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Biogeography based optimization (BBO) algorithm tominimise non-productive time during hole-makingprocessMehran Tamjidya, Shahla Paslara, B.T. Hang Tuah Baharudina, Tang Sai Honga & M.K.A.Ariffina

a Department of Mechanical and Manufacturing Engineering, University Putra Malaysia,Serdang, MalaysiaPublished online: 06 Oct 2014.

To cite this article: Mehran Tamjidy, Shahla Paslar, B.T. Hang Tuah Baharudin, Tang Sai Hong & M.K.A. Ariffin (2014):Biogeography based optimization (BBO) algorithm to minimise non-productive time during hole-making process, InternationalJournal of Production Research, DOI: 10.1080/00207543.2014.965356

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Biogeography based optimization (BBO) algorithm to minimise non-productive time duringhole-making process

Mehran Tamjidy*, Shahla Paslar, B.T. Hang Tuah Baharudin, Tang Sai Hong and M.K.A. Ariffin

Department of Mechanical and Manufacturing Engineering, University Putra Malaysia, Serdang, Malaysia

(Received 5 March 2014; accepted 3 September 2014)

Tool path optimization in today’s manufacturing systems is one of the crucial issues in holes-making machining. Thispaper presents an evolutionary optimization algorithm based on geographic distribution of biological organism to dealwith hole-making process problem. The proposed approach tackles the sequencing problem when several holes must bedrilled by means of different tools to reach their desired size. The aim of this study is to minimise the non-productivetime, including tool travelling time and tool switching time, by employing biogeography based optimization algorithm,since the problem is considered as NP-hard. The performance of proposed algorithm is evaluated based on various testproblems adopted from the literature. The obtained results demonstrate that the proposed algorithm can efficientlyimprove the solution quality in terms of minimising non-productive time.

Keywords: hole-making process; non-productive time; biogeography based optimization (BBO)

1. Introduction

Hole-making is a type of machining processes that are specifically used to cut a hole into a part. Drilling, tapping, ream-ing and punching are typical hole making operations and they compose a large portion of machining processes for vari-ous industries such as electronic and plastic injection mould (Ghaiebi and Solimanpur 2007). In some cases, a plasticinjection mould could have more than 100 holes with dissimilar diameters, depths, tolerance, and different surface con-ditions, in presence of various tool requirements and a large number of tool switches (Kolahan and Liang 2000). In realproduction systems, some small holes can be executed to its desired size with only one tool, or by using a sequence oftools with different diameters. Moreover, a tool can be used to perform several holes in different position as the final orintermediate operation (Liu et al. 2013). In fact, most of hole-making problems are NP-hard.

The costs of hole-making process is directly associated with the total production time that consists of travelling time,the time required to move the tool between two holes, switching time, the time to change the tool for next operation,cutting time, and the time to execute a hole (Onwubolu and Clerc 2004). According to the Merchant (1985) report, onaverage 70% of the total time takes by the part and tool movements in the manufacturing process. One of the crucialissues in current mass production industry is to minimise non-productive time such as travelling and switching time bymeans of automated machining.

Due the aforementioned complexities of hole-making problem metaheuristic approaches have found relatively morefavour in literature to optimise the hole-making process. Kolahan and Liang (2000) employed a tabu-search approach tominimise the total processing cost for hole-making operations. They considered four issues, including tool travel time,tool switching time, tool selection and machining speed specification and used the tabu-search algorithm to find thesolution. Onwubolu and Clerc (2004) proposed a new heuristic approach based on particle swarm optimization (PSO) tooptimise the operation path of automated or computer numerical control (CNC) drilling process. In their study, firstly,the tool path of a CNC drilling machine has been modelled as a travelling salesman problem (TSP); then a model forapproximate prediction of drilling time is developed. Finally, an adaptive PSO algorithm is used to solve the TSP.Abbas, Aly, and Hamza (2010) developed an approach based on ant colony optimization (ACO) to solve the CNC dril-ling tool path optimization problem as TSP in special case of production with large number of holes. Oysu and Bingul(2009) addressed three algorithms as approached of genetic algorithm (GA), simulate annealing (SA) and hybrid GASAto minimise the tool path on three-axis milling robot on wood materials. Their experiments were compared betweenthese algorithms based on minimum airtime. Liu et al. (2013) presented a new method to optimise the process planning

