Distributed Energy-Efficient Assembly Scheduling Problem with Transportation Capacity

Abstract

The real-life assembly production often has transportation between fabrication and assembly, and the capacity of transportation machine is often considered; however, the previous works are mainly about two-stage distributed assembly scheduling problems. In this study, a distributed energy-efficient assembly scheduling problem (DEASP) with transportation capacity is investigated, in which dedicated parallel machines with symmetry under the given conditions, transportation machines and an assembly machine are used. An adaptive imperialist competitive algorithm (AICA) is proposed to minimize makespan and total energy consumption. A heuristic and an energy-saving rule are used to produce initial solutions. An adaptive assimilation with adaptive global search and an adaptive revolution are implemented, in which neighborhood structures are chosen dynamically, and revolution probability and search times are decided by using the solution quality. The features of the problem are also used effectively. Computational experiments are conducted on a number of instances. The computational results demonstrate that the new strategies of AICA are effective and efficient and AICA can provide promising results for the considered DEASP.

1. Introduction

In recent years, single factory or centralized production has been continuously replaced by multi-factory production or distributed manufacturing with the further development of globalization. Distributed manufacturing enables manufacturers to be closer to their customers and suppliers, to produce and market their products more effectively, to respond to market changes more quickly, to achieve better product quality, lower production cost and reduced management risk. As an important part of manufacturing systems, scheduling is shifted from single factory scheduling to distributed scheduling with the change of mode of production. Distributed scheduling problems have received much attention in the past decade [1,2,3,4,5].

Assembly scheduling problem (ASP) is an effective way to balance batch production and production flexibility, and has attracted much attention after the pioneering works of Lee et al. [6] and Potts et al. [7], along with a number of related works that have been obtained. In recent years, Framinan et al. [8] described ASP as ${\alpha }_1\to {\alpha }_2\left|\beta \right|\gamma$, where ${\alpha }_1$ and ${\alpha }_2$ indicate machine environments of the fabrication stage and the assembly stage, respectively, and provided a full review of the previous works and the future topics. Komaki et al. [9] implemented a consolidated survey of ASP and proposed salient research opportunities. Assembly production systems are also shifted from single factory to multiple factories in recent years, and the distributed assembly scheduling problem (DASP) has been considered extensively.

As stated by Framinan et al. [8], two-stage ASP with $DPm\to 1$ layout, in which m dedicated parallel machines are for fabrication and one assembly machine is for assembly, has been extensively considered, and various methods including exact method, heuristic and meta-heuristic have been successfully used [10,11,12,13,14]. Meta-heuristics have become the main method because meta-heuristic performs better than an exact algorithm on large-scale ASP and often can produce better results than heuristics.

Allahverdi and Al-Anzi [15] presented tabu search (TS) and particle swarm optimization (PSO) for the problem for the first time. Shokrollahpour et al. [16]) proposed an imperialist competitive algorithm (ICA) for a bi-objective problem. Navaei et al. [17] handled the problem with sequence-dependent setup time (SDST) at two stages and developed four hybrid meta-heuristics with ICA, simulated annealing (SA) and heuristic. In the hybrid algorithm, SA and ICA are presented to find a sequence of jobs at the first stage, and a heuristic and SA are developed to assign addressed jobs to assembly machines at the second stage. Kazemi et al. [18] studied the problem with batching and delivery by a ICA and a hybrid ICA with the dominance relations. For two-stage ASP with $DP2\to 1$ layout, Wu et al. [11] developed some dominance propositions with a lower bound used in a branch-and-bound algorithm and presented six versions of hybrid PSO algorithms to find approximate solutions, they also presented four meta-heuristic including genetic algorithm (GA), cloud theory-based SA and iterated greedy [12], and Wang et al. [19] provided some dominance rules and opposition-based PSO to solve the problem with a cumulated learning effect.

Unlike two-stage ASP with $DPm\to 1$ layout, three-stage ASP with the above layout is not considered fully, in which the transportation stage is added between the fabrication stage and assembly stage. Hatami et al. [20] presented a mathematical model and two meta-heuristics, namely TS and SA, for the problem with SDST. Shoaardebili and Fattahi [21] developed two multi-objective meta-heuristics based on SA and GA for the problem with SDST and machine availability. Komaki et al. [22] proposed an improved discrete cuckoo optimization algorithm, which incorporates novel features such as clustering, egg laying and immigration of the cuckoos based on a discrete representation scheme.

As a typical distributed scheduling problem, DASP in multiple factories is considered fully. For DASP with F factories, each of which has $DPm\to 1$ layout, Xiong et al. [23] developed some hybrid algorithms named hybrid GA with reduced variable neighborhood search and a hybrid discrete differential evolution with reduced variable neighborhood search to minimize total completion time. Deng et al. [24] presented a mixed integer linear programming model and a competitive memetic algorithm based on a ring-based neighborhood structure and some knowledge-based local search operators to minimize makespan. Zhang and Xing [25] proposed a memetic social spider optimization algorithm with two improvement techniques, problem-special local search and self-adaptive restart strategy to minimize total completion time. Lei et al. [26] developed a cooperated teaching–learning-based optimization algorithm with class cooperation to minimize the makespan. A DASP with F processing factories and an assembly factory has also been investigated [27,28,29,30,31,32,33]; however, the above works mainly deal with two stages: the fabrication and assembly stages, and seldom focus on a DASP with three stages: fabrication, transportation and assembly.

As stated above, the previous works are mainly about two-stage ASP with $DPm\to 1$ layout. It is often assumed that the time for collecting and transporting components to the assembly stage is negligible. However, in many realistic situations, the collecting and transporting time is considerably large and cannot be ignored, in this case, transportation as a middle stage should be added between the fabrication stage and the assembly stage; moreover, the neglecting of transportation results in more loss in multiple factories. In the existing works [22], transportation time is frequently adopted and the capacity of the transportation machine is hardly included; on the other hand, although energy-efficient scheduling has been well-studied in various environments, it is hardly investigated in an assembly production environment. Generally, the energy efficiency of multiple factories is more important than that of a single factory, so it is necessary to deal with a distributed energy-efficient assembly scheduling problem (DEASP) with three stages.

ICA is inspired by the sociopolitical behaviors and has a good neighborhood search ability, effective global search property, good convergence rate, minor chance of trapping into local optima and flexible structure [34]. It has been used to solve various production scheduling problems [17,18,35,36,37,38,39,40,41,42]. Regarding distributed scheduling, Marandi and Fatemi Ghomi [43] studied distributed scheduling together with a vehicle routing problem and developed an improved ICA after a mathematical model was provided. Lei et al. [41] proposed a ICA with memory and new revolution for distributed unrelated parallel machine scheduling. Li et al. [44] presented a ICA with empire cooperation for solving a fuzzy DASP with $DPm\to 1$ layout. Its search ability and advantages are tested and proved. ICA is used to solve ASP; however, to the best of our knowledge, it is not applied to deal with a DEASP. The above features and extensive applications of ICA to scheduling problems motivate us to cope with the DEASP by ICA.

In this study, a DEASP with three stages is considered, in which assignment of the transportation machine is done based on capacity. An adaptive imperialist competitive algorithm (AICA) is proposed to minimize the makespan and total energy consumption simultaneously. A heuristic and an energy-saving rule are used to produce initial solutions. An adaptive assimilation with an adaptive global search is implemented. An adaptive revolution is given, in which neighborhood structures are chosen dynamically, and revolution probability and search times are decided based on solution quality. A number of experiments are conducted. The computational results demonstrate that the new strategies of AICA are effective and efficient and AICA can provide promising results for the considered DEASP.

The remainder of the paper is organized as follows. Problem description is given in Section 2 followed by an introduction to ICA in Section 3. Section 4 shows the proposed AICA for the problem. Numerical experiments on AICA are reported in Section 5, the conclusions are summarized in the final section and some topics of future research are provided.

2. Problem Description

A DEASP with three stages is described below. n independent jobs $J_1,J_2,\cdots ,J_n$ are produced in F homogeneous factories with different sites. Each factory f has m dedicated parallel machines $M_{f1},M_{f2},\cdots ,M_{fm}$ for fabrication, h identical transportation machines $T{M}_{f1},T{M}_{f2},\cdots ,T{M}_{fh}$ and an assembly machine $A{M}_f$. Each $T{M}_{fl}$ has capacity Q and is used to move components of one job at a time. There is a set V of d different processing speeds for machines $M_{fk}$ and $A{M}_f$. $V=\left\{v_1,v_2,\cdots ,v_d\right\}$.

It is often assumed that energy consumption increases and processing time decreases when a job is processed on a machine at a higher speed [45]. This assumption is also used on assembly time. Machines $M_{fk}$ and $A{M}_f$ have two modes: working mode and stand-by mode. $E{F}_{fk}$($E_f$) is the energy consumption per unit time in working mode and $SE{F}_{fk}$($S{E}_f$) is the energy consumption per unit idle time on machine $M_{fk}$($A{M}_f$). $ET$ indicates energy consumption for a back and forth trip of a transportation machine and $TT$ is time of a back and forth trip.

Each job $J_i$ has m components $co{m}_{i1},co{m}_{i2},\cdots ,co{m}_{im}$. $w_{ik}$ is the weight of $co{m}_{ik}$. These components are first processed on machines $M_{f1},M_{f2},\cdots ,M_{fm}$ at the fabrication stage, then they are collected and transferred by transportation machine $T{M}_{fl}$ to the assembly machine $A{M}_f$ and assembled together, and then the job is obtained. Each job $J_i$ has the same processing and assembly times in each factory. $p_{ik}$ and $a{s}_i$ indicate processing time on machine $M_{fk}$ and assembly time for $J_i$. $st{p}_{ik}$ and $st{a}_i$ are setup times for fabrication of $co{m}_{ik}$ and assembly of $J_i$. When $T{M}_{fl}$ transports the components of a job, the total weight of the components cannot exceed Q.

The following constraints are considered.

• All machines are available at time 0.
• Each machine can fabricate, transport or assemble at most one job at a time.
• Each job can be fabricated, transported or assembled at most one machine at a time.
• $T{M}_{fl}$ is used to move components of at most one job at a time.

