A Multi-Objective Rice Supply Chain Model under Uncertainty: Minimizing Cost and Environmental Impact of Soil Erosion

Agriculture is a critical sector for a nation’s economic, political, and social stability. However, the lack of integrated information systems and functional supply chain models has limited the effectiveness of this sector despite its potential [1], [2]. Agriculture contributes approximately 13% of food demand, 11% of raw materials for the food industry, 11% of gross national product, 22% of national employment, and 23% of non-oil exports in many developing economies [3]. Sustainable agriculture improves environmental quality and resource use while meeting human food needs [4].

Among agricultural products, rice holds a strategic position due to its high consumption and the environmental intensity of its cultivation. Proper design of the rice supply chain is essential for producing countries. This study proposed a multi-objective mathematical model for the rice supply chain that addressed both economic and environmental goals. The model sought to answer the following questions:

1. What is the optimal amount of unhusked rice production per region considering costs, demand, and environmental factors?

2. How much rice should be produced domestically versus imported?

3. What share of rice should go directly to markets or through distribution centres?

4. What is the amount of water consumption to reduce environmental impacts?

5. What is the optimal cultivated area for rice?

The proposed model was linear and multi-objective, thus minimizing total costs and environmental damage from water use. An extended goal programming approach was applied, while uncertainty in water availability was handled via stochastic programming. A case study from Iran, featuring four major rice-producing provinces and import options, validated the model.

Supply chain management integrates activities from raw material procurement to final product distribution [2], [5]. The agricultural supply chain has received growing attention due to its complexity, which includes cultivation, harvesting, processing, and distribution under uncertain conditions [6], [7].

Agriculture faces two key challenges: Meeting growing demand and balancing production with environmental sustainability [8]. The main components of an agricultural or rice supply chain include: (1) production and storage, (2) processing in mills, (3) distribution to wholesalers or centres, and (4) delivery to customers [6].

Uncertainty is inherent in agricultural supply chains due to weather, soil conditions, water availability, and market fluctuations [9]. Operations research has addressed agricultural problems since the 1940s, but uncertainty modeling gained prominence only in the last two decades [10], [11]. Common uncertainty-handling techniques include stochastic programming, robust optimization, and simulation [12].

Early work by Sorensen and Gilheany [13] simulated sugarcane harvest under stochastic weather conditions. Arnout and Maatouk [14] minimized harvest and transportation costs for processing facility location. Gorton et al. [15] examined environmental factors in fresh produce supply chains, including government policies and resource impacts. Masud et al. [16] modeled water consumption and soil damage for barley cultivation in Canada under uncertain rainfall.

Sustainability in agricultural supply chains nowadays integrates economic, environmental, and social objectives. Reidsma et al. [17] reviewed development and use of farm models for policy assessment in European Union. Stochastic programming is the most frequently used uncertainty method. Sazvar et al. [18] optimized a three-objective sustainable supply chain for organic products, to minimize costs and environmental damage while maximizing consumers’ health.

Recent studies (2019–2025) have extended these themes. For example, Achmad et al. [19] developed an agent-based modelling and robust optimization for food supply chains during COVID-19 pandemic. McClements [20] developed a roadmap for future agriculture research designing a sustainable and environment friendly food supply chain. Mortazavizadeh et al. [21] applied machine learning to predict water demand in agriculture water management. Gonçalves et al. [22] proposed a blockchain-enabled traceability system for sustainable rice supply chains. Douaioui et al. [23] developed an innovative goal programming and advanced metaheuristic techniques in optimizing supply chain efficiency.

Several recent studies have explicitly incorporated water consumption as a key performance metric in agricultural supply chain design namely, Hajimirzajan et al. [24] fuzzy framework for water-land allocation, Accorsi et al. [25] ecosystem carbon and water balance, Siyal et al. [26] water footprint for supply chain irrigation, and Baghizadeh et al. [27] water-energy-food nexus.

To handle uncertainty in agricultural supply chains, Hosseini and Goli [28] proposed a robust optimization model, while Alinezhad et al. [29] developed a fuzzy multi-objective optimization. Hosseini-Motlagh et al. [30] presented a hybrid sustainability and resiliency in the wheat network.

Seyedzadeh et al. [31] designed a multi-objective sustainable network, while Daneshvar et al. [32] proposed a multi-objective integrated with meta-heuristic algorithms for agricultural sustainable supply chain. Reidsma et al. [17] reviewed farm models for policy impact assessment.

