This paper reviews static scheduling research on single machine for the last three decades One hundred seventy one papers were found on this topic after an extensive search in the English language periodicals These papers are classified on the basis of the objective functions and constraints Contribution of each paper is analyzed in terms of its theoretical and computational results An analysis of the literature on the basis of number of articles published in each year, in each category, in different journals and by different authors has also been provided This reveals several interesting facts about the developments in single machine scheduling (SMS) research and its future

Single machine scheduling research

Some Real-Time systems may need a multivalence description of tasks. A generic way to achieve it consists in characterizing each task by a Time Value Function (TVF), which gives the contribution of the related task at its actual completion time. This approach can be viewed as a complementary paradigm to the bivalent deadline-driven paradigm, especially in the case of overload. In this paper, we investigate the problem of scheduling a set of TVF-based tasks under the criterion of maximizing the sum of tasks' contributions. To deal with this NP-hard problem, we use a decomposition approach. The latter consists in partitionning the initial task set into several subsets for which we can establish a scheduling order (on the subset level). We propose an algorithm which yields sub-optimal scheduling and finds out the optimal decomposition. This O(n

 3
 ) on-line scheduling algorithm yields sequences which are equal or statistically very close to the optimum, as suggested by our simulation study for various scenarii.

A scheduling algorithm for tasks described by time value function

Sequencing jobs on a single machine with a common due date and stochastic processing times

We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith’s rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k(k − 1)s(k p 1), denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time.

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On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

In this paper we are concerned with the problem of sequencing a given set of jobs without preemption on a single machine so as to minimize total cost, where associated with each job is a processing time and a differentiable cost function defined on the completion time of the job. The problem, in general, is NP-complete and, therefore, there is unlikely to be an algorithm to solve the problem in reasonable time, thus a heuristic algorithm is desirable. We present two heuristic algorithms to solve the problem. The first algorithm is based on the differential of the cost functions, and the second algorithm is based on the least square approximation of the cost functions. Computational experiences for the case of quadratic, cubic, and exponential cost functions are presented.

https://link.springer.com/content/pdf/10.1057%2Fjors.1993.24.pdf

Numerical Methods for Single Machine Scheduling with Non-linear Cost Functions to Minimize Total Cost

In this note, a node elimination procedure has been suggested in case the two sequences obtained by using Townsend's Townsend, W. 1978. The single machine problem with quadratic penalty function of completion times: A branch and bound solution. Management Sci.24 5 530-534. sufficient conditions for solving the single machine quadratic penalty function scheduling problem contain a subset Jr of r jobs in the first r positions. Numerical illustrations and computational experience has been given in the end.

K. R. Kalra

Papers

Note-A Node Elimination Procedure for Townsend's Algorithm for Solving the Single Machine Quadratic Penalty Function Scheduling Problem