OPTIMAL SEQUENCE OF HOLE-MAKING OPERATIONS USING PARTICLE SWARM OPTIMISATION AND SHUFFLED FROG LEAPING ALGORITHM

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1 Egieerig Review, Vol. 36, Issue, , OPTIMAL SEQUENCE OF HOLE-MAKING OPERATIONS USING PARTICLE SWARM OPTIMISATION AND SHUFFLED FROG LEAPING ALGORITHM Amol M. Dalavi 1* - Padmakar J. Pawar - Tejider P. Sigh 1 1 Departmet of Mechaical Egieerig, Symbiosis Istitute of Techology, Symbiosis Iteratioal Uiversity, Gram Lavale, Mulshi, Pue, Idia Departmet of Productio Egieerig, K. K. Wagh Istitute of Egieerig Educatio ad Research, Nashik, Idia ARTICLE INFO Article history: Received: Received i revised form: Accepted: Keywords: Ijectio mould Hole-makig operatios Particle swarm optimisatio Shuffled frog leapig algorithm 1 Itroductio I the process of machiig several idustrial parts such as dies ad moulds, operatios like drillig, reamig ad tappig accout for a huge segmet of processig. Usually a part, for e.g., a plastic ijectio mould, may have several holes of differet diameters, surface fiishes, ad maybe differet depths. Differet combiatios of tools ca be used to drill a Abstract: Tool travel ad tool switch schedulig are two major issues i hole-makig operatios. It is ecessary to fid the optimal sequece of operatios to reduce the total processig cost of hole-makig operatios. I this work therefore, a attempt is made to use both a recetly developed particle swarm optimisatio algorithm ad a shuffled frog leapig algorithm demostratig i this way a eample of plastic ijectio mould. The eact value of the miimum total processig cost is obtaied by cosiderig all possible combiatios of sequeces. The results obtaied usig particle swarm optimisatio ad shuffled frog leapig algorithm are compared with the miimum total processig cost results obtaied by cosiderig all possible combiatios of sequeces. It is observed that the results obtaied usig particle swarm optimisatio ad shuffled frog leapig algorithm are closer to the results of the miimum total processig cost obtaied by cosiderig all possible combiatios of sequeces preseted i this work. This clearly shows that particle swarm optimisatio ad shuffled frog leapig algorithm ca be effectively used i optimisatio of large scale ijectio mould hole-makig operatios. hole, which cosists of a pilot tool, oe or more itermediate tools ad a fial tool to achieve the fial hole size. E.g., for drillig the hole H 3, show i Fig. 1, there could be four differet combiatios of tools; {T 1, T, T 3}, { T 1, T 3}, { T, T 3}, ad { T 3}. Tool travel, tool switch ad toolig & machiig cost is directly iflueced by combiatios of tools used for machiig a hole [1]. Tool travel takes a cosiderable * Correspodig author. Tel.: address: amol.dalavi83@gmail.com,

