Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions

We consider the following scheduling problem. We are given a set S of jobs which are to be scheduled sequentially on a single processor. Each job has an associated processing time which is required for its processing. Given a particular permutation of the jobs in S, the jobs are processed in that order with each job started as soon as possible, subject only to the following constraint: For a fixed integer \(B \ge 2\), no unit time interval \([x, x+1)\) is allowed to intersect more than B jobs for any real x. There are several real world situations for which this restriction is natural. For example, suppose in addition to the jobs being executed sequentially on a single main processor, each job also requires the use of one of B identical subprocessors during its execution. Each time a job is completed, the subprocessor it was using requires one unit of time to reset itself. In this way, it is never possible for more than B jobs to be worked on during any unit interval. In Braun et al. (J Sched 17: 399–403, 2014a) it is shown that this problem is NP-hard when the value B is variable and a classical worst-case analysis of List Scheduling for this situation has been carried out. We prove a tighter bound for List Scheduling for \(B\ge 3\) and we analyze the worst-case behavior of the makespan \(\tau _\mathrm{LPT}(S)\) of LPT (longest processing time first) schedules (where we rearrange the set S of jobs into non-increasing order) in relation to the makespan \(\tau _o(S)\) of optimal schedules. We show that LPT ordered jobs can be processed within a factor of \(2-2/B\) of the optimum (plus 1) and that this factor is best possible.     

An efficient imperialist competitive algorithm for scheduling in the two-stage assembly flow shop problem

This paper considers a two-stage assembly flow shop problem where m parallel machines are in the first stage and an assembly machine is in the second stage. The objective is to minimise a weighted sum of makespan and mean completion time for n available jobs. As this problem is proven to be NP-hard, therefore, we employed an imperialist competitive algorithm (ICA) as solution approach. In the past literature, Torabzadeh and Zandieh (2010 Torabzadeh, E., and M. Zandieh. 2010. “Cloud theory-based Simulated Annealing Approach for Scheduling in the Two-stage Assembly Flow Shop.” Advances in Engineering Software 41: 1238–1243.[Crossref], [Web of Science ®] , [Google Scholar]) showed that cloud theory-based simulated annealing algorithm (CSA) is an appropriate meta-heuristic to solve the problem. Thus, to justify the claim for ICA capability, we compare our proposed ICA with the reported CSA. A new parameters tuning tool, neural network, for ICA is also introduced. The computational results clarify that ICA performs better than CSA in quality of solutions.     

On bilevel machine scheduling problems

Bilevel scheduling problems constitute a hardly studied area of scheduling theory. In this paper, we summarise the basic concepts of bilevel optimisation, and discuss two problem classes for which we establish various complexity and algorithmic results. The first one is the bilevel total weighted completion time problem in which the leader assigns the jobs to parallel machines and the follower sequences the jobs assigned to each machine. Both the leader and the follower aims to minimise the total weighted completion time objective, but with different job weights. When the leader’s weights are arbitrary, the problem is NP-hard. However, when all the jobs are of unit weight for the leader, we provide a heuristic algorithm based on iterative LP-rounding along with computational results, and provide a sufficient condition when the LP-solution is integral. In addition, if the follower weights induce a monotone (increasing or decreasing) processing time order in any optimal solution, the problem becomes polynomially solvable. As a by-product, we characterise a new polynomially solvable special case of the MAX m-CUT problem, and provide a new linear programming formulation for the P||?_j C_j{P||\sum_j C_j} problem. Finally, we present some results on the bilevel order acceptance problem, where the leader decides on the acceptance of orders and the follower sequences the jobs. Each job has a deadline and if a job is accepted, it cannot be late. The leader’s objective is to maximise the total weight of accepted jobs, whereas the follower aims at minimising the total weighted job completion times. For this problem, we generalise some known single-level machine scheduling algorithms.     

Single machine scheduling problems with generalised due-dates and job-rejection

We study single machine scheduling problems. Generalised due dates are assumed, i.e. job due dates are specified according to the positions of the jobs in the sequence, rather than their identity. Thus, assuming that due dates are numbered in a non-decreasing order, the jth due date refers to the job assigned to the jth position. In addition, we allow the option of job rejection, i.e. not all jobs must be processed. In this case, the scheduler is penalised for each rejected job, and the total rejection cost becomes part of the objective function. Two objective functions are considered: maximum tardiness plus rejection cost, and total tardiness plus rejection cost. Both problems are proved to be NP-hard. Pseudo-polynomial dynamic programmes and efficient heuristics are introduced and tested numerically.     

Minimisation of total tardiness for identical parallel machine scheduling using genetic algorithm

In recent years research on parallel machine scheduling has received an increased attention. This paper considers minimisation of total tardiness for scheduling of n jobs on a set of m parallel machines. A spread-sheet-based genetic algorithm (GA) approach is proposed for the problem. The proposed approach is a domain-independent general purpose approach, which has been effectively used to solve this class of problem. The performance of GA is compared with branch and bound and particle swarm optimisation approaches. Two set of problems having 20 and 25 jobs with number of parallel machines equal to 2, 4, 6, 8 and 10 are solved with the proposed approach. Each combination of number of jobs and machines consists of 125 benchmark problems; thus a total for 2250 problems are solved. The results obtained by the proposed approach are comparable with two earlier approaches. It is also demonstrated that a simple GA can be used to produce results that are comparable with problem-specific approach. The proposed approach can also be used to optimise any objective function without changing the basic GA routine.     

