a max–min ant colony system for assembly sequence planning

ORIGINAL ARTICLE

A max–min ant colony system for assembly sequenceplanning

Jiapeng Yu & Chengen Wang

Received: 7 June 2012 /Accepted: 20 December 2012 /Published online: 11 January 2013# Springer-Verlag London 2013

Abstract An improved ant colony optimization (ACO)-based assembly sequence planning (ASP) method for com-plex products that combines the advantages of ant colonysystem (ACS) and max–min ant system (MMAS) and inte-grates some optimization measures is proposed. The optimi-zation criteria, assembly information models, and componentsnumber in case study that reported in the literatures of ACO-based ASP during the past 10 years are reviewed and com-pared. To reduce tedious manual input of parameters andidentify the best sequence easily, the optimization criteria suchas directionality, parallelism, continuity, stability, and auxilia-ry stroke are automatically quantified and integrated into themulti-objective heuristic and fitness functions. On the precon-dition of geometric feasibility based on interference matrix,several strategies of ACS andMMAS are combined in a max–min ant colony system (MMACS) to improve the conver-gence speed and sequence quality. Several optimization meas-ures are integrated into the system, among which theperformance appraisal method transfers the computing re-source from the worst ant to the better one, and the groupmethod makes up the deficiency of solely depending onheuristic searching for all parallel parts in each group. An

assembly planning system “AutoAssem” is developed basedon Siemens NX, and the effectiveness of each optimizationmeasure is testified through case study. Compared with themethods of priority rules screening, genetic algorithm, andparticle swarm optimization, MMACS is verified to havesuperiority in efficiency and sequence performance.

Keywords Assembly sequence planning . Ant colonyoptimization algorithm .Max–min ant system . Extendedinterference matrix

1 Introduction

Assembly sequence planning (ASP) and disassembly se-quence planning (DSP) are the two opposite and uniform sidesof computer-aided assembly planning and green manufactur-ing. They are typical combinatorial optimization problemswith strong constraints in practice engineering. To avoid theinfluence of combination explosion in sequences caused byincrease of components, several heuristic optimization algo-rithms have been applied, and the three most dominant onesare genetic algorithm (GA), particle swarm optimization(PSO), and ant colony optimization (ACO).

GA is the most widely used heuristic algorithm in solvingASP. The researches of GA-based ASP are mainly focused onthe chromosome encoding of assembly sequence, the form ofgenetic operation, multi-objective evaluation, and the improve-ment of local search capability [1–4]. The precedence matrix(PM) is frequently used to determine feasible assemblysequences that satisfy precedence constraints [2, 3]. But PMcannot describe which direction the precedence exists, and thegeneration of PM is a hard manual work for large-scale assem-bly. The GA also tends to generate infeasible offspring thatviolates geometric precedence constraint because of crossoverandmutation operators, and the binary strings in chromosomesare less suitable for complex combinatorial problem [5].

Electronic supplementary material The online version of this article(doi:10.1007/s00170-012-4695-x) contains supplementary material,which is available to authorized users.

J. Yu (*) :C. WangState Key Laboratory of Synthetical Automation for ProcessIndustries, Northeastern University, 254 Mail Box No.11,Lane 3, WenHua Road, HePing District,Shenyang, Liaoning, China 110819e-mail: [email protected]

J. Yu : C. WangLiaoning Province Key Laboratory of Multidisciplinary OptimalDesign for Complex Equipment, Northeastern University,254 Mail Box No.11,Lane 3, WenHua Road, HePing District,Shenyang, Liaoning, China 110819

Int J Adv Manuf Technol (2013) 67:2819–2835DOI 10.1007/s00170-012-4695-x

http://dx.doi.org/10.1007/s00170-012-4695-x

PSO is quite similar to GA, in which the system is initial-ized with a population of random solutions. But it can yield aneasy implementation and requires fewer computational mem-ory because it only use a single velocity formula to evolve [6].This has motivated researchers [6–8] to apply PSO to ASP/DSP optimization. However, the original PSO is not suitableto be directly applied to ASP problems. The assembly prece-dence constraint may be violated in PSO. Moreover, PSO isnot favorable for the discrete problem such as ASP, where thesolution is in discrete integer space [9]. Another importantissue is that PSO is easily trapped in local optimum.

Inspired by the foraging behavior of real ant colony innature, a novel heuristic optimization algorithm based on antcolony is proposed by Dorigo [10]—that is ACO. A series ofcombinational optimization problems, such as traveling sales-man problem (TSP), have been solved with it. ACO has nowbecome one of the best potential combinatorial optimizationalgorithms in various fields. Failli and Dini [11] introducedACO into the study of ASP/DSP firstly in 2000 and analyzedthe influence of each parameter. However, the interferencematrix that they previously proposed [12] was not utilized todecide the geometric feasibility of sequence. Instead, thenetworkmodel was applied to search feasible sequence nodes,which is hard to describe and construct automatically. Besides,only taking the changes of tool and direction as optimizationcriteria results in the difficulty in differentiating the goodsequences from the bad ones. Based on their study, subsequentresearchers focused on ACO-based ASP/DSP mainly in con-structing feasible sequence and integrating other optimizationcriteria or algorithms [13–25].

Wang et al. [13] applied disassembly completed graph(DCG) and disassembly matrix (DM) to solve geometric fea-sibility, considering the criterion of reorientation. But the nodesnumber of DCG (6n) and the elements of pheromone matrix(6n×6n) are too excessive to express information and calculatesequence. ACO is also applied to selective disassembly plan-ning and described the dynamic constructive process of selec-tive disassembly sequence, by considering the criterion of thenumber of disassembled parts [14]. By modeling the disassem-bly hybrid graph that describes contact and precedence rela-tionship, Zhang et al. [15] searched feasible disassemblysequence and constructed ACO with geometric reasoning.However, the hybrid graph is the compound of two indepen-dent graphs or matrices, which also need interactive data input.

Xie et al. [16] stored the precedence relationship betweencomponents in a directed graph-based assembly model. Whilepath simulation and interference detection are being executedon the assembly tools in the whole assembly model, interfer-ence information of the surrounding components are recordedto decide whether the sequence satisfies the space demand oftool operation and to lead ACO algorithm obtaining assemblysequence. Literatures [18–20] construct three similar-structured assembly models separately, including disassembly

feasibility information graph (DFIG), disassembly hierarchyinformation graph (DHIG) and feasibility graph, to convertDSP to the problem of path optimization in graph. However,all the three types of graphs cannot be constructed automati-cally. Moreover, as literature [18] said, too many inefficientants were produced and then suicide in the process, whichwasted computational source and led to low efficiency. Be-sides, too many cost parameters need manually input in liter-ature [20]. Shan et al. [21] tried to improve the searchefficiency of satisfied disassembly sequence under complexconstraints, which is similar to literature [11]. The assemblyinformation and disassembly interference matrix cannot beacquired automatically and disassembly tools need manualinput. Sharma et al. [22] reflected the assembly cost to anenergy function associated with the assembly sequence andutilized ACO to generate robotic assembly sequence. However,the basic data is hard to obtain automatically and the componentthat must disassemble first must be decided first. In addition, thescale of pheromone matrix (5n×5n) is too large and the casestudy is too simple. In order to allow amore effective search forfeasible nondominated solutions, Lu et al. [23] investigated amulti-objective searching algorithm with uniform design toguide the ants searching the routes along the uniformly scat-tered directions towards the Pareto frontier. However, the scaleof the pheromone matrix (6×n×n) is too large.

