At the tactical level, uncertainties related to demand, transit times, capacity and costs are studied. However, the studied types of uncertainty differ between NFP and SND problems. Demand uncertainty is the most studied type of uncertainty for SND problems because the networks are usually set up in advance for longer periods of time. Therefore, complete demand information is not available yet. In contrast, stochastic demand is rarely studied in NFP problems since it is often assumed that demand is already known before routes are determined. Transit times and capacity are the most studied uncertainties for NFP, but are rarely included in studies on SND. Capacity uncertainty is modelled by lowering the capacities of links, nodes and intermodal terminals following disruptions. For severe disruptions such as disasters, capacities can also be set to zero. Only one study on SND considers capacity uncertainty, since the other studies assume that once a service is scheduled, it will never fail. One study on SND accounts for cost uncertainty by treating transport costs per arc as random variables within an interval [
46].
Although the objective of tactical planning problems is often to minimise costs, multi-objective optimisations are also used. Other elements in the objective function can include total transport time and emissions [
19,
49]. Demir et al. [
19] minimise costs, time and greenhouse gas emissions by assigning different weights to each objective. The authors find that when only emissions are minimised, penalty costs and transportation costs are very high. Trucks must wait for electric trains, which have insufficient capacity to transport all loads at once. As an alternative, Sun et al. [
47] include emissions in their objective function by adding emission costs. A different objective which does not include costs at all is to maximise the fraction of demand that can be satisfied following a disaster [
12,
14]. The remainder of this section is divided between studies on SND (
Section 4.1.1) and NFP (
Section 4.1.2).
4.1.1. Service Network Design
Studies on service network design with uncertainty are listed in
Table 3.
Andersen and Christiansen [
50] study travel time uncertainty in the European freight train network connecting Scandinavia with Italy. The proposed model supports strategic investment decisions in infrastructure for new railway lines and also models tactical scheduling decisions. Although infrastructure investments are required, these are not incorporated in the study, making it more of a tactical problem. Freight trains experience many delays in Germany due to congestion and lower priority than passenger trains, whereas truck transport through Switzerland and Austria is restricted leading to high travel times. The study assesses potential benefits on service quality of using the Polcorridor, a rail corridor through Poland. As a result of different track gauges between Poland and Czechia, two distinct locomotive fleets are needed as well as extra handling for container transfers. The required decisions are the selection of routes, train schedules and the number of stops at intermediate terminals. The fixed costs for the scheduled trains and used services are included. The variable costs are included for unit flows, repositioning moves, transport time based on the value of time and a penalty for the variability of the transport time. The last two costs are to include service quality in the objective. Demand varies but schedules must remain identical over all time periods. A trade-off needs to be made between excess capacity and lost revenue. Profits are maximised and solutions are obtained with a commercial solver. Fewer deliveries are performed when service quality is considered, because deliveries with high transport time or variability are rejected. Moreover, routes with less variability are chosen despite being longer and more expensive to operate.
Lium et al. [
51] study demand stochasticity in service network design. A deterministic model is compared with a stochastic one. Both models include time-windows as a hard constraint, meaning orders must be delivered on time. The authors conclude that using deterministic formulations for stochastic models causes extra costs. Moreover, stochastic models generate solutions with more consolidation possibilities. As a result, the orders can be consolidated faster if the demand is lower than expected. Regarding demand correlation, as long as demand is not perfectly positively correlated, consolidation could be possible. The more negatively correlated, the higher the potential consolidation opportunities. For the same reason, hub-and-spoke networks are favoured over other network structures when dealing with stochasticity. Hoff et al. [
52] study a realistically sized service network design problem in which demand becomes known gradually over time. The objective is to minimise the sum of service costs and the expected cost of sudden changes. Their model is based on the one from Lium et al. [
51] but a metaheuristic is proposed instead of an exact solution method, which makes it possible to solve larger problem instances.
