Study on Controllable Processing Time and Minmax Group Scheduling with Common Due-Window Assignment

Abstract

We considerthe single-machine group scheduling problem with controllable processing times (i.e., resource allocation) under a common due-window ($condw$) assignment. The objective is to minimize a total cost composed of earliness, tardiness, due-window-related penalties, and resource consumption. Motivated by realistic production settings such as aerospace component machining and electronics batch assembly, the study addresses the joint optimization of group sequence, job sequence, due-window placement, and resource allocation. For linear and convex resource models, we propose a branch-and-bound ($\widehat{BaB}$) algorithm and efficient heuristics. Numerical experiments show that the $\widehat{BaB}$ algorithm can solve instances with up to 250 jobs and 16 groups. The heuristics ($\widehat{UB}$), including a simulated annealing ($\widehat{SA}$) algorithm, obtain near-optimal solutions with an average error below 0.05% much faster, demonstrating their practical usefulness for real-time scheduling.

1. Introduction

Single-machine group scheduling is an important class of production scheduling problems in manufacturing systems, where jobs are partitioned into groups according to their similarities, and setup times are required between different groups. In practical production, the processing times of jobs can often be compressed by allocating additional resources, which leads to the so-called controllable processing time scheduling problem. Meanwhile, to balance customer satisfaction and inventory costs, due-window assignment strategies are widely adopted, among which the common due-window requires all jobs belonging to the same group to share a unified due-window ($condw$). This problem finds broad applications in scenarios such as aerospace component machining, electronics assembly, and batch logistics distribution, and thus possesses significant theoretical value and practical relevance.

With the increasing demands for resource optimization and scheduling in the manufacturing and service industries (see Nowicki and Zdrzalka [1], Shabtay and Steiner [2], Pinedo [3], Oron [4], Lu and Liu [5], Li et al. [6], Liu and Jiang [7], Li et al. [8], Sun et al. [9], Mao et al. [10], Lv and Wang [11], Bosman et al. [12], Zhu et al. [13], Lv and Wang [14], Cohen and Shapira [15], and Sun et al. [16]), controllable processing time (denoted by $cpt$, resource allocations) scheduling problems have been the focus of many scholars. Jiang et al. [17] studied seru production scheduling with the $cpt$. For the total processing cost plus scheduling measures, they showed that some problems are polynomially solvable. Gao et al. [18] considered no-wait job shop scheduling with the $cpt$. Under subcontracting and due date constraints of minimizing the makespan and subcontracting cost, they proposed two mathematical models. Pang and Meng [19] studied robust project scheduling with the $cpt$. Geng and Yuan [20] considered single-machine scheduling with the $cpt$. Under multiple projects, they proved that some problems have NP-hardness. Weng et al. [21] considered scheduling with the multi-resource operations. For the makespan minimization, they proposed a mixed-integer program model. Wang et al. [22] studied single-machine scheduling with the $cpt$ and deterioration effects. For a convex resource model, they presented a bicriteria analysis for the total weighted completion time and total resource consumption cost. To solve the problem, they proposed some heuristics and a branch-and-bound algorithm. Zhang et al. [23] investigated single-machine scheduling with the $cpt$ and time-dependent learning effects.

Furthermore, group technology (denoted by $gt$) scheduling problems have gradually become a focal point in the fields of operations research and production management (see Liu et al. [24], Huang [25], Bajwa et al. [26], Liao et al. [27]). Recently, Chen et al. [28] considered $gt$ scheduling with learning effect and due-window assignment. For the total cost including the due-window penalty and investment, they showed that the problem is polynomially solvable. Yin et al. [29] and Yin and Gao [30] investigated the single-machine $gt$ scheduling in which the setup times of groups have linear deterioration effects and processing times of jobs have learning effects. Yin and Gao [30] showed that the makespan minimization is polynomially solvable. For the total weighted completion time minimization, Yin et al. [29] proposed some solution algorithms. Li and Goossens [31] studied the multi-league $gt$ scheduling. For two objectives, i.e., the travel distance and capacity violations of venues shared by teams, they proposed a mixed-integer programming model and a two-layer heuristic. Zhao [32] considered the single-machine $gt$ scheduling with learning effects and $cpt$. Under a convex resource model of considering the makespan and total resource consumption, they proved that three versions are polynomially solvable. Yuraszeck et al. [33] considered $gt$ shop and mixed shop problems of minimizing the makespan. Huang et al. [34] studied the single-machine $gt$ scheduling with learning effects and $cpt$. For the convex resource model of minimizing the total resource consumption under the makespan when bounded, they proposed a branch-and-bound and a heuristic.

For many modern industrial processes, logistics and supply chain management, the due-window assignments are very important (see Janiak et al. [35], Geng et al. [36], Wang and Liu [37], Yue and Zhou [38], Tian [39]). In general, there are three types of due-window assignment, i.e., the common due-window (denoted by $condw$, see Wang et al. [40], Hoffmann et al. [41], Qian et al. [42], Bai et al. [43]), the slack due-window (denoted by $slkdw$, see Bai et al. [43], Mor and Mosheiov [44], Mor [45], Sun et al. [46], Qiu and Wang [47]), and the different due-window (denoted by $difdw$, see Lu et al. [48], Sun et al. [49], Wang et al. [50]). Wang et al. [51] studied the $condw$ and $slkdw$ scheduling with the $cpt$. For the convex resource model of minimizing the general earliness–tardiness cost subject to total resource consumption if bounded, they proposed a branch-and-bound and some heuristics. Sun et al. [52] considered the $condw$ and $slkdw$ no-wait flow shop scheduling with the $cpt$. The objective is to minimize the sum of general earliness–tardiness cost and total resource consumption, and they proved that the problem is polynomially solvable under some optimal solution properties.

Recently, Liu and Wang [53] and Chen et al. [54] considered $gt$ scheduling with due-date assignment and $cpt$. Under linear and convex resource models, they proved that some special cases of the problem are polynomially solvable. Wang and Liu [55] and Zhang and Yin [56] studied the single-machine $gt$ scheduling with $cpt$ under different due-date assignments aimed at minimizing earliness–tardiness cost, and proposed several solution algorithms. Lv and Wang [57] examined $gt$ scheduling with $cpt$ and $slkdw$, where the objective is to minimize the maximal value of earliness, tardiness, and due-window assignment cost. Under constant processing time, linear, and convex resource models, they presented relevant results. Zhang and Wang [58] investigated single-machine $gt$ scheduling with $cpt$ and $difdw$ under a minmax-type scheduling cost, also considering constant, linear, and convex resource models. Ren and Yang [59] studied the single-machine $gt$ scheduling with $cpt$ and $condw$, with the objective of minimizing the sum of maximal earliness–tardiness cost and total resource allocation cost. However, they focused on a special case (with group-position-dependent penalty, $gpp$) and proved it to be polynomially solvable.

In this paper, we extend the work of Ren and Yang [59] by studying the general case without the group-position-dependent penalty ($gpp$) assumption, which is proven to be NP-hard (as indicated in

Table 1. Problems studied.

Problem	Time Complexity	Reference
$1\left|gt,slkdw\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{q}_1^a+{\kappa }_{a4}{D}_a\right\}$	$O(max(n,mlogm\left)\right)$	Lv and Wang [57]
$1\left|gt,slkdw,linear,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{q}_1^{\left[a\right]}+{\kappa }_{a4}^{\prime }{D}_{\left[a\right]}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Lv and Wang [57]
$1\left|gt,slkdw,convex,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{q}_1^{\left[a\right]}+{\kappa }_{a4}^{\prime }{D}_{\left[a\right]}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Lv and Wang [57]
$1\left|gt,slkdw,linear\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{q}_1^a+{\kappa }_{a4}{D}_a\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	NP-hard	Lv and Wang [57]
$1\left|gt,slkdw,convex\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{q}_1^a+{\kappa }_{a4}{D}_a\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	NP-hard	Lv and Wang [57]
$1\left|gt,difdw\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^{a,b}+{\kappa }_{a4}{D}_{a,b}\right\}$	$O(max(n,mlogm\left)\right)$	Zhang and Wang [58]
$1\left|gt,difdw,linear,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{d}_1^{a,b}+{\kappa }_{a4}^{\prime }{D}_{a,b}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Zhang and Wang [58]
$1\left|gt,difdw,convex,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{d}_1^{a,b}+{\kappa }_{a4}^{\prime }{D}_{a,b}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Zhang and Wang [58]
$1\left|gt,difdw,linear\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^{a,b}+{\kappa }_{a4}{D}_{a,b}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	NP-hard	Zhang and Wang [58]
$1\left|gt,difdw,convex\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^{a,b}+{\kappa }_{a4}{D}_{a,b}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	NP-hard	Zhang and Wang [58]
$1\left|gt,condw\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}$	$O(max(n,mlogm\left)\right)$	Ren and Yang [59]
$1\left|gt,condw,linear,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{d}_1^{\left[a\right]}+{\kappa }_{a4}^{\prime }{D}_{\left[a\right]}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Ren and Yang [59]
$1\left|gt,condw,convex,gpp\right|{\sum }_{a=1}^m{max}_{1\le b\le n_{\left[a\right]}}\left\{max\left\{{\kappa }_{a1}^{\prime }{E}_{\left[a\right],b},{\kappa }_{a2}^{\prime }{T}_{\left[a\right],b}\right\}+{\kappa }_{a3}^{\prime }{d}_1^{\left[a\right]}+{\kappa }_{a4}^{\prime }{D}_{\left[a\right]}\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	$O(max(mn,m^3))$	Ren and Yang [59]
$1\left|gt,condw,linear\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$	NP-hard	This article
$1\left|gt,condw,convex\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b},$	NP-hard	This article

$gpp$ denotes group-position-dependent penalty, ${\kappa }_{a1}^{\prime }$ (${\kappa }_{a2}^{\prime }$, ${\kappa }_{a3}^{\prime }$, ${\kappa }_{a4}^{\prime }$) denotes the coefficient of $gpp$ for $G_{\left[a\right]}$; for $slkdw$, $d_1^{a,b}=P_{a,b}^A+q_1^a$, $d_2^{a,b}=P_{a,b}^A+q_2^a$, $D_a=q_2^a-q_1^a$, $q_1^a$ and $q_2^a$ are decision variables; for $difdw$, $d_1^{a,b}$ and $d_2^{a,b}$ are decision variables.