*Corresponding author. Email: mehrantamjidy@gmail.com

© 2014 Taylor & Francis

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of hole-making with several operations by means of different tool sets. Their optimization objectives were to minimisethe total time of the airtime and the tool switching time by aid of ACO algorithm. Medina-Rodríguez et al. (2012)employed a parallel ACO algorithm to find the best sequence of hole-making to create G code programing for the short-est cutting path. Ghaiebi and Solimanpur (2007) proposed an ant algorithm to deal with the optimization of hole-makingoperations in which a hole can be completed by using several tools. Moreover, a 0–1 nonlinear mathematical model isformulated to minimise the summation of tool travel time and switch time.

The main contribution of this study is to solve the hole-making problem by applying an efficient metaheuristic algo-rithm to minimise tool travel time and switch time. In 2008, a new population-based evolutionary algorithm based ongeographic distribution of biological organisms was firstly introduced by Simon (2008), entitled biogeography basedoptimization (BBO). He stated that this method is novel method for solving the NP hard problem. Although BBO is anaturally inspired algorithm, it has some fundamental distinction from common natural algorithms such as GA, PSO orACO. In the BBO, the initial population is not discarded among different generation. Instead, the migration concept isused to modify the population. As another distinction, in each generation, the fitness function is not used directly tomodify the population in which BBO uses fitness to determine the immigration and emigration rates.

Regarding the application of BBO algorithm in manufacturing system, BBO has been employed to solve schedulingproblem (Rahmati and Zandieh 2012; Attar, Mohammadi, Tavakkoli-Moghaddam 2013) and to design optimal place-ment of phasor measurement units (Jamuna and Swarup 2012). According to their comparative study of BBO with otherpopular metaheuristic algorithms; they concluded that BBO is capable to obtain the better results in terms of solutionquality and convergence characteristics.

Herein we propose a linear mathematical model for hole-making problem where the machining process of a holeconsists of several individual operations with various machining tools and different constraints such as precedence andassignment are imposed on the process. This model is developed based on the models presented by Ghaiebi andSolimanpur (2007) and Liu et al. (2013). Since this problem is considered to be NP-hard, a BBO algorithm is employedto find the optimum/near optimum solution within short amount of time. Our algorithms are implemented to minimisethe summation of two objective functions, namely, tool travelling time and tool switching time. The performance of theproposed algorithm is compared with some common meta-heuristic algorithms which have been widely used in the liter-ature such as GA and ACO reported in Liu et al. (2013) and Ghaiebi and Solimanpur (2007). Rest of this paper isorganised as follows: Section 2 describes the problem environment, Section 3 presents the mathematical model, Section 4develops the proposed BBO algorithm, Section 5 evaluates the performance of proposed BBO algorithm, and Section 6provides the final conclusion.

2. Problem environment

In hole-making process, some small holes can be machined to their final size through only one operation, but most ofthe holes need to be machined by several tools with different diameters, which is inevitable when the hole-diameter isrelatively large. In this case, a pilot hole can be drilled by means of a smaller tool and then enlarge the hole to itsdesired size by using a tool of larger diameter, probably completed by reaming or tapping when needed.

This paper considers the hole-making problem that includes a set of holes i 2 I that each hole is supposed to beaccomplished by a number of tools t 2 T . Each tool-hole combination is considered as an operation j 2 ni. Each opera-tion is supposed to be placed in a position p 2 N (N = total number of operation) in a sequence.