In aspect of integration with other heuristic algorithms,Ning et al. [24] and Shuang et al. [25] integrated GA and theidea of PSO into ACO separately to solve ASP. However,the integrated GA cannot insure that the sequence producedby genetic operations is still feasible. At the same time, themoving wedge matrix of the latter is hard to be acquiredautomatically.

As shown in Table 1, several characteristics in the liter-atures above are summarized, including optimization criteriain heuristic information and evaluation function, assemblyinformation model, components number in case study, etc.To sum up, in ACO-based ASP/DSP, geometric feasibility ismainly ensured through a matrix-based model (such as DM,interference matrix, and moving wedge matrix) or a graph-based model (such as DCG, DFIG, DHIG, and hybridgraph). Assembly cost is generally considered in changesof tool, direction and operation type, and tool reachability.

The above methods have several shortcomings: (1) Thematrices that used to decide geometric feasibility are all in theprincipal directions (±x, ±y, ±z), which leads to the incapabil-ity of deducing complex product that contains some tilt disas-sembly directions. (2) The nodes of the graph-based modelincreases exponentially with the complexity of the product,which results in the difficulty to construct the graph automat-ically and the decrease in algorithm efficiency. (3) Since mostof the optimization criteria only focus on the changes of tooland reorientation, it is difficult to distinguish and evaluateheuristic and fitness function. Therefore, the generated

2820 Int J Adv Manuf Technol (2013) 67:2819–2835

sequence cannot comprehensively reflect the practice. (4)Some criteria and parameters need tedious manual input,which blocks the automation of assembly evaluation andplanning. (5) Most case studies are too simple to verify thealgorithm efficiency as for complex equipment. Our work isdevoted to overcoming all these defects.

In our previous works, three kinds of assembly relationmatrices have been proposed and automatically constructed,based on which the method for discriminating geometricdisassembly feasibility is studied [26–28], and an ASPalgorithm using priority rules screening (PRS) is presented[29]. The method for generating hierarchical exploded viewautomatically is then brought forward [30].

In this paper, aiming at the particularities of ASP forcomplex products such as aircraft components, more optimi-zation criteria and measures are integrated into the improvedACO-based ASP/DSP solution (named max–min ant colonysystem (MMACS)). The remainder of this paper is as follows:In Section 2, three ACO algorithms are briefly introduced. InSection 3, four particularities of ACO-based ASP are summa-rized. In Section 4, two kinds of matrix-based assembly

models are established. In Section 5, the ASP approach withMMACS is presented. In Section 6, a case study is provided toverify each optimization measure and discuss the difference ofPRS, GA, PSO ,and MMACS in ASP. Section 7 concludeswith some advantages and limitation of our algorithm andpoints out some remaining future work.

2 Three forms of ACO algorithms

Real ants are capable of finding the shortest path from afood source to their nest without using visual cues byexploiting pheromone information. The double-bridge ex-periment further tested the self-catalysis behavior of the antcolony and revealed the characteristics of positive feedbackmechanism and self-organization.

2.1 Ant system

Ant system (AS) is the progenitor of all ACO algorithmsand was first applied to the TSP. Provided that there are n

Table 1 Comparison of ASP/DSP research in optimization criteria, assembly models and part numbers

Reference Optimization criteria Assembly informationmodel

Number of partsin case study

Algorithm based

Tool changes Reorientations Other criteria

[1] T T Similar operations grouped IM 11 GA

[2] T T PM 19 GA

[3] T F Assembly cost PM 13 GA

[4] T T IM 11 BC&GA

[6] F T Stability and geometric feasibility IM 15 PSO

[7] F T Interference times IM 22 DPSO

[9] T T Stability and accessibility andweight effect

PM 25 PSO

[11] T T Network 8 AS

[13] F T DM, DCG 16 AS

[14] F T Number of disassembled parts DM, DCG 8 AS

[15] T T Hybrid graph 12 AS

[16] T T DM 7 AS

[17] T T Directed graph 16 AS

[18] F F Disassembly cost DFIG 6 AS

[19] T T Number of disassembled parts DHIG 11 AS

[20] F F Cost and time and quality Feasibility graph 26 MMAS

[21] T T DM 12 AS

[22] F T Precedence constraint andconnection

DM 5 AS

[23] T T Operation type changes IM 18 AS

[24] T T Operation type changes IM 16 ACS

[25] F T MWM 9 AS

T adopted, F non-adopted, IM interference matrix, PM precedence matrix, DM disassembly matrix, DCG disassembly completed graph, DFIGdisassembly feasibility information graph, DHIG disassembly hierarchy information graph,MWMmoving wedge matrix, GA genetic algorithm, BCbacterial chemotaxis, PSO particle swarm optimization, DPSO discrete PSO, AS ant system, MMAS max–min ant system, ACS ant colony system

Int J Adv Manuf Technol (2013) 67:2819–2835 2821

cities and m ants. Each ant is initially located on a differentcity node i, and the pheromone intensity on the edgesbetween each two cities are the same. In moment t of thesearching process of ant k (Antk) (k=1, 2, …m), the proba-bilistic state transition rule Pk

ijðtÞ is calculated according to

the amount of pheromone and the heuristic information onthe edges, given by (1), which is called the random propor-tional rule.

PkijðtÞ ¼

t aij ðtÞ�η bij ðtÞP

s2allowedkt aisðtÞ�ηbisðtÞ

; j 2 allowedk

0; otherwise

8<

:ð1Þ

where allowedk={0, 1, …, n−1} is the cities that Antk isallowed to choose next, t ij is the pheromone intensity onedge E(i, j), ηij(t)=1/dij is the heuristic function, dij is thedistance between city i and j, α and β are the parameterswhich determine the relative importance of pheromone ver-sus distance.

Once all ants have built their tours after n moments,pheromone is updated on all edges according to

t ij ¼ 1� ρð Þt ij þXm

k¼1Δtkij ð2Þ

where 0<ρ<1 is a pheromone decay parameter, Δt kij is the

pheromone increment that deposited by Antk on E(i, j).Although AS was useful for discovering good or optimal

solutions for small TSPs (up to 30 cities), the time requiredmade it infeasible for larger problems. Ant colony system[10] and max–min ant system [31] are two typical improvedAS algorithms.

2.2 Ant colony system

Ant colony system (ACS) improves its performance mainlyfrom the three changes:

1. The state transition rule. The pseudo-random-proportionalrule is adopted in ACS, given by (3), which can balancebetween exploration of new edges and exploitation of apriori and accumulated knowledge about the problem.

s ¼ arg maxu2allowedk

t iu½ �a � ηiu½ �bn o

; if q � Q

S; otherwise

(

ð3Þ

where q is a random number uniformly distributed in [0, 1],0≤Q≤1 is a parameter that determines the relative impor-tance of exploitation versus exploration, and S is a randomvariable selected according to the probability distributiongiven in Eq. 1. This state transition rule favors transitionstoward nodes connected by short edges and with a largeamount of pheromone.

