months like March, suggests that caution should be exercised when using these forecasts for decision-making.

Table 3 95% C.I for price prediction.

Fig. 3

Price forecasting using the VAR model.

Fuzzy vector autoregressive models (FVAR) via fuzzy triangular numbers (FTN)

Based on Table 4, the value of R square, which is 0.471034, can be deduced that the model accounts for about 47.1% of the variability in production. The adjusted R-square value of 0.680930, which considers the total number of predictors, indicates that the relationships are moderately effective. Nevertheless, if the adjusted R-square is greater than the R-square, this suggests that there may be an anomaly, which frequently implies that there is a requirement for a more in-depth examination of the model’s variables or the computing process. Given that the F-statistic is significant (p-value = 0.009183), it may be inferred that the model possesses a level of predictive capacity for production that is statistically significant. The fact that the F-test is significant for all the delays of TPd indicates that the lags of TPd do contribute to the prediction.

Table 4 Production modelling by FVAR-FTN.

By Table 5, with an R-square value of 0.446610, it can be deduced that the model accounts for about 44.7% of the price fluctuation that exists. Because the number of predictors was taken into consideration, the adjusted R-square value of 0.630841 indicates that the fit is moderate. There is a significant disparity between the values of R square and adjusted R square, which is not typical and may suggest that there are problems with the model specification or the data. The fact that the F-test is significant for all of the delays of TPd indicates that the lags of TPd do contribute to the prediction.

Table 5 Price modelling by FVAR-FTN.

According to Table 6; Fig. 4, the 95% confidence intervals provide a range within which the actual TPr values are expected to fall with 95% confidence for 2023-24. The intervals are relatively wide, reflecting the uncertainty inherent in the predictions. For example, the confidence interval for January 2023 is (817.016, 1512.30), indicating substantial uncertainty around the point prediction of 1164.66. The FVAR model using FTNs provides a set of forecasted TPr values for 2023-24 with associated uncertainties. While the point predictions give a specific expected value, the confidence intervals highlight the range of possible outcomes, emphasizing the uncertainty in the forecasts.

Table 6 95% C.I for price prediction.

Fig. 4

Price forecasting by FVAR-FTN.

Fuzzy vector autoregressive models (FVAR) via fuzzy triangular numbers estimated (FTNest)

In Table 7, there is roughly 68.84% of the variability in production that can be attributed to the model, as indicated by the R-square value of 0.688360. Taking into consideration the total number of predictors in the model, the adjusted R-square value of 0.69872 indicates that the model is a good match for the data. That all of the delays are statistically significant is what the F-statistic is suggesting.

Table 7 Production modelling by FVAR-FTNest.

By observing Table 8, approximately 70.34% of the price variance can be attributed to the model, according to the R square value of 0.703396, which indicates that the model. With an adjusted R-square value of 0.719148, which considers the total of predictors, it can be concluded that the model provides a satisfactory match. The F-statistic for the model is 0.047979, and the p-value that corresponds to it is 0.001029. The fact that this is the case suggests that the model is statistically significant, which in turn suggests that at least one of the predictors is strongly connected to the variance in price.

Table 8 Price modelling by FVAR-FTNest.

By Table 9, Fig. 5, the forecasted values and their confidence intervals provide valuable insights into the expected price dynamics for the upcoming year. These predictions by the FVAR model through FTNs could provide a means for stakeholders to arrive at informed decisions pegged on expected price levels and uncertainty ranges.

Table 9 95% C.I for price prediction.

Fig. 5

Price forecasting by FVAR-FTNest.

Fuzzy vector autoregressive models (FVAR) via fuzzy trapezoidal numbers (FTrN)

In Table 10, the FVAR model using Fuzzy Trapezoidal Numbers provides a reasonable fit for the production variable (TrPd) with an adjusted R-square of 0.712136. The model is statistically significant overall, as indicated by the low p-value of the F-statistic. However, the individual lagged variables of TrPd and TrPr are not significant, except for the combined effect at lag 12. Despite the non-significance of individual lags, the overall model demonstrates that Fuzzy Trapezoidal Numbers can effectively capture the underlying dynamics of the production process. The significant combined effect at lag 12 suggests that longer-term dependencies and interactions among variables may play a crucial role in production behavior.

Table 10 Production by FVAR-FTrN.

By observing Table 11, the Fuzzy Vector Autoregressive (FVAR) model using Fuzzy Trapezoidal Numbers (FTrN) fits high price variable well with an R-square of 0.724070 which means 72.41% from a variation of rice prices depending on the mode it so, The adjusted R-square is 0.77447 which indicates our model in robust and dependable. The low p-value of the F-statistic in general signifies that it is significant, indicating overall significance for whether or not the predictors included collectively affect explaining rice prices. The significant p-values for the F-tests of zero restrictions for both TrPd and TrPr lagged variables determine the importance of considering past values in predicting future prices.

Table 11 Price by FVAR-FTrN.