). Our contribution is threefold. Theoretically, we address this more general and realistic problem setting, thereby filling a gap in the analysis of unrestricted group scheduling. Methodologically, we develop new lower bounds ($L{B}_{\mathrm{linear}}$, $L{B}_{\mathrm{convex}}$) and heuristic upper bounds, and propose a dedicated branch-and-bound algorithm along with other heuristic methods, providing a systematic solution framework for this challenging problem. Practically, the model better reflects real-world production environments such as aerospace component machining and electronics batch assembly, and the proposed algorithms can support effective decision-making for real-time scheduling and resource optimization. The remainder of the paper is structured as follows: Section 2 formulates the problem, Section 3 analyzes the linear and convex resource models, Section 4 reports the numerical experiments, and Section 5 concludes.

2. Problem Description

There are n jobs to be processed on a single machine, and the jobs are divided into m groups (i.e., groups $G_1,G_2,\dots ,G_m$) based on their similarity characteristics. Let n ($n\ge 1$) be the number of jobs in group $G_a\left(a=1,2,\dots ,m\right)$, we have ${\sum }_{a=1}^m{n}_a=n$. During the machining process, there is no idle time for the machine. Furthermore, the machine can only process one job at a time, and it continues to do so until the job is completed. Before group $G_a$ is processed, there is a setup (i.e., preparation) time $S_a$.

Let $O_{a,b}\left(b=1,2,\dots ,n_a\right)$ be the bth job in group $G_a$, and under linear resource model, the actual processing time of $O_{a,b}$ is

$$P_{a,b}^A=P_{a,b}-{ϵ}_{a,b}{u}_{a,b},0\le u_{a,b}\le {\overline{u}}_{a,b}\le {\displaystyle \frac{P_{a,b}}{{ϵ}_{a,b}}}$$

(1)

where $P_{a,b}$ indicates the normal processing time of $O_{a,b}$, ${ϵ}_{a,b}({ϵ}_{a,b}\ge 0)$ is its compression factor, $u_{a,b}$ is its resource allocation amount, and ${\overline{u}}_{a,b}$ is its upper bound of $u_{a,b}$. Under the convex resource model:

$$P_{a,b}^A={\left({\displaystyle \frac{w_{a,b}}{u_{a,b}}}\right)}^{\varrho }$$

(2)

where $w_{a,b}$ is job workload of $O_{a,b}$, and $\varrho >0$ is a constant.

Additionally, let $\left[d_1^a,d_2^a\right]$ be the due-window of $O_{a,b}$, where $d_1^{a,b}$ (resp. $d_2^{a,b}$) is the starting time (resp. completing) time of the due-window, then the due-window size of $O_{a,b}$ is $D_{a,b}=d_2^{a,b}-d_1^{a,b}$. Under common due-window ($condw$) assignment, all jobs belonging to each group $G_a$$\left(a=1,2,\dots ,m\right)$ are assigned a unified due-window $\left[d_1^a,d_2^a\right]$$\left(d_1^a\le d_2^a\right)$ (i.e., $d_1^{a,b}=d_1^a$ $d_2^{a,b}=d_2^a$), where $d_1^a$ and $d_2^a$ are the starting time and completing time of the due-window, $D_a=d_2^a-d_1^a$ denotes size of the due-window, and $d_1^a$ and $d_2^a$ are decision variables. Let $E_{a,b}=max\{d_1^a-C_{a,b},0\}$ (resp. $T_{a,b}=max\left\{0,C_{a,b}-d_2^a\right\}$) be the earliness (resp. tardiness) cost of job $O_{a,b}$, where $C_{a,b}$ is the completion of $O_{a,b}$. The goal is to find the group-sequence $\Psi$, the job-sequence ${\psi }_a$ within $G_a$, $d_1^a$, $d_2^a$, $\left(\widehat{\Psi }=(\Psi ,{\psi }_1,{\psi }_2,\dots ,{\psi }_m)\right)$ and the resource allocation amount $u_{a,b}$ for job $O_{a,b}$ such that the following objective cost is minimized, i.e.,

$$\begin{array}{ccc}\hfill OC& =& {\displaystyle \sum _{a=1}^m}{\displaystyle \underset{1\le b\le n_a}{max}}\left\{max\left\{{\kappa }_{a1}{E}_{a,b}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a,{\kappa }_{a2}{T}_{a,b}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}\right\}\hfill \\ & & +{\displaystyle \sum _{a=1}^m}{\displaystyle \sum _{b=1}^{n_a}}{\tau }_{a,b}{u}_{a,b}\hfill \\ & =& {\displaystyle \sum _{a=1}^m}{\displaystyle \underset{1\le b\le n_a}{max}}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+{\displaystyle \sum _{a=1}^m}{\displaystyle \sum _{b=1}^{n_a}}{\tau }_{a,b}{u}_{a,b}\hfill \end{array}$$

where ${\kappa }_{a1}$, ${\kappa }_{a2}$, ${\kappa }_{a3}$, and ${\kappa }_{a4}$, indicate the unit costs of earliness, tardiness, due-window starting time and size, respectively, and ${\tau }_{a,b}$ denotes the unit cost of resources consumed by $O_{a,b}$. Adopting the three-field notation (Graham et al. [60], Cai et al. [61] and  Chen and Hall [62]), the problems are denoted as follows:

$$1\left|gt,condw,linear\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

and

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

where $linear$ (resp. $convex$) denotes the linear (resp. convex) resource model. The problems studied with the $gt$ are given in Table 1.

3. Main Results

Mosheiov and Sarig [63] proposed an optimal solution for the problem:

$$1\left|gt,condw\right|\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}$$

Based on their conclusions, Lemma 1 is proposed as follows:

Lemma 1.

For group $G_a\left(a=1,2,\dots ,m\right)$, the objective cost

$$O{C}_a=\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}$$

can be minimized by the following

Table 2. The optimal strategy of condw.

	Case	Optimal Due-Window		${\mathit{OC}}_{\mathit{a}}$
		${\mathit{d}}_{\mathbf{1}}^{*}$	${\mathit{d}}_{\mathbf{2}}^{*}$
1	${\kappa }_{a2}\ge {\kappa }_{a3}$ and ${\kappa }_{a4}\le {\kappa }_{a3}$	0	$C_{a,\left[n_a\right]}$	${\kappa }_{a4}{C}_{a,\left[n_a\right]}$
2	${\kappa }_{a2}\ge {\kappa }_{a3}$, ${\kappa }_{a4}>{\kappa }_{a3}$			${\displaystyle \frac{{\kappa }_{a1}\left({\kappa }_{a3}-{\kappa }_{a2}\right)}{\left({\kappa }_{a1}+{\kappa }_{a2}\right)}}{C}_{a,\left[1\right]}+$
	and ${\kappa }_{a4}\ge {\displaystyle \frac{{\kappa }_{a2}\left({\kappa }_{a1}+{\kappa }_{a3}\right)}{{\kappa }_{a1}+{\kappa }_{a2}}}$	${\displaystyle \frac{{\kappa }_{a1}{C}_{a,\left[1\right]}+{\kappa }_{a2}{C}_{a,\left[n_a\right]}}{{\kappa }_{a1}+{\kappa }_{a2}}}$	${\displaystyle \frac{{\kappa }_{a1}{C}_{a,\left[1\right]}+{\kappa }_{a2}{C}_{a,\left[n_a\right]}}{{\kappa }_{a1}+{\kappa }_{a2}}}$	${\displaystyle \frac{{\kappa }_{a2}\left({\kappa }_{a1}+{\kappa }_{a3}\right)}{\left({\kappa }_{a1}+{\kappa }_{a2}\right)}}{C}_{a,\left[n_a\right]}$
3	${\kappa }_{a2}\ge {\kappa }_{a3}$, ${\kappa }_{a4}>{\kappa }_{a3}$
	and ${\kappa }_{a4}<{\displaystyle \frac{{\kappa }_{a2}\left({\kappa }_{a1}+{\kappa }_{a3}\right)}{{\kappa }_{a1}+{\kappa }_{a2}}}$	$C_{a,\left[1\right]}$	$C_{a,\left[n_a\right]}$	$\left({\kappa }_{a3}-{\kappa }_{a4}\right)C_{a,\left[1\right]}+{\kappa }_{a4}{C}_{a,\left[n_a\right]}$
4	${\kappa }_{a2}<{\kappa }_{a3}$ and ${\kappa }_{a4}\ge {\kappa }_{a2}$	0	0	${\kappa }_{a2}{C}_{a,\left[n_a\right]}$
5	${\kappa }_{a2}<{\kappa }_{a3}$ and ${\kappa }_{a4}<{\kappa }_{a2}$	0	$C_{a,\left[n_a\right]}$	${\kappa }_{a4}{C}_{a,\left[n_a\right]}$

:

Proof. 

