# Minimizing Cost of Assembly of an Interrelated Dimensional Chain Product Using ABC Algorithm

Authors: Mahalingam Sivakumar; Nagarajan Lenin; Kandasamy Jayakrishna; Natarajan Eswara Prasath
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Selective assembly is a method where components made with wider tolerance are grouped into a number of bins. Based on the best combination of the bin, the corresponding group components are randomly selected and matched together to make an assembly. Existing techniques focused on equal group number partitioning of components, equal probability, equal group width, and equal area methods to minimize either clearance variation or surplus parts using different optimization techniques. Mostly, simple assemblies with two or three components are worked by various authors in the literature without considering their original dimension by considering only their component’s tolerance. In the present work, components are classified into different unequal group numbers based on their tolerance values. The interrelated dimensional assemblies are made in a single stage by matching the parts based on the best bin combination obtained by the artificial bee colony algorithm. A simple linear assembly and a three-armed knuckle joint assembly are considered examples of problems to demonstrate the effectiveness of the proposed method by minimizing the manufacturing cost.


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## Body

## 1. Introduction

Manufacturing of products involved various manufacturing processes to ensure the products’ geometry, shape, size, and accuracy. A part cannot be manufactured precisely to the nominal dimension due to inherent natural variation in men, machine, material, and manufacturing processes. Tolerances of a mechanical assembly have a good bonding with the cost of manufacturing, quality, and reliability of a product. From the view of a designer, tolerance decides the fits, quality, and working of the final assembly. From the manufacturers’ perspective, tolerance involves selecting machines, jigs and fixtures, workmanship ability level, machine setup costs, inspection methods, amount of scrap, and rework. Hence, the identification of scientific methods for tolerance allocation to the components of the assembly in view of achieving the manufacturer’s objectives is the great motivation to the researchers.The precise assembly needs close tolerance components to perform the product’s required functioning, leading to secondary operations, increasing manufacturing costs. Selective assembly plays a vital role in making precise assemblies from wider tolerance components by classifying and matching parts randomly based on the best bin combinations. The researchers in the different period had used different grouping methods such as equal group number, equal probability, equal group width, and equal area methods to categorize the components into bins. The cost of product/assembly obtained by these methods is mainly based on two factors: clearance variation and surplus parts. As per the existing literature, these methods were contributed to minimizing the manufacturing cost of the product with minimum clearance variation and minimum surplus parts. Still, an effective method is required to minimize the manufacturing cost of a product to a further extent by minimizing the clearance variation and keeping zero surplus parts. This alternate method, namely, the unequal grouping method, has been introduced in this work for the stated purpose. This is the novelty and contribution of the proposed work.Various authors had dealt with the above-said problem environment using nontraditional optimization tools. Important kinds of the literature among them are presented in the next section.


## 2. Literature Review

Precise assembly requires precise components for the smooth functioning of the product. It requires stringent tolerance in the interchangeable assembly, whereas wider tolerance components can make precise assembly in selective assembly. In the literature, most selective assembly techniques focused on either minimizing surplus parts or minimizing the clearance variation in the assembly. Few works on minimizing surplus parts are given below.The authors handled the surplus parts minimization along with the clearance variation minimization at different periods. Chan and Linn [1] proposed a grouping method for parts of the different distribution with a cumulative distribution function in selective assembly to minimize surplus parts. Thesen and Jantayavichit [2] introduced small buffer storage of parts to evaluate a high-speed station for selective assembly of high-precision automotive components. Kannan and Jayabalan [3] proposed manufacturing mean of the mating parts before production based on the clearance for complex assemblies to minimize surplus parts where the group tolerance was designed to satisfy the clearance specifications of the assembly. Kern [4] developed a closed-form equation for various selective assembly techniques to minimize variation and surplus parts with a general approach applicable to different distribution variations. Matsuura and Shinozaki [5] introduced an optimal mean shift to minimize surplus parts for equal width partitioning schemes where the component with smaller variance was manufactured at three shifted means. Few works on minimizing clearance variation are described below.Various researchers had used different approaches and algorithms to solve the selective assembly problems. A few of them are discussed here. Pugh [6] proposed an assignment algorithm to reduce the variability by alternatively matching small to large partitions using an enhanced BASIC program that generates desired group numbers for the given component distributions. Pugh [7] studied the performance of the selective assembly when a component exhibits dissimilar variances. The author proposed truncating the component distributing more significant variances so that the resulting component variances are equal and discarded the components, which falls at the distribution extremes. Fang and Zhang [8] adopted the concepts of set theory and probability method to develop a predictive model for quantitatively estimating the matchable degree between mating parts for selective assembly. Babu and Asha [9] used Taguchi’s loss function to find the difference from the mean values of obtained minimum clearance variation. Very few authors focused on manufacturing cost in selective assembly. It is given below.The researchers in different time span made the manufacturing cost reduction by solving the selective assembly problems using GA. Kumar and Kannan [10] used GA for solving problems associated with the fan shaft assembly. The appreciable percentage of savings in the manufacturing cost was obtained through GA. Manickam and De [11] proposed a new method to find the combinations of best groups of components for reducing manufacturing cost using GA. Aderiani et al. [12] discussed a multistage approach using GA to solve hole and shaft assembly with the objectives of minimizing the clearance variation and zero surplus parts. Hui et al. [13] used a data-driven modelling approach using the combination of GA and oversampling technique to evaluate the quality of the linear axis assembly of the machine tool. Good results were obtained using the proposed approach than the previous literature.The authors had used different optimization tools during different periods for solving the selective assembly problems. Authors have used optimization tools such as GA, parallel population GA [14], and simulated annealing [15] in the literature to determine the best bin combination. The artificial bee colony (ABC) algorithm has not yet been implemented in selective assembly techniques. Several applications of the ABC algorithm in different fields are explained below. Singh [16] addressed an ABC algorithm for the leaf-constrained minimum spanning tree problem and compared the performance of the ABC approach. Computational results showed the superiority of the new ABC approach over all the other approaches. The developed approach obtained better quality solutions in a shorter time. Karaboga and Akay [17] developed a modified ABC algorithm for solving constrained optimization problems. 13 sets of benchmarking problems were considered to evaluate the modified ABC algorithm, and the developed method outperforms with other methods. The performance of the ABC algorithm also tested for real engineering problems existing in the literature. Karaboga and Ozturk [18] presented a novel clustering approach using the ABC algorithm. The ABC algorithm is used to cluster the data on benchmark problems. The performance of the ABC algorithm is compared with the PSO algorithm and the other nine classification techniques from the literature. The simulation results indicated that the ABC algorithm could efficiently be used for multivariate data clustering.Asadzadeh [19] proposed a parallel ABC algorithm-based solution methodology to solve job shop scheduling problems. Tian et al. [20] discussed the effective way to identify the disassembly sequence. Tian et al. [21] solved the disassembly problems using the ABC algorithm to maximize the profit and minimize the energy consumption. Li et al. [22] planned the robot path using an improved ABC algorithm. Chen [23] made few improvements to the existing ABC algorithm using an escaped foraging strategy. Xu et al. [24] proposed an improved multiobjective ABC algorithm to solve multistage resource levelling problems. Furthermore, He et al. [25] proposed a pseudolabel guided ABC algorithm to eliminate the redundant and noise bands from the hyperspectral images. Wang et al. [26] used a multiobjective feature selection algorithm based on the reduction of samples and ABC algorithm to reduce computation cost. Yong et al. [27] introduced a mixture of different strategies and multiobjective ABC algorithm to select the unsupervised band of hyperspectral images. Gao et al. [28] proposed three different strategies for the improvement of the ABC algorithm.It is concluded from the literature that authors have made significant efforts in minimizing clearance variation and surplus parts using various optimization methods such as GA, parallel population GA, PSO, and simulated annealing. Still, the manufacturers are searching for new methods to reduce the cost of manufacturing to sustain themselves in the global market. In this paper, a new method of grouping concept is introduced to make interrelated dimensional chain product with minimum manufacturing cost without affecting the assembly clearance variation or assembly specification. A concept of different group/bin number for each component according to their tolerance value is introduced. The repetition of group numbers is introduced based on the number of parts available in the particular group. In the existing method, an equal group number is assigned to all components. The best combination of the bin is obtained by implementing the ABC algorithm. The parts of the best bin combination are matched together to form assembly without any surplus parts with less manufacturing cost.


