Single machine scheduling problem with controllable processing times and resource dependent release times

doi:10.1016/j.ejor.2006.07.005

European Journal of Operational Research

Volume 181, Issue 2, 1 September 2007, Pages 645-653

https://doi.org/10.1016/j.ejor.2006.07.005 Get rights and content

Abstract

We consider two single machine scheduling problems with resource dependent release times and processing times, in which the release times and processing times are linearly decreasing functions of the amount of resources consumed. The objective is to minimize the total cost of makespan and resource consumption function that is composed of release time reduction and processing time reduction. In the first problem, the cost of reducing a unit release time for each job is common. We show that the problem can be solved in polynomial time. The second problem assumes different reduction costs of job release times. We show that the problem can be reduced polynomially from the partition problem and thus, is NP-complete.

Introduction

For single machine scheduling problems, it is generally assumed that release times and processing times of jobs are known and constant. However, in many cases, the release times and the processing times can be made earlier and shorter by using additional resources such as manpower, fuels, raw materials and so on. Many researchers have focused on problems with controllable processing times and problems with resource dependent release times, independently.

A single machine scheduling problem with controllable processing times was initiated by Vickson [16]. He assumed that the job processing times were a linear function of the resources consumed and the objective was to minimize the total weighted flow cost plus controllable job processing cost. Janiak [5] considered single machine scheduling problems with a linear resource consumption function in which the objective was to minimize regular performance measures, e.g. the maximal job completion time (makespan), maximum lateness and so on. He proposed several algorithms that could solve the problems in polynomial time. Janiak and Kovalyov [7] studied a single machine scheduling problem where job processing times depended linearly on the amount of resources consumed, in which the objective was to minimize the total weighted resource consumption subjective to deadlines. They showed that the problem with continuously divisible resource could be solved in polynomial time, but the problem with discrete resource was NP-complete. Hoogeveen and Woeginger [3] considered a single machine sequencing problem with controllable job processing times, in which the objective was to minimize the cost of the total weighted job completion time plus compression cost of job processing times that was a linear function of the processing times. They showed that the problem was NP-complete. Kaspi and Shabtay [11] considered a single machine scheduling problem with job processing times that assumed a convex decreasing resource consumption function, in which the objective was to minimize the makespan subjective to a common limited resource. They proposed two polynomial time algorithms for the cases of identical and non-identical job release times. Shatay and Kaspi [15] considered a single machine scheduling problem with convex resource consumption function to minimize the total weighted flow time. Several researchers [2], [9], [13] considered single machine group scheduling problems with controllable processing times, in which jobs were partitioned into groups and the machine processed jobs of the same group contiguously. Janiak et al. [10] considered the single machine scheduling problem with controllable job processing times to minimize a linear combination of the total weighted job completion time and the total weighted processing time compression. They showed that the problem was polynomially equivalent to the positive half-product minimization problem. Nowicki and Zdrzalka [14] made a survey of single machine scheduling problems with controllable job processing times.

For a few decades, single machine scheduling problems with resource dependent release times have been studied extensively. Janiak [4] considered single machine scheduling problems in which the objective was to minimize the makespan subjective to the total resource consumption. He assumed that all jobs had a common resource consumption function. He showed that the problem could be efficiently solved by ordering jobs according to non-increasing job processing times. Cheng and Janiak [1] considered a problem that exchanged the objective function with the resource constraint in the Janiak problem [4]. Janiak [6] generalized the research of Janiak [4] by considering different resource consumption functions, and showed that the problem was NP-complete in the strong sense. For the problem of Janiak [6], Janiak [8] presented some polynomially solvable cases and proposed some approximation algorithms with good worst-case performance ratios. Li [12] considered a single machine scheduling problem in which the objective was to minimize the total resource consumption cost plus the makespan, and showed that the problem could be solved in polynomial time.

Recently, Wang and Cheng [17] addressed a single machine scheduling problem, in which both release times and processing times could be controlled by the amount of the resource consumed. The objective was to minimize the makespan plus total resource consumption cost. They mentioned the problem was the generalization of one problem in Janiak [6] that they thought was NP-complete. However, the Janiak’s problem was known to be polynomially solvable [6], and Wang and Cheng’s claim that their problem was NP-complete might be wrong.

