Modelling and Simulation of the Single-Period Vehicle Routing Problem in the Agriculture Industry

The agricultural industry, a key driver of global food supply, faces increasing pressure to enhance efficiency while reducing environmental impacts. The growing global population and the need for an adequate supply of safe, high-quality agri-food products are putting pressure on the agriculture sector today. In addition, there has been a global increase in the reliance on imported food of almost 50% between 2006 and 2020 [1], and the existing food supply chain has become more globalized [2].

The flow and diversity of goods have expanded in the last few decades. Goods transportation and trade have expanded more quickly than countries' GDPs [3]. Due to the growth in international trade in food from far-off sources, food shipments have travelled longer and carried more weight globally in recent decades [4]. For instance, food travelled 50% further in the UK and 25% further in the USA at the start of the twenty-first century than it did in the 1980s [5].

One of the most critical challenges lies in optimizing logistics, particularly in the routing of vehicles for transporting goods. This issue, commonly known as the Vehicle Routing Problem (VRP), is especially significant in agriculture, where the perishability of goods, dispersed production areas, and variable road conditions in rural settings pose unique constraints. Transportation costs represent a substantial portion of total expenses in agricultural supply chains. According to a report by the Food and Agriculture Organization (FAO), transportation costs in agricultural supply chains can account for as much as 40-60% of total distribution expenses. This is particularly true in regions where agricultural production occurs in remote areas, far from consumer markets or processing centers. For instance, in Latin America, rural transport costs can increase significantly due to poor infrastructure, exacerbating supply chain inefficiencies and raising the cost of food for consumers [6]. Optimizing vehicle routing could therefore lead to substantial cost savings for producers and logistics companies, while making agricultural products more affordable for consumers.

The environment is impacted by the rise in the amount and distance that food and food animals’ travel. About 25% of greenhouse gas (GHG) emissions associated with global energy consumption are produced by the transportation sector, and 75% of these emissions are related to road transportation [7]. According to projections made by the study [8], the CO₂ emissions from all vehicle categories in Sweden are expected to rise from 18.5 million tons in 1998 to 25 million tons in 2020, suggesting that the freight transport industry may play a larger role in global warming. Reducing emissions from the transportation sector, especially transportation within the agriculture sector, should receive more emphasis in the endeavours to lessen the danger of climate change [9].

In addition to cost considerations, the environmental impact of transportation in agriculture is an increasingly critical concern. According to the International Energy Agency (IEA), the transport sector accounts for approximately 24% of global CO₂ emissions, and agricultural logistics are a significant contributor within this sector due to the high fuel consumption of heavy vehicles transporting goods over long distances [10]. Research has demonstrated that optimizing vehicle routes can reduce fuel consumption by up to 10-20%, depending on the efficiency of the routing algorithms and the specific logistics network being optimized [11]. In the European agricultural sector, for example, simulation models have shown that a reduction in fuel consumption from optimized routing can lead to a 15% decrease in greenhouse gas emissions without compromising delivery times [12].

In general, there has been an increasing amount of focus on the food supply chains. The primary causes of this are the following: consumers' decreased interest in chemical-based food production; growing societal awareness of sustainable food production, processing, and transportation; and the growing impact of food transportation on the environment, logistics costs, and animal welfare. In an endeavour to tackle these problems, the local food supply chain is making a comeback as a substitute food supply network. Despite the growing demand for locally grown food, the main logistical obstacles facing local food systems necessitate a thorough analysis to pinpoint the main bottlenecks and devise solutions. For the local food supply chain, this calls for the creation of effective and efficient logistical systems. Research projects aimed at enhancing the local food systems' logistical performance are still uncommon. Nonetheless, the idea of local food logistics may gain traction, and this paper can make a significant contribution to it by enhancing the logistical capabilities of regional food distributors and producers.

Furthermore, route optimization is one of the most extensively studied areas within transportation research, focusing on creating the most cost-effective and efficient distribution patterns to serve dispersed clients [13]. This approach has been applied in various fields, including forest harvesting, solid waste collection, and agricultural goods transport, to reduce operational costs and emissions. The primary goals of route optimization are to minimize travel distances and reduce the fleet size of vehicles used [14]. Therefore, route optimization analysis is crucial for enhancing goods distribution systems in the agriculture industry. An integrated strategy for supply chain management and logistics should be created to lessen the limitations and enhance the commodities distribution system [15, 16].

