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

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.


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.

Mathematical Problems in Engineering
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.

Problem Definition and Notation
By the terminology of Hall and Potts [15], our researchful problem can be stated as follows.Let  0 = { 1 , . . .,   0 } 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  * is an optimal job sequence with no idle time between the jobs.Let   = {  0 +1 , . . .,   0 +  } 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  0 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  0 are removed from the problem; any partly processed jobs are processed to completion, and  0 and  0 are updated accordingly.Let  =  0 ⋃   and  =  0 +   .Each job   ∈  has an integral normal processing time   and a deteriorating rate  > 0; the actual processing time of job   is   ( +   ), where   (≥0) is the starting time of job   and  (>0) is constant.For any schedule  of the jobs in , we define the following variables:   () is the time at which job   ∈  starts its processing in schedule .  () is the due date of job When there is no ambiguity, we simplify the above symbols and write   ,     ,   ,   ( * ), and Δ  ( * ), respectively.
In this paper, we consider a special case: the processing time and due date of jobs are agreeable; that is,   ≤   ⇒   ≤   for all jobs   and   .
Using the three-field notation [20], the considered problems can be denoted as

Minimum Tardiness Problem with Agreeable Job Parameters
We start with the following result by Kononov and Gawiejnowicz [21].
. .,   } and the starting time of the job  1 is , then makespan is sequence independent, and For notational convenience, we assume that the jobs are indexed by agreeable order; that is,  1 ≤ ⋅ ⋅ ⋅ ≤   0 and  1 ≤ ⋅ ⋅ ⋅ ≤   0 .Thus,  * = { 1 , . . .,   0 } with no idle time between jobs.We now show that the EDD or SPT rule applies to serval of the rescheduling problems we consider.
The total tardiness of jobs   and   in   is as follows: To compare the total tardiness of jobs   and   in  * and in   , we divide it into two cases.In the first case, when   ( * ) ≤   , we have   ( * ) +   ( * ) = max{  ( * ) −   , 0}.Suppose that neither   (  ) nor   (  ) is zero.Note that this is the most restrictive case since it comprises the case that either one or both   (  ) and   (  ) are zero.From Lemma 1 and In the second case, when   ( * ) >   , we have   ( * ) +   ( * ) =   ( * )+  ( * )−  −  .Suppose that neither   (  ) nor   (  ) is zero.From Lemma 1 and   ≤   ,   ≤   , we have Now, we have proved that the total tardiness of   is less than or equal to that of  * .
Let the position of job   in  * be  1 and let the position of job   in  * be  2 and in   be , where ℎ is the difference between the position of job   and   in  * .So,   ( * ,   ) <   ( * ,  * ).Hence, we have  max ( * ,   ) <  max ( * ,  * ).In either case, because Thus, we have Δ max ( * ,   ) < Δ max ( * ,  * ).In either case, because Δ  ( * ,   ) = Δ  ( * ,  * ) − ℎ  and Δ  ( * ,   ) ≤ Δ  ( * ,  * ) + ℎ  , where ℎ  =   ( * ) −   (  ).Then we deduce that ∑ Δ  ( * ,   ) ≤ ∑ Δ  ( * ,  * ).Thus, for either problem,   is feasible and optimal.We can show that there exists an optimal schedule in which the jobs of  0 are sequenced in the EDD or SPT order as in  * by finite numbers of repetitions of the argument.A similar interchange argument establishes that the jobs of   can also be obtained by sequencing in the EDD or SPT order.The same EDD or SPT ordering of the jobs of  0 in  * 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 From Lemmas 2 and 3, there are at most  jobs of   that can be sequenced before the last job of  0 , and these jobs have the smallest due dates.Thus, we propose the following algorithm under the maximum sequence disruption constraint.(see Box 1).Algorithm 4. Consider the following steps.
Step 1. Index the job of   in the EDD order.

Theorem 5. For the 1 | 𝑝
Proof.From Lemmas 2 and 3, the constraint  max ( * ) ≤  allows at most  jobs of   to be sequenced before the final of  0 , and these are the jobs of   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   −  jobs of   are also sequenced in the EDD order.
Next, we note that the Step 1 for the jobs of   requires (  log   ) time.Step 2 is executed in () time by merging the first  jobs of the EDD ordered jobs of   with the jobs of  0 as sequenced in  * and then placing the last   −  jobs of the EDD order ordered jobs of   at the end of the schedule.