The problem consists of a factory assignment sub-problem, speed selection used to decide speed of machines for fabrication and assembly, and scheduling sub-problem. There are strong coupled relations among them. The factory assignment notably affects results of the scheduling sub-problem, and optimal solutions can be obtained after the solutions of three sub-problems are effectively combined together.

The scheduling sub-problem in a factory is a three-stage ASP with $DPm\to 1$ layout, the time complexity of which is $O(n!)$. Obviously, the time complexity of DEASP is notably greater than that of ASP because of factory assignment, so the problem is NP-hard because the ASP is NP-hard [8]. On the other hand, in the general parallel machine scheduling with makespan, symmetry exists, that is, two jobs on the same machines are exchanged, and makespan is not changed. In a DEASP with $DPm\to 1$ layout, symmetry on the fabrication stage is still kept when the maximum completion times of jobs $J_i$ and $J_j$ at the fabrication stage are not changed, that is, the last fabricated components of two jobs are not swapped with any other components, for $J_i$ and $J_j$, their all components except the last one on the same parallel machine still have symmetry.

A DEASP with three stages is a multi-objective optimization problem with makespan and total energy consumption and all the above described constraints.

$$C_{max}=\underset{i=1,2,\cdots ,n}{max}\left\{C_i\right\}$$

(1)

$$TEC=TE{C}_{fab}+TE{C}_{tr}+TE{C}_{ass}$$

(2)

where $C_i$ is completion time of job $J_i$, $C_{max}$ denotes maximum completion time of all jobs, $TE{C}_{fab}$, $TE{C}_{tr}$ and $TE{C}_{ass}$ are total energy consumption at fabrication stage, transportation stage and assembly stage, respectively.

$$TE{C}_{fab}={\sum }_{f=1}^F{\sum }_{k=1}^m{\sum }_{u=1}^d{\int }_0^{C_{max}}E{F}_{fk}\times y_{fku}\left(t\right)dt+{\sum }_{f=1}^F{\sum }_{k=1}^m{\int }_0^{C_{max}}SE{F}_{fk}\times z_{fk}\left(t\right)dt$$

(3)

$$TE{C}_{tr}=ET\times {\sum }_{f=1}^F{\sum }_{l=1}^{h}b{f}_{fl}$$

(4)

$$TE{C}_{ass}={\sum }_{f=1}^F{\sum }_{u=1}^d{\int }_0^{C_{max}}{E}_f\times {\overline{y}}_{fu}\left(t\right)dt+{\sum }_{f=1}^F{\int }_0^{C_{max}}S{E}_f\times {\overline{z}}_f\left(t\right)dt$$

(5)

where $b{f}_{fl}$ indicates the total number of back and forth trips of $T{M}_{fl}$, $y_{fku}\left(t\right)$ is 1 if $M_{fk}$ works at speed $v_u$ at time t and 0 otherwise. $z_{fk}$ is 1 if $M_{fk}$ is idle at time t and 0 otherwise, ${\overline{y}}_{fu}\left(t\right)$ is 1 if $A{M}_f$ works at speed $v_u$ at time t and 0 otherwise, ${\overline{z}}_f$ is 1 if $A{M}_f$ is idle at time t and 0 otherwise.

Table 1. Processing data of the example.

i	1	2	3	4	5	6	7	8
$p_{i1}$	61.56	58.61	23.30	56.36	64.58	37.37	22.74	90.20
$p_{i2}$	58.90	49.61	50.06	84.99	77.50	69.30	56.73	75.53
$p_{i3}$	21.51	83.73	84.69	58.06	55.76	86.40	61.14	98.92
$p_{i4}$	47.17	41.24	70.47	99.91	90.04	62.05	40.12	50.00
$st{p}_{i1}$	9.81	7.66	9.02	6.23	5.18	8.32	6.94	8.29
$st{p}_{i2}$	7.75	7.80	8.92	6.35	6.41	6.57	8.71	5.16
$st{p}_{i3}$	6.01	7.93	6.73	9.50	5.48	8.89	8.06	9.34
$st{p}_{i4}$	6.45	8.21	6.95	9.30	6.66	5.19	7.01	5.63
$a{s}_i$	72.38	43.00	49.39	93.68	87.92	48.20	28.69	63.05
$st{a}_i$	6.57	7.39	9.61	6.88	7.35	5.12	7.07	9.29
$w_{i1}$	19.48	21.35	46.09	21.80	27.48	35.27	24.34	39.23
$w_{i2}$	35.86	45.30	10.27	27.66	11.57	32.76	38.75	49.96
$w_{i3}$	45.36	43.14	20.21	43.95	41.37	30.99	28.85	15.00
$w_{i4}$	32.00	37.84	43.11	27.07	16.10	34.63	18.45	23.44

shows an example with 2 factories, 8 jobs and 4 components of each. $Q=100$, $V=\left\{1,1.3,1.55,1.75,2.10\right\}$, data on energy consumption and transportation are shown in Section 5. A schedule of the example is shown in Figure 1. In this Figure, take job $J_8$ as an example, 8 on $M_{1k}$ denotes the k-th component of $J_8$, 8 on $T{M}_{1h}$ denotes the movement of components by $T{M}_{1h}$, 8 on $A{M}_f$ indicates that $J_8$ is assembled on $A{M}_f$. $C_1=266.99$, $C_2=302.68$, $C_8=163.74$, so $C_{max}=302.68$, $TE{C}_{fab}=9549.91$, $TE{C}_{tr}=240$, $TE{C}_{ass}=3092.39$, as a result, $TEC=12882.3$.

3. Introduction to ICA

In ICA, a country represents a solution of the problem, and solutions in population P are categorized into two parts: imperialists and colonies, the former are some of the best solutions in P and the latter are all solutions of P except imperialists.

ICA is shown below.

(1)

Initialization. Randomly produce an initial population P and calculate the cost of each solution in P.

(2)

Initial empires. Choose $N_{im}$ solutions with the smallest cost as imperialists, calculate the normalized cost ${\overline{c}}_k$ and the number $N{C}_k$ of colonies and randomly allocate $N{C}_k$ colonies for each imperialist k.

(3)

Assimilation. In each empire, each colony moves toward its imperialist and is replaced with the newly generated solution if possible.

(4)

Revolution. Perform revolution according to revolution probability $U_R$.

(5)

Exchange. In each empire, compare each colony with its imperialist and replace the imperialist with the colony with a smaller cost than its imperialist.

(6)

Imperialist competition. Calculate total cost $T{C}_k$, normalized total cost ${\overline{TC}}_k$ and power $PO{W}_k$ for each empire k, construct the following vector $[PO{W}_1-rand,PO{W}_2-rand,\cdots ,PO{W}_{N_{im}}-rand]$, decide an empire g with the biggest $PO{W}_g-r_g$ and allocate the weakest colony of the weakest empire into empire g.

(7)

If the termination condition is not met, go to (3); otherwise, stop the search.

Where $rand$ is a random number following uniform distribution in [0,1].

With respect to cost, the smaller the cost of a solution is, the better the solution is. When a new solution is better than a colony in assimilation and revolution, the colony is replaced with the new solution; otherwise, the colony is kept invariant. ${\overline{c}}_k$, $N{C}_k$, $T{C}_k$, ${\overline{TC}}_k$ and $PO{W}_k$ are defined by Hosseini and Khaled [34].

When ICA is used to solve production scheduling, the equation used in assimilation is hard to produce feasible solutions, so ICA is often discretized [38,39,40,41].

4. AICA for DEASP with Three Stages

To solve DEASP, a heuristic and an energy-saving rule are used to produce initial solutions, and adaptive search mechanisms are added into assimilation and revolution. These steps of AICA are described in the following.

4.1. Coding and Decoding

Lei et al. [26] proposed a representation method for DASP with $DPm\to 1$ layout. Based on this representation, a coding method is adopted to describe solution of DEASP in this study.

For DEASP with three stages, an assignment and scheduling string $[{\pi }_1,\cdots ,{\pi }_{q_1},{\delta }_1,{\pi }_{q_1+1},\cdots ,{\pi }_{q_1+q_2},{\delta }_2,\cdots ,{\delta }_{F-1},{\pi }_{H_F+1},\cdots ,{\pi }_n]$ and a speed selection string $[{\theta }_{11},\cdots ,{\theta }_{1m},{\theta }_{1(m+1)},\cdots ,{\theta }_{n1},\cdots ,{\theta }_{nm},{\theta }_{n(m+1)}]$ are used, where $q_f$ is the number of all allocated jobs ${\pi }_{H_f+1},\cdots ,{\pi }_{H_f+q_f}$ in factory f, ${\delta }_g\in \left\{n+1,n+2,\cdots ,n+F-1\right\}$, $H_1=0$, $H_f={\sum }_{l=1}^{f-1}{q}_l$, $f>1$, ${\theta }_{ik},k\le m$ is the chosen speed of $M_{fk}$ used to fabricate the k-th component of $J_i$. ${\theta }_{i(m+1)}$ is the speed of assembly machine $A{M}_f$ for $J_i$.

The decoding procedure is shown below. The first string is divided into F segments, for segment ${\pi }_{H_f+1},\cdots ,{\pi }_{H_f+q_f}$ of factory f, start with job $J_{{\pi }_{H_f+1}}$, for each job ${\pi }_g$, assign its m components on m machines at fabrication stage sequentially and deal with its fabrication on $M_{fk}$ with speed $v_{{\theta }_{{\pi }_{g}k}}$, then transfer them to assembly machine by $T{M}_f$, and complete assembly on assembly machine $A{M}_f$ on speed $v_{{\theta }_{{\pi }_g(m+1)}}$, where $g\ge (H_f+1)$.

