Rescheduling Problems with Agreeable Job Parameters to Minimize the Tardiness Costs under Deterioration and Disruption

Abstract

This paper considers single-machine rescheduling problems with agreeable job parameters under deterioration and disruption. Deteriorating jobs mean that the processing time of a job is defined by an increasing function of its starting time. Rescheduling means that, after a set of original jobs has already been scheduled, a new set of jobs arrives and creates a disruption. We consider four cases of minimization of the total tardiness costs with agreeable job parameters under a limit of the disruptions from the original job sequence. We propose polynomial-time algorithms or some dynamic programming algorithms under sequence disruption and time disruption.

1. Introduction

Scheduling problems are very important in manufacturing systems. Hence, numerous scheduling problems have been studied for many years. In the classical scheduling theory, job processing times are assumed to be known and fixed from the first job to be processed until the last job to be completed. However, there are many situations in which a job that is processed later consumes more time than the same job when processed earlier. Scheduling in this setting is known as scheduling deteriorating jobs.

Significant contributions towards addressing or solving deteriorating job scheduling problems on a single machine include, among others, the following: Browne and Yechiali [1] cited applications concerning the control of queues and communication systems where jobs deteriorate as they await processing. Kunnathur and Gupta [2] and Mosheiov [3] gave several other real-life situations where deteriorating jobs occur. These include the search for an object under worsening weather or performance of medical treatments under deteriorating health conditions. Comprehensive discussion of scheduling problems with time-dependent processing times of jobs can be found in Cheng et al. [4] and Gawiejnowicz [5]. Recently, Biskup and Herrmann [6] observed that the sum of the processing times of the jobs processed before a job contributes to the actual processing time of the job, and they cite equipment wearout (e.g., a drill) as a real-life example of their observation. Wang and Guo [7] considered a single-machine scheduling problem with the effects of learning and deterioration. The goal is to determine an optimal combination of the due date and schedule so as to minimize the sum of earliness, tardiness, and due-date costs. Ng et al. [8] considered a two-machine flow shop scheduling problem with linearly deteriorating jobs to minimize the total completion time. Cheng et al. [9] considered scheduling with deteriorating jobs in which the actual processing time of a job is a function of the logarithm of the total processing time of the jobs processed before it (to avoid the unrealistic situation where the jobs scheduled lately will incur excessively long processing times), and the setup times are proportional to the actual processing times of the already scheduled jobs.

Rescheduling involves adjusting a previously planned, possibly optimal, scheduling to account for a disruption. Examples of common disruptions include: the arrival of new orders, order cancelations, changes in order priority, processing delays, changes in release dates, machine breakdowns, and the unavailability of raw materials, personnel, or tools. There are several papers on rescheduling approaches in manufacturing systems. Raman et al. [10] developed a branch-and-bound procedure to reschedule a flexible manufacturing system in the presence of dynamic job arrivals. Church and Uzsoy [11] addressed a similar problem for which they describe periodic rescheduling policies and analyze their error bounds. Jain and Elmaraghy [12] used genetic algorithms to develop heuristic approaches for rescheduling a flexible manufacturing system. Vieira et al. [13] provided an extensive review of rescheduling problems. Yang [14] studied the single-machine rescheduling with new jobs arrivals and processing time compression. Hall and Potts [15] considered the problem of rescheduling of a single machine with newly arrived jobs to minimize the maximum lateness and the total completion time under a limit of the disruption from the original scheduling. Yuan and Mu [16] considered the rescheduling problem for jobs on a single machine with release dates to minimize makespan under a limit on the maximum sequence disruption. Zhao and Tang [17] presented two single-machine rescheduling problems with linear deteriorating jobs under disruption. deteriorating jobs mean that the actual processing time of the job is an increasing function of its starting time. They considered the rescheduling problem to minimize the total completion time under a limit of the disruption from the original scheduling. Hoogeveen et al. [18] tackled several simple setup time configurations yielding different scheduling problems for which they propose optimal polynomial time algorithms or provide NP-hardness proofs. They also present the problem of enumerating the set of strict Pareto optima for the sum of setup times and disruption cost criteria.

Based on the motivation of Hall and Potts [15] and Zhao and Tang [17], we consider some rescheduling problems with the criterion minimizing the total tardiness costs under a limit of the disruption from the original schedule in this paper. The rest of the paper is organized as follows. In the next section, we give the problem description. In Section 3, we consider single-machine scheduling problems. The last section is the conclusion.