See Mosheiov and Sarig [63] and Ren and Yang [59].    □

According to Lemma 1, the objective cost $O{C}_a$ of $G_a$ can be reformulated as follows:

$$O{C}_a=X_a{C}_{a,\left[1\right]}+Y_a{C}_{a,\left[n_a\right]}$$

(3)

where

$$X_a=\left\{\begin{array}{cc}0,\hfill & \mathrm{Case}1\hfill \\ {\displaystyle \frac{{\kappa }_{a1}\left({\kappa }_{a3}+{\kappa }_{a2}\right)}{\left({\kappa }_{a1}+{\kappa }_{a2}\right)}},\hfill & \mathrm{Case}2\hfill \\ \left({\kappa }_{a3}-{\kappa }_{a4}\right),\hfill & \mathrm{Case}3\hfill \\ 0,\hfill & \mathrm{Case}4\hfill \\ 0,\hfill & \mathrm{Case}5\hfill \end{array}\right.$$

(4)

and

$$Y_a=\left\{\begin{array}{cc}{\kappa }_{a4},\hfill & \mathrm{Case}1\hfill \\ {\displaystyle \frac{{\kappa }_{a2}\left({\kappa }_{a1}+{\kappa }_{a3}\right)}{\left({\kappa }_{a1}+{\kappa }_{a2}\right)}},\hfill & \mathrm{Case}2\hfill \\ {\kappa }_{a4},\hfill & \mathrm{Case}3\hfill \\ {\kappa }_{a2},\hfill & \mathrm{Case}4\hfill \\ {\kappa }_{a4}\hfill & \mathrm{Case}5\hfill \end{array}\right.$$

(5)

3.1. The $Linear$ Model

By the linear resource model (1) and Equation (3), we have

$$\begin{array}{cc}\hfill OC=& \sum _{a=1}^{m}O{C}_a+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,\left[b\right]}{u}_{a,\left[b\right]}\hfill \\ \hfill =& \sum _{a=1}^m\left(X_a{C}_{a,\left[1\right]}+Y_a{C}_{a,\left[n_a\right]}\right)+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,\left[b\right]}{u}_{a,\left[b\right]}\hfill \\ \hfill =& \sum _{a=1}^m\left\{X_a\left[\sum _{r=1}^{a-1}\left(S_r+{\tilde{P}}_r^A\right)+S_a+P_{a,\left[1\right]}^A\right]+Y_a\left[\sum _{r=1}^{a-1}\left(S_r+{\tilde{P}}_r^A\right)+S_a+{\tilde{P}}_a^A\right]\right\}+\hfill \\ & \sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,\left[b\right]}{u}_{a,\left[b\right]}\hfill \\ \hfill =& \sum _{a=1}^m\left\{\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]P_{a,\left[1\right]}^A+\left[\sum _{h=a+1}^m\left(X_h+Y_h\right)+Y_a\right]\sum _{j=2}^{n_a}{P}_{a,\left[j\right]}^A\right\}\hfill \\ & +\sum _{a=1}^m\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]S_a+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,\left[b\right]}{u}_{a,\left[b\right]}\hfill \end{array}$$

(6)

By substituting Equation (1) into Equation (6), we have

$$OC=\sum _{a=1}^m\left[\sum _{h=a}^m(X_h+Y_h)\right]S_a+\sum _{a=1}^m\sum _{b=1}^{n_a}{\displaystyle \frac{{\tau }_{a,\left[b\right]}{P}_{a,\left[b\right]}}{{ϵ}_{a,\left[b\right]}}}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\chi }_{a,\left[b\right]}{P}_{a,\left[b\right]}^A$$

(7)

where

$${\chi }_{a,\left[b\right]}=\left\{\begin{array}{cc}\sum _{h=a}^m(X_h+Y_h)-{\displaystyle \frac{{\tau }_{a,\left[1\right]}}{{ϵ}_{a,\left[1\right]}}},\hfill & b=1\hfill \\ \left[\sum _{h=a+1}^m(X_h+Y_h)+Y_a\right]-{\displaystyle \frac{{\tau }_{a,\left[b\right]}}{{ϵ}_{a,\left[b\right]}}},\hfill & b=2,3,\dots ,n_a\hfill \end{array}\right.fora=1,2,\dots ,m$$

(8)

Lemma 2.

For the given job-sequence ${\psi }_a$ within $G_a$ of the $linear$ model, the optimal resource $u_{a,\left[b\right]}^{*}$ ($a=1,2,\dots ,m$ and $b=1,2,\dots ,n_a$) is

$$u_{a,\left[b\right]}^{*}=\left\{\begin{array}{cc}0,\hfill & {\chi }_{a,\left[b\right]}<0\hfill \\ u_{a,\left[b\right]}\in [0,{\overline{u}}_{a,\left[b\right]}],\hfill & {\chi }_{a,\left[b\right]}=0\hfill \\ {\overline{u}}_{a,\left[b\right]},\hfill & {\chi }_{a,\left[b\right]}>0\hfill \end{array}\right.fora=1,2,\dots ,mandb=1,2,\dots ,n_a$$

(9)

Proof. 

See Ren and Yang [59].    □

Let

$${\chi }_{a,\left[b\right]}=\left\{\begin{array}{cc}{\chi }_{a,\left[b\right]}^1,\hfill & ifjob{O}_{a,\left[b\right]}isinthefirstpositionof{\psi }_a\hfill \\ {\chi }_{a,\left[b\right]}^2,\hfill & otherwise\hfill \end{array}\right.forb=1,2,3,\dots ,n_a$$

(10)

and

$$P_{a,\left[b\right]}^A=\left\{\begin{array}{cc}{P}_{a,\left[b\right]},\hfill & {\chi }_{a,\left[b\right]}<0\hfill \\ P_{a,\left[b\right]}^A\in \left[P_{a,\left[b\right]}-{ϵ}_{a,\left[b\right]}{\overline{u}}_{a,\left[b\right]},P_{a,\left[b\right]}\right],\hfill & {\chi }_{a,\left[b\right]}=0\hfill \\ P_{a,\left[b\right]}-{ϵ}_{a,\left[b\right]}{\overline{u}}_{a,\left[b\right]},\hfill & {\chi }_{a,\left[b\right]}>0\hfill \end{array}\right.forb=1,2,\dots ,n_a$$

(11)

where ${\chi }_{a,\left[b\right]}^1={\sum }_{h=a}^m(X_h+Y_h)-{\displaystyle \frac{{\tau }_{a,\left[1\right]}}{{ϵ}_{a,\left[1\right]}}}$ and ${\chi }_{a,\left[b\right]}^2=\left[{\sum }_{h=a+1}^m(X_h+Y_h)+Y_a\right]-{\displaystyle \frac{{\tau }_{a,\left[b\right]}}{{ϵ}_{a,\left[b\right]}}}$. As in Ren and Yang [59], for $G_a$ of the given group-sequence, the optimal job-sequence can be obtained in $O\left(n_a\right)$ time, which is obtained by Algorithm 1, where ${\varpi }_{a,\left[b\right]}={\chi }_{a,\left[b\right]}^1{P}_{a,\left[b\right]}^A+{\chi }_{a,\left[b\right]}^2{\sum }_{j=2}^{n_a}{P}_{a,\left[b\right]}^A$ and ${\varpi }_{a,\left[b\right]}$ denotes the minimum value of the term ${\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\chi }_{a,\left[b\right]}{P}_{a,\left[b\right]}^A$ resulting from scheduling $O_{a,\left[b\right]}$ in the first position in ${\psi }_a$. Obviously, the optimal job-sequence of all groups can be determined in ${\sum }_{a=1}^{m}O\left(n_a\right)=O\left(n\right)$ time.

