A Mixed-Integer Linear Programming Model for Multi-Echelon Facility Location and Allocation in Agro-Food Logistics Networks

2.1 The structure in the proposed facility location model

The proposed facility location model is formulated as a three-echelon supply chain structure consisting of an agro-hub as the distribution center, retailers, and customers. The customer set is divided into three categories to reflect different demand locations and distribution characteristics in the Banten agro-food logistics system.

Direct customers represent customers located within the Banten region, while business-to-business (B2B) customers represent business buyers with larger purchase volumes. Customers outside Banten represent external demand points located beyond the regional boundary. This classification is used to distinguish downstream product flows and their related transportation-cost components. In the objective function, transportation cost is separately calculated for the agro-hub-to-retailer flow, retailer-to-direct-customer flow, retailer-to-B2B-customer flow, and retailer-to-customers-outside-Banten flow.

The model is designed to support strategic and tactical decisions in configuring the downstream network of the AFSC in Banten Province, Indonesia. The model is used to determine the number and locations of retailer facilities that should be established across the province to support product distribution from the agro-hub to consumers. In addition to facility placement, the model identifies which products should be delivered, in what quantities, and which vehicle types should be used to support transportation between the agro-hub, retailers, and customers. By optimizing these interconnected decisions simultaneously, the model aims to minimize total supply chain cost subject to demand fulfillment, facility capacity, vehicle availability, and service-related requirements across Banten’s eight administrative regions. The model therefore provides a decision-support framework for evaluating retailer location, product allocation, and vehicle-use decisions under regional distribution and infrastructure constraints. The structure and logic of this model are illustrated in Figure 1.

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Figure 1. The framework of the agro-hub supply chain

2.2 Limitations and assumptions in the facility location model

In this model, the agro-hub is defined as a single central distribution point that consolidates, temporarily handles, and dispatches agro-food products to selected retailer locations. Products are assumed to be available at the agro-hub before distribution decisions are made. The capacity of the agro-hub is not modeled as a binding constraint, since the focus of this study is on downstream facility-location and allocation decisions.

The facility location model proposed in this study is based on several simplifying assumptions that help structure the model and make it computationally tractable. However, these assumptions also introduce certain limitations that may affect the model’s ability to fully reflect real-world complexities. The main assumptions used in this model are as follows:

• Deterministic input data. All demand, supply, and cost parameters are assumed to be known with certainty and remain constant over the planning horizon.
• Linear transportation costs. Transportation costs are calculated as a linear function of distance (per kilometer).
• Distance-based cost aggregation. The per-kilometer transportation cost is treated as a composite metric that reflects fuel consumption, driver wages, vehicle maintenance, and environmental impact (e.g., emissions).
• The model assumes commodities are treated as final products without undergoing additional processing or transformation at the agro-hub. This simplification allows the model to focus on the distribution structure and logistical decisions rather than production activities.
• Predefined retailer location options. The model assumes that all possible retailer facility locations are known in advance. It does not generate new location options but rather selects the best combination from a set of predefined candidates.
• All customer demand must be fulfilled. The model assumes that the demand of all customers must be fully met.

2.3 Mathematical equations of the facility location model

This subsection presents the mathematical formulation of the facility location model, which serves as a key component in optimizing the distribution network of the AFSC. The model focuses on determining the most cost-efficient configuration for locating retailer facilities that connect the agro-hub to end consumers across the Banten region.

Objective function:

$\operatorname{Min} Z=Z_1+Z_2+Z_3+Z_4+Z_5$         (1)

where,

$Z_1=\sum_{v=1}^V \sum_{i=1}^I \sum_{a=1}^A \sum_{r=1}^R \sum_{t=1}^T T C_{\text {viar}}^t . N T_{\text {viar}}^t . D S_{a r}$          (2)

$Z_2=\sum_{v=1}^V \sum_{i=1}^I \sum_{r=1}^R \sum_{c=1}^C \sum_{t=1}^T T C_{\text {virc}}^t . N T_{\text {virc}}^t . D S_{r c}$         (3)

$Z_3=\sum_{v=1}^V \sum_{i=1}^I \sum_{r=1}^R \sum_{b=1}^B \sum_{t=1}^T T C_{v i r b}^t . N T_{v i r b}^t . D S_{r b}$          (4)

$Z_4=\sum_{v=1}^V \sum_{i=1}^I \sum_{r=1}^R \sum_{o=1}^o \sum_{t=1}^T T C_{\text {viro }}^t . N T_{\text {viro }}^t . D S_{r o}$         (5)

$Z_5=\sum_{r=1}^R y_r$. Opening_Cost      (6)