Research Gap and Innovations

According to the reviewed agricultural supply chains, most research focused on economic objectives (cost minimization or profit maximization). Environmental objectives appear to be less frequently and typically address pollution, resource use, or waste. Few studies specifically examined water consumption and soil erosion in rice supply chains under uncertainty.

Innovations of This Study

• Dual objective: minimize total costs + minimize water-related soil erosion.

• Explicit modeling of surface water (rainfall) and irrigation water as separate sources.

• Rice-specific case study with four Iranian provinces.

• Extended goal programming combined with stochastic programming.

• Three water availability scenarios (bad, medium, and good).

The proposed rice supply chain in Figure 1 consists of six levels: farms, rice mills, distribution centres, import centres, by-product markets, and final consumer markets. Unhusked rice is harvested at farms and transported to rice mills, where it is processed into white rice and by-products (rice bran and broken rice). White rice may be sent directly to markets or routed through distribution centres for packaging and redistribution. Imported rice enters through import centres and moves to distribution centres, then to markets. Market demand is proportional to regional population. Figure 1 illustrates the six-level supply chain structure: (1) farms (production sites); (2) rice mills (processing facilities); (3) distribution centres (storage and repackaging); (4) import centres (foreign supply sources); (5) by-product markets (rice bran and broken rice); and (6) final consumer markets. Arrows indicate product flows including direct farm-to-market shipments, farm-to-distribution centre flows, and import-to-market flows. The multi-objective model proposed by Kazemi et al. [33] serves as the foundation for this study. We extend their work by incorporating water consumption as a third objective, transforming the problem into a tri-objective formulation. The inclusion of water consumption as a separate objective is motivated by recent studies highlighting the critical role of water scarcity in agricultural supply chains [24], [26], [27]. As noted by Hajimirzajan et al. [24], integrated water-land allocation is essential for sustainable crop planning, and Baghizadeh et al. [27] emphasized the water-energy-food nexus in supply chain design.

The model is linear with three objective functions: (1) minimize total costs, and (2) minimize environmental damage from water use in cultivation, and (3) minimize water consumption. The indices are given in Table 1, parameters are defined in Table 2, and decision variable are shown in Table 3.

Objective Functions

Minimize Total Costs:

Minimize Environmental Impact (Soil Erosion):

$ \operatorname{Min} \text { Env}=\sum_i\left(c f e_i \times C w r_i\right) $

Under uncertainty, the environmental objective becomes

where, $p_c$ is the probability of scenario $c$.

Minimize Total Water Consumption:

$ \operatorname{Min} Z_3=\sum_i C w_i=\sum_i\left(W g_i+W b_i\right) $

where, water consumption is already calculated in the model via $C w_i=W g_i+W b_i$.

Under uncertainty, the water consumption objective becomes

Constraints:

Equations (4)–(9) define the rice mill balance (no waste), distribution centre balance, farm production limit, rice mill capacity, distribution centre capacity and cultivated area limit constraints, respectively. All decision variables are non-negative.

The proposed model involved two complexities: (1) uncertainty in water availability (rainfall and irrigation), and (2) three conflicting objectives (cost vs. environmental impact and water consumption). We addressed uncertainty with stochastic programming and the multi-objective nature with extended goal programming. Multi-objective optimization under uncertainty has been extensively studied in recent agricultural supply chain literature. Hosseini and Goli [28] employed a robust optimization approach, while Alinezhad et al. [29] used fuzzy multi-objective programming. The extended goal programming method adopted in this study is consistent with these recent advances and has been successfully applied in similar contexts [17].

In real-world agricultural problems, parameters such as rainfall and groundwater availability are uncertain. Stochastic programming provides a framework for optimizing under uncertainty by defining a set of scenarios, each with a probability of occurrence.

Let $\Omega=\left\{\omega_1, \omega_2, \ldots, \omega_s\right\}$ represent the set of scenarios. Each scenario $\omega_k$ occurs with probability $p\left(\omega_k\right)$, where:

In this study, three scenarios were defined for water availability:

Good scenario: high rainfall and low irrigation need.

Medium scenario: moderate rainfall and irrigation.

Bad scenario: low rainfall and high irrigation need.

The deterministic equivalent of the stochastic model replaced the original environmental objective with the expected value across scenarios:

Water-related constraints become scenario-dependent:

All other constraints remain deterministic.

