Solving a dock assignment problem as a three-stage ... - CiteSeerX

Multidisciplinary International Conference on Scheduling : Theory and Applications (MISTA 2009) 10-12 August 2009, Dublin, Ireland

MISTA 2009

Solving a dock assignment problem as a three-stage flexible flow-shop problem Lotte Berghman · Roel Leus

1 Introduction Toyota is one of the world’s largest automobile manufacturers, selling over 7.5 million models (including Hino and Daihatsu) annually on all five continents, and generating almost 130 billion euro in net revenues. Since 1999, the total warehouse space floor of the European Distribution Centre TPCE (Toyota Parts Centre Europe) located in Diest (Belgium) has been expanded to 67.700 m2 . Toyota’s Distribution Centre delivers to 28 European distributors on a daily basis. At TPCE, the warehouse has some fifty gates, each with a capacity of one trailer, where goods can either be loaded on an empty trailer or be unloaded from a loaded trailer. Besides the warehouse with the gates, the site also has two parking lots, which can be seen as a buffer where trailers can be temporarily parked. All transportation activities of uncoupled trailers between these parking lots and the gates are done by two terminal tractors, which are tractors designed for use in ports, terminals and heavy industry. Obviously, if a trailer is coupled, there is no need for a terminal tractor since the trucker can transport the trailer. Because of the numerous loading, unloading and transportation activities, TPCE needs a schedule specifying the starting time and the assigned gate or terminal tractor for each activity. Each arriving trailer, for which the planned arrival time is known, is either dropped off by the trucker at a parking lot and afterwards transported to a gate by the terminal tractor, or it is delivered immediately at a gate. After unloading or loading at the gate, the trailer is either picked up by a trucker, or transported to a parking space by a tractor, where it will be picked up by a trucker later on. In practice, every good has a fixed place in the warehouse and is preferably loaded or unloaded on a gate near this location. The situation at TPCE can be modeled as a three-stage flexible flow shop. The first stage is the transportation from the parking lot to a specific gate, the second stage is the loading or unloading activity and the third stage is the movement back to a parking space. Two identical parallel machines execute both the first and the third stage. In these stages, the processing times are not dependent on the distance because the actual Lotte Berghman, Roel Leus ORSTAT, K.U.Leuven, Naamsestraat 69, 3000 Leuven, Belgium E-mail: {lotte.berghman,roel.leus}@econ.kuleuven.be

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driving time of the terminal tractor is small compared to the time it takes the driver to follow the safety instructions and attach the trailer to the tractor. In the first stage, the processing time is a non-zero constant when a task starts later than the ready time and in the third stage, the processing time is a non-zero constant when a task ends before the due date. Otherwise, the task is a dummy task with processing time equal to zero, because the tractor is not used. We will neglect the preference for certain gates in the remainder of this paper and model each gate as an identical machine; some fifty identical parallel machines execute the second stage. The processing time depends only on the job and is independent of the machine.

2 Parallel machines When we only look at the second stage and do not take the terminal tractors into P account, the problem at hand is close to Pm |rj , dj | wj Tj in the standard three-field notation, although some tasks have deadlines rather than due dates. We define T2 with |T2 | = n as the set of all unloading and loading activities. T2U , T2L is a partition of T2 : T2U is the set of unloading tasks, while T2L is the set of loading tasks. All tasks t ∈ T2 have a ready time rt . For the unloading tasks t ∈ T2U , this ready time equals the planned arrival time of the trailer. For the loading tasks t ∈ T2L , rt = 0 because we assume that all the goods to be loaded on the trailers are in the warehouse and that the empty trailer is on site. All tasks t ∈ T2 have a weight wt . All unloading tasks t ∈ T2U have a due date dt equal to their ready time because the aim is to avoid tardiness. All loading tasks t ∈ T2L have a deadline d¯t based on the estimated driving time to the customer and the agreed arrival time at the customer. The number of identical parallel machines in stage two is m < n. Each machine can process at most one job at a time. The processing time pt of a job t ∈ T2 is the time needed to load or unload the trailer at the gate. Our problem consists of scheduling the tasks in such a way that the total weighted lateness (or tardiness, since dt = rt ) of the unload operations minus the total weighted earliness of the load operations is minimized. We will use mathematical programming to solve this problem. The first formulation tested is based on Dessouky [3], although Dessouky’s objective is to minimize the maximum lateness. Tasks are assigned to positions at gates and additional sequencing variables are introduced to linearize the model. The job starting times constitute an extra set of decision variables. Our second formulation has much in common with the ones presented by Jain and Grossman [6] and by Sadykov and Wolsey [8], where the objective is to minimize the summed task-to-machine assignment costs. The main difference with the first formulation is that individual machine positions are no longer decided upon, but we directly establish a (transitive) predecessor relation between jobs on the same machine. The third model is based on Bard and Rojanasoonthon [1]; our setting is dissimilar from [1] in that the reference considers two time windows per task, setup times and priority classes. We let each task be the immediate successor of another task on the same machine and add dummy tasks as the first task on each machine. The final formulation is a time-indexed model based on Dyer and Wolsey [4], van den Akker et al. [10] and Bigras et al. [2]; contrary to the foregoing three continuoustime formulations, we now no longer need starting times as separate decision variables.

