Study on Scheduling Problems with Learning Effects and Past Sequence Delivery Times

: In this paper, we study a single-machine green scheduling problem with learning effects and past-sequence-dependent delivery times. The problem can be properly applied to tackle green manufacturing where production and delivery time are variable and highly subject to process-reengineering. Our goal is to determine the optimal sequence such that total weighted completion time and maximum tardiness are minimized. For the general case, we provide the analysis procedure of lower bound, and also propose the heuristic and branch-and-bound algorithms. Furthermore, computational experiments are conducted to demonstrate the effectiveness of our algorithms.


Introduction
For general scheduling problems, the processing times of the jobs are often assumed to be fixed.However, in actual production, due to the continuous repeated processing of a product by machines or workers, the processing time decreases with the increase in number of products and tends to be stable.This phenomenon is called the scheduling problem with "learning effect" (such as Biskup [1,2]; Cheng et al. [3], Cheng et al. [4]; Lu et al. [5], Lu et al. [6], Azzouz et al. [7], Geng et al. [8], Liang et al. [9], Wang et al. [10]).More recently, Liu and Jiang [11] addressed single-machine resource allocation scheduling problems with learning effects, where learning effects mean job-dependent position-based learning effects.Additionally, Muştu and Eren [12] addressed the optimization of a singlemachine scheduling cases with setup times under an extension of the general learning and forgetting effects, and the objective function is to minimize the maximum completion time.They proposed an integer non-linear programming model and a dynamic programming model, which can be executed in pseudo-polynomial time.While Lin [13] addressed the parallel-machine scheduling with controllable processing times and job-dependent learning effects, the objective function is to minimize the weighted sum of total completion time, total compression cost, and total load.They showed that the problem can be solved in O(n m+2 ) time, where n (resp.m) is the number of jobs (resp.machines).In addition, Jiang et al. [14] considered single-machine scheduling problems with general truncated sum-of-actual processing-time-based learning effect.They showed polynomial solvable cases and approximation algorithms for these problems.Furthermore, Sun et al. [15] considered the flow shop problem of minimising the total weighted completion time in which the processing times of jobs are variable according to general position weighted learning effects.They proposed some heuristics and analysed the worst-case error bounds.While Zhao [16] addressed resource allocation flowshop scheduling with learning effect and slack due window assignment.He combined learning effect and controllable processing times, in which the flowshop has a two-machine no-wait setup, and provided a bi-criteria analysis for the scheduling and resource consumption costs.Subsequently, Zhao [17] studied scheduling problems with general truncated learning effects and past-sequencedependent setup times on a single-machine.They demonstrated that some regular objective function minimizations can be solved in polynomial time.Yan et al. [18] illustrated the single-machine resource allocation scheduling problem with learning effects and group technology.For some special cases of the problem, they can be solved in polynomial time.For the general case, they proposed branch-and-bound algorithms.
Electronic components manufacturing is a high energy consuming industry.In practical application, an electronic component waiting to be processed may be exposed to a certain electromagnetic field and is required to neutralize the effect of electromagnetism.In this case, it needs extra time to eliminate adverse effects.As for green manufacturing environment, proces-reengineering is often introduced to reduce energy consumption and hence leads to the production as well as extra delivery time subject to learning effects.This extra time is modeled as past-sequence-dependent delivery times and is performed immediately after the component has been processed on the machine to provide a delivery to customer.In recent years, the scheduling problems with delivery times have also attracted widespread attention (see Koulamas and Kyparisis [19], Mateo et al. [20], Wang et al. [21], Pan et al. [22]).In addition, Rostami et al. [23] examined the single-machine scheduling problem of minimizing the total weighted completion and batch delivery times, and proposed a brand-and-bound algorithm as well as heuristic algorithm to specifically solve the studied problem,.Their experimental results proved that the proposed heuristic algorithm is more effective.Subsequently, Qian and Zhan [24] illustrated a single-machine scheduling problem with past-sequence-dependent delivery times and the truncated sumof-processing-times-based learning effect.The goal is to minimize the total costs that comprise the number of early jobs, the number of tardy jobs and due date.Under the common due date, slack due date and different due date, they demonstrated that these problems are polynomial time solvable, while Qian and Han [25] studied the due date assignment with the delivery time and deteriorating jobs of scheduling problem, the goal is to minimize the total costs, they proved that these problems are polynomial time solvable and proposed the corresponding algorithms.Moreover, Toksari et al. [26] illustrated both exponential past-sequence-dependent delivery times and learning effect where the job processing time is a function based on the sum of the logarithm of processing times of jobs already processed and showed that the single-machine scheduling problems to minimize makespan, total weighted completion time, total completion time and maximum tardiness have polynomial time solutions.Wang et al. [27] addressed single-machine due-date assignment scheduling with delivery times and truncated learning effect.Ren et al. [28] studied single-machine scheduling with exponential delivery times and learning effects.Recently, Lei et al. [29] illustrated the scheduling problems where each job has a past-sequence-dependent delivery time, and they proved that the makespan and total completion time minimizations can be solved in polynomial time.Under the agreeable due date condition and agreeable weight condition, they showed that the maximum tardiness and total weighted completion time minimizations remain polynomially solvable.Hence, in this paper, we continue the work of Lei et al. [29] and the main contributions are summarized as follows:

•
For the general case of the maximum tardiness and total weighted completion time minimizations, we provide the procedure of analyzing the lower bound of the total weighted completion time and maximum tardiness; • Furthermore, we give the heuristic, simulated annealing, branch-and-bound algorithms and conduct numerical experiments.
The structure of this paper is arranged as follows.The scheduling model is presented in Section 2. In Section 3, we present the branch-and-bound algorithm and heuristic algorithms.Concluding remarks are provided in Section 4.

Problem Description
We formally state the problem as follows: There is a set of independent jobs J to be processed on a single machine.The machine can handle at most one job at a time and job preemption is not allowed.All the jobs are available for processing at time 0. The actual processing time of job j scheduled at position r is denoted by p j[r] , j = 1, 2, ..., n, and p j[r] is given as follows: where p j is the normal processing time of job j, and a(a < 0) is learning effect.The delivery time after the processing of job p j[r] is denoted by q j[r] , j = 1, 2, ..., n.As in Lei et al. [29], the q j[r] is given as follows: where 0 ≤ γ ≤ 1 is a normalizing constant, and b(b > 0) is delivery times index.We study each job has a past-sequence-dependent delivery time to reduce the inventory costs for enterprises in this paper.Our goal is to determine the optimal schedule such that the maximum tardiness (i.e., T max = max{T j |j = 1, 2, . . ., n}, where T j = max{0, C j − d j }, C j is the completion time of job j and d j is the due date of job j) and the total weighted completion time (i.e., ∑ n j=1 w j C j , where w j is the weight of job j) are to be minimized.By the three-field notation of Graham et al. [30], the problem can be denoted as where the first field 1 denotes a single-machine, the second field LE is learning effects (1) and Q psd denotes delivery times (2), the third field Z ∈ {∑ n j=1 w j C j , T max } refers to the optimized criterion.

Solution Algorithms
To analyze the lower bound of ∑ n j=1 w j C j (T max ), we give the following lemmas as the preliminaries.
Lemma 1 (The SPT rule, Lei et al. [29]).For problem 1|LE, Q psd | ∑ n j=1 C j (C max ), an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of p j .
Lemma 2 (The WSPT rule, Lei et al. [29]).For problem 1|LE, Q psd | ∑ n j=1 w j C j , if p i ≤ p j implies w i ≥ w j for all jobs J i and J j , the optimal schedule can be obtained by sequencing the jobs in non-decreasing order of p j /w j .
Lemma 3 (The EDD rule, Lei et al. [29]).For problem 1|LE, Q psd |T max , if p i ≤ p j implies d i ≤ d j for all jobs J i and J i , the optimal schedule can be obtained by sequencing the jobs in non-decreasing order of d j .
To solve the problem 1|LE, Q psd |Z, a branch-and-bound (B&B) algorithm and heuristic algorithms are presented as follows.