2. Problem Definition and Notation

By the terminology of Hall and Potts [15], our researchful problem can be stated as follows. Let 𝐽₀={𝐽₁,…,𝐽_𝑛0} denote a set of original jobs to be processed non preemptively on a single machine. In the presented model, we assume that these jobs have been scheduled optimally to minimize some classical objective and that 𝜋^∗ is an optimal job sequence with no idle time between the jobs. Let 𝐽_𝑁={𝐽_𝑛0+1,…,𝐽_{𝑛0+𝑛𝑁}} denote a set of new jobs that arrive together. We assume that these jobs arrive at time zero after a schedule for the jobs of 𝐽₀ has been determined, but before processing begins. There is no loss of generality in this assumption: if the jobs arrive after time zero, then the fully processed jobs of 𝐽₀ are removed from the problem; any partly processed jobs are processed to completion, and 𝐽₀ and 𝑛₀ are updated accordingly. Let 𝐽=𝐽₀⋃𝐽_𝑁 and 𝑛=𝑛₀+𝑛_𝑁. Each job 𝐽_𝑗∈𝐽 has an integral normal processing time 𝑝_𝑗 and a deteriorating rate 𝑏>0; the actual processing time of job 𝐽_𝑗 is 𝑝_𝑗(𝑎+𝑏𝑠_𝑗), where 𝑠_𝑗 (≥0) is the starting time of job 𝐽_𝑗 and 𝑎 (>0) is constant. For any schedule 𝜎 of the jobs in 𝐽, we define the following variables:𝑠_𝑗(𝜎) is the time at which job 𝐽_𝑗∈𝐽 starts its processing in schedule 𝜎. 𝑑_𝑗(𝜎) is the due date of job 𝐽_𝑗∈𝐽 in schedule 𝜎. 𝐶_𝑗(𝜎) is the time at which job 𝐽_𝑗∈𝐽 is completed in schedule 𝜎. 𝑇_𝑗(𝜎)=max{𝐶_𝑗(𝜎)−𝑑_𝑗,0} is the tardiness value of job 𝐽_𝑗∈𝐽. 𝐷_𝑗(𝜋^∗;𝜎) is the sequence disruptions of job 𝐽_𝑗∈𝐽₀; that is, if 𝐽_𝑗 is the 𝑥th job in 𝜋^∗ and the 𝑦th job in 𝜎, respectively, then 𝐷_𝑗(𝜋^∗;𝜎)=|𝑦−𝑥|. Δ_𝑗(𝜋^∗;𝜎)=|𝐶_𝑗(𝜎)−𝐶_𝑗(𝜋^∗)| is the time disruption of job 𝐽_𝑗∈𝐽₀.

When there is no ambiguity, we simplify the above symbols and write 𝑠_𝑗, 𝑑_𝑗𝐶_𝑗, 𝑇_𝑗, 𝐷_𝑗(𝜋^∗), and Δ_𝑗(𝜋^∗), respectively.

We consider only the single-scheduling problems with the following constraints on the amount of disruption, where 𝑘≥0 is a known integer: 𝐷_max(𝜋^∗)≤𝑘: max{𝐷_𝑗(𝜋^∗)𝐽_𝑗∈𝐽₀}≤𝑘; the maximum sequence disruption of the jobs cannot exceed 𝑘. ∑𝐷_𝑗(𝜋^∗)≤𝑘: ∑_{𝐽𝑗∈𝐽0}𝐷_𝑗(𝜋^∗)≤𝑘; the total sequence disruption of the jobs cannot exceed 𝑘. Δ_max(𝜋^∗)≤𝑘: max{Δ_𝑗(𝜋^∗)𝐽_𝑗∈𝐽₀}≤𝑘; the maximum time disruption of the jobs cannot exceed 𝑘. ∑Δ_𝑗(𝜋^∗)≤𝑘: ∑_{𝐽𝑗∈𝐽0}Δ_𝑗(𝜋^∗)≤𝑘; the total time disruption of the jobs cannot exceed 𝑘.

Since the 1∣∑𝑇_𝑗 problem is NP-hard in Du and Leung [19], the 1∣Γ≤𝑘∣∑𝑇_𝑗 problem is also NP-hard, where Γ∈{𝐷_max(𝜋^∗),∑𝐷_𝑗(𝜋^∗),Δ_max(𝜋^∗),∑Δ_𝑗(𝜋^∗)}. In this paper, we consider a special case: the processing time and due date of jobs are agreeable; that is, 𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗 for all jobs 𝐽_𝑖 and 𝐽_𝑗.

Using the three-field notation [20], the considered problems can be denoted as 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, 𝐷_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗, 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, ∑𝐷_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗; 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Δ_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗; 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, ∑Δ_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗.

3. Minimum Tardiness Problem with Agreeable Job Parameters

We start with the following result by Kononov and Gawiejnowicz [21].