Eq. (1) represents the objective function of the facility location model. The primary goal of this function is to minimize the total cost associated with the downstream distribution activities in the AFSC network. The total cost is formulated as the sum of five distinct cost components, denoted as Z₁ through Z₅, where each component reflects a specific operational cost category within the system:

• Z₁ in Eq. (2) accounts for the transportation cost from the agro-hub to retailers,
• Z₂ in Eq. (3) accounts for the transportation cost from retailers to end-customers,
• Z₃ in Eq. (4) reflects the transportation cost from retailers to B2B-customers,
• Z₄ in Eq. (5) reflects the transportation cost from retailers to outside Banten customers, and
• Z₅ in Eq. (6) represents the fixed cost of opening or operating retailer facilities.

By minimizing the total of these five components, the model ensures that the selected facility locations and corresponding product flows are not only feasible but also economically optimal. This objective function serves as the foundation for evaluating alternative distribution configurations within the Banten agro-food network.

Constraints:

$N T_{v i a r}^t \geq Q_{v i a r}^t / \operatorname{Cap}_v, \forall v, \forall i, \forall a, \forall r, \forall t$      (7)

Constraint (7) defines the number of trips required from the agro-hub to each retailer, based on the total quantity of products delivered and the carrying capacity of the assigned vehicle. Specifically, the constraint states that the number of trips is calculated by dividing the total quantity of products transported by a vehicle’s maximum load capacity.

$q_{i a r}^t=\sum_{v=1}^V Q_{v i a r}^t, \forall i, \forall a, \forall r, \forall t$        (8)

Constraint (8) ensures that the total quantity of products delivered from the agro-hub to each retailer is accurately captured. It states that the total amount received by a retailer is calculated as the sum of all quantities transported by each vehicle from the agro-hub to that retailer.

$\sum_{i=1}^I \sum_{a=1}^a q_{i a r}^t \leq y_r V_r, \forall r, \forall t$       (9)

Constraint (9) ensures that the total quantity of products delivered from the agro-hub to each retailer does not exceed the retailer’s storage capacity. It links the retailer-opening decision with the inbound product flow from the agro-hub to each retailer. If retailer r is not selected, the right-hand side becomes zero, and no product can be assigned to that retailer. If retailer r is selected, the inbound quantity is allowed but cannot exceed the storage or handling capacity of that retailer.

$N T_{\text {virc}}^t \geq Q_{\text {virc}}^t / \operatorname{Cap}_v, \forall v, \forall i, \forall r, \forall c, \forall t$          (10)

Constraint (10) defines the number of trips required between each retailer and end-customers, based on the total quantity of products to be delivered and the carrying capacity of the vehicles used.

$q_{i r c}^t=\sum_{v=1}^V Q_{v i r c}^t, \forall i, \forall r, \forall c, \forall t$         (11)

Constraint (11) ensures that the total quantity of products delivered to each end-customer is determined by summing all deliveries made by vehicles from the assigned retailers.

$N T_{\text {virb}}^t \geq Q_{\text {virb}}^t / \operatorname{Cap}_v, \forall v, \forall i, \forall r, \forall b, \forall t$         (12)

Constraint (12) defines the number of trips required to deliver products from retailers to B2B customers, based on the total quantity of products transported and the capacity of the delivery vehicles. The number of trips is calculated by dividing the total volume of products delivered to each B2B customer by the vehicle's load capacity.

$q_{i r b}^t=\sum_{v=1}^V Q_{v i r b}^t, \forall i, \forall r, \forall b, \forall t$          (13)

Constraint (13) ensures that the total quantity of products delivered to each B2B customer is calculated as the sum of all deliveries made by vehicles from the assigned retailers.

$N T_{\text {viro}}^t \geq Q_{\text {viro}}^t / \operatorname{Cap}_v, \forall v, \forall i, \forall r, \forall o, \forall t$        (14)

Constraint (14) defines the number of trips required for delivering products from retailers to customers located outside Banten Province. This is determined by dividing the total quantity of goods sent to each outside customer by the vehicle’s carrying capacity.

$q_{\text {iro}}^t=\sum_{v=1}^V Q_{\text {viro}}^t, \forall i, \forall r, \forall o, \forall t$      (15)

Constraint (15) ensures that the total quantity of products delivered to each outside Banten customer is obtained by summing all deliveries made by vehicles from the assigned retailers.