Algorithm 1: Optimal job-sequence ${\psi }_a$: $linear$

1 Step 1. Initialization of values.   2              Compute $X_a$ and $Y_a$ for $G_a$← Equations (4) and (5)   3 Step 2. Iterative position analysis.   4              Set the initial group counter: $a\leftarrow 1$   5              While $a\le m$ do   6                       Set the initial position counter: $r\leftarrow 1$   7                       While $r\le n_a$ do   8                               For each job $O_{a,\left[b\right]}\left(b=1,2,\dots ,n_a\right)$   9                                      Compute ${\chi }_{a,\left[b\right]}\leftarrow$ Equation (10) 10                                      Compute $P_{a,\left[b\right]}^A\leftarrow$ Equation (11) 11                               Update the position counter: $r\leftarrow r+1$ 12                       Update the group counter: $a\leftarrow a+1$ 13              Identify the first processed job: $min\left\{{\varpi }_{a,\left[b\right]},b=1,2,\dots ,n_a\right\}$ 14              Generate the optimal sequence ${\psi }_a^{*}$ 15 Step 3. Optimal resource allocation and due-window. 16              Determine optimal resource allocation: $u_{a,\left[b\right]}^{*}\leftarrow$ Equation (9) 17              Determine optimal $d_1^{*}$ and $d_2^{*}\leftarrow$ Lemma 1

3.1.1. An Upper Bound

For the general case, we suspect

$$1\left|gt,condw,linear\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

is NP-hard; hence, an exact branch-and-bound algorithm can be proposed. From Equation (7), Zhang and Wang [58], ${\sum }_{a=1}^m\left[{\sum }_{h=a}^m(X_h+Y_h)\right]S_a$ is minimized by the descending order of ${\displaystyle \frac{X_a+Y_a}{S_a}}$. ${\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\chi }_{a,\left[b\right]}{P}_{a,\left[b\right]}^A$ is minimized by the descending order of ${\displaystyle \frac{X_a+Y_a}{{\tilde{P}}_a^A}}$, where ${\tilde{P}}_a^A={\sum }_{b=1}^{n_a}{P}_{a,\left[b\right]}^A$. Hence, the following heuristic (i.e., Algorithm 2) is proposed as an upper bound $\widehat{UB}$ for the group-sequence of $1\left|gt,condw,linear\right|{\sum }_{a=1}^m{max}_{1\le b\le n_a}$$\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+{\sum }_{a=1}^m{\sum }_{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$.

3.1.2. A Lower Bound

Let $\Psi =\left({\Psi }_{group-s},{\Psi }_{group-u}\right)$ be a group-sequence, where ${\Psi }_{group-s}\left({\Psi }_{group-u}\right)$ is the set of scheduled (unscheduled) part of groups, and there is g groups in the set ${\Psi }_{group-s}$. From Equation (7), we have

$$\begin{array}{cc}\hfill OC& =\sum _{a=1}^g\left[\sum _{r=a}^g\left(X_r+Y_r\right)+\sum _{r=g+1}^m\left(X_{\left[r\right]}+Y_{\left[r\right]}\right)\right]S_a+\sum _{a=g+1}^m\left[\sum _{r=a}^m{X}_{\left[r\right]}+Y_{\left[r\right]}\right]S_{\left[a\right]}\hfill \\ & +\sum _{a=1}^g\sum _{b=1}^{n_a}{\displaystyle \frac{{\tau }_{a,b}{P}_{a,b}}{{ϵ}_{a,b}}}+\sum _{a=g+1}^m\sum _{b=1}^{n_a}{\displaystyle \frac{{\tau }_{\left[a\right],\left[b\right]}{P}_{\left[a\right],\left[b\right]}}{{ϵ}_{\left[a\right],\left[b\right]}}}+\sum _{a=1}^g\{\left[\sum _{r=a}^g\left(X_r+Y_r\right)\right]\hfill \\ & +\left[\sum _{r=g+1}^m\left(X_{\left[r\right]}+Y_{\left[r\right]}\right)\right]-{\displaystyle \frac{{\tau }_{a,1}}{{ϵ}_{a,1}}}\}{P}_{a,1}^A+\sum _{a=g+1}^m\left\{\left[\sum _{r=a}^m\left(X_{\left[r\right]}+Y_{\left[r\right]}\right)\right]-{\displaystyle \frac{{\tau }_{\left[a\right],\left[1\right]}}{{ϵ}_{\left[a\right],\left[1\right]}}}\right\}{P}_{\left[a\right],\left[1\right]}^A\hfill \\ & +\sum _{a=1}^g\sum _{b=2}^{n_a}\left[\sum _{r=a+1}^g\left(X_r+Y_r\right)+\sum _{r=g+1}^m\left(X_{\left[r\right]}+Y_{\left[r\right]}\right)+Y_a-{\displaystyle \frac{{\tau }_{a,b}}{{ϵ}_{a,b}}}\right]P_{a,b}^A\hfill \\ & +\sum _{a=g+1}^m\sum _{b=2}^{n_a}\left[\sum _{r=a+1}^m\left(X_{\left[r\right]}+Y_{\left[r\right]}\right)+Y_{\left[a\right]}-{\displaystyle \frac{{\tau }_{\left[a\right],\left[b\right]}}{{ϵ}_{\left[a\right],\left[b\right]}}}\right]P_{\left[a\right],\left[b\right]}^A\hfill \end{array}$$

(12)

Algorithm 2: Upper bound algorithm: $\widehat{UB}-linear$

1 Step 1. Calculate $X_a$ and $Y_a$← Equations (4) and (5)   2              Calculate ${\tilde{P}}_a^A$←${\tilde{P}}_a^A={\sum }_{b=1}^{n_a}{P}_{a,\left[b\right]}^A$   3 Step 2. Determine job-sequence for each group.   4              The optimal job-sequence ${\psi }_a$ of $G_a$← Algorithm 1   5 Step 3. Determine group-sequence.   6              Step 3.1: Arrange groups in descending order of ${\displaystyle \frac{X_a+Y_a}{S_a}}$   7                              Compute $O{C}_A\leftarrow$ Equation (7)   8              Step 3.2: Arrange groups in descending order of ${\displaystyle \frac{X_a+Y_a}{{\tilde{P}}_a^A}}$   9                              Compute $O{C}_B\leftarrow$ Equation (7) 10              Step 3.3: Arrange groups in increasing order of $S_a$ 11                              Compute $O{C}_C\leftarrow$ Equation (7) 12              Step 3.4: Arrange groups in descending order of $X_a+Y_a$ 13                              Compute $O{C}_D\leftarrow$ Equation (7) 14              Step 3.5: Arrange groups in increasing order of ${\tilde{P}}_a^A$ 15                              Compute $O{C}_E\leftarrow$ Equation (7) 16 Step 4. Best group-sequence selection. 17              Compute $O{C}_{best}\leftarrow O{C}_{best}=min\left\{O{C}_A,O{C}_B,O{C}_C,O{C}_D,O{C}_E\right\}$

    From (12), ${\sum }_{a=1}^g\left[{\sum }_{r=a}^g\left(X_r+Y_r\right)\right]S_a$, ${\sum }_{a=1}^g{\sum }_{b=1}^{n_a}{\displaystyle \frac{{\tau }_{a,b}{P}_{a,b}}{{ϵ}_{a,b}}}$, ${\sum }_{a=g+1}^m{\sum }_{b=1}^{n_a}{\displaystyle \frac{{\tau }_{\left[a\right],\left[b\right]}{P}_{\left[a\right],\left[b\right]}}{{ϵ}_{\left[a\right],\left[b\right]}}}$,

${\sum }_{a=1}^g\left\{\left[{\sum }_{r=a}^g\left(X_r+Y_r\right)\right]-{\displaystyle \frac{{\tau }_{a,1}}{{ϵ}_{a,1}}}\right\}{P}_{a,1}^A$, and ${\sum }_{a=1}^g{\sum }_{b=2}^{n_a}\left[{\sum }_{r=a+1}^g\left(X_r+Y_r\right)+Y_a-{\displaystyle \frac{{\tau }_{a,b}}{{ϵ}_{a,b}}}\right]P_{a,b}^A$ are known and constants. Hence, the first lower bound (denoted by $L{B}_{linear}$) is obtained as follows:

$$\begin{array}{cc}\hfill {LB}_{linear}^1& ={\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}\left(X_r+Y_r\right)+{\displaystyle \sum _{r=g+1}^m}\left(X_{<r>}+Y_{<r>}\right)\right]S_a+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}{X}_{<r>]}+Y_{<r>}\right]S_{min}\hfill \\ & +{\displaystyle \sum _{a=1}^g}{\displaystyle \sum _{b=1}^{n_a}}{\displaystyle \frac{{\tau }_{a,b}{P}_{a,b}}{{ϵ}_{a,b}}}+{\displaystyle \sum _{a=g+1}^m}{\displaystyle \sum _{b=1}^{n_a}}{\displaystyle \frac{{\tau }_{\left[a\right],\left[b\right]}{P}_{\left[a\right],\left[b\right]}}{{ϵ}_{\left[a\right],\left[b\right]}}}+{\displaystyle \sum _{a=1}^g}\{\left[{\displaystyle \sum _{r=a}^g}\left(X_r+Y_r\right)\right]\hfill \\ & +\left[{\displaystyle \sum _{r=g+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)\right]-{\displaystyle \frac{{\tau }_{a,1}}{{ϵ}_{a,1}}}\}{P}_{a,1}^A+{\displaystyle \sum _{a=g+1}^m}\left\{\left[{\displaystyle \sum _{r=a}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)\right]-{\displaystyle \frac{{\tau }_{\left(a\right),\left[1\right]}}{{ϵ}_{\left(a\right),\left[1\right]}}}\right\}{P}_{\left(a\right),\left[1\right]}^A\hfill \\ & +{\displaystyle \sum _{a=1}^g}{\displaystyle \sum _{b=2}^{n_a}}\left[{\displaystyle \sum _{r=a+1}^g}\left(X_r+Y_r\right)+{\displaystyle \sum _{r=g+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)+Y_a-{\displaystyle \frac{{\tau }_{a,b}}{{ϵ}_{a,b}}}\right]P_{a,b}^A\hfill \\ & +{\displaystyle \sum _{a=g+1}^m}{\displaystyle \sum _{b=2}^{n_a}}\left[{\displaystyle \sum _{r=a+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)+Y_{\left[a\right]}-{\displaystyle \frac{{\tau }_{\left(a\right),\left[b\right]}}{{ϵ}_{\left(a\right),\left[b\right]}}}\right]P_{\left(a\right),\left[b\right]}^A\hfill \end{array}$$

