Approximation algorithms for some position-dependent scheduling problems

Abstract

The general position-dependent scheduling problems are studied in this paper. A 2-approximation algorithm is proposed for the learning scheduling problem on a single machine with release time to minimize the makespan. To minimize the total completion time on a single machine with release time and learning effect, we design a 2-approximation algorithm and a $PTAS$. Moreover, for the deteriorating scheduling problem to minimize the makespan on parallel machines, a 2-approximation algorithm is provided.

Introduction

For a traditional scheduling problem, the processing time of a job is assumed to be fixed. However, with the development of mechanization, this assumption seems inappropriate for modelling certain applications where the processing time changes during the processing. The model with varied processing times was first proposed by Browne and Yechiali [3], Kunnathru and Gupta [6] independently. In their papers, the processing time of job $J_j$ is denoted by $p_j=a_j+f\left(s_j\right)$, which means that in a schedule, the actual processing time $p_j$ of job $J_j$ depends on both the original processing time $a_j$ and a start time related function $f\left(s_j\right)$. The problem in which job’s actual processing time depends on its start time in the schedule is time-dependent scheduling problem. Another model is the position-dependent scheduling problem where job’s actual processing time is related with the job’s position $k$ in the schedule and denoted by $p_{jk}=f(a_j,k)$. If the function $f$ decreases with the start time or the position, the problem is a learning scheduling problem. If the function $f$ increases with the start time or the position, the problem is a deteriorating scheduling problem. Kovalyov and Kubiak [5] presented a fully polynomial approximation scheme for a deteriorating scheduling problem to minimize the makespan on a single machine. They showed that sequencing the jobs in increasing order in the ratio of processing time to deterioration rate of jobs minimizes the makespan on single machine scheduling problem. In this paper we mainly focus on the position-dependent scheduling problem.

Bachman and Janiak [2] gave a review for the position-dependent scheduling problem. $1\left|p_{jk}\right|C_{max}$ and $1\left|p_{jk}\right|\sum C_j$ can be solved in $O\left(n^3\right)$ time by the assignment integer programming, where $p_{jk}$ denotes the actual processing time of job $J_j$ at the position $k$. In the same paper, the authors also proved that $1|r_j,p_{jk}=a_j-b_{j}k|C_{max}$ and $1|r_j,p_{jk}=a_j{k}^{-b}|C_{max}$ are strongly NP-hard, where $r_j$ is the release time of job $J_j$, $a_j$ is the original processing time, $b_j$ is the parameter that depends on job $J_j$ and $b$ is a fixed parameter. Rudek [10] proved that $1|p_{jk}=a_{j}k|L_{max}$ and $1|r_j,p_{jk}=a_j(n+1-k)|C_{max}$ are strongly NP-hard. In another paper, Rudek [11] proved some solvable problems became NP-hard with a piecewise function and a learning effect and gave some properties for the branch and bound algorithm. Eren [4] and Lee et al. [7] considered the learning scheduling problems with release time. The actual processing time of job $J_j$ at the position $k$ is $p_{jk}=a_j{k}^b$ where $b$ is a constant. The objective is to minimize the makespan or the total weighted completion time. Both papers were mainly focused on the computational experiments. For the multi-machine problems, Wang [16] considered the two-machine flowshop scheduling problem $F_2\left|p_{i,j,k}\right|C_{max}$, where $p_{i,j,k}=a_{i,j}(a-bk)$ or $p_{i,j,k}=a_{i,j}{k}^b$. $a_{i,j}$ is the original processing time of job $J_j$ on machine $M_i$ and $a$, $b$ are constants. He proved that the Johnson’s Algorithm is no longer the optimal to minimize the makespan and gave the worst case ratio. Some special cases to minimize the sum of weighted completion time and lateness were also considered in his paper. Xu et al. [17] considered the uniform machines scheduling problem to minimize the total tardiness where the actual processing time is $p_{ij\left[k\right]}=a_{i,j}{k}^b$. They proposed a branch and bound algorithm and several heuristic algorithms.

Problems mentioned before are based on some specific functions. However, the learning or deteriorating effect may be too complicated to follow such regular functions in the real life. Veerle and Tjark [15] studied a single machine scheduling problem where preemptive was allowed and the speed of the machine varied with the number of completed jobs. They showed that it is NP-hard to minimize the total weighted completion time if both of the weight and processing time are arbitrary. When the weight or the original processing time of the job is unit, the problem can be solved by a greedy algorithm. A. Rudek and R. Rudek [13] studied a flowshop scheduling problem. They proved that the two-machine flowshop problem is NP-hard even if the processing time only varies on one machine. If the processing time varies on both of the machines, then the problem becomes strongly NP-hard. There are several papers that considered parallel-machine scheduling problems. For the problem to minimize the total load on parallel machines, namely, $P_m|p_{jk},DE|TL$, Mosheiov [9] gave an integer programming of the assignment problem and the algorithm can be extended to the uniform or unrelated machines. The total load is the sum of the completion time of the last job on every machine. Yu et al. [18] also studied the multi-machine scheduling problem to minimize the total load and the actual processing time depends on both of the position and the machine. Rudek [12] studied a parallel-machine scheduling problem with varied processing time to minimize the makespan. He designed a dynamic programming algorithm to obtain the optimal solution and a FPTAS to obtain the near optimal solution for some scheduling problems with learning or deteriorating effects. To solve the instances in a large scale, the author also designed a fast parallel dynamic programming algorithm.

In this paper, we focus on the general position-dependent processing time scheduling problems. In Section 2, a $2$-approximation algorithm is proposed to minimize the makespan and the total completion time for the learning scheduling problem with release time and learning effect on a single machine. A $PTAS$ is designed for the problem to minimize the total completion time. In Section 3, we design a $2$-approximation algorithm for the deteriorating scheduling problem on parallel machines to minimize the makespan. Conclusions and remarks are given in Section 4.