This thesis considers a variant of the traveling salesman problem by incorporating order delivery into the problem setting. In classical scheduling problems, the identities in the objective functions are jobs, which may consist of several tasks or operations.
The concept of order delivery lies in the fact that many applications define the objective functions in terms of higher-level entities through aggregation. In the scheduling problem addressed in this thesis, jobs belonging to the same order or group will be delivered as a single batch, and the objective function considers order completion times instead of job completion times. The completion of an order is determined by the completion of the last job of that order. A set of jobs are grouped, in a disjoint sense, as different orders in a single machine scheduling environment. There exists a sequence-dependent setup time between any two consecutive jobs, and job processing times are negligible. The objective is to determine a job processing sequence so as to minimize the weighted sum of completion times over all groups.
The problem setting originates from a simplified model of the satellite imaging problem, although applications in many other areas are possible. Several requests/orders of various numbers of photographs need to be fulfilled via satellite imaging operations.
Each order may consist of several photos, and is associated with a weight that characterizes the importance or the degree of urgency. The satellite should adjust the imaging angle of the camera for the next photo shooting upon the completion of the current photo shooting. Such alternation of camera lens requires great accuracy with technical concerns, and takes a much longer processing time compared with the photo shooting time. Therefore, various setup times are indispensable between any two consecutive imaging operations, while the shooting time may be ignored. For example, the Ministries of Defense, Agriculture, and Energy may place orders which demand various numbers of photos. The objective is to accomplish all requests according to a photo shooting schedule which minimizes the weighted sum of completion times over
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all orders.
This model may be adopted by some other applications under different scenarios, such as in vehicle routing problems (VRP). The customers to be served may actually belong to different companies, and the service provided to a company is fulfilled only when all the subsidiary sites of that company are served. In addition, such concept involving change-over setup costs with batch delivery may also be applied as a variation of the traveling salesman problem. If the constraint of the order delivery is removed, the studied problem is equivalent to the classical deliveryman problem or the minimum latency problem, which determines the visitation sequence of nodes with the objective of minimizing the sum of completion times of nodes on the network. Nevertheless, the incorporation of order delivery is essential to certain practical applications. For example, an order may make sense to a customer only when all of the components are accomplished; or some orders may have much higher degree of urgency than the rest of the orders do. Some other transportation problems, such as VRP, may also incorporate with the concept of order delivery in order to capture and interpret complicated down-to-earth problems in real life.
In this thesis, we denote the studied by the TSP-WOCT (Weighted Order Completion Times). The TSP-WOCT conjoins the classical deliveryman problem with batch delivery scheduling on single machine to compose a new model. The deliveryman problem is similar to the well-known traveling salesperson problem with differences in the objective functions. The total delay over all nodes is deliberated in the deliveryman problems while the TSP considers only the total length. The TSP is studied from an aspect of internal efficiency and the deliveryman problem focused instead on customer satisfaction. It is interesting that deliveryman problem has been studied under different titles, such as “the traveling repairman problem” (Afrati et al., 1986; Garcia et al., 2002) and “deliveryman problem” (Minieka, 1989; Fischetti et al., 1993). Some recent researches about the minimum latency problems actually cope with the same problems (Blum et al., 1994; Wu 2000). Such minimum latency problems are also proven to be NP-hard, and the polynomial time algorithms can only be applied for some specific graphs, such as paths (Afrati et al., 1986, Garcia et al., 2002), edge-unweighted trees
(Minieka, 1989), tree of diameter 3 (Blum et al., 1994). Wu et al. (2004) proposed an exact algorithm by applying a dynamic programming algorithm along with branch and bound technique for the small-scale problems. In addition, some researches developed approximation algorithms for the minimum latency problems (Archer and Williamson, 2003; Goemans and Kleinberg, 1998).
In single machine scheduling with batch delivery, a set of jobs are to be scheduled and the jobs may be grouped as batches, and a batch contains contiguously scheduled jobs. All jobs in the same batch are delivered to the customer as a whole upon the completion of the last job in the batch. Cheng and Kahbacher (1993) first considered the problem with the objective to minimize the sum of the total weighted batch delivery time. They also proposed another objective of minimizing the total weighted earliness and a batch delivery penalty depending on the number of batches. The general version of the proposed problem was demonstrated to be ordinary NP-hard in the same article.
For the same problem, Cheng et al. (1996) later proved that it is strongly NP-hard. They also proposed polynomial algorithms for special cases with equal processing times or equally weighted batches. Although a dynamic programming algorithm was applied by Cheng and Gordon (1994) to solve the general version of the single machine with batch delivery scheduling problem, Cheng et al. (1996) further clarified that this problem can be formulated as a classical parallel machine scheduling problem. Therefore, the exact/approximate algorithms and complexity analyses for the corresponding parallel machine scheduling problem can be easily extended to the problem. Cheng et al. (1997) proposed another objective function to the batch delivery scheduling problem by minimizing the total weighted earliness and mean batch delivery time. They proved the strong NP-hardness of the problem and provided polynomial algorithms for some special cases. Yang (2000) studied the single machine scheduling problems with generalized batch delivery dates and earliness penalties, which is proven to be strongly NP-hard. He also suggested a polynomial time solvable case, even for general earliness penalty function, when all processing times are equal. An exact algorithm for such case, while weighted earliness functions being considered, was provided by Yang in the same
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article. Recently, Ji et al. (2007) considered a scheduling problem with batch delivery property, that is, jobs are delivered in batches and the delivery date of a batch determined by the completion time of the last job in the batch. The objective is to minimize the sum of the total weighted flow time and delivery cost. They proved the problem to be strongly NP-hard and applied a dynamic programming algorithm that runs in pseudo-polynomial time for the cases with bounded numbers of batches. They also proposed optimal algorithms for two special cases.
The rest of this thesis is organized as follows. In Chapter 2, we formally define the studied scheduling problem. A binary integer programming model will be presented to describe the problem. In Chapter 3, an exact algorithm is developed by the dynamic programming approach. A special case is also studied. In Chapter 4, we design a tabu search algorithm, an iterated local search algorithm and a genetic algorithm to produce approximate solutions within an acceptable period of time. In Chapter 5, a series of computational experiments are conducted to scrutinize the performance of the proposed algorithms. Finally, some conclusions and suggestions for further research are addressed in Chapter 6.