(13)

where $S_{min}=min\left\{S_a\in {\Psi }_{group-u}\right\}$, $X_{<g+1>}+Y_{<g+1>}>X_{<g+2>}+Y_{<g+2>}>\cdots >X_{<m>}+Y_{<m>}$ and ${\displaystyle \frac{X_{(g+1)}+Y_{(g+1)}}{{\tilde{P}}_{(g+1)}^A}}>{\displaystyle \frac{X_{(g+2)}+Y_{(g+2)}}{{\tilde{P}}_{(g+2)}^A}}>\cdots >{\displaystyle \frac{X_{\left(m\right)}+Y_{\left(m\right)}}{{\tilde{P}}_{\left(m\right)}^A}}$. Similarly, the second $L{B}_{linear}$ is

$$\begin{array}{cc}\hfill {LB}_{linear}^2& ={\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}\left(X_r+Y_r\right)+{\displaystyle \sum _{r=g+1}^m}\left(X_{<r>}+Y_{<r>}\right)\right]S_a+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}{X}_{<r>]}+Y_{<r>}\right]S_{<a>}\hfill \\ & +{\displaystyle \sum _{a=1}^g}{\displaystyle \sum _{b=1}^{n_a}}{\displaystyle \frac{{\tau }_{a,b}{P}_{a,b}}{{ϵ}_{a,b}}}+{\displaystyle \sum _{a=g+1}^m}{\displaystyle \sum _{b=1}^{n_a}}{\displaystyle \frac{{\tau }_{\left[a\right],\left[b\right]}{P}_{\left[a\right],\left[b\right]}}{{ϵ}_{\left[a\right],\left[b\right]}}}+{\displaystyle \sum _{a=1}^g}\{\left[{\displaystyle \sum _{r=a}^g}\left(X_r+Y_r\right)\right]\hfill \\ & +\left[{\displaystyle \sum _{r=g+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)\right]-{\displaystyle \frac{{\tau }_{a,1}}{{ϵ}_{a,1}}}\}{P}_{a,1}^A+{\displaystyle \sum _{a=g+1}^m}\left\{\left[{\displaystyle \sum _{r=a}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)\right]-{\displaystyle \frac{{\tau }_{\left(a\right),\left[1\right]}}{{ϵ}_{\left(a\right),\left[1\right]}}}\right\}{P}_{\left(a\right),\left[1\right]}^A\hfill \\ & +{\displaystyle \sum _{a=1}^g}{\displaystyle \sum _{b=2}^{n_a}}\left[{\displaystyle \sum _{r=a+1}^g}\left(X_r+Y_r\right)+{\displaystyle \sum _{r=g+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)+Y_a-{\displaystyle \frac{{\tau }_{a,b}}{{ϵ}_{a,b}}}\right]P_{a,b}^A\hfill \\ & +{\displaystyle \sum _{a=g+1}^m}{\displaystyle \sum _{b=2}^{n_a}}\left[{\displaystyle \sum _{r=a+1}^m}\left(X_{\left(r\right)}+Y_{\left(r\right)}\right)+Y_{\left[a\right]}-{\displaystyle \frac{{\tau }_{\left(a\right),\left[b\right]}}{{ϵ}_{\left(a\right),\left[b\right]}}}\right]P_{\left(a\right),\left[b\right]}^A\hfill \end{array}$$

(14)

where ${\displaystyle \frac{X_{<g+1>}+Y_{<g+1>}}{S_{<g+1>}}}>{\displaystyle \frac{X_{<g+2>}+Y_{<g+2>}}{S_{<g+2>}}}>\cdots >{\displaystyle \frac{X_{<m>}+Y_{<m>}}{S_{<m>}}}$ and ${\displaystyle \frac{X_{(g+1)}+Y_{(g+1)}}{{\tilde{P}}_{(g+1)}^A}}>{\displaystyle \frac{X_{(g+2)}+Y_{(g+2)}}{{\tilde{P}}_{(g+2)}^A}}$$>\cdots >{\displaystyle \frac{X_{\left(m\right)}+Y_{\left(m\right)}}{{\tilde{P}}_{\left(m\right)}^A}}$. Note that ${\displaystyle \frac{X_{<a>}+Y_{<a>}}{S_{<a>}}}$ and ${\displaystyle \frac{X_{\left(a\right)}+Y_{\left(a\right)}}{{\tilde{P}}_{\left(a\right)}^A}}\left(a=g+1,g+2,\dots ,m\right)$ do not mean the same group. To make the $L{B}_{linear}$ tighter, we choose the largest value of ${LB}_{linear}^1$ and ${LB}_{linear}^2$, i.e.,

$${LB}_{linear}=max\left\{{LB}_{linear}^1,{LB}_{linear}^2\right\}$$

(15)

3.2. The $Convex$ Model

For the $convex$ function of

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

from Ren and Yang [59], we have

$$\begin{array}{cc}\hfill OC=& \sum _{a=1}^m\left\{\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]{\left({\displaystyle \frac{w_{a,\left[1\right]}}{u_{a,\left[1\right]}}}\right)}^{\varrho }+\left[\sum _{h=a+1}^m\left(X_h+Y_h\right)+Y_a\right]\sum _{j=2}^{n_a}{\left({\displaystyle \frac{w_{a,\left[j\right]}}{u_{a,\left[j\right]}}}\right)}^{\varrho }\right\}\hfill \\ & +\sum _{a=1}^m\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]S_a+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,\left[b\right]}{u}_{a,\left[b\right]}\hfill \end{array}$$

(16)

Lemma 3.

For the given job-sequence ${\psi }_a$ within $G_a$ of $convex$ model, the optimal resource $u_{a,\left[b\right]}^{*}$ ($a=1,2,\dots ,m$ and $b=1,2,\dots ,n_a$) is

$$u_{a,\left[b\right]}^{*}\left(\Psi ,{\psi }_1,{\psi }_2,\dots ,{\psi }_m\right)=$$

$$\left\{\begin{array}{cc}{\left[{\displaystyle \frac{\varrho {\sum }_{h=a}^m\left(X_h+Y_h\right)}{{\tau }_{a,\left[1\right]}}}\right]}^{{\displaystyle \frac{1}{\varrho +1}}}{\left(w_{a,\left[1\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}},\hfill & a=1,2,\dots ,m;b=1\hfill \\ {\left[{\displaystyle \frac{\varrho \left({\sum }_{h=a+1}^m\left(X_h+Y_h\right)+Y_a\right)}{{\tau }_{a,\left[b\right]}}}\right]}^{{\displaystyle \frac{1}{\varrho +1}}}{\left(w_{a,\left[b\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}},\hfill & a=1,2,\dots ,m;b=2,3,\dots ,n_a\hfill \end{array}\right.$$

(17)

According Equations (16) and (17), we have

$$\begin{array}{cc}\hfill OC=& \sum _{a=1}^m\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]S_a+\left({\varrho }^{-{\displaystyle \frac{\varrho }{\varrho +1}}}+{\varrho }^{{\displaystyle \frac{1}{\varrho +1}}}\right)\sum _{a=1}^m\{{\left[\sum _{h=a}^m\left(X_h+Y_h\right)\right]}^{{\displaystyle \frac{1}{\varrho +1}}}{\left({\tau }_{a,\left[1\right]}{w}_{a,\left[1\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}}\hfill \\ & +{\left[\sum _{h=a+1}^m\left(X_h+Y_h\right)+Y_a\right]}^{{\displaystyle \frac{1}{\varrho +1}}}\sum _{j=2}^{n_a}{\left({\tau }_{a,\left[j\right]}{w}_{a,\left[j\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}}\}\hfill \end{array}$$

(18)

Let ${\xi }_{a,\left[b\right]}={\left({\tau }_{a,\left[b\right]}{w}_{a,\left[b\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}}$, $\left(a=1,2,\dots ,m,b=1,2,\dots ,n_a\right)$, as in Ren and Yang [59], for $G_a$ of the given group-sequence, the optimal job-sequence can be obtained in $O\left(n_a\right)$ time, which is obtained by Algorithm 3. Obviously, the optimal job-sequence of all groups can be determined in ${\sum }_{a=1}^{m}O\left(n_a\right)=O\left(n\right)$ time.

