| Summary: | 博士 === 國立交通大學 === 資訊管理研究所 === 101 === This study proposes a novel scheduling model based on the production of media objects. A media product comprises various types of components or objects, including text, audio, and video clips. Unlike components or parts in the manufacturing of substantial products, media objects can be shared and reused by several media products. This type of production necessitates to a scheduling model consisting of supporting operations and regular jobs that respectively correspond to media objects and media products. Because a regular job is complete when all of its supporting operations are finished, the objective functions account for only the completion times of regular jobs. The completion times of supporting operations are not considered in the calculation of objective values. Because the completion of a job must follow the completion of all its supporting operations, this scheduling model can be regarded as scheduling under a particular type of precedence constraints. This thesis consists of three parts. The rest part is focused on the manufacturing setting in which a single machine is available, and all of the supporting operations and regular jobs are processed on this machine. We examined two objective functions, i.e. the total weighted job completion time and the number of late jobs. To minimize the total weighted job completion time, the problem remains strongly NP-hard even if the processing times of all operations and all jobs are one and all job weights are one. For operations in which the processing sequence is given and fixed, we propose a polynomial time algorithm to determine an optimal schedule. In considering the objective function of the number of late jobs, we provide a strong NP-hardness proof for the special case in which the processing times of all operations and all jobs are one, and each operation supports at most three jobs and each job is supported by at most one operation. Another case, in which each operation supports at most two jobs and each job is supported by at most one operation, is proven to be ordinary NP-hard.
The second part of this thesis addresses a generalized model involving parallel dedicated machines. A job is complete when all of its operations on each dedicated machines are complete. The objective function considered is the total job completion time. Four integer programming models equipped with different types of decision variables are proposed to formulate this problem. Model SV uses sequential variables that describe the relative positions between each pair of operations on the machines. Model PV uses positional variables for assigning operations to positions on the machines. Model TV uses time-indexed variables that indicate if an operation is complete at a special time point. Model ETV introduces another set of time-indexed variables into Model TV to enhance the constraint speciation. Based on four proposed integer programs, we developed four corresponding Lagrangian relaxations for determining the lower and upper bounds on the optimal solution values.
The final part concerns different features of four proposed integer programming models. We suggest three solution procedures in the Lagrangian relaxations. The first solution procedure is based on two heuristics to obtain the upper bounds of the original problem and update the Lagrange multipliers by using the obtained upper bounds. The second involves decomposing the model into two sub-problems. One of the two sub-problems can obtain the lower and upper bounds through a polynomial-time pre-processing procedure before invoking the Lagrangian relaxations. The other sub-problem can acquire optimal solutions by applying a polynomial time solvable algorithm. The last procedure reduces the relaxed problem into m+1 sub-problems that are relatively easy to address.
To examine the effectiveness and efficiency of the proposed integer programming models and the Lagrangian relaxations, we designed a series of computational experiments. Test instances in this study were generated by applying various combinations of processing times and supporting relations. We investigated the elapsed execution times and solution quality of the proposed models and approaches. Analyzing the numerical results reveals the advantages and disadvantages of the integer programming models and the Lagrangian relaxations.
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