Crainic et al. [
53], Bai et al. [
55], Meng et al. [
56] and Zhao et al. [
58] develop two-stage stochastic programming formulations for the SND problem with stochastic demand. The services are selected in the first stage while demand is unknown. The demand materialises in the second stage, which is where routing decisions are made. Such an approach appears as “recourse” in
Table 3. If insufficient capacity is available in the second stage, the excess demand can be outsourced at a higher cost.
Bai et al. [
55] extend the study of Lium et al. [
51] by considering rerouting to mitigate the costs of stochastic demand. Similarly to Lium et al. [
51] and Hoff et al. [
52], a scenario tree is used to account for stochasticity. Hard time constraints are kept and it is assumed that excess demand can be outsourced to an external network. In the first stage, the sum of the network setup costs and expected additional costs due to uncertainty are minimised. In the second stage, the optimal flow distribution between the network obtained in the first stage and the external network is generated. The vehicles are also switched between arcs in the model with rerouting. The second stage takes place once the disruption becomes known. As such, the study combines tactical with operational decisions. The stochastic SND model with rerouting is compared against the deterministic and stochastic models in Lium et al. [
51]. The results indicate that including rerouting leads to larger cost reductions for correlated demand and requires fewer orders to be outsourced compared to the other models. A drawback of the proposed model is the high computational time. For large instances, the stochastic model without rerouting finds better solutions faster. Both stochastic models lead to the highest savings over the deterministic model when demand is correlated and highly uncertain. The total costs are lowest for all models when demand is clustered in space and time.
The two-stage model proposed by Zhao et al. [
58] performs scheduling decisions in the first stage and routing decisions in the second stage. The model accounts for both stochastic demand and transit times. It is applied on a realistic network with 17 railway stations and three ports, of which two are rail–ship transfer hubs. The results indicate an increase in costs and decrease in punctuality as transit time variability increases.
Zhao et al. [
48] study sea–rail intermodal container routing with stochastic travel and transfer times. The objective is to minimise total costs, which are composed of: (1) transport costs which depend on the mode and container quantity; (2) transfer costs for loading and unloading containers as well as changing modes; (3) inventory costs for containers waiting to be picked up in transfer hubs or at destination ports; (4) late delivery penalty costs which are proportional to the delay time; and (5) nonfulfilment penalty costs if delays lead to missed transfers. Lower limits on the probabilities of late arrivals at transfer hubs and deliveries are imposed, making it a chance-constrained stochastic problem. The model is applied on a realistically sized network between China and Singapore. The authors find that higher variability leads to higher total costs and more late deliveries. Compared to the deterministic case, routes are chosen with larger time buffers between mode changes to prevent missed transfers. The results of a sensitivity analysis indicate that it is always beneficial to have high on-time arrival probabilities at transfer hubs so as not to miss the next mode. In contrast, higher limits for on-time delivery probabilities lead to higher costs as variability increases, because the costs required to maintain the same service level outweigh penalty costs.
Meng et al. [
56] investigate the amount of train, truck and barge capacity to procure under stochastic demand. The study is performed in the context of car manufacturers which need to ship cars between factories or to storage centres. Train and barge services have a limited capacity and fixed weekly schedules, whereas an unlimited number of trucks is available. The probability function of the demand is known and at the start of each week the demand becomes known. The amounts of train and barge capacity to book are determined for each week in the first stage. In the second stage, it is assumed that the demand of the whole week is known and cars are assigned to the services that were selected in the first stage. If the booked capacity is insufficient, the excess cars are shipped by truck. A case study is performed on a realistic Chinese network with 19 nodes, 17 train routes and 8 barge routes. The results indicate that only accounting for the expected value of demand leads to higher costs.