Lower Bounds
3.1.1.For the Total Weighted Completion Time (i.e., ∑ n j=1 w j C j ) We assume that π = [ π PS , π US ] be a sequence, where π PS ( π US ) is the scheduled (unscheduled) part, and there are r jobs already scheduled.So, for the r + 1th (r + jth) job, we have We assume a is a fixed constant, and b can be minimized by Lemma 1.Hence, the first lower bound of 1|LE, Q psd | ∑ n j=1 w j C j is: where p <r+1> ≤ p <r+2> ≤ . . .≤ p <n> .Similarly, if p i ≤ p j implies w i ≥ w j for all the jobs J i and J j , b is minimized by Lemma 2, we have where w (r+1) ≥ w (r+2) ≥ . . .≥ w (n) and p <r+1> ≤ p <r+2> ≤ . . .≤ p <n> (note that w (j) and p <j> do not necessarily correspond to the same job)).
To make the lower bound tighter, combining LB i (T max ) and LB ii (T max ), the lower bound of 1|LE, Q psd |T max is

Algorithms
The NEH algorithm is a high-performing heuristic for solving the flow shop problem, hence, the NEH heuristic (Nawaz et al. [31]) can be adopted for the the general problems 1|LE, Q psd | ∑ n j=1 w j C j and 1|LE, Q psd |T max , where NEH with an initial solution sorted by the WSPT/EDD rule (WSPT for Min ∑ n j=1 w j C j , and EDD for Min T max ).Therefore, the pseudo code of the NEH (WSPT) algorithm is summarized as follows (i.e., Algorithm 1).Simulated annealing (SA) is a global optimization algorithm whose basic idea is to avoid falling into a local optimum solution by accepting bad solutions with a certain probability.Therefore, the SA is also a solution to solve 1|LE, Q psd | ∑ n j=1 w j C j and 1|LE, Q psd |T max .The pseudo code of the SA is summarized as follows (i.e., Algorithm 2).Branch-and-bound (B&B) algorithm is a type of algorithm for solving optimization problems.It improves the efficiency of the algorithm by constantly breaking the problem down into smaller subproblems and setting bounds for each subproblem to prune search paths that are unlikely to be optimal solutions.The pseudo code of the B&B algorithm is summarized as follows (i.e., Algorithm 3).

Algorithm 1: Heuristic
Input: p j , w j (d j ) Output: the best sequence and minimizes ∑ n j=1 w j C j (T max ) Begin sort n jobs are arranged according to the WSPT(EDD) rule define a provisional sequence with the first two jobs in the sorted sequence for j = 3 to n do for k = 1 to j do insert job j in the provisional sequence at position k compute the sequence in order to find the ∑ n j=1 w j C j (T max ) end select the position that makes the ∑ n j=1 w j C j (T max ) minimum insert job j in the provisional sequence at the selected position end return Algorithm 2: Simulated Annealing Input: π 0 , p j , w j (d j ), g, r, T 0 , T f , S 0 Output: ∑ n j=1 w j C j (T max ) Begin S ← S 0 , S best ← S // Initialize the current solution and the optimal solution T ← T 0 , L ← 1000 // Initialize temperature and number of cycles

Computational Experiments
In order to assess the efficiency of the proposed heuristic and B&B algorithms, we write code in C++ and run on a PC with 4 GB of RAM on a Windows 10 OS.Processing time p j is generated from a uniform discrete distribution with an interval [5,20]; the weight w j is uniformly distributed over [1,10]; the due dates d j is generated from a uniform discrete distribution with an interval [1, 1  2 C max ] (where C max can be obtained by the SPT rule); The index of delivery times b = 1.1, 1.3, 1.5 and the learning index a = −0.35,−0.3, −0.25.For each parameters combination (n, a, b), 20 randomly generated instances were evaluated.