## 3. Problem Definition and Methodology

Tolerance on critical dimension controls the functioning of the product. Stringent tolerance on the functioning dimension creates the necessity of going for a secondary operation, leading to the increased manufacturing cost of the product. Selective assembly converts wider tolerance components into close tolerance assemblies/precision assemblies by matching the classified parts based on the best bin combinations. Two-component assembly with three partitions will have 36 (3! ∗ 3!) combination of bins, whereas five components with six partitions have (6! ∗ 6! ∗ 6! ∗ 6! ∗ 6!) 1.93492E + 14 combinations. This makes the situation an NP-hard problem. Reducing group numbers according to the tolerance of the components may reduce the combination of bins to some extent without affecting the functional tolerance. In this work, components are classified into different group numbers rather than the same group number as traditional methods according to their tolerance value. Furthermore, specific group numbers are allowed to repeat based on the number of parts that falls in that particular bin. Hence, the assemblies are made in a single stage. This is the novelty of this work. The ABC algorithm has been adopted to obtain the best bin combinations.The proposed work has been carried out in two stages; in the first stage, steps a, b, and c are carried out. The definitions of symbols used in this manuscript are given in Table1.(a)
One thousand random dimensions of each component are generated for the known mean and standard deviation. Equation (1) is used to estimate the standard deviation where the tolerance ± Ti is considered:(1)σi=Ti3.(b)
According to tolerance, each component’s bin can be decided, and parts are classified, counted, and calculated for each bin’s group width using(2)gwi=2Tinbi.(c)
The length of combination of bin and the approximate repetition of bin number of each component are calculated using the equations (3) and (4). The possible bin numbers can be formed as per equation (5):(3)lb=maxnbii=1nc+G∗nc,(4)rbij=lb∗npijNt,(5)pbi=1,…,rbi12,…,rbi2,…,nbi,…,rbinbi.(d)
In the second stage, the best bin combinations are determined using the ABC algorithm. The minimum and maximum clearance values are obtained when mating the components available in thekth position of the corresponding group using equations (6) and (7), respectively. The clearance variation when mating the components available in the kth position of the corresponding group is obtained by calculating the difference between the maximum and minimum clearance values. This is expressed in equation (8). The number of assemblies made when mating the components available in the kth position of the corresponding group is obtained using equation (9). The number of surplus parts of ith component available is calculated by subtracting the number of assemblies made as per equation (9) from the number of parts of the ith component in the kth position group. This is expressed in equation (10). The clearance variation of the assembly is calculated by taking the difference between the maximum and minimum clearance variation. This is explained using equation (11). The total number of assemblies made is calculated by considering the number of assemblies made when mating the components available in all the positions of the corresponding group, and it is determined using equation (12). The combination of bins, cbik, is considered bees where k takes the values from 1 to lb. The objective of the algorithm is to produce assemblies without surplus parts to meet the assembly specification:(6)CMk=gwi∗cbik−1+gwi+1∗cbi+1k−1+⋯+gwn∗cbnk−1,(7)CXk=gwi∗cbik+gwi+1∗cbi+1k+⋯+gwn∗cbnk,(8)CVk=CXk−CMk,(9)NAk=minNiki=1n,(10)Sij=Nik−NAk,(11)Ca=maxCXkk=1lb−minCMkk=1lb,(12)H=∑k=1lbNAk.Table 1
Symbol definition.
SymbolDefinitioniComponent’ index;i = 1 to ncµiMean value ofith componentTiTolerance ofith component in the proposed worknbiNumber of bins ofith componentσiStandard deviation ofith componentdmiRandomly generated dimension ofith componentnpijNumber of parts ofith component in jth bin numberlbLength of bin combinationGConstant varies from 1 to 2rpijNumber of repeatedjth bin number for ith componentpbiPossible bin numbers ofith componentAiAllocated tolerance ofith componentMCiaManufacturing cost ofith component for allocated toleranceTimaxMaximum tolerance ofith componentTiminMinimum tolerance ofith componentF0i, F1i, F2iManufacturing cost model’s coefficients ofith componentMCiManufacturing cost ofith componentDimaxMaximum dimension ofith componentDiminMinimum dimension ofith componentA, B, CDimension of gears A, B, and CX1, X2a, X2b, X3a, X3b, X4, X5Dimension of KJA with three armsTX1, TX2a, TX2b, TX3a, TX3b, TX4, TX5Tolerance of KJA with three armsY1, Y2, Y3Critical dimensionw1,w2,w3Weights for critical dimensionSYSum of critical dimension after considering weightsTamaxMaximum tolerance of the assemblyTaminMinimum tolerance of the assemblyCVClearance variation of the assemblyrbijNumber of repetition ofith component’s jth grouppbiPossible bin combination ofith componentNjNumber of components available injth group of ith componentNtTotal number of parts manufactured for a componentgniNumber of group ofith componentgwiGroup width ofith componentcbiRandomly generated selective group combination ofith componentcbikGroup number ofith component in kth position of combination of selective groupsCXkMaximum clearance by matingkth position selective groupCMkMinimum clearance by matingkth position selective groupCVkClearance variation obtained by matingkth position selective groupsNAkNumber of assemblies made by matingkth position selective groupsNikNumber of parts ofith component in kth position groupSijNumber of surplus parts available injth group of ith componentCaClearance variation of the entire assemblyHTotal number of assemblies madenbiNumber of bins ofith component


## 4. ABC Algorithm

As per the literature survey [29], it is noted that the ABC algorithm is a well-known algorithm due to its simplicity in solving optimization problems, high level of flexibility, good robustness, limited control factors, ready to combine with other methods, ease handling of the objective with stochastic nature, quick convergence, and both exploration and exploitation. All these parameters are essential for solving complex problems in selective assembly. Hence, the ABC algorithm is selected in this work.The intelligent foraging behaviour of artificial honey bees [30] is simulated in the ABC algorithm. In general, the following types of bees are in any hive: employed, unemployed, and scouts/onlooker bees. The number of employed bees depends on the number of food sources around the hive, where the employed bees are named as scout bees once foods are vanished [31]. In this stage, they start to search for new food sources, described as follows.(a)
All employed bees are assigned food sources in the initial stage.(b)
The search neighbourhood food source by employed bees after reaching the food sources in the memory determines the nectar amount, and according to that, they will dance.(c)
This dance will be identified by unemployed bees within the hive regarding food sources, and they determine the nectar amounts.(d)
Replacement of empty food sources with new food sources by scout bees.(e)
Within steps (a) to (d), the best food source is recorded. The steps starting (b) to (e) are repeated to obtain the top best. Figure1 shows the flowchart of the ABC algorithm.Figure 1
Flowchart of the ABC algorithm.In this work, the combination of bins of all components is considered an artificial bee in the colony. The top out of best food source is considered the total number of assemblies made within the specified assembly specifications. The ABC parameters are assumed based on Nagarajan et al. [32] and are listed in Table 2.Table 2
ABC parameters [32].
ABC parameterValue/methodAnalogy/coding processBeeA combination of bins of all componentsAn assembly with components A, B, and C considering 3, 4, and 5 bin partition in which a combination of bins will be considered a bee. For example, {(2,1,3),(4,3,1,2),(3,5,1,2,4)}Total size of bee colony4040 different combinations of the bin of components A, B, and CSize of employed bees50% of total beesTop 20 bees based on the ascending order of the clearance variation of an assemblySize of unemployed bees50% of total beesThe remaining 20 bees out of the total size of bee colonySelection of food by onlooker beesRoulette wheel selection methodSelection of best-employed bees based on clearance variationSize of scout bees50% of unemployed beesRandomly generated ten bees out of 20 unemployed beesTermination criteriaThe specific number of iteration or no change in the best value for the last 50 generations is consideredNo change in the clearance variation for the consecutive 50 iterations