In their literature review on intermodal transport, Macharis and Bontekoning [
23] conclude that the research focused on multiple decision makers is very limited, even though coordination is required for large networks. One such study from Puettmann and Stadtler [
54] presents a cooperation scheme between an intermodal operator and two carriers, one for first-mile and one for last-mile drayage operations. Without cooperation, the different parties have no knowledge of the other parties’ capacities, costs and existing orders. In the studied cooperation method the, cost proposals are iteratively exchanged between the carriers and the intermodal operator. Due to long travel times for the long-haul, it is assumed that demand at the destination is not known yet at the departure time. The study estimates the effects of this demand uncertainty on cooperation. Performance is evaluated against the optimal situation, which is characterised by a fully centralised decision maker. With the proposed cooperation scheme, optimality gaps between 4% and 7% are obtained. This is a reduction of 65% to 75% compared to the optimality gaps of up to 20% in scenarios without cooperation. Compared to a deterministic setting, cooperative savings are only slightly lower with uncertain demand and decrease as uncertainty increases.
Differently from previous tactical studies with stochastic demand, Demir et al. [
19] and Hrušovský et al. [
57] include costs, total delivery times and emissions in the objective function. Both studies consider stochastic transit times, while Demir et al. [
19] is one of the few studies which combines stochastic demand and transit times. The authors consider a network with road, rail and inland waterway transport. Demir et al. [
19] experiment on a dataset with 10 terminals connected by 32 services. To account for stochasticity, the sample average approximation (SAA) approach is used to obtain a deterministic problem. The results indicate that rail and barge modes have a higher share in a deterministic setting compared to a stochastic one. This is caused by the stricter departure times of those modes as opposed to trucks that do not have fixed schedules. If a larger weight is given to delivery times in the objective function, the share of trucks also increases because they are faster.
Hrušovský et al. [
57] assume departure times of rail and barge services are fixed, but travel times vary. A simulation optimisation approach is proposed in which solutions obtained from deterministic optimisation are used as input for the simulation step. Uncertainty is only considered in the simulation step where the performance of the deterministic solutions are evaluated over multiple scenarios. Experiments are conducted on the same instances as in Demir et al. [
19]. The authors conclude that the SAA approach is much more sensitive to the instance size than the simulation–optimisation approach. Therefore, simulation–optimisation is better suited at solving large instances.
Similarly to Hrušovský et al. [
57], Layeb et al. [
59] apply a simulation–optimisation approach. However, both stochastic demand and travel times are included. The model is validated with a deterministic real-world case study from Demir et al. [
19] using road, rail and inland waterway transport. It is modelled with Arena software and solved with Optquest. The simulation–optimisation model reaches 90% on time and in full deliveries. As demand and travel time distributions are skewed with fat tails, lognormal distributions are used for demand and railway travel times. For travel times, it is assumed that congestion occurs with a given probability of 20%. In case of congestion, the average travel times are assumed to be 20% longer. The main finding is that only the mean and variance of demand and transit times are not sufficient to set up reliable schedules, especially when empirical data displays skewness. The results indicate that ignoring stochasticity will lead to significantly higher costs, even when demand variability is low. This differs from Bai et al. [
55], who found that costs are only slightly higher without accounting for stochasticity if demand variability is low.
Yang et al. [
46] study how to plan intermodal hub-and-spoke networks with cost and travel time uncertainty. They present a MIP model which optimises both expected costs and maximum travel time. The network is uncapacitated and includes road, rail and air transport for long-haul transport between hubs. First and last mile deliveries are always performed by trucks. It is assumed that all hubs are directly connected to each other and shipments must pass through at least one hub. The required decisions are to select services and route freight on those services. The results indicate a trade-off between cost and maximum transport time. As the maximum travel time increases, air and truck transport decrease in favour of the cheaper but slower rail transport.
Sun et al. [
60] examine the effect of rail capacity uncertainty on intermodal routing decisions. Trains follow a fixed schedule, whereas trucks are flexible and uncapacitated. Rail capacity constraints are modelled as fuzzy chance constraints, which are integrated in a MILP. A linear reformulation is performed on this MILP such that it can be solved with exact solution methods. This model is applied on a real large-scale Chinese rail–road intermodal network with 40 terminals and 118 arcs. The authors conclude that there is a trade-off between reliability and costs, because minimising costs results in infeasible decisions. The decisions are infeasible if the actual capacity is lower than what is required. All the test results lead to infeasible decisions when capacity uncertainty is not considered.
