Group Scheduling Problems with Time-Dependent and Position-Dependent DeJong’s Learning Effect

In classical scheduling problems, the scheduling models routinely assume that job processing times are known and fixed throughout the period of production process. However, this assumption may be unrealistic in many situations that the processing time of jobs may be shortened due to learning effect over time. Production scheduling problems with learning effect have been paid much attention in recent years. To the best of our knowledge, there is little research that considers position-dependent and time-dependent processing time group scheduling model. Besides, the existing learning effect scheduling model suffers the drawback that when a job’s position or the starting processing time is sufficiently large in a schedule, its actual processing time is close to zero (infinity). )is paper is to introduce a new scheduling model with position-dependent and timedependent processing time, which overcomes the above shortcomings and is more general and realistic than the models existing in the literature. )e remaining part of this paper is organized as follows. In Section 2, we present a brief review of the existing scheduling model with learning effect. In Section 3, a precise formulation of the problem is given. Section 4 considers several single-machine scheduling problems with positiondependent and time-dependent DeJong’s learning effect to minimize makespan, the total completion time, and the total weighted completion time, respectively. )e last section contains some conclusions of our model.


Introduction
In classical scheduling problems, the scheduling models routinely assume that job processing times are known and fixed throughout the period of production process. However, this assumption may be unrealistic in many situations that the processing time of jobs may be shortened due to learning effect over time. Production scheduling problems with learning effect have been paid much attention in recent years. To the best of our knowledge, there is little research that considers position-dependent and time-dependent processing time group scheduling model. Besides, the existing learning effect scheduling model suffers the drawback that when a job's position or the starting processing time is sufficiently large in a schedule, its actual processing time is close to zero (infinity). is paper is to introduce a new scheduling model with position-dependent and timedependent processing time, which overcomes the above shortcomings and is more general and realistic than the models existing in the literature. e remaining part of this paper is organized as follows. In Section 2, we present a brief review of the existing scheduling model with learning effect. In Section 3, a precise formulation of the problem is given. Section 4 considers several single-machine scheduling problems with positiondependent and time-dependent DeJong's learning effect to minimize makespan, the total completion time, and the total weighted completion time, respectively. e last section contains some conclusions of our model. Wright [1]; Biskup [2]; and Cheng and Wang [3] are among the pioneers that brought the learning effect into the field of scheduling research. Since then, learning effect has been widely employed in management science. To overcome the shortcoming associated with Wright's [1] definition of learning effect that the improvement of learning effect is infinite, DeJong [4] introduced a new learning scheduling model, which is more realistic. Wang et al. [5] studied timedependent DeJong's learning effect model p A jr � p j (αa
In many production processes, the efficiency can be improved by grouping various parts and products with similar designs.
is phenomenon is known as group technology in the literature. Many advantages have been claimed through the wide applications of group technology, for instance, Ji et al. [23] and Wang and Liu [24]. For the latest results on group scheduling problems, we refer the reader to the studies of Zhang et al. [18,25]; Wang and Wang [26]; and Lu et al. [27] among others.

Problem Formulation
We first define the notations which are used throughout this paper, followed by the descriptions of the problems: m: the number of groups (m ≥ 2) n i : the number of jobs in group G i , i � 1, 2, . . . , m n: the total number of jobs, i.e., n 1 + n 2 + · · · + n m � n ere are n independent jobs. All jobs are classified into m groups G 1 , G 2 , . . . , G m by the similarities and to be processed on a single machine. All the jobs are available at time zero, and job preemption is not allowed. A group setup time is required if the machine switches to process from one group to another. Jobs in the same group are processed consecutively and need no setup time.
e machine can handle one job at a time. Each job J ij has a normal processing time p ij . e actual processing time of job will be shorter than its normal processing time due to the learning effect.
In this paper, we consider the following new time-dependent and position-dependent model: where 0 < a < 1, b < 0 is the learning index, t [ . M represents "the factor of incompressibility" (0 ≤ M ≤ 1). If M � 0, the model simplifies to the classical learning model p jr � p j r a . If M � 1, the processing time of job is constant. e objectives are to find the optimal job sequence in each group and the optimal group sequence such that makespan, the total completion time, and the total weighted completion time are minimized, respectively. Using the three-field notation for scheduling problem, we denote our problems as Before proving the problems, some lemmas are introduced as follows.
Proof. Taking the first derivative of F(x, r) with respect to x, the following is obtained: Taking the first derivative of f(x, r) with respect to x, the following is obtained: Since Proof. From the Lagrange mean value theorem, the lemma can be easily obtained.

