Scheduling of waterways with tide and passing box

Due to the increasing amount of goods transported by vessels and the resulting increased size of the vessels, waterway scheduling becomes a challenging task. Waterways can often only be expanded with enormous costs and environmental damage. Therefore, this paper investigates a scheduling problem on a restricted waterway. Wide vessels are only allowed to pass in a passing box and vessels with deep draught can only pass the waterway in a time window around high tide. We present a mixed‐integer program (MIP) for the problem setting and develop techniques which allow us to fix variables and reduce the number of variables and constraints of the model. The resulting model formulations are evaluated in a comprehensive computational study on a real‐world setting at the river Elbe next to Hamburg (Germany).


INTRODUCTION
In March 2021, the Ever Given, a container vessel with a length of 400 m (Vesselfinder, 2021), blocked the Suez Canal for 6 days. The Suez Canal is passed through by over 19000 ships a year, which represents roughly 13% of world's trade such that the blockade resulted in costs of several billions of dollars (Allianz, 2021). The blockade showed that due to the enormous amount of goods transported by vessels and accordingly their increasing number and size, waterway scheduling becomes a bottleneck problem. This holds especially for highly frequented canals like the Suez Canal, the Kiel Canal (Meisel & Fagerholt, 2019), the Strait of Istanbul (Uluscu et al., 2009), or the Panama Canal (Canal de Panamá, 2020) and for waterways which are used as accesses to ports. Expanding a waterway, however, leads to massive costs and usually to environmental damage. Therefore, efficient waterway scheduling is important.
One way to solve the trade-off between economic interests and environmental damage is the implementation of passing boxes. A passing box is a segment within the waterway which is wider than the other segments of the waterway. Within the passing box accommodating vessels can pass each other which are in total too wide to pass outside the passing box. A well-known example for a waterway with passing boxes is the Kiel Canal (Lübbecke et al., 2019). However, there are further examples of real world applications like the Göta River in Sweden (Holm & Grundevik, 2013).
Moreover, the access to waterways can be restricted due to tide such that deep vessels can access the waterway only around high tide. An example for a tide dependent waterway is the river Elbe, which is also the access to the Port of Hamburg, where around 9 million containers are handled per year (Port of Hamburg, 2020a). The way between Elbe delta and Port of Hamburg is 107 km long (Hamburg Tourism, 2020). Recently, a passing box has been implemented in between (Gill, 2020). In this paper, we analyze a tide dependent waterway with a passing box like the river Elbe is. We develop a mixed-integer programming formulation for the problem setting and introduce techniques which allow us to fix variables and reduce the number of variables and constraints in the model formulation.
The paper is structured as follows: First, we review the related literature in Section 2. Then, we give a detailed problem description (Section 3). Afterwards, our mixedinteger programming approach to solve the considered waterway scheduling problem is presented (Section 4). The model is further improved by techniques explained in Section 5. In the following Section 6, we test our approach in a comprehensive computational study. Finally, the paper closes with a conclusion in Section 7.