Algorithm 3: Optimal job-sequence ${\psi }_a$: $convex$

1 Step 1. Initialization of values 2              Compute $X_a$ and $Y_a$← Equations (4) and (5) 3 Step 2. Generate sequence 4              Compute ${\xi }_{a,\left[b\right]}$←${\xi }_{a,\left[b\right]}={\left({\tau }_{a,\left[b\right]}{w}_{a,\left[b\right]}\right)}^{{\displaystyle \frac{\varrho }{\varrho +1}}}$ 5              Select the job $O_{a,\left[1\right]}$ with largest ${\xi }_{a,\left[b\right]}$ value 6              Other jobs in any order 7 Step 3. Optimal resource allocation and due-window. 8              Determine optimal resource allocation: $u_{a,\left[b\right]}^{*}\leftarrow$ Equation (17) 9              Determine optimal $d_1^{*}$ and $d_2^{*}\leftarrow$ Lemma 1

3.2.1. An Upper Bound

For the general case, we suspect

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

is NP-hard; hence, an exact branch-and-bound algorithm can be proposed. Similarly, the following heuristic (Algorithm 4) is proposed as an upper bound for the group-sequence of

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}.$$

Algorithm 4: Upper bound algorithm: $\widehat{UB}-convex$

1 Step 1. Calculate $X_a$ and $Y_a$← Equations (4) and (5)   2              Calculate ${\xi }_a\leftarrow {\xi }_a={\sum }_{b=1}^{n_a}{\xi }_{a,\left[b\right]}$   3 Step 2. Determine job-sequence for each group.   4              The optimal job-sequence ${\psi }_a$ of $G_a$← Algorithm 3   5 Step 3. Determine group-sequence.   6              Step 3.1: Arrange groups in descending order of ${\displaystyle \frac{X_a+Y_a}{S_a}}$   7                              Compute $O{C}_A\leftarrow$ Equation (18)   8              Step 3.2: Arrange groups in descending order of ${\displaystyle \frac{X_a+Y_a}{{\xi }_a}}$   9                              Compute $O{C}_B\leftarrow$ Equation (18) 10              Step 3.3: Arrange groups in increasing order of $S_a$ 11                              Compute $O{C}_C\leftarrow$ Equation (18) 12              Step 3.4: Arrange groups in descending order of $X_a+Y_a$ 13                              Compute $O{C}_D\leftarrow$ Equation (18) 14              Step 3.5: Arrange groups in increasing order of ${\xi }_a$ 15                              Compute $O{C}_E\leftarrow$ Equation (18) 16 Step 4. Best group-sequence selection. 17              Compute $O{C}^{*}\leftarrow O{C}^{*}=min\left\{O{C}_A,O{C}_B,O{C}_C,O{C}_D,O{C}_E\right\}$

3.2.2. A Lower Bound

Similar to Section 3.1.2, from Equation (18), we have

$$\begin{array}{ccc}\hfill OC& =& {\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{\left[r\right]}+Y_{\left[r\right]})\right]S_a+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{\left[r\right]}+Y_{\left[r\right]})\right]S_{\left[a\right]}\hfill \\ & & +\left({\varrho }^{-{\displaystyle \frac{\varrho }{\varrho +1}}}+{\varrho }^{{\displaystyle \frac{1}{\varrho +1}}}\right)\left\{{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{\left[r\right]}+Y_{\left[r\right]})\right]{\xi }_{a,\left[1\right]}\right.\hfill \\ & & +{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a+1}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{\left[r\right]}+Y_{\left[r\right]})+Y_a\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{a,\left[j\right]}\hfill \\ & & +\left.{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{\left[r\right]}+Y_{\left[r\right]})\right]{\xi }_{\left[a\right],\left[1\right]}+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a+1}^m}(X_{\left[r\right]}+Y_{\left[r\right]})+Y_{\left[a\right]}\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{\left[a\right],\left[j\right]}\right\}\hfill \end{array}$$

(19)

From Equation (19), ${\sum }_{a=1}^g\left[{\sum }_{r=a}^g(X_r+Y_r)\right]S_a$, ${\sum }_{a=1}^g\left[{\sum }_{r=a}^g(X_r+Y_r)\right]{\xi }_{a,\left[1\right]}$ and

${\sum }_{a=1}^g\left[{\sum }_{r=a+1}^g(X_r+Y_r)\right]{\sum }_{j=2}^{n_a}{\xi }_{a,\left[j\right]}$ are constants. Hence, the first lower bound (denoted by $L{B}_{convex}$) is obtained as follows:

$$\begin{array}{ccc}\hfill & & {LB}_{convex}^1\hfill \\ & =& {\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{<r>}+Y_{<r>})\right]S_a+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{<r>}+Y_{<r>})\right]S_{min}\hfill \\ & & +\left({\varrho }^{-{\displaystyle \frac{\varrho }{\varrho +1}}}+{\varrho }^{{\displaystyle \frac{1}{\varrho +1}}}\right)\left\{{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{\left(r\right)}+Y_{\left(r\right)})\right]{\xi }_{a,\left[1\right]}\right.\hfill \\ & & +{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a+1}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{\left(r\right)}+Y_{\left(r\right)})+Y_a\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{a,\left[j\right]}\hfill \\ & & +\left.{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{\left(r\right)}+Y_{\left(r\right)})\right]{\xi }_{\left(a\right),\left[1\right]}+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a+1}^m}(X_{\left(r\right)}+Y_{\left(r\right)})+Y_{\left(a\right)}\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{\left(a\right),\left[j\right]}\right\}\hfill \end{array}$$

(20)

where $S_{min}=min\left\{S_a\in {\Psi }_{group-u}\right\}$, $X_{<g+1>}+Y_{<g+1>}>X_{<g+2>}+Y_{<g+2>}>\cdots >X_{<m>}+Y_{<m>}$ and ${\displaystyle \frac{X_{(g+1)}+Y_{(g+1)}}{{\xi }_{(g+1)}}}>{\displaystyle \frac{X_{(g+2)}+Y_{(g+2)}}{{\xi }_{(g+2)}}}>\cdots >{\displaystyle \frac{X_{\left(m\right)}+Y_{\left(m\right)}}{{\xi }_{\left(m\right)}}}$.

Let $X_{min}+Y_{min}=min\left\{X_a+Y_a\in {\Psi }_{group-u}\right\}$, similarly, the second $L{B}_{convex}$ is

$$\begin{array}{ccc}\hfill & & {LB}_{convex}^2\hfill \\ & =& {\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{min}+Y_{min})\right]S_a+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{min}+Y_{min})\right]S_{min}\hfill \\ & & +\left({\varrho }^{-{\displaystyle \frac{\varrho }{\varrho +1}}}+{\varrho }^{{\displaystyle \frac{1}{\varrho +1}}}\right)\left\{{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{min}+Y_{min})\right]{\xi }_{a,\left[1\right]}\right.\hfill \\ & & +{\displaystyle \sum _{a=1}^g}\left[{\displaystyle \sum _{r=a+1}^g}(X_r+Y_r)+{\displaystyle \sum _{r=g+1}^m}(X_{min}+Y_{min})+Y_a\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{a,\left[j\right]}\hfill \\ & & +\left.{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a}^m}(X_{min}+Y_{min})\right]{\xi }_{\left(a\right),\left[1\right]}+{\displaystyle \sum _{a=g+1}^m}\left[{\displaystyle \sum _{r=a+1}^m}(X_{min}+Y_{min})+Y_{\left(a\right)}\right]{\displaystyle \sum _{j=2}^{n_a}}{\xi }_{\left(a\right),\left[j\right]}\right\}\hfill \end{array}$$

(21)

where ${\xi }_{\left(g+1\right)}<{\xi }_{\left(g+2\right)}<\cdots <{\xi }_{\left(m\right)}$. Note that ${\xi }_{\left(a\right)}$, and $X_{\left(a\right)}+Y_{\left(a\right)}\left(a=g+1\right.$, $\left.g+2,\dots ,m\right)$ do not mean the same group. Similar to the $linear$ model, we choose the largest value of ${LB}_{convex}^1$ and ${LB}_{convex}^2$, i.e.,

$${LB}_{convex}=max\left\{{LB}_{convex}^1,{LB}_{convex}^2\right\}$$

(22)

3.3. Solution Algorithm

3.3.1. Branch-And-Bound

For the $linear$ and $convex$ models, based on the lower bounds Equations (15) and (22), and upper bounds Algorithms 2 and 4, we propose an exact branch-and-bound ($\widehat{BaB}$) to address the problems

$$1\left|gt,condw,linear\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

and

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}.$$

The polynomial-time algorithm for the special case cannot be applied to the general case. To address this, we developed a novel Branch-and-Bound ($\widehat{BaB}$) framework and designed tight lower bounds for effective pruning, which is crucial for tackling this NP-hard problem (Algorithm 5).