Figure 1 illustrates the basic concept of the goods distribution system in Malaysia's agriculture industry. It connects farmers, collection centers (CC), distribution centers (DC), and retailers into a cohesive supply chain system. This system requires efficient coordination of logistics activities through an integrated approach, where the distribution of agricultural products is managed from the farmers, through the CCs and DCs, and finally reaches the retailers as the final stage.

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Figure 1. Goods distribution system in the agriculture sector

Therefore, the focus of this paper is to improve the goods distribution system which is the collection and transportation of agricultural products. This comprises product assessment from each CC to the DC and determining the best routes with regard to travel distance and time as well as trips. We developed a linear programming model and applied appropriate constraint optimization techniques. To solve the mathematical problems and analyze the results, we used A Mathematical Programming Language (AMPL). Figure 2 shows the simulation of the location for each DC and CC scattered around Kedah Darul Aman, Malaysia. The triangle symbol on the Map of Kedah stands as DC and the circle symbol stands as DC.  

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Figure 2. Map of Kedah, locations of the DC and CC

Given the economic and environmental stakes, the importance of addressing the VRP in the agriculture industry cannot be overstated. By implementing advanced modeling and simulation techniques to optimize vehicle routes, agricultural supply chains can not only reduce costs but also contribute to global efforts to mitigate climate change. The need for such optimizations is especially urgent in the context of growing global demand for sustainable agricultural practices and climate-resilient supply chains.

In agricultural supply chains, many products, particularly perishable goods such as fruits, vegetables, and dairy, have short shelf lives. This requires deliveries to be made within narrow time windows to prevent spoilage. As a result, the logistics involved often focus on short-term, day-to-day operational decisions, where each delivery period is relatively independent of the next. By modeling the problem as a single-period vehicle routing problem (SP-VRP), we can simulate these daily logistics more realistically, allowing for fine-tuned, efficient routing solutions that address the immediate needs of the supply chain.

A single-period horizon also simplifies the complexity of the problem, enabling a more detailed exploration of routing strategies within one time period. This allows us to focus on specific factors such as vehicle capacity, route optimization, and fuel consumption without the added complexity of considering multiple periods. Given that each day’s routing problem in agriculture can vary due to changes in demand, weather, or vehicle availability, a single-period approach is well-suited to capture these variations effectively.

The SP-VRP as described in this paper, involves a single distribution center (DC) utilizing a fleet of uniform vehicles to gather agricultural products from several geographically spread collection centers (CC) within a specified planning horizon. It is assumed that the demand rates at the collection centers are known and constant, and travel times remain unchanged over time. The assumptions of known and constant demand rates and unchanging travel times are made primarily for the sake of model simplification and to focus on the key challenges of vehicle routing in agriculture. These assumptions allow for a more tractable model and are reasonable for a SP-VRP, particularly in rural or developing regions where traffic patterns are more predictable and demand at collection centers is relatively stable. While these assumptions may limit the model's applicability to highly dynamic or unpredictable environments, they provide a solid foundation for optimizing agricultural logistics.

The goal of this SP-VRP is to optimize the quantities collected from each CC, determine the delivery schedules, and plan the vehicle routes in order to minimize the overall distribution costs. To formulate our model for the SP-VRP, the following assumptions are made:

• The model excludes the amount of time needed to load and unload the vehicle.
• It is considered that the cost of transportation is related to the distance travelled.
• Split deliveries are prohibited; only one vehicle is permitted to fully replace each CC.

The following subsections detail the pertinent variables, parameters, and a linear mixed-integer formulation of the SP-VRP:

Let H = {1, 2, … T} represent the set of consecutive periods in the planning horizon, indexed by t, and H⁺ = H∪{0}. Let τ_t denote the duration of period t in time units, such as 8 working hours. Define S as the set of CC, indexed by i and j, with S⁺ = S∪{r}, which represents the DC. A fleet of vehicles V is employed to service these CCs. The remaining relevant parameters of the model are provided below:

• φ_jt: the fixed handling cost (in $) per delivery at location $j \in S^{+}$(CC and DC) in period $t \in H$;
• η_jt: the per unit per period holding cost of the product at location $j \in S^{+}$(in $ per tons per period);
• ψ ^v: the fixed operating cost of vehicle $v \in V$ (in $ per vehicle);
• δ_v: travel cost of vehicle $v \in V$ (in $ per km);
• κ^v: the capacity of vehicle $v \in V$ (in tons);
• ν_v: average speed of vehicle $v \in V$ (in km per hour);
• θ_ij: duration of a trip from $\mathrm{CC} i \in S^{+}$to $\mathrm{CC} j \in S^{+}$(in hour);
• d_jt: the stationary demand rate at CC j (in tons per hour) in period $t \in H$.