Small-Sized Instances
For small-scale examples (n = 8, 9, 10, 11, 12), it is defined that "CPU time (s)" as the running time of the WSPT, EDD, NEH, SA, B&B algorithm, time unit is second, and the error of the solution produced by heuristics was calculated as T max (G * ) ) for ∑ n j=1 w j C j (T max ), respectively, where G is a schedule obtained by the WSPT(EDD), NEH, SA and the optimal schedule G * is obtained by the B&B algorithm.The corresponding results were summarized in Tables 1-4.
For ∑ n j=1 w j C j , from Table 1, we can see that the CPU time of WSPT changes very little as n increases, but B&B increases rapidly, i.e., grows exponentially.Table 2 shows that the error of the SA algorithm is somewhat smaller than that of the WSPT and NEH algorithms, indicating that SA appears to be more accurate than the WSPT and NEH (WSPT).In conclusion, in terms of both time and error dimensions, algorithm B&B is more accurate although takes longer to execute.
For T max , again the CPU time of B&B grows exponentially as n increases.and it can be seen from Table 4 that the algorithm NEH appears to be more accurate than SA.
T max (Best) , where G was a sequence obtained by the WSPT, EDD, NEH [SPT], NEH [WSPT], NEH [EDD] and SA, and Best denotes the best solution found by any of the heuristic procedures in each of the problem instances.The corresponding results were summarized in Tables 5-8.
For ∑ n j=1 w j C j , it can be seen from Table 5 that the CPU times of NEH (SPT) and NEH (WSPT) are longer than that of SA.When n = 100, the CPU times of NEH (SPT), NEH (WSPT) and SA grow faster, and when n = 200, their CPU times have reached more than 800 s, while SA is only about 270 s, indicating that the algorithm SA is slightly more accurate than NEH in terms of CPU time when calculating large-scale.From Table 6, we can see that the error of WSPT, NEH (SPT) is relatively large and unsuitable for computing this problem.the error of NEH (WSPT) and SA is relatively smaller, but there are still differences between them, and obviously the error of NEH (WSPT) is a little smaller, and at this time, the NEH (WSPT) algorithm is better.
For T max , as n increases, the CPU time of the NEH algorithm is somewhat longer than that of SA, but the difference is not very large relative to ∑ n j=1 w j C j .And it is obvious from Table 8 that the error of NEH (EDD) is significantly smaller than the other algorithms.In short, from the two dimensions of time and error, NEH (EDD) is superior compared to NEH (SPT) and SA.

Conclusions
This article studied scheduling problems with general learning effects and where each job has a past-sequence-dependent delivery time on a single machine.For the total weighted completion time and maximum tardiness minimizations, we proposed the heuristic, simulated annealing, branch-and-bound algorithms and conducted numerical experiments.Future research might focus on analyzing the above problems with m-machine flow shops the node represents a complete solution then update the best solution found so far prune the node's descendants end else select and remove the node with the smallest lower bound from Q, calculate the initial UB by WSPT for ∑ n j=1 w j C j and EDD for T max , calculate the LB (see Equation (8) for ∑ n j=1 w j C j , and Equation (12) for T max ) for the node if LB ≥ UB then eliminate the node and all the nodes following it in the branch

Table 1 .
CPU time results of the algorithms for ∑ n j=1 w j C j .

Table 2 .
Error results of the algorithms for ∑ n j=1 w j C j .

Table 3 .
CPU time results of the algorithms for T max .

Table 4 .
Error results of the algorithms for T max .= 100, 125, 150, 175, 200), it is defined that "CPU time" (s) as the running time of the WSPT (EDD), NEH, SA, B&B algorithm, time unit is second, and the performance of the heuristics WSPT, EDD, NEH [SPT], NEH [WSPT], NEH [EDD] and SA was verified by the ratios

Table 5 .
CPU time of the algorithms for ∑ n j=1 w j C j .

Table 6 .
Error results of the algorithms for ∑ n j=1 w j C j .

Table 7 .
CPU time of the algorithms for T max .

Table 8 .
Error results of the algorithms for T max .