2. The global updating rule. In AS, all ants are allowed todeposit pheromone, which causes it cannot focus thesearch behavior on the optimal tour quickly and candecrease the search efficiency of the best tour. On thecontrary, the global updating rule of ACS is given as

t i; jð Þ 1� að Þ � t ij þ a � Δt ij ð4Þ

Δt ij ¼ Lgb� ��1

; E i; jð Þ 2 global best tour0; otherwise

�

ð5Þ

where 0<α<1 is the pheromone decay parameter, andLgb is the length of the global best tour from the begin-ning of the trial.

Only those edges that belong to the global best tourwill receive reinforcement, while the pheromone on otheredges will decrease because of the decay mechanism.This rule increases the difference of the pheromone be-tween the best tour and the worst tour and then makes thesearch quickly concentrated near the optimal tour.

3. The local updating rule. While building a solution of theTSP, each edge E(i, j) is visited by ants and the phero-mone on the edge is changed with the local updatingrule according to

t i; jð Þ 1� ρð Þ � t ij þ ρ � t0 ð6Þwhere 0<ρ<1 is a parameter, t0=(nLnn)

−1 is the tourlength constructed according to the nearest neighborheuristic method.

Here, the role of ρ is to shuffle the tours, by whichants can make a better use of pheromone information.Without this local updating, all ants would search in anarrow neighborhood of the previous best tour.

2.3 Max–min ant system

Max–min ant system (MMAS) improves its performancemainly in the following three aspects:

1. Pheromone trail updating. In MMAS, only the singleant that gets the iterative optimal tour is used to updatethe pheromone trails after each iteration. The modifiedupdating rule is given by (7)

t ij t þ 1ð Þ ¼ ρt ijðtÞ þ Δt best ð7Þwhere Δtbest=1/f(sbest), and f(sbest) denotes the solutioncost of either the iteration best solution (sib) or theglobal best solution (sgb). By this choice, the solutionelements which frequently occur in the best found sol-utions get a large reinforcement.

2. Pheromone trail limits. By limiting the influence of thepheromone trails in [tmin, tmax], the relative differences


between the pheromone trails can be easily avoidedfrom becoming too extreme to explore the solution insearch space during the run of the algorithm.

3. Pheromone trail initialization. Initializing pheromonetrails by tmax other than tmin or 0 can slow down theevolution of selection probabilities and lead to the in-crease of exploration for solutions during the first iter-ations of the algorithm to improve its performance.

From the comparison in Table 1, we can also findthat most existing researches on ASP/DSP are based onAS other than ACS or MMAS. However, the combina-tion of ACS and MMAS is one of the focuses of thispaper.

3 Particularities of ACO-based ASP

From graph theory point of view, given the graph G=(V, E),where V is the collection of components and E is the col-lection of the edges that connect each two components, andthe constraints and assembly cost between each contactedcomponent, then the core of ASP is to seek a Hamiltoncircuit that all parts are assembled only once with the min-imum cost, which is similar to TSP.

It is a trend of development that the universal ACOalgorithm should be specialized to solve problem in prac-tice. The ASP for complex products has the followingparticularities:

1. Strict constraints. In the combination optimization prob-lems such as TSP, quadratic assignment, and networkrouting, the objects of problem are usually simplified ascentroids and then modeled and optimized based ongraph theory. Components are the objects of ASP prob-lem. However, since assembly is a compact-structuredintegrality with components interacting and constrain-ing with each other, a sequence with the path withoutany collision must be found. Otherwise, the sequence isnot a feasible one, not to speak of the optimum one.Thus, components cannot just be simplified as a seriesof centroids. Owing to the strict constraints, initialnodes of ants cannot be randomly selected or selectedaccording to uniform distribution law. Furthermore, inthe process of ant constructing its tour, the next compo-nent to be selected belongs to the candidates that havegeometric feasibility and not disassembled. This strictconstraint can also decrease the total sequences. Thebase part, which is taken as the benchmark of assemblyoperations and the kernel part to assure the assemblystability, is the last one to be disassembled and shouldbe paid more attention to.

2. Multi-objective optimization. Several objectives andcriteria directly or indirectly influence the cost, time,

and efficiency of assembly, which are the main purposeof ASP. Since current ACO-based ASP algorithmsmainly focus on tool changes and reorientation, whileother criteria are hard to be automatically quantified orcalculated, it is difficult to distinguish the syntheticallybest sequences. Thus, the algorithms are prone to pre-mature convergence. Therefore, it is necessary to intro-duce and integrate more objectives that can beautomatically quantified.

3. Huge computational. Besides calculating the state tran-sition proportion, the global and local updating, ACO-based ASP also need constructing candidate list andmatrix operations resulting from the strict constraint,together with the calculation of multi-objective heuristicand fitness functions. The time and space complexity ofACO is exponential to the complexity and scale of theproduct. Therefore, it is urgent to get the optimum orsuboptimum solutions in acceptable time by integratingthe advantages of various algorithms.

4. Dynamic changes of information. The heuristic infor-mation between each node in TSP is determined before-hand and remains unchanged in the optimizationprocess, and the candidate list of each ant is the com-plementary set of the disassembled components. Whilein ASP/DSP, since the operation directions are influ-enced by the sequence and geometric constraints, heu-ristic information and interference matrix is changeddynamically, and the candidate list of each ant must becounted in real time.

4 Assembly relation matrix

Since matrix-based assembly models are more easier thangraph-based models to store and express the static or dy-namic relationships between components, they are widelyused to solve the problems of ASP/DSP.

4.1 Extended interference matrix

Interference matrix describes the collision interference rela-tionships between the moving component and the othercomponents on the way from the original assembly positionof the moving component to the infinite distance along theaxes of Cartesian coordinates (±x, ±y, ±z), and is widelyused to determine geometric assembly feasibility of compo-nent. However, since assembly directions along the nonstan-dard axes are seldom taken into account, existinginterference matrix cannot support the sequence/path plan-ning for products with complex structure. As shown inFig. 1, parts 2, 3 and parts 4, 5 are assembled onto part 6along the slanting directions, respectively. With the tradi-tional interference matrix in direction ±x, ±y, ±z, parts 2, 3


and parts 4, 5 are all constrained to part 6 and are all non-removable along the standard global coordinate system (GCS).

Besides the uniform GCS, here assembly coordinate sys-tems are extended to the non-uniform local coordinate system(LCS) of each component, as shown in Fig. 1. Each LCScorresponds to the main modeling feature of each component,which is also the mating feature to other mating componentsin the assembly space and finally decides the feasible assem-bly direction (usually the normal direction). As shown inFig. 1, the axial direction of the cylinder feature of part 5(direction zl) is proven to be just the feasible disassemblydirection. Based on the standard axial directions of GCS ±x,±y, ±z, the novel interference matrix is extended to additionalsix axial directions of LCS / \x, / \y, / \z, where / and \denote the positive and negative axial direction of theLCS, respectively.

Furthermore, the friction between contacted faces of twocomponents will influence their roughness and assemblycost. Thus, the assembly direction without friction shouldbe preference. As shown in Fig. 1, direction −zg and −zlshould be the preferred assembly directions for part 4,compared with directions ±xg and ±xl. Interference can bedivided into touch interference and hard interference. Touchinterference will lead to friction. Here, they are distin-guished and integrated into the novel interference matrixto find a frictionless direction.