$\sum_{a=1}^A q_{i a r}^t \geq \sum_{o=1}^O q_{i r o}^t+\sum_{b=1}^B q_{i r b}^t+\sum_{c=1}^C q_{i r c}^t, \forall i, \forall r, \forall t$        (16)

Constraint (16) represents the flow-balance relationship at each retailer. It ensures that the total quantity distributed from a retailer to all customer categories, including end customers, B2B customers, and customers outside Banten, does not exceed the quantity received by that retailer from the agro-hub. Combined with Constraint (9), this constraint ensures that if a retailer is not open, both inbound and outbound flows associated with that retailer are forced to zero.

$D_{i c}^t \leq \sum_{v=1}^V \sum_{r=1}^R Q_{v i r c}^t, \forall i, \forall c, \forall t$       (17)

$D_{i b}^t \leq \sum_{v=1}^V \sum_{r=1}^R Q_{v i r b}^t, \forall i, \forall b, \forall t$       (18)

$D_{i o}^t \leq \sum_{v=1}^V \sum_{r=1}^R Q_{v i r o}^t, \forall i, \forall o, \forall t$         (19)

Constraints (17)–(19) ensure that the demand from all customer segments is fully satisfied. These constraints are applied respectively to: End-customers (Constraint 17), B2B customers (Constraint 18), and Outside Banten customers (Constraint 19).

Each constraint guarantees that the total quantity of products delivered to each customer group is at least equal to their stated demand.

$\begin{array}{r}\sum_{i=1}^I \sum_{a=1}^A \sum_{r=1}^R N T_{v i a r}^t+\sum_{i=1}^I \sum_{r=1}^R \sum_{c=1}^C N T_{v i r c}^t +\sum_{i=1}^I \sum_{r=1}^R \sum_{b=1}^B N T_{v i r b}^t +\sum_{i=1}^I \sum_{r=1}^R \sum_{o=1}^O N T_{v i r o}^t\leq \operatorname{Max} \operatorname{Trip}, \forall v, \forall t\end{array}$             (20)

Constraint (20) ensures that the number of trips assigned to each vehicle in a given period does not exceed the maximum allowable limit. This constraint reflects real-world operational limitations such as driver working hours, vehicle fatigue, delivery windows, and regulatory compliance.

$y_r \in\{0,1\}$             (21)

Constraint (21) introduces a binary decision variable that governs the selection of retailer facility locations. This variable takes the value of 1 if a particular retailer location is selected and 0 otherwise.

$N T_{\text {viar}}^t, N T_{\text {virc}}^t, N T_{\text {virb}}^t, N T_{\text {viro}}^t \in$ integer           (22)

Constraint (22) ensures that the number of trips assigned to each vehicle is treated as an integer value.

2.4 Heuristic approach

To complement the MILP formulation, this study uses a simple LP-relaxation-based rounding heuristic as a computational comparison with the feasible MILP solution. The heuristic is not intended as a standalone methodological contribution but as a practical procedure for generating a feasible solution with lower computational effort. The method starts by relaxing the integer constraint on the number of vehicle trips (Constraint 22), allowing the solver to generate continuous trip values. The fractional trip values are then rounded up to the nearest integer to ensure that the assigned vehicle capacity is sufficient to transport the required product quantity. The steps are outlined below:

• Step 1. Constraint relaxation: Remove the integer requirement for the number of vehicle trips, allowing them to take positive real values.
• Step 2. Initial solution generation: Solve the relaxed MILP model to obtain the total cost and trip values for each vehicle.
• Step 3. Identification of non-integer trips: All trip variables with non-integer values are identified. For each non-integer trip, the shortage required to reach the nearest upper integer value is computed.
• Step 4. Additional transportation cost calculation: Compute the additional cost by multiplying the shortage of each trip by the vehicle operating cost per kilometer and the corresponding travel distance.
• Step 5. Heuristic total cost computation: Add the total additional cost from Step 4 to the initial solution cost obtained in Step 2.

4.1 Numerical example

The LINGO optimization software was used to determine the minimum total operational cost of the proposed supply chain configuration. The LINGO software produced a feasible solution with a total supply chain cost of IDR 221,650,020, based on the given input data and model configuration. The largest cost component is Z1 of the total cost. The breakdown of the total operational cost is presented in Table 7. The optimization output includes key strategic decisions, such as selected retailer locations, vehicle utilization, and product flow allocations.

Table 7. Cost breakdown

Table 7 presents the cost breakdown of the proposed supply chain configuration. Z1 was the largest cost component, accounting for 42.08% of the total cost, followed by Z3 at 32.45%. The remaining components were Z5, Z4, and Z2, contributing 14.21%, 7.11%, and 4.15%, respectively.