Drilling machine has a worktable can move in two directions x and y. Measuring the distance between two sequen-tial holes is based on how the worktable moves. The movement of the two-axis drill machine can be in both directionsimultaneously or just in one direction at each time (Ghaiebi and Solimanpur 2007). The mentioned worktable move-ments can be measured by the Euclidean and rectilinear distance function. In this study both distance functions, Euclid-ean and rectilinear, are considered. For two adjacent hole-operation positions, dii0 denotes the distance between twodifferent holes i and i0 located at coordinates ðxi; yiÞ and ðxi0 ; yi0 Þ respectively which executed by same tool. The dis-tance between hole i and hole i0 can be expressed as follows.(a) Euclidean distance

dii0 ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixi � xi0ð Þ2 þ yi � yi0ð Þ2

q(1)

(b) Rectilinear distance

dii0 ¼ xi � xi0j j þ yi � yi0j j (2)

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By considering the speed of worktable in x direction as vx and y direction as vy in case of rectilinear distance and v inEuclidean distance, the travelling time between two sequential hole i and hole i0, denoted by TTii0, can be calculated byEquations (3) and (4).(a) Euclidean distance

TTii0 ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixi � xi0ð Þ2 þ yi � yi0ð Þ2

qv

(3)

(b) Rectilinear distance

TTii0 ¼ xi � xi0j jvx

þ yi � yi0j jvy

(4)

Hole-making decision for each tool-hole combination is made by considering some assumptions, which are made asfollow:

(1) Each hole can be completed in multi-passes if there is particular tool sequence required for completing thehole.

(2) When a particular tool completes two sequential operations the tool switching time will be zero (0).(3) In this study the start point of spindle is located in point (0, 0).(4) Tool life of each tool type is adequate for executing entire holes assigned to it.(5) The model considers based on both Euclidean and rectilinear distance during the movement of spindle from

one hole to the next.

The subsequent example shows the simple part with 3 holes and their required number of tools in order to completethese holes. Figure 1(a) depicts the example part with their required number of tools. Based on the hole-operation infor-mation given in Table 1, initially the first operation of each hole must be selected for machining. In this case, first, oneof the operations o11; o21 and o31 must be executed before their successive operations if there is any. At each stage ofhole-making the sequence of hole-operations must be maintained. For better visualisation Figure 1(b) shows a technicalsequence of operations for each hole and a possible sequence of tool travel path.

The aim of this study is to optimise the tool-hole sequence by employing BBO algorithm in order to reduce thenon-productive time including tool travelling time and tool switching time. A tool switching time can be defined assummation of tool to tool changing time and a tool travelling time that occurs due to the tool trip from the executedhole to the tool magazine and then trip from the tool magazine to the next hole.

3. Model formulation

In this section, a mathematical model is presented to clearly specify the key parameters and their influence on the hole-making problem. The present model modifies the model presented in Ghaiebi and Solimanpur (2007) and Liu et al.(2013).

Figure 1. (a) Example part of tool sequence for hole-making. (b) Technical sequence of operations.

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3.1 Notation

Subscriptsi; i0 hole indices; 1� i; i0 � Ij; j0 operation indices; 1� j; j0 � nit; t0 tool indices; 1� t; t0 � Tp; p0 position in the sequence; 1� p; p0 �N

Parameters and setsTTii0 travelling time between hole i and hole i0

TToi travelling time between starting point and hole iTTiM travelling time between hole i and coordination of tool magazineN total number of positions in a sequence and or total number of operationsTS tool to tool changing time

Decision variablesLijtp i0j0t0pþ1ð Þ equal to one if the spindle moves between two adjacent position of sequence to process operation j and j0

of hole i and i0 by tool t and t0 respectively; zeros otherwiseXijtp equal to one if operation j of hole i is processed by using tool t on pth position of the sequence; zero

otherwise

3.2 Mathematical model

The mathematical model which is able to take into account an objective function and various constraints arising fromthe problem environment can be stated as follows.