Definition 1: extended interference matrix.Assuming assembly P is composed of N parts, an

integer edij represents the relation of collision interferencebetween part i (pi) and part j (pj) while pi moving to

infinity in direction d, where d=(+x, +y, +z, /x, \x, /y, \y,/z, \z), provided that:

edij ¼0; pi does not interfere with pj;1; pi touch interferes with pj;2; pi hard interferes with pj:

8<

:

Thus,E ¼ ½edij�9�N�N is termed the extended interfer-ence matrix (EIM). EIM can be divided into global inter-ference matrix (GIM, denoted asG) in GCS direction (d=+x, +y, +z) and local interference matrix (LIM, denoted asL) in LCS direction (d=/x, \x, /y, \y, /z, \z). The GIM indirection +z and the LIM in direction/z of the assembly inFig. 1 are shown as Fig. 2a, b, respectively.

In GCS direction (d=±x, ±y, ±z), if all elements in the ithrow (or column) of GIM are less than 2, that is max(G(d, i,j))<2 (ormax(G(d, j, i))<2), (j∊ [1,N]), part pi is removablein +d (or −d). In LCS directions (d=/ \ x, / \ y, / \ z), if allelements in the ith row of LIM are less than 2, that is max(L(d, i, j))<2, (j∊ [1, N]), then part pi is removable in d.

Therefore, from the matrices above, part 1 can be dis-assembled with friction in the direction +z and /z (+zcoincides with its /z) in the initial time. Parts 3 and 5 canbe disassembled with friction in the direction /z of theirown.

4.2 Contact–connection matrix

Parts can be divided into functional parts (such as shaft,gear), and connectors (such as screw, rivet, and key). Thus amatrix-based assembly model is presented to distinguish thetwo types of parts. Taking the integer cij as the contact andconnection relationship between pi and pj, provided that

cij ¼0; pi does not contact pj1; pi is a functional part and contacts pj2; pi is a connector and connects pj

8<

:

The matrix C ¼ ½cij�N�N is termed contact–connectionmatrix (CCM). The CCM of the assembly in Fig. 1 is shownin Fig. 2c. The part type can be automatically identifiedfrom the part name. For example, the part “bolt B” isidentified to be a connector because of the hidden characterstring “bolt”. Therefore, CCM can distinguish the general

3

4

5

6

1

2

LCS

GCS

Fig. 1 Two kinds of coordinate systems of an assembly; 1 lifting lug, 2block A, 3 bolt B, 4 block C, 5 bolt D, 6 base

+ 1 2 3 4 5 6

1 0 0 0 0 0 1

2 0 0 2 0 0 0

3 0 2 0 0 0 2

4 0 0 0 0 2 0

5 0 0 0 2 0 0

6 2 2 2 2 2 0

z / 1 2 3 4 5 6

1 0 0 0 0 0 1

2 0 0 2 0 0 0

3 0 0 0 0 0 1

4 0 0 0 0 2 0

5 0 0 0 0 0 1

6 2 2 2 2 2 0

z 1 2 3 4 5 6

1 0 0 0 0 0 1

2 0 0 1 0 0 1

3 0 2 0 0 0 2

4 0 0 0 0 1 1

5 0 0 0 2 0 2

6 1 1 1 1 1 0

CC

(a)Interference matrix in +z (b) Interference matrix in /z (c) Contact-connection matrix

Fig. 2 a Interference matrixin +z, b interference matrix in /z,c contact–connection matrix


contact from fastening connection in a compact structureand can be used to quantify the continuity and stability ofassembly operation.

Assembly products are mainly built on the platform of3D CAD such as SIEMENS NX, SolidWorks, Pro/Engi-neering, etc. Digital assembly models contain comprehen-sive geometric information and design intent. Therefore, thetechnology of further development based on 3D CAD plat-form can be an efficient method for automatically andrapidly generating EIM and CCM [26–28].

5 MMACS-based ASP

Either ACS or MMAS alone has some deficiency in solvingASP for complex products. Therefore, ACS and MMAS arehere combined into a MMACS to solve ASP, and somespecific measures are integrated into it. Based on the as-sumption that disassembly sequence is the reverse of assem-bly sequence, ASP is solved indirectly by starting with DSP.

5.1 Determine the number of ants

Each ant occupies some computer memory with a set ofCCM, EIM, and a list of sequence that contains part namesand directions. At the same time, a number of matrix oper-ations are performed by each ant when constructing a se-quence, which is very time consuming. Therefore, forcomplex products that contain a large number of compo-nents, too many ants will occupy a huge amount of storageand increase the computational cost, which is inefficient forASP. Otherwise, if the ants are too few to cover all solutionspace, the best sequence will be lost.

Let assembly P consist of n components and set thenumber of ants m to be the number of the components thatcan be disassembled initially. To initialize the algorithm,each ant is assigned on a different detachable componentas in (8).

Antstartk ¼ PfeasibleðkÞ; k ¼ 1; . . . ;mð Þ ð8Þ

where Antstartk is the start component of Antk and Pfeasible isthe set of the detachable components in the beginning,which can be counted with EIM, as aforementioned. There-fore, DSP is converted to a conditional path optimizationproblem that starts with m different detachable componentsand ends with the base part.

5.2 Multi-objective heuristic function

In this paper, reorientation and changes of tool changes arequantified in more detail, together with extra three criteria,including continuity, stability, and auxiliary stroke, which

are not well studied before. A multi-objective heuristicfunction is thus constructed.

1. Reorientation. The changes of assembly direction willincrease the difficulty and cost of assembly. However,the changes of direction are roughly classified into thesame and the different in existing researches, withoutconsidering the opposite one. In fact, a pair of compo-nents that possess the opposite assembly directions usu-ally have a mating relation or a symmetric relation.Thus, the paired component should be prior assembledthan other components that only possess different mainassembly directions. By this means, the stability of theassembly can be increased, the sequence can be moreunderstandable and the simulation can have a bettervisual effect. The heuristic function of reorientation(D) is given by (9).

Dij ¼1; di ¼ dj0:5; di ¼ �dj0:1; di 6¼ �dj

8<

:ð9Þ

where di is the disassembly direction of the ith compo-nent (pi). The disassembly direction of pi and pj can bedecided by EIM.

2. Parallelism. The parallelism here refers to the degreeof similar operations being accomplished in a batchmode, which is measured by the changes of assem-bly tool/gripper. The fewer the changes, the morethe parallelism and the shorter the assembly time is;the more compact structure of exploded view is.Manual input of assembly tools is needed in exist-ing researches, which is a tedious work. Generally,components of the same type (such as nut or screw)need assembly tools of the same type (wrench orscrewdriver, respectively). Components of the sameweight (or volume) is the same model number,which the assembly tool of the same type andmodel number are generally used. The weight andvolume of components can be acquired by APIfunctions from 3D models, while the types of com-ponents can be identified by their part names.Therefore, whether the types and model numbersof assembly tools are the same can be determinedmainly by component weight (or volume) and sup-plemented by component type. The heuristic func-tion of parallelism (P) is given by (10).

