Single Machine Total Tardiness Problem Research Articles

The proportionate flow shop total tardiness problem

We consider the proportionate flow shop total tardiness problem and show how to implement Lawler's (1977) pseudo-polynomial dynamic programming (DP) algorithm for the single-machine total tardiness problem to the multi-machine environment of proportionate flow shop. We also present solvable special cases including one with small/big jobs that has not been considered for the corresponding single-machine problem. We then convert the DP algorithm for the proportionate flow shop to a fully polynomial time approximation scheme (FPTAS). Finally, we show by a counterexample that Pinedo's (2002, p. 140) statement that “the elimination criteria for the single-machine total weighted tardiness problem also apply to the proportionate flow shop total weighted tardiness problem” does not always hold and present an appropriately revised statement.

Solving Composite MultiobjectiveSingle Machine Scheduling ProblemUsingBranch and Bound and Local SearchAlgorithms

This paper present algorithms for solving a single machine scheduling problem to minimize the sum of total completion times, total tardiness,maxim-um tardiness,and maximum earliness.The single machine total tardiness problem is already NP-hard, so the consider problem is strongly NP-hard, and several algorithms are used to solve it. Branch and bound algorithmwith dominance ruleand local search algorit- hms are proposed for the problem. For the Branch and bound algorithm results- show that using dominance rule improve the performance of the algorithm in both computation times and optimal values,but it need longer times.Thus we tackle the problemof large sizes with local search algorit- hms descent method, simulated annealing and tabusearch. The perfomance of these algorithms is evaluated on a large set of test problems and the results are compared.The computational results show that simulated annealing algorithm and Tabu search algorithm are better than Descent method with preference to simulated annealing algorithm,and show that the three algorithms find optimal or near optimal solutions inreasonable times.

An exact exponential branch-and-merge algorithm for the single machine total tardiness problem

This paper proposes an exact exponential algorithm for the single machine total tardiness problem. It exploits the structure of a basic branch-and-reduce framework based on the well known Lawler's decomposition property that solves the problem with worst-case complexity in time O⁎(3n) and polynomial space. The proposed algorithm, called branch-and-merge, is an improvement of the branch-and-reduce technique with the embedding of a node merging operation. Its time complexity converges to O⁎(2n) keeping the space complexity polynomial. This improves upon the best-known complexity result for this problem provided by dynamic programming across the subsets with O⁎(2n) worst-case time and space complexity. The branch-and-merge technique is likely to be generalized to other sequencing problems with similar decomposition properties.

Local Search Algorithms for Multi-criteria Single Machine Scheduling Problem

Real life scheduling problems require the decision maker to consider a number of criteria before arriving at any decision. In this paper, we consider the multi-criteria scheduling problem of n jobs on single machine to minimize a function of five criteria denoted by total completion times (∑), total tardiness (∑), total earliness (∑), maximum tardiness () and maximum earliness (). The single machine total tardiness problem and total earliness problem are already NP-hard, so the considered problem is strongly NP-hard.
 We apply two local search algorithms (LSAs) descent method (DM) and simulated annealing method (SM) for the 1// (∑∑∑) problem (SP) to find near optimal solutions. The local search methods are used to speed up the process of finding a good enough solution, where an exhaustive search is impractical for the exact solution. The two heuristic (DM and SM) were compared with the branch and bound (BAB) algorithm in order to evaluate effectiveness of the solution methods.
 Some experimental results are presented to show the applicability of the (BAB) algorithm and (LSAs). With a reasonable time, (LSAs) may solve the problem (SP) up to 5000 jobs.

Hybrid metaheuristic approaches for the single machine total stepwise tardiness problem with release dates

This paper presents two hybrid metaheuristic approaches, viz. a hybrid genetic algorithm and a hybrid artificial bee colony algorithm for a single machine scheduling problem where tardiness cost of a job increases stepwise with various due dates and the objective is to minimize the total tardiness cost. This kind of tardiness cost occurs in several real life scenarios particularly in transportation. Two versions of the scheduling problem are considered. In the first version, all jobs are assumed to be available for processing at the beginning, whereas in the latter version jobs have release dates. For both versions, we have employed a local search to further improve the solutions obtained through our metaheuristic approaches. To the best of our knowledge, our approaches are the first metaheuristic approaches for the latter version of the problem. For the first version, we have compared our approaches with the state-of-the-art approaches available in the literature. Computational results show the superiority of our approaches over previous approaches in terms of solution quality and running time both. For the latter version, hybrid artificial bee colony algorithm based approach outperformed the hybrid genetic algorithm based approach in terms of solution quality and running time both.

Erratum—One Machine Sequencing to Minimize Total Tardiness: A Fourth Theorem for Emmons

We clarify that Theorem 4 in Kanet (2014) [Kanet JJ (2014) One-machine sequencing to minimize total tardiness: A fourth theorem for Emmons. Oper. Res. 62(2):345–347] is not incremental to Theorem 3 of Emmons (1969) [Emmons H (1969) One-machine sequencing to minimize certain functions of job tardiness . Oper. Res. 17(4):701–715.] for the single-machine total tardiness problem.

