Research Article Group Scheduling with Learning Effect and Random Processing Time

In this paper, we establish a stochastic grouping scheduling model. In the model, there is no installation time between the jobs in the same group, but each group has an installation time before processing. There are learning eﬀects between groups and within groups, and the completion time of jobs is a random variable. We take the long expected schedule and the expected total completion time as the objective function, and the noninterruptible static priority strategy is obtained. At the same time, heuristic algorithms and examples are given.


Introduction
Since Biskup [1] researched on a single-machine scheduling which is based on position learning effect, this kind of problem has become a hot research topic, and the research results have been very fruitful. Moshiov [2] studied a singlemachine scheduling problem and proved that the SPT ordering rules is the optimal solution to minimize makespan and EDD ordering rules is optimal order to minimize the maximum delay. e authors in [3][4][5] have also carried out a lot of research studies on this problem. Mosheiov and Sidney [6] proved that model 1|p jr � p j r a |C max , 1|p jr � p j r a | C j , and 1|p jr � p j r a i , d j � d|( ω 1 C j + ω 2 T j + ω 3 d) can be transformed into assignment problem, time complexity is O(n 3 ). Mosheiov and Sidney [7] studied the problem 1|p jr � p j r a i , d j � d| U j . ey transformed the problem into a classic assignment problem, and the time complexity of the problem is O(n 3 log n). Lin [8] proved the problem to minimize the tardy jobs when learning rates of the jobs. Bachman and Janiak [9] discussed the problem in special cases 1|p jr � p j r a , d j � d| ω j C j , and 1|p jr � p j r a | ωp j C j . e former is based on the nonincreasing order, and the latter, according to the SPT sequence, can get the optimal solution. Zhao et al. [10] gave a solution to question 1|p jr � p j r a | ω j C j . When p i ≤ p j ⇒w i ≥ w j , the WSPT order rule is used. When p i ≤ p j ⇒d i ≤ d j , by using EDD ordering rules for the problem of minimizing the maximum tardiness problem, we can get optimal solutions. Liu and Cao [11] solve a kind of difficult problem with the structure network of N prism. Cheng et al. [12] found that there is no gap between the jobs in the process of the jobs, that is to say, the jobs are continuous processing. In [13][14][15], the authors continue to study this kind of problem and get many results.
As people pay attention to the processing time, time of arrival, and uncertainty of machine breakdown, the problem of stochastic scheduling has been paid more and more attention. Pinedo and Hadavi [16], Pinedo and Rammouz [17], Frenk [18], and Liu et al. [19] have carried out a lot of pioneering work on stochastic scheduling problems, and their objective function are mostly the normal function.
en, the content of the research is also more and more diversified. In addition, Zhang et al. [20] continued to study this question. Li [21] considered a stochastic scheduling problem. However, in real life, the machining time is affected by many factors, the randomness of the universal existence. e learning effect of the machine has not changed. e authors in [22][23][24][25][26][27][28] continued to study this kind of problem and obtained good results in many aspects.
In the classical scheduling problems, most of the research studies on the problem that the installation time is needed to replace the jobs are assumed to be independent of the scheduling of machines and jobs, so the installation time is usually incorporated into the processing time of the jobs. Ham et al. [29] first proposed the "group technology," but found that the production of products according to a certain same attribute or grouping can greatly improve the production efficiency, thus breaking through the classical assumption of sorting and scheduling. In group technology, the workpiece is divided into several groups according to the similar production technology. e workpiece in the group is processed continuously, and there is no installation time or cost between the parts in the group. Lee and Wu [30] proposed a model: 1|p r,k ij � p ij r a i k a 2 , G, s r i � s i r a 1 |C max and 1|p r,k ij � p ij r a i k a 2 , G, s r i � s i r a 1 | C j , where p r,k ij represents the processing time of J ij in group G i and r and k represent the position of groups and jobs, respectively, and a 1 ( < 0) and a 2 ( < 0) represent the learning factor. Yang and Chand [31] constructed three basic models: there is no forgetting, partial forgetting, and total forgetting in job processing, and the processing time of jobs is related to the location. e authors in [32][33][34][35] continue to study this problem and obtain many important results.
In this paper, we establish a stochastic grouping scheduling model, and we take the long expected schedule and the expected total completion time as the objective function.