Pij ¼1; Wi ¼ Wj;0:5; Wi 6¼ Wj&Ti ¼ Tj;0:1; Wi 6¼ Wj&Ti 6¼ Tj:

8<

:ð10Þ

where Wi is the weight of pi, and Ti is the part typeof pi.


3. Continuity. Continuity refers to the physical and logicalmating relation and connection between two continu-ously assembled components, which makes assemblerconcentrate on finishing the assembly in his workingarea. Several classical algorithms based on liaison graphhave considered the importance of contact or connectionrelationship in assembly modeling and sequencing.However, as shown in Table 1, continuity has not beentaken into account in existing ASP/DSP researches thatbased on ACO. The heuristic information of continuity(U) can be determined by CCM, given by (11).

Uij ¼1; C i; j½ � ¼ 20:5; C i; j½ � ¼ 10:1; C i; j½ � ¼ 0

8<

:ð11Þ

4. Stability. The stability of assembly influences thereliability of assembly operations and the complex-ity of fixtures. Owning the difficulty in acquiringinterference matrix, existing researches assume allcomponents are of weightlessness, which is not co-incide with the practice engineering that any com-ponent without the support of other componentsneed extra fixtures to prevent it from dropping.Moreover, the components with less connectionshould be disassembled first to avoid extra resis-tance and mechanical damage. The heuristic func-tion of stability (S) is given by (12).

Sij ¼1;

Pn

k¼0EIMG k; jð Þ ¼ 0

CCDmax�CCDj

CCDmax; otherwise

8<

:ð12Þ

where EIMG is the EIM in gravity direction (−z as

default), CCDj ¼Pn

k¼1C k; j½ � is the remained contact-

connection degree (CCD) of pj, and CCDmax is themaximum CCD of all components.

5. Auxiliary stroke. Frequent changes of operation sta-tion will generate auxiliary stroke, which will in-crease the auxiliary assembly time. Thus the ASPproblem contains a TSP problem, to some extent.For convenience of description, the 3D ASP issimplified to a 2D problem, as shown in Fig. 3,

where the nodes denote four components withoutany restrict geometric constraint.

The distance of auxiliary stroke in sequence①→②→③→④ (like the dotted path) is 623 mm, whilethe distance in sequence③→①→②→④ (like the solidpath) is 383 mm. However, this TSP-like auxiliary strokethat concealed in ASP has not been noticed in existingresearches. The heuristic function of auxiliary stroke (A)is given by (13).

Aij ¼ Lmax � Lij� �

=Lmax ð13Þ

where Lmax is the maximum distance between the boundingbox centers of each two components, and Lij is the distancebetween the bounding box centers of pi and pj. With thisfunction, the ants are inclined to select the components thatare closer to the just disassembled component.

Therefore, the multi-objective heuristic function is givenby (14).

ηij ¼ w1Dij þ w2Pij þ w3Uij þ w4Sij þ w5Aij ð14Þ

where w1−w5 (P5

i¼1wi ¼ 1) are the weights of each objec-

tive, according to the structure particularity and the assem-bly resource availability of the specific product.

5.3 Multi-objective fitness function

To reflect the assembly cost of the sequence, each singlefitness function is inversely related to the correspondingheuristic function that mentioned above to some extent.The meaning of each variable can be referred to the afore-mentioned functions, for lack of space.

1. Reorientation. The increment of assembly cost causedby direction changes from the (i−1)th sequence compo-nent to the ith one is given by (15).

fD i� 1; ið Þ ¼0:1; di�1 ¼ di;0:5; di�1 ¼ �di;1; di�1 6¼ �di:

8<

:ð15Þ

where di is the disassembly direction of the ith compo-nent of the sequence. The function indicates that thecost is less in the same disassembly direction than indifferent main directions.

2. Parallelism. The increment of assembly cost caused bytool changes from the (i−1)th sequence component tothe ith one is given by (16).

fP i� 1; ið Þ ¼0:1; Wi�1 ¼ Wi;0:5; Wi�1 6¼ Wi&Ti�1 ¼ Ti;1; Wi�1 6¼ Wi&Ti�1 6¼ Ti:

8<

:ð16Þ

1 2

3 4

100

100

100

Fig. 3 TSP-like auxiliary strokes in ASP


The function indicates that the cost is less by usingthe tool of the same type and model number than usingthe tools of different model number or types.

3. Continuity. The increment of assembly cost caused bycontinuity changes from the (i−1)th sequence componentto the ith one is given by (17).

fU i� 1; ið Þ ¼0:1; C i� 1; i½ � ¼ 20:5; C i� 1; i½ � ¼ 11; C i� 1; i½ � ¼ 0

8<

:ð17Þ

4. Stability. The increment of assembly cost caused bystability of the ith sequence component is given by (18).

fSðiÞ ¼0:1;

Pn

k¼0EIMG k; ið Þ ¼ 0;

n�ið Þ�CCDiPn

k¼0CCDk

; otherwise:

8>>><

>>>:

ð18Þ

The function indicates that the cost is less byearlier disassembling the components with smallerCCD.

5. Auxiliary stroke. The increment of assembly costcaused by auxiliary stroke from the (i−1)th sequencecomponent to the ith one is given by (19).

fA i� 1; ið Þ ¼ Li�1;i=Lavg ð19Þ

where Lavg=TNN/n is the average value of the auxiliarystrokes (from bounding box centers) of each pair ofcomponents generated by nearest neighbor greedy algo-rithm.

Therefore, the multi-objective fitness function is givenby (20).

F ¼Xn

i¼2w1fD i� 1; ið Þ þ w2fP i� 1; ið Þ þ w3fU i� 1; ið Þ þ w4fSðiÞ þ w5fA i� 1; ið Þ½ �=n ð20Þ

The fitness function identifies the quality of the currentsequence to a greater degree of distinction and enhances theexploration of ant colony for high-quality sequences.

5.4 Max–min pheromone limits

As for ASP of complex products, certain edges may seldombe visited by some ants, because of the strict constraints andthe economy limit of ants number. After several globalupdating at the beginning, the pheromone of these edgesmay even decay to zero, while that of other edges areabundant. Then the algorithm will be convergent premature-ly. Therefore, these ants are useless to some extent. To avoidthe differences between the pheromone trails from being tooextreme to explore the solution and to save the computa-tional resource, the measure of max–min pheromone limit ofMMAS is introduced to limit the influence of the phero-mone trails to [tmin, tmax], given by (21).

tmax ¼ a Fnnð Þ�1; tmin ¼ aFnnð Þ�1 ð21Þ

where a>1 is a coefficient to control the interval scalabilityand set to be a ≤ 1/α, 0<α<1 is the pheromone decayparameter, and Fnn is the optimal solution value with thenearest neighbor greedy algorithm based on auxiliary stroke.

Set tij(0)=tmax, therefore, at any time, ðaFnnÞ�1 � t ijðtÞ � a

ðFnnÞ�1.