2 188 A. M. Dalavi, P. J Pawar, T. P. Sigh: Optimal sequece amout of time, as a result of the poit-to-poit machiig aspect i hole-makig. Kolaha ad Liag [1] report a tabu-search (TS) approach to reduce the total machiig cost for holemakig operatios. I order to reduce the total processig cost, the correct sequece of operatios ad the related machiig speeds used to carry out every operatio are importat. Figure 1. Diagram showig various combiatios of tool used for machiig the holes [1]. I machiig processes, it takes more machiig time for tool switchig ad table movemet from oe positio to aother.. Curret idustry sceario for machiig a hole is to use the same tool required for all possible holes, which icreases the tool travel cost. O the other side, Carryig out all drillig, elargig ad if required tappig or reamig operatios o each hole at a time icreases tool switch cost. Luog ad Speddig [] report process schedulig i hole-makig operatios usig a geeric kowledge based method. Qudeiri ad Hidehiko [3] itroduced a geetic algorithm (GA) to achieve the least cuttig path possible. Ghaiebi et al. [4] applied the proposed at coloy optimisatio (ACO) algorithm for optimizig the sequece of holemakig operatios i a idustrial part. Hsieh et al. [5] ivestigated the optimal sequece of hole-makig operatios usig a immue based evolutioary approach. Alam et al. [6] preseted a practical applicatio of a computer-aided process plaig method to reduce the overall machiig time of ijectio moulds usig geetic algorithm. Tamjidy et al. [7] preseted a evolutioary algorithm to reduce tool travel ad tool switchig time durig holemakig operatios based o geographic distributio of a biological orgaism. Rajkumar ad Aamalai [8] ivestigated assembly fiture desig cost usig the geetic algorithm. Dalavi et al. [9] used particle swarm optimisatio to optimise hole-makig operatios for plastic ijectio mouldig of upper holder. Srivatsava et al. [10] preseted a firefly algorithm (FA) for achievig optimal test sequece geeratio. Mariakis Y ad Mariaki M [11] used bumble bees matig optimisatio (BBMO) algorithm for the ope vehicle routig problem. Narooei et al. [1] used ACO algorithm for optimizig the tool path of case study ivolvig multiple holes. Oscar et al. [13] preseted a methodology to optimize the maufacturig time by usig ACO. Liu et al. [14] used ACO algorithm for process plaig optimisatio of hole-makig operatios. Lim et al. [15] used a hybrid cuckoo search-geetic algorithm (CS-GA) for hole-makig sequece optimisatio. Lim et al. [16] used Cuckoo Search (CS) algorithm for optimisatio of sequece i PCB Holes drillig process. It is foud i the literature related to this area that the advaced optimisatio techiques such as tabusearch, geetic algorithm, at coloy optimisatio, firefly algorithm, ad immue based evolutioary approach were used i fidig the optimal path of drillig operatios. A frequetly used optimisatio method is geetic algorithm (GA) which requires more parameters [17]. Covergece of ACO algorithms is slow due to pheromoe evaporatio ad due to high CPU time availability requiremet [17]. Immue based evolutioary approach requires more parameters for solvig optimisatio problem ad it is perhaps difficult to deal with multi-objective optimizatio problem. Hece, it is required to use a algorithm which gives more correct results [18]. Therefore, to reduce the total processig cost of a applicatio eample cosidered, a attempt is made by usig particle swarm optimisatio [19] ad a shuffled frog leapig algorithm [0, 1]. The et sectio briefly describes formulatio of a optimisatio model. Formulatio of a optimisatio model With the objective of reducig the total processig cost of hole-makig operatios, the followig optimisatio model is formulated based o aalysis

3 Egieerig Review, Vol. 36, Issue, , give by Kolaha ad Liag cosiderig followig compoets of total costs [1]: a) Tool travel cost: It occurs whe tool travels from oe place to aother place. b) Tool switch cost: It occurs wheever a differet tool is used for et operatio. If tool type required for operatio is ot available o spidle, the the required tool must be loaded o the spidle prior to performig operatio. c) Toolig ad machiig costs: It icludes the ew tool cost ad the cost of machie dow time required to replace the tool. Machiig cost comprises the operatig cost ad the machie overhead cost. The combied toolig ad machiig costs whe tool type m is used o hole ca be epressed as Eq. (1): Z m tm Ym Z tm (1) T m Where, m, tool type ide i ascedig order accordig to the tool diameters, m=1,...,m; hole ide, =1,...,N; m, ide for the previous tool to be used o hole ; Z m, collective toolig ad machiig costs whe tool type m is used o hole ; t m, machiig time ecessary by tool m for hole ; T m life of tool type m related with cuttig operatio o hole ; Y m, cost of tool type m; Z, machiig cost per uit time. Machiig time, t m, is determied by Eq. (): d m L tm () 1000 U m f m Where, d m, diameter of tool m; L, depth of hole, with the clearace; U m, cuttig speed of tool m related with a operatio o hole ; f m,suggested feed rate for tool type m; I ormal practice, depth of cut ad feed rate of cuttig speed i drillig operatios is kept fied ad costat. Hece the optimum cuttig speed, U m, for the costat feed rate ca be obtaied by solvig the followig differetial Eq. (3): dz du m m 0 (3) The cuttig speed obtaied from Eq. (3) reduces the sum of toolig ad machiig costs for a sigle operatio. Cosiderig all aspects metioed above, the fial optimisatio model ca be epressed as give by Eqs. (4)- (6). Mi P pk t mm' G( s) Mi b q Ym tmm Z mm ' (4) T N N N l ' a mm m l k mm ' m M m M l1 1 k 1 mm' Subject to: N N mm m M l 1 k 1 m m l k ' 1 (5) l mm m l k l, k, m M ' mm m kl ',, k, m, m, m ' ', m M 1;, m M j (6) Where, s order ide, idicatig a specific permutatio of operatios; G(s), total processig cost related with operatios i order s; k,l, hole ide, k=1,...,n l=1,...,n; a, cost per uit o-productive travellig distace;