LITERATURE REVIEW
In this section, we review the related waterway scheduling literature. While the minimization of vessels' waiting times is common practice (cf. Hill et al., 2019;Lalla-Ruiz et al., 2018;Lübbecke et al., 2019;B. Zhang et al., 2020;J. Zhang et al., 2017;X. Zhang et al., 2019), different restrictions regarding depth and width are assumed in the literature. Therefore, our literature review focuses on these two aspects.
Causes for variable depths of waterways are tides, which influence the possible draught of a vessel on a waterway. Large vessels with deep draughts may experience delays in entering or leaving tidal ports, such as Antwerp, Hamburg or Shanghai, as they are only allowed to pass canals or rivers within certain time windows (B. Zhang et al., 2020). Moreover, an oversized width leads to conflicts of accommodating vessels because they are not allowed to pass each other due to the width restrictions of the waterway. Lalla-Ruiz et al. (2018) and Hill et al. (2019) presented the waterway ship scheduling problem (WSSP) that takes-inspired by the Yangtze river's estuary-restrictions of a constant width and a time-dependent depth of waterways into account. The WSSP considers several waterways each with an available time window and discrete time periods. Tide is modeled by a time-varying depth of the waterway. Hill et al. (2019) used a multimode resource-constrained project scheduling reformulation to solve the WSSP. Kelareva et al. (2012) took also a time-dependent depth with discrete time periods into account in their models for planning vessels in ports. Du et al. (2015) and Ernst et al. (2017) considered the tide in their modeling of the berth allocation problem. Zhen et al. (2017) also investigated the berth allocation problem. They took the tide into account by determining feasible berthing and departing time windows for tide-dependent vessels. In addition, the model comprises restrictions on canal flow control. These include locks, limited dimensions of canals or rivers, and bottleneck resources such as a limited number of pilot vessels. B. Zhang et al. (2020) presented a model and an algorithm for vessel scheduling through a two-way tidal canal. In their approach, the tide is modeled individual for all positions of the waterway. It is evaluated according to real data from Yangtze river.
In addition to the limited depth of a waterway, the width is also a significant restriction. In the case of two-way waterways frequented by large vessels, bottlenecks and long waiting times can occur, as the vessels are not allowed to pass each other on the entire waterway but only in certain areas. Lübbecke et al. (2019) developed a ship scheduling model for the Kiel Canal with constant depth and variable width. The width is variable because the Kiel Canal has passing boxes, where wide ships can meet. The problem has also been considered in the dissertation by Lübbecke (2015). Meisel and Fagerholt (2019) and Andersen et al. (2021) considered the Kiel Canal as well. Meisel and Fagerholt (2019) included variable ship speeds, capacities of passing boxes, and a maximum waiting time for ships. Andersen et al. (2021) assumed uncertain arrival times for the ships. They introduced time-corridors such that the schedule is valid as long as the ships arrive within their corresponding time-corridor. However, the Kiel Canal is not tide-dependent because it has locks at both sides. Mavrakis and Kontinakis (2008) presented a queueing model for the maritime traffic in the Bosporus Strait. Uluscu et al. (2009) also investigated the Bosporus Strait regarding transit vessel scheduling. The basic idea of their planning algorithm is to give priority to vessels with the highest waiting times and large vessels carrying dangerous goods. The primary objective of the work is a maritime risk analysis. Holm and Grundevik (2013) presented the project GOTRIS (Göta älv River information system), which aims to optimize the routes for ships on the Göta River taking into account bridges, lock openings, and ship meeting points. Disser et al. (2015) investigated the problem of scheduling bidirectional traffic along a path composed of multiple segments in general.
In addition to waterway scheduling problems, there are other maritime planning problems with the objective of minimizing waiting times of vessels. These include the planning of the passage through locks (Campbell et al., 2007;Passchyn et al., 2016) and the berth allocation problem (BAP) (Buhrkal et al., 2011;Ernst et al., 2017;Xu et al., 2012). Fatemi-Anaraki et al. (2020) solved the waterway scheduling simultaneously with berth allocation and quay crane assignment. They modeled the combined problem on the basis of a hybrid flow shop scheduling problem. They also integrated tidal impacts. Muñuzuri et al. (2018) considered the 90 km long Guadalquivir waterway which connects Seville Port with the Atlantic Ocean. Guadalquivir waterway is a restricted tidal two-way waterway with varying width and depth. Critical points regarding the width and depth of the waterway are identified and determined. One assumption is that tide-dependent vessels can anchor in certain areas of the waterway to wait for the next high tide. In addition, speed variations are allowed to avoid meeting or overtaking of vessels at critical points. The authors presented a heuristic solution approach to optimize a multi-criteria objective function including the number of served vessels, the total waiting time, the waterway Like Muñuzuri et al. (2018), but in contrast to the other presented papers, our paper focuses on vessel traffic planning on restricted waterways with tide and an area where accommodating vessels can pass each other. We provide an exact solution approach to minimize the total waiting time in a setting with tide and a single passing box.