5.5 Global updating rule with mixed strategies

In the stage of global updating, to avoid the blindnesscaused by simply using the iteration best solution (Fib) andto avoid the loss of exploration caused by simply using theglobal best solution (Fgb), the algorithm uses the mixedstrategies by choosing Fib as a default for updating thepheromone and using Fgb only every five iterations, givenby (22) and (23).

t ij t þ 1ð Þ 1� að Þ � t ijðtÞ þ a � Δt ijðtÞ ð22Þ

Δt ijðtÞ ¼Fgb

� ��1; Mod t; 5ð Þ ¼ 0 andE i; jð Þ 2 global best tour

Fibð Þ�1; Mod t; 5ð Þ 6¼ 0 andE i; jð Þ 2 iteration best tour0; otherwise

8<

:

ð23Þ

where Δt ijðtÞ is the increment of pheromone at time t, andMod is the complementation operation. The mixed strategiescan well balance between the exploration and the guidanceof ASP for complex products.

5.6 Performance appraisal for initial allocation

The m ants are initially allocated to the m different detach-able components. However, after several iterations, Antkmaybe still unable to obtain any Fib or Fgb, and may even


construct the iteration-worst solution (Fiw) for severaltimes. This is due to the birth defect of the initial-allocated component possessed by Antk. Therefore, thiskind of ants is useless for the global optimization. Toreduce the full life cycle occupation of the computingresources by these ants and make full use of the limitedant colony, the mechanism of appraising employee byperformance in human society is introduced: The em-ployee with the best performance in the current weekwill be rewarded, and then let the bygones bad recordbe bygones. However, the employee with the worstperformance for several weeks will be required to learnfrom the best one or he will be retired. To make ananalogy to employee, each Antk is allowed to get Fiw

for Wallowed times. Worst(k) is used to count the timesthat Antk gets Fiw. If Worst(k) is more than Wallowed, theinitial allocation of Antk is changed to that of Fgb, i.e.Antstartk ¼ Antstartgb . If Antk obtains Fib, then its previousbad records is cleared and it is provided with morechance of exploration, given by (24).

WorstðkÞ ¼ WorstðkÞ þ 1; FðkÞ ¼ Fiw

0; FðkÞ ¼ Fib or WorstðkÞ �Wallowed

�

ð24Þ

where F(k) is the fitness value of Antk, Wallowed=Ncmax/m, and Ncmax is the total iterations. That is to say, allthe iterations are evenly assigned to the m ants, andeach ant is not allowed to repeat too much mistakes.The measure of performance appraisal for initial alloca-tion (PAIA) can transfer the computing resource to theadvanced ants and accelerate the convergence speed.

5.7 Group method for same components

Components of the same model make up a large proportion(about 60 %) over the total components of complex prod-ucts. Most same components are fasteners (such as bolts andnuts) and are often disassembled synchronously. ACO algo-rithm cannot insure to entirely find each group of samecomponents, only with the heuristic function on parallelism.Thus, the generated sequence will be too disordered tosimulate assembly process or generate exploded viewefficiently.

This deficiency can be overcome by the group methodthat forcing the same components to be disassembled simul-taneously: (1) After pi has just been disassembled, the com-ponents that same to pi (in name or weight) are searched toform a collection Ps, from the components that have notbeen disassembled and can be disassembled on the premiseof geometric feasibility based on EIM. (2) According to the

similarity of the disassembly direction to pi (the same, theopposite or otherwise), the components in Ps are classifiedinto three subcollections, namely Pd1, Pd2, and Pd3. (3) Thecomponents in each subcollection are sorted, according tothe nearest neighbor algorithm based on auxiliary stroke. (4)The same components are forced to be disassembled by thegeneral order of Pd1, Pd2, and Pd3.

The group method makes up the deficiency of purelydepending on heuristic search for all the parallel parts ineach group. Furthermore, since the measure of sorting dis-tance replaces the calculation of heuristic information andpheromone updating for the same components, plenty oftime is saved and the sequence combination is reduced, sothat the algorithm can be fast convergent to the globaloptimum.

5.8 Algorithm steps

The advantages of pseudo-random proportional rule and thelocal updating rule of ACS are combined with the globalupdating rule with mixed strategies and max–min phero-mone limits of MMAS, and some specific measures areintegrated in the following MMACS algorithm for ASP/DSP:

Step 1 Initialization of the basic data. The initializationincludes the control parameters (such as Ncmax, α,w1~w5) and the large-scale data decided by productstructure (such as the heuristic and fitness matricesof parallelism, continuity, and auxiliary stroke).

Step 2 Initialization of the ant information. The base part isdeduced byCCD and ant numberm is determined byEq. (1). Each ant is assigned with a set of CCM andEIM, which are automatically acquired by interfer-ence detection beforehand. Fnn, tmax, and tmin arecalculated with Eq. (8), and the elements in thepublic pheromone matrix are initialized with tmax.

Step 3 Inner iteration in sequence construction. In processof each ant constructing its tour in current iteration,candidate components with geometric feasibilityare determined with EIM. Heuristic informationis calculated with Eq. (14) and the component tobe disassembled is decided with (3). Once a com-ponent is disassembled, the CCM and EIM of eachant are updated and the pheromone is shuffled withthe local updating rule in Eq. (4). The multi-objective fitness value is then calculated. Thegroup method for same components is performedwhen necessary.

Step 4 Outer iteration in sequence construction. Aftereach iteration, the best ant and the worst one


are obtained, according to the sequence fitnessvalue of each ant. As for the best ant, the publicpheromone matrix is updated with the globalupdating rule with mixed strategies in Eqs. (22)and (23), and the elements are limited to theinterval [tmin, tmax]. The performance of theworst ant is appraised with Eq. (24), and itsinitial allocation is changed when necessary.

Step 5 End of the routine. The algorithm terminates afterall the iterations are completed. The global bestsequence that output in the last time is the finalsolution. The global convergence of the algorithmis drawn with colorful curves, and the rationality ofthe sequence is verified by automatic simulationon 3D CAD platform in AutoAssem.

The flow chart of MMACS-based ASP is shown inFig. 4.

6 Case study and analysis

6.1 Case study

AnASP system named AutoAssem has been developed basedon Siemens NX, which can realize automatic generation ofCCM, EIM, exploded view, and ASP/DSP algorithms basedon MMACS and priority rules screening (PRS) [32, 33]. Avalve that consists of 29 parts is taken as the example, asshown in Fig. 5. The part numbers are consistent with those inthe tree structure.

AutoAssem extracts the information of parts first andexecutes interference detection in GCS directions. TheEIM in direction +z is shown in Fig. 6, which is consistentwith the fact. All matrices are saved to the database.

The parameters are chosen as follows: Ncmax=100, β=4.0, α=0.1, ρ=0.1, Q=0.2, w1=2, and w2~w5=1. Part 17 isautomatically determined to be the base part, and 13 ants are

Initialize heuristic and fitnessmatrices of parallelism,

continuity, auxiliary stroke

CCMEIM

Determine ants number m;Locate ants on parts with (8);

Initialize iteration Nc=0

Clear Sequencek

move to Antkstart

Calculate τmax , τmin with (21);Initialize pheromone with

τmax; Determine the base part

1 5

,,Q,Ncmax

,

~ ...w w

Determine Candidatek(i)with EIM

Choose next part with(3) from Candidatek(i);Local updating with (6)

Calculate fitness valueFk(i); Update CCM, EIM

Realtime output convergencecurve & global-best sequence

k ≥ m

Nc ≥ Ncmax

i ≥ n

Yes

Search for Antib, Antiw

and Antgb

For Antib& Antgb: Globalupdating with (22) ;

Limit τij to [τmin , τmax]

For Antiw: Appraise theperformance with (24);If Worst(k) > Wallowed,then Antiw

start=Antgbstart

Yes

Acquirematrices

No

Nc = Nc+1

Output the global-bestsequence;

Algorithm terminate;

Yes

Assignmatrices

Use the group method todisassemble the samecomponents together

k++No

i++

No

Inner interation

Outer interation

Fig. 4 Flow chart of MMACS-based ASP


needed. During the iterative process, seven temporary opti-mal sequences are displayed dynamically, as shown inFig. 7. The final global best disassembly sequence is listedin Table 2, the reverse of which is the assembly sequence.