About the allocation of transportation machine, for job $J_{{\pi }_g}$ allocated in factory f, (1) choose transportation machine $T{M}_{fl}$ with earliest available time ${\eta }_{fl{\pi }_g}$ for $J_{{\pi }_g}$. If more than one transportation machines with the same ${\eta }_{fl{\pi }_g}$, a transportation machine with the smallest subscript is chosen. If more than one $T{M}_{fl}$ is required, the above steps are repeated until all required transportation machines are decided; (2) at the fabrication stage, each fabricated component of $J_{{\pi }_g}$ is loaded into $T{M}_l$ according to first completion first transportation rule; if the sum of weight of all loaded components exceeds Q when a component is loaded, it is added into another transportation machine.

For the example in Table 1, a solution is $[8,4,1,2,9,5,6,3,7]$ and $[2,2,2,1,5,1,2,1,1,3,1,1,2,3,2,2,2,1,1,4,1,1,1,1,1,1,2,2,1,5,1,1,2,1,5,2,1,3,3,1]$ and the obtained schedule is shown in Figure 1. In Figure 1, $J_8$ needs two machines $T{M}_{11}$ and $T{M}_{12}$, which are selected sequentially. Components 1, 3 and 4 of $J_8$ are loaded into $T{M}_{11}$, when component 2 is added, the capacity is exceeded, so component 2 is moved by $T{M}_{12}$; moreover, $T{M}_{12}$ is just used to transfer component 2 and does not move the components of other jobs.

4.2. Initialization and Initial Empires

Heuristic 1 is shown below. Suppose that all jobs processed on a two-machine flow shop, the processing time of $J_i$ on the first machine is $\underset{k=1,2,\cdots ,m}{max}\left\{p_{ik}\right\}$ and the processing time on the second machine is $a{s}_i$, then apply NEH and obtain a permutation of all jobs.

Heuristic 2 is described as follows. A job permutation is first obtained by using heuristic 1, then start with the first job on the permutation, for each job, decide a new factory and a new position on the assignment and scheduling string that can lead to the smallest makespan.

An energy-saving rule is used. For each factory $f=1,2,\cdots ,F$, decide the smallest speed of each fabrication machine and assembly machine for each job under the condition that the makespan is kept invariant.

N initial solutions are produced. For solution $x_1$, randomly produce a speed selection string, and then apply heuristic 2 and the energy-saving rule sequentially. For solution $x_2$, each ${\theta }_{ij}$ is first set to be $v_d$, then heuristic 2 is used to produce an assignment and scheduling string and energy-saving rule is applied to adjust speeds of fabrication and assembly. For solution $x_3$, produce speed selection string by letting each ${\theta }_{ij}$ be $v_1$ and construct an assignment and scheduling string by using heuristic 2. Other $N-3$ initial solutions are randomly produced.

After population P with N initial solutions is produced, non-dominated sorting and crowding distance computation are done according to Deb et al. [46]; then, for each solution $x_i\in P$, its rank value $ran{k}_i$ and crowding distance $dis{t}_i$ are obtained, and its normalized cost ${\overline{c}}_i$ is defined; $N_{im}$ solutions with the biggest ${\overline{c}}_i$ are chosen as imperialists and other solutions are colonies; finally, compute $po{w}_k$ and $N{C}_k$ for each imperialist k and randomly select $N{C}_k$ solutions as colonies of imperialist k.

$${\overline{c}}_i=\underset{j=1,2,\cdots ,N}{max}\left\{ran{k}_j\right\}-ran{k}_i+dis{t}_i/\phantom{dis{t}_i\underset{j=1,2,\cdots ,N}{max}\left\{dis{t}_j\right\}}\underset{all{x}_{j}ran{k}_j=ran{k}_i}{max}\left\{dis{t}_j\right\}$$

(6)

$$po{w}_k={\overline{c}}_k/\phantom{{\overline{c}}_k{\sum }_{l\in Imp}{\overline{c}}_l}{\sum }_{l\in Imp}{\overline{c}}_l$$

(7)

$$N{C}_k=round(po{w}_k\times N_{col})$$

(8)

where $dis{t}_i$ indicates crowding distance of $x_i$, $dis{t}_i$ is set to 2 for boundary solution, $ran{k}_i$ is rank value of $x_i$, the set $Imp$ is the set of all imperialists, $N_{col}=N-N_{im}$, $round\left(x\right)$ is a function that gives the nearest integer of x, $po{w}_k$ indicates power of imperialist k.

$N_{im}$ initial empires are produced. The normalized total cost ${\overline{TC}}_k$ of empire k is defined by

$${\overline{TC}}_k={\overline{c}}_k+\zeta {\sum }_{x_i\in E_k}{\overline{c}}_i/\phantom{\zeta {\sum }_{x_i\in E_k}{\overline{c}}_{i}N{C}_k}N{C}_k$$

(9)

where $\zeta$ is a real number and often set to be 0.1, $E_k$ indicates the set of all colonies possessed by imperialist k.

4.3. Adaptive Assimilation

Assimilation is often executed by making each colony move toward its imperialist, and the movement is implemented by global search operator between colony and its imperialist [40,47]. In this study, an adaptive assimilation is proposed, in which assimilation probability $U_A^{x_i}$ of $x_i$ is computed by

$$U_A^{x_i}=\left\{\begin{array}{cc}\hfill 1\hfill & ifran{k}_i>1\hfill \\ 1-dis{t}_i/\phantom{dis{t}_i\underset{forall{x}_{j}withran{k}_j=ran{k}_i}{max}\left\{dis{t}_j\right\}}\underset{forall{x}_{j}withran{k}_j=ran{k}_i}{max}\left\{dis{t}_j\right\}\hfill & otherwise\hfill \end{array}\right.$$

(10)

The detailed steps of adaptive assimilation are shown below.

(1)

Let $k=1$, decide the weakest empire g with the smallest ${\overline{TC}}_k$.

(2)

For each colony $x_i\in E_k$, if random number $rand<U_A^{x_i}$, then if $k=g$ and rank value of its imperialist is greater than 1, then randomly select an imperialist l with rank value of 1; otherwise, directly use imperialist k; if $rand<S{m}_{x_i}/(S{m}_{x_i}+S{e}_{x_i})$, then execute order crossover (OX) between $x_i$ and the chosen imperialist; otherwise, implement a two-point crossover between $x_i$ and the selected imperialist, a new solution z is obtained, if z is not dominated by $x_i$, z substitutes for $x_i$ and update external archive $\mathrm{\Omega }$.

(3)

$k=k+1$, if $k<N_{im}$, go to step (2); otherwise, stop the assimilation.

Where $S{m}_{x_i}$ is obtained by sorting all solutions in P in the ascending order of makespan and $S{e}_{x_i}$ is computed by sorting all solutions in the ascending order of $TEC$. Archive $\mathrm{\Omega }$ is used to store non-dominated solutions produced by AICA.

OX [24] acts on assignment and scheduling string, and two-point crossover is performed on a speed selection string.

When $x_i$ is used to update $\mathrm{\Omega }$, $x_i$ is included into $\mathrm{\Omega }$, all solutions of $\mathrm{\Omega }$ are compared and all dominated solutions are removed from $\mathrm{\Omega }$.

In the above assimilation, colony $x_i\in E_k$ is given $U_A^{x_i}$ according to its rank value, the colony of the weakest empire may choose the imperialist of other empires, and OX and two-point crossover are chosen by $S{m}_{x_i}/(S{m}_{x_i}+S{e}_{x_i})$. If $S{m}_{x_i}=S{e}_{x_i}$, two crossovers are selected with the same probability, if $S{m}_{x_i}<(>)S{e}_{x_i}$, then the speed selection string (assignment and scheduling string) is given higher priority than the assignment and scheduling string (speed selection string) to diminish $TEC$($C_{max}$). The above selection on two crossovers is done because of the following features of the problem: the makespan is mainly decided by the assignment and scheduling string and $TEC$ has a high correlation with the speed selection string.

These adaptive mechanisms are seldom adopted in the previous ICAs [38,39,47]. They can result in good exploration ability and high search efficiency because of the usage of adaptive $U_A^{x_i}$ and the above features.

4.4. Adaptive Revolution

Revolution is about an unexpected change in characteristics of some colonies. In general, a colony for revolution is chosen according to revolution probability, and the revolution of the chosen colony is implemented by using neighborhood search [40]. In this study, an adaptive revolution is presented, in which revolution probability $U_R^{x_i}$ and search times R are defined adaptively and an adaptive multiple neighborhood search is proposed based on eight neighborhood structures.

$$U_R^{x_i}=\left\{\begin{array}{cc}\hfill 0.1\hfill & \hfill ran{k}_i>5\hfill \\ \hfill 1-0.1\times ran{k}_i\hfill & 1<ran{k}_i\le 5\hfill \\ \hfill 1\hfill & \hfill ran{k}_i=1\hfill \end{array}\right.$$

(11)

Revolution probability is often set to be 0.1 in many ICAs. When $ran{k}_i$ of $x_i$ is greater than 5, $x_i$ has bad convergence performance and low solution quality, its $U_R^{x_i}=0.1$ and $U_A^{x_i}=1$, these settings are used to make $x_i$ execute global search mainly; on the contrary, when $ran{k}_i$ of $x_i$ is 1, lower $U_A^{x_i}$ and higher $U_R^{x_i}$ are also applied to make $x_i$ execute a neighborhood search mainly, thus, these adaptive settings on $U_A^{x_i}$ and $U_R^{x_i}$ can effectively make a good balance between exploration and exploitation. Adaptive revolution is shown below. In each empire $k=1,2,\cdots ,N_{im}$, for each colony $x_i\in E_k$, if random number $rand<U_R^{x_i}$, then execute an adaptive multiple neighborhood search.

An adaptive multiple neighborhood search of $x_i$ is described as follows.

(1)

Decide factory $fa{c}_1$ with biggest completion time and factory $fa{c}_2$ with biggest energy consumption

(2)

Let $l=1$, repeat the following steps until $l>R$:

If$fa{c}_1=fa{c}_2$, then $g=1$, repeat the following steps until $g>3$: produce $z\in {\mathcal{N}}_g\left(x\right)$; $l=l+1$, if z is not dominated by $x_i$, then replace $x_i$ with z and update $\mathrm{\Omega }$; otherwise, $g=g+1$.