MinXi

Xi0

i0 6¼ i

Xj

Xj0

Xt

Xt0

t0 ¼ t

XN�1p¼1

Lijtp i0j0t0pþ1ð ÞTTii0 þXi

Xi0

Xj

Xj0

Xt

Xt0

t0 6¼ t

XN�1p

Lijtp i0j0t0pþ1ð Þ TS þ TTiM þ TTi0Mð Þ½ �

þXi

Xj

Xt

Xijt1TToi

(5)

Subject to:

Xp

Xijtp ¼ 1 8i; j; t 2 ComTij (6)

Xi

Xj

Xt2ComTij

Xijtp ¼ 1 8p (7)

Table 1. Sequence of operation for holes.

Hole (i) Tool sequence (t) Sequence of operation (oij)

Hole 1 t1 o11Hole 2 t1; t2 o21 � o22Hole 3 t1; t2; t3 o31 � o32 � o33

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Xijtp�XN

p0¼pþ1Xi jþ1ð Þtp0 8i; j 6¼ ni; t 2 ComTij (8)

Xijtp þ Xi0j0t0pþ1 � 2Lijtp i0j0t0pþ1ð Þ � 0

8i; i0; j; j0; t 2 ComTij; t0 2 ComTi0j0 ; t 6¼ t0; p 2 1; 2; . . .;N � 1f g or i 6¼ i0; t ¼ t0(9)

Xijtp þ Xi0j0t0pþ1 � Lijtp i0j0t0pþ1ð Þ � 1

8i; i0; j; j0; t 2 ComTij; t0 2 ComTi0j0 ; t 6¼ t0; p 2 1; 2; . . .;N � 1f g or i 6¼ i0; t ¼ t0(10)

Xijtp 2 0; 1f g 8i; j; t; p (11)

Lijtp i0j0t0pþ1ð Þ 2 0; 1f g

8i; i0; j; j0; t 2 ComTij; t0 2 ComTi0j0 ; t 6¼ t0; p 2 1; 2; . . .;N � 1f g or i 6¼ i0; t ¼ t0 (12)

Equation (5) prescribes the objective function that minimise the summation of the tool travelling time between holes,tool switching time and tool travelling time of first position respectively. Tool travelling time can be calculated by sum-mation of tool travelling between different holes in the adjacent position that processed with the same tool. The secondcomponent in objective function, tool switching time, is formulated by total summation of tool to tool changing timeand tool travelling time, occurs when spindle has to move to tool magazine to change the tool for the next operation.The tool changing point is located at zero in x-axis and value of y-axis coordination of last hole i which is executed bytool t. The last component represents the first tool travelling time between starting point with coordination value of(0, 0) and hole in first position of sequence.

Constraint (6) indicates that for each tool-hole operation only one position of the sequence can be assigned. Con-straint (7) ensures that for each position in the sequence only one tool-hole operation can be occupied. Constraint (8)prescribes the precedence constrain between operations of each hole. Constraints (9) and (10) state the relationship oftwo consecutive position of sequence. The result of Equations (9) and (10) should be equal to one if the mentionedpositions have been processed by two different tools or when two different holes processed with same tool, otherwise iszero. Constraints (11) and (12) restrict the decision variables into zero-one values.

4. Proposed algorithms

4.1 BBO algorithm

The BBO is a new evolutionary algorithm among the popular meta-heuristic approaches which have arisen as attractiveoptimization algorithms due to their competitive results (Simon 2008). This population-based algorithm is a naturallyinspired algorithm in which mimics the migration process of species for solving engineering problems (Rahmati andZandieh 2012). This algorithm has revealed notable performance on many well-known case studies (Dawei, Simon, andErgezer 2009). BBO algorithm starts the optimization process with a number of candidate solutions, called habitats orislands. Each island feature is considered by a suitability index variable (SIV). Each habitat is characterised by a quanti-tative performance index, named habitat suitability index (HSI).

The main principal of BBO is based on immigration and emigration of species in a habitat, known as migra-tion. With probabilistic migration, BBO is able to share more information from good solutions to poor ones. Inother words, this algorithm prevents the good solutions to be demolished during the evolution. Thus, it can effi-ciently utilise the characteristics and information of population in per iteration. Without mutation operator that canincrease the diversity among the population, the solutions with high HSI have tendency to be more dominant inpopulation.

