Minimum-cost flow optimization of road freight delivery in a mountainous transport network: a case study of Kyrgyzstan

2.1. Network representation of the freight system

The freight delivery system is modeled as a directed transport network $G=(N,A)$, where $N$ is the set of nodes and $A$ is the set of arcs.

The nodes represent major logistics points, including distribution centers, regional demand points, and border-related transport hubs. In the case study of Kyrgyzstan, the network includes Bishkek, Osh, Karakol, and Jalal-Abad as domestic nodes, with cross-border extensions toward Almaty and Tashkent for international freight scenarios. The arcs represent road connections between nodes and are characterized by distance, travel time, and route capacity.

Let $x_{ij}$ denote the freight flow transported from node $i$ to node $j$, measured in tons. Each arc $(i,j)\in A$ has an associated transportation distance $L_{ij}$, normal travel time $t_{ij}$, additional delay time ${\delta }_{ij}$, and maximum carrying capacity $u_{ij}$. The arcs represent road connections between nodes and are characterized by distance, normal travel time, additional delay, and route capacity.

This representation makes it possible to formulate the freight delivery problem as a minimum-cost flow optimization task under infrastructure and operational constraints.

2.2. Objective function

The purpose of the model is to determine the optimal freight flows that minimize the total logistics cost in the network. The total cost includes transportation cost, handling cost at logistics nodes, and additional delay-related cost on the corridors.

The objective function is formulated as:

where: $Z$ is the total logistics cost, USD; $c_{ij}$ is the basic transportation cost coefficient on arc $\left(i,j\right)$, USD/ton; $d_{ij}$ is the delay cost coefficient on arc $\left(i,j\right)$, USD/(ton·hour); ${\delta }_{ij}$ is the additional delay time on arc $\left(i,j\right)$, h; $x_{ij}$ is the freight flow on arc $\left(i,j\right)$, tons; $h_i$ is the handling cost at node $i$, USD/ton; $y_i$ is the processed cargo volume at node $i$, tons.

In this formulation, the transportation component and the delay component are separated explicitly. This makes the model more suitable for mountainous and border-constrained freight systems, where delay is an important cost driver independent of transport distance.

2.3. Transportation cost specification

The basic transportation cost coefficient on each arc is defined as a function of route length and normal travel time:

where: $\alpha$ is the transportation cost per ton-kilometre, USD/(ton·km); $\beta$ is the time-related operating cost coefficient, USD/(ton·hour); $L_{ij}$ is the distance between nodes $i$and $j$, km; $t_{ij}$ is the normal travel time on arc $\left(i,j\right)$, h.

Here, $t_{ij}$ represents the planned travel time under standard operating conditions, whereas ${\delta }_{ij}$ in Eq. (1) represents additional delay associated with border procedures, congestion, or other non-standard operational constraints. Such separation avoids double counting of time-related costs and improves the interpretability of the model.

2.4. Model constraints

The optimization model is subject to the following constraints.

1) Flow conservation constraint. For each node $i\in N$, the balance between incoming and outgoing freight flows must be satisfied:

where $b_i$ is the net supply or demand at node $i$.

If $b_i>0$, the node is a supply node; if $b_i<0$, it is a demand node; if $b_i=0$, it is a transit node.

2) Node-processing volume definition. To account for handling cost at logistics nodes, the processed cargo volume is defined as:

This expression represents the average throughput processed at node $i$ and avoids double counting of transit freight flows in the handling-cost component.

3) Capacity constraint. The freight flow on each route must not exceed the available carrying capacity:

where $u_{ij}$ is the maximum admissible freight capacity of arc $\left(i,j\right)$, tons.

4) Arc-level travel-time constraint. For time-sensitive shipments, each arc included in the feasible freight network must satisfy an admissible travel-time bound:

where $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum allowable travel time on an individual route segment, including both normal travel time and additional delay. This constraint excludes excessively time-consuming links from the feasible solution space and reflects operational requirements for delay-sensitive cargo.

2.5. Input parameters and case-study assumptions

The model was calibrated using recent transport statistics, corridor reports, and regional logistics assessments for Kyrgyzstan and Central Asia. The main input parameters include freight demand, route distance, travel time, handling cost, vehicle capacity, and delay penalties.

For the domestic scenario, the total freight demand was set at 500 tons and distributed across the main nodes of the Kyrgyz network. For the extended scenario, cross-border demand toward Almaty and Tashkent was added, increasing the total freight volume to 800 tons.

The baseline parameter values were defined as follows:

– $\alpha =$ 0.045 USD/(ton·km), representing the average road transport cost under regional operating conditions.

– $\beta =$ 2.5 USD/(ton·hour), reflecting time-related operating losses under normal travel conditions.

– Handling cost $h_i=15$ USD/ton for major logistics hubs and $20$ USD/ton for secondary nodes;

– Delay cost coefficient $d_{ij}=1.8$ USD/(ton·hour).

– Truck payload capacity was set at 20 tons per vehicle, while the arc capacity $u_{ij}$ was defined as the maximum aggregate freight throughput that could be served on corridor $\left(i,j\right)$ during the planning period, taking into account the assumed number of available truck trips.

