Scheduling Jobs with Variable Job Processing Times on Unrelated Parallel Machines

Abstract

m unrelated parallel machines scheduling problems with variable job processing times are considered, where the processing time of a job is a function of its position in a sequence, its starting time, and its resource allocation. The objective is to determine the optimal resource allocation and the optimal schedule to minimize a total cost function that dependents on the total completion (waiting) time, the total machine load, the total absolute differences in completion (waiting) times on all machines, and total resource cost. If the number of machines is a given constant number, we propose a polynomial time algorithm to solve the problem.

1. Introduction

In classical scheduling theory, it is assumed that the job processing times are fixed and constant values. In practice, however, we often encounter settings in which job processing times may be subject to change due to the deterioration effects and/or learning effects and/or controllable processing times. Extensive surveys of different scheduling models and problems involving deteriorating jobs can be found in Alidaee and Womer [1], Cheng et al. [2], and Gawiejnowicz [3]. A recent survey of the scheduling problems with learning effects could be found in Biskup [4]. A recent survey of the scheduling problems with controllable processing times was given by Shabtay and Steiner [5]. More recent papers that have considered scheduling problems with deterioration effects and/or learning effects and/or controllable processing times include Bai et al. [6, 7], Cheng et al. [8], Gorczyca and Janiak [9], Hsu and Yang [10], Huang and Wang [11], Lee et al. [12], Leyvand et al. [13], Nian and Mao [14], Shabtay and Steiner [15], Wang et al. [16], J. B. Wang and M. Z. Wang [17], J. B. Wang and J. J. Wang [18], Wang et al. [19–21], Wei et al. [22], Wu and Lee [23], Wu and Liu [24], Yang and Kuo [25], S. J. Yang and D. L. Yang [26], Yang et al. [27], Yin et al. [28–31], and Zhao and Tang [32]. Mosheiov and Sidney [33] considered the single machine model: 𝑝_𝑗=𝑎_𝑗𝑟^𝑏𝑗, (1) where 𝑎_𝑗 is the original (normal) processing time of job 𝐽_𝑗, 𝑝_𝑗 is the actual processing time of job 𝐽_𝑗, 𝑟 is the position of job 𝐽_𝑗 when scheduled on the machine, and 𝑏_𝑗≤0 is the job-dependent learning index of job 𝐽_𝑗. Yang and Kuo [25] considered single machine scheduling with the deterioration and learning effect in which the job processing times are 𝑝_𝑗=𝑎_𝑗𝑟^𝑏+𝛼𝑡, (2) where 𝑡 is the starting time of job 𝐽_𝑗, 𝑏≤0 is a learning index, and 𝛼≥0 is a common deterioration rate for all the jobs. Shabtay and Steiner [15] considered single machine scheduling with the resource allocation model in which the job processing times are 𝑝_𝑗=𝑎_𝑗−𝛽_𝑗𝑢_𝑗, (3) where 𝑢_𝑗 is the amount of a nonrenewable resource allocated to job 𝐽_𝑗, with 0≤𝑢_𝑗≤𝑢_𝑗<𝑎_𝑗/𝛽_𝑗, where 𝑢_𝑗 denote the maximum amount of resource allocated to job 𝐽_𝑗 and 𝛽_𝑗 is the positive compression rate of job 𝐽_𝑗.

Wang et al. [19] considered single machine scheduling problem where the job processing time is a function of position in a sequence, starting time, and resource allocation to this job on a single machine; that is, the model 𝑝_𝑗=𝑎_𝑗𝑟^𝑏+𝛼𝑡−𝛽_𝑗𝑢_𝑗, (4) where 0≤𝑢_𝑗≤𝑢_𝑗<𝑎_𝑗𝑛^𝑏/𝛽_𝑗. For minimizing a cost function containing makespan, total completion (waiting) time, total absolute differences in completion (waiting) times, and total resource cost, they proved that the problem can be solved in polynomial time.