PROBLEM DESCRIPTION
We consider the static problem of scheduling vessels in a waterway with restricted width b max 1 and tide-dependent depth (d max is the depth at low tide). We assume that all vessels can pass the waterway at high tide. For each vessel i ∈ I the width b i , the length l i , and the depth d i are given. Moreover, we know for each vessel its driving direction (left to right or vice versa). If its depth exceeds d max , the amount of time before and after high tide it can use the waterway is denoted by g i . The first high tide is at time h while all successive high tides occur n time units after the previous one. All vessels pass the waterway with a unitary speed s. A maximum speed is often given due to safety reasons and resulting waves, as too high waves may destroy the shore. A vessel follows its precursor with a safety distance of at least Δ d time units. Vessels can have a length of around 400 m which means that bow and tail enter the waterway with a certain time lag. We measure all times at the middle of the vessel and assume that Δ d is large enough such that each pair of vessels holds the safety distance between the tail of the first and the bow of the second vessel. Vessels are not allowed to overtake another one inside the waterway.
The waterway has a passing box with width b max 2 and length l box such that vessels of an accumulated length of l box (including safety distances) can occupy it at the same time if they drive in the same direction. Since the vessels wait at the end of the passing box for accommodating vessels, we assume a shorter safety distance of Δ d box distance units if vessels wait inside the box. In the passing box, vessels can pass each other if their cumulated width is too large to pass outside the box. However, there might be pairs of vessels which cannot pass each other even in the box. These vessels are not allowed to meet each other anywhere in the waterway. Figure 1 shows the problem under consideration. We consider the part with continuous lines in the middle. We assume that there are no restrictions regarding width and depth outside this area such that each vessel can pass each other vessel there. Our modeled area consists of three parts-two parts of normal width and a passing box with an extended width somewhere in between. Vessels drive on the right side of the waterway according to their driving direction. Vessels A and B are too wide to pass each other outside the passing box. Therefore, vessel B is waiting inside the passing box until vessel A passes. Vessels A and C or D, however, can pass outside the passing box. Tides are not pictured in the figure. Nevertheless, the water level is time dependent.
Note that waiting within the waterway but outside the passing box is not useful. Assuming that a vessel i would wait in the first segment before the passing box, it could enter the waterway later instead such that it reaches the passing box at the same point in time without waiting. Then, accommodating vessels are more flexible, as vessel i utilizes the small part of the waterway for a shorter time. Following vessels in the same direction could also postpone their entry in the waterway if necessary without being later at the passing box because they could still drive with the same gap to vessel i which does not reach the passing box later.
Although our problem setting does not explicitly include a time horizon, we need an endpoint for our model formulation.

MIXED-INTEGER PROGRAM
We present our mixed-integer program for the problem setting in this section. Section 4.1 presents the model. In Section 4.2, our choices for the Big Ms are discussed.