Lemma 1. For the 1∣𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣𝐶_max problem, if 𝜋={𝐽₁,…,𝐽_𝑛} and the starting time of the job 𝐽₁ is 𝑡, then makespan is sequence independent, and 𝐶_max(𝜋)=𝑡^𝑛_𝑖=11+𝑏𝑝_𝑖+𝑎𝑏^𝑛_𝑖=11+𝑏𝑝_𝑖−1. (1)

For notational convenience, we assume that the jobs are indexed by agreeable order; that is, 𝑝₁≤⋯≤𝑝_𝑛0 and 𝑑₁≤⋯≤𝑑_𝑛0. Thus, 𝜋^∗={𝐽₁,…,𝐽_𝑛0} with no idle time between jobs. We now show that the EDD or SPT rule applies to serval of the rescheduling problems we consider.

Lemma 2. Problems 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, 𝐷_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗, and 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Δ_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 have an optimal schedule with no idle time between jobs, and(a) a schedule for problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, 𝐷_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 is feasible if the number of jobs of 𝐽_𝑁 scheduled before the last job of 𝐽₀ is less than or equal to 𝑘;(b) a schedule for problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Δ_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 is feasible if the total actual processing time of jobs of 𝐽_𝑁 scheduled before the last job of 𝐽₀ is less than or equal to 𝑘.

Proof. The proof is similar to that of Lemma 1 in Hall and Potts [15].

Lemma 3. For problems 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Γ≤𝑘 and 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗, where Γ∈{𝐷_max(𝜋^∗),∑𝐷_𝑗(𝜋^∗),Δ_max(𝜋^∗),∑Δ_𝑗(𝜋^∗)}, there exists an optimal schedule in which the jobs of 𝐽₀ are sequenced in the EDD or SPT rule as in 𝜋^∗, the jobs of 𝐽_𝑁 are sequenced in the EDD or SPT rule, and there is no idle time between jobs.