## 5. Numerical Illustration

The effectiveness of the proposed method is demonstrated on both linear assembly and interrelated assembly. Initially, the concept of different group number and approximate repetition of group number based on the probability of parts available in each bin is introduced in linear assembly dealt by Wang et al. [33]. Later, an interrelated dimensional chain product, Knuckle joint assembly with three arms dealt by Sivakumar et al. [34], is considered to reduce the manufacturing cost by implementing the new concept.


### 5.1. Linear Assembly

A simple gear assembly consists of three gears, A, B, and C, which is considered as an example problem 1 [33]. These gears are assembled linearly in a gear box which is shown in Figure 2.Figure 2
Linear gear box assembly.The dimension of the gears A, B, and C isφ10−0.000+0.012, φ12−0.000+0.015, and φ14−0.000+0.018, respectively. If the gears are assembled interchangeably, the dimension of the assembly will be φ36−0.000+0.045. A new selective assembly technique used by Wang et al. [33] reduces the assembly clearance variation from 0.045 to 0.0175. Gears A, B, and C are classified into 4, 5, and 6 groups, and the dimensional distribution of one thousand random numbers of gears A, B, and C is shown in Table 3. The length of bin combination and the approximate repetition of group numbers are calculated using equations (3) and (4). The possible combination of the bin is formed using equation (5) and represented in Table 4. The best bin combinations obtained by implementing the ABC algorithm is presented in Table 5. The evaluation details of the best bin combination are given in Table 6. Furthermore, the calculation of the number of assemblies through equations (6)–(12) is explained in Table 6:(13)lb=maxgnii=1nc+G∗nc=max4,5,6+1.5∗6=6+9=15,rbij=lb∗npij1000=rbA1=15∗1311000=1.965=2.Table 3
Dimensional distribution of gears A, B, and C fornb = [4 5 6].
B. no.Number of partsNumber of approximate repetition of group numberABCABC11311812211252216912373233164834115654312913551455399311661Table 4
The possible combination of the bin (pbi).
C. no.Possible bin combinationA112222222333334B122233333344445C122333334444456Table 5
Best bin combination produced 15µm.
C. no.Possible bin combinationA212333222433332B345434332212123C533123344464443Table 6
Evaluation of best bin combination.
kcbkCMkCXkNAkABC12352130932143152413132532130394341152412533215241236343213014872331524938234182717492241524123104242130311131621306123241827151331415241214324182701523315240(14)CM1=3∗2−1+3∗3−1+3∗5−1=21,CX1=3∗2+3∗3+3∗5=30,CV1=30−21=9,NA1=min522,483,93=93,Ca=max30,24,…,27,24−min21,15,…,18,15=30−15=15,H=93+131+39+12+123+148+93+174+123+31+6+15+12+0+0=1000.With reference to the above calculations, the clearance variation (Ca = 15) found using the proposed method is less when compared with Wang et al. [33] (Ca = 17.5); i.e., 14.3% reduction in the clearance variation is obtained. Furthermore, the number of surplus parts is zero in the proposed method. Hence, the results showed the effectiveness of the proposed method.


### 5.2. Interrelated Dimensional Chain Assembly/Product

Knuckle joint assembly with three arms dealt with by Sivakumar et al. [34] has been considered as an example problem in this work. The dimensional details are shown in Figure 3. It consists of seven dimensions and three critical dimensions with interrelated dimensional chains that decide the product’s functioning. Table 7 illustrates the allocated tolerance, tolerance cost coefficients, process tolerance limits, and manufacturing cost of each dimension obtained by Sivakumar et al. [34]. The assumed mean dimensions and manufacturing tolerance values of each dimension are represented in Table 8. The minimum, maximum, and standard deviation of each dimension are calculated using equations (15)–(17). All dimensions and tolerances are in mm:(15)Dimin=μi−Ti,(16)Dimax=μi+Ti,(17)σi=Ti3.Figure 3
KJA with three arms.Table 7
Manufacturing details of KJA with three arms in the existing work [34].
C. no.P. no.AiTiminTimaxF0iF1iF2iMCiaX130.137620.010.15296.4019.523.8244.07X2a, X2b30.048390.010.15296.4019.523.82139.19X3a,X3b30.046650.010.15296.4019.523.82143.18X420.030980.020.2082.4316.721.0070.14X520.025130.010.10160.4386.729.2047.35C. no.: component number; P. no.: process number.Table 8
Manufacturing details of KJA with three arms in the proposed work.
C. no.µi±TiDiminDimaxσiMCiX125.000.1524.8525.150.0539.73X2b25.500.1525.3525.650.0539.73X2a18.750.1518.6018.900.0539.73X3b63.500.1563.3563.650.0539.73X3a18.750.1518.6018.900.0523.92X4107.750.20107.55107.950.06666729.23X56.250.106.156.350.03333329.22Y1 = X2b − X10.500.200.300.70Y2 = X3b − 2X2a − X2b0.500.200.300.70Y3 = X4 − X5 − 2X3a − X3b0.500.200.300.70The manufacturing cost of the components is estimated using equation (18) where the cost coefficients are taken from Sivakumar et al. [34]. The interrelated critical dimensions are calculated using equations (19)–(21):(18)Fi=F0i∗e−C1iti+F2i,(19)Y1=X2b−X1,(20)Y2=X3b−2X2a+X2b,(21)Y3=X4−2X3a+X3b+X5.The tolerances on critical dimensions are considered the same values as taken by Sivakumar et al. [34] which are given in the following equations:(22)TY1=TX2b+TX1≤0.2,(23)TY21=TX3b+2TX2a+TX2b≤0.2,(24)TY3=TX4+2TX3a+TX3b+TX5≤0.2.


