Table 1
Primary Transportation Assumptions.
| (T1) | Rail is unavailable (or already used to capacity). |
| (T2) | Ferries (e.g., across the Baltic Sea and Tysfjord) are unavailable (or already used to capacity). |
| (T3) | At this time, there are no adversarial actions affecting the remaining transportation network. |
| (T4) | When a vehicle is dispatched to a destination, the path with the shortest travel time is always chosen. |
| (T5) | Vehicles always return to their home base (SPOD) after delivery. |
| (T6) | Vehicles can only be dispatched every sixth hour, meaning four times per day. |
| (T7) | No interim facilities (like convoy support centres) or resources along the distribution routes are required. |
| (T8) | Once a road arc is disrupted, the arc cannot be used for at least 24 hours. |
Table 2
Correlated Set: Historical Data. Each row represents a day of the transportation network. Each column contains the status of an arc in the set of all arcs.
| DAY | ROAD 1 | ROAD 2 | ROAD 3 | ROAD 4 |
|---|---|---|---|---|
| 1 | ||||
| 2 | Closed | |||
| 3 | Closed | Closed | ||
| 4 | Closed | Closed | Closed | Closed |
| 5 | Closed | |||
| Sum | 2 | 2 | 2 | 2 |
Table 3
Round Trip Travel Timetable (Hours) Between SPODs (S1, S2) and Destinations (D1, D2). An infinite travel time means no path exists between the SPOD and the destination in the given day (of 24 hours).
| DAY | S1–D1 | S1–D2 | S2–D1 | S2–D2 |
|---|---|---|---|---|
| 1 | 19 | 22 | 23 | 21 |
| 2 | 19 | 22 | 27 | 21 |
| 3 | 19 | ∞ | 23 | ∞ |
| 4 | ∞ | ∞ | ∞ | ∞ |
| 5 | 24 | 22 | 23 | 21 |
Figure 1
Graph illustration of quickest paths through the network between SPODs (S1, S2) and destinations (D1, D2). The same is done for the returning path (from destinations to SPODs).
Table 4
Generation of Randomized Network States from the Correlated Set. Each column of the randomized set is created from random sampling without replacement from the corresponding column in the correlated set.
Table 5
Time Period Discretization of Travel Times. Four rows representing night, morning, afternoon and evening of the same day. Travel time in hours.
Table 6
Stochastic Mixed Integer Linear Program.
| SYMBOL | DESCRIPTION |
|---|---|
| Sets | |
| S | Set of SPODs indexed by s |
| D | Set of destinations indexed by d |
| F | Set of all days indexed by f (later split into time periods) |
| K | Set of time periods of equal time duration indexed by k |
| N | Set of whole numbers {0, …, |N|} indexed by n |
| Parameters | |
| tk,s,d | Round trip travel time between s and d at time k excluding waiting time |
| bk,s,d,n | |
| ek,s,d | |
| ci | Time penalty for safety stock backlog (per vehicle per time period) |
| cw | Time cost of assigning a vehicle to a SPOD |
| cu | Time penalty for not meeting demand (per vehicle) |
| Maximum safety stock capacity at destination d | |
| hx | Maximum number of SPODs allowed to be opened |
| hν | Maximum number of vehicles available for assignment |
| M | Arbitrarily large value ≥ hν (for binary constraints) |
| qk,d | Number of vehicles requested at destination d at time k (demand) |
| Variables for the Strategic Decision-Maker | |
| xs | |
| ws | Number of vehicles assigned to SPOD s as home base (integer) |
| Variables for the Dispatcher | |
| yk,s,d | Number of vehicles dispatched from SPOD s to destination d at time k |
| vk,s,d,n | Number of vehicles returning to s from d with n periods remaining at time k |
| ik,d | Safety stock backlog at destination d at time k |
| uk,d | Unmet demand (vehicles) at destination d at time k |
Table 7
Stochastic Mixed Integer Linear Constraint Explanations.