Wang et al. [19] considered single machine scheduling problem with effect of deterioration, learning, and resource allocation simultaneously. The example of the phenomena of deterioration, learning effect, and resource allocation occurring simultaneously can be found in Wang et al. [19]. However, the parallel machine scheduling is interesting and closer to real problems in practice (Cheng et al. [8], Hsu and Yang [10], Huang and Wang [11], Yang et al. [27], and Yin et al. [30]). This paper extends the single machine scheduling results of Wang et al. [19], by considering unrelated parallel machine scheduling problems that include the one given in Wang et al. [19] as a special case. The remainder of this paper is organized as follows. In Section 2 we formulate the model. In Sections 3 and 4, we show that the problem can be solved in polynomial time for two different cost functions. The last section contains some conclusions.

2. Problem Formulation

The model is described as follows. There are 𝑛 independent jobs {𝐽₁,𝐽₂,…,𝐽_𝑛} to be processed on 𝑚 unrelated parallel machines {𝑀₁,𝑀₂,…,𝑀_𝑚}. Each of them is available at time 0. The machine can handle one job at a time, and preemption is not allowed. Let 𝑛_𝑖 denote the number of jobs assigned to 𝑀_𝑖(𝑖=1,2,…,𝑚) and 𝑃(𝑛,𝑚)=(𝑛₁,𝑛₂,…,𝑛_𝑚) denote a job-allocation vector, where (𝑛₁+𝑛₂+⋯𝑛_𝑚=𝑛). We assume, as in most practical situations, that 𝑚<𝑛 and 𝑚 are a given constant. Each job can be processed on any one of the 𝑚 unrelated parallel machines. Associated with each job 𝐽_𝑗(𝑗=1,2,…,𝑛) on machine 𝑀_𝑖, there is a normal processing time 𝑎_𝑖𝑗. Let 𝑝_𝑖𝑗 denote the actual processing time for job 𝐽_𝑗 on machine 𝑀_𝑖. In this paper, we consider a general unrelated parallel machines model stemming from Yang and Kuo [25], Shabtay and Steiner [15], and Wang et al. [19]; that is, 𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗, (5) where 𝑟 is the position of job 𝐽_𝑗 when scheduled on machine 𝑀_𝑖, 𝑡 is the starting time of job 𝐽_𝑗 on machine 𝑀_𝑖, 𝑏_𝑖𝑗≤0 is a job-dependent learning index, 𝛼≥0 is a common deterioration rate for all the jobs, 𝜃_𝑖𝑗≥0 is the positive compression rate of job 𝐽_𝑗 on machine 𝑀_𝑖, and 𝑢_𝑖𝑗 is the amount of resource that can be allocated to job 𝐽_𝑗 on machine 𝑀_𝑖, with 0≤𝑢_𝑖𝑗≤𝑢_𝑖𝑗<𝑎_𝑖𝑗𝑛^𝑏𝑖𝑗/𝜃_𝑖𝑗, where 𝑢_𝑖𝑗 is the upper bound on the amount of resource that can be allocated to job 𝐽_𝑗 on machine 𝑀_𝑖.

Let 𝐽_𝑖[𝑗] denote the 𝑗th job on machine 𝑀_𝑖 and 𝐶_𝑖[𝑗](𝑊_𝑖[𝑗]) denote the completion (waiting) time of job 𝐽_𝑖[𝑗]; 𝑝_𝑖[𝑗], 𝑎_𝑖[𝑗], 𝑏_𝑖[𝑗], 𝜃_𝑖[𝑗], 𝑢_𝑖[𝑗], and 𝑢_𝑖[𝑗] are defined similarly, where 𝑊_𝑖[𝑗]=𝐶_𝑖[𝑗]−𝑝_𝑖[𝑗]. Let 𝐿_𝑖=max{𝐶_𝑖𝑗∣𝑗=1,2,…,𝑛_𝑖}, 𝑇𝐶_𝑖=∑^𝑛𝑖_𝑗=1𝐶_𝑖𝑗 (𝑇𝑊_𝑖=∑^𝑛𝑖_𝑗=1𝑊_𝑖𝑗), and 𝑇𝐴𝐷𝐶_𝑖=∑^𝑛𝑖_𝑗=1∑^𝑛𝑖_𝑙=𝑗|𝐶_𝑖𝑗−𝐶_𝑖𝑙| (𝑇𝐴𝐷𝑊_𝑖=∑^𝑛𝑖_𝑗=1∑^𝑛𝑖_𝑙=𝑗|𝑊_𝑖𝑗−𝑊_𝑖𝑙|) be the load, the total completion (waiting) times, and the total absolute differences in completion (waiting) times of machine 𝑀_𝑖. Then, the total machine load, the total completion (waiting) time, and the total absolute deviation of job completion (waiting) time on all machines are ∑^𝑚_𝑖=1𝐿_𝑖, ∑^𝑚_𝑖=1𝑇𝐶_𝑖 (∑^𝑚_𝑖=1𝑇𝑊_𝑖), and ∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖 (∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖), respectively. The objective is to determine the optimal resource allocations and the optimal schedule on the machines so that the corresponding value of the following cost functions is optimal: 𝑍₁(𝜋,𝑢)=𝛼₁^𝑚_𝑖=1𝐿_𝑖+𝛼₂^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗,𝑍₂(𝜋,𝑢)=𝛼₁^𝑚_𝑖=1𝐿_𝑖+𝛼₂^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗, (6)(7) where weights 𝛼₁≥0, 𝛼₂≥0, 𝛼₃≥0, and 𝛼₄≥0 are given constants and 𝐺_𝑖𝑗 is the per time unit cost associated with the resource allocation. Using the three-field notation introduced by Graham et al. [34], the corresponding scheduling problem is denoted by 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 and 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗.