Otherwise, if$rand<Sm\left(x_i\right)/(Sm\left(x_i\right)+Se\left(x_i\right)$, then define $a_1=4,a_2=5,a_3=1$, $a_4=2$, let $g=1$, repeat the following steps until $g>4$: produce $z\in {\mathcal{N}}_{a_g}\left(x_i\right)$, if z is not dominated by $x_i$, then replace $x_i$ with z and renew $\mathrm{\Omega }$; otherwise, $g=g+1$.

Otherwise, let $g=6$, repeat the following steps until $g>8$: generate $z\in {\mathcal{N}}_g\left(x_i\right)$, if z is not dominated by $x_i$, then z substitutes for $x_i$ and is used to renew $\mathrm{\Omega }$; otherwise, $g=g+1$.

$$R=\left\{\begin{array}{cc}\hfill \alpha -\beta \hfill & \hfill ran{k}_i>5\hfill \\ \hfill \alpha \hfill & \hfill 1<ran{k}_i\le 5\hfill \\ \hfill \alpha +\beta \hfill & \hfill ran{k}_i=1\hfill \end{array}\right.$$

(12)

where $\alpha ,\beta$ are integer, $a_1,a_2,a_3,a_4$ are defined to decide a using sequence of ${\mathcal{N}}_{1-4}$.

${\mathcal{N}}_1$ is applied to produce new solutions by removing a randomly chosen job from factory $fa{c}_1$ to a randomly chosen factory $f\ne fa{c}_1$. In ${\mathcal{N}}_2$, a factory $f\ne fa{c}_1$ is randomly selected and a job from factory $fa{c}_1$ and a job randomly chosen from factory f are swapped. ${\mathcal{N}}_3$ is shown below. Two randomly selected jobs from factory $fa{c}_1$ are exchanged.

Insertion and swap in ${\mathcal{N}}_1,{\mathcal{N}}_2,{\mathcal{N}}_3$ are done in the assignment and scheduling string, for example, job $J_6$ in factory 2 and $J_4$ in factory 1 are exchanged, then the assignment and scheduling string becomes $[8,6,1,2,9,5,4,3,7]$.

After assembly of job $J_i$ is finished on $A{M}_f$, if $A{M}_f$ is idle in a time interval, then $J_i$ is a type-II job; otherwise, $J_i$ is a type-I job. In Figure 1, job $J_5$ is type-I and $J_2$ is type-II.

${\mathcal{N}}_4$–${\mathcal{N}}_8$ make full use of the feature of the DEASP. ${\mathcal{N}}_4$ is done in the following way. In factory $fa{c}_1$, decide the last type-II job $J_i$, randomly decide a job $J_j$ after $J_i$, for the last fabricated component of $J_j$, if the fabrication speed is $v_l,l<d$, then the speed becomes $v_{l+1}$. ${\mathcal{N}}_5$ is similar to ${\mathcal{N}}_4$. In ${\mathcal{N}}_5$, $J_i,J_j$ are selected as done in ${\mathcal{N}}_4$, then if the assembly speed of $J_j$ is $v_l,l<d$, then the new speed is $v_{l+1}$. In Figure 1, $fa{c}_1=2$, $J_3$ is the last type-II job, $J_7$ is decided, the speed of the last fabricated component or the assembly speed increases, and makespan can diminish likely.

${\mathcal{N}}_6$ is shown below. Randomly select a type-II job that is not in factory $fa{c}_1$ and let its assembly speed $v_l,l>1$ diminish to $v_{l-1}$. ${\mathcal{N}}_7$ is used to reduce fabrication speed of the first fabricated component for a randomly chosen type-II job not in factory $fa{c}_1$. ${\mathcal{N}}_8$ is similar to ${\mathcal{N}}_7$. In ${\mathcal{N}}_8$, a type-I job $J_i$ is randomly selected and a job $J_j$ is considered, which meets the following condition: ${\pi }_{l+1}=j$ if ${\pi }_l=i$ on assignment and scheduling string, then the fabrication speed of the first fabricated component of $J_j$ is reduced as done in ${\mathcal{N}}_7$. When ${\mathcal{N}}_6$–${\mathcal{N}}_8$ are used, the reduction of speed can lead to smaller $TEC$ and a fixed makespan, as a result, a dominating solution z can be generated.

4.5. Algorithm Description

The detailed steps of AICA are shown below.

(1)

Initialization. Produce initial population P by using heuristics and random way.

(2)

Sort all solutions in the ascending order of makespan and $TEC$, respectively, and obtain $U_A^{x_i}$, $U_R^{x_i}$, $S{m}_{x_i}$ and $S{e}_{x_i}$ for each $x_i\in P$.

(3)

Execute adaptive assimilation.

(4)

Perform adaptive revolution and exchange imperialist and colony if possible in each empire.

(5)

Implement imperialist competition.

(6)

If the termination condition is not met, go to (2); otherwise, stop the search.

Flow chart of AICA is shown in Figure 2.

When the imperialist competition is done, the normalized total cost ${\overline{TC}}_k$ in Equation (9) is used to compute $PO{W}_k$, and the winning empire is decided and the weakest colony of the weakest empire is moved to the winning empire, as done in Section 3.

AICA has some different features from the previous ICAs [18,39,40,41,42]. Adaptive assimilation is implemented by using adaptive assimilation probability and adaptive global search based on the characteristics of the problem. In adaptive revolution, $U_R^{x_i}$ is decided by solution quality, and an adaptive multiple neighborhood search with adaptive parameter R is adopted; moreover, features of DEASP are also used in assimilation and revolution.

5. Computational Experiments

Extensive experiments are conducted to test the performance of AICA for the considered DEASP. All experiments are implemented by using Microsoft Visual Studio 2019 C++ and run on a desktop computer with i5-7200U CPU 2.50 GHz 8 GB RAM.

5.1. Instances, Comparative Algorithms and Metrics

Overall, 127 instances are used and denoted as $F\times n\times m$ [23]. The related data are shown as follows. $p_{ik},a{s}_i\in [20,100]$, $st{p}_{ik},st{a}_i\in [5,20]$, $w_{ig}\in [10,50]$, $Q=100$, $ET=16$, $TT=20$, $V=\left\{1.0,1.3,1.55,1.75,2.10\right\}$, $E{F}_{fl}=E_f=4\times v_l^2$, $SE{F}_{fl}=S{E}_f=1$, $h=3$.

Wang et al. [48] proposed a multi-objective whale swarm algorithm (MOWSA) to solve a distributed permutation flow shop scheduling problem with the objectives of makespan and energy consumption. Li et al. [49] handled energy-efficient scheduling of distributed permutation flow shop with total flow time and total energy consumption and presented an improved non-dominated sorting genetic algorithm-II (INSGA-II). In recent years, a competitive memetic algorithm (CMA) has been applied to solve distributed two-stage ASP [24], multi-objective distributed permutation flow shop scheduling [20] and carbon-efficient scheduling of distributed flow shop with makespan and carbon emissions [50]. In this study, CMA [51] is chosen to compare with AICA.

The above algorithms can be directly applied to solve our DEASP after fabrication, transportation and assembly are executed in the decoding procedure; moreover, these algorithms deal with two objectives: a time-related objective and an energy-related objective, so they are chosen as comparative algorithms.

ICA is obtained by deleting adaptive mechanisms from AICA, that is, the difference between AICA and ICA just lies in that $U_A^{x_i}=1$ and $U_R^{x_i}=0.1$. The comparisons between AICA and ICA are used to show the effect of adaptive mechanisms.

Metric $\mathcal{C}$ [52] is applied to compare the approximate Pareto optimal set, respectively, obtained by algorithms. $\mathcal{C}\left(L,B\right)$ measures the fraction of members of B that are dominated by members of L.

$$\mathcal{C}\left(L,B\right)=\frac{\left|\left\{b\in B:\exists h\in L,h\succ b\right\}\right|}{\left|B\right|}$$

(13)

Metric $\rho$ [53] is the ratio of the number of the elements in the set $\left\{x\in {\mathrm{\Omega }}_l\left|x\in {\mathrm{\Omega }}^{*}\right.\right\}$ to $\left|{\mathrm{\Omega }}^{*}\right|$, where the reference set ${\mathrm{\Omega }}^{*}$ is composed of the non-dominated solutions in the union of non-dominated sets of all algorithms.

Metric $TS$ [50] calculated as follows is used to evaluate the distribution of the obtained non-dominated solutions. The smaller the metric is, the more uniformly the solutions are distributed.

$$TS=\sqrt{\frac{1}{\left|{\mathrm{\Omega }}_l\right|}{\sum }_{x_i\in {\mathrm{\Omega }}_l}{\left(D_i-\overline{D}\right)}^2/\phantom{{\left(D_i-\overline{D}\right)}^2\overline{D}}\overline{D}}$$

(14)

where $D_i$ is the Euclid distance in the objective space between solution $x_i\in {\mathrm{\Omega }}_l$ and its nearest solution and $\overline{D}={\sum }_{x_i\in {\mathrm{\Omega }}_l}{D}_i/\phantom{{\sum }_{x_i\in {\mathrm{\Omega }}_l}{D}_i\left|{\mathrm{\Omega }}_l\right|}\left|{\mathrm{\Omega }}_l\right|$.

5.2. Parameters Settings

AICA has following parameters: N, $N_{im}$, $\alpha$, $\beta$ and stopping condition. We first tested stopping condition and found that AICA can converge well when $0.5n$ seconds CPU times reaches. We also found that $0.5n$ seconds CPU times also can be used as stopping condition of comparative algorithms.

Taguchi method [54] is used to decide the settings for other parameters. We select instance $4\times 200\times 6$. The levels of each parameter are shown in

Table 2. Parameters and their levels.

Parameters	Factor Level
	1	2	3	4
N	60	80	100	120
$N_{im}$	6	7	8	9
$\alpha$	8	9	10	11
$\beta$	2	3	4	5

. AICA with each combination runs 10 times independently for the chosen instance. The orthogonal array $L_{16}\left(4^4\right)$ is in

Table 3. The orthogonal $L_{16}\left(4^4\right)$.