Five parallel groups, including 19 parts and eight changesof direction, are found. The multi-objective fitness value ofthe sequence is 3.034. The total running time is 5.2 s, on aHP8400 work station (2GHz CPU and 2GB RAM), and thetime to get the final global-best solution is 2 s in the 45thiteration. The convergence curves are shown in Fig. 8.Based on the method in [30], the exploded view of the valve

is generated automatically with the above sequence, asshown in Fig. 9.

6.2 Comparison of optimization measures

The algorithm that only contains the group method is takenas the basis of comparison. The contribution of each opti-mization measure is verified through ten experiments, on theminimum, maximum and average fitness value of the globalbest sequence and the iterations taken for the sequence (Nt),as shown in Table 3.

Fig. 5 Example of valveassembly; 1–6 nut A; 7–12 boltA; 13, 29 pipe; 14 bolt B, 15nut B; 16 rod; 17 house; 18cover; 19 shaft; 20 key; 21 arm;22 valve; 23 nut C; 24–25screw A; 26–28 screw B

Fig. 6 The interface for acquiring EIM in direction +z


The following conclusions can be obtained through theexperimental data:

1. With any single optimization measure, the minimumfitness value 3.034 can be found at least once in the

Fig. 7 Output global best sequences and fitness values

Table 2 Sequence of the valve generated by MMACS-based ASP

No. Part Direction Parallel No. Part Direction Parallel

1 10 +y 1 16 2 −x 4

2 11 +y 1 17 1 −x 4

3 12 +y 1 18 15 −z 0

4 9 −y 1 19 14 +z 0

5 7 −y 1 20 16 +z 0

6 8 −y 1 21 23 +z 0

7 29 −y 2 22 21 +z 0

8 13 +y 2 23 28 +z 5

9 25 +y 3 24 26 +z 5

10 24 +y 3 25 27 +z 5

11 22 +y 0 26 20 −x 0

12 3 +x 4 27 18 +z 0

13 6 +x 4 28 19 +z 0

14 4 −x 4 29 17 +z 0

15 5 −x 4

Iteration-average

Iteration-best

Global-best

Fitn

ess

valu

e

Iteration

Fig. 8 The convergence curves of MMACS-based ASP


10 experiments, although there are some differencesbetween each optimization measure in the overall qual-ity of sequence and the speed of convergence.

2. With the max–min limits of pheromone, the overallquality of the solution is improved and the convergencerate is slowed down, which means the measure canprevent premature convergence.

3. By using the global updating rule with mixed strategies,the overall quality of the solution is improved and theconvergence rate is slowed down, which means the mea-sure can balance well between the exploration of iteration-best solution and the guidance of global best solution.

4. With the measure of PAIA, the overall quality of thesolution is slightly improved and convergence rate isspeeded up, which means the measure can transfer thecomputing resource from the worst ant to the better oneto accelerate the process of algorithm.

5. By utilizing all the four measures simultaneously, Fgb

equals to 3.034 at each time, and the convergence rate is

also significantly improved. Thus additional enhance-ments have risen without any conflict.

6. Without the group method, although minimum Fgb=3.032 can be got by only utilizing the three other meas-ures, the quality of the maximum and average Fgb hasfallen, and more iterations are needed. The gap betweenthe iteration best and the iteration average and thedifference between iterations all become wider, asshown in Fig. 10. Purely using the heuristic methodand treating all the five objectives fairly, global bestsequence is obtained at the expense of losing completeparallelism, which will decrease the effect of the se-quence simulation and the generated exploded view.

6.3 Comparison with PRS-ASP

The algorithm of PRS-ASP generates sequence by screeninga single optimum component in every cyclic process, fromthe five criteria and objectives, to ultimately comprise anoptimized sequence automatically, using the above matrices[29]. By applying PRS-ASP to the same instance, thestrength and weakness between the heuristic method andthe direct method are compared and analyzed, from aspectsof running time, occupied space and sequence quality, froman intuitive and objective view. The generated sequence isshown in Table 4.

Through 10 experiments, the results of running time andoccupied space of the two algorithms, counted by the win-dows task manager, are listed in Table 5.

1. Running time. The average running time to get theglobal best sequence of MMACS-ASP is 20 times asmuch as that of PRS-ASP. This indicates that the effi-ciency of heuristic algorithm is lower than that of directmethod, which is determined by the essence of theiterative optimization of heuristic algorithm.

2. Occupied space. The occupied space includes storagespace and CPU resources. As for storage space,MMACS-ASP occupies about eight times as much asPRS-ASP. The relation between storage space M, num-ber of ants m and number of components n in MMACS-

Fig. 9 The exploded view of valve generated by AutoAssem

Table 3 Comparison of fouroptimization measures inMMACS-based ASP

Optimization measure Fgb Nt

Min Max Average Min Max Average

Group method (①) 3.034 3.078 3.040 11 150 44

Max-Min limit (②)+① 3.034 3.041 3.036 18 180 56

Mixed strategies (③)+① 3.034 3.041 3.036 14 146 47

PAIA (④)+① 3.034 3.072 3.040 11 84 32

②+③+④ 3.032 3.092 3.070 17 147 60

②+③+④+① 3.034 3.034 3.034 17 70 33


ASP is approximate to M∝(m×n2), while the relation-ship in PRS-ASP is approximate to M∝n2. Therefore, itis the number of ants that determines the storage spaceoccupied of MMACS-ASP is inevitable more. As forCPU resources, MMACS-ASP occupies about fourtimes the space of PRS-ASP, which shows that thecomplexity of the former is much larger than the latter.

3. Intuitive sequence quality. The sequence quality of thetwo algorithms can be directly compared by Tables 3and 4, which are different mainly in parallelism andreorientation. As for parallelism, although five groupsof parallel components are both found by the two algo-rithms, the total number of same components in

MMACS-ASP are one more than that in PRS-ASP(component 28 in group 5). Thus MMACS-ASP is morelogical and overcomes the incapability of the latter inexploring the optimal solution in depth. As for reorien-tation, the exact direction changes eight times inMMACS-ASP while 11 times in PRS-ASP, and themain direction changes four times in the former whilesix times in the latter. Therefore, the sequence quality ofMMACS-ASP outperforms PRS-ASP in parallelismand reorientation, and the global search capability ofheuristic algorithm is reflected.

4. Objective sequence quality. With the fitness functions ofMMACS-ASP, the fitness values of each objective in thetwo algorithm can be calculated, by which the objectivesequence quality is reflected, as shown in Table 6.

