Journal ArticleDOI

# A Combinatorial Benders' decomposition for the lock scheduling problem

01 Feb 2015-Computers & Operations Research (Pergamon)-Vol. 54, pp 117-128
TL;DR: The results indicate that the decomposition approach significantly outperforms other exact approaches presented in the literature, in terms of solution quality and computation time.
Abstract: The Lock Scheduling Problem (LSP) is a combinatorial optimization problem that represents a real challenge for many harbours and waterway operators. The LSP consists of three strongly interconnected subproblems: scheduling lockages, assigning ships to chambers, and positioning the ships inside the chambers. These should be interpreted respectively as a scheduling, an assignment, and a packing problem. By combining the first two problems into a master problem and using the packing problem as a subproblem, a decomposition is achieved that can be solved efficiently by a Combinatorial Benders' approach. The master problem is solved first, thereby sequencing the ships into a number of lockages. Next, for each lockage, a packing subproblem is checked for feasibility, possibly returning a number of combinatorial inequalities (cuts) to the master problem. The result is an exact approach to the LSP. Experiments are conducted on a set of instances that were generated in correspondence with real world data. The results indicate that the decomposition approach significantly outperforms other exact approaches presented in the literature, in terms of solution quality and computation time.

### 1. Introduction

• It is a major hub for both inland and intercontinental cargo traffic.
• These locks entail a complex optimization problem: the vast number of ships entering and leaving the harbour every day have to be assigned to lock chambers, their exact position inside the locks need to be determined, and the lockages have to be scheduled.
• Experiments are conducted on a large number of instances based on data obtained from different Belgian locks.

### 2. Problem Outline

• The Lock Scheduling Problem consists of three interconnected sub problems: an assignment, a packing and a scheduling problem.
• Each problem comes with a large number of constraints, mainly resulting from safety and nautical regulations.
• The problem has been described in detail by Verstichel et al. (2014a); here the authors only sketch the general outline.
• The main lock scheduling specific terms used throughout this paper are elucidated in Figure 1, which shows a lock with two identical parallel chambers.

### 2.1. Scheduling, Assignment and Packing

• A lock consists of one or more chambers, which can perform lockage operations independently of each other.
• The ships must be grouped into a number of batches (i.e. lockages), where each batch contains the ships that are transferred in a single lockage operation.
• A chamber is always in one of it’s two possible states, which handle either downstream or upstream transfer.
• A first-come-firstserved (FCFS) policy is often enforced by the lock authorities.
• Therefore ship i must depart from the lock no later than ship j should ship i arrive at the lock before ship j.

### 3. Literature review

• LSP was first introduced in an inland setting by Verstichel and Vanden Berghe (2009), who presented a heuristic approach.
• Due to the complex nature of LSP, only relatively small instances were solved to optimality.
• In more recent work, e.g. Geoffrion (1972) and Hooker and Ottoson (2003), the Benders’ decomposition approach has been generalized to a broader class of problems, no longer requiring the sub problem to be linear.
• Stronger Combinatorial cuts may be obtained by identifying small subsets of variables responsible for the infeasibility of the sub problem, and expressing cuts in terms of these variables.
• Tran and Beck (2012) solve a Parallel Machine Scheduling Problem (PMSP) with machine and sequence dependent setup times through Logic Based Benders’ decomposition.

### 4. A Combinatorial Benders’ Decomposition

• Verstichel et al. (2014a) attempted to solve the LSP via a single, large, Mixed Integer Linear Programming problem.
• In addition, efficient dedicated algorithms can be employed to solve the master and sub problem, whereas there may not exist an algorithm capable of tackling the entire problem at once.
• The presented method is based on Codato and Fischetti (2006)’s original algorithm.
• Subsequently, the sub problem verifies, for each subset, whether the ships in this set can be transferred simultaneously, i.e. whether they fit to- gether inside the lock chamber.
• The LSP is solved whenever an optimal MP schedule is determined in which each subset satisfies the packing constraints of the sub problem.

### 4.1. Master problem

• The following Mixed Integer Linear Programming problem defines the master problem.
• The parameters are defined in Table 1; the variables (marked in bold) are discussed below the model.
• Other values would favor, for example, the number of lockages, maximum waiting time, etc. Constraints (2)-(4) assign ships to lockage operations.
• Constraints (9), (10) describe the actual scheduling restrictions on the lockages per chamber.
• A lockage cannot commence before all ships have arrived at the lock (Constraints (11)).

### 4.2. Sub problem

• Once the master problem has assigned the ships to a number of lockages, the feasibility of these lockages needs to be verified.
• Whenever a configuration is considered infeasible, a combinatorial cut will be generated and added to the master problem.
• The latter will be elaborated in the next section.
• Nk are located within the chamber’s dimensions.
• The remaining constraints ensure that the ships do not overlap.

### 4.3. Combinatorial Benders’ cuts

• When an infeasible sub problem is encountered, one or more combinatorial Benders’ cuts are generated and added to the master problem, effectively preventing the master problem from assigning specific ships to the same lockage.
• The strongest cuts are based on minimum infeasible subsets (MIS).
• It would require solving the sub problem from Section 4.2 for every possible subset of N ′. Section 4.4 presents approaches to computing strong cuts requiring far less computational effort.
• Combinatorial Benders’ cuts can be generated at different times in the process: Initial cuts are added to strengthen the MP before the first MPSP iteration.
• Applying initial cuts reduces the number of infeasible MP lockages generated.

