Group Technology Scheduling with Due-Date Assignment and Controllable Processing Times

: This paper investigates common (slack) due-date assignment single-machine scheduling with controllable processing times within a group technology environment. Under linear and convex resource allocation functions, the cost function minimizes scheduling (including the weighted sum of earliness, tardiness, and due-date assignment, where the weights are position-dependent) and resource-allocation costs. Given some optimal properties of the problem, if the size of jobs in each group is identical, the optimal group sequence can be obtained via an assignment problem. We then illustrate that the problem is polynomially solvable in O ( ℘ 3 ) time, where ℘ is the number of jobs.


Introduction
Classical scheduling problems consider fixed job processing times. However, scheduling problems with controllable processing times ( CPT, i.e., resource allocation) have received extensive attention (see Lu et al. [1], Liu et al. [2]). In 2018, Li and Wang [3] studied single-machine scheduling with deteriorating jobs and CPT. For the general linear deterioration function, they proved that the weighted sum minimization of the makespan and total resource consumption costs can be solved in polynomial time. Lu and Liu [4] delved into single-machine scheduling with CPT and position-dependent workloads. For scheduling and total resource consumption costs, they performed bicriterion analysis for the problem. In 2019, Geng et al. [5], and Sun et al. [6] investigated two-machine flow-shop problems with learning effects and CPT. Under common due-date assignment and no-wait constraints, Geng et al. [5] proved that irregular objective minimization is solved in polynomial time. Under slack due-date assignment and no-wait constraints, Sun et al. [6] proved that irregular objective minimization is solved in polynomial time. In 2020, Liu and Jiang [7] studied scheduling with learning effects and CPT on a two-machine no-wait flow-shop setting. Under common and slack due date assignments, they provided bicriterion analysis for scheduling and resource-consumption costs. In 2021, Lu et al. [8] considered a singlemachine due-date assignment problem with CPT and learning effects. Zhao [9], and Lv and Wang [10] revisited no-wait flow-shop problems with learning effects and CPT. Under a slack (different) due-window assignment, Zhao [9] (Lv and Wang [10]) performed bicriterion analysis of scheduling (including earliness-tardiness penalties, due-window starting times, and the due-window size of all jobs) and resource-consumption costs. Zhao [9], and Lv and Wang [10] proved that several scheduling and resource-consumption costs can be solved in polynomial time. In 2022, Tian [11] addressed single-machine due-window assignment scheduling with CPT. Under linear and convex resource allocation functions, the objective is to minimize generalized earliness and tardiness penalties. For common and slack due-window assignments, they demonstrated that the problem could be solved in polynomial time. In 2023, Wang et al. [12] explored single-machine scheduling with CPT. Under linear and convex resource allocation functions, the objective was to minimize the weighted sum of general earliness-tardiness and resource-consumption costs where weights are position-dependent. They demonstrated that the problem was polynomially solvable.
In addition, the study of group technology (GT) is very important (see Liu [13]). In 2018, Wang et al. [14] considered single-machine scheduling with shortened job processing times. Under GT and ready times, they proved that some special cases of the makespan minimization could be solved in polynomial time. In 2019, Huang [15] scrutinized the scheduling with deteriorating jobs and GT, and proved that bicriterion single-machine minimization is polynomially solvable, where primary (secondary) criterion is the total weighted completion time (maximal cost). Liu et al. [16] focused on single-machine scheduling GT and deterioration effects. For makespan minimization with ready times, they proposed heuristic and branch-and-bound algorithms, and tested them via randomly instances. In 2021, Wang et al. [17] examined single-machine scheduling with GT and due-date assignment. For common, slack, and different due-date assignments, they proved that irregular objective minimization could be solved in O(n log n) time, where n is the number of jobs. In 2022, Wang et al. [18] investigated single-machine scheduling with GT and a shortened proportional linear processing time. For the general problem of makespan minimization, they proposed a heuristic algorithm and a branch-and-bound algorithm to solve the problem. In 2023, Chen et al. [19] scrutinized the single-machine problem with GT and a controllable learning effect. In due-window assignments, the objective is to minimize the total cost comprising due-window related penalties and investment costs. They proved that the problem could be solved in polynomial time.
To our knowledge, scheduling with GT and CPT are concurrently widely reflected in real production (see Shabtay et al. [20], Zhu et al. [21], Wang et al. [22]). Wang and Liang [23], and Liang et al. [24] explored single-machine scheduling with GT, convex CPT, and deterioration effects. In 2023, Yan et al. [25] studied single-machine scheduling with GT and CPT. Under learning effects and limited resource availability, the goal is to minimize the total completion time. These authors proved that some special cases of the problem could be solved in polynomial time. For a general case of the problem, they also proposed heuristic and branch-and-bound algorithms. Chen et al. [26] worked on single-machine scheduling with GT and CPT. In different due-date assignments and for a special case, they proved that the problem could be solved in polynomial time. In view of the importance of GT and CPT, in this article, we continue the work of the concurrent single-machine scheduling with GT and CPT for a common due-date assignment (Ξ CON ; for details, see Gordon et al. [27]) and slack due-date assignment (Ξ SLK ; see Gordon et al. [28], Liu et al. [29]). Our objective is to minimize the sum of scheduling (including the weighted sum of earliness, tardiness, and due-date assignments where weights are position-dependent ( PDW; see Wang et al. [30], and Wang et al. [31])) and resource-allocation costs. This paper's contributions are as follows: • We scrutinize the single-machine due-date assignment problem with the group technology and controllable processing times. • Under Ξ CON , and Ξ SLK , the goal is to minimize the sum of scheduling (including the weighted sum of earliness, tardiness, and due-date assignment, where weights are PDW) and resource-allocation costs. • The optimal properties of a special case are presented, and we prove that the problem could be solved in polynomial time.
The rest of this article is organized as follows: In Section 2, we present the model. In Sections 3 and 4, we analyze the linear and convex resource functions, respectively. In Section 5, a numerical example is presented. In Section 6, we conclude the paper.