Experiment Number	Factors				$\mathit{\rho }$
	$\mathit{N}$	${\mathit{N}}_{\mathit{im}}$	$\mathit{\alpha }$	$\mathit{\beta }$
1	1	1	1	1	0.04092
2	1	2	2	2	0.06138
3	1	3	3	3	0.15857
4	1	4	4	4	0.01023
5	2	1	2	3	0.08696
6	2	2	1	4	0.02788
7	2	3	4	1	0.02558
8	2	4	3	2	0.03069
9	3	1	3	4	0.10627
10	3	2	4	3	0.06790
11	3	3	1	2	0.14974
12	3	4	2	1	0.06035
13	4	1	4	2	0.02035
14	4	2	3	1	0.07046
15	4	3	2	4	0.03069
16	4	4	1	3	0.05205

and the related results on each level are listed in

Table 4. Related results on level of parameters.

Level	N	${\mathit{N}}_{\mathit{im}}$	$\mathit{\alpha }$	$\mathit{\beta }$
1	0.06778	0.06362	0.06765	0.04933
2	0.04277	0.05691	0.05984	0.06554
3	0.09606	0.09114	0.0915	0.09137
4	0.04339	0.03833	0.03101	0.04377
Delta	0.05329	0.05282	0.06048	0.0476
Rank	2	3	1	4

.

The results of $\rho$ and $S/N$ ratio are shown in Figure 3, in which $S/N$ ratio is defined as $-10\times {log}_{10}\left({\rho }^2\right)$. When the metrics are computed, all archives of an algorithm obtained in 10 runs are formed into a non-dominated set and the algorithms are compared to each other according to their non-dominated set.

It can be found from Figure 3 that AICA with the combination $N=100$, $N_{im}=8$, $\alpha =10$ and $\beta =4$ can obtain better results than AICA with other combinations, so the above combination is adopted.

ICA has the same parameters with AICA.

We directly use parameter settings of three comparative algorithms except the stopping condition and test these settings for each comparative algorithm. The computational results show that those settings of each comparative algorithm are still effective, and comparative algorithms with those settings can produce better results than MOWSA, INSGA-II and CMA with other settings.

5.3. Results and Discussions

Five algorithms are tested and compared, which are AICA, ICA, MOWSA, INSGA-II and CMA. Each algorithm randomly runs 10 times for each instance. The corresponding results of all algorithms are shown in

Table 5. Computational results of five algorithms on $\mathcal{C}$.

Instance	$\mathcal{C}$(A,C)	$\mathcal{C}$(C,A)	$\mathcal{C}$(A,M)	$\mathcal{C}$(M,A)	$\mathcal{C}$(A,N)	$\mathcal{C}$(N,A)	$\mathcal{C}$(A,I)	$\mathcal{C}$(I,A)
2 × 20 × 2	0.553	0.171	0.484	0.034	0.391	0.270	0.505	0.398
2 × 20 × 4	0.466	0.236	0.431	0.227	0.741	0.064	0.699	0.283
2 × 20 × 6	0.689	0.114	0.398	0.093	0.791	0.040	0.495	0.414
2 × 20 × 8	0.792	0.044	0.422	0.343	0.825	0.000	0.599	0.291
2 × 50 × 2	0.282	0.663	0.408	0.029	0.981	0.000	0.764	0.153
2 × 50 × 4	0.874	0.025	0.310	0.060	1.000	0.000	0.739	0.209
2 × 50 × 6	0.895	0.041	0.311	0.140	0.898	0.000	0.720	0.198
2 × 50 × 8	0.804	0.052	0.323	0.081	0.694	0.000	0.828	0.110
2 × 100 × 2	0.844	0.073	0.305	0.003	0.970	0.000	0.774	0.145
2 × 100 × 4	0.635	0.113	0.317	0.020	0.886	0.000	0.822	0.082
2 × 100 × 6	0.788	0.057	0.211	0.034	0.878	0.000	0.757	0.183
2 × 100 × 8	0.675	0.069	0.464	0.107	0.760	0.000	0.743	0.198
2 × 200 × 2	0.540	0.095	0.198	0.086	1.000	0.000	0.795	0.155
2 × 200 × 4	0.700	0.066	0.210	0.036	0.933	0.000	0.783	0.137
2 × 200 × 6	0.578	0.074	0.192	0.123	0.861	0.000	0.851	0.136
2 × 200 × 8	0.570	0.054	0.308	0.050	0.882	0.000	0.824	0.119
2 × 500 × 2	0.523	0.061	0.125	0.000	0.970	0.000	0.846	0.089
2 × 500 × 4	0.364	0.069	0.150	0.053	0.889	0.000	0.787	0.134
2 × 500 × 6	0.339	0.054	0.684	0.035	0.854	0.000	0.672	0.219
2 × 500 × 8	0.286	0.036	0.913	0.020	0.760	0.000	0.753	0.175
3 × 20 × 2	0.645	0.156	0.587	0.067	0.842	0.090	0.480	0.538
3 × 20 × 4	0.789	0.033	0.470	0.262	0.931	0.006	0.608	0.241
3 × 20 × 6	0.868	0.052	0.375	0.374	0.974	0.000	0.474	0.407
3 × 20 × 8	0.894	0.028	0.957	0.039	0.720	0.039	0.533	0.391
3 × 50 × 2	0.434	0.446	0.433	0.021	0.948	0.000	0.508	0.416
3 × 50 × 4	0.882	0.027	0.387	0.077	0.859	0.000	0.651	0.265
3 × 50 × 6	0.773	0.078	0.352	0.074	0.718	0.000	0.829	0.156
3 × 50 × 8	0.648	0.080	0.508	0.178	0.622	0.002	0.689	0.238
3 × 100 × 2	0.827	0.064	0.024	0.414	1.000	0.000	0.735	0.203
3 × 100 × 4	0.833	0.047	0.304	0.252	0.780	0.000	0.674	0.231
3 × 100 × 6	0.838	0.053	0.093	0.155	0.686	0.000	0.616	0.220
3 × 100 × 8	0.799	0.040	0.172	0.147	0.780	0.000	0.740	0.138
3 × 200 × 2	0.398	0.266	0.016	0.254	0.917	0.000	0.717	0.163
3 × 200 × 4	0.622	0.048	0.134	0.155	0.939	0.000	0.869	0.074
3 × 200 × 6	0.500	0.081	0.212	0.142	0.529	0.000	0.790	0.146
3 × 200 × 8	0.543	0.048	0.366	0.075	0.559	0.000	0.803	0.200
3 × 500 × 2	0.573	0.058	0.026	0.171	0.830	0.003	0.874	0.071
3 × 500 × 4	0.360	0.061	0.097	0.102	0.863	0.000	0.746	0.130
3 × 500 × 6	0.460	0.042	0.452	0.049	0.640	0.000	0.866	0.086
3 × 500 × 8	0.449	0.025	0.983	0.000	0.667	0.000	0.812	0.083
4 × 20 × 2	0.495	0.201	0.536	0.095	0.914	0.067	0.621	0.273
4 × 20 × 4	0.989	0.006	0.500	0.208	1.000	0.000	0.603	0.247
4 × 20 × 6	0.742	0.054	0.723	0.092	0.920	0.017	0.871	0.061
4 × 20 × 8	0.769	0.113	0.677	0.258	0.918	0.008	0.481	0.477
4 × 50 × 2	0.750	0.157	0.448	0.030	0.614	0.039	0.583	0.341
4 × 50 × 4	0.852	0.047	0.388	0.290	0.899	0.000	0.769	0.120
4 × 50 × 6	0.798	0.059	0.445	0.052	0.809	0.000	0.583	0.290
4 × 50 × 8	0.785	0.045	0.504	0.196	0.618	0.000	0.450	0.441
4 × 100 × 2	0.839	0.127	0.331	0.027	0.859	0.011	0.745	0.192
4 × 100 × 4	0.824	0.065	0.362	0.061	0.804	0.000	0.717	0.203

,

Table 6. Computational results of five algorithms on $\mathcal{C}$.