### 4.4. Cut separation

• The different cut separation methods are clarified using the example from Figure 2, where the MP proposes a solution in which ships 1 through 7 are assigned to a single lockage.
• The feasible lockages for this example (under a first-come-first-served policy) are displayed on the right side of this figure.
• For the example from Figure 2, the following weak cut is generated: 7∑ i=1 fik ≤ 6, ∀k (21) Minimal infeasible subsets (MIS) can be found by applying the following constructive procedure.
• An efficient approach to identifying small infeasible subsets of ships is based on surface calculations: any set of ships having a combined surface that exceeds the total surface of the lock chamber is infeasible.
• Whenever surface calculations are used to identify infeasible subsets, it will be denoted as follows: ‘subsurf’ (subset based) and ‘surf’ (order based).

### 5. Experiments

• To assess the quality of the combinatorial Bender’s approach, a number of experiments have been conducted on instances based on real-world data originating from the Albertkanaal in Belgium (Verstichel, 2013).
• The characteristics of the locks, also known as 2. lock data.
• The ship inter arrival times have been selected from a uniform distribution between 0 and 2σ (Table 2).
• A comparison of the Benders’ procedure and the monolithic approach from Verstichel et al. (2014a) is presented.

### 5.1. FCFS single chamber lock

• The first series of experiments is performed on a single chamber lock with a first-come-first served policy for the ships.
• The x-axis of each figure displays the different instances, which are ordered, from left to right, based on (1) increasing number of ships (2) increasing inter arrival time and (3) traffic ratio (first 70/30, then 50/50).
• A slightly larger variation in the number of required iterations and cuts for equal instance size can be observed compared to the small chamber settings.
• The difference between generated cuts for the initial order cuts and the order feasibility cuts is also more significant.
• Finally, for the 48 instances consisting of 70 to 90 ships, the decomposition method solves 21 instances to optimality while for the remaining instances, feasible solutions were found.

### 5.2. No FCFS single chamber lock

• The second series of experiments is conducted on the same instances, but without the FCFS policy.
• Figures 6 (c) and (d) compare the performance of various methods.
• For the small chamber lock, initial subset cuts appear to be the best approach.
• The subship+order cut method on the other hand needed 27 cuts for the same instance, while computing only 12 seconds.
• From the above results, it is apparent that the absence of the FCFS rule has a significant impact on the computation times.

### 5.3. FCFS parallel identical chamber lock

• For the identical parallel chamber instances, the results for instances with 10 and ≥ 30 ships were omitted.
• The difference between the initial cuts is limited for a lock with two small parallel chambers ).
• The decomposition method is almost always faster for the other instances, with a total computation time of 6.5 for the monolithic approach and only 2 hours for the feasibility subset cuts.
• Applying one of the other methods (no initial cuts, subset based surface initial cuts or subset based ship placement initial cuts) results in an average of 570− 1007 cuts per instace.
• While the monolithic approach is the best choice when facing large inter arrival times.

### 5.4. FCFS multi chamber type lock

• The results for the multi chamber type lock are summarized in Figure 9.
• Here only the ≥ 30 ship instances were omitted.
• Similar to the SSC results it appears that, aside from the number of ships, the ship inter arrival time has the largest influence on the required computation time.
• Applying feasibility subset cuts combined with initial subset based surface cuts (subship+subsurf) appears to be the best way of solving LSP for this multi chamber type setting.
• This result is noteworthy, as the subship+subsurf method produces significantly more cuts than the other approaches ).

### 5.5. Heuristic sub problem approach

• The last experiment considers the effects of applying the multi-order bestfit heuristic for the ship placement problem (Verstichel et al., 2014b) to the SP.
• For the larger cases the heuristic approach matched the exact results on all but one instance, where a gap of 0.03% remained after 12 hours of computation time.
• For the single large chamber lock, the average gap for the heuristic approach on the instances with less than 60 ships was 0.10%, with a maximum of 1.19%.
• By increasing the computation time limit for these 12 instances, the authors were able to determine that in 3 cases the exact results could be matched by the heuristic approach.
• All PSC instances were solved to optimality by the heuristic decomposition method when applying initial subset cuts.

### 5.6. Summary of the experiments

• The experimental results show that the proposed Combinatorial Benders’ decomposition is very effective for the lock scheduling problem.
• This is especially the case for instances with a complex packing aspect, i.e. where several ships can be transferred in a single lockage operation, and (very) large instances.
• Indeed, for the single large chamber experiments, the authors find the largest decrease in computation time for the instances with short inter arrival times, with computation time differences of several orders of magnitude on several instances.
• Furthermore, the decomposition approach was able to produce feasible solutions, and often attest optimality, on a large number of instances that could not be tackled by the existing ‘monolithic’ approach.
• The same advantages are seen when applying a heuristic method to the sub problem.

### 6. Conclusion

• An exact Combinatorial Benders’ decomposition to the lock scheduling problem was proposed.
• The scheduling and assignment problems, whereas the packing problem is dealt with in the sub problem.
• Especially instances having ships that can be transferred simultaneously in a single lockage operation benefit from the new approach, as is shown in the experiments where the ship inter arrival times are short.
• Finally, applying a heuristic to the sub problem instead of an exact algorithm results in high quality solutions on all instances, with a maximal optimality gap of 2.93%, while the maximal time spent in the sub problem is reduced from 527 to 0.5 seconds.

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##### References
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TL;DR: In this paper, the extremal value of the linear program as a function of the parameterizing vector and the set of values of the parametric vector for which the program is feasible were derived using linear programming duality theory.
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Abstract: Paper presented to the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961.

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