Single-machine scheduling to minimize total convex resource consumption with a constraint on total weighted flow time

In this paper, we consider single-machine scheduling problem in which processing time of a job is described by a convex decreasing resource consumption function. The objective is to minimize the total amount of resource consumed subject to a constraint on total weighted flow time. The optimal resource allocation is obtained for any arbitrary job sequence. The computational complexity of the general problem remains an open question, but we present and analyze some special cases that are solvable by using polynomial time algorithms. For the general problem, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm proposed to solve the problem. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions. The results show that the average percentage error of the proposed heuristic algorithm from optimal solutions is less than 3%.     

Two-machine flow shop problem with effects of deterioration and learning

In this paper we consider a two-machine flow shop scheduling problem with effects of deterioration and learning. By the effects of deterioration and learning, we mean that the processing time of a job is a function of its execution starting time and its position in a sequence. The objective is to find a sequence that minimizes the total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived, which are used to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed, which is shown by computational experiments to perform effectively and efficiently in obtaining near-optimal solutions.     

A branch and bound algorithm for minimizing makespan on a single machine with unequal release times under learning effect and deteriorating jobs

We present a single-machine problem with the unequal release times under learning effect and deteriorating jobs when the objective is minimizing the makespan. In this study, we introduced a scheduling model with unequal release times in which both job deterioration and learning exist simultaneously. By the effects of learning and deterioration, we mean that the processing time of a job is defined by increasing function of its execution start time and position in the sequence. A branch-and-bound algorithm incorporating with several dominance properties and lower bounds is developed to derive the optimal solution. A heuristic algorithm is proposed to obtain a near-optimal solution. The computational experiments show that the branch-and-bound algorithm can solve instances up to 30 jobs, and the average error percentage of the proposed heuristic is less than 0.16%.     

A branch-and-bound algorithm for solving a two-machine flow shop problem with deteriorating jobs

In this paper we consider a two-machine flow shop scheduling problem with deteriorating jobs. By a deteriorating job we mean that the job's processing time is an increasing function of its starting time. We model job deterioration as a function that is proportional to a linear function of time. The objective is to find a sequence that minimizes the total completion time of the jobs. For the general case, we derive several dominance properties, some lower bounds, and an initial upper bound by using a heuristic algorithm, and apply them to speed up the elimination process of a branch-and-bound algorithm developed to solve the problem.     

Minimizing makespan in three-machine flow shops with deteriorating jobs

In this paper, a three-machine permutation flow shop scheduling problem with time-dependent processing times is considered. By the time-dependent processing times we mean that the job's processing time is an increasing function of its starting time. The objective is to find a sequence that minimizes the makespan. This problem is well known to be NP-hard. Several dominance properties and a lower bound are derived to speed up the elimination process of a branch-and-bound algorithm. Moreover, two heuristic algorithms are proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational experiments on randomly generated problems are conducted to evaluate the branch-and-bound algorithm and heuristic algorithm. Computational results show that the proposed heuristic algorithm M-NEH perform effectively and efficiently.     

Deteriorating job scheduling to minimize the number of late jobs with setup times

Deteriorating jobs scheduling problems have been widely studied recently. However, research on scheduling problems with deteriorating jobs has rarely considered explicit setup times. With the current emphasis on customer service and meeting the promised delivery dates, we consider a single-machine scheduling problem to minimize the number of late jobs with deteriorating jobs and setup times in this paper. We derive some dominance properties, a lower bound, and an initial upper bound by using a heuristic algorithm to speed up the search process of the branch-and-bound algorithm. Computational experiments show that the algorithm can solve instances up to 1000 jobs in a reasonable amount of time.     

A tabu method for a two-agent single-machine scheduling with deterioration jobs

In recent 10 years, the multi-agent idea applied in scheduling issues has received continuing attention. However, the study of the multi-agent scheduling with deteriorating jobs is relatively limited. In light of this, this paper deliberates upon a two-agent single-machine scheduling problem with deteriorating jobs. Taking the proposed model, the actual processing time of a job from both the first agent and the second agent is modeled as a linearly increasing function of its starting time. The goal of this paper is to minimize the total weighted number of tardy jobs of the first agent subject to the condition that the maximum lateness of the second agent is allowed to have an upper bound. The complexity of the model concerned in the paper is claimed as an NP-hard one. Following that, several dominance rules and a lower bound are proposed to be applied in a branch-and-bound algorithm for the optimal solution, and a tabu algorithm is applied to find near-optimal solutions for the problem. The simulation results obtained from all the proposed algorithms are also reported.     

