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 Information Technology Journal , 2012, Abstract: We consider the problem of scheduling deteriorating jobs and common due window location on a single machine. The actual processing time of a job is a linear increasing function of its starting time. The problem is to determine the optimal earliest due date, the due window size and the job schedule simultaneously to minimize costs for earliness, tardiness, earliest due date assignment and due window size penalties. A polynomial time optimal algorithm is presented to solve the problem.
 Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/928312 Abstract: In many real scheduling environments, a job processed later needs longer time than the same job when it starts earlier. This phenomenon is known as scheduling with deteriorating jobs to many industrial applications. In this paper, we study a scheduling problem of minimizing the total completion time on identical parallel machines where the processing time of a job is a step function of its starting time and a deteriorating date that is individual to all jobs. Firstly, a mixed integer programming model is presented for the problem. And then, a modified weight-combination search algorithm and a variable neighborhood search are employed to yield optimal or near-optimal schedule. To evaluate the performance of the proposed algorithms, computational experiments are performed on randomly generated test instances. Finally, computational results show that the proposed approaches obtain near-optimal solutions in a reasonable computational time even for large-sized problems.
 Journal of Chongqing Normal University , 2011, Abstract: In this paper, we consider an unrelated parallel machine scheduling problem with setup time and deteriorating jobs simultaneously.The setup time is proportional to the length of the already processed jobs, that is , the setup time is past-sequence-dependent(p-s-d). It is assumed that the job processing time is defined by functions dependent on their starting times. The objective is to minimize the total completion time. If the number Ni of jobs for each machine Mi is known is advance, the problem of the total completion time minimization on unrelated parallel machines can be formulated as an assignment problem. The assignment problem is solved in O(n3)time and Ni(i=1,2,,m) can be determined in O(nm-1)time. So we show that the proposed problem can be solved in polynomial time. We also discuss two special cases of the problem.First, each job has the same deterioration and setup parameter b on different machines. The normal processing time of job j is aj no matter what machine the job is assigned to . That is all jobs are processed on identical parallel machines. Second, based on the first case, we also consider all jobs with no deterioration ,that is &i=O. For the two cases, we show that they can be optimally solved by lower order algorithms.
 International Journal of Industrial Engineering Computations , 2010, Abstract: This paper addresses a bi-criteria scheduling problem with deteriorating jobs on a single machine. We develop a model for a single machine bi-criteria scheduling problem (SMBSP) with the aim of minimizing total tardiness and work in process (WIP) costs. WIP cost increases as a job passes through a series of stages in the production process. Due to the uncertainty involved in real-world scheduling problems, it is sometimes unrealistic or even impossible to acquire exact input data. Hence, we consider the SMBSP under the hypothesis of fuzzy L-R processing time's knowledge and fuzzy L-R due date. The effectiveness of the proposed model and the denoted methodology is demonstrated through a test problem.
 Mathematics , 2014, Abstract: This paper addresses the scheduling problem of minimizing the total weighted tardiness on a single machine with step-deteriorating jobs. With the assumption of deterioration, the job processing times are modeled by step functions of job starting times and pre-specified job deteriorating dates. The introduction of step-deteriorating jobs makes a single machine total weighted tardiness problem more intractable. The computational complexity of this problem under consideration was not determined. In this study, it is firstly proved to be strongly NP-hard. Then a mixed integer programming model is derived for solving the problem instances optimally. In order to tackle large-sized problems, seven dispatching heuristic procedures are developed for near-optimal solutions. Meanwhile, the solutions delivered by the proposed heuristic are further improved by a pair-wise swap movement. Computational results are presented to reveal the performance of all proposed approaches.
 Journal of Applied Sciences , 2008, Abstract: In the classical scheduling problems, processing time was assumed to be constant and takes predefined values. In many realistic environments, such as machine maintaining or crisis event management, processing time on each machine depends on the position of jobs on the machine sequence or their starting time on that machine. We assume that processing time was an increasing linear function of its start time, in other words pij = αijtij, in the literature these jobs were called deteriorating jobs. This problem addresses classic nmfCmax scheduling problem with a new assumption on the processing time of the jobs and it was surmised in the format of nmp,pij = αijtijCmax. We have used new Electro Magnetic meta-heuristic algorithm to find near optimum solution and compared the results with the results obtained from modified classical algorithms like CDS and Palmer.