A new graphical approach for solving single-machine scheduling problems approximately

Often the problem of determining an optimal or approximate production schedule in a company can be reduced to the problem of solving a scheduling problem on a bottleneck machine. However, even the majority of the resulting single-machine scheduling problems are NP-hard from a computational point of view and, therefore, it is difficult to solve large instances of such problems exactly. In this paper, we consider five such single-machine problems, where a dynamic programming algorithm of pseudo-polynomial complexity exists. The running time of such an algorithm can often be improved by so-called graphical algorithms which do not need to consider all states in a dynamic programming algorithm separately. Based on such graphical algorithms, we present fully polynomial-time approximation schemes (FPTASes) for five single-machine total tardiness problems. The new FPTASes have the best running time among the known approximation schemes for these problems. The presented approach is rather general and can be applied to many other scheduling and combinatorial optimisation problems as well.

An electromagnetism-like mechanism for the single machine total stepwise tardiness problem with release dates

In recent years, a new type of tardiness cost, called stepwise tardiness, has received attention. To the authors’ knowledge, only a few studies have investigated this type of tardiness in the scheduling problem. This study considered the single machine total stepwise tardiness problem with release dates, which is strongly NP hard. Because of the essential complexity of the problem, heuristics were first developed to quickly generate initial solutions. Subsequently, a new electromagnetism-like mechanism (EM), which is a novel metaheuristic, was proposed to improve the solution quality. The new EM includes a natural encoding scheme, a new distance measure between solutions, and effective attraction and repulsion operators. Comparisons with a current EM and other metaheuristics were performed to verify the proposed EM. The computational results show that the proposed EM exhibits good performance for the considered problem.

An Improved Tabu Search Algorithm for a Type of Single Machine Sequencing Problem

Machine scheduling is a central task in production planning. In general it means the problem of scheduling job operations on a given number of available machines. In this paper we consider a machine scheduling problem with one machine, or the Single Machine Total Tardiness Problem. To solve this NP-hard problem, we develop an improved Tabu Search Algorithm, which is tested to have the ability to find good results by an example.

A drastic hybrid heuristic algorithm to approach to JIT policy considering controllable processing times

Job scheduling has always been a challenging task in modern manufacturing and the most real life scheduling problems which involves multi-criteria, multi-machine environments. In this research, the single-machine scheduling problem is studied in which job processing times are controllable, namely, they may vary within a specified interval. The goal of this research is to minimize total tardiness and earliness on a single machine, simultaneously. In this context, we first propose a mathematical model for the considered problem and then a net benefit compression–net benefit expansion heuristic is presented for obtaining the set of amounts of compression and expansion of jobs processing times in a given sequence. Two meta-heuristic approaches are then employed to solve medium-to-large-sized problems as local search methods. Thereafter, we apply a hybrid method based on our heuristic as well as these two meta-heuristics in order to obtain solutions with higher quality within lesser computational time. The addressed problem is NP-hard since the single machine total tardiness problem is already NP-hard. The computational results show that our proposed heuristics can effectively solve such Just-In-Time problem with a high-quality solution.

The single-machine total tardiness problem with unequal release times and a linear deterioration

Recently, machine scheduling problems with deteriorating jobs have received more attention from the scheduling research community. In this paper we consider a single-machine scheduling problem with a linear deteriorating effect and unequal release times. The objective is to minimize the total tardiness. We propose a branch-and-bound algorithm incorporating with several dominance properties and two lower bounds to search for the optimal solution. In addition, we propose a marriage in honey-bees optimization algorithm (MBO) to provide a near-optimal solution. The computational experiment is also conducted to evaluate the impacts of the parameters over the performances of the proposed algorithms.

A Graphical Approach for Solving Single Machine Scheduling Problems Approximately

For five single machine total tardiness problems a fully polynomial-time approximation scheme (FPTAS) based on a graphical algorithm is presented. The FPTAS has the best running time among the known approximation schemes for these problems.

A theoretical development for the total tardiness problem and its application in branch and bound algorithms

This paper deals with the single machine total tardiness problem, and proves that if the job sequences produced by two heuristics, named as Time Forward and Time Backward algorithms, have the same starting and ending job subsequences, then there exists an optimal job sequence with the starting and ending job subsequences. The computation experiments show that there is a significant improvement of the running time of a branch and bound algorithm with the incorporation of the new property.

Single machine scheduling with controllable processing times to minimize total tardiness and earliness

In most deterministic scheduling problems job processing times are considered as invariable and known in advance. Single machine scheduling problem with controllable processing times with no inserted idle time is presented in this study. Job processing times are controllable to some extent that they can be reduced or increased, up to a certain limit, at a cost proportional to the reduction or increase. In this study, our objective is determining a set of compression/expansion of processing times in addition to a sequence of jobs simultaneously, so that total tardiness and earliness are minimized. A mathematical model is proposed firstly and afterward a net benefit compression–net benefit expansion (NBC–NBE) heuristic is presented so as to acquire a set of amounts of compression and expansion of jobs processing times in a given sequence. Three heuristic techniques in small problems and in medium-to-large instances two meta-heuristic approaches, as effective local search methods, as well as these heuristics are employed to solve test examples. The single machine total tardiness problem (SMTTP) is already NP-hard, so the considered problem is NP-hard obviously. The computational experiments demonstrate that our proposed heuristic is efficient approach for such just-in-time (JIT) problem, especially equipped with competent heuristics.