Model formulation
The notation is described in Table 1.
capacity passing box departure time out of the passing box meeting of wide ships only in the passing box variable ranges Objective function (1) minimizes the sum of the waiting times of all vessels. Waiting times are computed in constraints (2) as the difference between the time when the vessel leaves the passing box and its driving time into (e i ⋅ Δ 1 + (1 − e i ) ⋅ Δ 2 ) and through the passing box (Δ box ) and its arrival time at the waterway (a i ). Thanks to constraints (3) no vessel is allowed to enter the waterway before its arrival. Constraints (4) and (5) determine the unique (cf. (5)) sequence of the ingoing vessels. The next block of constraints, (6)-(10), considers vessels which drive in the same direction through the waterway. First of all, there is a safety distance between two consecutive vessels which must be fulfilled before and inside the passing box (constraints (6)) and after leaving it (constraints (7)). The box itself has a limited length l box . Hence, we must ensure that the capacity restriction in constraints (9) is always fulfilled. For this, we have to know which vessels are simultaneously in the passing box and drive in the same direction as vessel i. The corresponding variable y ij is determined in constraints (8). Variable y ij equals 1 if the departure time of vessel i out of the passing box, t box i , is later than the arrival time of vessel ⋅ Δ 2 , in the passing box if i enters the waterway before j (z ji = 0). The vessel requires Δ 1 time units for the first segment and Δ 2 time units for the second while e j = 1 if the vessel starts with the first segment and e j = 0 if it drives the other way round through the waterway. Vessels are not allowed to wait within the segments. However, they can wait at the end of the passing box. Thus, the arrival time at the end of the passing box, t i + e i ⋅ Δ 1 + (1 − e i ) ⋅ Δ 2 + Δ box , is less or equal to the departure time out of the passing box, t box i , (constraints (10)). Vessels driving in opposite directions may meet somewhere in the waterway. If they do, x ij = 1 due to constraints (11) and (12). Constraints (11) ensure that x ij = 1 if vessel i enters the waterway before vessel j but has not left it (leaving time: . It is sufficient to define x ij for i < j. Therefore, constraints (12) consider the case where i > j.
If vessels i and j meet (x ij = 1) but are in total too wide to meet outside the passing box, both vessels are not allowed to leave the passing box (at time t box j ) before the other one entered it (at time t i + e i ⋅ Δ 1 + (1 − e i ) ⋅ Δ 2 ). Constraints (13) and (14) take care for this. However, two vessels i, j with b i + b j > b max 2 are not allowed to meet anywhere in the waterway at all (x ij is fixed to zero; cf. description of x ij ).
Some vessels can only use the waterway at high tide. Constraints (15) and (16) set time windows for them. Variable f i decides for vessel i with a sufficiently large draught which high tide it uses. Then, constraints (15) ensure that the vessel does not enter the waterway before the water is deep enough. Constraints (16) take care that the vessel leaves the waterway before the water level is too flat again.
Finally, constraints (17)-(21) restrict the variable domains. Note that x ii , y ii , z ii are not defined for all i ∈ I. Moreover, x ij is only defined if e i ≠ e j and y ij only if e i = e j . Thus, effectively less than 2|I| 2 discrete variables have to be fixed in the branch-and-bound procedure. Nevertheless, off-the-shelf solvers may struggle for larger instances with the quantity of binary and integer variables as well as with the Big Ms in constraints (4), (6)-(8), (11)-(14) (cf. Section 5.1 and Disser et al. (2015) relating to the NP-hardness of the problem setting). Therefore, we search for small Big Ms in the following subsection. Afterwards, we present techniques to reduce the number of binary variables in Section 5.

Choice of Big Ms
We need an endpoint T of the considered time horizon for our Big Ms. If for example the vessels of 1 day are scheduled, T would be 1440 (minutes per day). We set The Big M is necessary if vessel j enters the waterway after vessel i, that is, if t j ≥ t i . Vessel j has to pass the whole waterway between t j and T.
⋅ Δ 2 has to hold. As already shown is t box

IMPROVEMENT OF THE MODEL FORMULATION
In this section, we improve the model formulation. First, some variables can be fixed, which is investigated in Section 5.1. Second, the number of considered variables (Section 5.2) and constraints (Section 5.3) can be reduced.

Fixation of variables
The following theorem shows that for a pair of vessels i and j with sufficiently similar sizes (length, width, and draught) which drive in the same direction it is optimal to schedule i before j if a i < a j . Thus, we can set z ij = 1.
Theorem 1 An optimal solution with z ij = 1 for all i, j ∈ I ∶ e i = e j ∧ max , that is, the set of all vessels which drive in the same direction (e i = e j ), can pass the waterway at the same time , have the same conflicts with accommodating vessels , require the same space in the passing box (l i = l j ), and have a later arrival time at the waterway than vessel i.
Let (i 1 , i 2 , … , i n ) be the sequence of vessels according to their departure times into the waterway in an optimal solution, that is, vessel i 1 is the first vessel which enters the waterway, i 2 the second one, and so on (t i 1 ≤ t i 2 ≤ · · · ≤ t i n ). This means that z i j ,i k = 1 iff k > j. We can assume that there is at least one vessel i such that z ji = 1 for a vessel j ∈ I i . Otherwise, there is nothing to do. Because a i < a j , vessel i can also depart at the position of vessel j and vice versa, as both of them can interfere with the same accommodating vessels and can drive at the same time (tide) due to the choice of I i . Moreover, i and j have the same length. Thus, a swap does not violate the capacity of the passing box. Hence, swapping the departure times of vessels i and j does not increase the total waiting time (the reduction in the waiting time of vessel i equals the increase in the waiting time of vessel j) and must, therefore, be optimal.
We can repeat the aforementioned argumentation for each vessel i and all vessels j ∈ I i with z ji = 1. Thus, there is an optimal solution with z ij = 1 for all Theorem 1 leads to compatibility classes C with i ∈ C ⇒ i ∈ I j or j ∈ I i for all i ≠ j ∈ C. Lübbecke et al. (2019) group vessels into compatibility classes as well. Disser et al. (2015) mention further that vessels inside a compatibility class can be scheduled according to a first come first serve (FCFS) policy. Lübbecke (2015) speaks of conflicting pairs if a pair of accommodating vessels cannot pass each other on a segment.
In real-world applications it can be justified to assume that the length of the passing box is not restrictive. The passing box on the Elbe is, for example, 7.6 km long (Wasserstraßenund Schifffahrtsverwaltung des Bundes, 2020) while the longest vessels entering the Port of Hamburg are up to 400 m long (Port of Hamburg, 2020b). Thus, a couple of vessels can occupy the box simultaneously. If the assumption of a nonrestrictive passing box is justified, the problem complexity can be reduced (the knapsack problem is NP-hard; Garey & Johnson, 1979) and the vessel's length is not restrictive in Theorem 1. Thus, the following corollary follows immediately.