Proof. We first analyze the jobs of 𝐽₀. Consider an optimal schedule 𝜎^∗ in which the jobs of 𝐽₀ are not sequenced in the EDD or SPT rule as in 𝜋^∗. Let 𝐽_𝑖 be the job with the smallest index that appears later relative to the other jobs of 𝐽₀ in 𝜎^∗ than in 𝜋^∗, and let 𝐽_𝑗(𝑗>𝑖) be the last job of 𝐽₀ that precedes job 𝐽_𝑖 in 𝜎^∗. Because 𝜋^∗ is an optimal sequence, 𝑝_𝑗 and 𝑑_𝑗 are agreeable. Assume that the starting time of job 𝐽_𝑗 in 𝜎^∗ is 𝑡₀, then 𝐶_𝑗(𝜎^∗)=𝑡₀+𝑝_𝑗(𝑎+𝑏𝑡₀). Perform an interchange on jobs 𝐽_𝑗 and 𝐽_𝑖, and get a new schedule 𝜎^′. In 𝜎^′, the starting time of job 𝐽_𝑖 is 𝑡₀, then 𝐶_𝑖(𝜎^′)=𝑡₀+𝑝_𝑖(𝑎+𝑏𝑡₀)<𝑡₀+𝑝_𝑗(𝑎+𝑏𝑡₀)=𝐶_𝑗(𝜎^∗). From Lemma 1, 𝐶_𝑗(𝜎^′)=𝐶_𝑖(𝜎^∗). Thus, the jobs between job 𝐽_𝑖 and 𝐽_𝑗 are completed earlier in 𝜎^′ than in 𝜎^∗. Next, we consider the total tardiness of jobs 𝐽_𝑖 and 𝐽_𝑗 in 𝜎^′ and in 𝜎^∗.
The total tardiness of jobs 𝐽_𝑖 and 𝐽_𝑗 in 𝜎^∗ is as follows: 𝑇_𝑖𝜎^∗+𝑇_𝑗𝜎^∗=max𝐶_𝑖𝜎^∗−𝑑_𝑖𝐶,0+max_𝑗𝜎^∗−𝑑_𝑗,0. (2)
The total tardiness of jobs 𝐽_𝑖 and 𝐽_𝑗 in 𝜎^′ is as follows: 𝑇_𝑖𝜎^′+𝑇_𝑗𝜎^′=max𝐶_𝑖𝜎^′−𝑑_𝑖𝐶,0+max_𝑗𝜎^′−𝑑_𝑗,0. (3)
To compare the total tardiness of jobs 𝐽_𝑖 and 𝐽_𝑗 in 𝜎^∗ and in 𝜎^′, we divide it into two cases. In the first case, when 𝐶_𝑗(𝜎^∗)≤𝑑_𝑗, we have 𝑇_𝑖(𝜎^∗)+𝑇_𝑗(𝜎^∗)=max{𝐶_𝑖(𝜎^∗)−𝑑_𝑖,0}. Suppose that neither 𝑇_𝑖(𝜎^′) nor 𝑇_𝑗(𝜎^′) is zero. Note that this is the most restrictive case since it comprises the case that either one or both 𝑇_𝑖(𝜎^′) and 𝑇_𝑗(𝜎^′) are zero. From Lemma 1 and 𝑝_𝑖≤𝑝_𝑗, 𝑑_𝑖≤𝑑_𝑗, we have {𝑇_𝑖(𝜎^∗)+𝑇_𝑗(𝜎^∗)}−{𝑇_𝑖(𝜎^′)+𝑇_𝑗(𝜎^′)}=𝐶_𝑖(𝜎^∗)−𝐶_𝑖(𝜎^′)−𝐶_𝑗(𝜎^′)+𝑑_𝑗=𝑑_𝑗−𝐶_𝑖(𝜎^′)≥𝑑_𝑗−𝐶_𝑗(𝜎^∗)≥0.
In the second case, when 𝐶_𝑗(𝜎^∗)>𝑑_𝑗, we have 𝑇_𝑖(𝜎^∗)+𝑇_𝑗(𝜎^∗)=𝐶_𝑖(𝜎^∗)+𝐶_𝑗(𝜎^∗)−𝑑_𝑖−𝑑_𝑖. Suppose that neither 𝑇_𝑖(𝜎^′) nor 𝑇_𝑗(𝜎^′) is zero. From Lemma 1 and 𝑝_𝑖≤𝑝_𝑗, 𝑑_𝑖≤𝑑_𝑗, we have 𝑇_𝑖𝜎^∗+𝑇_𝑗𝜎^∗−𝑇_𝑖𝜎^′+𝑇_𝑗𝜎^′=𝐶_𝑖𝜎^∗+𝐶_𝑗𝜎^∗−𝐶_𝑗𝜎^′−𝐶_𝑖𝜎^′≥0. (4) Now, we have proved that the total tardiness of 𝜎^′ is less than or equal to that of 𝜎^∗.
Let the position of job 𝐽_𝑖 in 𝜋^∗ be 𝑘₁ and let the position of job 𝐽_𝑗 in 𝜋^∗ be 𝑘₂ and in 𝜎^′ be 𝑘₃. If 𝑘₃≥𝑘₂, then 𝐷_𝑗(𝜋^∗,𝜎^′)=𝑘₃−𝑘₂, 𝐷_𝑖(𝜋^∗,𝜎^∗)=𝑘₃−𝑘₁. Since 𝑖<𝑗 implies 𝑘₁<𝑘₂, we have 𝐷_𝑗(𝜋^∗,𝜎^′)<𝐷_𝑖(𝜋^∗,𝜎^∗). If 𝑘₃<𝑘₂, then 𝐷_𝑗(𝜋^∗,𝜎^′)=𝑘₂−𝑘₃ and 𝐷_𝑗(𝜋^∗,𝜎^∗)=𝑘₂−(𝑘₃−ℎ), where ℎ is the difference between the position of job 𝐽_𝑖 and 𝐽_𝑗 in 𝜎^∗. So, 𝐷_𝑗(𝜋^∗,𝜎^′)<𝐷_𝑗(𝜋^∗,𝜎^∗). Hence, we have 𝐷_max(𝜋^∗,𝜎^′)<𝐷_max(𝜋^∗,𝜎^∗). In either case, because 𝐷_𝑖(𝜋^∗,𝜎^′)=𝐷_𝑖(𝜋^∗,𝜎^∗)−ℎ and 𝐷_𝑗(𝜋^∗,𝜎^′)≤𝐷_𝑗(𝜋^∗,𝜎^∗)+ℎ. Hence, ∑𝐷_𝑗(𝜋^∗,𝜎^′)≤∑𝐷_𝑗(𝜋^∗,𝜎^∗).
Moreover, if 𝐶_𝑗(𝜎^′)≥𝐶_𝑗(𝜋^∗), then Δ_𝑗(𝜋^∗,𝜎^′)=𝐶_𝑗(𝜎^′)−𝐶_𝑗(𝜋^∗). Since Δ_𝑖(𝜋^∗,𝜎^∗)=𝐶_𝑖(𝜎^∗)−𝐶_𝑖(𝜋^∗)=𝐶_𝑗(𝜎^′)−𝐶_𝑖(𝜋^∗) and 𝐶_𝑖(𝜋^∗)<𝐶_𝑗(𝜋^∗); therefore, Δ_𝑗(𝜋^∗,𝜎^′)<Δ_𝑖(𝜋^∗,𝜎^∗). If 𝐶_𝑗(𝜎^′)<𝐶_𝑗(𝜋^∗), then Δ_𝑗(𝜋^∗,𝜎^′)=𝐶_𝑗(𝜋^∗)−𝐶_𝑗(𝜎^′) because Δ_𝑗(𝜋^∗,𝜎^∗)=𝐶_𝑗(𝜋^∗)−𝐶_𝑗(𝜎^∗), Δ_𝑗(𝜋^∗,𝜎^′)<Δ_𝑗(𝜋^∗,𝜎^∗). Thus, we have Δ_max(𝜋^∗,𝜎^′)<Δ_max(𝜋^∗,𝜎^∗). In either case, because Δ_𝑖(𝜋^∗,𝜎^′)=Δ_𝑖(𝜋^∗,𝜎^∗)−ℎ^′ and Δ_𝑗(𝜋^∗,𝜎^′)≤Δ_𝑗(𝜋^∗,𝜎^∗)+ℎ^′, where ℎ^′=𝐶_𝑖(𝜎^∗)−𝐶_𝑖(𝜎^′). Then we deduce that ∑Δ_𝑗(𝜋^∗,𝜎^′)≤∑Δ_𝑗(𝜋^∗,𝜎^∗). Thus, for either problem, 𝜎^′ is feasible and optimal. We can show that there exists an optimal schedule in which the jobs of 𝐽₀ are sequenced in the EDD or SPT order as in 𝜋^∗ by finite numbers of repetitions of the argument. A similar interchange argument establishes that the jobs of 𝐽_𝑁 can also be obtained by sequencing in the EDD or SPT order. The same EDD or SPT ordering of the jobs of 𝐽₀ in 𝜋^∗ and an optimal schedule show that there is no idle time in this optimal schedule. Otherwise, removing this idle time maintains feasibility and decreases the total tardiness.