An inequality for single machine total tardiness problem

AbstractWhen solving single machine total tardiness problem with decomposition method, the key issue is evaluating the performance for two decomposition positions. An inequality is proposed in this note to deal with this problem. Such result gives a new insight into the relationship between two decomposition positions.

Solving the single-machine sequencing problem using integer programming

Various integer programming models have been proposed for sequencing problems. However, little is known about the practical value of these models. This paper reports a comparison of six different integer programming formulations of the single-machine total tardiness problem. We created a set of especially difficult test problems and attempted to solve them with each of the formulations, using CPLEX software. We found that one formulation performs much more effectively than the others. A generic integer programming approach is still not capable of solving problems with hundreds of jobs, so in that respect, it does not compete with state-of-the-art tardiness algorithms. However, the integer programming approach remains viable for problems containing as many as 40 or 50 jobs and may be the better algorithmic choice when convenience in implementation is considered.

A faster fully polynomial approximation scheme for the single-machine total tardiness problem

Abstract Lawler [E.L. Lawler, A fully polynomial approximation scheme for the total tardiness problem, Operations Research Letters 1 (1982) 207–208] proposed a fully polynomial approximation scheme for the single-machine total tardiness problem which runs in O n 7 e time (where n is the number of jobs and e is the desired level of approximation). A faster fully polynomial approximation scheme running in O ( n 5 log n + n 5 e ) time is presented in this note by applying an alternative rounding scheme in conjunction with implementing Kovalyov’s [M.Y. Kovalyov, Improving the complexities of approximation algorithms for optimization problems, Operations Research Letters 17 (1995) 85–87] bound improvement procedure.

Algorithms for special cases of the single machine total tardiness problem and an application to the even–odd partition problem

The scheduling problem of minimizing total tardiness on a single machine is known to be N P -hard in the ordinary sense. In this paper, we consider the special case of the problem when the processing times p j and the due dates d j of the jobs j , j ∈ N = { 1 , 2 , … , n } , are oppositely ordered: p 1 ≥ p 2 ≥ ⋯ ≥ p n and d 1 ≤ d 2 ≤ ⋯ ≤ d n . It is shown that already this special case is N P -hard in the ordinary sense, too. The set of jobs N is partitioned into k , 1 ≤ k ≤ n , subsets M 1 , M 2 , … , M k , M ν ⋂ M μ = 0̸ for ν ≠ μ , N = M 1 ⋃ M 2 ⋃ ⋯ ⋃ M k , such that max i , j ∈ M ν | d i − d j | ≤ min j ∈ M ν p j for each ν = 1 , 2 , … , k . We propose algorithms which solve the problem: in O ( k n ∑ p j ) time if 1 ≤ k < n ; in O ( n 2 ) time if k = n ; and in O ( n 2 ) time if max i , j ∈ N | d i − d j | ≤ 1 . The polynomial algorithms do neither require the conditions p 1 ≥ p 2 ≥ ⋯ ≥ p n mentioned above nor integer processing times to construct an optimal schedule. Finally, we apply the idea of the presented algorithm for the case k = 1 to the even–odd partition problem.

Sequencing single-machine tardiness problems with sequence dependent setup times using an iterated greedy heuristic

The single-machine tardiness problem with sequence dependent setup times is a core topic for scheduling studies. Tardiness is actually a difficult criterion to deal with, even in a relatively simple manufacturing system, such as a single-machine is strongly NP-hard. Motivated by the computational complexity of this problem, a simple iterated greedy (IG) heuristic is proposed to solve it. To validate and verify the proposed IG heuristic, computational experiments were conducted on three benchmark problem sets that included weighted and un-weighted tardiness problems. The experiment results clearly indicate that the proposed IG heuristic is highly effective as compared to the state-of-the-art meta-heuristics on the same benchmark instances. In terms of both solution quality and computational expense, this study successfully develops an effective and efficient approach for single-machine total tardiness problems with sequence dependent setup times.

Minimizing total tardiness on a single machine with controllable processing times

The concept of time-cost trade-off is commonly considered in PERT/CPM, but it is seldom considered in the scheduling area. Such concept implies that the processing times of jobs are controllable by increasing or decreasing the available resources, such as manpower and equipment. In this paper, we focus on the single machine total tardiness problem with controllable processing times. First, a mixed-integer programming (MIP) model is formulated to find the optimal solution. Then, we propose both a linear programming model and a net benefit of compression (NBC) algorithm to obtain a set of optimal amounts of compression for a given sequence. To solve medium- to large-size problem instances, we develop a heuristic based on the NBC algorithm. To verify the proposed heuristic, the MIP model is used as a comparison for small-size problem instances, whereas for medium- to large-size instances the variable neighborhood search, a useful local search method, is employed. Computational results show that the proposed heuristic has a good performance.