It is shown that MMACS-ASP is better than the PRS-ASP in terms of overall performance. The fitness values ofparallelism and reorientation further confirmed the differ-ences between the two algorithms. As for auxiliary stroke,the fitness value of MMACS-ASP (0.653) is also smallerthan that of PRS-ASP (0.675), which indicates that byintegrating the idea of TSP optimization into ASP, a betterrouting performance can be achieved. As for continuity andstability, since both algorithms have made good use ofcontact and connection relation between components, thetwo algorithms are not very different.

In summary, all criteria of sequence quality of MMACS-ASP are superior to that of PRS-ASP, in aspects of intuitivesimulation and objective numerical analysis. On the contrary,PRS-ASP has its obvious advantages in running time andoccupying space and the generated sequence is also logical.Therefore, the two algorithms can complement each other.

6.4 Comparison with other heuristic algorithms

Other heuristic algorithms, such as GA and PSO, are alsoapplied to solve ASP by the members of our group. Thisfacilitates the analysis of the performance of MMACS-ASPsynchronously. The valve is adopted in both case studies.

1. Comparison with GA. Wang et al. [32, 33] presented aGA-based ASP, where reorientation and interference aretaken as optimization objectives. Randomly initializedpopulations of 200 chromosomes are used and the

Iteration-average

Iteration-best

Global-best

Fitn

ess

valu

e

Iteration

Fig. 10 Convergence curve of MMACS-based ASP without the groupmethod

Table 4 Disassembly sequence of the valve generated by PRS-ASP

No. Part Direction Parallel No. Part Direction Parallel

1 23 +z 0 16 4 +z 3

2 24 +y 1 17 29 −y 4

3 25 +y 1 18 13 +y 4

4 22 +y 0 19 26 +z 5

5 11 +y 2 20 27 +z 5

6 12 +y 2 21 15 −z 0

7 10 +y 2 22 14 +z 0

8 7 −y 2 23 16 +z 0

9 9 −y 2 24 21 +z 0

10 8 −y 2 25 28 +z 0

11 2 −y 3 26 20 −x 0

12 5 −y 3 27 18 +z 0

13 6 +z 3 28 19 +z 0

14 3 +z 3 29 17 +z 0

15 1 +z 3

Table 5 Efficiency comparison of MMACS and PRS in running timeand occupied space

Algorithm Running time Storage space CPU resourcest/s M/kb c (%)

MMACS-ASP 2.0 7200 50

PRS-ASP 0.1 900 13


crossover and mutation rates are set to 0.95 and 0.9. Thealgorithm was convergent at the 220th generations, andthe computation time was 25 s. Among the global bestsequences, some run counter to the feasibility of collision-free assembly path. Thus MMACS-ASP only consumes10 % as much as GA-based ASP, with the additionalassurance of all geometric feasibility. Therefore, MMACSis superior to GA in efficiency and accuracy, when solv-ing ASP problems.

The difference is caused by the following reasons: (1)The optimization performance of GA are significantlyaffected by the initial population of sequences. With theinitial sequences of poor quality, GA cannot assure toconverge to better sequence within an acceptable time.(2) The two main genetic operators of GA, includingcrossover and mutation, are iterative searching the globalbest solution randomly and blindly, under a certain opera-tion probability. The operators do not consider the essentialcharacteristic of the strong constraint in ASP. Thus, thealgorithm is prone to randomly search solutions in laterstage, which is no good to improve the quality of assemblysequence. (3) While constructing a whole feasible se-quence, each ant in MMACS-ASP is making a strategicdecision of choosing the next component and accumulat-ing the fitness value. However, in GA, sequence is ran-domly constructed and evaluated afterward, and a largenumber of invalid solutions are generated in the process.Therefore, the convergence speed and the execution speedof GA-ASP is much slower than that of MMACS-ASP

2. Comparison with PSO. In our research group, Yu et al[6] designed a PSO-based ASP system, where reorien-tation and contact relationship were taken as optimiza-tion objectives. The population size was set to 100, thetermination generation number was set to 400, the localand global optimum coefficients are set to 0.8, and theinertia weight was set to 0.9. The initial population wasgenerated automatically, and the algorithm took 28 s to

be convergent to the optimal, after a 245-generationevolution. However, some global best sequences alsorun counter to the feasibility of collision-free assemblypath. Therefore, MMACS-ASP only consumes 1/14 asmuch as PSO-based ASP, together with the obviousadvantages in efficiency and accuracy.

To sum up, the comparison of ASP based on GA,PSO, and MMACS with the valve assembly is shown inTable 7.

7 Conclusion and future work

The presented MMACS-ASP has the advantages of bothACS and MMAS. The integrated optimization measureshad obviously improved the performance of MMACS-ASP. The five optimization criteria, including reorienta-tion, parallelism, continuity, stability, and auxiliarystroke, can all be automatically quantified and integratedinto the multi-objective heuristic and fitness functions,so that tedious manual input of criteria and index isavoided and the synthetical best sequence is identified.The EIM extends the directions for describing the inter-ference information to the axes of local coordinate sys-tem in each component, and geometric disassemblyfeasibility of MMACS-ASP in the slanting directionscan be determined with it. Although the slanting direc-tions do not appear in the assembly of the valve, theycan be found in our prior case study [29].

Based on the developed assembly planning systemAutoAssem, each optimization measure is verified to beeffective in sequence quality and convergence rate. Themeasure of performance appraisal for initial allocation ofcomponents can transfer the computing resource from theworst ant to the better one and accelerate the convergence.The group method for same components makes up thedeficiency of purely depending on heuristic search for allparallel parts in each group and therefore insures a goodeffect on assembly simulation and exploded view. Com-pared with PRS-ASP, MMACS-ASP is superior to the direct(or exact) method in intuitive sequence quality and objectivefitness value, but is certainly inferior in running time andoccupying space. Comparing with the GA or PSO-basedalgorithm, MMACS-ASP has obvious advantages in effi-ciency and geometric feasibility. The defects raised in priorwork are overcome to some extent.

Table 6 Comparison ofMMACS and PRS in fitnessvalue

Algorithm Sequence Parallelism Continuity Stability Reorientation Stroke

MMACS 3.034 0.531 0.686 0.357 0.276 0.653

PRS 3.195 0.562 0.686 0.358 0.352 0.675

Table 7 Comparison of GA, PSO, and MMACS with the valveassembly

Algorithm Populationsize

Convergencegeneration

CPU time t/s Interferencesin sequence

GA 200 220 25.0 4

PSO 100 245 28.0 2

MMACS 13 45 5.2 0


In another case study of MMACS-ASP, it takes 300 s tosolve a gear reducer that consists of 103 parts. The efficien-cy is not very satisfied for some aircraft components thatconsist of over 1,000 parts. How to improve the efficiencyby combining MMACS with PRS to solve the ASP ofcomplex products, and how to use Pareto optimal approachto deal with this multi-objective optimization are our workin the future. The automatic generation of hierarchical ex-ploded view based on assembly sequence for complex prod-ucts with the tree structure of more than four levels isanother interesting topic.

Acknowledgments The support of National Natural Science Foun-dation of China (no. 51105069) in carrying out this research is grate-fully acknowledged.

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http://dx.doi.org/10.1007/s00170-012-4615-0

a max–min ant colony system for assembly sequence planning

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