#### 5.2.1. Implementation of ABC Algorithm

The general structure and the behaviour of bees in the ABC algorithm are detailed in Section4. The stepwise procedure of the ABC algorithm is presented in this section.(a)
Population initializationGeneration of a different combination of bins of each component is used for population initialization. A set of combinations of all components of the bin is treated as a bee. For example, a set of possible combinations of bins of the components A, B, and C listed in Table3 is called a bee. This is the way of encoding a bee.(b)
Initialized bees: evaluationThe clearance variation of the assembly is obtained by evaluating the initialized bees and the total number of assemblies made by matching the parts as per the combination of bins. The example calculation is given in Section5.1.(c)
Identification of search space by employed beesEquation (25) gives the value of the addition of critical dimensions of the assembly. Based on the increasing order of these values, the bees are listed. The first half count of bees among the list are considered employed bees, and the unemployed bees are the balance:(25)SY=w1Y1+w2Y2+w3Y3.Equation (26) is used to identify the fresh position of the employed bees in the solution space, i.e., the combinations of bins. The number of new bees based on the fresh positions is identical to the number of employed bees. The randomly generated food source (xk,j) and existing food source (xi,j) are considered to identify the new solution space (Ni,j):(26)Ni,j=xi,j+φi,jxi,j−xk,j.The replacement of employed bees is based on the evaluation results of randomly generated bees. The greedy selection process stated in Karaboga and Basturk [30] is used for this purpose.(d)
Identification of new solutions instead of unemployed beesThe new bees are randomly generated based on roulette wheel selection stated in Nagarajan et al. [32] and are equal in numbers concerning unemployed bees, followed by the updation of unemployed bees with the newly generated bees using the greedy selection process.(e)
Best bee out of the first iteration and generation of next iteration beesThe best bee based on evaluating the updated employed and unemployed bees is selected from the first iteration. The same list of employed and unemployed bees concerning the minimum value of assembly specification is selected as the next iteration bees.(f)
Best solution: identificationThe above steps are repeated till the termination criteria as stated in Table2 to identify the best solution. Furthermore, the proposed ABC algorithm’s sorted list of possible solutions is presented in Tables 9 and 10 for linear assembly. Tables 11–19 represent the best values obtained for KJA with three arms.Table 9
The best combination of bin,CV, and NAk for various bin numbers from 3 to 6, linear assembly.
G. no.C. no.Best combination of binsCVNos. of assemblies made by matchingkth position bins (NAk)3A21113322232215.6188841518216432507312722004B2423313322335C4353132333334A2312422223441729116913139161091519491471235B4235331333315C3432135422225A4523313223323331513611167129381123111568664981426B4356234643133216C3331542343644444A13234323222233216131123224631161231215541156412206B4431246553322136C4246333133555554A21233322243333215931313912123148931741233161512005B3454343322121236C5331233444644435A235423414221132163916911123186265126118951461005B5233344331343326C3432431635355446A641533333232333172266627939123926122172524419125B3342533144222216C3353324614344556A2241331653355241727192266253920037241131202018005B2333544244221414C4323223313333235A233233122445433161882271615231688170658321196006B4453635531322445C4215233333333335A4532323142254331741837423271673381679949328006B4146253333222334C122243232333323Table 10
The best combination of bin andCV for various bin numbers from 5 to 8, linear assembly.
G. no. (gni)C. no.Best combination of binsCV5A42313332332425443315.216B2544441363633331537C5434247615454444445A32522213443334444321316.456B3544644444422122223448C453634551238744737585Table 11
Assembly specifications for various bin numbers, KJA with three arms.
Pr. no.Bin number (nb)ClbY1Y2Y3H1[3 3 3 3 3 4 3]1170.40.60.633333310000.50.60.6666667100021.5210.40.50.666666710000.60.60.666666710000.50.60.666666710003[4 4 4 4 4 5 3]1.5240.5250.5250.676666710000.3750.60.673333310000.60.60.69510000.450.5250.6166667100041.8260.60.6750.681666710000.5250.60.686666710000.450.6750.676666710000.3750.5250.681666710005[5 4 5 4 4 4 5]1.5250.540.690.6610000.480.5550.6810000.420.570.65510006[4 4 4 4 4 4 6]1.5260.3750.5250.67510000.5250.5250.608333310000.450.6750.610007[5 5 4 4 4 4 6]1.5260.480.4950.691666710000.420.5550.62510000.4650.60.633333310000.4650.6750.608333310008[5 4 5 4 4 4 6]1.5270.360.60.666666710000.480.5550.658333310000.420.660.658333310009[5 4 5 4 5 4 6]1.5270.420.510.663333310000.480.5250.671666710000.480.480.6516667100010[5 5 5 5 5 6 4]1.5270.480.60.691000111.8280.60.60.6810000.480.540.6910000.480.480.6110000.420.540.66333331000122290.480.540.6210000.540.540.6310000.420.540.666666710000.540.60.563333310000.360.540.5566667100013[6 6 6 6 6 4 7]2320.450.550.610000.350.550.6714286100014[6 6 5 5 6 4 7]2320.440.520.617142910000.420.570.691428610000.390.590.675714310000.480.580.63100015[5 5 6 6 6 4 7]2320.370.610.68571431000Pr. no.: problem number.Table 12
Best bin combinations for dimensionX1.
Pr. no.Best bin combinations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able 13
Best bin combinations for dimensionX2a.
Pr. no.Best bin combinations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able 14
Best bin combinations for dimensionX2b.
Pr. no.Best bin combinations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able 15
Best bin combinations for dimensionX3a.
Pr. no.Best bin combinations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able 16
Best bin combinations for dimensionX3b.
Pr. no.Best bin combinations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able 17
Best bin combinations for dimensionX4.
Pr. no.Best bin combinations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able 18
Best bin combinations for dimensionX5.
Pr. no.Best bin combinations112233222222212222232233212122222122312222322223333323221322222222222211122222223212322222232323232313222222212222221222222322322323121122223311112222232221222233233333333132122222233322232333222242123223222322222222222222232222232213232233333333223223133322222222222222222212132322222332232322222213254145344223534234344333223331332453532454424444444441434335555542444442442446433444444456555555221225253354356424444314443344436444533124633444532455333155754345134434234465555555555553345544654144444442442224345234441365535523333332163544345454255514424444224855444462334343361433323333154534412544546663333323311133645454455244124444442222594354516434444444444442442225264534344543434434144444355443234614444544553345455551024231333333333233322222222211432333332233333123323223233333343332321323344222222222223233232313443333223234442222322232333234334233334333313112333312323223433344323333322223322132433433433333333333333343333323333232133223222221222423332333332223422311343444111333333233223433422342322222213574552634345535644444413666663353444425647555575666666565555521114365444555441567254554555554444425623455441444453364753555555553343445555471563757356366632333322644543573345355535555555532313331545414636554755545654544555525514Table 19
Number of assemblies made by mating parts as per best combination of bin.
Pr. no.12345678910111213141516171819202122232425262728293031321472247918460451354738630711605220131135141504168106163837431215319550279581561415045477221987184284910000024479184871353947151841766752528000004547842158791083413927837313734131800000347244464412610203254362211667797836193920005435946662242038132791431469241314382223002044475425395141581722552175521830397168000476058202614652835629410252456132451236142000420584986212441573916741364014332431831000000602124620720884910105812634434812101295211000047884658168201091965485226631844411548142000086584934602017888143202844103146101731106161500005797100472875137919925117928175196125253816150244575447604749345238667210354265100413212626797536475127319172620305322212281698625434679120129464711245864301531102928902021241542400010472065449731725347131121725251273331690229220025310150355313047313212461726326623501302851840072454477947511861066535318183643114243610172815974760791704145543723171295201272242110271681425840024106039374530343314525433053333191145311231238532447111947978314531783551719515185211088812512492124711475394040472015205424611164223160100272458878131474733021531624151528176332613021138314516469219483448396255262221921772263009544519126225312520434516424171782615133028173145151688533126798647530514674596734944192857233191921010079843053191543151121031151433091214141851323153051027923253171374101264782131296572523245551664307271192119100241482712662244241592634171327322714212224320100138535124458329121171164521221922646271993721297101002716326306727531725215675752651152412302121793511332516727245326100156891230413721314185134143613584791221250653183394156756850620333553532175172768415269537913491221786322763078143491922463251002351102791321173518724299462312155112271414451181911253262721268611746100185630122234254241185175110027117181483224251226674531452173111357539827251713792243471502668274915862702823121533196863101102011124914014313274720213113154214130397843155472310229840141712513121425222052323555521252163042618615676663425622321133238631724616351510066997934771141332659325304135414242711242265141013111255512824615222217829125112901315234255185564426235104151524152211641434728532722281015322148227123119814192629710