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4.1.1 Habitat encoding and the decoding

Defining an appropriate habitat representation plays a key role in designing and implementing an optimization algo-rithm. To solve hole-making problem, one of the main issue must be taken into account is sequencing the hole-opera-tions. One of the most extensively used encoding schemes for sequencing problem in the literature is operation-baserepresentation. The main advantage of this representation lies in the ability to create feasible sequence with any permu-tation of the habitat. This habitat contains a number of elements or SIVs, equal to the number of hole-operations thatare to be processed. The structure of this habitat is depicted in Figure 2. The habitat in Figure 2 represents a typicalsequence of tool-hole operations. In this habitat, each hole i occurs exactly ni times. The sequence chromosome is avector like [3 1 2 3 2 3] (according to example presented in Section 2) in which three holes executed with differentsequence of tools.

4.1.2 Initialize the habitat

To start the BBO algorithm, an initial population of solutions based on the above habitat representation must be gener-ated. In this article a population of these habitats is produced at random.

4.1.3 Migration

In biogeography, migration is the adaptive process which is used to move the species between different habitats. Themigration process is occurred based on a probabilistic operator. In BBO, the probability to select the solution Hi as theimmigrating habitat is related to its immigration rate ki and solution Hj is related to its emigration rate lj. Migration canbe express as Equation (13).

Hi SIVð Þ Hj SIVð Þ (13)

The immigration and emigration rates are functions of the solutions’ fitness. They can be evaluated by Equations(14) and (15) respectively.

ki ¼ I 1� kin

� �(14)

li ¼ Ekin

� �(15)

In Equations (14) and (15), I and E represent the maximum possible immigration and emigration rate respectively;ki is the rank of habitat i after sorting all habitats according to their HSI; and n is the number of solutions in the popula-tion. It is clear that the better solution has higher emigration and lower immigration rate, while the converse is true fora poor solution. Often, I and E set equal to one or slightly less than one (Ma and Simon 2011).

After determining the immigrating and emigrating habitats, the migration process can be performed like crossover inevolutionary algorithms. In this study, to do the migration, improved precedence operation crossover (IPOX) is adoptedfrom Zhang et al. (2007) for operation sequence. This effective migration operator is described as follows.

Step 1: Divide the set of holes 1; 2; 3; . . .nf g, into two non-empty groups H1 and H2 randomly.Step 2: Direct copies (same position) those numbers in H1 from immigrating habitat to the modified habitat.Step 3: Indirect copies (same order) those numbers in H2 from emigrating habitat to the modified habitat.

Figure 2. An example of habitat representation for tool-hole operation sequence.

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Figure 3 illustrates an example of the IPOX of three holes 1; 2; 3f g with 1, 2 and 3 operations respectively. The setof hole is divided into two sets H1 ¼ 3f g and H2 ¼ 1; 2f g randomly. The migration of immigrating habitat and emigrat-ing habitat generates the modified habitat 3; 2; 1; 3; 2; 3f g. It can be seen that modified habitat preserves the positionand order of hole 3f g in immigrating habitat and the order of hole 1; 2f g in emigrating habitat respectively. ThereforeIPOX is excellent in the characteristics-preservingness.

4.1.4 Mutation

In BBO, mutation rate is inversely related to the solution probability and can be calculated by Equation (16).

mi ¼ mmax � 1� Pi

Pmax

� �(16)

In Equation (16), mmax is a user-defined maximum probability; Pmax ¼ argmax Pi; i ¼ 1; 2; . . .; n (n is populationsize), and Pi is the solution probability (Ma and Simon 2011). Based on the mutation probability, the mutation operatorscan be done as follow. For each operation sequence habitat, one of the three mutation operators; swap, insertion, andreversion can be implemented with equal probability at each iteration. Figure 4 depicts the mutation operator for opera-tion sequence.

