Single-Machine Group Scheduling Problems with Deterioration to Minimize the Sum of Completion Times

Abstract

We consider two single-machine group scheduling problems with deteriorating group setup and job processing times. That is, the job processing times and group setup times are linearly increasing (or decreasing) functions of their starting times. Jobs in each group have the same deteriorating rate. The objective of scheduling problems is to minimize the sum of completion times. We show that the sum of completion times minimization problems remains polynomially solvable under the agreeable conditions.

1. Introduction

In classical scheduling problems, the scheduling models routinely assume that job processing times are known and fixed throughout the period of job processing. However, this assumption may be unrealistic in many situations that the processing times of jobs may be prolonged due to deterioration or shortened due to learning over time. Scheduling with deteriorating jobs was first independently introduced by J. N. D. Gupta and S. K. Gupta [1] and Browne and Yechiali [2]. Since then, related models of scheduling with deteriorating jobs have been extensively studied from a variety of perspectives. Cheng et al. [3] give a detailed review of scheduling problems with deteriorating jobs. More recent papers which have considered scheduling jobs with deteriorating jobs include Wang [4], Voutsinas and Pappis [5], Lee et al. [6], Wang et al. [7], Zhu et al. [8], Cheng et al. [9, 10], Cheng et al. [11], Sun et al. [12], Wei et al. [13], Zhao and Tang [14], and Yin et al. [15, 16].

On the other hand, the production efficiency can be increased by grouping various parts and products with similar designs and/or production processes. This phenomenon is known as group technology in the literature. To the best of our knowledge, only a few results concerning scheduling problems with deteriorating effect under group technology are given. Wei and Wang [17] consider single-machine scheduling problems with group technology (GT) and deteriorating jobs. Yang [18] investigate group scheduling problems with simultaneous considerations of learning and deterioration effects on a single-machine setting. The learning phenomenon is implemented to model the setup time of groups. Three models of deteriorating for the job processing time within a group are examined. Bai et al. [19] consider single-machine group scheduling problems with effects of learning and deterioration at the same time. Lee and Lu [20] discuss a single-machine scheduling problem of minimizing the total weighted number of late jobs with deteriorating jobs and setup times. A branch-and-bound with several dominance properties and a lower bound is developed to solve the problem optimally. S. J. Yang and D. L. Yang [21] study two models of learning for the job processing time, they provide polynomial time solutions for the makespan minimization problems. Huang et al. [22] present a single-machine group scheduling problems with both learning effects and deteriorating jobs. Wu and Lee [23] consider a situation where both setup times and job-processing times are lengthened as jobs wait to be processed. Two single-machine group-scheduling problems are investigated where the group setup times and the job-processing times are both increasing functions of their starting times.

In this paper, we study the group scheduling model proposed by Lee and Wu [24], who have considered the makespan problems. In our study, we investigate the sum of completion times minimization problem under the same model as Lee and Wu [24]. In addition, we suppose jobs in each group have the same deteriorating rate. Hence, the problems considered will be noted by 1/GT𝑝_𝑖𝑗=𝑎_𝑖𝑗±𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖±𝛿_𝑖𝑡/∑𝐶_𝑖𝑗. We show that the problems remain polynomially solvable under the agreeable conditions.

The remainder of this paper is organized in four sections. The solution procedure for the linear increasing function problem is given in the next section, and the solution procedure for the linear decreasing function is described in Section 3. Then, some examples are offered in Section 4. The conclusion is given in the last section.

2. The 1/𝐺𝑇, 𝑝_𝑖𝑗=𝑎_𝑖𝑗+𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖+𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 Problem

In this section, we consider the single-machine group scheduling problem with deterioration to minimize the sum of completion times of all jobs. In our model, we follow the notation and terminology used by Lee and Wu [24]. In addition, 𝑎_𝑖[𝑗] is the basic (normal) processing time of job 𝐽_𝑖[𝑗] scheduled in the 𝑗th position of group 𝐺_𝑖, and 𝐶_𝑖[𝑗] is the completion time scheduled in the 𝑗th position of group 𝐺_𝑖. We suppose the jobs in each group have the same deterioration rate noted by 𝑏_𝑖.

