Wlodzimierz Szwarc

Single-machine scheduling to minimize absolute deviation of completion times from a common due date

The article considers a single-machine n-job scheduling problem to minimize the sum of absolute lateness given a common due date. Two models are defined depending on whether the start time t0 of schedules is arbitrary or fixed. Conditions are provided when those two models coincide. The developed branch-and-bound procedure is tested on nine known examples from the literature (6 ⩽ n ⩽ 14) and 90 medium-size random problems (15 ⩽ n ⩽ 25) with a fixed t0.

Solution of the single machine total tardiness problem

The paper deals with the solution of the single machine total tardiness model. It improves and generalizes an important rule to decompose the model into two subproblems. It also provides a O(n2) procedure to implement this rule and its generalization. Those two rules, along with some known results, are incorporated in a branch and bound algorithm that efficiently handles instances with up to 300 jobs and uses the original and maximally increased due dates to solve the original problem. Several properties that justify the modified due date version of our algorithm and produce an easy-to-implement new lower bound are established. The paper also provides an explanation why using the increased due dates may improve the efficiency of certain algorithms. Copyright

Optimal timing schedules in earliness-tardiness single machine sequencing

The article deals with a single machine earliness-tardiness scheduling model where idle times are permitted in job processing. Based on a cluster concept we develop properties of the model that lead to a very fast algorithm to find an optimal timing schedule for a given sequence of jobs. The performance of this algorithm is tested on 480 randomly generated problems involving 100, 200, 400 and 500 jobs. It takes less than two seconds to solve a 500 job problem on a PC.

Optimal Elimination Methods in the m × n Flow-Shop Scheduling Problem

For solving the flow-shop scheduling problem, this paper examines elimination techniques that reduce the set of solutions to a subset that must contain the optimal solution being sought. The paper shows 1 that the elimination method of Szwarc [Naval Res. Log. Quart. 18, 295-305 1971] removes at least as many solutions as any other method, and is therefore optimal, and 2 how to construct a general counterexample to any procedure that removes more sequences than this optimal method.

Decomposition of the single machine total tardiness problem

The paper deals with the single machine total tardiness problem. It develops a new decomposition rule and presents a special and very fast branch and bound algorithm based on pure decomposition. This algorithm is tested on 2400 problems whose sizes vary from 100 to 150 jobs.

Minimizing the weighted sum of quadratic completion times on a single machine

This article discusses the scheduling problem of minimizing the weighted sum of quadratic completion times on a single machine. It establishes links between orderings of adjacent and nonadjacent jobs that lead to a powerful branch and bound method. Computational results show that this method clearly outperforms the state of the art algorithm.

Parametric precedence relations in single machine scheduling

The paper provides a theoretical background to solve a variety of single machine scheduling problems with quadratic separable functions of completion times including waiting time and due date models. The approach is based on a parametric ordering as well as an adjacent precedence matrix concept. Rules are presented to decompose the problem into separate subproblems. Those rules are incorporated in a branch and bound procedure that also utilizes other properties of the precedence matrix. Necessary optimality conditions are given for two common due date models where lower bounds are not yet available.

Optimal Two-Machine Orderings in the 3 × n Flow-Shop Problem

Consider the 3 × n flow-shop problem with machines A, B, and C. By AB and BC optimum we mean optimal solutions produced by Johnsons method for the two-machine problems AB and BC, respectively. Although not every permutation that is an AB and BC optimum simultaneously is an optimal solution of the entire problem, the set P of such permutations does contain an ABC optimum whenever it is not empty. We summarize known analytical results and solution procedures for the 3 × n flow-shop problem.

The weighted common due date single machine scheduling problem revisited

Abstract This paper deals with the earliness-tardiness common due date model of minimizing the sum of earliness and tardiness penalties. We assume that those penalties vary from job to job and that the ratios of processing times and penalties satisfy certain (called agreeable) conditions. Three types of schedules (V0, V1, and W) are identified where the complexity of W schedules far exceeds the complexity of the remaining types. Conditions are derived that significantly improve the existing methods to find the best W schedule. This improvement is demonstrated on 1500 test problems of the sizes n = 10, 20, 50, 70 and 100.

Technical Note-Dominance Conditions for the Three-Machine Flow-Shop Problem

This note examines the mathematical structure and applicability of a dominance condition of Gupta and Reddi for the three-machine flow-shop problem. It shows that this condition may eliminate fewer sequences than another dominance condition. A sequential procedure of the combined strength of the two conditions is presented. We also derive necessary conditions for the Gupta-Reddi condition.