| (2) | The sum of vehicles in transit between SPOD s and any destination d in either direction at any time period cannot exceed the number of vehicles assigned to SPOD s as a home base ws. |
| (3) | The sum of assigned vehicles cannot exceed the number of vehicles available for assignment. |
| (4) | Vehicles dispatched from SPOD s to any destination d in time period k cannot exceed the available vehicles present at SPOD s (vehicles with n = 0). |
| (5) | The number of vehicles dispatched to destination d in time period k must be sufficient to meet demand. Any deficit will add to the backlog, while any excess will replenish the safety stock. If dispatched vehicles and safety stock are insufficient, the unmet demand is recorded and penalized in the objective. |
| (6) | Constraint 5 is also valid in the first time period and there is no backlog at the start of the first time period. |
| (7) | Vehicles can only be assigned to SPODs that are open. |
| (8) | Unused SPODs remain closed. |
| (9) | The number of SPODs opened cannot exceed the number of SPODs allowed. |
| (10) | For each time period k, we keep track of the number of vehicles in transit between SPOD s and destination d as well as the number of remaining time periods n each vehicle has until returning to their home SPOD s. If vehicles are dispatched in time period k the parameter b add the vehicles to the correct vehicle tracking variable ν. |
| (11) | Vehicles available at SPOD s in the first time period k = 0 are all vehicles assigned to SPOD s except vehicles dispatched from SPOD s in the first time period. |
| (12) | Ensures that no vehicles are on route in the first time period, except those being dispatched. |
| (13) | The safety stock backlog (use) cannot exceed the safety stock capacity. |
| (14) | Whenever SPOD s becomes isolated from destination d, it can be beneficial to dispatch vehicles even though they cannot return to SPOD s without waiting for a path to open. Therefore, if SPOD s becomes isolated from destination d in time period k, and there was a path in k–1, we assume vehicles are still able to reach the destination, however, they have to wait for the path to open in order to return to the SPOD. If there is no path for multiple time periods in a row, vehicles cannot reach the destination other than in the first time period of the sequence. In other words, vehicles can only be dispatched from s to d if a path exist in time period k or in k–1. Vehicles need n time periods to return to SPOD s (where bk,s,d,n = 1). |
| (15) | Technical constraint to stop vehicles from appearing in |N|. |
| (16) | Non-negativity constraints |
Table 8
Primary information (I) and modeling (A) assumptions.
| (I1) | Dispatcher knows the safety stock levels and demand of every destination. |
| (I2) | Dispatcher knows the number of vehicles available at each SPOD and when the others will return. |
| (I3) | Dispatcher’s weather forecast provides perfect foresight of round trip travel times to every destination over the relevant time horizon. |
| (A1) | All open SPODs hold sufficient stocks to supply every destination. |
| (A2) | Vehicles dispatched arrive at the destination instantaneously. |
| (A3) | Vehicles dispatched in the first time period of a day, when no path exists between a SPOD and a destination, can reach the destination but cannot return until a path is reestablished, as explained (14). |
| (A4) | When a vehicle is dispatched, its return time to the SPOD is known and captured by the parameter bk,s,d,n = 1. |
| (A5) | Vehicle dispatches can be non-integer and evens out over time. |
Figure 2
Arcs where historical road condition data has been collected.
Note. The arcs in Norway and Sweden are classified by name, while Finland has coordinates. This has no practical implication other than the visual difference in density. Extracted using OSMNX (Boeing, 2017). World base map from Esri (2024).
Figure 3
Potential SPODs (Set S in blue) and arbitrary destinations (Set D in green).
Table 9
Primary Network Assumptions.
| (N1) | SPOD locations in Denmark are too far away from the destinations and are therefore excluded. |
| (N2) | All SPODs have been expanded as needed to accommodate the required supply throughput capacity. |
| (N3) | All arcs in the transportation network, remaining after road class filtering in OpenStreetMap, have sufficient capacity to accommodate all transport demands. |
| (N4) | Multiple vehicle crews work on shifts, allowing for continuous travel. The only exception is the time interval between the vehicles’ return to its SPOD and the next dispatch window. |
| (N5) | Other risks regarding safety and security beyond what is captured in the dataset are considered insignificant. |
Figure 4
Heat map of travel-time correlations between SPODs and Alta (Norway) and Lappeenranta (Finland), sorted by latitude.