Corollary 1 If the length of the passing box is not restrictive, an optimal solution with z ij
Proof As the length of the passing box is not restrictive, the lengths of the vessels are not restrictive. Thus, the corollary follows immediately from Theorem 1 by assuming that all vessels have an identical length. ▪ Beside the length of the vessels, their draughts and widths are restrictive in Theorem 1. Thus, Theorem 1 implies, analogous to Corollary 1, that FCFS is optimal if the waterway is not tide dependent and wide enough that all accommodating vessels can pass each other on the whole waterway. Nevertheless, Disser et al. (2015) proved that switching between the two directions, that is the width, makes the problem already NP-hard.
Note further that-although we do not minimize the maximum waiting time of a vessel-Theorem 1 balances the waiting times among the vessels with similar sizes which drive in the same direction by using FCFS among them.

Reduction of the number of variables
Due to the introduced end of the time horizon T (Section 4.2), we also have a last high tide in the time horizon. Because of that, Moreover, we saw in Theorem 1 that accommodating vessels are independent of each other if their added width does not exceed the waterway's width b max 1 . This inside allows us to reduce the number of x and z variables in the model formulation. x variables must only be defined for pairs i, j if b max 1 < b i + b j , i < j, and e i ≠ e j . Thus, we replace (18) by With the same argumentation we define z ij only if e i = e j or if e i ≠ e j and b max Beside, we implement the findings of Section 5.1 and set z ij = 1 and z ji = 0 for all i, Independence of all accommodating vessels implies further that it can never be optimal to wait as the first vessel in the passing box. Hence, y ij is only defined for all

Reduction of the number of constraints
The aforementioned changes lead to some changes in the definition of the constraints. Therefore, we denote the lower bound of z ij by z l ij and the upper bound of z ij by z u ij . This means if z ij is already fixed to 0 or 1, z l ij = z u ij = 0 and z l ij = z u ij = 1, respectively. Otherwise, z l ij = 0 and z u ij = 1 if z ij is defined. Constraints (4) are useless if z l ij = 1. Keeping (23) in mind, we replace (4) by Constraints (5) are also only helpful if z l ij ≠ z u ij . Otherwise, we fix z ji in the preprocessing. Thus, we replace (5) by We need constraints (6) and (7) if z ji = 0 but not if z ji = 1. We replace them by It is sufficient to define (9) only for vessels which are not independent of accommodating vessels, as it cannot be optimal for all other vessels to wait at the front of the passing box. Thus, (9) is replaced by Finally, (11) and (12) are replaced by