3. Problem 𝑅𝑚|𝑝_𝑖𝑗 = 𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗 + 𝛼𝑡 − 𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖 + 𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖 + 𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖 + 𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗

If the number of jobs on machine 𝑀_𝑖 is known in advance, from Wang et al. [19], we have 𝐶_𝑖[𝑗]=^𝑗_𝑙=1(1+𝛼)^𝑗−𝑙𝑎_𝑖[𝑙]𝑙^{𝑏𝑖[𝑙]}−𝛽_𝑖[𝑙]𝑢_𝑖[𝑙],𝑝_𝑖[𝑗]=𝑎_𝑖[𝑗]𝑗^{𝑏𝑖[𝑗]}−𝛽_𝑖[𝑗]𝑢_𝑖[𝑗]+𝛼^𝑗−1_𝑙=1(1+𝛼)^{𝑗−1−𝑙}𝑎_𝑖[𝑙]𝑙^{𝑏𝑖[𝑙]}−𝛽_𝑖[𝑙]𝑢_𝑖[𝑙]. (8)

From Kanet [35], we have 𝑇𝐴𝐷𝐶_𝑖=∑^𝑛𝑖_𝑗=1(𝑗−1)(𝑛_𝑖−𝑗+1)𝑝_𝑖[𝑗]; in addition, 𝐶_𝑖[𝑗]=∑^𝑗_𝑙=1𝑝_𝑖[𝑙], 𝐿_𝑖=∑^𝑛𝑖_𝑗=1𝑝_𝑖[𝑗], and 𝑇𝐶_𝑖=∑^𝑛𝑖_𝑗=1𝐶_𝑖[𝑗]=∑^𝑛𝑖_𝑗=1(𝑛_𝑖−𝑗+1)𝑝_𝑖[𝑗]; hence, we have 𝑍₁(𝜋,𝑢)=𝛼₁^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑝_𝑖[𝑗]+𝛼₂^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑛_𝑖−𝑗+1𝑝_𝑖[𝑗]+𝛼₃^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑗−1𝑛_𝑖−𝑗+1𝑝_𝑖[𝑗]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝛼₁+𝛼₂𝑛_𝑖+1−𝑗+𝛼₃𝑗−1𝑛_𝑖−𝑗+1𝑝_𝑖[𝑗]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝜔_𝑖𝑗𝑝_𝑖[𝑗]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝜔_𝑖𝑗𝑎_𝑖[𝑗]𝑗^{𝑏𝑖[𝑗]}−𝛽_𝑖[𝑗]𝑢_𝑖[𝑗]+𝛼^𝑗−1_𝑙=1(1+𝛼)^{𝑗−1−𝑙}𝑎_𝑖[𝑙]𝑙^{𝑏𝑖[𝑙]}−𝛽_𝑖[𝑙]𝑢_𝑖[𝑙]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚_𝑖=1𝜔_𝑖1𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝜔_𝑖2𝑎_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+𝛼𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝜔_𝑖3𝑎_𝑖[3]3^𝑏𝑖[3]−𝛽_𝑖[3]𝑢_𝑖[3]+𝛼𝑎_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