Theorem 2.1. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗+𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖+𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, the optimal schedule is obtained by sequencing the jobs in each group in nondecreasing order of 𝑎_𝑖[𝑗], that is, 𝑎_𝑖[1]≤𝑎_𝑖[2]≤𝑎_𝑖[3]≤𝑎_𝑖[4]≤⋯≤𝑎_{𝑖[𝑛𝑖]}, 𝑖=1,2,3,…𝑚, (the SPT rule).

Proof. In the same group, the result can be easily obtained by using interchanging technology and is omitted.

Theorem 2.2. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗+𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖+𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, the groups are arranged in nondecreasing order of 𝜔(𝐺_𝑖), 𝜔(𝐺_𝑖)=𝑏_𝑖((1+𝑏_𝑖)^𝑛𝑖(1+𝛿_𝑖)−1)/(1+𝑏_𝑖)((1+𝑏_𝑖)^𝑛𝑖−1)(1+𝛿_𝑖), if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable conditions, that is, 𝜔(𝐺_𝑖)≥𝜔(𝐺_𝑗) if and only if 𝐵_𝑖/𝐴_𝑖≥𝐵_𝑗/𝐴_𝑗, 𝐴_𝑖=(1+𝑏_𝑖)(1−(1+𝑏_𝑖)^𝑛𝑖)(1+𝛿_𝑖)/(1−(1+𝑏_𝑖)), 𝐵_𝑖=∑^𝑛𝑖_𝑘=1(1+𝑏_𝑖)^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+(1+𝑏_𝑖)^𝑛𝑖𝜃_𝑖.

Proof. Let 𝜋₁ and 𝜋₂ be two job schedules where the difference between 𝜋₁ and 𝜋₂ is a pairwise interchange of two adjacent groups 𝐺_𝑖 and 𝐺_𝑗. That is, 𝜋₁=[𝑆₁,𝐺_𝑖,𝐺_𝑗,𝑆₂], 𝜋₂=[𝑆₁,𝐺_𝑗,𝐺_𝑖,𝑆₂], where 𝑆₁ and 𝑆₂ are partial sequences and 𝑆₁ and 𝑆₂ may be empty. Furthermore, we assume that 𝑡 denotes the completion time of the last job in 𝜋₁. To show 𝜋₁ dominates 𝜋₂, it suffices to show that ^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁+^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁≤^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂. (2.1) By definition, the completion times of jobs in 𝜋₁ are given by 𝐶_𝑖[1]𝜋₁=𝑎_𝑖[1]+1+𝑏_𝑖𝜃_𝑖+1+𝑏_𝑖1+𝛿_𝑖⋮𝐶𝑡,_{𝑖[𝑛𝑖]}𝜋₁=^𝑛𝑖_𝑘=11+𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1+𝑏_𝑖^𝑛𝑖𝜃_𝑖+1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖𝐶𝑡,_𝑗[1]𝜋₁=𝑎_𝑗[1]+1+𝑏_𝑗𝜃_𝑗+1+𝑏_𝑗1+𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁,⋮𝐶_{𝑗[𝑛𝑗]}𝜋₁=^𝑛𝑗_𝑘=11+𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1+𝑏_𝑗^𝑛𝑗𝜃_𝑗+1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁. (2.2) And the completion times of jobs in 𝜋₂ are given by 𝐶_𝑗[1]𝜋₂=𝑎_𝑗[1]+1+𝑏_𝑗𝜃_𝑗+1+𝑏_𝑗1+𝛿_𝑗⋮𝐶𝑡,_{𝑗[𝑛𝑗]}𝜋₂=^𝑛𝑗_𝑘=11+𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1+𝑏_𝑗^𝑛𝑗𝜃_𝑗+1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗𝐶𝑡,_𝑖[1]𝜋₂=𝑎_𝑖[1]+1+𝑏_𝑖𝜃_𝑖+1+𝑏_𝑖1+𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂,⋮𝐶_{𝑖[𝑛𝑖]}𝜋₂=^𝑛𝑖_𝑘=11+𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1+𝑏_𝑖^𝑛𝑖𝜃_𝑖+1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂. (2.3) Therefore, we have ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁=1+𝑏_𝑗1−1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗𝑡1−1+𝑏_𝑗+1+𝑏_𝑖1−1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂1−1+𝑏_𝑖−1+𝑏_𝑖1−1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖𝑡1−1+𝑏_𝑖−1+𝑏_𝑗1−1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁1−1+𝑏_𝑗. (2.4) By substituting 𝐴_𝑖=1+𝑏_𝑖1−1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖1−1+𝑏_𝑖,𝐴_𝑗=1+𝑏_𝑗1−1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗1−1+𝑏_𝑗,𝐵_𝑖=^𝑛𝑖_𝑘=11+𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1+𝑏_𝑖^𝑛𝑖𝜃_𝑖,𝐵_𝑗=^𝑛𝑗_𝑘=11+𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1+𝑏_𝑗^𝑛𝑗𝜃_𝑗, (2.5) we can obtain ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁=𝐴_𝑖1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗−1−𝐴_𝑗1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖−1𝑡+𝐴_𝑖𝐵_𝑗−𝐴_𝑗𝐵_𝑖. (2.6) To have ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁≥0, (2.7) when 𝑡≥0, if and only if 𝐴_𝑖1+𝑏_𝑗^𝑛𝑗1+𝛿_𝑗−1−𝐴_𝑗1+𝑏_𝑖^𝑛𝑖1+𝛿_𝑖𝐴−1≥0,_𝑖𝐵_𝑗−𝐴_𝑗𝐵_𝑖≥0. (2.8)(2.9) From the result of (2.8), we have ((1+𝑏_𝑖)^𝑛𝑖(1+𝛿_𝑖)−1)/𝐴_𝑖≤((1+𝑏_𝑗)^𝑛𝑗(1+𝛿_𝑗)−1)/𝐴_𝑗, that is to say 𝜔(𝐺_𝑖)≤𝜔(𝐺_𝑗).
And from the case of (2.9), we can obtain 𝐵_𝑖/𝐴_𝑖≤𝐵_𝑗/𝐴_𝑗.
Therefore, if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable conditions, the optimal sequence between groups are arranged in nondecreasing order of 𝜔(𝐺_𝑖), This completes the proof.