Problem Formulation
A set of ℘ jobs to be processed on a single-machine are divided into ℵ groups G 1 , G 2 , . . . , G ℵ , where a number of jobs belong to group G i is ℘ i and ℘ 1 + ℘ 2 + . . . + ℘ ℵ = ℘. All jobs and the machine are available at Time 0. Machine setup time s i is incurred before the jobs are processed in G i . There is not setup time between jobs in the same group, and jobs within each group must be processed consecutively. Let J ih be the hth job in G i , i = 1, . . . , ℵ; h = 1, . . . , ℘ i . For a linear resource function, the actual processing time of J ih is where a ih and b ih are the normal processing time and positive compression rate of job J ih , respectively (the normal processing time means that the processing time without any resource allocation), u ih is the amount of a nonrenewable resource allocated to J ih , andū ih denotes the maximal amount of the resource allocated to J ih . For a convex resource function, where θ ih is a workload of J ih ( > 0 is a given constant). Let C ih = and d ih be the completion time and due date, respectively, of J ih in G i . For the Ξ CON assignment, we assumed that d ih = d i ; for the Ξ SLK assignment, we assumed that d ih = p ih + q i , where q i denotes common flow allowance for G i . Let E ih = max{0, d ih − C ih } and T ih = max{0, C ih − d ih } denote the earliness and tardiness, respectively, of job J ih . Let [r] be a scheduled job (group) in the rth position, J i[r] a scheduled job in the rth position in G i . Our goal was to find group sequence and internal job sequence ϕ i within G i , the set of due-datesd = {d 1 , d 2 , . . . , d m } (flow allowancesq = {q 1 , q 2 , . . . , q m }) and the resource allocationũ = {u ih |i = 1, . . . , ℵ; h = 1, 2, . . . , ℘ i } such that cost function is minimized, where α ih and β ih are position-dependent weights for earliness and tardiness costs, i.e., α ih and β ih are not related to job J ih , but to position h in group G i , γ ≥ 0 is a given constant, and g ih is the cost of one unit of the allocated resource to job J ih . With three field notations, this problem is denoted as follows: and where 1 stands for the single-machine, field {p ih , Ξ CON /Ξ SLK , GT, PDW} denotes the characteristics of jobs and groups, and is the cost function. The notations and symbols used in this article are listed in Table 1.

Symbol Definition
℘ (resp. ℵ) number of jobs (resp. groups) Normal processing time and compression rate, respectively, of J ih θ ih Workload of J ih u ih andū ih Amount and maximal amount, respectively, of the assigned resource to J ih p ih Actual processing time of J ih s i setup time of G i C ih and d ih Completion time and due date, respectively, of J ih E ih and T ih Earliness and tardiness, respectively, of J ih d i and q i Common due date and flow allowance, respectively, of G i α ih and β ih ) Position-dependent weights of earliness and tardiness, respectively, in the hth position in G i g ih Unit resource cost for J ih

Linear Resource Function
There exists an optimal sequence that does not include idle machine times. Let S i be the starting time of G i , for a given job sequence Lemma 1. For a given job sequence ϕ i and resource allocation within group G i , under Ξ CON and Ξ SLK assignments, if the values of d i and q i , respectively, are within the starting and ending times of G i , there exists an optimal value at which d i and q i are equal to the completion time of some job (i = 1, . . . , ℵ).
Proof. For the Ξ CON assignment, it was assumed that If Then and For the Ξ SLK method, this result can be similarly obtained.