+()×𝑎1+𝛼_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝜔_𝑖4𝑎_𝑖[4]4^𝑏𝑖[4]−𝛽_𝑖[4]𝑢_𝑖[4]+𝛼𝑎_𝑖[3]3^𝑏𝑖[3]−𝛽_𝑖[3]𝑢_𝑖[3]+()×𝑎1+𝛼_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+(1+𝛼)²𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+⋯+𝜔_{𝑖,𝑛𝑖−1}𝑎_{𝑖[𝑛𝑖−1]}𝑛_𝑖−1^{𝑏𝑖𝑖[𝑛−1]}−𝛽_{𝑖[𝑛𝑖−1]}𝑢_{𝑖[𝑛𝑖−1]}+𝛼𝑎_{𝑖[𝑛𝑖−2]}𝑛_𝑖−2^{𝑏𝑖𝑖[𝑛−2]}−𝛽_{𝑖[𝑛𝑖−2]}𝑢_{𝑖[𝑛𝑖−2]}+()×𝑎1+𝛼_{𝑖[𝑛𝑖−3]}𝑛_𝑖−3^{𝑏𝑖𝑖[𝑛−3]}−𝛽_{𝑖[𝑛𝑖−3]}𝑢_{𝑖[𝑛𝑖−3]}+()+⋯1+𝛼^𝑛𝑖−4𝑎_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+(1+𝛼)^𝑛𝑖−3𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝜔_𝑖𝑛𝑖𝑎_{𝑖[𝑛𝑖]}𝑛^{𝑏𝑖]𝑖[𝑛}_𝑖−𝛽_{𝑖[𝑛𝑖]}𝑢_{𝑖[𝑛𝑖]}+𝛼𝑎_{𝑖[𝑛𝑖−1]}𝑛_𝑖−1^{𝑏𝑖𝑖[𝑛−1]}−𝛽_{𝑖[𝑛𝑖−1]}𝑢_{𝑖[𝑛𝑖−1]}+(1+𝛼)𝑎_{𝑖[𝑛𝑖−2]}𝑛_𝑖−2^{𝑏𝑖𝑖[𝑛−2]}−𝛽_{𝑖[𝑛𝑖−2]}𝑢_{𝑖[𝑛𝑖−2]}+()+⋯1+𝛼^𝑛𝑖−3𝑎_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+(1+𝛼)^𝑛𝑖−2𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚_𝑖=1𝜔_𝑖1+𝛼𝜔_𝑖2+𝛼(1+𝛼)𝜔_𝑖3+⋯+𝛼(1+𝛼)^𝑛−2𝜔_𝑖𝑛𝑖×𝑎_𝑖[1]1^𝑏𝑖[1]−𝛽_𝑖[1]𝑢_𝑖[1]+𝜔_𝑖2+𝛼𝜔_𝑖3+𝛼(1+𝛼)𝜔_𝑖4()+⋯+𝛼1+𝛼^𝑛𝑖−3𝜔_𝑖𝑛𝑖𝑎_𝑖[2]2^𝑏𝑖[2]−𝛽_𝑖[2]𝑢_𝑖[2]+𝜔_𝑖3+𝛼𝜔_𝑖4+𝛼(1+𝛼)𝜔_𝑖5()+⋯+𝛼1+𝛼^𝑛𝑖−4𝜔_𝑖𝑛𝑖𝑎_𝑖[3]3^𝑏𝑖[3]−𝛽_𝑖[3]𝑢_𝑖[3]+⋯+𝜔_{𝑖,𝑛𝑖−1}+𝛼𝜔_𝑖𝑛𝑖𝑎_{𝑖[𝑛𝑖−1]}−𝛽_{𝑖[𝑛𝑖−1]}𝑢_{𝑖[𝑛𝑖−1]}+𝜔_𝑖𝑛𝑖𝑎_{𝑖[𝑛𝑖]}𝑛^{𝑏𝑖]𝑖[𝑛}_𝑖−𝛽_{𝑖[𝑛𝑖]}𝑢_{𝑖[𝑛𝑖]}+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚𝑛_𝑖=1^𝑖_𝑗=1Ω_𝑖𝑗𝑎_𝑖[𝑗]𝑗^{𝑏𝑖[𝑗]}+^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗𝑢_𝑖[𝑗], (9) where 𝜔_𝑖𝑗=𝛼₁+𝛼₂(𝑛_𝑖+1−𝑗)+𝛼₃(𝑗−1)(𝑛_𝑖−𝑗+1) and Ω_𝑖1=𝜔_𝑖1+𝛼𝜔_𝑖2+𝛼(1+𝛼)𝜔_𝑖3+⋯+𝛼(1+𝛼)^𝑛𝑖−2𝜔_𝑖𝑛𝑖,Ω_𝑖2=𝜔_𝑖2+𝛼𝜔_𝑖3+𝛼(1+𝛼)𝜔_𝑖4+⋯+𝛼(1+𝛼)^𝑛𝑖−3𝜔_𝑖𝑛𝑖,Ω_𝑖3=𝜔_𝑖3+𝛼𝜔_𝑖4+𝛼(1+𝛼)𝜔_𝑖5+⋯+𝛼(1+𝛼)^𝑛𝑖−4𝜔_𝑖𝑛𝑖,⋮Ω_{𝑖,𝑛𝑖−1}=𝜔_{𝑖,𝑛𝑖−1}+𝛼𝜔_𝑖𝑛𝑖,Ω_𝑖𝑛𝑖=𝜔_𝑖𝑛𝑖. (10)