To solve the multi-objective model, we used the extended goal programming method proposed by Romero [34]. This method balances the achievement of multiple goals and allows the decision-maker to control the trade-off between optimization and goal satisfaction through a parameter $\alpha$.

The general formulation is:

where,

$q$ = number of objectives (here $q$ = 3)

$b_i$ = target value for objective $i$ (obtained by solving each objective separately)

$n_i$, $p_i$ = negative and positive deviations from the target

$u_{j}$, $ v_i$ = weights for deviations

$\lambda$ = maximum weighted deviation

$\alpha \in[ 0,1]$ = trade-off parameter ($\alpha$ = 1 emphasizes goal achievement; $\alpha$ = 0 emphasizes optimization)

For this study, the extended goal programming model became

subject to the model constraints.

The weights $u_1$, $v_1$ for the economic objective, $u_2$, $v_2$ for the environmental objective and $u_3$, $v_3$ for the water consumption objective reflect policy priorities. In the base case, we set ($u_1$, $v_1$) = (0.6, 0.2), ($u_2$, $v_2$) = (0.2, 0.4) and ($u_3$, $v_3$) = (0.2, 0.4), but sensitivity analyses varied these weights.

The model was validated using data from Iran, where 17 out of 31 provinces produce rice, with annual white rice production of approximately 2.5 million tons. Four major producing regions were considered: Mazandaran, Gilan, Khuzestan, and Golestan. The supply chain included 4 farms (one per region), 4 rice mills (one per farm location), 6 distribution centres, 4 import centres, 32 market regions (provinces), and 2 by-products (rice bran and broken rice). Water scenarios are given in Table 4.

Key Assumptions

• One rice mill per farm; inter-region transfers allowed.

• Transport vehicles are identical; costs are known and fixed.

• Processing coefficients for white rice and by-products are constant across regions.

• Import volume is limited to a percentage of total demand.

• Water consumption is measured in m$^3$ per hectare.

• Environmental damage is calculated based on surface water (rainfall + irrigation).

Probabilities: equal (1/3 each) unless stated otherwise.

Further, key parameter of the model are extracted from the data as shown in Table 5.

Rice mill processing cost: 300–315 (1,000 Tomans/ton)

Import cost: 1,080–2,280 (1,000 Tomans/ton)

Distribution centre capacities: 190,000–350,000 tons

Annual rice demand: 36 kg per capita × provincial population

The multi-objective model was coded in Linear Interactive and General Optimizer (LINGO) optimization software and solved using extended goal programming with base weights: Economic objective weight = 0.7, environmental weight = 0.3. Table 6 compares the results of solving the multi-objective model using three different approaches: (1) minimizing only total costs (economic objective); (2) minimizing only environmental impact; and (3) extended goal programming balancing both objectives. The table reports cost objective (Z1), environmental impact objective (Z2), and water objective (Z3), production percentage, import percentage, and deviations from individual optima.

The goal programming solution balances both objectives, with each deviating 15% from its individual optimum. This represents a practical trade-off for policymakers.

Figure 2 shows the optimal production quantities for the four major rice-producing regions in Iran: Mazandaran (1,400,000 tons), Gilan (1,170,000 tons), Khuzestan (0 ton), and Golestan (approximately 689,000 tons). The figure demonstrates that Khuzestan produces no rice due to its high environmental damage coefficient (0.05) and water scarcity, to be consistent with actual government policy.

Khuzestan produces no rice due to high environmental damage coefficient (0.05 vs. 0.006 for northern regions) and water scarcity, to be consistent with actual government policy banning rice cultivation there.

Figure 3 illustrates the distribution of rice shipments from each rice mill to the six distribution centres (DC). The figure shows that distribution centres 4, 5 and 6 receive 0 shipment from domestic mills, indicating that these centres are served primarily by import channels.

Table 7 shows the distribution of imported rice from four import centres to six distribution centres. The data indicate the linkage between import centres and their respective serving distribution centres as well as the corresponding shipment volumes. For example, Import Centre 1 supplies DC3 (350,000 tons), DC4 (60,000 tons), and DC5 (190,000 tons). DCs stand for distribution centers.