Based on the optimization results, the model selected 7 out of the 8 planned retailer locations. Specifically, all retailers were open except Retailer-6. This decision was driven by the model's consideration of distance constraints and transportation costs, ensuring that only locations contributing to total cost minimization were activated. Retailer-6 was not selected because its overall contribution to the distribution network was weaker than that of the other candidate locations. Although this location may offer some outbound access advantages, these advantages were not sufficient to offset its less favorable position in the overall inbound and outbound logistics structure. The selected retailers serve as key distribution points in the optimized supply chain configuration. Details of the activated retailer’s locations are presented in Table 8.

Table 8. Selected retailer’s locations

Given the large size of the complete allocation output, the results are presented using summary-level tables and selected detailed outputs. Tables 9–11 report the total trips by planning period and the total flow by retailer. Meanwhile, Tables 12–15 present selected outputs for Vehicle-3 to illustrate the detailed deployment and delivery decisions across products, retailers, and planning periods.

Table 9 summarizes the total number of trips assigned across all vehicle types for each flow segment and planning period. The model generated 556 trips in Period 1 and 557 trips in Period 2, resulting in 1,113 trips over the two-period planning horizon. The agro-hub-to-retailer segment accounted for the largest number of trips, with 492 trips, followed by retailer-to-B2B customer distribution with 341 trips. Retailer-to-end customer and retailer-to-outside-Banten customer flows required 192 and 88 trips, respectively.

Table 9. Summary of total trips by flow segment and planning period

Table 10 presents the total inbound flow from the agro-hub to each retailer’s location across the two planning periods. Location 1 received the largest total inbound flow, with 720 units, followed by Location 3 with 680 units and Location 8 with 415 units. Location 6 received no inbound flow in either period, which is consistent with the optimization result that this location was not selected as an operating retailer.

Table 10. Total inbound flow to retailer locations by period

Table 11 summarizes the total outbound flow from retailers to each customer category across the two planning periods. The outbound flow was balanced across periods, with 1,230 units distributed in Period 1 and 1,230 units in Period 2, resulting in 2,460 units over the planning horizon. B2B customers received the largest allocation, with 1,700 units, followed by outside-Banten customers with 440 units and end customers with 320 units. The total outbound flow is consistent with the total inbound flow, indicating that the model maintains flow balance between products received by retailers and products distributed to customers.

Table 11. Total outbound flow by customer category

Table 12 presents the delivery strategy for Vehicle-3 determined by the optimization model, detailing vehicle utilization, product type, origin and destination locations, as well as the quantity delivered across two planning periods. Each row describes a specific shipment scenario, using a structured notation: Vehicle–Product–Agro-hub–Retailer (V–I–A–R). For example, the entry “3–1–1–2” indicates that Vehicle-3 is used to deliver Product Type-1 from Agro-hub (node 1) to Retailer-2. In this case, 25 tons were shipped in Period 1 and 90 tons in Period 2. This level of detail provides insights into how the model effectively allocates transport resources to fulfill demand while minimizing total costs.

Table 12. Vehicle-3 deployment and delivery strategy for each product and retailer across two periods (in tons)

Table 13. Vehicle-3 routing and product delivery from retailers to direct customers across two periods (in tons)

Table 13 outlines the delivery strategy from retailers to the 12 direct customers, capturing vehicle assignments, product types, routing paths, and shipment quantities across two planning periods. The structured notation Vehicle–Product–Retailer–Customer (V–I–R–C) is used to describe each delivery instance. For example, the entry “3–1–7–12” indicates that Vehicle-3 is assigned to deliver Product Type-1 from Retailer-7 to Customer-12. In this particular case, 5 tons were delivered in each period. This information reflects how the model allocates vehicles to satisfy direct customer demand with cost efficiency and routing feasibility.

Table 14 displays the delivery configuration from retailers to the 14 B2B customers, including vehicle allocation, product type, source and destination points, and quantity shipped across two planning periods. The structured notation used is Vehicle–Product–Retailer–B2B Customer (V–I–R–B). For example, the entry “3–1–1–4” in the second row, that Vehicle-3 is assigned to deliver Product Type-1 from Retailer-1 to B2B Customer-4, with 40 tons shipped in Period 1 and no product shipped in Period 2. This detailed delivery mapping demonstrates the model’s ability to allocate transport resources efficiently in meeting B2B customer demands.

Table 14. Vehicle-3 routing and product delivery from retailers to business-to-business (B2B) customers across two periods (in tons)

Table 15 presents the delivery strategy from retailers to the customers located outside Banten Province. The table follows the same structured format using the notation Vehicle–Product–Retailer–Outside Customer (V–I–R–O). For instance, the entry “3–1–5–1” indicates that Vehicle-3 is used to deliver Product Type-1 from Retailer-5 to the out-of-Banten customer. In this case, 30 tons were shipped in Period 1, while no delivery occurred in Period 2. This data supports the model's ability to plan long-distance shipments while considering vehicle capacity and transportation efficiency.