Instance	$\mathcal{C}$(A,C)	$\mathcal{C}$(C,A)	$\mathcal{C}$(A,M)	$\mathcal{C}$(M,A)	$\mathcal{C}$(A,N)	$\mathcal{C}$(N,A)	$\mathcal{C}$(A,I)	$\mathcal{C}$(I,A)
4 × 100 × 6	0.872	0.064	0.362	0.178	0.733	0.000	0.765	0.135
4 × 100 × 8	0.732	0.072	0.433	0.133	0.514	0.000	0.688	0.231
4 × 200 × 2	0.674	0.130	0.039	0.300	0.520	0.035	0.732	0.171
4 × 200 × 4	0.734	0.040	0.188	0.184	0.859	0.000	0.738	0.152
4 × 200 × 6	0.567	0.089	0.348	0.149	0.833	0.000	0.678	0.232
4 × 200 × 8	0.327	0.112	0.567	0.075	0.471	0.011	0.805	0.146
4 × 500 × 2	0.691	0.018	0.020	0.136	0.874	0.000	0.761	0.141
4 × 500 × 4	0.553	0.030	0.217	0.144	0.717	0.000	0.872	0.066
4 × 500 × 6	0.360	0.057	0.351	0.044	0.774	0.000	0.841	0.102
4 × 500 × 8	0.543	0.034	0.945	0.000	0.655	0.000	0.834	0.099
5 × 20 × 2	0.506	0.265	0.452	0.146	0.896	0.030	0.619	0.413
5 × 20 × 4	0.927	0.032	0.466	0.209	1.000	0.000	0.605	0.342
5 × 20 × 6	0.939	0.016	0.652	0.123	0.944	0.006	0.754	0.205
5 × 20 × 8	0.716	0.062	0.754	0.084	0.766	0.007	0.730	0.193
5 × 50 × 2	0.952	0.015	0.553	0.045	0.898	0.020	0.424	0.454
5 × 50 × 4	0.806	0.048	0.352	0.238	0.942	0.000	0.568	0.335
5 × 50 × 6	0.745	0.071	0.514	0.238	0.618	0.000	0.444	0.498
5 × 50 × 8	0.680	0.113	0.730	0.108	0.791	0.000	0.556	0.298
5 × 100 × 2	0.800	0.117	0.412	0.097	0.880	0.025	0.619	0.262
5 × 100 × 4	0.840	0.049	0.289	0.288	0.828	0.000	0.750	0.189
5 × 100 × 6	0.835	0.067	0.384	0.133	0.863	0.000	0.696	0.242
5 × 100 × 8	0.690	0.102	0.339	0.151	0.656	0.000	0.770	0.246
5 × 200 × 2	0.770	0.074	0.113	0.308	0.786	0.018	0.674	0.226
5 × 200 × 4	0.701	0.057	0.146	0.241	0.529	0.000	0.632	0.211
5 × 200 × 6	0.791	0.034	0.516	0.064	0.787	0.000	0.772	0.141
5 × 200 × 8	0.436	0.052	0.809	0.040	0.688	0.000	0.699	0.173
5 × 500 × 2	0.589	0.027	0.059	0.163	0.961	0.000	0.751	0.185
5 × 500 × 4	0.575	0.027	0.456	0.117	0.773	0.000	0.834	0.092
5 × 500 × 6	0.376	0.059	0.741	0.033	0.739	0.000	0.715	0.174
5 × 500 × 8	0.402	0.022	0.951	0.010	0.536	0.000	0.683	0.213
6 × 20 × 2	0.610	0.132	0.247	0.410	1.000	0.005	0.604	0.327
6 × 20 × 4	1.000	0.000	0.500	0.240	0.962	0.000	0.667	0.265
6 × 20 × 6	0.841	0.098	0.672	0.215	0.846	0.014	0.658	0.285
6 × 20 × 8	1.000	0.000	0.929	0.052	0.787	0.000	0.489	0.477
6 × 50 × 2	0.738	0.101	0.039	0.600	0.773	0.046	0.519	0.330
6 × 50 × 4	0.714	0.063	0.472	0.294	0.854	0.000	0.656	0.256
6 × 50 × 6	0.913	0.018	0.486	0.178	0.651	0.000	0.654	0.284
6 × 50 × 8	0.775	0.036	0.589	0.202	0.675	0.000	0.400	0.465
6 × 100 × 2	0.870	0.053	0.403	0.036	0.871	0.003	0.719	0.198
6 × 100 × 4	0.895	0.031	0.405	0.104	0.942	0.000	0.726	0.149
6 × 100 × 6	0.742	0.092	0.674	0.093	0.848	0.000	0.658	0.249
6 × 100 × 8	0.752	0.060	0.457	0.147	0.679	0.000	0.752	0.126
6 × 200 × 2	0.824	0.078	0.080	0.287	0.967	0.000	0.703	0.209
6 × 200 × 4	0.778	0.065	0.294	0.194	0.851	0.000	0.807	0.119
6 × 200 × 6	0.720	0.033	0.454	0.137	0.689	0.000	0.762	0.140
6 × 200 × 8	0.686	0.061	0.748	0.086	0.603	0.000	0.591	0.316
6 × 500 × 2	0.646	0.032	0.133	0.175	0.833	0.007	0.800	0.085
6 × 500 × 4	0.568	0.034	0.174	0.124	0.556	0.001	0.790	0.130
6 × 500 × 6	0.459	0.034	0.774	0.018	0.696	0.000	0.845	0.113
6 × 500 × 8	0.243	0.052	0.987	0.000	0.667	0.004	0.814	0.143

,

Table 7. Computational results of five algorithms on $\rho$.

Instance	A	C	M	N	I	Instance	A	C	M	N	I
2 × 20 × 2	0.227	0.157	0.450	0.060	0.107	4 × 100 × 6	0.680	0.036	0.131	0.010	0.143
2 × 20 × 4	0.445	0.120	0.315	0.060	0.060	4 × 100 × 8	0.638	0.043	0.096	0.020	0.202
2 × 20 × 6	0.334	0.050	0.322	0.035	0.259	4 × 200 × 2	0.515	0.037	0.240	0.024	0.185
2 × 20 × 8	0.417	0.046	0.268	0.007	0.262	4 × 200 × 4	0.707	0.024	0.127	0.002	0.140
2 × 50 × 2	0.176	0.320	0.445	0.000	0.059	4 × 200 × 6	0.649	0.044	0.074	0.007	0.226
2 × 50 × 4	0.490	0.014	0.283	0.000	0.213	4 × 200 × 8	0.754	0.056	0.059	0.024	0.107
2 × 50 × 6	0.552	0.031	0.266	0.009	0.142	4 × 500 × 2	0.714	0.019	0.127	0.000	0.139
2 × 50 × 8	0.575	0.044	0.239	0.031	0.111	4 × 500 × 4	0.831	0.023	0.062	0.004	0.079
2 × 100 × 2	0.518	0.062	0.293	0.002	0.124	4 × 500 × 6	0.818	0.033	0.035	0.004	0.109
2 × 100 × 4	0.632	0.064	0.222	0.003	0.080	4 × 500 × 8	0.852	0.028	0.000	0.009	0.111
2 × 100 × 6	0.648	0.029	0.164	0.005	0.154	5 × 20 × 2	0.469	0.000	0.379	0.009	0.142
2 × 100 × 8	0.646	0.058	0.106	0.013	0.176	5 × 20 × 4	0.715	0.000	0.138	0.000	0.148
2 × 200 × 2	0.618	0.092	0.144	0.000	0.146	5 × 20 × 6	0.801	0.000	0.074	0.000	0.125
2 × 200 × 4	0.703	0.028	0.136	0.001	0.133	5 × 20 × 8	0.683	0.046	0.053	0.039	0.180
2 × 200 × 6	0.803	0.028	0.100	0.005	0.064	5 × 50 × 2	0.295	0.003	0.303	0.016	0.383
2 × 200 × 8	0.774	0.035	0.063	0.001	0.127	5 × 50 × 4	0.414	0.048	0.202	0.002	0.333
2 × 500 × 2	0.808	0.040	0.082	0.000	0.069	5 × 50 × 6	0.380	0.063	0.105	0.035	0.417
2 × 500 × 4	0.785	0.020	0.061	0.000	0.134	5 × 50 × 8	0.557	0.074	0.060	0.014	0.294
2 × 500 × 6	0.764	0.040	0.004	0.003	0.189	5 × 100 × 2	0.443	0.060	0.292	0.002	0.203
2 × 500 × 8	0.800	0.053	0.002	0.010	0.135	5 × 100 × 4	0.608	0.047	0.155	0.008	0.183
3 × 20 × 2	0.253	0.004	0.498	0.031	0.214	5 × 100 × 6	0.615	0.045	0.126	0.007	0.207
3 × 20 × 4	0.511	0.037	0.256	0.019	0.178	5 × 100 × 8	0.667	0.056	0.093	0.023	0.161
3 × 20 × 6	0.435	0.003	0.189	0.000	0.372	5 × 200 × 2	0.552	0.057	0.246	0.002	0.142
3 × 20 × 8	0.556	0.000	0.007	0.066	0.372	5 × 200 × 4	0.641	0.019	0.150	0.025	0.165
3 × 50 × 2	0.243	0.158	0.403	0.005	0.191	5 × 200 × 6	0.762	0.032	0.059	0.015	0.132
3 × 50 × 4	0.422	0.008	0.283	0.018	0.269	5 × 200 × 8	0.735	0.062	0.014	0.018	0.170
3 × 50 × 6	0.496	0.033	0.282	0.047	0.141	5 × 500 × 2	0.692	0.029	0.116	0.003	0.161
3 × 50 × 8	0.584	0.088	0.107	0.027	0.193	5 × 500 × 4	0.789	0.033	0.048	0.001	0.130
3 × 100 × 2	0.459	0.054	0.336	0.000	0.151	5 × 500 × 6	0.751	0.029	0.009	0.005	0.206
3 × 100 × 4	0.611	0.045	0.165	0.007	0.171	5 × 500 × 8	0.704	0.041	0.002	0.013	0.240
3 × 100 × 6	0.538	0.021	0.136	0.020	0.284	6 × 20 × 2	0.503	0.000	0.340	0.000	0.157
3 × 100 × 8	0.644	0.028	0.164	0.017	0.147	6 × 20 × 4	0.691	0.000	0.138	0.004	0.167
3 × 200 × 2	0.532	0.088	0.264	0.001	0.115	6 × 20 × 6	0.730	0.000	0.098	0.039	0.132
3 × 200 × 4	0.752	0.045	0.158	0.000	0.045	6 × 20 × 8	0.500	0.000	0.015	0.064	0.421
3 × 200 × 6	0.697	0.044	0.125	0.013	0.121	6 × 50 × 2	0.284	0.069	0.446	0.059	0.142
3 × 200 × 8	0.752	0.034	0.077	0.018	0.119	6 × 50 × 4	0.538	0.064	0.130	0.011	0.257
3 × 500 × 2	0.791	0.035	0.112	0.003	0.059	6 × 50 × 6	0.585	0.002	0.090	0.032	0.291
3 × 500 × 4	0.767	0.031	0.067	0.001	0.135	6 × 50 × 8	0.312	0.030	0.091	0.030	0.538
3 × 500 × 6	0.880	0.026	0.025	0.011	0.057	6 × 100 × 2	0.468	0.040	0.294	0.002	0.196
3 × 500 × 8	0.812	0.043	0.001	0.015	0.130	6 × 100 × 4	0.577	0.028	0.235	0.002	0.158
4 × 20 × 2	0.347	0.020	0.446	0.020	0.168	6 × 100 × 6	0.604	0.060	0.079	0.013	0.245
4 × 20 × 4	0.683	0.000	0.121	0.000	0.196	6 × 100 × 8	0.591	0.070	0.115	0.020	0.203
4 × 20 × 6	0.855	0.000	0.067	0.010	0.067	6 × 200 × 2	0.597	0.041	0.208	0.002	0.152
4 × 20 × 8	0.496	0.017	0.068	0.017	0.403	6 × 200 × 4	0.686	0.040	0.161	0.006	0.107
4 × 50 × 2	0.358	0.037	0.337	0.051	0.217	6 × 200 × 6	0.669	0.044	0.093	0.021	0.172
4 × 50 × 4	0.566	0.061	0.224	0.005	0.145	6 × 200 × 8	0.630	0.046	0.029	0.040	0.256
4 × 50 × 6	0.412	0.046	0.225	0.029	0.288	6 × 500 × 2	0.736	0.032	0.126	0.001	0.105
4 × 50 × 8	0.435	0.048	0.135	0.027	0.355	6 × 500 × 4	0.779	0.026	0.081	0.009	0.105
4 × 100 × 2	0.533	0.043	0.322	0.005	0.097	6 × 500 × 6	0.828	0.036	0.015	0.007	0.114
4 × 100 × 4	0.595	0.018	0.211	0.008	0.168	6 × 500 × 8	0.657	0.040	0.076	0.016	0.211

and

Table 8. Computational results of five algorithms on $TS$.