## 5.1. Linear Assembly

A simple gear assembly consists of three gears, A, B, and C, which is considered as an example problem 1 [33]. These gears are assembled linearly in a gear box which is shown in Figure 2.Figure 2
Linear gear box assembly.The dimension of the gears A, B, and C isφ10−0.000+0.012, φ12−0.000+0.015, and φ14−0.000+0.018, respectively. If the gears are assembled interchangeably, the dimension of the assembly will be φ36−0.000+0.045. A new selective assembly technique used by Wang et al. [33] reduces the assembly clearance variation from 0.045 to 0.0175. Gears A, B, and C are classified into 4, 5, and 6 groups, and the dimensional distribution of one thousand random numbers of gears A, B, and C is shown in Table 3. The length of bin combination and the approximate repetition of group numbers are calculated using equations (3) and (4). The possible combination of the bin is formed using equation (5) and represented in Table 4. The best bin combinations obtained by implementing the ABC algorithm is presented in Table 5. The evaluation details of the best bin combination are given in Table 6. Furthermore, the calculation of the number of assemblies through equations (6)–(12) is explained in Table 6:(13)lb=maxgnii=1nc+G∗nc=max4,5,6+1.5∗6=6+9=15,rbij=lb∗npij1000=rbA1=15∗1311000=1.965=2.Table 3
Dimensional distribution of gears A, B, and C fornb = [4 5 6].
B. no.Number of partsNumber of approximate repetition of group numberABCABC11311812211252216912373233164834115654312913551455399311661Table 4
The possible combination of the bin (pbi).
C. no.Possible bin combinationA112222222333334B122233333344445C122333334444456Table 5
Best bin combination produced 15µm.
C. no.Possible bin combinationA212333222433332B345434332212123C533123344464443Table 6
Evaluation of best bin combination.
kcbkCMkCXkNAkABC12352130932143152413132532130394341152412533215241236343213014872331524938234182717492241524123104242130311131621306123241827151331415241214324182701523315240(14)CM1=3∗2−1+3∗3−1+3∗5−1=21,CX1=3∗2+3∗3+3∗5=30,CV1=30−21=9,NA1=min522,483,93=93,Ca=max30,24,…,27,24−min21,15,…,18,15=30−15=15,H=93+131+39+12+123+148+93+174+123+31+6+15+12+0+0=1000.With reference to the above calculations, the clearance variation (Ca = 15) found using the proposed method is less when compared with Wang et al. [33] (Ca = 17.5); i.e., 14.3% reduction in the clearance variation is obtained. Furthermore, the number of surplus parts is zero in the proposed method. Hence, the results showed the effectiveness of the proposed method.


## 5.2. Interrelated Dimensional Chain Assembly/Product

Knuckle joint assembly with three arms dealt with by Sivakumar et al. [34] has been considered as an example problem in this work. The dimensional details are shown in Figure 3. It consists of seven dimensions and three critical dimensions with interrelated dimensional chains that decide the product’s functioning. Table 7 illustrates the allocated tolerance, tolerance cost coefficients, process tolerance limits, and manufacturing cost of each dimension obtained by Sivakumar et al. [34]. The assumed mean dimensions and manufacturing tolerance values of each dimension are represented in Table 8. The minimum, maximum, and standard deviation of each dimension are calculated using equations (15)–(17). All dimensions and tolerances are in mm:(15)Dimin=μi−Ti,(16)Dimax=μi+Ti,(17)σi=Ti3.Figure 3
KJA with three arms.Table 7
Manufacturing details of KJA with three arms in the existing work [34].
C. no.P. no.AiTiminTimaxF0iF1iF2iMCiaX130.137620.010.15296.4019.523.8244.07X2a, X2b30.048390.010.15296.4019.523.82139.19X3a,X3b30.046650.010.15296.4019.523.82143.18X420.030980.020.2082.4316.721.0070.14X520.025130.010.10160.4386.729.2047.35C. no.: component number; P. no.: process number.Table 8
Manufacturing details of KJA with three arms in the proposed work.
C. no.µi±TiDiminDimaxσiMCiX125.000.1524.8525.150.0539.73X2b25.500.1525.3525.650.0539.73X2a18.750.1518.6018.900.0539.73X3b63.500.1563.3563.650.0539.73X3a18.750.1518.6018.900.0523.92X4107.750.20107.55107.950.06666729.23X56.250.106.156.350.03333329.22Y1 = X2b − X10.500.200.300.70Y2 = X3b − 2X2a − X2b0.500.200.300.70Y3 = X4 − X5 − 2X3a − X3b0.500.200.300.70The manufacturing cost of the components is estimated using equation (18) where the cost coefficients are taken from Sivakumar et al. [34]. The interrelated critical dimensions are calculated using equations (19)–(21):(18)Fi=F0i∗e−C1iti+F2i,(19)Y1=X2b−X1,(20)Y2=X3b−2X2a+X2b,(21)Y3=X4−2X3a+X3b+X5.The tolerances on critical dimensions are considered the same values as taken by Sivakumar et al. [34] which are given in the following equations:(22)TY1=TX2b+TX1≤0.2,(23)TY21=TX3b+2TX2a+TX2b≤0.2,(24)TY3=TX4+2TX3a+TX3b+TX5≤0.2.