4 190 A. M. Dalavi, P. J Pawar, T. P. Sigh: Optimal sequece b, cost per uit tool switch time; M, set of tools that ca be used to drill hole to its fial size; P k, o-productive travellig distace betwee curret hole ad followig hole k; P l, o-productive travellig distace betwee curret hole l ad followig hole ; q mm, tool switch time betwee curret tool type, m, ad tool m required by hole ; t mm machiig time required by curret tool type, m', ad tool m required by hole ; T mm life of curret tool type, m', ad tool m associated with cuttig operatio o hole j; a 0-1 iteger variable, =1 if tool ' mmm l k ', m chages tool m to drill hole which is situated i the path betwee previous hole l ad followig hole k ad has bee drilled by tool m'; 0, otherwise. Similarly ' ' mmm kl mmm l k, a 0-1 iteger variable, ' ' mmm kl if tool m chages tool m to drill hole which is situated i the path betwee previous hole k ad followig hole l ad has bee drilled by tool m'; 0, otherwise. The idices m, m, m i the 0-1 decisio variable are used to determie the proper tool switch order durig operatio. The 0-1 decisio variables, ' simultaeously decide the order of holes to mmm l k be processed as well as the order of tools to be used to process each hole. The fuctio of this variable is to make sure the each operatio should be carried out oce. This particular coditio has bee take care by costrait Eq. (5). Eq. (6) makes sure that, the backward movemet of spidle is ot allowed uless the tool switch is required [1]. Mathematical model give i Eqs. (4)- (6), which requires large amout of computatioal time due to higher umber of 0-1 decisio variables i order to miimise the total processig cost of hole-makig operatios. Hece, to solve this model, two efficiet solutio procedures, amely, particle swarm optimisatio algorithm ad shuffled frog leapig algorithm are proposed. Net sectio discusses particle swarm optimisatio algorithm. 3 Particle swarm optimisatio (PSO) algorithm Particle swarm optimisatio (PSO) is a evolutioary computatio method developed by Keedy ad Eberhart [19]. This method starts with iitializatio of populatio of radom solutios called particles. Optimal solutio is obtaied by updatig geeratios =1 by Eqs. (7)- (8) [19]. Particle searches the optimum solutio through the problem space by comparig the curret optimum particles. This algorithm cosists of two best values. First is p best the best fitess values of idividual particles achieved so far. Secod oe is g best which are the best values betwee all particles. I PSO, velocity of particles is chaged at every geeratio towards the p best ad g best. Velocity ad positio of idividual particles are obtaied usig followig Eqs. (7)- (8): Vi 1 wvi C1 r1 ( pbesti Xi) (7) C r ( g X ) besti i1 i i1 i X X V (8) Where, V i+1= New velocity of each particle usig Eq. (7); X i+1= New positio of each particle usig Eq. (8); C l = cogitive parameter of each particle; C = social parameter of each particle; W = Iertia weight. This process of iteratios usig Eqs. (7) - (8) cotiues till covergece criteria are satisfied. 4 Shuffled frog leapig algorithm The shuffled frog leapig algorithm is a metaheuristic optimisatio techique, origially developed by Eusuff ad Lasey i 003, which is based o the coduct of a group of frogs while searchig for the maimum amout of food site [0]. The most well-kow beefit of shuffled frog leapig algorithm is its fast covergece speed [17]. The various steps i SFL algorithm with modificatio are as follows [0]: 1. Geerate virtual frog radomly called populatio p;. Evaluate the fitess of populatio; 3. Sort the populatio i descedig order; 4. Partitio the populatio i m memeplees; 5. Frogs i is epressed as X i = (X i1, X i,..x is) where S represets umber of variables; 6. Idetify the worst X w ad the best frog X b withi each memeplees;