Two Stochastic Grouping Scheduling Problems with Uniform Processing Time and Learning Effect
For the group scheduling and scheduling problem of this model, we assume that all jobs can be processed at zero; the processing time of jobs is a random variable, which obeys uniform distribution and has location-based learning effect; there is no installation time between jobs in the same group, but each group has an installation time before processing; the group installation time obeys the classical learning effect hypothesis. e assumptions and symbols of the model are as follows: m: the number of groups in the sequence, m ≥ 2 G i : group i, i � 1, 2, . . . , m n i : number of jobs in group G i n: number of total jobs n 1 + n 2 + · · · + n i � n J ij : the jth job in group G i , i � 1, 2, . . . , m; j � 1, 2, . . . , n i a: learning rate for group installation, a < 0 s i : normal installation time of group G i , i � 1, 2, . . . , m s r i : actual installation time of group G i in number r, s r i � s i r a a i : learning rates of jobs in group G i , a i < 0 p ij : the random processing time of the job J ij , which obeys the uniform distribution in the interval (0, λ ij ) p k ij : random processing time of job J ij in position k of group G i , p k ij � p ij k a i E(C ij ): expected completion time of job J ij f(C j ): function of processing time of job J j , such as C max , C j Now, we consider the stochastic problem of group scheduling, discuss the optimal group sequence and job sequence under the noninterruptible static priority strategy, and give their algorithms. is problem can be expressed by three parameters: (1) Proof. Firstly, we prove that the optimal algorithm of noninterruptible static priority strategy by the nonsubtraction order of parameters λ ij , that is, the ESPT order, can be obtained. Considering a group, this problem can be attributed to Suppose S and S ′ are two sequences and J is and J it (s < t) are two jobs; denote S � (π 1 , J is , J it , π 2 ) and S ′ � (π 1 , J it , J is , π 2 ). π 1 and π 2 are job sequences except job J is and J it . J is and J it are processing in S at rth and (r + 1)th positions, respectively, but in S ′ , the case is reverse. E(t 0 ) is the expected completion time in addition to the jobs J is and J it in the sequence. Here, we only need to compare E[C it (S)] and E[C is (S ′ )].
By assuming conditions, we obtain So, Secondly, we prove the group sequence rule. Let s i < s j , is satisfied, it can be proved that sequence Q is superior to sequence Q ′ . According to the hypothesis, we can obtain where where E(t 0 ) is the expected completion time of the last job of group G i and G j . It is noted that s i < s j and a < 0, by the difference between (1) and (2), we can obtain So, sequence Q is superior to sequence Q ′ . erefore, by repeating such a swap operation, all groups can be arranged in nonsubtractive order of group installation time. eorem 1 is proved.
By proving eorem 1, we can design a simple algorithm for problem e time complexity of Algorithm 1 is O(n log n). Next, we give an example of eorem 1.
Solution. e solution of Example 1 is as follows: Step 1: in group G 1 , the optimal order of jobs is J 12 ⟶ J 11 ; in group G 2 , the optimal order of jobs is J 21 ⟶ J 23 ⟶ J 22 Step 2: because of s 1 � 3 < s 2 � 4, the groups are sorted as G 1 ⟶ G 2 erefore, the expected minimum maximum completion time of this example is E(C max ) � 11.1086.
Next, we consider another stochastic problem of group scheduling. We have the following conclusion. Proof. First of all, we prove that the optimal algorithm under the noninterruptible static priority strategy can be obtained by arranging the jobs in the group according to the nondecreasing order of the parameters λ ij . When only one group G i is considered, the problem can be transformed into Suppose S and S ′ are two sequences and J is and J it (s < t) are two jobs; denote S � (π 1 , J is , J it , π 2 ) and S ′ � (π 1 , J it , J is , π 2 ). π 1 and π 2 are job sequences, except job J is and J it . e meaning of J is , J it , and E(t 0 ) are the same in eorem 1. In order to prove that sequence S is superior to sequence S ′ , According to the definition, we can obtain By hypothesis a i < 0 and when s < t, there is λ is < λ it ; therefore, Journal of Mathematics 3 at is, In the same way, the jobs in the group are arranged according to the nondecreasing order of parameter λ ij through the operation of pairwise exchange, and the optimal algorithm under the noninterruptible static priority strategy can be obtained.
Secondly, we prove the group sequence rule. Suppose s i < s j , Q � σ 1 , G i , G j , σ 2 , and Q ′ � σ 1 , G j , G i , σ 2 . Let the expected total completion times of jobs in group G i and G j of sequence Q be E[ According to the same assumption in the definition and eorem 1, we can obtain And, we can also obtain E n j l�1 C jl Q ′ ⎡ ⎣ ⎤ ⎦ � n j E t 0 + n j s j r a + E n j k�1,l�1 Notice (s i /n i ) ≤ (s j /n j ) and ( λ jl k a j /n j ); from (5)-(16), we can obtain � n i + n j · s i − s j · r a − (r + 1) a + n j s i − n i s j · (r + 1) a + 1 2 n j n i l,k�1 Step 1: according to the nondecreasing order of parameter λ ij , we arrange the processing of the job in the group Step 2: arrange the groups according to the nondecreasing order of the basic installation time s i of the groups ALGORITHM 1: Heuristic algorithm.

Journal of Mathematics
By repeating the similar exchange procedure, we can make the group satisfy the nondecreasing order of n i l,k�1 λ il k a i /n i , thus completing the proof of group sequence rule. erefore, we have completed the proof of eorem 2. We can design an algorithm for problem: e intragroup G i time complexity in step 1 is O(n i log n i ) and the total time complexity is m i�1 O(n i log n i ); the optimal order time complexity of step 2 is O(m log m).
e time complexity of Algorithm 2 is O(n log n).
Next, we give an example of eorem 2.

Conclusions
For the expected total completion time problem, the optimal solution can be obtained in polynomial time when there is a consistent relationship between the number of jobs in the group and the sum of the expected processing times of the jobs in the group. When the consistency relation is not satisfied, if the number of groups is small and the number of jobs in the group is small, the optimal algorithm can be found by comparison enumeration; however, when the number of groups is large, the polynomial time algorithm cannot be given, and the complexity of the algorithm is unknown.
Considering the randomness of machine learning effect and the randomness of job processing time, we can further discuss the possibility that machine processing time obeys different types of random functions and machine has the random learning index.

Data Availability
All data included in this study are available from the corresponding author upon request.

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