 Bulletin of the Polish Academy of Sciences: Technical Sciences , 2009, DOI: 10.2478/v10175-010-0110-7 Abstract: In the paper we investigate four single processor scheduling problems, which deal with the process of the negotiation between a producer and a customer about delivery time of final products. This process is modeled by a due window, which is a generalization of well known classical due date and describes a time interval, in which a job should be finished. Due window assignment is a new approach, which has been investigated in the scientific literature for a few years. In this paper we consider various models of due window assignment. To solve the formulated problems we have to find such a schedule of jobs and such an assignment of due windows to each job, which minimizes a given criterion dependent on the maximum or total earliness and tardiness of jobs and due window parameters. One of the main results is the mirror image of the solutions of the considered problems and other problems presented in the scientific literature. The wide survey of the literature is also given.
 Mathematics , 2013, Abstract: In this article, we study a single-machine scheduling problem of minimizing the total tardiness for a set of independent jobs. The processing time of a job is modeled as a step function of its starting time and a specific deteriorating date. A mixed integer programming model was applied to the problem and validated. Since the problem is known to be NP-hard, we proposed a heuristic named simple weighted search procedure (SWSP) and a general variable neighborhood search algorithm (GVNS). A perturbation procedure with 3-opt is embedded within the GVNS process in order to explore broader spaces. Extensive numerical experiments are carried out on some randomly generated test instances so as to investigate the performance of the proposed algorithms. By comparing to the results of the CPLEX optimization solver, the heuristic SWSP and the standard variable neighborhood search, it is shown that the proposed GVNS algorithm can provide better solutions within a reasonable running time.
 Mathematics , 2013, Abstract: This article considers the parallel machine scheduling problem with step-deteriorating jobs and sequence-dependent setup times. The objective is to minimize the total tardiness by determining the allocation and sequence of jobs on identical parallel machines. In this problem, the processing time of each job is a step function dependent upon its starting time. An individual extended time is penalized when the starting time of a job is later than a specific deterioration date. The possibility of deterioration of a job makes the parallel machine scheduling problem more challenging than ordinary ones. A mixed integer programming model for the optimal solution is derived. Due to its NP-hard nature, a hybrid discrete cuckoo search algorithm is proposed to solve this problem. In order to generate a good initial swarm, a modified heuristic named the MBHG is incorporated into the initialization of population. Several discrete operators are proposed in the random walk of L\'{e}vy Flights and the crossover search. Moreover, a local search procedure based on variable neighborhood descent is integrated into the algorithm as a hybrid strategy in order to improve the quality of elite solutions. Computational experiments are executed on two sets of randomly generated test instances. The results show that the proposed hybrid algorithm can yield better solutions in comparison with the commercial solver CPLEX with one hour time limit, discrete cuckoo search algorithm and the existing variable neighborhood search algorithm.
 Computer Science , 2015, Abstract: We consider the job assignment problem in a multi-server system consisting of $N$ parallel processor sharing servers, categorized into $M$ ($\ll N$) different types according to their processing capacity or speed. Jobs of random sizes arrive at the system according to a Poisson process with rate $N \lambda$. Upon each arrival, a small number of servers from each type is sampled uniformly at random. The job is then assigned to one of the sampled servers based on a selection rule. We propose two schemes, each corresponding to a specific selection rule that aims at reducing the mean sojourn time of jobs in the system. We first show that both methods achieve the maximal stability region. We then analyze the system operating under the proposed schemes as $N \to \infty$ which corresponds to the mean field. Our results show that asymptotic independence among servers holds even when $M$ is finite and exchangeability holds only within servers of the same type. We further establish the existence and uniqueness of stationary solution of the mean field and show that the tail distribution of server occupancy decays doubly exponentially for each server type. When the estimates of arrival rates are not available, the proposed schemes offer simpler alternatives to achieving lower mean sojourn time of jobs, as shown by our numerical studies.
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