COMPUTATIONAL STUDY
We evaluate our model formulation (2), (3), (10), (13)-(17), (21)-(32) with objective function (1) in this section. We refer to this model as reduced. Moreover, we set z ij = 1 and We compare the reduced model with a model formulation where we assume that the passing box is not restrictive. Therefore, we omit y variables and constraints (24), (29), and (30). Furthermore, we set z ij = 1 and z ji = 0 for all ) ∧ e i = e j (i.e., we do not require  l i = l j ; cf. Corollary 1). We refer to this model as relaxed because it might lead to infeasible solutions regarding the length of the passing box. In contrast to the relaxed model, a feasible solution returned by the reduced model is always feasible for the standard model (2)-(21) with objective function (1). Nevertheless, we also solve the standard model to evaluate the benefit of the reduced model formulation in the reduced model. The models were implemented in GAMS (version 32.2) and solved by CPLEX (version 12.10). The computational study was executed on a single AMD EPYC 7302 core with 2.99GHz. Section 6.1 describes the composition and Section 6.2 the results.

Composition
We varied the number of considered vessels |I| and the fraction of wide and deep vessels in the computational study, as the fraction of wide vessels is a measure for the complexity of the problem instance (cf. Section 5.1), the fraction of deep vessels indicates how tide-dependent the instance is, and a higher number of vessels makes the instance also more difficult.
We orient ourselves in the choice of our parameters at the river Elbe, which is the access to the Port of Hamburg. At the Elbe vessels with a combined width of 92 m (b max 1 ) can pass each other on the whole waterway while vessels with a combined width of up to 98 m (b max 2 ) can pass each other only in the passing box. Moreover, vessels with a draught of up to 13.5 m can pass the Elbe independent of the tide. At high tide vessels with a draught of up to 14.5 m can pass (Wasserstraßen-und Schifffahrtsverwaltung des Bundes, 2020). We set the widest width of all vessels to 50 m. Thus, we determined the width of the wide vessels uniformly between 42.1 m (an accommodating vessel with width 50 m can only be passed in the passing box) and 50 m. For the deep vessels the draught is determined uniformly between 13.6 and 14.5 m. As the time between two high tides is approximately 12 h (n = 720 min), the earliest point in time a deep vessel can enter the waterway is 359 min before high tide. The latest point in time it has to leave the waterway is accordingly 359 min after high tide. Therefore, we set +Δ 1 +Δ 2 +Δ box (because of 14.5 − 13.5 = 1, it is not necessary to divide d i − 13.5 by the range to obtain the fraction). Width and draught of all smaller vessels are not relevant for the model. Therefore, we set d i = 10 and b i = 35 for the remaining vessels. Since the longest vessels entering the Port of Hamburg are 400 m long (Port of Hamburg, 2020b), we choose the length uniformly between 100 and 400 m. Arrival times a i , i ∈ I, are generated uniformly in a time horizon of 1 day (1440 min). The vessel's direction e i , i ∈ I, is also determined uniformly.
All remaining parameters are not instance specific and are fixed for the whole computational study. The first high tide is assumed to be after 360 min, that is, a deep vessel can use the first high tide if it arrives at time 1. The allowed speed on the Elbe varies in dependency of time and tide. Therefore, we choose our constant speed as an average value of 22 km per hour, that is, around 12 knots. Thus, s = 22∕60. We assumed the length of the waterway to be 62 km like the part of the Elbe next to the Port of Hamburg which is currently extended. The Elbe is wider on the remaining segment till its estuary. The width is not restricting there which is the reason why we excluded this segment. The passing box is around 10 km before the Port of Hamburg and is 7.6 km long (Wasserstraßen-und Schifffahrtsverwaltung des Bundes, 2020). Thus, we set l 1 = 44, l box = 7.6, and l 2 = 10. We assumed a minimum time interval of 5 min between two vessels outside the passing box (Δ d = 5) and a safety distance for  waiting vessels inside the box of 200 m (Δ d box = 0.2). Finally, the endpoint of our time horizon is T = 2⋅1440 = 2880 which means that four high tides are within the planning horizon.
We considered the following numbers of vessels: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 120, 150, and 200. The fractions of wide and deep vessels were set for each number of vessels to 0, 0.1, 0.2, 0.3, and 0.5. This means that there is no wide or deep vessel in the case with a fraction of 0 and each vessel has a probability of 10% to be wide and deep, respectively, in the case with a fraction of 0.1. Wide and deep vessels are independent of each other. Thus, a wide vessel has the same probability to be a deep vessel like a vessel of normal width. For each of these settings we ran 30 independently generated instances and set the time limit to 3600 s (1 h).