We refer to the (EDD, EDD) property when a schedule is constructed using Lemmas 2 and 3. We first consider problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, 𝐷_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗. From Lemmas 2 and 3, there are at most 𝑘 jobs of 𝐽_𝑁 that can be sequenced before the last job of 𝐽₀, and these jobs have the smallest due dates.Thus, we propose the following algorithm under the maximum sequence disruption constraint. (see Box 1).

Box 1 

Algorithm 4. Consider the following steps.
Step 1. Index the job of 𝐽_𝑁 in the EDD order.
Step 2. Schedule jobs 1,…,𝑛₀+𝑘 in the EDD rule in the first 𝑛₀+𝑘 position and schedule jobs 𝑛₀+𝑘+1,…,𝑛₀+𝑛_𝑁 in the EDD order in the final 𝑛_𝑁−𝑘 positions.

Theorem 5. For the 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, 𝐷_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 problem, Algorithm 4 finds an optimal schedule in 𝑂(𝑛+𝑛_𝑁log𝑛_𝑁) time.

Proof. From Lemmas 2 and 3, the constraint 𝐷_max(𝜋^∗)≤𝑘 allows at most 𝑘 jobs of 𝐽_𝑁 to be sequenced before the final of 𝐽₀, and these are the jobs of 𝐽_𝑁 with the smallest due dates. Classical schedule theory shows that the jobs of this first group are sequenced in the EDD order, while Lemma 3 establishes that the remaining 𝑛_𝑁−𝑘 jobs of 𝐽_𝑁 are also sequenced in the EDD order.
Next, we note that the Step 1 for the jobs of 𝐽_𝑁 requires 𝑂(𝑛_𝑁log𝑛_𝑁) time. Step 2 is executed in 𝑂(𝑛) time by merging the first 𝑘 jobs of the EDD ordered jobs of 𝐽_𝑁 with the jobs of 𝐽₀ as sequenced in 𝜋^∗ and then placing the last 𝑛_𝑁−𝑘 jobs of the EDD order ordered jobs of 𝐽_𝑁 at the end of the schedule.

Next, we consider problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, ∑𝐷_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗. From Lemmas 1, 2, and 3, there is the total sequence disruption of the jobs of 𝐽_𝑁 which is less than or equal to 𝑘 and can be sequenced before the last job of 𝐽₀, and these jobs have the smallest due dates. Thus, we propose the following algorithm under the total sequence disruption constraint. (see Box 2).

Box 2 

Let 𝑓(𝑖,𝑗,𝛿) be minimum total tardiness value of a partial schedule for jobs 𝐽₁,…,𝐽_𝑖 and 𝐽_𝑛0+1,…,𝐽_𝑛0+𝑗, where the total sequence disruption is equal to 𝛿. The dynamic programming procedure can now be stated as follows.