4.1.2. Network Flow Planning
Table 4 lists studies on network flow planning with the types of uncertainty and solution methods. Minimising costs during disruptions is the main objective of almost all studies on NFP. Li et al. [
61] perform a multiobjective optimisation of an intermodal routing problem with cost and transit time uncertainty while accounting for risks. They identify three major objectives in the previous literature, namely costs, time and reliability, where reliability is measured by lost and damaged goods. Only the first two objectives are considered in their study. For each of these objectives, weighted subobjectives are added to account for risks. For the cost objective, these are to minimise the mean and standard deviation of the total cost and maximise the probability that the total cost of each selected route is within an acceptable threshold. The included costs are transport costs, transfer costs, holding costs and drayage costs. The subobjectives related to time are to minimise the mean and standard deviation of delivery times and maximise the probability of arriving within specified time-windows for just-in-time deliveries. The included transport modes are truck, train and barge. Different priorities are assigned to the objectives with an Analytic Hierarchy Process. The results indicate that the inventory costs are low at less than 3.7%. Despite accounting for only 2.7% of the total delivery time, transfers make up around 22.5% of total costs. The authors conclude that the share of barge transport is highest when delivery time windows are long because of the lower cost and speed compared to other transport modes.
In addition to costs, Huang et al. [
49] also minimise delays in their multiobjective optimisation problem. It is assumed that several carriers are involved in intermodal transport and successively handle goods. When a link becomes disrupted, its capacity is set to zero. The duration of this disruption is then estimated. No adjustments are needed if the duration is lower than the tolerance of the next carrier. If readjustments are needed, routes with the smallest deviation are chosen. Deviations are measured as the weighted sum of proportional changes in time and cost compared to the original route, with weights chosen depending on preferences. The total costs are the sum of the transport and transfer costs. The network is modelled as a state space which is solved with a depth-first search strategy. The method is applied on a small theoretical network, but no comparison is given with a cost minimisation objective.
Meng et al. [
62] study liner ship fleet planning with stochastic demand. The objective is to determine the number and types of ships in the fleet and then assign them to routes such that profits are maximised. The routes must be decided under stochastic demand. It is possible to charter additional ships or charter out owned ships in case of overcapacity. The experiments are performed on eight ship routes operated by a global liner container shipping company and 36 ports. The results indicate that higher demand variability leads to higher costs.
Chen and Miller-Hooks [
12] and Miller-Hooks et al. [
14] consider disasters in an intermodal network, which are modelled by reducing arc capacities and increasing their transit time. On top of routing decisions, recovery actions can be used to mitigate the impact of disruptions. Miller-Hooks et al. [
14] consider a given budget to allocate between pre- and postdisaster actions to maximise resilience, whereas Chen and Miller-Hooks [
12] only consider postdisaster actions. Predisaster actions are performed before disasters occur and mitigate their impact. Examples include additional fire stations and retrofitting bridges to enhance their durability. Postdisaster actions are performed after a disaster. The objective is to maximise resilience, which the authors define as the fraction of demand that can be satisfied postdisaster. The authors conclude that postdisaster actions are more effective than predisaster actions and combining both is the most effective. Larger mitigations are obtained at the same cost with predisaster actions but low probabilities of individual scenarios render it inefficient to invest in them.