In this paper, we consider the Wang and Cheng’s problem for the cases of common or different reduction costs of release times, in which reduction costs of job processing times are not limited. The rest of this paper is organized as follows. In Section 2, we define our problems. Notations and assumptions used in the paper are also presented there. In Section 3, we show that the problem with a common reduction cost of release times can be solved in polynomial time. In Section 4, we consider the problem with different reduction costs of release times and prove that the problem is NP-complete. Finally, we provide the summary and concluding remarks.

Section snippets

Notations and problem definition

The following notations will be used through the paper:

J_j

job j

a_j

initial processing time of job j

x_j

amount of processing times reduced for job j

p_j

actual processing time of job j

c_j

cost of reducing a unit processing time of job j

initial release time of jobs

w_j

cost of reducing a unit release time of job j

u_j

cost of the resource consumed with respect to w_j (the calculation of u_j is defined below)

r_j

actual release time of job j

σ = {σ(1), … , σ(n)}

job sequence, where σ(j) = k implies that job k is positioned j

Some optimality conditions for Problem SP

The following three lemmas discuss necessary optimal conditions for Problem SP. Let the objective function of Problem SP be K₁(x, r, σ).

Lemma 1

There exists no intermediate idle time in an optimal solution.

Proof

It follows from Wang and Cheng [17]. □

By Lemma 1, K₁(x, r, σ) can be written as

K_{1} (x, r, σ) = \max_{1 ⩽ j ⩽ n} \{r_{σ (j)} + \sum_{i = j}^{n} (a_{σ (i)} - x_{σ (i)})\} + \sum_{i = 1}^{n} c_{σ (i)} x_{σ (i)} + \sum_{i = 1}^{n} w (v - r_{σ (i)}) = r_{σ (1)} + \sum_{i = 1}^{n} (a_{σ (i)} - x_{σ (i)}) + \sum_{i = 1}^{n} c_{σ (i)} x_{σ (i)} + \sum_{i = 1}^{n} w (v - r_{σ (i)}) .

Lemma 1 implies Lemma 2.

Lemma 2

Let (n − k) jobs be totally compressed in an optimal solution (i.e., the

NP-completeness of Problem GP

In this section, we show that Problem GP is NP-complete by reducing Partition Problem to it. Since Lemma 1, Lemma 2, Lemma 3 hold for Problem GP, the objective function K(x, r, σ) of Problem GP can be written as $K (x, r, σ) = r_{σ (1)} + \sum_{i = 1}^{n} (a_{σ (i)} - x_{σ (i)}) + \sum_{i = 1}^{n} c_{σ (i)} x_{σ (i)} + \sum_{i = 1}^{n} w_{σ (i)} (v - r_{σ (i)}),$ where $\sum_{j = 1}^{n} a_{j} < v$ .

Recognition version of Problem GP

Given an initial processing time a_i, a cost of reducing a unit processing time c_i, a cost of reducing a unit release time w_i, an initial release time v and a threshold value L, is there a solution (x, r, σ)

Summary and conclusion

In this paper, we consider two single machine scheduling problems with resource dependent release times and processing times, in which the objective is to minimize the total cost of makespan and resource consumption function. The first problem (Problem SP) assumes that costs of reducing job release times are common. We show that the problem can be solved in O(n⁵). The second problem (Problem GP) assumes different costs of reducing job release times. We show that the problem is NP-complete. The

Acknowledgement

This work was supported by the Soongsil University Research Fund.

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There are more references available in the full text version of this article.

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Discrete OptimizationSingle machine scheduling problem with controllable processing times and resource dependent release times

Abstract

Introduction

Section snippets

Notations and problem definition

Some optimality conditions for Problem SP

NP-completeness of Problem GP

Summary and conclusion

Acknowledgement

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Computers and Operations Research

Discrete Applied Mathematics

Computers and Operations Research

Discrete Optimization
Single machine scheduling problem with controllable processing times and resource dependent release times