Algorithm 6. Consider the following steps.
Step 1 (Initialization). Set 𝑓(𝑖,𝑗,𝛿)=0 for (𝑖,𝑗,𝛿)=(0,0,0) and 𝑓(𝑖,𝑗,𝛿)=∞ for (𝑖,𝑗,𝛿)≠(0,0,0), 𝑖=1,…,𝑛₀, 𝑗=1,…,𝑛_𝑁, 𝑖=0,…,𝑘.
Step 2 (Recurrence Relation). 𝑓𝑓𝐶𝑖,𝑗,𝛿=min𝑖−1,𝑗,𝛿−𝑗+max_𝑖−𝑑_𝑖;𝑓𝐶,0𝑖,𝑗−1,𝛿+max_𝑛0+𝑗−𝑑_𝑛0+𝑗,0, (5) where 𝐶_𝑗 denotes the completion time of job 𝐽_𝑗.
Step 3 (Optimal Solution). Calculate the optimal solution value min_{0≤𝛿≤𝑘}{𝑓(𝑛₀,𝑛_𝑁,𝛿)}.

In the recurrence relation, the first term in the minimization corresponds to the case where the partial schedule ends with job 𝐽_𝑖∈𝐽₀. Because 𝐽_𝑗 jobs of 𝐽_𝑁 appear before job 𝐽_𝑖 in such a partial schedule, the increase in total sequence disruption is equal to 𝑗. The second term corresponds to the case where the partial schedule ends with job 𝐽_𝑛0+𝑗∈𝐽_𝑁.

In addition, we demonstrate the result of Algorithm 6 in the following example.

Example 7. 𝑛₀=3, 𝑛_𝑁=3, 𝑡₀=0, 𝐽₀={𝐽₁,𝐽₂,𝐽₃}, 𝐽_𝑁={𝐽₄,𝐽₅,𝐽₆}, 𝑝₁=3.2, 𝑑₁=4, 𝑝₂=1, 𝑑₂=2.4, 𝑝₃=2, 𝑑₃=3,𝑝₄=1.5, 𝑑₄=3, 𝑝₅=4, 𝑑₅=5, 𝑝₆=3, 𝑑₆=4.7, 𝑎=0.2, 𝑏=0.4, and 𝑘=5.
Solution: According to Algorithm 4 and Lemmas 2 and 3. Because the total sequence disruption of the jobs 𝐽₁, 𝐽₂, 𝐽₃ can not exceed 𝑘=5. By dynamic programming algorithm, we obtain job sequence and the total tardiness cost as follows: if  𝑘=0, the optimal sequence is [𝐽₂→𝐽₃→𝐽₁→𝐽₄→𝐽₆→𝐽₅], and the total tardiness cost is 26.8007; if  𝑘=0, the job sequence is [𝐽₂→𝐽₃→𝐽₁→𝐽₄→𝐽₆→𝐽₅], and the total tardiness cost is 26.8007; if  𝑘=1, the job sequence is [𝐽₂→𝐽₃→𝐽₄→𝐽₁→𝐽₆→𝐽₅], and the total tardiness cost is 25.8007; if  𝑘=2, the job sequence is [𝐽₂→𝐽₄→𝐽₃→𝐽₁→𝐽₆→𝐽₅], and the total tardiness cost is 25.8007; if  𝑘=3, the job sequence is [𝐽₄→𝐽₂→𝐽₃→𝐽₁→𝐽₆→𝐽₅], and the total tardiness cost is 25.8007; if  𝑘=4, the job sequence is [𝐽₄→𝐽₂→𝐽₃→𝐽₆→𝐽₁→𝐽₅], and the total tardiness cost is 26.8223; if  𝑘=5, the job sequence is [𝐽₄→𝐽₂→𝐽₆→𝐽₃→𝐽₁→𝐽₅], and the total tardiness cost is 28.5223;
Furthermore, if total sequence disruption 𝑘 is equal to 1, 2, 3, then minimizing total tardiness cost is 25.8007, and optimal job sequence is [𝐽₂→𝐽₃→𝐽₄→𝐽₁→𝐽₆→𝐽₅], [𝐽₂→𝐽₄→𝐽₃→𝐽₁→𝐽₆→𝐽₅], and [𝐽₄→𝐽₂→𝐽₃→𝐽₁→𝐽₆→𝐽₅].

Theorem 8. For the 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, ∑𝐷_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 problem, Algorithm 6 finds an optimal schedule in 𝑂(𝑛²₀𝑛²_𝑁) time.