From Theorems 2.1 and 2.2, if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable condition, the problem 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗+𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖+𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 can be solved by the following algorithm.

Algorithm 2.3. Step  1. Jobs in each group are scheduled in nondecreasing order of the basic processing time 𝑎_𝑖[𝑗], that is, 𝑎_𝑖[1]≤𝑎_𝑖[2]≤𝑎_𝑖[3]≤𝑎_𝑖[4]≤⋯≤𝑎_{𝑖[𝑛𝑖]}, 𝑖=1,2,3,…𝑚, (the SPT rule).
Step  2. Calculate 𝜔(𝐺_𝑖)=𝑏_𝑖((1+𝑏_𝑖)^𝑛𝑖(1+𝛿_𝑖)−1)/((1+𝑏_𝑖)(1+𝑏_𝑖)^𝑛𝑖−1)(1+𝛿_𝑖).
Step  3. Groups are scheduled in nondecreasing order of 𝜔(𝐺_𝑖), that is, 𝜔(𝐺₁)≤𝜔(𝐺₂)≤𝜔(𝐺₃)≤⋯≤𝜔(𝐺_𝑚).
Obviously, It is easy to show that the total time for Algorithm 2.3 is 𝑂(𝑛log𝑛).

3. The 1/𝐺𝑇, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 Problem

In order to discuss the problem 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 conveniently, we suppose 0≤𝑏_𝑖≤1 and 0≤𝜃_𝑖≤1.

Theorem 3.1. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, the optimal schedule is obtained by sequencing the jobs in each group in nondecreasing order of basic processing time 𝑎_𝑖[𝑗], that is, 𝑎_𝑖[1]≤𝑎_𝑖[2]≤𝑎_𝑖[3]≤𝑎_𝑖[4]≤⋯≤𝑎_{𝑖[𝑛𝑖]}, 𝑖=1,2,3,…𝑚, (the SPT rule).