From (9), for any given sequence on all machines, the optimal resource allocation on all machines can be obtained by the following.

Lemma 1. For a given sequence, the optimal resource allocation of the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 can be determined by 𝑢^∗_𝑖[𝑗]=𝑢_𝑖[𝑗],𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗𝑢<0,_𝑖[𝑗]∈0,𝑢_𝑖[𝑗],𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗=0,0,𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗>0, (11) where 𝑢^∗_𝑖[𝑗],𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛_𝑖, represents the optimal resource allocation of the job in position 𝑗 on machine 𝑀_𝑖.

Proof. For the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗, taking the derivative by 𝑢_𝑖[𝑗] to (9), we have 𝑑𝑓(𝜋,𝑢)/𝑢_𝑖[𝑗]=𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗 for 𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛_𝑖. Hence, if 𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗>0, we should not allocate any resource to job 𝐽_𝑖[𝑗]; if 𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗<0, we will allocate the maximal feasible amount of resource to job 𝐽_𝑖[𝑗]; and if 𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ω_𝑖𝑗=0, any feasible resource allocation can be optimal.

We define 𝑥_𝑖𝑗𝑟=1 if job 𝐽_𝑗 is scheduled in position 𝑟 on machine 𝑀_𝑖, and 𝑥_𝑖𝑗𝑟=0 otherwise. If the number of jobs on machine 𝑀_𝑖 is known in advance, then we formulate the 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 problem as the following assignment problem: Min𝑍=^𝑚𝑛_𝑖=1^𝑖𝑛_𝑟=1𝑗=1𝜆_𝑖𝑗𝑟𝑥_𝑖𝑗𝑟 (12) subject to ^𝑚𝑛_𝑖=1^𝑖_𝑟=1𝑥_𝑖𝑗𝑟=1,𝑗=1,2,…,𝑛,_𝑛𝑗=1𝑥_𝑖𝑗𝑟=1,𝑖=1,2,…,𝑚;𝑟=1,2,…,𝑛_𝑖,𝑥_𝑖𝑗𝑟=0or1,𝑖=1,2,…,𝑚;𝑟=1,2,…,𝑛_𝑖;𝑗=1,2,…,𝑛, (13)(14)(15) where 𝜆_𝑖𝑗𝑟=Ω_𝑖𝑟𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗,if𝛼₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ω_𝑖𝑟Ω≥0,_𝑖𝑟𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ω_𝑖𝑟𝑢_𝑖𝑗,if𝛼₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ω_𝑖𝑟<0. (16) Constraint (13) makes sure that each job is scheduled exactly once. Constraint (14) makes sure that each position on each machine is taken by one job.