Proof. From Lemmas 1, 2, and 3, it only remains to enumerate all possible ways of merging the EDD ordered lists of jobs of 𝐽₀ and 𝐽_𝑁. Algorithm 6 does so by comparing the cost of all possible state transitions and therefore finds an optimal schedule.
Because 𝑖≤𝑛₀, 𝑗≤_𝑛𝑁 and 𝛿≤𝑘≤𝑛₀𝑛_𝑁, there are 𝑂(𝑛²₀𝑛²_𝑁) values of the state variables. Step 1 requires 𝑂(𝑛_𝑁log𝑛_𝑁). Step 2 requires constant time for each set of values of the state variables. Thus, the overall time complexity of Algorithm 6 is 𝑂(𝑛²₀𝑛²_𝑁).

Now, we consider problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Δ_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗. From Lemmas 1, 2, and 3, the maximum time disruption of jobs of 𝐽₀ is at most 𝑘, and jobs of 𝐽_𝑁 before the last job of 𝐽₀ have the smallest due dates. Thus, we propose the following algorithm under the maximum time disruption constraint. (see Box 3).

Box 3 

Let 𝑓(𝑖,𝑗,𝛿) be minimum total tardiness value of a partial schedule for jobs 𝐽₁,…,𝐽_𝑖 and 𝐽_𝑛0+1,…,𝐽_𝑛0+𝑗, where the maximum time disruption is equal to 𝛿. The dynamic programming procedure can now be stated as follows.

Algorithm 9. Consider the following steps.
Step 1 (Initialization). Set 𝑓(𝑖,𝑗,𝛿)=0 for (𝑖,𝑗,𝛿)=(0,0,0) and 𝑓(𝑖,𝑗,𝛿)=∞ for (𝑖,𝑗,𝛿)≠(0,0,0), 𝑖=1,…,𝑛₀, 𝑗=1,…,𝑛_𝑁, and 𝑖=0,…,𝑘.
Step 2 (Recurrence Relation). 𝑓𝑓𝑖,𝑗,𝛿=min𝑖−1,𝑗,𝛿−𝑃_ℎ+max𝐶_𝑖−𝑑_𝑖;𝑓𝐶,0𝑖,𝑗−1,𝛿+max_𝑛0+𝑗−𝑑_𝑛0+𝑗,0, (6) where 𝑃_ℎ is the sum of actual processing time of the new jobs of 𝐽_𝑁 between 𝐽_𝑖−1 and 𝐽_𝑖 and 𝐶_𝑗 denotes the completion time of job 𝐽_𝑗.
Step 3 (Optimal Solution). Calculate the optimal solution value min_{0≤𝛿≤𝑘}{𝑓(𝑛₀,𝑛_𝑁,𝛿)}.

In the recurrence relation, the first term in the minimization corresponds to the case where the partial schedule ends with job 𝐽_𝑖∈𝐽₀. Because 𝐽_𝑗 jobs of 𝐽_𝑁 appear before job 𝐽_𝑖 in such a partial schedule, the increase in the maximum time disruption is equal to 𝑃_ℎ. The second term corresponds to the case where the partial schedule ends with job 𝐽_𝑛0+𝑗∈𝐽_𝑁.

Similar to Example 7, by Algorithm 9, we have(i)the maximum time disruption 𝑘 is in [1.7237,25.7919]; the job sequence and the total tardiness cost are the same to Example;(ii)the maximum time disruption 𝑘 is in [0,1.7237), the job sequence is [𝐽₂→𝐽₃→𝐽₁→𝐽₄→𝐽₆→𝐽₅], and the total tardiness cost is 26.8007.

Theorem 10. For the 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, Δ_max(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 problem, Algorithm 9 finds an optimal schedule in 𝑂(𝑛₀𝑛_𝑁𝐶_max+𝑛_𝑁log𝑛_𝑁) time.

Proof. From Lemmas 1, 2, and 3, Δ≤𝑘 means that the total actual processing time of the new jobs of 𝐽_𝑁 before the last job of 𝐽₀ is at most 𝑘, and these are the jobs of 𝐽_𝑁 with the smallest due dates. Hence, Algorithm 9 schedules the jobs according to the (EDD, EDD) property.
Because 𝑖≤𝑛₀, 𝑗≤𝑛_𝑁 and 𝛿≤𝑘<𝐶_max, there are 𝑂(𝑛₀𝑛_𝑁𝐶_max) values of the state variables. Step 1 requires 𝑂(𝑛_𝑁log𝑛_𝑁). Step 2 requires constant time for each set of values of the state variables. Thus, the overall time complexity of Algorithm 9 is 𝑂(𝑛₀𝑛_𝑁𝐶_max+𝑛_𝑁log𝑛_𝑁).