Next, we consider problem 1 | 𝑝
From Lemmas 1, 2, and 3, there is the total sequence disruption of the jobs of   which is less than or equal to  and can be sequenced before the last job of  0 , and these jobs have the smallest due dates.Thus, we propose the following algorithm under the total sequence disruption constraint.(see Box 2).
Let (, , ) be minimum total tardiness value of a partial schedule for jobs  1 , . . .,   and   0 +1 , . . .,   0 + , where the total sequence disruption is equal to .The dynamic programming procedure can now be stated as follows.Algorithm 6.Consider the following steps.
In the recurrence relation, the first term in the minimization corresponds to the case where the partial schedule ends with job   ∈  0 .Because   jobs of   appear before job   in such a partial schedule, the increase in total sequence disruption is equal to .The second term corresponds to the case where the partial schedule ends with job   0 + ∈   .
In addition, we demonstrate the result of Algorithm 6 in the following example.Solution: According to Algorithm 4 and Lemmas 2 and 3.Because the total sequence disruption of the jobs  1 ,  2 ,  3 can not exceed  = 5.By dynamic programming algorithm, we obtain job sequence and the total tardiness cost as follows: if  = 0, the optimal sequence is [ 2 →  Let (, , ) be minimum total tardiness value of a partial schedule for jobs  1 , . . .,   and   0 +1 , . . .,   0 + , where the maximum time disruption is equal to .The dynamic programming procedure can now be stated as follows.
Algorithm 9. Consider the following steps.

𝑓 (𝑖, 𝑗
where  ℎ is the sum of actual processing time of the new jobs of   between  −1 and   and   denotes the completion time of job   . Step 3 (Optimal Solution).Calculate the optimal solution value min 0≤≤ {( 0 ,   , )}.
In the recurrence relation, the first term in the minimization corresponds to the case where the partial schedule ends with job   ∈  0 .Because   jobs of   appear before job   in such a partial schedule, the increase in the maximum time disruption is equal to  ℎ .The second term corresponds to the case where the partial schedule ends with job   0 + ∈   .
Similar to Example 7, by Algorithm 9, we have Proof.From Lemmas 1, 2, and 3, Δ ≤  means that the total actual processing time of the new jobs of   before the last job of  0 is at most , and these are the jobs of   with the smallest due dates.Hence, Algorithm 9 schedules the jobs according to the (EDD, EDD) property.
Because  ≤  0 ,  ≤   and  ≤  <  max , there are ( 0    max ) values of the state variables.Step 1 requires (  log   ).Step 2 requires constant time for each set of values of the state variables.Thus, the overall time complexity of Algorithm 9 is ( 0    max +   log   ).Now, we consider problem 1 |   ≤   ⇒   ≤   , ∑ Δ  ( * ) ≤ ,   ( +   ) | ∑   .From Lemmas 1, 2, and 3, there is the total time disruption of jobs of   which is less than or equal to  and can be sequenced before the last job of  0 , and these jobs have the smallest due dates.The following dynamic programming algorithm performs an optimal merging of jobs of  0 and   in a way similar to Algorithm 6. (see Box 4).
and 1 |   ≤   ⇒   ≤   , Δ max ( * ) ≤ ,   (+  ) | ∑   have an optimal schedule with no idle time between jobs, and(a) a schedule for problem 1 |   ≤   ⇒   ≤   , max ( We first analyze the jobs of  0 .Consider an optimal schedule  * in which the jobs of  0 are not sequenced in the EDD or SPT rule as in  * .Let   be the job with the smallest index that appears later relative to the other jobs of  0 in  * than in  * , and let   ( > ) be the last job of  0 that precedes job   in  * .Because  * is an optimal sequence,   and   are agreeable.Assume that the starting time of job   in  * is  0 , then   ( * ) =  0 +   ( +  0 ).Perform an interchange on jobs   and   , and get a new schedule   .In   , the starting time of job   is  0 , then   ( The total tardiness of jobs   and   in  * is as follows: * ) ≤ ,   ( +   ) | ∑   is feasible if the number of jobs of   scheduled before the last job of  0 is less than or equal to ; (b) a schedule for problem 1 |   ≤   ⇒   ≤   , Δ max ( * ) ≤ ,   ( +   ) | ∑   is feasible if the total actual processing time of jobs of   scheduled before the last job of  0 is less than or equal to .Proof.The proof is similar to that of Lemma 1 in Hall and Potts [15].Lemma 3.For problems 1 |   ≤   ⇒   ≤   , Γ ≤  and   ( +   ) | ∑   , where Γ ∈ { max ( * ), ∑   ( * ), Δ max ( * ), ∑ Δ  ( * )}, there exists an optimal schedule in which the jobs of  0 are sequenced in the EDD or SPT rule as in  * , the jobs of   are sequenced in the EDD or SPT rule, and there is no idle time between jobs.Proof. ) =  0 +   ( +  0 ) <  0 +   ( +  0 ) =   ( * ).From Lemma 1,   (  ) =   ( * ).Thus, the jobs between job   and   are completed earlier in   than in  * .Next, we consider the total tardiness of jobs   and   in   and in  * .