By Table 12; Fig. 6, the FVAR model with Fuzzy Trapezoidal Numbers provides a detailed yet nuanced forecast for rice prices in the coming years of 2023 through 2024, explicitly representing our best assessments of both the anticipated market values as well as the permeating unpredictability. According to the findings of this model, the cost of rice is expected to fluctuate meaningfully across the yearly timeline, with conspicuous highs and lows that likely reflect ever-changing influences such as seasonal supply variations or the complex interplay of industry forces.

Table 12 95% C.I for price prediction.

Fig. 6

Price forecasting By FVAR-FTrN.

Fuzzy vector autoregressive models (FVAR) via fuzzy trapezoidal numbers estimated (FTrNest)

In Table 13, the Fuzzy Vector Autoregressive Model with Fuzzy Trapezoidal Numbers Estimation for predicted total production demonstrated a robust fit with meaningful factors. The exceptionally high R-square and adjusted R-square values imply that the model describes a considerable amount of unpredictability in amounts created. The F-tests validate the importance of the delayed factors, strengthening the reliability of the model. The investigation spotlights the effectiveness of applying Fuzzy Trapezoidal Numbers Estimation in anticipating production, giving valuable insights for stakeholders in making educated choices about future output levels. The lack of significant autocorrelation adds to the credibility of the model by ensuring forecasts are not impacted by serial connections among the residuals.

Table 13 Production by FVAR- FTrNest.

By Table 14, the Fuzzy Vector Autoregressive Model leveraging Fuzzy Trapezoidal Numbers (Estimated) for pricing forecasts demonstrates a strong fit with multiple meaningful factors. The exceedingly high R-square and adjusted R-square values unambiguously indicate that the model elucidates a considerable amount of the variability in prices. The F-tests unreservedly validate the importance of prior fluctuations, solidifying the model’s integrity. However, the Durbin-Watson statistic implies some positive autocorrelation amidst the residuals which may necessitate supplementary examination. Despite this minor anomaly, the model provides precious insights into the dynamics of pricing, helping stakeholders in forming sagacious choices moving forward. The statistically significant predictors coupled with the sterling goodness-of-fit measures accentuate the potency of applying Fuzzy Trapezoidal Numbers (Estimated) in modeling pricing patterns into the future.

Table 14 Price by FVAR- FTrNest.

By analyzing Table 15; Fig. 7, describing FTrNest in the FVAR model, the predicted unit price in 2023-24 shows that there is substantial variation and uncertainty. Wider confidence intervals and varying standard errors underscore just how unpredictable price forecasts are at present. Nevertheless, the model does provide useful clues as to likely price trends and periods when it is either worse or better than usual. Careful thought and constant observation will be needed if the prices turn out in such a way as this analysis indicates. The stakeholders can make informed judgments regarding market interventions, pricing strategies, and resource allocation.

Table 15 95% C.I for price prediction.

Fig. 7

Price forecasting by FVAR-FTrNest.

Comparative analysis in Table 16, shows that the proposed fuzzy models have an extraordinary advantage over traditional Vector Autoregressive (VAR) models in both accuracy and efficiency. Using performance measures from key performance metrics such as Mean Absolute Percentage Error (MAPE), forecast for December 2023, 95% confidence interval, and Relative Efficiency (RE), the study gives convincing proof of the superiority in terms of accuracy and efficiency of the fuzzy logic based forecasting methods. The addition of fuzzy logic into forecasting models brings about sizable gains in both accuracy and efficiency. The TrFN with optimized parameters, the fuzzy model that stood head and shoulders above all the others was the most efficient, having an error rate lower than any other and relatively precise forecasts. These results indicate that fuzzy logic-based prediction systems are very suitable for applications where the utmost accuracy and reliability are needed. Fusing the strengths of fuzzy logic, these models provide a robust framework capable of dealing with the complexities and uncertainties intrinsic to time-series data and thereby yield substantial advances in traditional predictive techniques.

Table 16 Overall performance of the models.

Conclusion

The analysis of the data from 2018 to 2012 on rice prices and production as given by the Malaysian Agricultural Research & Development Institute (MERDI) leverages anyone of numerous models employed other than this one to give critical insights into market dynamics for rice.

Production by VAR model

Employing a conventional production model, the R-square reached 0.505956, indicating that the model can explain about 50.6% of production variability. This outcome resulted from analyzing data. The total F-statistic was not significant (P = 0.519089). However, we see that the adjusted R-square is 0.579569, which shows some improvement when the number of predictors included in the model is taken into consideration. This lack of significance implies that the model might not be reliable for predicting production, as neither the lags of production nor the lags of price were found to significantly contribute to the model.

Price by VAR model

The price model, given an R-square at 0.807495. This outcome means that the model can explain about 80.7% of price variation. Adjusted R-square for 0.606620, provides additional evidence that we have a robust model on our hands. The significant overall F-statistic (p-value = 0.000691) is further proof of the model’s reliability in predicting prices. Furthermore, F-tests show that both the lags of production and price were significant contributors to the model. While the price forecasts for 2023-24 indicate a general upward trend, as the years’ progress forecasts are marked by increasing uncertainty. This implies that while we should take guidance from the predictions, the stakeholders must be careful due to high levels of uncertainty in particular months, notably March.