### 5.2.1. Implementation of ABC Algorithm

The general structure and the behaviour of bees in the ABC algorithm are detailed in Section4. The stepwise procedure of the ABC algorithm is presented in this section.(a)
Population initializationGeneration of a different combination of bins of each component is used for population initialization. A set of combinations of all components of the bin is treated as a bee. For example, a set of possible combinations of bins of the components A, B, and C listed in Table3 is called a bee. This is the way of encoding a bee.(b)
Initialized bees: evaluationThe clearance variation of the assembly is obtained by evaluating the initialized bees and the total number of assemblies made by matching the parts as per the combination of bins. The example calculation is given in Section5.1.(c)
Identification of search space by employed beesEquation (25) gives the value of the addition of critical dimensions of the assembly. Based on the increasing order of these values, the bees are listed. The first half count of bees among the list are considered employed bees, and the unemployed bees are the balance:(25)SY=w1Y1+w2Y2+w3Y3.Equation (26) is used to identify the fresh position of the employed bees in the solution space, i.e., the combinations of bins. The number of new bees based on the fresh positions is identical to the number of employed bees. The randomly generated food source (xk,j) and existing food source (xi,j) are considered to identify the new solution space (Ni,j):(26)Ni,j=xi,j+φi,jxi,j−xk,j.The replacement of employed bees is based on the evaluation results of randomly generated bees. The greedy selection process stated in Karaboga and Basturk [30] is used for this purpose.(d)
Identification of new solutions instead of unemployed beesThe new bees are randomly generated based on roulette wheel selection stated in Nagarajan et al. [32] and are equal in numbers concerning unemployed bees, followed by the updation of unemployed bees with the newly generated bees using the greedy selection process.(e)
Best bee out of the first iteration and generation of next iteration beesThe best bee based on evaluating the updated employed and unemployed bees is selected from the first iteration. The same list of employed and unemployed bees concerning the minimum value of assembly specification is selected as the next iteration bees.(f)
Best solution: identificationThe above steps are repeated till the termination criteria as stated in Table2 to identify the best solution. Furthermore, the proposed ABC algorithm’s sorted list of possible solutions is presented in Tables 9 and 10 for linear assembly. Tables 11–19 represent the best values obtained for KJA with three arms.Table 9
The best combination of bin,CV, and NAk for various bin numbers from 3 to 6, linear assembly.
G. no.C. no.Best combination of binsCVNos. of assemblies made by matchingkth position bins (NAk)3A21113322232215.6188841518216432507312722004B2423313322335C4353132333334A2312422223441729116913139161091519491471235B4235331333315C3432135422225A4523313223323331513611167129381123111568664981426B4356234643133216C3331542343644444A13234323222233216131123224631161231215541156412206B4431246553322136C4246333133555554A21233322243333215931313912123148931741233161512005B3454343322121236C5331233444644435A235423414221132163916911123186265126118951461005B5233344331343326C3432431635355446A641533333232333172266627939123926122172524419125B3342533144222216C3353324614344556A2241331653355241727192266253920037241131202018005B2333544244221414C4323223313333235A233233122445433161882271615231688170658321196006B4453635531322445C4215233333333335A4532323142254331741837423271673381679949328006B4146253333222334C122243232333323Table 10
The best combination of bin andCV for various bin numbers from 5 to 8, linear assembly.
G. no. (gni)C. no.Best combination of binsCV5A42313332332425443315.216B2544441363633331537C5434247615454444445A32522213443334444321316.456B3544644444422122223448C453634551238744737585Table 11
Assembly specifications for various bin numbers, KJA with three arms.
Pr. no.Bin number (nb)ClbY1Y2Y3H1[3 3 3 3 3 4 3]1170.40.60.633333310000.50.60.6666667100021.5210.40.50.666666710000.60.60.666666710000.50.60.666666710003[4 4 4 4 4 5 3]1.5240.5250.5250.676666710000.3750.60.673333310000.60.60.69510000.450.5250.6166667100041.8260.60.6750.681666710000.5250.60.686666710000.450.6750.676666710000.3750.5250.681666710005[5 4 5 4 4 4 5]1.5250.540.690.6610000.480.5550.6810000.420.570.65510006[4 4 4 4 4 4 6]1.5260.3750.5250.67510000.5250.5250.608333310000.450.6750.610007[5 5 4 4 4 4 6]1.5260.480.4950.691666710000.420.5550.62510000.4650.60.633333310000.4650.6750.608333310008[5 4 5 4 4 4 6]1.5270.360.60.666666710000.480.5550.658333310000.420.660.658333310009[5 4 5 4 5 4 6]1.5270.420.510.663333310000.480.5250.671666710000.480.480.6516667100010[5 5 5 5 5 6 4]1.5270.480.60.691000111.8280.60.60.6810000.480.540.6910000.480.480.6110000.420.540.66333331000122290.480.540.6210000.540.540.6310000.420.540.666666710000.540.60.563333310000.360.540.5566667100013[6 6 6 6 6 4 7]2320.450.550.610000.350.550.6714286100014[6 6 5 5 6 4 7]2320.440.520.617142910000.420.570.691428610000.390.590.675714310000.480.580.63100015[5 5 6 6 6 4 7]2320.370.610.68571431000Pr. no.: problem number.Table 12
Best bin combinations for dimensionX1.
Pr. no.Best bin combinations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able 13
Best bin combinations for dimensionX2a.
Pr. no.Best bin combinations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able 14
Best bin combinations for dimensionX2b.
Pr. no.Best bin combinations133222122222222223222222312223311132232321232222222232222223321222221222223222123223322222222223222323332133423334233333332433333224431132422222223243332334314343333333113333433223323132432442223243332333423343443444411144333224431334333333333333334343323433234332133333113234324333322331332333322332254424353344344334144444442354433443135433344324444323134544334343443244444446242333233333133333333333341223443343343333333333333321333344434333333333333333734333333142333232332222222332333433433332332143423233212234333332334424222222223343223333333412441444443824233344133434345454444444324434355353333333313333333434244233553344314344333333393344432335444452332333221224233443135423433333333332222343443443343335553331333331043434443543333254244142442211342445334543333333333333311224354433333441343333443333332435334345344144344443334333353344423343443434134444444412534344432444343443313333232223441234233344334535353332333234412354334444333443333333333244323333435333535532333333313523333344444344344344414444413452434365555541444444444444444443364554545444442343441444444444414321343544343434433443434443444443233534433143344433434443333333351234343334433344443444444444444415234324444334334443444442222221545214343234344536555464466555555Table 15
Best bin combinations for dimensionX3a.
Pr. no.Best bin combinations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able 16
Best bin combinations for dimensionX3b.
Pr. no.Best bin combinations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able 17
Best bin combinations for dimensionX4.
Pr. no.Best bin combinations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able 18
Best bin combinations for dimensionX5.
Pr. no.Best bin combinations112233222222212222232233212122222122312222322223333323221322222222222211122222223212322222232323232313222222212222221222222322322323121122223311112222232221222233233333333132122222233322232333222242123223222322222222222222232222232213232233333333223223133322222222222222222212132322222332232322222213254145344223534234344333223331332453532454424444444441434335555542444442442446433444444456555555221225253354356424444314443344436444533124633444532455333155754345134434234465555555555553345544654144444442442224345234441365535523333332163544345454255514424444224855444462334343361433323333154534412544546663333323311133645454455244124444442222594354516434444444444442442225264534344543434434144444355443234614444544553345455551024231333333333233322222222211432333332233333123323223233333343332321323344222222222223233232313443333223234442222322232333234334233334333313112333312323223433344323333322223322132433433433333333333333343333323333232133223222221222423332333332223422311343444111333333233223433422342322222213574552634345535644444413666663353444425647555575666666565555521114365444555441567254554555554444425623455441444453364753555555553343445555471563757356366632333322644543573345355535555555532313331545414636554755545654544555525514Table 19
Number of assemblies made by mating parts as per best combination of bin.
Pr. no.12345678910111213141516171819202122232425262728293031321472247918460451354738630711605220131135141504168106163837431215319550279581561415045477221987184284910000024479184871353947151841766752528000004547842158791083413927837313734131800000347244464412610203254362211667797836193920005435946662242038132791431469241314382223002044475425395141581722552175521830397168000476058202614652835629410252456132451236142000420584986212441573916741364014332431831000000602124620720884910105812634434812101295211000047884658168201091965485226631844411548142000086584934602017888143202844103146101731106161500005797100472875137919925117928175196125253816150244575447604749345238667210354265100413212626797536475127319172620305322212281698625434679120129464711245864301531102928902021241542400010472065449731725347131121725251273331690229220025310150355313047313212461726326623501302851840072454477947511861066535318183643114243610172815974760791704145543723171295201272242110271681425840024106039374530343314525433053333191145311231238532447111947978314531783551719515185211088812512492124711475394040472015205424611164223160100272458878131474733021531624151528176332613021138314516469219483448396255262221921772263009544519126225312520434516424171782615133028173145151688533126798647530514674596734944192857233191921010079843053191543151121031151433091214141851323153051027923253171374101264782131296572523245551664307271192119100241482712662244241592634171327322714212224320100138535124458329121171164521221922646271993721297101002716326306727531725215675752651152412302121793511332516727245326100156891230413721314185134143613584791221250653183394156756850620333553532175172768415269537913491221786322763078143491922463251002351102791321173518724299462312155112271414451181911253262721268611746100185630122234254241185175110027117181483224251226674531452173111357539827251713792243471502668274915862702823121533196863101102011124914014313274720213113154214130397843155472310229840141712513121425222052323555521252163042618615676663425622321133238631724616351510066997934771141332659325304135414242711242265141013111255512824615222217829125112901315234255185564426235104151524152211641434728532722281015322148227123119814192629710