Instance	A	C	M	N	I	Instance	A	C	M	N	I
2 × 20 × 2	5.190	6.529	5.398	5.333	5.274	4 × 100 × 6	8.645	12.994	12.764	12.161	9.947
2 × 20 × 4	6.091	6.864	7.352	6.128	6.311	4 × 100 × 8	9.227	11.257	16.931	9.900	9.456
2 × 20 × 6	6.161	9.739	7.905	7.040	6.565	4 × 200 × 2	6.285	9.833	9.779	8.319	8.161
2 × 20 × 8	7.260	10.781	8.631	11.311	7.308	4 × 200 × 4	7.717	11.919	13.675	9.499	9.617
2 × 50 × 2	6.402	6.259	5.793	5.987	6.164	4 × 200 × 6	9.321	10.644	17.441	10.409	9.921
2 × 50 × 4	6.991	8.121	8.110	7.756	6.734	4 × 200 × 8	11.072	15.809	18.024	10.201	10.115
2 × 50 × 6	6.990	8.519	9.122	7.888	7.716	4 × 500 × 2	6.205	10.246	11.921	9.283	8.257
2 × 50 × 8	8.400	9.822	11.702	8.815	8.696	4 × 500 × 4	6.922	11.079	18.621	10.508	10.140
2 × 100 × 2	5.421	7.754	6.253	6.952	7.036	4 × 500 × 6	11.524	13.682	20.208	13.914	11.887
2 × 100 × 4	7.437	7.947	9.317	8.082	7.760	4 × 500 × 8	13.062	14.823	21.708	13.629	14.612
2 × 100 × 6	8.972	10.416	11.342	9.411	9.377	5 × 20 × 2	5.679	7.519	6.684	8.064	5.810
2 × 100 × 8	9.852	9.120	11.921	5.890	8.952	5 × 20 × 4	5.729	8.031	8.688	7.014	6.677
2 × 200 × 2	7.525	7.888	9.567	5.975	8.045	5 × 20 × 6	7.028	8.666	10.247	9.753	7.691
2 × 200 × 4	8.562	8.275	12.213	8.298	8.886	5 × 20 × 8	6.624	8.558	13.580	9.552	7.204
2 × 200 × 6	9.472	10.325	12.736	10.844	9.825	5 × 50 × 2	6.475	7.838	8.121	6.051	6.831
2 × 200 × 8	6.959	9.956	17.132	10.748	10.517	5 × 50 × 4	7.307	9.023	10.403	9.507	7.450
2 × 500 × 2	8.660	7.972	11.546	8.436	8.316	5 × 50 × 6	7.768	10.812	12.576	9.786	8.133
2 × 500 × 4	6.531	11.824	17.538	10.043	10.147	5 × 50 × 8	8.434	10.265	12.659	11.626	8.870
2 × 500 × 6	11.716	12.801	20.216	10.169	12.410	5 × 100 × 2	7.058	9.603	8.584	7.425	7.450
2 × 500 × 8	10.569	13.476	32.782	11.702	14.662	5 × 100 × 4	8.711	9.342	10.961	8.011	7.932
3 × 20 × 2	5.270	7.700	6.007	5.351	5.416	5 × 100 × 6	7.966	11.393	13.672	8.857	10.190
3 × 20 × 4	6.022	10.318	8.461	7.459	6.078	5 × 100 × 8	9.556	18.366	15.123	11.665	9.748
3 × 20 × 6	6.728	8.654	9.429	6.642	6.752	5 × 200 × 2	8.291	10.055	10.269	9.211	8.713
3 × 20 × 8	7.210	9.901	11.099	12.096	7.669	5 × 200 × 4	9.544	9.385	13.267	9.783	9.192
3 × 50 × 2	6.219	7.658	7.092	6.844	6.501	5 × 200 × 6	8.804	12.043	17.984	10.896	11.078
3 × 50 × 4	7.157	9.833	8.118	9.340	7.315	5 × 200 × 8	9.490	12.364	19.295	17.865	10.756
3 × 50 × 6	8.013	10.058	10.915	8.997	7.673	5 × 500 × 2	8.718	10.723	12.915	5.528	9.097
3 × 50 × 8	8.002	8.957	11.808	9.966	8.595	5 × 500 × 4	11.356	11.789	19.571	12.896	12.002
3 × 100 × 2	6.874	8.823	7.576	7.919	7.410	5 × 500 × 6	12.790	17.475	19.994	24.109	12.852
3 × 100 × 4	7.724	9.805	10.541	8.451	8.225	5 × 500 × 8	9.851	14.998	26.560	12.615	13.596
3 × 100 × 6	6.934	9.864	11.485	9.320	8.797	6 × 20 × 2	5.565	5.647	7.007	6.628	5.845
3 × 100 × 8	7.119	10.797	13.822	10.240	9.335	6 × 20 × 4	6.030	7.914	9.284	6.837	6.435
3 × 200 × 2	6.660	9.256	7.967	8.392	8.464	6 × 20 × 6	6.906	11.461	11.915	9.479	7.356
3 × 200 × 4	5.913	10.618	11.887	9.619	8.420	6 × 20 × 8	7.339	12.652	16.704	10.010	7.355
3 × 200 × 6	12.220	10.995	16.795	11.043	10.558	6 × 50 × 2	7.485	8.302	8.411	7.746	7.615
3 × 200 × 8	7.424	13.954	19.622	11.832	10.991	6 × 50 × 4	7.253	10.133	10.263	10.181	7.216
3 × 500 × 2	6.953	10.813	12.029	8.930	8.401	6 × 50 × 6	8.337	8.323	14.594	13.754	8.678
3 × 500 × 4	9.448	9.511	15.153	11.278	11.715	6 × 50 × 8	7.832	13.426	14.591	10.282	9.270
3 × 500 × 6	12.223	14.136	20.142	12.934	13.166	6 × 100 × 2	7.458	9.699	9.485	8.878	7.661
3 × 500 × 8	12.534	13.044	20.069	15.700	13.712	6 × 100 × 4	7.823	10.354	11.630	9.834	8.462
4 × 20 × 2	6.381	6.805	7.633	6.675	6.615	6 × 100 × 6	9.696	13.975	12.420	9.028	9.365
4 × 20 × 4	7.111	9.360	9.864	8.749	6.079	6 × 100 × 8	8.763	12.577	15.983	14.355	10.177
4 × 20 × 6	6.291	8.036	10.851	7.820	6.625	6 × 200 × 2	8.420	11.299	10.042	6.452	8.430
4 × 20 × 8	6.947	12.988	12.931	10.303	7.789	6 × 200 × 4	8.730	11.134	13.646	14.548	10.579
4 × 50 × 2	5.240	8.167	7.653	6.240	6.597	6 × 200 × 6	9.387	11.179	17.115	9.660	11.147
4 × 50 × 4	6.437	9.392	11.135	7.988	7.557	6 × 200 × 8	9.627	12.172	19.227	10.885	10.445
4 × 50 × 6	9.930	14.371	13.708	8.276	8.495	6 × 500 × 2	9.614	10.488	13.042	7.376	9.857
4 × 50 × 8	7.483	9.414	12.838	8.325	8.586	6 × 500 × 4	9.507	11.117	18.098	12.345	11.374
4 × 100 × 2	7.107	10.206	7.898	7.332	7.843	6 × 500 × 6	10.642	15.477	24.991	30.280	12.663
4 × 100 × 4	8.821	9.825	11.710	9.505	8.354	6 × 500 × 8	12.383	14.276	24.702	14.560	12.506

, and small scale examples are listed in

Table A1. Computational results about small-scale instances of five algorithms on $\mathcal{C}$.