Proof. In the same group, the result can be easily obtained by using interchanging technology and is omitted.

Theorem 3.2. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, the groups are arranged in nondecreasing order of 𝜔(𝐺_𝑖), 𝜔(𝐺_𝑖)=𝑏_𝑖((1−𝑏_𝑖)^𝑛𝑖(1−𝛿_𝑖)−1)/(1−𝑏_𝑖)(1−(1−𝑏_𝑖)^𝑛𝑖)(1−𝛿_𝑖), if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable conditions, that is, 𝐵_𝑖/𝐴_𝑖≥𝐵_𝑗/𝐴_𝑗 if and only if 𝜔(𝐺_𝑖)≥𝜔(𝐺_𝑗), 𝐴_𝑖=(1−𝑏_𝑖)(1−(1−𝑏_𝑖)^𝑛𝑖)(1−𝛿_𝑖)/(1−(1−𝑏_𝑖)), and 𝐵_𝑖=∑^𝑛𝑖_𝑘=1(1−𝑏_𝑖)^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+(1−𝑏_𝑖)^𝑛𝑖𝜃_𝑖.

Proof. Here, we still use the same notations as that in Theorem 2.2. To show 𝜋₁ dominates 𝜋₂, it suffices to show that ^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁+^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁≤^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂. (3.1) By definition, the completion times of jobs in 𝜋₁ are given by 𝐶_𝑖[1]𝜋₁=𝑎_𝑖[1]+1−𝑏_𝑖𝜃_𝑖+1−𝑏_𝑖1−𝛿_𝑖⋮𝐶𝑡,_{𝑖[𝑛𝑖]}𝜋₁=^𝑛𝑖_𝑘=11−𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1−𝑏_𝑖^𝑛𝑖𝜃_𝑖+1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖𝐶𝑡,_𝑗[1]𝜋₁=𝑎_𝑗[1]+1−𝑏_𝑗𝜃_𝑗+1−𝑏_𝑗1−𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁,⋮𝐶_{𝑗[𝑛𝑗]}𝜋₁=^𝑛𝑗_𝑘=11−𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1−𝑏_𝑗^𝑛𝑗𝜃_𝑗+1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁. (3.2) And the completion times of jobs in 𝜋₂ are given by 𝐶_𝑗[1]𝜋₂=𝑎_𝑗[1]+1−𝑏_𝑗𝜃_𝑗+1−𝑏_𝑗1−𝛿_𝑗⋮𝐶𝑡,_{𝑗[𝑛𝑗]}𝜋₂=^𝑛𝑗_𝑘=11−𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1−𝑏_𝑗^𝑛𝑗𝜃_𝑗+1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗𝐶𝑡,_𝑖[1]𝜋₂=𝑎_𝑖[1]+1−𝑏_𝑖𝜃_𝑖+1−𝑏_𝑖1−𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂,⋮𝐶_{𝑖[𝑛𝑖]}𝜋₂=^𝑛𝑖_𝑘=11−𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1−𝑏_𝑖^𝑛𝑖𝜃_𝑖+1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂. (3.3) Therefore, we have ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁=1−𝑏_𝑗1−1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗𝑡1−1−𝑏_𝑗+1−𝑏_𝑖1−1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖𝐶_{𝑗[𝑛𝑗]}𝜋₂1−1−𝑏_𝑖−1−𝑏_𝑖1−1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖𝑡1−1−𝑏_𝑖−1−𝑏_𝑗1−1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗𝐶_{𝑖[𝑛𝑖]}𝜋₁1−1−𝑏_𝑗. (3.4) By substituting 𝐴_𝑖=1−𝑏_𝑖1−1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖1−1−𝑏_𝑖,𝐴_𝑗=1−𝑏_𝑗1−1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗1−1−𝑏_𝑗,𝐵_𝑖=^𝑛𝑖_𝑘=11−𝑏_𝑖^{𝑛𝑖−𝑘}𝑎_𝑖[𝑘]+1−𝑏_𝑖^𝑛𝑖𝜃_𝑖,𝐵_𝑗=^𝑛𝑗_𝑘=11−𝑏_𝑗^{𝑛𝑗−𝑘}𝑎_𝑗[𝑘]+1−𝑏_𝑗^𝑛𝑗𝜃_𝑗, (3.5) we can obtain ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁=𝐴_𝑖1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗−1−𝐴_𝑗1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖−1𝑡+𝐴_𝑖𝐵_𝑗−𝐴_𝑗𝐵_𝑖. (3.6) To have ^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₂+^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₂−^𝑛𝑖_𝑙=1𝐶_𝑖[𝑙]𝜋₁−^𝑛𝑗_𝑙=1𝐶_𝑗[𝑙]𝜋₁≥0, (3.7) when 𝑡≥0 if and only if 𝐴_𝑖1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗−1−𝐴_𝑗1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖𝐴−1≥0,_𝑖𝐵_𝑗−𝐴_𝑗𝐵_𝑖≥0. (3.8)(3.9) From the result of (3.8), we have 1−𝑏_𝑖^𝑛𝑖1−𝛿_𝑖−1𝐴_𝑖≤1−𝑏_𝑗^𝑛𝑗1−𝛿_𝑗−1𝐴_𝑗, (3.10) that is to say 𝜔(𝐺_𝑖)≤𝜔(𝐺_𝑗).
And from the case of (3.9), we can obtain 𝐵_𝑖/𝐴_𝑖≤𝐵_𝑗/𝐴_𝑗.
Therefore, if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable conditions, the optimal sequence between groups are arranged in nondecreasing order of 𝜔(𝐺_𝑖), This completes the proof.