Next, the question is how many 𝑃(𝑛,𝑚)=(𝑛₁,𝑛₂,…,𝑛_𝑚) vectors exist. Note that 𝑛_𝑖 may be 0,1,2,…,𝑛 for 𝑖=1,2,…,𝑚. So, if we get the numbers of jobs on the first 𝑚−1 machines, the number of jobs processed on the last machine is then determined uniquely due to 𝑛₁+𝑛₂+⋯+𝑛_𝑚=𝑛. Therefore, the upper bound of the number of 𝑃(𝑛,𝑚) vectors is (𝑛+1)^𝑚−1. Based on the above analysis, we have the following result.

Theorem 2. The problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 can be solved in 𝑂(𝑛^𝑚+2) time; that is, the problem is polynomially solvable because 𝑚 is a constant.

Base on the above analysis, we can determine the optimal solution for the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 via the following algorithm.

Algorithm 3. Consider the following.
Step  1. For all the possible vectors (𝑛₁,𝑛₂,…,𝑛_𝑚), solve the assignment problems ((12)–(16)). Then, obtain the optimal schedule and the corresponding total cost 𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 for each possible vector (𝑛₁,𝑛₂,…,𝑛_𝑚).
Step  2. The optimal solution for the problem is the one with the minimum value of the total cost 𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗.
Step  3. Calculate the optimal resources allocation by using (11).

The following example illustrates the working of Algorithm 3 for the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗.

Example 4. Data.𝑛=4, 𝑚=2, 𝛼=0.05, and 𝛼₁=𝛼₂=𝛼₃=𝛼₄=1, and the other corresponding parameters are shown in Table 1.

Table 1 

The date of Example 4.

Solution. When 𝑛₁=0,𝑛₂=4, Ω₂₁=5.9931,Ω₂₂=7.6125,Ω₂₃=7.2500,Ω₂₄=5, then the optimal schedule on machine 𝑀₂ is [𝐽₄,𝐽₁,𝐽₂,𝐽₃] and 𝑍₁=155.2178.

When 𝑛₁=1,𝑛₂=3, Ω₁₁=2,Ω₂₁=4.4600,Ω₂₂=5.2000,Ω₂₃=4, the optimal schedule on machine 𝑀₁ is [𝐽₃] and on machine 𝑀₂ is [𝐽₄,𝐽₁,𝐽₂], and 𝑍₁=102.7967.

When 𝑛₁=2,𝑛₂=2,Ω₁₁=3.1500,Ω₁₂=3,Ω₂₁=3.1500,Ω₂₂=3, the optimal schedule on machine 𝑀₁ is [𝐽₂,𝐽₃] and on machine 𝑀₂ is [𝐽₄,𝐽₁], and 𝑍₁=99.9928.

When 𝑛₁=3,𝑛₂=1, Ω₁₁=4.4600,Ω₁₂=5.2000,Ω₁₃=4,Ω₂₁=2, the optimal schedule on machine 𝑀₁ is [𝐽₂,𝐽₃,𝐽₄] and on machine 𝑀₂ is [𝐽₁], and 𝑍₁=115.8243.

When 𝑛₁=4,𝑛₂=0, Ω₁₁=5.9931,Ω₁₂=7.6125,Ω₁₃=7.2500,Ω₁₄=5, the optimal schedule on machine 𝑀₁ is [𝐽₄,𝐽₃,𝐽₂,𝐽₁], and 𝑍₁=168.1833.

Hence, the optimal schedule on machine 𝑀₁ is [𝐽₂,𝐽₃] and on machine 𝑀₂ is [𝐽₄,𝐽₁], 𝑢^∗_1[1]=4,𝑢^∗_1[2]=3,𝑢^∗_2[1]=2,𝑢^∗_2[2]=3 and optimal cost is 𝑍₁=99.9928.