Table 10
SPOD Round Trip Travel Times (in Hours). Deterministic travel time: m1; average travel time when paths exist: m2; maximum recorded waiting and travel time: m3. Bold font indicates the best performance in one of the criteria. SPODs are sorted by latitude.
| PORT | DESTINATIONS | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ALTA | LAKSELV | SALLA | KAJAANI | LAPPEENRANTA | |||||||||||
| m1 | m2 | m3 | m1 | m2 | m3 | m1 | m2 | m3 | m1 | m2 | m3 | m1 | m2 | m3 | |
| Tromsø | 10 | 11 | 62 | 15 | 15 | 67 | 17 | 17 | 65 | 21 | 21 | 69 | 30 | 30 | 78 |
| Harstad | 14 | 15 | 66 | 19 | 19 | 71 | 17 | 18 | 69 | 21 | 21 | 73 | 29 | 29 | 82 |
| Narvik | 13 | 13 | 64 | 16 | 17 | 70 | 15 | 16 | 67 | 19 | 19 | 71 | 27 | 27 | 80 |
| Bodø | 25 | 25 | 102 | 27 | 27 | 105 | 21 | 21 | 98 | 21 | 21 | 98 | 29 | 29 | 106 |
| Mo i Rana | 24 | 24 | 34 | 26 | 26 | 36 | 20 | 20 | 30 | 21 | 21 | 31 | 28 | 29 | 38 |
| Trondheim | 33 | 33 | 37 | 34 | 35 | 36 | 28 | 28 | 31 | 29 | 29 | 31 | 37 | 37 | 39 |
| Tjeldbergodden | 36 | 36 | 40 | 38 | 38 | 40 | 32 | 32 | 34 | 32 | 32 | 34 | 40 | 40 | 43 |
| Oslo | 39 | 39 | 44 | 41 | 41 | 42 | 34 | 34 | 37 | 35 | 35 | 37 | 43 | 43 | 45 |
| Göteborg | 39 | 39 | 45 | 41 | 41 | 42 | 35 | 35 | 37 | 35 | 35 | 37 | 43 | 43 | 46 |
Figure 5
Average and maximal recorded round trip travel time to the nearest destination for each SPOD. A 24-hour delay is added to the travel time for every subsequent day where no path exists.
Table 11
SPOD Selection Criteria and Results.
| NAME | SPOD SELECTION CRITERION | SPOD SELECTED | DESTINATIONS SUPPLIED |
|---|---|---|---|
| Benchmark | Marked as key areas for defence logistics in the region by the Nordic Chiefs of Defence. | Narvik Trondheim Göteborg | Not specified |
| M1 | Lowest deterministic travel time to at least one of the destinations, i.e. travel time under perfect road conditions. | Tromsø Narvik | Alta and Lakselv Salla, Kajaani and Lappeenranta |
| M2 | Lowest average travel time (when path exists) to at least one of the destinations across all scenarios in the correlated set. | Tromsø Narvik | Alta and Lakselv Salla, Kajaani and Lappeenranta |
| M3 | Lowest recorded maximum waiting and travel time to at least one of the destinations across all scenarios in the correlated set. | Mo i Rana | All destinations |
Figure 6
Average vehicle deficit (units of unmet demand) for different SPOD selections. Total demand per day is 500 (left) and 600 (right).
Figure 7
Highest recorded vehicle deficit (unmet demand) across any week for different SPOD selections. Total demand per day is 500 (left) and 600 (right).
Figure 8
Optimal vehicle allocation (left panels) and vehicle usage (right panels) for the MILP SPOD selection. The lower panels depict winter-season results.
Figure 9
Quickest paths between Narvik and destinations in Finland. Map to the right contains all used paths when route E10 is open.
Figure 10
Optimal vehicle allocation (left) and vehicle usage (right) during winter when E10 is open.
Note. Safety stock capacity is 100 per destination.