Branch and bound for the cutwidth minimization problem

The cutwidth minimization problem consists of finding a linear arrangement of the vertices of a graph where the maximum number of cuts between the edges of the graph and a line separating consecutive vertices is minimized. We first review previous approaches for special classes of graphs, followed by lower bounds and then a linear integer formulation for the general problem. We then propose a branch-and-bound algorithm based on different lower bounds on the cutwidth of partial solutions. Additionally, we introduce a Greedy Randomized Adaptive Search Procedure (GRASP) heuristic to obtain good initial solutions. The combination of the branch-and-bound and GRASP methods results in optimal solutions or a reduced relative gap (difference between upper and lower bounds) on the instances tested. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the small instances (up to 32 vertices) as well as some of the large instances tested (up to 158 vertices) using less than 30 minutes of CPU time. We compare the results of our method with previous lower bounds, and with the best previous linear integer formulation solved using Cplex. Both comparisons favor the proposed procedure.     

Scheduling deteriorating jobs on a single machine with release times

This paper investigates a single-machine deteriorating job scheduling problem with job release times where its objective is to minimize the makespan. The problem is known to be NP-hard. Therefore, a branch-and-bound algorithm incorporating with several dominance properties and lower bounds is proposed to derive the optimal solution for the problem. In addition, easy-implemented heuristic algorithms are also provided to obtain the near-optimal solution. The computational experiments indicate that the branch-and-bound algorithm can solve most of the medium-job-sized problems within a reasonable time, and the heuristic is quite accurate with an average error percentage of less than 0.3%.     

Algorithms for the boundary selection problem

User-oriented clustering schemes enable the classification of documents based upon the user's perception of the similarity between documents, rather than on some similarity function presumed by the designer to represent the user's criteria. In an earlier paper it was shown that such a classification scheme can be developed in two stages. The first stage involves the accumulation of relevance judgements provided by users,vis-à-vis past query instances, into a suitable structure. The second stage consists of cluster identification. When the structure chosen, in the first stage, for the accumulation of corelevance characteristics of documents is a straight line, the second stage can be formulated as a function optimization problem termed the Boundary Selection Problem (BSP). A branch-and-bound algorithm with a good bounding function is developed for the BSP. Although significant pruning is achieved due to the bounding function, the complexity is still high for a problem of a large size. For such a problem a heuristic that divides it into a number of subproblems, each being solved by a branch-and-bound approach, is developed. Then the overall problem is mapped to an integer knapsack problem and solved by the use of dynamic programming. The tradeoff between accuracy and complexity can be controlled, giving the user a preference of one over the other. Assuming that the heuristic which divides the overall problem introduces no errors and is given sufficient time, the branch and bound with dynamic programming (BBDP) approach will converge to the optimal solution. Two other heuristic approaches, one with the application of a polynomial dynamic programming algorithm and the other which works in a greedy way, are also proposed for the BSP and an experimental comparison of all these approaches is provided. Experimental results indicate that all proposed algorithms show better performance compared with the existing algorithm.     

A Best-first Branch-and-bound Algorithm for Orthogonal Rectangular Packing Problems

In this paper we discuss the problem of packing a set of small rectangles (pieces) in an enclosing final rectangle. We present first a best-first branch-and-bound exact algorithm and second a heuristic approach in order to solve exactly and approximately this problem. The performances of the proposed approaches are evaluated on several randomly generated problem instances. Computational results show that the proposed exact algorithm is able to solve small and medium problem instances within reasonable execution time. The derived heuristic performs very well in the sense that it produces high-quality solutions within small computational time.     

Ant colony algorithms for a two-agent scheduling with sum-of processing times-based learning and deteriorating considerations

This paper addresses a two-agent single-machine scheduling problem with the co-existing sum-of-processing-times-based learning and deteriorating effects. In the proposed model, it is assumed that the actual processing time of a job of the first (second) agent is a decreasing function of the sum-of-processing-times-based learning (or increasing function of the sum-of-processing-times-based deteriorating effect) in a schedule. The aim of this paper is to find an optimal schedule to minimize the total weighted completion time of the jobs of the first agent with the restriction that no tardy job is allowed for the second agent. For the proposed model, we develop a branch-and-bound and some ant colony algorithms for an optimal and near-optimal solution, respectively. Besides, the extensive computational experiments are also proposed to test the performance of the algorithms.     

Permutation flowshop problems with bi-criterion makespan and total completion time objective and position-weighted learning effects

In this paper, we study the problem of minimizing the weighted sum of makespan and total completion time in a permutation flowshop where the processing times are supposed to vary according to learning effects. The processing time of a job is a function of the sum of the logarithms of the processing times of the jobs already processed and its position in the sequence. We present heuristic algorithms, which are modified from the optimal schedules for the corresponding single machine scheduling problem and analyze their worst-case error bound. We also adopt an existing algorithm as well as a branch-and-bound algorithm for the general m-machine permutation flowshop problem. For evaluation of the performance of the algorithms, computational experiments are performed on randomly generated test problems.     