Therefore, the optimization variable $x_{ij}$ represents the total freight volume assigned to corridor $\left(i,j\right)$ over the planning horizon, rather than the load of a single vehicle:

– Maximum allowable travel time $T_{\mathrm{m}\mathrm{a}\mathrm{x}}=48$ hours.

– Additional delay time ${\delta }_{ij}$ was assigned for border-affected and operationally constrained arcs.

These assumptions reflect the practical conditions of freight delivery in Kyrgyzstan, where road transport dominates and logistics performance is strongly affected by terrain and border delays.

2.6. Solution procedure

The optimization problem was solved as a linear programming minimum-cost flow model. The computational implementation was carried out in Python using the PuLP package. The solver generated optimal freight flows $x_{ij}$ for all routes and the corresponding minimum total logistics cost $Z$.

The numerical output was then analyzed for two scenarios:

1) A domestic freight distribution network within Kyrgyzstan.

2) An extended cross-border network including international demand nodes.

This procedure made it possible to compare the effect of geography, handling operations, and border delays on the total logistics cost and delivery performance.

2.7. Computational workflow and extraction of route-level optimization results

The computational workflow consisted of four sequential stages. First, the freight network was encoded as a directed graph with nodes, arcs, capacities, travel times, and delay parameters. Second, the generalized arc cost coefficients were calculated from the transportation, handling, and delay components defined in Eqs. (1-6). Third, the linear programming problem was solved in Python using the PuLP package to obtain the optimal arc flows $x_{ij}$ and the minimum total logistics cost Z. Fourth, the optimization output was post-processed at the arc level to identify active routes and summarize the principal route-level results for the domestic and cross-border scenarios. For each corridor $\left(i,j\right)$, the optimal freight flow $x_{ij}$, admissible corridor capacity $u_{ij}$, utilization ratio $x_{ij}/u_{ij}$, and generalized transport cost coefficient were extracted from the solved model. Based on these outputs, the selected routes were classified as active ($x_{ij}>$ 0) or inactive ($x_{ij}=$ 0) and analyzed separately for the domestic and cross-border scenarios.

3.1. Domestic freight optimization scenario

The proposed minimum-cost flow model was first applied to a domestic freight distribution network in Kyrgyzstan. The case-study network included Bishkek as the main logistics hub, Osh as the principal southern demand node, Karakol as an eastern regional node, and Jalal-Abad as an intermediate southern logistics point. This structure reflects the practical dominance of road freight transport in Kyrgyzstan and the importance of inter-regional connections in a mountainous environment.

For the domestic scenario, the total freight volume was set at 500 tons under balanced supply and demand conditions. After solving the optimization problem, the minimum total logistics cost was obtained as 18,750 USD. The cost structure showed that transportation costs formed the largest component, accounting for approximately 68 % of the total value, while handling costs represented 17 % and delay-related costs accounted for 15 %. The average effective logistics cost was 0.058 USD/ton-km.

The optimal solution demonstrated that the largest direct flow was assigned to the Bishkek-Osh corridor, where approximately 280 tons were transported. This route remained the most significant cost contributor because of its long mountainous alignment and strategic role in north-south freight connectivity. The Bishkek-Karakol route carried approximately 120 tons, while the use of intermediate transshipment through secondary nodes was minimized. This result indicates that, under the selected parameter set, direct routing is more cost-efficient than repeated handling and redistribution through intermediate nodes.

From an engineering perspective, the domestic scenario confirms that mountainous geography increases not only transportation distance but also the time-dependent part of the generalized logistics cost. Therefore, route selection in Kyrgyzstan cannot be based on distance alone; it must also account for lower speed, corridor geometry, and operational delays.

3.2. Cross-border freight scenario

In the second stage, the model was extended to include cross-border freight demand toward Almaty and Tashkent, increasing the total freight volume to 800 tons. Under this configuration, the optimized total logistics cost rose to 32,400 USD, while the average delivery time on international routes increased to approximately 36-42 hours.

The results show that cross-border operations significantly reduce transport efficiency because part of the generalized logistics cost is generated by border crossing procedures, inspection time, and administrative delays. In the optimized solution, international routes accounted for approximately 45 % of the total system cost, despite the relatively moderate geographic distances of some corridors. This confirms that time-related penalties are a major determinant of freight system performance in landlocked transport networks.

The comparison between the domestic and international scenarios clearly indicates that the addition of border-related delays changes the structure of optimal routing decisions. When time-dependent costs are incorporated into the objective function, routes with lower nominal distance are not always the most efficient if they are affected by severe customs and procedural delays. Thus, the model demonstrates the need to evaluate freight corridors using a combined spatial-temporal cost function rather than a conventional distance-only approach.