Algorithm 5: $\widehat{BaB}-linear$/$convex$

1 Step 1. Initial upper bound ($\widehat{UB}-linear$/$\widehat{UB}-convex$) ${\Psi }^0\leftarrow$ Algorithm 2/ Algorithm 4   2              Calculate $OC\left({\Psi }^{*}\right)=OC\left({\Psi }^0\right)\leftarrow$ Equation (7)/Equation (18)   3 Step 2. For each node (group) $G_a$:   4                     Calculate ${LB}_{linear}\left(G_a\right)$/${LB}_{convex}\left(G_a\right)$← Equation (15)/(Equation (22)   5                     If ${LB}_{linear}\left(G_a\right)$/${LB}_{convex}\left(G_a\right)$$\ge OC\left({\Psi }^{*}\right)$:   6                            Delete node $G_a$ and its subtree   7              For each unfathomed node $G_j$:   8                     Calculate ${LB}_{linear}\left(G_j\right)$/${LB}_{convex}\left(G_j\right)$← Equation (15)/(Equation (22)   9                     If ${LB}_{linear}\left(G_j\right)$/${LB}_{convex}\left(G_j\right)$$\ge OC\left({\Psi }^{*}\right)$: 10                            Delete node $G_j$ and its subtree 11                     Else: 12                            Complete the group-sequence to obtain $\Psi$ 13                            Compute $OC\left(\Psi \right)$ 14                            If $OC\left(\Psi \right)<OC\left({\Psi }^{*}\right)$: 15                                     Update: ${\Psi }^{*}\leftarrow \Psi$ 16                            Else: 17                                     Discard $\Psi$ 18 Step 3. If all nodes have been searched, terminate the algorithm 19              Output the optimal group-sequence ${\Psi }^{*}$ and $OC({\Psi }^{*}$)

The branching strategy expands a node by generating child nodes, each of which appends a distinct unscheduled group to the current partial sequence. Nodes are explored in the order they are generated, corresponding to a depth-first traversal of the tree. To enhance efficiency, a pruning rule is applied at each node: if the calculated lower bound for the partial sequence is greater than or equal to the objective value of the current best-known complete solution, the node and its entire subtree are discarded. The algorithm terminates when all nodes in the search tree have been either fully explored or pruned.

3.3.2. Simulated Annealing

As in Kirkpatrick et al. [64], Lv and Wang [65] and Sun et al. [66], the simulated annealing ($\widehat{SA}$) is proposed to solve

$$1\left|gt,condw,linear\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}$$

and

$$1\left|gt,condw,convex\right|\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}.$$

The $\widehat{SA}$ algorithm (Algorithm 6) is introduced to systematically search for and approximate optimal solutions within acceptable time, bridging the gap between the fast constructive heuristic ($\widehat{UB}$) and the time-consuming exact algorithm ($\widehat{BaB}$).

Algorithm 6: $\widehat{SA}-linear$/$convex$

1 Step 1. Initial feasible solution ${\Psi }^0\leftarrow$ Algorithm 2/Algorithm 4   2              Current solution $\Psi \leftarrow {\Psi }^0$   3              Compute $OC\left({\Psi }^{*}\right)=OC\left({\Psi }^0\right)\leftarrow$ Equations (7) and (18)   4              $ST\leftarrow 1.0$ (starting temperature)   5              $LTL\leftarrow 1\times {10}^{-16}$ (lower temperature limit)   6              $CR\leftarrow 0.99999999$ (cooling rate)   7              $IT\leftarrow 1000\times n$ (iterations)   8              $RDF\leftarrow U(0,1)$ (random decision factor)   9 Step 2. Iterative Optimization 10              While $IT>0$ and $\left|ST\right|>LTL$ do: 11                      Randomly select positions $\widehat{R1},\widehat{R2}$ 12                      If $\widehat{R1}=\widehat{R2}$: 13                              $IT\leftarrow IT+1$, continue 14                      Else: 15                              Generate neighbor ${\Psi }^{\prime }\leftarrow$ Swap $\widehat{R1}$ and $\widehat{R2}$ in $\Psi$ 16                      $\Delta E\leftarrow OC\left({\Psi }^{\prime }\right)-OC\left(\Psi \right)$ 17                      If $\Delta E<0$ or $exp(-\Delta E/ST)>RDF$: 18                              Accept $\Psi \leftarrow {\Psi }^{\prime }$ 19                              If $OC\left({\Psi }^{\prime }\right)<OC\left({\Psi }^{*}\right)$: 20                                     $OC\left({\Psi }^{*}\right)\leftarrow OC\left({\Psi }^{\prime }\right)$ 21                                     ${\Psi }^{*}\leftarrow {\Psi }^{\prime }$ 22                      Update temperature: $ST\leftarrow ST\times CR$ 23                      $IT\leftarrow IT-1$ 24 Step 3. Termination 25              Output best group-sequence ${\Psi }^{*}$ and $OC\left({\Psi }^{*}\right)$

4. Numerical Experiments

To analyze the effectiveness and performance of $\widehat{UB}$ (Algorithm 2 and Algorithm 4), $\widehat{BaB}$ (Algorithm 5) and $\widehat{SA}$ (Algorithm 6), a set of experiments were conducted. Random instances were generated, where n = 50, 100, 150, 200, 250; $m=$ 8, 10, 12, 14, 16; $P_{a,b}\in U[1,100]$; ${ϵ}_{a,b}\in U[1,100]$; $w_{a,b}\in U[1,100]$; $S_a\in U[1,10]$; ${\kappa }_{ai}\in U[1,10]$ ($i=1,2,3,4$); ${\tau }_{a,b}\in U[1,10]$, $\varrho =2$. These algorithms were run on a DELL personal computer with 3.10 GHz CPU, 8 GB RAM, and these algorithms were coded in C++ and compiled with Microsoft Visual Studio 2019.

For the $\widehat{BaB}-linear$ and $\widehat{BaB}-convex$), the average and maximum node numbers and CPU time (in millisecond, ms) were recorded. For the $\widehat{UB}-linear$, $\widehat{UB}-convex$, $\widehat{SA}-linear$ and $\widehat{SA}-convex$, the average and maximum CPU time (in seconds) were recorded. The error of the $\widehat{UB}-linear$, $\widehat{UB}-convex$, $SA-linear$ and $SA-convex$ is calculated as

$${\displaystyle \frac{OC\left(H\right)-O{C}^{*}}{O{C}^{*}}}\times 100\%,$$

where $OC\left(H\right)$ is the objective cost obtained by the algorithm H, $H\in \{\widehat{UB}-linear,\widehat{UB}-convex,\widehat{SA}-linear,\widehat{SA}-convex\}$ and $O{C}^{*}$ is the optimal cost obtained by $\widehat{BaB}$. For each combination of m and n, 20 random examples were generated. From

Table 3. CPU time (ms) of $condw$ for linear models.

n	m	UB-Linear		SA-Linear		$\mathit{BB}$		$\mathit{BB}$ Nodes
		Mean	Max	Mean	Max	Mean	Max	Mean	Max
50	8	14.9088	23.208	296.2808	346.56	49.74969	67.7076	1054.35	1452
	10	20.063	33.68	379.8795	512.31	302.095635	495.0918	6759.3	11,131
	12	24.5106	33.588	503.1198	740.412	2237.70766	3162.2199	45,319	67,463
	14	32.1279	47.222	705.8009	1067.066	23,078.83646	44,647.7313	407,422.4	754,785
	16	47.0688	78.912	870.7608	1636.832	195,306.0996	381,843.1225	3,473,117.7	6,588,502
100	8	22.9456	42.76	1058.9224	1806.856	81.30942	127.4241	1172.5	1881
	10	32.2715	103.07	1264.1235	1950.62	547.097055	884.3089	7575.05	12,345
	12	42.3954	135.084	1584.3606	2058.48	4164.310825	6437.5259	56,047.6	85,700
	14	54.3641	125.986	2121.0665	4143.468	32,222.81033	49,109.7604	419,441.7	644,204
	16	59.332	156.176	2420.5336	3440.784	356,116.6568	635,590.2297	4,376,056.7	7,738,105
150	8	28.4788	41.576	2014.8352	2453.64	95.8844	125.4708	1059.3	1345
	10	41.8725	57.38	2657.5895	3432.59	747.119815	1314.9022	7673.45	13,048
	12	58.3632	104.784	3153.7596	3749.568	5291.199145	9228.4548	53,505.95	91,720
	14	69.3007	151.298	4114.9143	5451.152	44,373.66308	96,211.7484	439,917.1	904,255
	16	78.0968	170.192	4819.2376	7734.832	423,181.6796	855,484.0617	3,974,097	7,923,352
200	8	41.396	75.528	3753.4344	4702.888	131.489335	200.906	1128.25	1552
	10	51.3975	74.72	4731.963	5372.66	983.114315	1249.2476	8074.95	10,178
	12	60.4698	83.544	5724.6174	7344.504	7362.903935	11,935.5246	59,416.4	92,948
	14	79.3401	132.79	6659.6999	9273.488	51,553.84987	90,554.1662	428,132.9	743,408
	16	119.2776	286.368	7647.4616	10,240.128	512,735.7826	926,534.6346	4,066,520	7,324,635
250	8	45.482	77.048	5022.452	5814.928	152.116975	240.403	1171.1	1956
	10	65.2595	124.33	6464.126	7764.13	1017.597095	1513.1619	7852.55	11,483
	12	76.275	148.284	8154.108	9426.096	8298.62114	14,924.36	59,375.75	103,418
	14	94.8668	205.324	10,188.8367	12,719.686	74,559.64571	111,269.9429	519,645.8	765,085
	16	102.8472	240.848	11,500.812	14,687.68	581,300.0625	985,967.6343	4,106,525.95	6,954,585

and

Table 4. CPU time (ms) of $condw$ for convex models.