Instance	$\mathcal{C}$(A,C)	$\mathcal{C}$(C,A)	$\mathcal{C}$(A,M)	$\mathcal{C}$(M,A)	$\mathcal{C}$(A,N)	$\mathcal{C}$(N,A)	$\mathcal{C}$(A,I)	$\mathcal{C}$(I,A)
2 × 8 × 2	0.615	0.147	0.345	0.329	0.827	0.154	0.850	0.161
2 × 8 × 3	0.891	0.057	0.366	0.213	0.938	0.036	0.743	0.258
2 × 8 × 4	0.859	0.052	0.425	0.165	1.000	0.000	0.656	0.294
2 × 10 × 2	0.581	0.145	0.049	0.459	0.902	0.047	0.539	0.401
2 × 10 × 3	0.566	0.182	0.320	0.286	0.794	0.143	0.626	0.355
2 × 10 × 4	0.585	0.197	0.453	0.071	0.916	0.059	0.593	0.283
2 × 12 × 2	0.654	0.210	0.479	0.107	0.572	0.239	0.534	0.403
2 × 12 × 3	0.660	0.095	0.073	0.439	0.897	0.061	0.502	0.481
2 × 12 × 4	0.500	0.230	0.090	0.540	0.930	0.010	0.667	0.275
3 × 8 × 2	0.451	0.265	0.375	0.197	0.955	0.029	0.623	0.382
3 × 8 × 3	0.738	0.176	0.419	0.304	1.000	0.000	0.824	0.287
3 × 8 × 4	0.974	0.034	0.833	0.162	1.000	0.007	0.792	0.216
3 × 10 × 2	0.794	0.138	0.349	0.344	1.000	0.000	0.873	0.094
3 × 10 × 3	0.986	0.032	0.317	0.372	0.987	0.000	0.869	0.229
3 × 10 × 4	0.753	0.106	0.565	0.245	1.000	0.005	0.734	0.191
3 × 12 × 2	0.590	0.102	0.247	0.404	0.877	0.060	0.702	0.235
3 × 12 × 3	0.526	0.142	0.309	0.278	1.000	0.006	0.547	0.341
3 × 12 × 4	0.753	0.103	0.391	0.210	1.000	0.004	0.695	0.262
4 × 8 × 2	0.544	0.270	0.348	0.327	0.945	0.028	0.714	0.312
4 × 8 × 3	0.983	0.051	0.434	0.182	1.000	0.010	0.813	0.253
4 × 8 × 4	0.843	0.148	0.352	0.344	1.000	0.041	0.725	0.344
4 × 10 × 2	0.464	0.215	0.339	0.331	0.986	0.015	0.556	0.485
4 × 10 × 3	0.812	0.087	0.526	0.150	0.836	0.044	0.729	0.300
4 × 10 × 4	0.917	0.071	0.500	0.264	1.000	0.007	0.632	0.336
4 × 12 × 2	0.606	0.124	0.187	0.352	0.947	0.026	0.723	0.202
4 × 12 × 3	0.464	0.145	0.382	0.295	1.000	0.000	0.495	0.430
4 × 12 × 4	0.706	0.102	0.514	0.194	1.000	0.000	0.571	0.369

,

Table A2. Computational results about small-scale instances of five algorithms on $\rho$.

Instance	A	C	M	N	I	Instance	A	C	M	N	I
2 × 8 × 2	0.517	0.000	0.367	0.041	0.075	3 × 10 × 4	0.721	0.000	0.168	0.000	0.112
2 × 8 × 3	0.604	0.000	0.359	0.000	0.036	3 × 12 × 2	0.443	0.000	0.389	0.000	0.168
2 × 8 × 4	0.497	0.000	0.420	0.000	0.083	3 × 12 × 3	0.521	0.000	0.318	0.000	0.161
2 × 10 × 2	0.225	0.000	0.519	0.000	0.257	3 × 12 × 4	0.705	0.000	0.172	0.000	0.123
2 × 10 × 3	0.520	0.000	0.309	0.046	0.126	4 × 8 × 2	0.641	0.023	0.250	0.000	0.086
2 × 10 × 4	0.444	0.047	0.325	0.026	0.158	4 × 8 × 3	0.520	0.000	0.439	0.000	0.041
2 × 12 × 2	0.329	0.008	0.386	0.041	0.236	4 × 8 × 4	0.492	0.000	0.344	0.000	0.164
2 × 12 × 3	0.290	0.100	0.338	0.010	0.262	4 × 10 × 2	0.385	0.000	0.333	0.009	0.274
2 × 12 × 4	0.451	0.013	0.405	0.008	0.122	4 × 10 × 3	0.675	0.006	0.135	0.055	0.129
3 × 8 × 2	0.468	0.000	0.377	0.000	0.156	4 × 10 × 4	0.654	0.008	0.211	0.000	0.128
3 × 8 × 3	0.603	0.017	0.314	0.000	0.066	4 × 12 × 2	0.524	0.042	0.296	0.011	0.127
3 × 8 × 4	0.704	0.000	0.173	0.000	0.123	4 × 12 × 3	0.505	0.011	0.242	0.000	0.242
3 × 10 × 2	0.689	0.000	0.277	0.000	0.034	4 × 12 × 4	0.605	0.005	0.174	0.000	0.215
3 × 10 × 3	0.556	0.000	0.350	0.006	0.087

and

Table A3. Computational results about small-scale instances of five algorithms on $TS$.

Instance	A	C	M	N	I	Instance	A	C	M	N	I
2 × 8 × 2	4.120	5.756	5.587	5.999	4.967	3 × 10 × 4	5.493	6.068	6.826	5.881	5.929
2 × 8 × 3	4.082	6.263	5.519	6.226	4.466	3 × 12 × 2	5.355	6.516	6.073	6.095	5.995
2 × 8 × 4	5.139	6.650	6.861	5.128	5.164	3 × 12 × 3	5.857	8.270	6.836	7.123	5.721
2 × 10 × 2	4.808	5.934	5.920	5.241	5.056	3 × 12 × 4	5.432	8.058	7.382	6.498	5.520
2 × 10 × 3	4.930	5.457	6.427	5.849	5.392	4 × 8 × 2	4.445	6.412	7.836	7.399	4.935
2 × 10 × 4	4.886	6.957	6.071	6.078	5.210	4 × 8 × 3	5.911	7.121	6.335	7.736	6.329
2 × 12 × 2	4.827	5.963	5.308	5.278	4.467	4 × 8 × 4	6.267	7.260	8.231	6.272	6.602
2 × 12 × 3	5.075	6.657	7.564	5.835	5.850	4 × 10 × 2	4.312	5.652	6.570	5.739	5.224
2 × 12 × 4	4.929	5.313	6.860	6.301	5.069	4 × 10 × 3	5.286	6.124	7.761	9.027	5.651
3 × 8 × 2	4.458	6.127	5.661	5.736	4.388	4 × 10 × 4	5.084	7.875	9.246	7.138	5.886
3 × 8 × 3	5.373	7.853	8.213	7.627	5.798	4 × 12 × 2	5.258	6.336	5.996	7.926	5.642
3 × 8 × 4	5.722	7.255	6.784	8.571	6.434	4 × 12 × 3	5.380	7.949	7.429	7.629	5.625
3 × 10 × 2	4.421	6.282	5.901	5.998	5.142	4 × 12 × 4	5.502	7.244	8.106	6.810	6.007
3 × 10 × 3	4.994	7.129	7.022	6.815	5.245

, in which A, I, C, M and N indicate AICA, ICA, CMA, MOWSA and INSGA-II, respectively. Figure 4 describes the box plot of five algorithms and Figure 5 shows non-dominated solutions produced by all algorithms.

As shown in Table 5 and Table A3, AICA produces better results than ICA. $\mathcal{C}(I,A)$ is less than $\mathcal{C}(A,I)$ on nearly all instances and $\mathcal{C}(A,I)$ is greater than 0.6 on more than 100 instances, that is, most of non-dominated solutions of ICA are dominated by the non-dominated set of AICA. On the other hand, $\rho$ of AICA is bigger than that of ICA on 124 instances and bigger than that of ICA by at least 0.4 on nearly all instances. The above analyses show that AICA performs better than ICA.

It can be found from Table 8 and Table A3 that AICA possesses smaller $TS$ than ICA on 116 instances. Non-dominated solutions of AICA are distributed more uniformly than those of ICA. Figure 4 also reveals the difference between AICA and ICA on $TS$. The notable performance between AICA and ICA demonstrates that adaptive mechanisms really have a positive impact on the performance of AICA.

As stated in Table 5, Table 6 and Table 7, AICA has smaller $\mathcal{C}(C,A)$ than $\mathcal{C}(A,C)$ on all instances; moreover, $\mathcal{C}(C,A)$ is less than 0.1 on most of the instances. Very few members of the non-dominated set of AICA are dominated by those of the non-dominated set of CMA. $\rho$ of CMA is smaller than 0.1 on nearly all instances and 0 on about 15 instances, that is, CMA cannot provide any members or very few members for ${\mathrm{\Omega }}^{*}$. AICA has better convergence performance than CMA. This conclusion also can be drawn from Figure 4 and Figure 5.

When AICA is compared with MOWSA and INSGA-II, it can be found that AICA also converges better than these two comparative algorithms. $\mathcal{C}(N,A)$ is 0 on most of the instances, and $\mathcal{C}(M,A)$ is less than $\mathcal{C}(A,M)$ on more than 100 instances. $\rho$ of AICA is bigger than that of MOWSA and that of INSGA-II on nearly all instances.

With respect to $TS$, AICA has a bigger value than its three comparative algorithms on very few instances, that is, the non-dominated solutions of AICA are distributed more uniformly than those of INSGA-II, CMA and MOWSA. The conclusion on $TS$ can also be found from Figure 4 and Figure 5.

AICA has adaptive mechanisms to decide $U_A^{x_i}$ and $U_R^{x_i}$ adaptively based on solution quality and adaptive multiple neighborhood search considering the features of the problem. Higher $U_A^{x_i}$ and smaller $U_R^{x_i}$ are given to solutions with bigger $ran{k}_i$ and vice versa. Neighborhood structures are dynamically selected according to features on two objectives, as a result, exploration and exploitation can be balanced well and search efficiency is improved greatly, thus, AICA can provide notably better results than its comparative algorithms and is a very competitive algorithm for solving DEASP with three stages.

6. Conclusions

Transportation is an important stage in real-life assembly production systems; however, the previous works are mainly about DASP with two stages. In this study, DEASP with fabrication, transportation and assembly is considered, and a new algorithm called AICA is proposed to minimize makespan and total energy consumption simultaneously. A heuristic and an energy-saving rule are used to produce initial solutions. Adaptive assimilation with adaptive global search and an adaptive revolution are implemented, in which neighborhood structures are chosen dynamically, and revolution probability and search times are decided based on solution quality. Features of the problem are also used in assimilation and revolution. A number of computational experiments are conducted. The computational results demonstrate that the new strategies of AICA are effective and efficient and AICA can provide promising results for a DEASP with three stages.

A DEASP with three stages does not attract sufficient attention. We will continue to cope with this problem with various processing constraints such as factory eligibility, which is an extended version of machine eligibility. DEASPs with other layouts, for example, $HFm\to 1$ layout, is also our future topic. We will also pay attention to DEASP with uncertainty. New optimization mechanisms such as the usage of machine learning are added into meta-heuristics such as imperialist competitive algorithm for obtaining high quality solutions. In recent years, there are some works on the integration of machine learning and meta-heuristics and the effectiveness of the integration is proved. These are also our main future works.