## 5.2.1. Implementation of ABC Algorithm

The general structure and the behaviour of bees in the ABC algorithm are detailed in Section4. The stepwise procedure of the ABC algorithm is presented in this section.(a)
Population initializationGeneration of a different combination of bins of each component is used for population initialization. A set of combinations of all components of the bin is treated as a bee. For example, a set of possible combinations of bins of the components A, B, and C listed in Table3 is called a bee. This is the way of encoding a bee.(b)
Initialized bees: evaluationThe clearance variation of the assembly is obtained by evaluating the initialized bees and the total number of assemblies made by matching the parts as per the combination of bins. The example calculation is given in Section5.1.(c)
Identification of search space by employed beesEquation (25) gives the value of the addition of critical dimensions of the assembly. Based on the increasing order of these values, the bees are listed. The first half count of bees among the list are considered employed bees, and the unemployed bees are the balance:(25)SY=w1Y1+w2Y2+w3Y3.Equation (26) is used to identify the fresh position of the employed bees in the solution space, i.e., the combinations of bins. The number of new bees based on the fresh positions is identical to the number of employed bees. The randomly generated food source (xk,j) and existing food source (xi,j) are considered to identify the new solution space (Ni,j):(26)Ni,j=xi,j+φi,jxi,j−xk,j.The replacement of employed bees is based on the evaluation results of randomly generated bees. The greedy selection process stated in Karaboga and Basturk [30] is used for this purpose.(d)
Identification of new solutions instead of unemployed beesThe new bees are randomly generated based on roulette wheel selection stated in Nagarajan et al. [32] and are equal in numbers concerning unemployed bees, followed by the updation of unemployed bees with the newly generated bees using the greedy selection process.(e)
Best bee out of the first iteration and generation of next iteration beesThe best bee based on evaluating the updated employed and unemployed bees is selected from the first iteration. The same list of employed and unemployed bees concerning the minimum value of assembly specification is selected as the next iteration bees.(f)
Best solution: identificationThe above steps are repeated till the termination criteria as stated in Table2 to identify the best solution. Furthermore, the proposed ABC algorithm’s sorted list of possible solutions is presented in Tables 9 and 10 for linear assembly. Tables 11–19 represent the best values obtained for KJA with three arms.Table 9
The best combination of bin,CV, and NAk for various bin numbers from 3 to 6, linear assembly.
G. no.C. no.Best combination of binsCVNos. of assemblies made by matchingkth position bins (NAk)3A21113322232215.6188841518216432507312722004B2423313322335C4353132333334A2312422223441729116913139161091519491471235B4235331333315C3432135422225A4523313223323331513611167129381123111568664981426B4356234643133216C3331542343644444A13234323222233216131123224631161231215541156412206B4431246553322136C4246333133555554A21233322243333215931313912123148931741233161512005B3454343322121236C5331233444644435A235423414221132163916911123186265126118951461005B5233344331343326C3432431635355446A641533333232333172266627939123926122172524419125B3342533144222216C3353324614344556A2241331653355241727192266253920037241131202018005B2333544244221414C4323223313333235A233233122445433161882271615231688170658321196006B4453635531322445C4215233333333335A4532323142254331741837423271673381679949328006B4146253333222334C122243232333323Table 10
The best combination of bin andCV for various bin numbers from 5 to 8, linear assembly.
G. no. (gni)C. no.Best combination of binsCV5A42313332332425443315.216B2544441363633331537C5434247615454444445A32522213443334444321316.456B3544644444422122223448C453634551238744737585Table 11
Assembly specifications for various bin numbers, KJA with three arms.
Pr. no.Bin number (nb)ClbY1Y2Y3H1[3 3 3 3 3 4 3]1170.40.60.633333310000.50.60.6666667100021.5210.40.50.666666710000.60.60.666666710000.50.60.666666710003[4 4 4 4 4 5 3]1.5240.5250.5250.676666710000.3750.60.673333310000.60.60.69510000.450.5250.6166667100041.8260.60.6750.681666710000.5250.60.686666710000.450.6750.676666710000.3750.5250.681666710005[5 4 5 4 4 4 5]1.5250.540.690.6610000.480.5550.6810000.420.570.65510006[4 4 4 4 4 4 6]1.5260.3750.5250.67510000.5250.5250.608333310000.450.6750.610007[5 5 4 4 4 4 6]1.5260.480.4950.691666710000.420.5550.62510000.4650.60.633333310000.4650.6750.608333310008[5 4 5 4 4 4 6]1.5270.360.60.666666710000.480.5550.658333310000.420.660.658333310009[5 4 5 4 5 4 6]1.5270.420.510.663333310000.480.5250.671666710000.480.480.6516667100010[5 5 5 5 5 6 4]1.5270.480.60.691000111.8280.60.60.6810000.480.540.6910000.480.480.6110000.420.540.66333331000122290.480.540.6210000.540.540.6310000.420.540.666666710000.540.60.563333310000.360.540.5566667100013[6 6 6 6 6 4 7]2320.450.550.610000.350.550.6714286100014[6 6 5 5 6 4 7]2320.440.520.617142910000.420.570.691428610000.390.590.675714310000.480.580.63100015[5 5 6 6 6 4 7]2320.370.610.68571431000Pr. no.: problem number.Table 12
Best bin combinations for dimensionX1.
Pr. no.Best bin combinations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able 13
Best bin combinations for dimensionX2a.
Pr. no.Best bin combinations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able 14
Best bin combinations for dimensionX2b.
Pr. no.Best bin combinations133222122222222223222222312223311132232321232222222232222223321222221222223222123223322222222223222323332133423334233333332433333224431132422222223243332334314343333333113333433223323132432442223243332333423343443444411144333224431334333333333333334343323433234332133333113234324333322331332333322332254424353344344334144444442354433443135433344324444323134544334343443244444446242333233333133333333333341223443343343333333333333321333344434333333333333333734333333142333232332222222332333433433332332143423233212234333332334424222222223343223333333412441444443824233344133434345454444444324434355353333333313333333434244233553344314344333333393344432335444452332333221224233443135423433333333332222343443443343335553331333331043434443543333254244142442211342445334543333333333333311224354433333441343333443333332435334345344144344443334333353344423343443434134444444412534344432444343443313333232223441234233344334535353332333234412354334444333443333333333244323333435333535532333333313523333344444344344344414444413452434365555541444444444444444443364554545444442343441444444444414321343544343434433443434443444443233534433143344433434443333333351234343334433344443444444444444415234324444334334443444442222221545214343234344536555464466555555Table 15
Best bin combinations for dimensionX3a.
Pr. no.Best bin combinations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able 16
Best bin combinations for dimensionX3b.
Pr. no.Best bin combinations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able 17
Best bin combinations for dimensionX4.
Pr. no.Best bin combinations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able 18
Best bin combinations for dimensionX5.
Pr. no.Best bin combinations112233222222212222232233212122222122312222322223333323221322222222222211122222223212322222232323232313222222212222221222222322322323121122223311112222232221222233233333333132122222233322232333222242123223222322222222222222232222232213232233333333223223133322222222222222222212132322222332232322222213254145344223534234344333223331332453532454424444444441434335555542444442442446433444444456555555221225253354356424444314443344436444533124633444532455333155754345134434234465555555555553345544654144444442442224345234441365535523333332163544345454255514424444224855444462334343361433323333154534412544546663333323311133645454455244124444442222594354516434444444444442442225264534344543434434144444355443234614444544553345455551024231333333333233322222222211432333332233333123323223233333343332321323344222222222223233232313443333223234442222322232333234334233334333313112333312323223433344323333322223322132433433433333333333333343333323333232133223222221222423332333332223422311343444111333333233223433422342322222213574552634345535644444413666663353444425647555575666666565555521114365444555441567254554555554444425623455441444453364753555555553343445555471563757356366632333322644543573345355535555555532313331545414636554755545654544555525514Table 19
Number of assemblies made by mating parts as per best combination of bin.
Pr. no.12345678910111213141516171819202122232425262728293031321472247918460451354738630711605220131135141504168106163837431215319550279581561415045477221987184284910000024479184871353947151841766752528000004547842158791083413927837313734131800000347244464412610203254362211667797836193920005435946662242038132791431469241314382223002044475425395141581722552175521830397168000476058202614652835629410252456132451236142000420584986212441573916741364014332431831000000602124620720884910105812634434812101295211000047884658168201091965485226631844411548142000086584934602017888143202844103146101731106161500005797100472875137919925117928175196125253816150244575447604749345238667210354265100413212626797536475127319172620305322212281698625434679120129464711245864301531102928902021241542400010472065449731725347131121725251273331690229220025310150355313047313212461726326623501302851840072454477947511861066535318183643114243610172815974760791704145543723171295201272242110271681425840024106039374530343314525433053333191145311231238532447111947978314531783551719515185211088812512492124711475394040472015205424611164223160100272458878131474733021531624151528176332613021138314516469219483448396255262221921772263009544519126225312520434516424171782615133028173145151688533126798647530514674596734944192857233191921010079843053191543151121031151433091214141851323153051027923253171374101264782131296572523245551664307271192119100241482712662244241592634171327322714212224320100138535124458329121171164521221922646271993721297101002716326306727531725215675752651152412302121793511332516727245326100156891230413721314185134143613584791221250653183394156756850620333553532175172768415269537913491221786322763078143491922463251002351102791321173518724299462312155112271414451181911253262721268611746100185630122234254241185175110027117181483224251226674531452173111357539827251713792243471502668274915862702823121533196863101102011124914014313274720213113154214130397843155472310229840141712513121425222052323555521252163042618615676663425622321133238631724616351510066997934771141332659325304135414242711242265141013111255512824615222217829125112901315234255185564426235104151524152211641434728532722281015322148227123119814192629710