The model variables are defined as follows:

• $Q_{i j t}^v$: The quantity (in tons) of product remaining in vehicle $v \in V$ when it travels directly to location $j \in S^{+}$in $i \in S^{+}$ from location $i \in S^{+}$ during period $t \in H$. This quantity is zero if the trip (i,j)  is not part of any route taken by vehicle $v \in V$ in period t;
• q_jt: The quantity (in tons) collected at location $j \in S$ in period $t \in H$, otherwise, it is zero;
• $x_{i j t}^v$: A binary variable set to 1 if location $j \in S^{+}$is visited immediately after location $i \in S^{+}$by vehicle $v \in V$ in period $t \in H$; otherwise, it is zero.
• $y_t^v$: A binary variable set to 1 if vehicle $v \in V$ is in use during period t; otherwise, it is zero.

SP-VRP: Minimize

$C V=\sum_{t \in H} \sum_{v \in V}\left[\psi^v y_t^v+\sum_{i \in S^{+}} \sum_{j \in S^{+}}\left(\delta_v v_v \theta_{i j}+\phi_{j t}\right) x_{i j t}^v\right]$        (1)

Subject to:

$\sum_{v \in V} \sum_{i \in S^{+}} x_{i j t}^v \leq 1, \forall j \in S, t \in H$        (2)

$\sum_{i \in S^{+}} x_{i j t}^v-\sum_{k \in S^{+}} x_{j k t}^v=0, \forall j \in S^{+}, t \in H, v \in V$            (3)

$\sum_{i \in S^{+}} \sum_{j \in S^{+}} \theta_{i j} x_{i j t}^v \leq \tau_t, t \in H, v \in V$       (4)

$\sum_{v \in V} \sum_{i \in S^{+}} Q_{i j t}^v-\sum_{v \in V} \sum_{k \in S^{+}} Q_{j t t}^v=q_{j t}, \forall j \in S, t \in H$          (5)

$Q_{i j t}^v \leq k x_{i j t}^v, \forall j \in S^{+}, t \in H, v \in V$         (6)

$q_{j t} \geq d_{j t} \tau_t, \forall j \in S, t \in H$        (7)

$x_{r j t}^v \leq y_t^v, \forall j \in S^{+}, t \in H, v \in V$          (8)

$x_{r j t}^v, y_t^v \in\{0,1\}, Q_{i j t}^v \geq 0, q_{j t} \geq 0, \forall j \in S^{+}, t \in H, v \in V$

Three cost components make up the objective function (1): the total fixed operational cost of employing the vehicle(s), the total cost of transportation, and the total cost of delivery handling after each period. Constraints (2) guarantee that every CC is visited no more than once during time t. A vehicle must depart from a CC after serving it to the DC or the next CC, according to Constraints (3). Constraints (4) make sure that vehicles finish their routes in a single travel period, therefore a vehicle's total travel duration shouldn't be longer than the number of scheduled working hours for each period. The amount collected to a CC is determined by Constraints (5), which also exclude sub-tours. The vehicle capacity constraints are given by (6) and ensure that the variables $Q_{i j t}^v$ cannot carry any cumulated flow unless $x_{i j t}^v$ equals 1. Constraints (7) is demand at the CC does not exceed the quantity collected from the CC. Constraints (8) indicate that a vehicle cannot be used to serve any CC unless it is selected.

We investigated the single-period vehicle routing problem (SP-VRP), in which a single distribution center (DC) collects agricultural products from a set of collection centers (CC) that consume them at fixed demand rates, using a fleet of homogeneous vehicles over a defined finite horizon. The objective is to determine the optimal quantities to collect from the CCs, establish collection times, and design vehicle delivery routes to minimize total transportation costs. The SP-VRP was formulated as a linear mixed-integer program. Results from a medium-sized example case show that vehicle capacity is utilized efficiently, achieving an average capacity utilization of 100% across all tours.

Moving forward, the current model and solution will be adapted for numerical experiments and real-world challenges, including scenarios involving large CCs, as part of our future research strategy. Additionally, while this study provides valuable insights into the SP-VRP in agriculture, future research should explore multi-period extensions to account for longer-term decision-making and to capture the full complexity of agricultural logistics. These extensions could lead to even greater cost savings, improved fleet management, and enhanced environmental outcomes, making them an important direction for continued exploration.