n	m	UB-Convex		SA-Convex		$\mathit{BB}$		$\mathit{BB}$ Nodes
		Mean	Max	Mean	Max	Mean	Max	Mean	Max
50	8	14.0092	25.504	213.8704	309.072	47.2445	81.9703	915.4	1685
	10	16.63	25.85	279.4435	308.62	281.884485	554.474	5383.3	10,601
	12	25.0812	63.804	387.4434	528.768	1995.445185	3384.0362	31,876.75	50,135
	14	41.7207	162.218	574.0336	922.418	19,324.33216	44,301.9899	280,893.2	685,144
	16	45.4176	77.728	610.2368	924.064	154,054.2619	302,089.3153	2,334,002.25	4,609,124
100	8	18.1308	34.68	636.4244	1175.192	78.056675	107.3817	1127.75	1551
	10	22.104	32.38	778.073	980.88	501.678595	812.902	6724.45	10,779
	12	29.031	44.244	1042.2702	1313.892	3896.99359	6146.9746	49,441.55	76,893
	14	46.1069	125.636	1343.6241	1807.666	29,904.56491	53,424.1858	360,634.9	644,945
	16	53.2976	116.64	1558.0792	1997.168	303,743.1704	508,514.5931	3,382,350.4	5,727,921
150	8	22.1508	39.712	1158.6384	1499.072	87.604695	139.8171	1065.8	1569
	10	29.144	40.14	1554.333	1927.48	661.687145	1135.0919	7426.2	12,400
	12	50.1834	153.684	1999.341	2300.208	4928.357055	8079.5484	51,338.3	82,531
	14	54.6539	97.79	2491.9146	3623.158	38,734.83223	73,870.4459	393,856.6	756,958
	16	52.9984	85.2	2749.6112	3666.56	355,472.9039	542,691.2738	3,408,258.4	5,164,454
200	8	24.4656	36.848	1846.0644	2136.664	110.255325	172.0278	1148.4	1800
	10	37.2655	77.33	2361.4815	3878.1	798.17429	1045.0738	7808	10,457
	12	42.7518	85.608	2993.4126	4620.96	5872.14315	9883.743	56,480	93,448
	14	56.3367	173.11	3584.9611	4886.35	49,372.92988	81,875.8839	441,629.35	729,433
	16	80.056	153.856	4444.7072	5434.544	442,259.9251	826,716.0385	3,680,805.9	6,892,847
250	8	29.6284	84.144	2746.1744	4058.76	125.26891	211.019	1193.45	1956
	10	38.445	75.74	3660.476	5377.82	846.657065	1114.7384	7648.55	10,338
	12	59.6256	119.952	4462.869	5602.236	7011.789075	10,914.5426	58,498.6	90,152
	14	61.2129	186.466	5241.8205	6640.886	57,949.30358	77,964.78	459,999.1	619,674
	16	74.1832	147.392	6112.5728	7533.376	497,255.3724	864,228.0558	3,845,406.95	6,697,976

, it can be seen that $\widehat{BaB}-linear$ and $\widehat{BaB}-convex$ can still be solved 250 jobs and 16 groups within the time (i.e., ≤) 985967.6343 ms. For the heuristics, the $\widehat{UB}-linear$ and $\widehat{UB}-convex$ have the shortest CPU time than $\widehat{SA}-linear$ and $\widehat{SA}-convex$. For the error of heuristics, from

Table 5. Error of $condw$.

n	m	UB-Linear		UB-Convex		SA-Linear		SA-Convex
		Mean	Max	Mean	Max	Mean	Max	Mean	Max
50	8	0.006997	0.030574	0.001819	0.019022	0.000290	0.003236	0.001670	0.017365
	10	0.007024	0.023686	0.000913	0.018251	0.001356	0.006647	0.000849	0.016983
	12	0.014013	0.048737	0.002965	0.022467	0.001727	0.007394	0.002501	0.019864
	14	0.017983	0.045223	0.007845	0.037220	0.002017	0.008469	0.006726	0.029332
	16	0.016463	0.038894	0.010157	0.030139	0.003160	0.009910	0.009238	0.030139
100	8	0.005491	0.029006	0.000000	0.000000	0.000492	0.003043	0.000000	0.000000
	10	0.007255	0.026654	0.001207	0.015567	0.000204	0.000813	0.000699	0.007010
	12	0.009145	0.035949	0.001432	0.016744	0.000650	0.003516	0.001016	0.011891
	14	0.007649	0.030009	0.004740	0.025903	0.000679	0.002896	0.004395	0.023051
	16	0.008909	0.031258	0.004135	0.028901	0.000650	0.001833	0.003608	0.020566
150	8	0.001950	0.012827	0.000000	0.000000	0.000326	0.002434	0.000000	0.000000
	10	0.003657	0.026714	0.000000	0.000000	0.000282	0.001257	0.000000	0.000000
	12	0.008587	0.021467	0.000000	0.000000	0.000412	0.001509	0.000000	0.000000
	14	0.008086	0.029781	0.001682	0.015398	0.000559	0.001255	0.001531	0.015062
	16	0.009259	0.030160	0.002680	0.016431	0.000458	0.001397	0.002430	0.016431
200	8	0.001939	0.016315	0.000000	0.000000	0.000240	0.001874	0.000000	0.000000
	10	0.004639	0.020718	0.000000	0.000000	0.000280	0.000835	0.000000	0.000000
	12	0.005200	0.020197	0.000000	0.000000	0.000238	0.001079	0.000000	0.000000
	14	0.005225	0.016524	0.000741	0.014817	0.000324	0.001547	0.000741	0.014817
	16	0.007382	0.017377	0.000336	0.006712	0.000356	0.001095	0.000336	0.006712
250	8	0.002052	0.008562	0.000000	0.000000	0.000090	0.000503	0.000000	0.000000
	10	0.002446	0.015276	0.000000	0.000000	0.000168	0.001366	0.000000	0.000000
	12	0.003926	0.030186	0.000000	0.000000	0.000241	0.001338	0.000000	0.000000
	14	0.007543	0.016428	0.000000	0.000000	0.000328	0.001399	0.000000	0.000000
	16	0.005247	0.016696	0.000000	0.000000	0.000275	0.000890	0.000000	0.000000

, it can be seen that the maximum error of all heuristics (i.e., $\widehat{UB}-linear,\widehat{UB}-convex,\widehat{SA}-linear,\widehat{SA}-convex$ ) is $0.048737\%$. From Table 5, we also obtain that the average error of $\widehat{SA}-linear$ and $\widehat{SA}-convex$ outperforms $\widehat{UB}-linear$ and $\widehat{UB}-convex$.

A comprehensive analysis of these tables yields the following main conclusions regarding algorithm performance: (1) Efficiency vs. Optimality: The exact $\widehat{BaB}$ algorithm guarantees optimality but exhibits rapidly increasing computation time with problem scale, confirming the problem’s inherent complexity. In contrast, both heuristics provide solutions within fractions of a second. (2) Quality of Heuristics: As noted, the $\widehat{SA}$ algorithm consistently delivers lower average errors than $\widehat{UB}$, demonstrating its superior search capability. The superior accuracy of $\widehat{SA}$ stems directly from its iterative refinement mechanism, which allows it to escape local optima and explore a broader solution space than the single-pass, rule-based $\widehat{UB}$. (3) Practical Utility: The results delineate a clear trade-off: $\widehat{UB}$ is the fastest heuristic for a quick, high-quality solution, while $\widehat{SA}$ offers the best balance, providing near-optimal solutions with only a modest increase in runtime.

5. Summary and Future Research

This paper considered $gt$ scheduling with the $cpt$ and $condw$. Under the $linear$ and $convex$ models, the goal of optimization is to minimize

$$\sum _{a=1}^m\underset{1\le b\le n_a}{max}\left\{max\left\{{\kappa }_{a1}{E}_{a,b},{\kappa }_{a2}{T}_{a,b}\right\}+{\kappa }_{a3}{d}_1^a+{\kappa }_{a4}{D}_a\right\}+\sum _{a=1}^m\sum _{b=1}^{n_a}{\tau }_{a,b}{u}_{a,b}.$$

To address this NP-hard optimization problem, we proposed a suite of solution methodologies. A dedicated branch-and-bound (B&B) algorithm was developed, incorporating newly designed lower bounds ($L{B}_{\mathrm{linear}}$, $L{B}_{\mathrm{convex}}$) and heuristic-based upper bounds to prune the search space efficiently. Complementing the exact algorithm, effective heuristic methods, including constructive rules and a simulated annealing metaheuristic, were provided for obtaining high-quality solutions within short computational times. Numerical experiments conducted on a wide range of test instances demonstrated the efficacy of our approaches. The proposed method provides production planners with a practical tool for generating near-optimal schedules in real time, which is crucial for dynamic manufacturing scenarios such as aerospace component machining.

Future research could explore several promising extensions of this work: (1) flow shop scheduling with group technology and controllable processing times, including Sun et al. [67], Marquez and Ribeiro [68], Liu and Shi [69], Sotskov et al. [70], and Fasihi et al. [71]; (2) group scheduling with deterioration effects, as studied by He et al. [72], Sun et al. [73], and Pei et al. [74]; (3) group scheduling with learning effects, investigated by Jiang et al. [75], Zhao [76], and Liu and Wang [77].