Minimizing total completion time for re-entrant flow shop scheduling problems

In this paper, we study re-entrant flow shop scheduling problems with the objective of minimizing total completion time. In a re-entrant scheduling problem, jobs may visit some machines more than once for processing. The problem is NP-hard even for machine number m=2. A heuristic algorithm is presented to solve the problem, in which an effective k-insertion technique is introduced as the improvement strategy in iterations. Computational experiments and analyses are performed to give guidelines of choosing parameters in the algorithm. We also provide a lower bound for the total completion time of the optimal solution when there are only two machines. Objective function values of the heuristic solutions are compared with the lower bounds to evaluate the efficiency of the algorithm. For randomly generated instances, the results show that the given heuristic algorithm generates solutions with total completion times within 1.2 times of the lower bounds in most of the cases.     

Generalized disjunctive constraint propagation for solving the job shop problem with time lags

This paper addresses the job-shop scheduling problem with time-lags. We propose an insertion heuristic and generalized resource constraint propagation mechanisms. Our propositions are embedded in a branch-and-bound algorithm to provide an experimental evaluation on some benchmark instances. The results obtained conclude that our heuristic achieves the best solutions on the instances, especially when problems involve tightened time lags. The results also prove the interest of the constraint propagation generalization when time lags are considered.     

Scheduling two-machine no-wait open shops to minimize makespan

This paper examines the problem of scheduling two-machine no-wait open shops to minimize makespan. The problem is known to be strongly NP-hard. An exact algorithm, based on a branch-and-bound scheme, is developed to optimally solve medium-size problems. A number of dominance rules are proposed to improve the search efficiency of the branch-and-bound algorithm. An efficient two-phase heuristic algorithm is presented for solving large-size problems. Computational results show that the branch-and-bound algorithm can solve problems with up to 100 jobs within a reasonable amount of time. For large-size problems, the solution obtained by the heuristic algorithm has an average percentage deviation of 0.24% from a lower bound value.     

A branch-and-bound algorithm to minimize total weighted completion time on identical parallel machines with job release dates

In this paper, we consider an identical parallel machine scheduling problem with release dates. The objective is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose some dominance properties and two lower bounds. We also present an efficient heuristic. A branch-and-bound algorithm, in which the heuristic, the lower bounds and the dominance properties are incorporated, is proposed and tested on a large set of randomly generated instances.     

Two-stage hybrid flow shop scheduling on parallel batching machines considering a job-dependent deteriorating effect and non-identical job sizes

In this paper, we investigate a specialized two-stage hybrid flow shop scheduling problem with parallel batching machines considering a job-dependent deteriorating effect and non-identical job sizes simultaneously. A novel concept of three-dimensional wasted volume based on the job normal processing time, job size, and job deteriorating rate is first proposed. Some structural properties, as well as a heuristic algorithm, are developed to solve the single parallel batching machine scheduling problem. Since the two-stage hybrid flow shop scheduling problem is NP-hard, a hybrid EDA-DE algorithm combining estimation of distribution algorithm (EDA) and differential evolution (DE) algorithm is proposed to tackle the studied problem. In addition, the Taguchi method of design of experiments (DOE) is implemented to tune the parameters of the EDA-DE. Finally, a series of computational experiments are carried out to compare the performance of the proposed hybrid EDA-DE algorithm and some recent existing algorithms from the literature, and the comparative results validate the effectiveness and efficiency of the proposed algorithm.     

Single machine total completion time minimization scheduling with a time-dependent learning effect and deteriorating jobs

In this article, we consider a single machine scheduling problem with a time-dependent learning effect and deteriorating jobs. By the effects of time-dependent learning and deterioration, we mean that the job processing time is defined by a function of its starting time and total normal processing time of jobs in front of it in the sequence. The objective is to determine an optimal schedule so as to minimize the total completion time. This problem remains open for the case of ?1?a?a denotes the learning index; we show that an optimal schedule of the problem is V-shaped with respect to job normal processing times. Three heuristic algorithms utilising the V-shaped property are proposed, and computational experiments show that the last heuristic algorithm performs effectively and efficiently in obtaining near-optimal solutions.     

Genetic algorithm for minimizing the total weighted completion time scheduling problem with learning and release times

This paper considers a single-machine problem with the sum-of-processing time based learning effect and release times. The objective is to minimize the total weighted completion times. First, a branch-and-bound algorithm incorporating with several dominance properties and two lower bounds are developed for the optimal solution. Then a genetic heuristic-based algorithm is proposed for a near-optimal solution. Finally, a computational experiment is conducted to evaluate the performances of the proposed algorithms. The results show that the branch-and-bound algorithm can solve instances up to 15 jobs, and the average error percentage of the genetic heuristic algorithm is less than 0.105%.