## 6. Results and Discussion

The new grouping method is applied to minimize the manufacturing cost of both linear and complex assemblies. In linear assembly, compared with the existing method, the length of the bin combination is reduced by introducing a new method. The proposed method has arrived at a clearance variation of 15µm, which is 2.5 µm less than the existing method dealt by Wang et al. [33]. Tables 9, 10, and 20 represent the best combination of bins, CV, and NAk for various bin numbers from 5 to 8. In Tables 9 and 11, the bolded value of zero indicates that the length of the bin combination can be reduced, and the improved length of the bin is shown in Table 21. Figure 4 represents the dimensional distribution of gears A, B, and C according to bin numbers from 3 to 8. Tables 22 and 23 represent the dimensional distribution/number of parts in each bin for all components of KJA with three arms. Figure 5 illustrates the tolerance of the components both in the existing and proposed method.Table 20
The value ofNAk for various bin numbers from 5 to 8, linear assembly.
G. no. (gni)C. no.Nos. of assemblies made by matchingkth position bins (NAk)5A10716726081611344178916211141144006B7C5A28016711129231981505368431782475600006B8CTable 21
Details of length of bin for linear assembly.
Bin numberlbEMPMIPM[5 6 6]1615[4 6 6]161514[4 5 6]161513[5 5 6]161513[4 5 5]139[5 6 7]1918[5 6 8]2221[6 5 6]1615[6 5 4]161513[5 6 5]161513[5 6 4]161513[3 4 5]131210EM: existing method; PM: proposed method; IPM: improvement in proposed method.Figure 4
Dimensional distribution of linear assembly. (a) Gear A. (b) Gear B. (c) Gear C.
(a)(b)(c)Table 22
Dimensional distribution ofX1, X2a, X2b, and X3b.
Bin no.X1X2aX2bX3a3456345634563456113146241419186533545105213549271526874142111176254132561566842531004372443821712031824554483291844134003082716174492181414594783524852723588823831712039546654246372545152531505124432126630342715Table 23
Dimensional distribution ofX3b, X4, and X5.
Bin no.X3bX4X5345634563456711054726157944215817753273942519112045626114869829012053263156468490352418463366244601442249143460263372472123359237544933153012620118562133506151231125722Figure 5
Existing and proposed tolerance of component’s dimension.The % of cost savings in each component of KJA with three arms is calculated using the following equation. These values are shown in Figure6.(27)%Si=100∗MCia−MCiMCia.Figure 6
Percentage of cost savings in the proposed method.For example, the % of cost saving of componentXi can be calculated as(28)%SX1=100∗44.07−39.7344.07=9.86%.Similarly, the % savings of other components are calculated and represented in Figure6. The overall % of cost savings of the assembly is estimated using(29)%TS=100∗TCEM−TCPMTCEM,where(30)TCEM=∑i=1ncMCia=MCX1a+2MCX2aa+MCX2ba+2MCX3aa+MCX3ba+MCX4a+MCX5a,TCEM=44.07+2∗139.19+139.19+2∗143.18+143.18+70.14+47.35=1008.66.(31)TCEM=∑i=1ncMCi=MCX1+2MCX2a+MCX2b+2MCX3a+MCX3b+MCX4+MCX5,TCEM=39.73+2∗39.73+39.73+2∗39.73+39.73+23.92+29.23=331.23,%TS=100∗1008.66−331.231008.66=67.16%.Table11 represents the length of bin (lb), assembly specification, and total assemblies made for various bin numbers of components of KJA with three arms. The best bin combinations which produce the assemblies with the required specification without surplus parts are represented in Tables 12–18. The number of assemblies made in each mating of the kth position of bin combinations is listed in Table 19. The details of the length of the bin based on the existing and proposed methods are presented in Table 24. Furthermore, the improved length of the bin is also shown.Table 24
Details of the length of the bin for KJA with three arms.
Pr. no.GEMPMIPM1126171621.526211631.533242341.833262251.533252461.540262371.540262581.540272691.5402726101.54027111.84028271224029132473231142473231152473231


## 7. Conclusion

In the present work, the interrelated dimensional chain product is manufactured with significantly less manufacturing cost by implementing the novel unequal grouping method. The proposed method is implemented to minimize the manufacturing cost of linear and complex assemblies. In the linear assembly, 14.3% reduction in clearance variation has been achieved when compared with the the method proposed by Wang et al. [33]. For KJA assembly, as compared to the existing method introduced by Sivakumar et al. [34], the present method can yield 67.16% of cost savings in manufacturing the assembly without compromising the assembly specification and with nil surplus parts in a single stage. Also, the proposed method reduced the length of bin value by 38.46% compared with the existing method. From these outcomes, the proposed novel unequal grouping method along with the ABC algorithm is most suitable to solve any linear or complex assembly with 100% inspection by considering the manufacturing objectives such as the minimization of surplus parts, clearance variation, and manufacturing cost. The work can be extended further to solve the problems associated with nonlinear assemblies using an equal area method or an unequal area method, or an unequal grouping method by considering the manufacturing objectives.



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*Source: 9811204-2021-07-30.xml*