Now, we consider problem 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗, ∑Δ_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗. From Lemmas 1, 2, and 3, there is the total time disruption of jobs of 𝐽_𝑁 which is less than or equal to 𝑘 and can be sequenced before the last job of 𝐽₀, and these jobs have the smallest due dates. The following dynamic programming algorithm performs an optimal merging of jobs of 𝐽₀ and 𝐽_𝑁 in a way similar to Algorithm 6. (see Box 4).

Box 4 

Let 𝑓(𝑖,𝑗,𝛿) be minimum total tardiness value of a partial schedule for jobs 𝐽₁,…,𝐽_𝑖 and 𝐽_𝑛0+1,…,𝐽_𝑛0+𝑗, where the total time disruption is equal to 𝛿. The dynamic programming procedure can now be stated as follows.

Algorithm 11.    
Step 1 (Initialization). Set 𝑓(𝑖,𝑗,𝛿)=0 for (𝑖,𝑗,𝛿)=(0,0,0) and 𝑓(𝑖,𝑗,𝛿)=∞ for (𝑖,𝑗,𝛿)≠(0,0,0), 𝑖=1,…,𝑛₀, 𝑗=1,…,𝑛_𝑁, and 𝑖=0,…,𝑘.
Step 2 (Recurrence Relation).𝑓𝑓𝑖,𝑗,𝛿=min𝑖−1,𝑗,𝛿−^𝑛0+𝑗_ℎ=𝑛0+1𝑝_[ℎ]+max𝐶_𝑖−𝑑_𝑖;𝑓𝐶,0𝑖,𝑗−1,𝛿+max_𝑛0+𝑗−𝑑_𝑛0+𝑗,0, (7)where 𝑝_[ℎ] is the actual processing time of job 𝐽_ℎ; 𝐶_𝑗 denotes the completion time of job 𝐽_𝑗.
Step 3 (Optimal Solution). Calculate the optimal solution value min_{0≤𝛿≤𝑘}{𝑓(𝑛₀,𝑛_𝑁,𝛿)}.

In the recurrence relation, the first term in the minimization corresponds to the case where the partial schedule ends with job 𝐽_𝑖∈𝐽₀. Because 𝐽_𝑗 jobs of 𝐽_𝑁 appear before job 𝐽_𝑖 in such a partial schedule, the increase in total time disruption is equal to ∑^𝑛0+𝑗_ℎ=𝑛0+1𝑝_[ℎ]. The second term corresponds to the case where the partial schedule ends with job 𝐽_𝑛0+𝑗∈𝐽_𝑁.

Theorem 12. For the 1∣𝑝_𝑖≤𝑝_𝑗⇒𝑑_𝑖≤𝑑_𝑗,∑Δ_𝑗(𝜋^∗)≤𝑘, 𝑝_𝑗(𝑎+𝑏𝑠_𝑗)∣∑𝑇_𝑗 problem, Algorithm 11 finds an optimal schedule in 𝑂(𝑛²₀𝑛_𝑁𝐶_max) time.

Proof. The proof of optimality of Algorithm 11 is similar to Theorem 8 regarding the time complexity. Because 𝑖≤𝑛₀, 𝑗≤𝑛_𝑁, and 𝛿≤𝑘≤𝑛₀𝐶_max, there are 𝑂(𝑛²₀𝑛_𝑁𝐶_max) values of the state variables. Step 1 requires 𝑂(𝑛_𝑁log𝑛_𝑁). Similar arguments to those in the proof of Theorem 8 show that the overall time complexity of Algorithm 11 is 𝑂(𝑛²₀𝑛_𝑁𝐶_max).

4. Conclusions

In this paper, we studied the issue of rescheduling to allow the unexpected arrival of new jobs and took into account the effect of the disruption on a previously planned optimal schedule. The main contribution of this paper is that we develop the machine rescheduling scheduling problems agreeable job parameters under deterioration and disruption. Rescheduling means to schedule the jobs again, together with a set of new jobs. Deteriorating job means that the actual processing time of a job is an increasing function of its starting time. When the processing time and due date of jobs are agreeable, we considered some problems to minimize total tardiness under a limit on two disruption constraints: sequence disruption and time disruption. We proposed polynomial time algorithms or some dynamic programming algorithms for each problem. Future research may stimulate rescheduling models to mitigate the effects of the disruptions that occur frequently in manufacturing practice.

Acknowledgments

The authors would like to thank the authors of the references for enlightening them. This paper was supported by National Natural Science Foundation of China (71272085), Tianyuan Fund for Mathematics (11226237), the Humanities and Social Sciences Program of the Ministry of Education of China (12YJA630135), the Foundation of Chongqing Education Commission (KJ120624), and the Key Project Fundation of Chongqing Normal University (2011XLZ05).