4. Problem 𝑅𝑚|𝑝_𝑖𝑗 = 𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗 + 𝛼𝑡 − 𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖 + 𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖 + 𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖 + 𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗

Similar to the analysis of the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 presented in the previous section, if we substitute 𝑊_𝑖[𝑗]=∑^𝑗−1_𝑙=1𝑝_𝑖[𝑙], 𝑇𝑊_𝑖= ∑^𝑛𝑖_𝑗=1𝑊_𝑖[𝑗], and 𝑇𝐴𝐷𝑊_𝑖 = ∑^𝑛𝑖_𝑗=1𝑗(𝑛_𝑖−𝑗)𝑝_𝑖[𝑗] (Bagchi [36]) into (7), we have 𝑍₂(𝜋,𝑢)=𝛼₁^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑝_𝑖[𝑗]+𝛼₂^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑛_𝑖−𝑗𝑝_𝑖[𝑗]+𝛼₃^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝑗𝑛_𝑖−𝑗𝑝_𝑖[𝑗]+𝛼₄^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝐺_𝑖[𝑗]𝑢_𝑖[𝑗]=^𝑚𝑛_𝑖=1^𝑖_𝑗=1Ψ_𝑖𝑗𝑎_𝑖[𝑗]𝑗^{𝑏𝑖[𝑗]}+^𝑚𝑛_𝑖=1^𝑖_𝑗=1𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ψ_𝑖𝑗𝑢_𝑖[𝑗], (17) where 𝜈_𝑖𝑗=𝛼₁+𝛼₂(𝑛_𝑖−𝑗)+𝛼₃𝑗(𝑛_𝑖−𝑗), Ψ_𝑖1=𝜈_𝑖1+𝛼𝜈_𝑖2+𝛼(1+𝛼)𝜈_𝑖3+⋯+𝛼(1+𝛼)^𝑛𝑖−2𝜈_𝑖𝑛𝑖,Ψ_𝑖2=𝜈_𝑖2+𝛼𝜈_𝑖3+𝛼(1+𝛼)𝜈_𝑖4+⋯+𝛼(1+𝛼)^𝑛𝑖−3𝜈_𝑖𝑛𝑖,Ψ_𝑖3=𝜈_𝑖3+𝛼𝜈_𝑖4+𝛼(1+𝛼)𝜈_𝑖5+⋯+𝛼(1+𝛼)^𝑛𝑖−4𝜈_𝑖𝑛𝑖,⋮Ψ_{𝑖,𝑛𝑖−1}=𝜈_{𝑖,𝑛𝑖−1}+𝛼𝜈_𝑖𝑛𝑖,Ψ_𝑖𝑛𝑖=𝜈_𝑖𝑛𝑖. (18)

Applying a similar analysis in the previous section, we have the following results.

Lemma 5. For a given sequence, the optimal resource allocation of the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 can be determined by 𝑢^∗_𝑖[𝑗]=𝑢_𝑖[𝑗],𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ψ_𝑖𝑗𝑢<0,_𝑖[𝑗]∈0,𝑢_𝑖[𝑗],𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ψ_𝑖𝑗=0,0,𝑖𝑓𝛼₄𝐺_𝑖[𝑗]−𝛽_𝑖[𝑗]Ψ_𝑖𝑗>0, (19) where 𝑢^∗_𝑖[𝑗],𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛_𝑖 represents the optimal resource allocation of the job in position 𝑗 on machine 𝑀_𝑖.

Lemma 6. For the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂𝑇𝑊_𝑖+𝛼₃𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗, if the vector 𝑃(𝑛,𝑚)=(𝑛₁,𝑛₂,…,𝑛_𝑚) is given, the optimal sequence can be determined by solving the following assignment problem: Min𝑍=^𝑚𝑛_𝑖=1^𝑖𝑛_𝑟=1𝑗=1𝜃_𝑖𝑗𝑟𝑥_𝑖𝑗𝑟𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜(13),(14)𝑎𝑛𝑑(15), (20) where 𝜃_𝑖𝑗𝑟=Ψ_𝑖𝑟𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗,𝑖𝑓𝛿₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ψ_𝑖𝑟Ω≥0,_𝑖𝑟𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ψ_𝑖𝑟𝑚_𝑖𝑗,𝑖𝑓𝛼₄𝐺_𝑖𝑗−𝛽_𝑖𝑗Ψ_𝑖𝑟<0. (21)

Theorem 7. The problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂𝑇𝑊_𝑖+𝛼₃𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 can be solved in 𝑂(𝑛^𝑚+2) time; that is, the problem is polynomially solvable because 𝑚 is a constant.

The optimal solution for the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂𝑇𝑊_𝑖+𝛼₃𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 can be obtained by the following algorithm.