A divide and conquer-based greedy search for two-machine no-wait job shop problems with makespan minimisation

This paper addresses a two-machine no-wait job shop problem with makespan minimisation. It is well known that this problem is strongly NP-hard. A divide-and-conquer approach (DC for short) is adopted to calculate the optimal timetable of a given sequence. It decomposes the given sequences into several independent parts and conquers them separately. A timetable enhancing method is introduced to further improve the timetable obtained by DC. It constructs a set of flow shop type jobs based on the result from DC and calculates the best timetable for these newly constructed jobs by the well-known Gilmore and Gomory method (GG for short). An efficient greedy search is proposed by integrating DC with GG to search for the best sequence. Experimental results show that the proposed algorithm can find the optimal solutions for 96% of the randomly generated test instances on average.     

Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model

In this paper, we introduce a novel approach in the nonconvex optimization framework for image restoration via a Markov random field (MRF) model. While image restoration is elegantly expressed in the language of MRF’s, the resulting energy minimization problem was widely viewed as intractable: it exhibits a highly nonsmooth nonconvex energy function with many local minima, and is known to be NP-hard. The main goal of this paper is to develop fast and scalable approximation optimization approaches to a nonsmooth nonconvex MRF model which corresponds to an MRF with a truncated quadratic (also known as half-quadratic) prior. For this aim, we use the difference of convex functions (DC) programming and DC algorithm (DCA), a fast and robust approach in smooth/nonsmooth nonconvex programming, which have been successfully applied in various fields in recent years. We propose two DC formulations and investigate the two corresponding versions of DCA. Numerical simulations show the efficiency, reliability and robustness of our customized DCAs with respect to the standard GNC algorithm and the Graph-Cut based method—a more recent and efficient approach to image analysis.     

A new approach for solution of vehicle routing problem with hard time window: an application in a supermarket chain

In this study, a vehicle routing problem with hard time windows (VRPHTW) that appears to meet demands of customers’ service within time intervals in a supermarket chain is solved. In VRPHTW, to reach a solution by an exact method is quite difficult and sometimes impossible if number of constraints is large enough (i.e., NP-hard), and solution time of a VRPHTW grows exponentially. As the size of the problem grows, a near optimal solution can be found using a heuristic method. A hierarchical approach consisting of two stages as “cluster-first route-second” is proposed. In the first stage, customers are assigned to vehicles using three different clustering algorithms (i.e., K-means, K-medoids and DBSCAN). In the second stage, a VRPHTW is solved using a MILP. The main contribution of the article is that the proposed hierarchical approach enables us to deal with a large size real problem and to solve it in a short time using the exact method. Finally, the proposed approach is employed on a supermarket chain. An instance of the algorithm is demonstrated to illustrate the applicability of the proposed approach and the results obtained are highly favourable.     

Batch scheduling in differentiation flow shops for makespan minimisation

This paper considers a two-stage three-machine differentiation flow shop that comprises a common machine at stage 1 and two independent dedicated machines at stage 2. Two types of jobs are to be processed. All jobs must visit the stage-1 machine, and then the jobs of each type proceed to their dedicated machine for stage-2 processing. The stage-1 machine processes the jobs in batches, each of which, whenever formed, requires a constant setup time. The objective is to find a schedule that attains the minimum makespan. While the problem is strongly NP-hard, we investigate the case where the processing sequences of the two types of jobs are given and fixed. A polynomial-time dynamic programming algorithm is designed to solve this problem. We then deploy this algorithm to compute the lower bounds of the general problem.     

Working time constraints in operational fixed job scheduling

In this study we consider the operational fixed job scheduling problem under working time limitations. The problem has several practical implications in both production and service operations; however the relevant research is scarce. We analyse pre-emptive and non pre-emptive versions of the problem and its special cases. We provide polynomial-time algorithms for some special cases. We show that the non pre-emptive jobs problem is strongly NP-hard, and propose a branch-and-bound algorithm that employs efficient bounding procedures and dominance properties. We conduct a numerical experiment to observe the effects of parameters on the quality of the solution. The results of our computational tests for the branch-and-bound algorithm reveal that our algorithm can solve the instances with up to 100 jobs in reasonable times. To the best of our knowledge our branch-and-bound algorithm is the first optimisation attempt to solve the problem.     

Spread time considerations in operational fixed job scheduling

In this study, we consider the operational fixed job scheduling problem on identical parallel machines. We assume that the jobs have fixed ready times and deadlines, and spread time constraints are imposed on machines. Our objective is to select a set of jobs for processing so as to maximise the total weight. We show that the problem is strongly NP-hard, and we investigate several special polynomially solvable cases. We propose a branch and bound algorithm that employs size reduction mechanisms, dominance conditions, and powerful lower and upper bounds. The computational results reveal that the branch and bound algorithm returns optimal solutions for problem instances with up to 100 jobs in reasonable solution times.