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3 Three-stage flexible flow-shop In the second part of this work, we introduce the terminal tractors as the resources that are shared by stages one and three. Each job is composed of three tasks, one for each stage, and each task has to be scheduled on exactly one machine in the appropriate stage. The facilities are disjunctive, in the sense that each machine may process only one task at a time. Task preemption is not allowed. We let T represent the set of all tasks; |T | = 3n. Set T is partitioned into T1 , T2 , T3 . The set T1 contains all transportation activities from the parking lot to a gate, T2 comprises all tasks on the gates (as in Section 2) and T3 are the movements from a gate to the parking lot. Each set Ti (i = 1, 2, 3) is further partitioned as TiU , TiL , with these two sets holding the unload and load tasks, respectively. All tasks t ∈ T1 have a ready time rt . For the unloading tasks t ∈ T1U , this ready time equals the planned arrival time of the trailer. For the loading tasks t ∈ T1L , rt = 0 because we assume that all the goods to be loaded are present in the warehouse and that the empty trailer is available. All tasks t ∈ T2U ∪T3L have a weight wt . All unloading tasks t ∈ T2U have a due date dt equal to the ready time of the corresponding task t ∈ T1U because the aim is to avoid tardiness. All loading tasks t ∈ T3L have a deadline d¯t based on the promised arrival time at the customer. All transportation activities between the parking lot and the gates have a constant duration C, independent of the distance. The remainder of the problem description, including the objective function, is similar to the one presented in Section 2. The mathematical formulations that are listed in Section 2 will be extended to incorporate the planning of stages one and three.

4 Implementation In our presentation, we will report extensive computational results on benchmark datasets, both for the parallel-machine case as well as for the three-stage flexible flowshop problem. We also provide some insights into the differences in computational efficiency between the formulations tested. A straightforward implementation of the mathematical-programming formulations allows the exact solution within reasonable CPU time of medium-sized scheduling instances only. In order to increase the number of tasks that can be scheduled, we investigate the use of Lagrangian relaxation. Lagrangian relaxation is an approach to handle computationally hard optimization problems, in which some sets of difficult constraints are dualized to create a relaxed problem that is easier to solve. Upper bounds, corresponding with feasible solutions, can usually be obtained via minor modifications of the Lagrangian solution [5, 7]. Competitive solutions for parallel-machine scheduling based on mathematical programming often rely on Dantzig-Wolfe decomposition in combination with a branchand-price algorithm, see for instance Bard and Rojanasoonthon [1] and Sadykov and Wolsey [8]. The problem is reformulated as a master problem and m subproblems, one per machine, where the master problem selects schedules and the subproblems search for promising (partial) schedules. van den Akker et al. [9] also find the best set of mutually distinct machine schedules that contain all jobs, where new columns are generated by a local-search procedure. These approaches are very difficult to extend

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to a flexible flow-shop setting, however, because the single-machine schedules are now interdependent due to the joint use of the terminal tractors.

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