Algorithm 8. Consider the following.
Step  1. For all the possible vectors (𝑛₁,𝑛₂,…,𝑛_𝑚), solve the assignment problems ((20), (13)–(15), (21)). Then, obtain the optimal schedule and the corresponding total cost 𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 for each possible vector (𝑛₁,𝑛₂,…,𝑛_𝑚).
Step  2. The optimal solution for the problem is the one with the minimum value of the total cost 𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗.
Step  3. Calculate the optimal resources allocation by using (19).

The following example illustrates the working of Algorithm 8 for the problem 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗.

Example 9. The same date in Example 4 is used.

Solution. When 𝑛₁=0,𝑛₂=4, Ψ₂₁=7.6676,Ψ₂₂=7.3025,Ψ₂₃=5.0500,Ψ₂₄=1, then the optimal schedule on machine 𝑀₂ is [𝐽₄,𝐽₁,𝐽₂,𝐽₃] and 𝑍₂=126.7347.

When 𝑛₁=1,𝑛₂=3, Ψ₁₁=1,Ψ₂₁=5.2525,Ψ₂₂=4.0500,Ψ₂₃=1, the optimal schedule on machine 𝑀₁ is [𝐽₃] and on machine 𝑀₂ is [𝐽₄,𝐽₁,𝐽₂], and 𝑍₂=84.13484.

When 𝑛₁=2,𝑛₂=2, Ψ₁₁=3.0500,Ψ₁₂=1,Ψ₂₁=3.0500,Ψ₂₂=1, the optimal schedule on machine 𝑀₁ is [𝐽₃,𝐽₄] and on machine 𝑀₂ is [𝐽₁,𝐽₂], and 𝑍₂=73.1692.

When 𝑛₁=3,𝑛₂=1, Ψ₁₁=5.2525,Ψ₁₂=4.0500,Ψ₁₃=1,Ψ₂₁=1, the optimal schedule on machine 𝑀₁ is [𝐽₃,𝐽₄,𝐽₂] and on machine 𝑀₂ is [𝐽₁], and 𝑍₂=95.8810.

When 𝑛₁=4,𝑛₂=0, Ψ₁₁=7.6676,Ψ₁₂=7.3025,Ψ₁₃=5.0500,Ψ₁₄=1, the optimal schedule on machine 𝑀₁ is [𝐽₄,𝐽₃,𝐽₂,𝐽₁], and 𝑍₂=144.1351.

Hence, the optimal schedule on machine 𝑀₁ is [𝐽₃,𝐽₄] and on machine 𝑀₂ is [𝐽₁,𝐽₂], 𝑢^∗_1[1]=3,𝑢^∗_1[2]=1,𝑢^∗_2[1]=3,𝑢^∗_2[2]=0 and optimal cost is 𝑍₂=73.1692.

5. Conclusions

In this paper, we have studied the problem of scheduling 𝑛 jobs on 𝑚 unrelated parallel machines with simultaneous consideration of learning effect, deteriorating jobs, and controllable processing times. We provide an 𝑂(𝑛^𝑚+2) time algorithm for the problems 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝐶_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝐶_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗 and 𝑅𝑚|𝑝_𝑖𝑗=𝑎_𝑖𝑗𝑟^𝑏𝑖𝑗+𝛼𝑡−𝜃_𝑖𝑗𝑢_𝑖𝑗|𝛼₁∑^𝑚_𝑖=1𝐿_𝑖+𝛼₂∑^𝑚_𝑖=1𝑇𝑊_𝑖+𝛼₃∑^𝑚_𝑖=1𝑇𝐴𝐷𝑊_𝑖+𝛼₄∑^𝑚_𝑖=1∑^𝑛𝑖_𝑗=1𝐺_𝑖𝑗𝑢_𝑖𝑗, respectively. The algorithms can also be easily applied to the cases 𝑏_𝑖𝑗>0 (aging effect) and 𝛼<0. Future research may focus on similar problems with more general processing time model and extend the problems to flow shop machine settings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful for three anonymous referees for their helpful comments on earlier version of the paper. This research was supported by the National Natural Science Foundation of China (Grant nos. 11001181 and 71271039), the New Century Excellent Talents in University (NCET-13-0082), the Changjiang Scholars and Innovative Research Team in University (IRT1214), and the Fundamental Research Funds for the Central Universities (DUT14YQ211).