3.3. Comparative analysis of selected Central Asian freight corridors

To demonstrate the regional applicability of the proposed minimum-cost flow model, a comparative analysis was carried out for three representative freight corridors in Central Asia: Bishkek-Almaty, Bishkek-Tashkent, and Almaty-Tashkent. These routes reflect different combinations of transport distance, border-related delay, and monthly freight volume, and therefore provide a suitable basis for evaluating the spatial and temporal efficiency of freight delivery in the region. The comparative characteristics of these corridors are summarized in Table 1.

As shown in Table 1, the Almaty-Tashkent corridor produces the highest absolute monthly logistics cost because it combines the greatest route length with a high shipment volume. In contrast, Bishkek-Almaty has the shortest physical distance, but its operational efficiency is strongly affected by border procedures, which account for approximately half of the total route time. The Bishkek-Tashkent corridor occupies an intermediate position in terms of distance, yet its performance is additionally constrained by mountainous sections and cross-border formalities, which increase the total generalized logistics cost.

The comparative results confirm that freight efficiency in Central Asia is determined not by distance alone, but by the combined influence of distance, delay, and freight volume. Routes with shorter geographic length may still become inefficient if they are affected by persistent customs waiting time and administrative frictions. Conversely, longer routes can remain operationally attractive when supported by better infrastructure and smoother corridor procedures. From an engineering point of view, this confirms the importance of using a generalized cost formulation that includes both spatial and time-dependent components.

Another important finding is that the relative contribution of border delay remains substantial across all three corridors. In the current model, the delay share ranges from 32 % to 50 % of the total transit time, which indicates that corridor modernization should not be limited to physical road improvement alone. Digital customs tools, coordinated checkpoint procedures, and queue management systems may significantly improve overall freight performance even without major infrastructure expansion.

Overall, the corridor comparison demonstrates that the proposed minimum-cost flow framework can be used not only for domestic freight allocation in Kyrgyzstan, but also for the engineering assessment of cross-border logistics efficiency in the wider Central Asian transport system. This makes the model applicable to route planning, corridor prioritization, and infrastructure management under landlocked and mountainous operating conditions.

To make the optimization output more explicit, Table 2 summarizes the principal reported route-level and corridor-level results of the model. The table presents the main domestic corridors identified in the optimized solution together with the key cross-border corridors discussed in the comparative analysis.

As shown in Table 2, the optimization model produces an interpretable corridor-level freight allocation for the domestic network, while the cross-border analysis highlights the strong influence of border-related delay on corridor efficiency. This makes the engineering implications of the model more transparent.

3.4. Sensitivity and engineering interpretation

The sensitivity analysis confirms that the proposed minimum-cost flow model is highly responsive to additional delay on international corridors. To examine this effect, an additional scenario analysis was carried out by varying the delay parameter ${\delta }_{ij}$ on the cross-border arcs while keeping all other model parameters unchanged. The analysis showed that a reduction in additional delay leads to a direct decrease in the generalized logistics cost and improves the attractiveness of cross-border routes in the optimized solution. This confirms that time-related penalties are among the dominant determinants of freight efficiency in landlocked and mountainous transport systems.

From a transport engineering point of view, the obtained results support three practical conclusions. First, direct road corridors with limited transshipment are preferable in mountainous systems when handling costs are significant. Second, time-dependent penalties should be explicitly incorporated into optimization models for freight delivery in landlocked regions. Third, border process improvement through customs digitalization, coordinated checkpoint procedures, and electronic queue management may generate substantial efficiency gains even without large-scale physical infrastructure expansion.

Overall, the results show that freight transport performance in Kyrgyzstan and adjacent Central Asian corridors is constrained not only by geography, but also by the operational organization of the logistics chain.

Therefore, minimum-cost flow optimization can serve as an effective decision-support tool for freight routing, corridor planning, and logistics modernization under landlocked and mountainous operating conditions.

Thus, the sensitivity analysis confirms that delay-related penalties are not merely an auxiliary parameter of the model, but one of the key factors governing optimal freight allocation and corridor efficiency in mountainous and landlocked transport systems.

3.5. Comparison with a distance-only baseline

To further demonstrate the contribution of the proposed formulation, an additional baseline experiment was considered in which delay-related penalties were excluded from the objective function and route selection was driven only by transportation and handling costs. Compared with this simplified formulation, the proposed model produced a different freight allocation for the cross-border scenario, because some geographically shorter routes became less attractive once border-related delay costs were included.

This comparison confirms that the explicit consideration of time-dependent penalties is not only a modeling detail, but a factor that changes the optimization outcome and improves the engineering realism of the solution. Therefore, freight corridors in mountainous and landlocked regions should be evaluated using a combined spatial-temporal cost formulation rather than a conventional distance-only approach.

3.6. Limitations of the study

Despite the usefulness of the proposed model, several limitations should be acknowledged. First, the parameterization is based on average corridor values and does not fully capture route-specific seasonal variability. Second, the model does not include stochastic disturbances such as extreme weather, temporary road closures, or unexpected border congestion. Third, rail and multimodal alternatives were not explicitly modeled in the current version.

These limitations do not invalidate the obtained results, but they indicate directions for further research. Future work may extend the model by introducing stochastic travel time, dynamic demand, and multimodal freight routing options.