4.1.5 Evaluating the HSI

In BBO, each habitat is evaluated based on its corresponding HSI. In this study, the fitness value is measured accordingto the objective function mentioned in Section 3.2, mathematical model.

4.1.6 Update habitat

To update population for the next generation three steps, merging, sorting, and truncating have been implemented. Thisscheme is used to preserve elite habitats for the next generation. Merging is related to combination of habitats, beforeand after applying BBO operators which makes the habitat population size twice a time. Then, the combined habitatmust be sorted based on their HSI in ascending order (in minimization problem). Finally, the best habitats are selectedfrom the combined and sorted habitat with the amount of original habitat size for the next generation.

Figure 3. IPOX migration operator for tool-hole operation sequence for H1: [3] and H2: [1, 2].

Figure 4. Example of mutation operator for tool-hole operation sequence, (a) swap, (b) reversion, (c) insertion.

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4.1.7 Stopping criteria

The optimization algorithm with various operators such as migration, and mutation is performed repeatedly until a stop-ping criterion is met. In this study, reaching to a maximum time limit of 1000 s of CPU time is considered as termina-tion criterion.

4.1.8 Implementation

Based on aforementioned BBO Operators, the BBO algorithm to solve the hole-making problem is described as fol-lows.

4.2 The genetic algorithm

As previously mentioned, to evaluate the performance of the proposed BBO algorithm much more clearly, it is com-pared with a GA. To do so, the operators of GA are considered just like the BBO’s operator to minimise the impact ofthe different operators on the performance of the algorithms. Therefore, in the proposed GA, the initialization method isthe same as the BBO, the crossover is like the migration of the BBO (IPOX), and mutation structures are also the same.It is worth reminding that their most difference is in their selection strategies. In GA, the selection strategy is roulettewheel to form donors’ mating-pool based on a selection probability given by:

Psel ¼ Find

Ftot; ind ¼ 1; 2; . . .;N (17)

where Psel is the probability of choosing the indth individual, N is the population size, Find is the indth individual fit-ness, and Ftot is the total fitness of all individuals in the current generation.

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5. Computational results

In order to examine the performance and effectiveness of proposed BBO algorithm, we use two test problems to con-duct computational tests and comparison on both small-scale and large-scale instances for hole-making process. Testproblem 1 addresses to the example adopted from Liu et al. (2013). Test Problem 2 corresponds the second exampleaddresses by Ghaiebi and Solimanpur (2007). The mentioned test problems are described in the following sections.

These test problems are implemented by the proposed BBO algorithm coded using Matlab R2013a and run onIntel® core™ 2 Duo CPU T8100 at 2.1 GHZ, 3 GB RAM computer with windows 7.

The BBO parameters are set in this study after number of careful runs as follows; the habitat size (N) = 100, maxi-mum migration and immigration rate of each habitat = 1 and mutation probability = 0.3.

5.1 Test problem 1

Test problem 1 considers a part with 42 (6 × 7) holes in which different number of tools is employed to complete thedesired size of each hole. The data about the distances between the holes, the diameter and type of holes are shown inFigure 6(a). Table 2 presents a short description of these tool-hole combinations. This test problem corresponds to theexample addressed by Liu et al. (2013). The objective of test problem 1 is to minimise the total non-productive time(auxiliary time). In this example the speed of spindle ðvÞ and tool to tool changing time ðTSÞ are set to 12 m

min and 5 srespectively. The tool magazine is located beside the column, so the tool changing point will move along the line paral-lel to y-axis, where x ¼ Xc. In this paper, it is assumed that Xc ¼ 0. Two measuring distance methods, Euclidian andrectilinear, are used to test the performance of proposed BBO algorithm in the following sections. The proposed BBOalgorithm was applied to determine sequences of operations where a hole consists of several individual operations withdifferent machining tools.

5.2 Test problem 2

The problem of finding the best path for traversing the points to be drilled can be modelled as a TSP in order to cutdown the non-productive time. Test problem 2 addresses a set of TSP from small to large-scale instances which is

Figure 5. (a) The method to determine the position of the ith hole in the workpiece. (b) The position of 10 holes in the workpiece.