Lemma 2. For a given job sequence
Proof. For the Ξ CON assignment, from Lemma 1, it was assumed that d i = C i[k i ] ; we then have With the technique of small perturbations, if Hence, For the Ξ SLK assignment, the result can similarly be obtained. For the Ξ CON assignment, from Equation (6), Lemma 2,and d

Remark 1. If k i does not satisfy inequality
where For the Ξ SLK assignment, from Equation (6), Lemma 2, and where Since k i values are independent of and ϕ i (i = 1, . . . , ℵ), from Lemmas 1 and 2, if p ih = a ih − b ih u ih , the cost objective can be expressed: where for the Ξ CON assignment, For the Ξ SLK assignment, Lemma 3. In the given group sequence and job sequence within each group, optimal resource allocationũ * ( , ϕ 1 , . . . , ϕ ℵ ) is Proof. Let the derivative of Equation (21) with respect to u [i][j] be equal to 0, and the result can be obtained.
For a given group order , from Equation (21), is independent of the internal job sequence within each group. Now, we prove that the optimal sequence within each group can be obtained with the following lemma.

Proof. For a given G
jh be a binary variable, i.e., if J As in Wang et al. [22], optimal job sequence ϕ * i ( ) can be obtained with the assignment problem ( AP): s.t.
The above AP is solvable in O(n 3 [i] ) time; hence, determining the total complexity of from Lemmas 1-4, the complexity of determining the optimal group sequence is still an open problem, so we discuss a special case, i.e., ℘ i = N, i = 1, . . . , ℵ.

Lemma 5.
For Proof. From Equation (21), cost function (3) is determined by both the group and job sequences. Optimal job sequence ϕ * i can be obtained with Lemma 4, and the cost function with ϕ * i is just dependent on the ith group position in . In . Let y ir be a binary variable. If group G i is assigned to rth position, y ir = 1; otherwise, y ir = 0, i, r = 1, · · · , ℵ. Let As in Shabtay et al. [20], optimal group sequence * was obtained with the following AP: The above AP is solvable in Via Lemmas 1-5 and the above analysis, for ℘ i = N (i = 1, . . . , ℵ), the following algorithm (i.e., Algorithm 1) is presented to solve

Algorithm 1: Linear resource function
Step 1. Calculate k i by Lemma 2.
Step 2. For each possible position of each group in , calculate ϑ [i] jh with Equation (25) for j, h = 1, . . . , N, where for the Ξ CON assignment, ξ [i]h is given by Equation (22), and for the Ξ SLK assignment, ξ [i]h is given by Equation (23).
Step 3. Solve AP (26)- (29) to find internal job sequence ϕ * ir within G i if this group is assigned to the rth position in .
Step 7. For the Ξ CON and Ξ SLK assignments, calculate d

Convex Resource Function
Similar to Section 3, for problem we have where for the Ξ CON assignment, ξ [i]h is given by Equation (22), and for the Ξ SLK assignment, ξ [i]h is given by Equation (23), i, h = 1, . . . , ℘ i .

Proof. From Equation (35), H is a convex function of u [i][h]
; hence, let and the result of Equation (36) can be obtained.
By ubstituting Equation (36) into Equation (35), it follows that For the Ξ CON assignment, let For the Ξ SLK assignment, let Lemma 7. Given group order , optimal job sequence ϕ * i (i = 1, . . . , ℵ) within G i can be obtained by matching the smallest and second smallest ξ ih to the job with the largest and second largest g ih θ ih , respectively, and so on. r] are independent of the internal job sequence within each group. According to Hardy et al. [32], the optimal job sequence for G i is obtained by matching the smallest and second smallest ξ ih to the job with the largest and second largest g ih θ ih , respectively, and so on.
the complexity of finding the optimal group sequence is still an open problem, but for the special case ℘ i = N, i = 1, . . . , ℵ, the optimal * is obtained in O(℘ 3 ) time.

Algorithm 2: Convex resource function
Step 1. Calculate k i by Lemma 2.
Step 6. For the Ξ CON and Ξ SLK assignments, calculate d * i = C i[k i ] and q * i = C i[k i −1] , respectively, using Lemma 2.

Conclusions
In this article, we investigated single-machine group technology scheduling with CPT. Under Ξ CON /Ξ SLK assignments, the goal is to find the job and group sequences, resource allocation, and due-date assignment, such that the sum of scheduling and resourceallocation costs is minimized. For ℘ i = N (i = 1, . . . , ℵ), we demonstrated that this problem is polynomially solvable. Future work could explore job (or flow) shop problems (see Guo et al. [33], and Karacan et al. [34]) with group technology and controllable processing times to study the general version of 1 p ih , Ξ CON /Ξ SLK , GT, PDW where p ih ∈ a ih − b ih u ih , θ ih u ih (e.g., the proposed cuckoo search algorithm of Xie et al. [35]).
Future work could also consider problems regarding maintenance activity (see Wu et al. [36]).
Author Contributions: Conceptualization, W.L. and X.W.; methodology, W.L. and X.W.; writing, review and editing, W.L. and X.W. All authors have read and agreed to the published version of the manuscript.
Funding: This research was supported by the National Natural Science Regional Foundation of China (72061029 and 71861031).