Li et al. [
26] study a dynamic intermodal network with both demand and transit time uncertainty. The scope only encompasses long-haul transport between deep-sea terminals and inland terminals. Rail, road, barge and deep-sea transport are considered. Barges and trains operate under fixed schedules and transfer times at terminals are included. A receding horizon control approach is used, which means optimisation is applied to determine which actions should be taken, but only the ones of the current time period are implemented. The optimisation is performed at each time period with estimates based on the latest information. The authors assume decisions are taken by a single decision maker with access to current vehicle locations at all times. The objective is to determine container flows on each outgoing link of each node at each time step such that total costs are minimised. These costs include transport costs, transfer costs, storage costs and penalty costs for unfulfilled demand at the end of the planning horizon. The receding horizon intermodal container flow control (RIFC) model is applied on an intermodal connection from Rotterdam to Venlo over a time period of 24 h with one hour steps. A longer prediction horizon leads to better solutions up to a certain point, after which solutions stabilise. This comes at the cost of higher computation time. Variations in demand have no significant impact on the average computation time of the RIFC model. Higher demand increases the proportion of freight shipped on trains and barges because of capacity limitations for trucks. As the optimisations are based on estimated data, a sensitivity analysis is performed on the proportion of erroneous predictions. This reveals that the RIFC model is very robust to prediction errors, although the assumption was made that predictions within a given horizon are completely accurate.
In a study by Uddin and Huynh [
17], the effects of disruptions on an intermodal network are investigated. This is done with a stochastic mixed-integer program in which total costs are minimised. Only the transfer costs at intermodal terminals, the transport costs and thepenalty costs for unsatisfied demand are considered. The model is applied on a small-scale theoretical network and on a large-scale real-life network in the US, which only includes freight transport on major highways and railroads. These networks consist of nodes, links and intermodal terminals where goods are transferred between the nodes. The types of examined disruption scenarios are: (1) link disruptions where several links have their capacity reduced by 50%; (2) node disruptions where all links connected to the disrupted node have capacity reductions of 80%; and (3) intermodal terminal disruptions with capacity reductions of 80%. For the theoretical network, node disruptions lead to the highest costs, closely followed by link disruptions. The total costs for terminal disruptions resulted in less than two thirds the cost of the other disruption types. This is explained by the network layout, which favours road transport, causing low terminal utilisation. In the large-scale network under link or node disruptions, most freight should be shipped by a combination of road and rail because of the lower rail costs and a robust network with sufficient excess capacity. With terminal disruptions, the majority of the freight is shipped by trucks at a slightly higher total cost.
Uddin and Huynh [
18] extend their previous model by including multiple commodities, which have different transport and transfer costs, and stochastic capacity. In the event of disruptions, the capacity uncertainty is increased. Tests are performed with different required confidence levels. The results indicate that more uncertainty and higher confidence levels lead to higher costs. The total costs are the highest in the scenario where all terminals are disrupted, in which case all freight must be shipped by trucks. In case of lower required confidence levels with link and node disruptions, most freight is shipped by intermodal rail transport. For high required confidence levels, most freight is shipped by trucks because they can more easily find another route. The authors conclude that trucks should be used if reliability must be maximised, whereas intermodal rail transport should be used to minimise costs.
Sun et al. [
47] include uncertain road travel times and rail capacity in their intermodal routing problem. The number of trucks is not restricted, but their transit times are uncertain because of traffic congestion. Trains do not suffer from congestion, but have a limited capacity. The included costs are the transport costs the, loading and unloading costs, the inventory costs for early deliveries, the penalty costs for delayed deliveries and the CO
2 emission costs. In contrast to Zhao et al. [
48], the loading and unloading times are omitted because they are low. Moreover, the CO
2 emission costs are considered and a second model is proposed with biobjective optimisation of total costs and emissions, in which different weights are assigned to the objectives. A fuzzy chance constraint is added such that the probability of insufficient rail capacity remains lower than a given upper limit. The model is applied on a real-life network with major Chinese cities. The conclusion regarding emission costs is that current values are too low to have an impact. Including emissions directly in the objective function is more effective and can lead to 0.94% higher costs for a decrease in emissions of 3.90%, after which costs start increasing at a much higher rate. As is the case in Zhao et al. [
48], higher reliability leads to higher costs.