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Figure 6. The optimal tool travel path for Euclidian distance.

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adopted from Ghaiebi and Solimanpur (2007). Each city in TSP indicates the position of each hole that must be drilledand visited once (only one operation for each hole), and the tool will return to the initial point after completing its tour.This test problem consists of 6 test instances, 5, 10, 15, 20, 25 and 50 holes respectively. To make the attempted prob-lems reproducible, the arrangement of holes in the tested problems is considered as follows:

� The number of rows in the workpiece is b ffiffiIp c where I is the total number of holes.

� The centre to centre distance of holes in each direction is assumed to be 2 cm.� The position of the ith hole is determined as shown in Figure 5(a).

For example, the position of holes in the workpiece when I ¼ 10 is shown in Figure 5(b). As seen in this figure,the number of rows is b ffiffiffiffiffi

10p c ¼ 3. The objective of test problem 2 is to minimise the tool path.

5.3 Performance of proposed BBO algorithm for test problem 1

Computational performance of proposed BBO algorithm is evaluated by considering the test problem 1. Table 3describes a comparison between solution quality among the best near optimal solutions acquired by proposed BBO algo-rithm, proposed GA and the results of ACO proposed by Liu et al. (2013) for ten times computation when consideringnon-productive time (auxiliary time) as objective function for two different measuring distance methods, Euclidian andrectilinear. The first column indicates the measuring distance methods name. In the second column, the best results ofproposed BBO algorithm are shown. The remaining columns reports the best results of the two algorithms, proposedGA and ACO proposed by Liu et al. (2013), we compare with, together with the relative deviation with respect to ourBBO algorithm. Relative deviation criterion is used to compare the results of proposed BBO with the results obtainedby the two mentioned algorithms in term of solution quality. Relative deviation is obtained as follows.

dev ¼ NPT compð Þ � NPT BBOð ÞNPT compð Þ

� �� 100 (18)

where NPT BBOð Þ is the best non-productive time obtained by our proposed BBO and NPT compð Þ is the best non-pro-ductive time of the solution methodology that we compared ours to.

The results of this table shows how this simple version of BBO algorithm has significant improvement to reach sub-optimal solution in comparison with GA and the results reported by Liu et al. (2013). As can be seen from the results,the proposed algorithm indicates 2.35 and 9.28% improvements in solution quality of proposed GA and ACO proposedby Liu et al. (2013) respectively for Euclidian distance and 0.36 and 6.26% respectively for Rectilinear distance. Theoverall result of the proposed BBO algorithm reveals an improvement in solution quality, 1.35 and 7.77% in averagewith respect to the two solution approaches, proposed GA and ACO proposed by Liu et al. (2013) respectively. It canbe stated that this improvement may be obtained due to the ability of proposed BBO algorithm to efficiently explore thesearch space based on two main operators, migration and mutation. It is worth mentioning that the main difference ofproposed BBO algorithm with other evolutionary algorithms is in its selection strategies, one for migration and one formutation. Details of these selection strategies are previously explained in Sections 4.1.3 and 4.1.4.

To better understanding of the results found by proposed BBO algorithm, this example has been considered withmore details. Results that were reached by BBO algorithm for this test problem are shown in Table 4 as well as

Table 2. Tool hole combination of example given by Liu et al. (2013).

Type of hole Number of holes Tool sequence

Bh 18 2–6Th 10 2–3–4–5Sh 14 1–8

Table 3. Computational results for test problem 1.

Measuring distance Proposed BBO Proposed GA dev (%) ACO Proposed by Liu et al. (2013) dev (%)

Euclidian distance 119.2 122.07 2.35 131.4 9.28Rectilinear distance 136.85 137.35 0.36 146 6.26

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Table

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parisonresults

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2–3–4–5–

1–8–6

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4–5–6–1–8

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Figures 6 and 7. Table 4 provides a detailed comparison between the results acquired by BBO algorithm, GA and ACOmodel proposed by Liu et al. (2013) for this test problem. Figures 6 and 7 illustrate the optimal tool path represented

Figure 7. The optimal tool travel path for rectilinear distance.