Results
Feasibility of the relaxed model can be checked for each vessel i by adding up the lengths (including the safety distance Δ d box ) of all vessels j which drive in the same direction like i, enter the waterway after i and reach the passing box before t box i . If this sum added to l i is at the most l box for all vessels i, the solution is feasible. More formally a solution of relaxed is feasible regarding the passing box restriction (9) if An optimal solution of relaxed which fulfills (33), is optimal for standard (because only passing box requirements are relaxed). All optimal solutions found by relaxed were feasible regarding (33) in our computational study. Thus, they are also optimal for standard. Table 2 presents the results in detail. The first two columns denote the parameter setting (number of vessels and share).
The share is always identical for width and draught. The next five columns present the results for relaxed (average time required, average objective value, average gap of all instances for which we obtained a solution within 1 h of computation time, number of optimal solutions, and number of instances where a feasible solution was found). The same classification numbers are shown for reduced in columns 8-12 and for standard in columns 13-17.
For a share of 0 relaxed finds all optimal solutions within a second on average, as all binary and integer variables are already fixed in the preprocessing. However, reduced and standard were not able to prove optimality for some instances with 200 vessels. This indicates that even the setting where both directions are independent from each other and which is tide-independent is challenging although a simple FCFS policy among all vessels which drive in the same direction is optimal. Nevertheless, the average objective values are identical for all three models. Thus, reduced and standard found always an optimal solution.
For an increasing share average computation times and gaps increase as well for reduced as for standard while the number of optimally solved instances and feasible solutions found decrease. For a share of 0.5 even instances with 70 vessels are challenging. However, reduced leads over almost all settings to higher numbers of feasible and optimal solutions and smaller gaps in less computation time. From 120 vessels on no instance could be solved to optimality by the three models. The obtained objective values for relaxed are in the most cases (beside 80, 90, and 120 vessels and share 0.5) not larger and the number of optimally and feasibly solved instances is always (beside the number of feasibly solved instances for 200 vessels and share 0.3) at least as high and mostly higher (if less than 30) than for reduced. Table 3 presents the average number of variables and constraints for all three models and all settings. As relaxed has no y variables, the corresponding column is omitted. The table shows that our reduction techniques strongly reduce the model complexity in comparison with the standard formulation. In comparison between reduced and standard, the model size can already be reduced significantly without missing the chance to find an optimal solution. The comparison between relaxed and reduced discloses that a further reduction depends on the share of wide and deep vessels (smaller reductions for larger shares). However, if the length of the passing box is not expected to be restrictive, it is profitable to use the relaxed model formulation, since feasibility can easily be checked by (33).

CONCLUSION
The paper presents a mixed-integer program for a tide-dependent waterway scheduling problem with a passing box. Moreover, we introduced several techniques to strengthen the model formulation. First, Big M values were minimized (Section 4.2). Afterwards, we fixed variables (Section 5.1) and reduced the number of variables (Section 5.2) and constraints (Section 5.3). The different model formulations were evaluated in a comprehensive computational study concerning a real-world setting at the river Elbe next to Hamburg. Our results indicate that the passing box in our setting is not restrictive. However, there is no consequential implication for the river Elbe, as we used artificial test instances and have no information on anticipated instance characteristics. Nevertheless, it is important to evaluate how long a passing box should be, as a smaller passing box leads to less costs and could reduce the environmental damage.
The passing box capacity was not restrictive in our setting. If it is restrictive, the model formulation might be strengthened by cover inequalities (cf. Gu et al., 1999). A further direction for future research is to investigate the benefit of an increased number of passing boxes. Following the real-world application at the river Elbe our model contains only a single passing box. However, our results show that large instances are challenging even for the relaxed model formulation although the number of variables and constraints were substantially reduced. This gives a hint that good heuristics and meta-heuristics are important to solve tide-dependent waterway scheduling problems especially if several passing boxes are considered. Nonetheless, our tightened model formulation may help researchers in the design of heuristic solution approaches.

ACKNOWLEDGMENTS
Open access funding enabled and organized by Projekt DEAL.