Result of Optimization
In this section, we address the application of our model to solve the single-machine group scheduling problems involving makespan minimization, the total completion time minimization, and the total weighted completion time minimization.

Makespan Minimization
the job sequence in each group is in the smallest normal processing time order (SPT) and (ii) the groups can be sequenced in any order.
Proof. Suppose a sequence is G [1] , G [2] , . . . , G [m] . erefore, the makespan of the sequence is obtained as follows: e first term of equation (7) is a constant. e value of the second term of equation (7) is determined by the job sequence in each group. erefore, the makespan minimization of the studied model is independent of the sequence of group. Hence, the optimal group sequence can be scheduled in any order, so (ii) follows.
Next, we determine the optimal job sequence in each group to minimize the makespan by implementing the adjacent job exchange argument. Let π 1 and π 1 ′ be two job schedules of a group G i where the difference between π 1 and π 1 ′ is a pairwise interchange of two adjacent jobs J ik and J il .
at is, where S 1 and S 2 are the partial sequences. e positions of J ik and J il in sequence π 1 are r and r + 1, respectively, which are reverse in sequence π 1 ′ . Furthermore, we assume that t denotes the completion time of the last job in S 1 and p ik ≤ p il . To prove π 1 dominates π 1 ′ , it suffices to show that C il (π 1 ) ≤ C ik (π 1 ′ ) and C iu (π 1 ) ≤ C iu (π 1 ′ ) for any job J iu in S 2 . By definition, the completion time periods of jobs J ik and J il in sequence π 1 and π 1 ′ are given by us, we can obtain From p ik ≤ p il and Lemma 1, it is easy to obtain that C ik (π 1 ′ ) − C il (π 1 ) ≥ 0. Hence, we have C il (π 1 ) ≤ C ik (π 1 ′ ).
Repeating the job interchange argument for all jobs in each group not sequenced in the SPT order yields part (i) of eorem 1 and we complete the proof of eorem 1. From eorem 1, the problem 1|GT, [r] r b )|C max can be solved by using Algorithm 1.

Algorithm 1
Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time (the SPT order) Step 2. Groups are scheduled in any order We address application of Algorithm 1 in the following example.

Example 1.
ere are five jobs classified into two groups G 1 and Step 1. In group G 1 , the optimal job sequence is J 11 ⟶ J 12 . In group G 2 , the optimal job sequence is J 23 ⟶ J 22 ⟶ J 21 .
Step 2. Groups are scheduled in any order. erefore, the optimal schedule is J 11 Hence, the minimum makespan is as follows:
Next, we determine the optimal group sequence. Let π 2 and π 2 ′ be two schedules where the difference between π 2 and π 2 ′ is a pairwise interchange of two adjacent groups G i and G j and job sequence in each group of π 2 and π 2 ′ is in SPT order.
at is, π 2 � [δ 1 , G i , G j , δ 2 ] and π 2 ′ � [δ 1 , G j , G i , δ 2 ], where δ 1 and δ 2 are the partial sequences. Furthermore, denote t as the completion time of the last job in δ 1 . To prove π 2 dominates π 2 ′ , it suffices to address . By definition, the completion times of jobs of groups G i and G j in π 2 are given by C i [1] π 2 � t + s i + p i [1] , e completion times of jobs of groups G j and G i in π 2 ′ are given by C j [1] π 2 ′ � t + s j + p j [1] , C j [2] π 2 ′ � t + s j + p j [1] + p j [2] M +(1 − M)a p j [1] [1] , us, we have We have is completes the proof. From eorem 2, the problem 1|GT, can be solved by the following algorithm:

Algorithm 2
Step 1. Jobs in each group are scheduled in a nondecreasing order of the normal processing time (the SPT order).
Step 2. To each group, calculate Step 3. Groups are scheduled in the nondecreasing order of θ i .
Obviously, it is easy to show that the total time for Algorithm 2 is O(n log n).
We show application of Algorithm 2 in the following example.
erefore, the optimal group sequence is G 2 ⟶ G 1 and the optimal schedule is J 23 Hence, the minimum total completion time is C ij � 79.65.

Total Weighted Completion Time
Minimization. In our model, we consider the group scheduling to minimize the total weighted completion time if jobs in each group satisfy an agreeable condition.
if jobs in each group have agreeable weighted normal processing time, i.e., p ij ≤ p il implies w ij ≥ w il , for all jobs J ij and J il in G i , i � 1, 2, . . . , m. e optimal schedule satisfies the following: (i) the job in each group is in nondecreasing order of p ij /w ij (the WSPT order) and (ii) the groups can be sequenced in the nondecreasing order of φ i , where Proof. First, we show the optimal job sequence in each group. Without loss of generality, we still use the notation of eorem 1. From eorem 1 and p ik ≤ p il , we have w ik ≥ w il . erefore, From Lemma 2, it is easy to show h ′ (ξ 1 ) ≥ h ′ (ξ 2 ). Since p ik ≤ p il and w ik ≥ w il , it can be obtained w ik C ik (π 1 ) + w il C il (π 1 ) ≤ w ik C ik (π 1 ′ ) + w il C il (π 1 ′ ). From eorem 1, we have C iu (π 1 ) ≤ C iu (π 1 ′ ) for any job J iu in S 2 . Hence, the job completion time of S 2 in π 1 is earlier than the same job completion time of S 2 in π 1 ′ . us, the job in each group is in nondecreasing order of p ij /w ij (the WSPT order) in the optimal schedule. is completes the proof of part (i).

Mathematical Problems in Engineering 5
Next, determine the optimal group sequence. We still use the notation of eorem 2. To prove π 2 dominates π 2 ′ , it suffices to show that We have . (22) Hence, the optimal groups can be sequenced in the nondecreasing order of φ i , and this completes the proof of the theorem. Step 1. Jobs in each group are scheduled in nondecreasing order of p ij /w ij (the WSPT order).
Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of φ i .
Obviously, it is easy to show that the total time for Algorithm 3 is O(n log n).
We present application of Algorithm 3 in the following example.
Example 3. Also turn to the Example 1, change the objective to the total weighted completion time minimization (w 11 � 2, w 12 � 1, w 21 � 1, w 22 � 2, and w 23 � 4). Obviously, jobs in each group of this instance have the agreeable weighted normal processing times in eorem 3. By Algorithm 3, we solve the problem as follows: Step 1: In group G 1 , the optimal job sequence is J 11 ⟶ J 12 . In group G 2 , the optimal job sequence is J 23 ⟶ J 22 ⟶ J 21 .
erefore, the optimal group sequence is G 2 ⟶ G 1 and the optimal schedule is J 23 ⟶ J 22 ⟶ J 21 ⟶ J 11 ⟶ J 12 .
Hence, the minimum total weighted completion time is w ij C ij � 122.28.

Conclusions
We study new group scheduling models with position-dependent and time-dependent learning effect. e actual processing time of a job is a function of the total normal processing time and the total number of jobs scheduled ahead of it, which is motivated by DeJong's learning schedule model. We show that the models can be solved polynomially when the objectives are the makespan minimization and the total completion time minimization. e total weighted completion time minimization is proved to be solved in polynomial time under an agreeable condition. It is suggested for future research to extend to other more practical learning effect scheduling models in the other machine environment, including multimachine and jobshop settings.

Data Availability
All data generated or analysed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.