This chapter presents a novel idea for the general case of the Common Due-Date (CDD) scheduling problem. The problem is about scheduling a certain number of jobs on a single or parallel machines where all the jobs possess different processing times but a common due-date. The objective of the problem is to minimize the total penalty incurred due to earliness or tardiness of the job completions. This work presents exact polynomial algorithms for optimizing a given job sequence for single and identical parallel machines with the run-time complexities of O(nlogn) for both cases, where n is the number of jobs. Besides, we show that our approach for the parallel machine case is also suitable for non-identical parallel machines. We prove the optimality for the single machine case and the run-time complexities of both. Henceforth, we extend our approach to one particular dynamic case of the CDD and conclude the chapter with our results for the benchmark instances provided in the OR library.
Cheng again considered the same problem with slight variations and presented a linear programming formulation @cite_13 . In @math Cheng and Kahlbacher and Hall studied the CDD problem extensively, presenting some useful properties for the general case @cite_10 @cite_1 . A pseudo polynomial algorithm of O( @math ) (where, @math is the common due-date) complexity was presented by Hoogeveen and Van de Velde for the restrictive case with one machine when the earliness and tardiness penalty weights are symmetric for all the jobs @cite_2 . In @math Hall studied the unweighted earliness and tardiness problem and presented a dynamic programming algorithm @cite_1 . Besides these earlier works, there has been some research on heuristic algorithms for the general common due date problem with asymmetric penalty costs. James presented a tabu search algorithm for the general case of the problem in @math @cite_18 .
@cite_13 @cite_18 @cite_1 @cite_2 @cite_10
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This paper considers the problem of scheduling jobs on single and parallel machines where all the jobs possess different processing times but a common due date. There is a penalty involved with each job if it is processed earlier or later than the due date. The objective of the problem is to find the assignment of jobs to machines, the processing sequence of jobs and the time at which they are processed, which minimizes the total penalty incurred due to tardiness or earliness of the jobs. This work presents exact polynomial algorithms for optimizing a given job sequence for single and parallel machines with the run-time complexities of O(n log n) and O(mn2 log n) respectively, where n is the number of jobs and m the number of machines. The algorithms take a sequence consisting of all the jobs (J_i, i = 1,2,...,n) as input and distribute the jobs to machines (for m>1) along with their best completion times so as to get the least possible total penalty for this sequence. We prove the optimality for the single machine case and the runtime complexities of both. Henceforth, we present the results for the benchmark problems and compare with previous work for both single and parallel machine cases, up to 200 jobs. A companion paper Part I considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date, d, which is unrestrictively late. This paper Part II considers the problem of minimizing the unweighted earliness and tardiness of jobs, allowing the possibility that d is early enough to constrain the scheduling decision. We describe several optimality conditions. The recognition version of the problem is shown to be NP-complete in the ordinary sense, confirming a well known conjecture. Moreover, this complexity definition is precise, since we describe a dynamic programming algorithm which runs in pseudopolynomial time. This algorithm is also extremely efficient computationally, providing an improvement over earlier procedures, of almost two orders of magnitude in the size of instance that can be solved. Finally, we describe a special case of the problem which is polynomially solvable. This article uses tabu search to solve the restricted, common-due-date, early tardy machine scheduling problem generalised earliness and tardiness penalties. Different forms of the tabu search are tested, including one based on a sequence of jobs solution space and another based on an early tardy solution space. Results show that a search which uses an early tardy solution space with a neighbourhood scheme which eliminates infeasible areas of the solution space is the most efficient and effective solution method. We consider a one-machine scheduling problem with earliness and tardiness penalties. All jobs are assigned a common due date and the objective is to minimize the total penalty due to job earliness and tardiness. We are interested in finding the optimal combination of the common due-date value and the job sequence. Despite the fact that this problem in general is very hard to solve, we prove that there exists at least a common property for all optimal solutions: The first job in an optimal sequence is one of the longest jobs. We also prove that this property holds for a general class of unimodal penalty functions.