Fuzzy Vector Autoregressive models (FVAR) via fuzzy triangular numbers (FTNs)

The FVAR models with triangular fuzzy numbers for production and price represent novel methods of modeling the dynamics in these variables. The adjusted R-square for the production model was 0.680930, which indicated that 68.09% of the variation is explained by this decreased model This is a good indication that the model performs well. The price model had an adjusted R-square of 0.630841, explaining the variability through a total of 63.08%. The lagged variables had statistically significant F-tests, suggesting that both models showed predictive power.

Fuzzy Vector Autoregressive models (FVAR) via fuzzy trapezoidal numbers (FTrNs)

The Fuzzy Trapezoidal Numbers-based FVAR models gave fairly good results with these studies. An R-square of 0.712136 (71.21% explained variability) indicated a good fit of the production model to the data. For having an adjusted R-square value of 0.77447, the pricing model revealed high predictive capabilities for explaining 77.44% of the variance present in the data under consideration. Our models reinforce that past values are necessary when forecasting future production and price because high F-tests in the lags indicate resources from previous periods.

Fuzzy Vector Autoregressive models (FVAR) via fuzzy trapezoidal numbers estimated (FTrNest)

Fuzzy Trapezoidal Numbers Estimated for FVAR models having fitted the best among the fuzzy models. The production model had an adjusted R-square of 0.844022 and the price model with adjusted R-square of 70.34%, again almost perfect models with very high prediction strength. They have in potential to be used for predicting rice production and prices as shown by significant predictors, and high goodness-of-fit measures of these models.

Final remarks

In general, the analysis shows that it is very difficult to model rice production and prices. Ancient models had trouble predicting production while also producing good results for prices. The integration of Fuzzy models especially established based on fuzzy trapezoidal numbers (estimated) resulted in improved predictive performance and better insight. These findings are useful for stakeholders to make well-informed decisions by considering the scenarios with uncertainties and as a demonstration that various modeling approaches should be integrated because of their characteristics in representing different aspects interacting within the rice market.

Recommendations for future perspectives

Because the agricultural sector overall, market valuation and fulfilling human needs is highly critical - hence a wide variety of insights for future investigation are rendered in this domain. Nonetheless, we have compiled an array of core features to inform and stimulate future research toward valuable progress.

Including additional features

Future studies could include more factors that may affect rice production and price like weather conditions, soil quality, pest attacks & new agricultural technological innovations. Including these variables can thus allow for a richer understanding of the factors driving the rice market.

Machine learning approaches

Use machine learning models including Random Forests, Gradient Boosting Machines (GBM), and Neural Networks to predict region-wise rice production and prices. These models can deal with big data and model intricate nonlinear relationships that the traditional economy-centered input-output relationship type 1 might overlook.

Spatial analysis

Perform spatial analysis to see the geographic distribution of rice growing and its correlation with prices. One should certainly consider consulting Geographic Information Systems (GIS) to map spatial trends, possibly areas of production efficiency, and those that may be done much better.

Longitudinal studies

Carry out long-term studies to see what trends develop over time. This can help to detect long-run patterns and cycles in rice production, and sales, which is important when making future estimates.

Policy impact analysis

Examine the effects of government policies, subsidies, and international trade agreements on rice production as well as prices. An understanding of the policy implications would help policymakers design interventions to stabilize the rice market.

Integration of economic indicators

Use macroeconomic indicators i.e. inflation, exchange rate, and GDP growth in the models They serve as barometers, which are good indicators of what goes on in the market and how wider global economic conditions impact rice markets.

Climate change adaptation

Explore the impact of climate on rice and its adaptive measures, such modeling can be used to understand how climate change might threaten rice yields so support strategies are in place that allow food security.

Data quality and granularity

Guarantee greater granularity; and high data quality better understanding of detailed data, with improved temporal and spatial resolution can help to improve the precision of model outcomes leading to more reliable models.

Collaboration with agricultural experts

Further model-building to embed domain-specific understanding from agronomists, agricultural economists, and others. This interdisciplinary approach can be a valuable addition to our ability to understand and model rice production as well as its prices.

Scenario analysis

Perform scenario analysis to assess how different future realities would affect rice production and prices. These could range from extreme weather events and shifts to the ways that food is grown, harvested, or transported right through to changes in consumer demand.

Continuous model validation

Always ensure that the data is fed back into the model so your Predictive Engagement Model constantly validates and updates to be as relevant yesterday as today. This is key to ensuring the predictive power of your models and that they are up to date with market changes.

Implementation of these recommendations would allow for richer empirical findings, more robust economic forecasts, and support stakeholder decisions to promote a rice industry that can withstand shocks.