From Theorems 3.1 and 3.2, if 𝜔(𝐺_𝑖) and 𝐵_𝑖/𝐴_𝑖 have agreeable condition, the problem 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 can be solved by the following algorithm.

Algorithm 3.3. Step  1. Jobs in each group are scheduled in nondecreasing order of the basic processing time 𝑎_𝑖[𝑗], that is, 𝑎_𝑖[1]≤𝑎_𝑖[2]≤𝑎_𝑖[3]≤𝑎_𝑖[4]≤⋯≤𝑎_{𝑖[𝑛𝑖]}, 𝑖=1,2,3,…𝑚, (the SPT rule).
Step  2. Calculate 𝜔(𝐺_𝑖)=𝑏_𝑖((1−𝑏_𝑖)^𝑛𝑖(1−𝛿_𝑖)−1)/(1−𝑏_𝑖)(1−(1−𝑏_𝑖)^𝑛𝑖)(1−𝛿_𝑖).
Step  3. Groups are scheduled in nondecreasing order of 𝜔(𝐺_𝑖), that is, 𝜔(𝐺₁)≤𝜔(𝐺₂)≤𝜔(𝐺₃)≤⋯≤𝜔(𝐺_𝑚).
Obviously, It is easy to show that the total time for Algorithm 3.3 is 𝑂(𝑛log𝑛).

4. Examples

Example 4.1. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗+𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖+𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, we consider six jobs are divided into three groups. The basic job and group processing times, the job and group deterioration rates for each group are given, respectively, as follows: 𝑝₁₁=5,𝑝₁₂=8,𝑝₂₁=4,𝑝₂₂=10,𝑝₂₃=6,𝑝₃₁𝑏=7,₁=0.04,𝑏₂=0.02,𝑏₃=0.01,𝜃₁=5,𝜃₂=4,𝜃₃𝛿=6,₁=0.01,𝛿₂=0.05,𝛿₃=0.04. (4.1)

Solution 1. According to Algorithm 2.3, we solve the example as follows.

Step 1. In group 𝐺₁, the optimal job sequence is 𝐽₁₁→𝐽₁₂.
In group 𝐺₂, the optimal job sequence is 𝐽₂₁→𝐽₂₃→𝐽₂₂.
In group 𝐺₃, the optimal job sequence is 𝐽₃₁.