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by BBO algorithm in case of Euclidian and rectilinear distance measuring respectively. These diagrams depict the tool-hole operations and tool sequences.

According to the results of Table 4, the solution obtained by proposed BBO algorithm is better than the two otheralgorithms GA and ACO reported by Liu et al. (2013), since tool travelling time decreased from 101.4 and 92.07 to 89.2for Euclidian distance and decreased from 116 and 107.35 to 106.85 for Rectilinear distance when the non-productivetime considered as objective function.

5.4 Performance of proposed BBO algorithm for test problem 2

In order to make a further investigation on the performance and effectiveness of proposed BBO algorithm, we ran theBBO and GA on more challenging instances, test problem 2, that range from small to large scale instances. The resultsof our algorithm are compared with ACO algorithm proposed by Ghaiebi and Solimanpur (2007) which is initially hasbeen compared with the dynamic programming (DP) and obtained optimum/near optimum solution in reasonableamount of time for all test instances. As reported in their study, DP method is not able to solve the problems for 25 and50 holes in a reasonable time due to the size of memory required for running DP. Table 5 describes a comparisonbetween solution quality among the best near optimal solutions acquired by proposed BBO algorithm, proposed GA andthe results of ACO proposed by Ghaiebi and Solimanpur (2007) for 25 trials when considering minimum tool path asobjective function. The first column indicates the number of holes for each test instance. In the second column, the bestresults of proposed BBO algorithm are shown. The remaining columns reports the best results of the two algorithms,proposed GA and ACO proposed by Ghaiebi and Solimanpur (2007) that we compare with, together with the relativedeviation with respect to our BBO algorithm. Results presented in Table 5 can be analysed from the point of view ofdifferent number of holes and their associated computational difficulty. As can be seen, the proposed BBO performswell by augment in the number of holes. The proposed algorithm indicates 23.53 and 7.14% improvements in solution

Table 5. Computational results for test problem 2.

Number of holes Proposed BBO Proposed GA dev (%) ACO proposed by Ghaiebi and Solimanpur (2007) dev (%)

5 12 12 0 12 010 24 24 0 24 015 32 32 0 32 020 40 40 0 40 025 52 52 0 52 050 104 112 7.14 136 23.53

Figure 8. The result of the test problem 2 with 50 holes.

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quality of proposed GA and ACO proposed by Ghaiebi and Solimanpur (2007) respectively for test instance with 50holes. The results of other test instances are equal to the best results obtained by proposed GA and ACO proposed byGhaiebi and Solimanpur (2007). The overall result of the proposed BBO algorithm reveals an improvement in solutionquality, 1.19 and 3.92% in average with respect to the two solution approaches, proposed GA and ACO proposed byGhaiebi and Solimanpur (2007) respectively. For visualisation, Figure 8 depicts the optimum tool travel path of testinstance 6 in which 50 holes must be drilled in a part.

6. Conclusion

This research work developed a mathematical programming model for hole-making problem. Whereas, this problem isconsidered as NP-hard, a newly meta-heuristic optimization algorithm, called BBO, is developed to solve the problemof minimising the non-productive time including tool travelling time and tool switching time. The hole-making problemin this study is concerned with the sequencing of various tool-hole combinations. The proposed solution methodologyhas been tested on various test problems and the results obtained are compared with those from some of the existingmetaheuristic algorithms. As can be seen from this comparative study, it has been observed that the proposed algorithmoffers better results for majority of the test problems. However, application of the proposed BBO algorithm is limited tocertain instances where there is a part with different number of holes which are executed by one or more tool types.This research can be further extended to see how the performance of BBO algorithm can be improved by adopting somelocal search methods or integration of different strategies for generating initial population. Moreover, the performance ofBBO algorithm can be explored for high-dimension real-world problem which is a challenging task for any algorithm.

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