Markets receive rice via two channels: Directly from rice mills or through distribution centres. Major producing provinces such as Mazandaran and Gilan had higher direct purchase coefficients (0.6–0.9), reflecting local supply chains. Table 8 presents sample direct supply quantities from rice mills to specific markets. The two major producing provinces had higher direct purchase coefficients (0.6–0.9), reflecting shorter supply chains and local distribution patterns. For example, Mazandaran mill supplied 118,080 tons to Mazandaran market, and 336,095 tons to Tehran market.

Table 9 quantifies water use and environmental impact for the four rice-producing regions. It shows rainfall water (m$^3$), irrigation water (m$^3$), total water consumption (m$^3$), and soil erosion (m$^2$ soil loss) calculated as total water multiplied by the environmental damage coefficient ($cfe_i$). Khuzestan showed zero production under the base solution due to its high damage coefficient (0.05 vs. 0.006 for northern regions).

Sensitivity Analyses:

Analysis 1 (Water Scenarios)

Figure 4 shows the quantities of rice bran and broken rice produced at each rice mill. These by-products were generated during the milling process and were sold to separate markets, thus contributing to overall supply chain revenue (or cost offset).

Table 10 compares three water scenarios (bad, medium, and good) across multiple performance metrics. The results showed that Khuzestan produced rice only under the good water scenario. As water conditions improved, domestic production increased from 61% to 71%, imports decreased from 39% to 29%, and environmental impact (Z2) increased (worsened) due to higher water use.

Only under the good water scenario did Khuzestan produce rice. Production increased and import decreased as water conditions improved.

Analysis 2: Importance of Objective Functions (weights $\boldsymbol{u}$, $\boldsymbol{v}$)

Figure 5 compares production and import percentages across three water scenarios. The figure demonstrated that as water conditions improved, domestic production increased and import dependence decreased.

Table 11 shows how production percentage, import percentage, and deviations from individual optima changed as the relative importance of the economic and environmental objectives varied.

As environmental and water importance increased, production decreased (import increased) to reduce water-related soil erosion.

Analysis 3: Import Limit

Figure 6 shows how production and import percentages changed as the environmental objective weight increased from 0.1 to 0.8. When environmental importance was low (weight 0.1), production was high (61%). As environmental weight increased, production decreased and imports increased, reaching a stable equilibrium at weight 0.5.

Table 12 demonstrates the effect of reducing the maximum allowable import limit on production ratio, import ratio, Khuzestan production status, and deviations from individual optima.

Reducing the import limit forced domestic production to increase, including that from Khuzestan. However, this increased environmental damage below the 27% import limit, the model became infeasible due to insufficient domestic production capacity.

Rice is a strategic crop providing food for over half of the world’s population. For producing countries, an efficient rice supply chain supports both food security and economic growth. This study presented a multi-objective mathematical model for a rice supply chain that minimized total costs (economic) and soil erosion caused by water used in cultivation (environmental) and total water consumption — making it one of the first rice supply chain models to explicitly include water minimization as a separate objective. Uncertainty in rainfall and irrigation water was handled through stochastic programming with three scenarios (bad, medium, and good); the multi-objective model was solved using extended goal programming.

A case study from Iran including 4 major rice-producing provinces validated the model. Under base conditions (equal scenario probabilities, economic weight 0.5, environmental weight 0.3, and water weight 0.2), the goal programming solution achieved a balanced trade-off with 15% deviation from the cost optimum, 15% from the environmental optimum, and 18% from the water optimum. Khuzestan province did not produce rice in the base solution due to its high environmental damage coefficient (0.05) and high water consumption, consistent with actual government policy. Under the “good” water scenario, production in Khuzestan becomes viable (215,000 tons) but at the cost of increased water consumption (138.7M m$^3$). The water minimization objective (Z3) significantly influences production decisions: when water weight increases to 0.5, production decreases to 60% and imports increase to 40%. Reducing the maximum import limit below 24% renders the model infeasible, revealing a practical constraint on import substitution. The model achieves significant water savings: compared to the cost-only solution, the balanced solution reduces water consumption by 17.9% (from 185.6M m$^3$ to 152.4M m$^3$) with only a 13% increase in cost.

Future Research Directions

1. Incorporate additional objectives such as profit maximization, resource utilization, or social welfare;

2. Add an export sector if domestic production exceeds demand;

3. Model demand uncertainty (e.g., population growth and consumption shifts) alongside supply uncertainty; and

4. Apply robust optimization or fuzzy programming to further address parameter uncertainty.