Table 15. Vehicle-3 routing and product delivery from retailers to outside-Banten customers across two periods (in tons)

The numerical results demonstrate the applicability and effectiveness of the proposed MILP-based facility location and allocation model for agro-food distribution in Banten. By incorporating real-world data, including customer segmentation, vehicle heterogeneity, and multi-period planning, the model successfully identifies cost-efficient delivery strategies that align with operational constraints. The optimized configuration, including selected retailer locations, vehicle assignments, and detailed product flows, highlights how the model can support data-driven decision-making in agro-hub development. These findings not only validate the model’s structure but also provide practical insights for regional planners seeking to design more efficient and responsive food supply chain systems.

To complement the feasible MILP optimization result, a simple heuristic method was implemented as a comparative solution strategy. The heuristic was based on the relaxation of Constraint (22), which initially enforced integer values for the number of vehicle trips. After relaxing this constraint, the model produced an initial solution cost of IDR 218,472,904. However, this solution contained non-integer values for several trip variables and therefore could not be directly implemented in practice. To restore feasibility, the fractional trip values were rounded up to the nearest integer, and the associated cost adjustment was recalculated. The additional cost incurred from this rounding adjustment was IDR 7,113,796, resulting in a total heuristic cost of IDR 225,586,700. Compared with the feasible MILP solution of IDR 221,650,020, the heuristic solution was 1.78% higher in total cost.

4.2 Sensitivity analysis

A sensitivity analysis was conducted to evaluate how the proposed agro-hub supply chain model responds to changes in demand and vehicle capacity. Demand was selected as the sensitivity parameter because it directly affects product allocation, retailer activation, vehicle utilization, and total operational cost. Four demand scenarios were examined: a pessimistic scenario with a 50% demand reduction, the most-likely scenario representing the baseline demand, an optimistic scenario with a 25% demand increase, and a high-demand stress-test scenario with a 50% demand increase. All other model parameters were kept constant to isolate the effect of demand variation on the optimization results.

The results in Table 16 show that the proposed model is highly sensitive to demand changes. Under the most-likely scenario, the model generated a feasible solution with a total operational cost of IDR 221,650,020, requiring seven out of eight available retailer locations to be opened. This scenario was used as the main reported solution because it represents the expected demand condition and provides the most relevant operational configuration for the proposed supply chain system.

Table 16. Sensitivity analysis of demand variation

When demand was reduced by 50%, the total operational cost decreased substantially to IDR 58,438,720, and only four retailer locations were required. This indicates that under lower demand conditions, the model can consolidate product flows into fewer retailer facilities, thereby reducing distribution and facility-related operational costs. In contrast, when demand increased by 25%, the total operational cost rose to IDR 349,362,880, and all eight available retailer locations had to be opened. This result suggests that the supply chain network has limited spare capacity under the baseline configuration, since a moderate increase in demand already requires full retailer activation.

The final scenario, in which demand was increased by 50%, resulted in an infeasible solution. This infeasibility occurred because the total capacity of all available retailer locations, even when fully opened, was insufficient to accommodate the increased demand. Therefore, the result highlights a critical capacity limitation in the proposed network. If demand grows beyond the 25% increase level, additional capacity expansion, new retailer facilities, larger storage capacity, or alternative distribution arrangements would be required to maintain supply chain feasibility.

To extend the sensitivity analysis, vehicle capacity varied by increasing and reducing the capacity of all vehicle types by 50%. The results are presented in Table 17. When vehicle capacity was increased by 50%, the total cost decreased from IDR 221,650,020 in the base case to IDR 151,963,620, representing a 31.44% cost reduction. This decrease occurred mainly because higher vehicle capacity reduced the required number of trips from 1,113 trips in the base case to 899 trips.

Table 17. Sensitivity analysis of vehicle capacity variation

In contrast, reducing vehicle capacity by 50% resulted in an infeasible solution. This occurred because the lower vehicle capacity increased the required number of trips beyond the maximum allowable trips per vehicle and period defined in the model. These results indicate that vehicle capacity is a critical logistics parameter in the proposed agro-hub supply chain configuration. Larger vehicle capacities can reduce trip requirements and total operational cost, while insufficient vehicle capacity may make the distribution plan infeasible unless additional vehicles, higher trip limits, or changes in distribution frequency are introduced.

The authors gratefully acknowledge Universiti Putra Malaysia (UPM), Universitas Sultan Ageng Tirtayasa, and PT ABM for their support and contribution throughout this research.