Step 2. calculate: 𝐵₁/𝐴₁=8.683, 𝐵₂/𝐴₂=7.483, 𝐵₃/𝐴₃=12.483, and 𝜔(𝐺₁)=0.043, 𝜔(𝐺₂)=0.035, 𝜔(𝐺₃)=0.048. It is easy to say 𝐵₂/𝐴₂<𝐵₁/𝐴₁<𝐵₃/𝐴₃ and 𝜔(𝐺₂)<𝜔(𝐺₁)<𝜔(𝐺₃). Therefore, the optimal group sequence is 𝐺₂→𝐺₁→𝐺₃ and the optimal schedule is [𝐽₂₁→𝐽₂₃→𝐽₂₂]→[𝐽₁₁→𝐽₁₂]→[𝐽₃₁]. The completion times of jobs are 𝐶₂₁=8.08, 𝐶₂₃=14.242, 𝐶₂₂=24.527, 𝐶₁₁=35.963, 𝐶₁₂=45.402, and 𝐶₃₁=60.750, and the sum of completion times is ∑𝐶_𝑖𝑗=8.08+14.242+24.527+35.963+45.402+60.750=188.964.

Example 4.2. For the 1/GT, 𝑝_𝑖𝑗=𝑎_𝑖𝑗−𝑏_𝑖𝑡, 𝑠_𝑖=𝜃_𝑖−𝛿_𝑖𝑡/∑𝐶_𝑖𝑗 problem, we consider six jobs are divided into three groups as Example 4.1, the job and group deterioration rates for each group are given, respectively, as follows: 𝑝₁₁=9,𝑝₁₂=7,𝑝₁₃=5,𝑝₂₁=10,𝑝₃₁=8,𝑝₃₂𝑏=6,₁=0.03,𝑏₂=0.05,𝑏₃=0.02,𝜃₁=4,𝜃₂=8,𝜃₃𝛿=5,₁=0.15,𝛿₂=0.01,𝛿₃=0.1. (4.2)

Solution 2. According to Algorithm 2.3, we solve the example as follows.

Step 1. In group 𝐺₁, the optimal job sequence is 𝐽₁₃→𝐽₁₂→𝐽₁₁.
In group 𝐺₂, the optimal job sequence is 𝐽₂₁.
In group 𝐺₃, the optimal job sequence is 𝐽₃₂→𝐽₃₁.

Step 2. calculate: 𝐵₁/𝐴₁=10.060, 𝐵₂/𝐴₂=18.713, 𝐵₃/𝐴₃=10.700, and 𝜔(𝐺₁)=−0.093, 𝜔(𝐺₂)=−0.063, 𝜔(𝐺₃)=−0.078. It is easy to say 𝐵₁/𝐴₁<𝐵₃/𝐴₃<𝐵₂/𝐴₂ and 𝜔(𝐺₁)<𝜔(𝐺₃)<𝜔(𝐺₂). Therefore, the optimal group sequence is 𝐺₁→𝐺₃→𝐺₂ and the optimal schedule is [𝐽₁₃→𝐽₁₂→𝐽₁₁]→[𝐽₃₂→𝐽₃₁]→[𝐽₂₁]. The completion times of jobs are 𝐶₁₃=8.88, 𝐶₁₂=15.614, 𝐶₁₁=24.146, 𝐶₃₂=34.326, 𝐶₃₁=41.639, 𝐶₂₁=53.201, and the sum of completion times is ∑𝐶_𝑖𝑗=8.88+15.614+24.146+34.326+41.639+53.201=177.806.

5. Conclusions

In this paper, we investigate the single-machine group scheduling problems with deterioration which is studied by Lee and Wu [24]. We consider the sum of completion times minimization problem, and we show the problem to minimize total completion time of all jobs is polynomially solvable under agreeable conditions in our study.

Acknowledgments

The authors thank the valuable comments of the referees for an earlier version of this paper. Their comments have significantly improved the paper. This work is supported by the National Natural Science Foundation of China (no. 71001025). Also, this research is partly supported by the Program for New Century Excellent Talents in University (no. NCET-10-0327) and the Ministry of Education of China: Grant-in-aid for Humanity and Social Science Research (no. 11YJCZH139).