5 Egieerig Review, Vol. 36, Issue, , Idetify the global best frog X g i etire populatio; 8. Apply the local search for ew positios (X i+1) by followig equatios: X i X r ( X X ) (9) 1 i b w If fitess of ew frog geerated by above Eq.(9) is better tha previous frog, the replace it with ew frog, where: X i1 = New positio of frog; X i = Previous positio of frog; r = Radom umber values betwee 0 to 1; X b = Positio of best frog amog the memeplees; = Positio of worst frog amog the memeplees. X w The et sectio briefly describes ijectio mould eample. 5 Ijectio mould eample The particle swarm optimisatio & shuffled frog leapig algorithm are applied to determie the optimal sequece of operatios ad correspodig cuttig speeds for the upper base of idustrial plastic ijectio mould as show i Fig., cosistig of overall 8 holes. 9. If ot, apply Eq.(10) to obtai better positio: X i X r ( X X ) (10) 1 i g w 10. If fitess of ew frog geerated by Eq.(10), is better tha previous frog, the replace it with ew frog, else replace the worst frog radomly, where X g = Positio of best frog amog the memeplees. Figure. Plastic ijectio mould upper hole plate. Distaces betwee each hole are give i Tab. 1. Tab. shows the tool switch times. Table 1. Distace betwee each holes i mm. Hole

6 19 A. M. Dalavi, P. J Pawar, T. P. Sigh: Optimal sequece Table. Tool switch times i miutes. Tool Predecessor tool Successor tool I this work it is assumed that the hole ca be machied usig sigle idividual tool. Iformatio of diameters of holes ad tools required to machie these holes are as show i Fig. 3 ad they are as follows: H =H 4=H 6= Ø 4, Tool 1 is required, H 1=H 3=H 8=Ø10=Tool is required, H 7= Ø 1=Tool 3 is required, H 5= Ø 16=Tool 4 is required. Table 3. Data related to feed rate, diameter of tool ad machiig cost. Tool f m, (mm/rev.) d m, mm Y m, Results & discussio The proposed particle swarm optimisatio algorithm ad shuffled frog leapig algorithm were coded i code blocks C++ ad ru o a Widows 8 PC with Itel core i GHz to determie the optimum sequece of operatios for a workpiece show i Fig. 3. Possible umber of sequeces by cosiderig all combiatios of sequeces for above eample ca be obtaied usig followig Eq. (11) [4]: i l l 1 i1 i i!! (11) Figure 3. Data of hole diameters. Tab. 3 shows data related to feed rate, diameter of tool & machiig cost. Possible umbers of sequeces = 4030, where, i is a total umber of operatios required for machiig. i stads for sigle machiig operatio. I stads for total umber of operatios required for etire part. To carry out this aalysis, iitially 4030 sequeces of operatios as give i Eq. (11) are geerated. The total processig cost of each sequece is obtaied as per the mathematical model give i Eqs. (4)-(6) by codig it i code blocks C++ software ad the sequece correspodig to the miimum value of total processig cost obtaied betwee all sequeces is cosidered for compariso with results of PSO & SFLA with modificatio. I et paragraph the results of PSO ad SFLA with modificatio are discussed. The followig costats a & b are used durig computatioal eperimets give i Eq. (4), i order to determie the total processig cost of hole-makig operatios. a= /mi; b= 0.666/mi; Z m=..3335;

7 Egieerig Review, Vol. 36, Issue, , Z, machiig cost per uit time= 1/mi Collective machiig ad toolig cost of all holes o ijectio mould are calculated usig Eq. (1). To calculate collective machiig ad toolig cost Z m, tool life ad optimum cuttig speeds are obtaied from Eq. (1) ad Eq. (13), respectively. The tool life epressio for these operatios [] is as follows i Eq. (1), for drillig a ew hole: T m 0.4 m 0.7 m fm 8d 5 ( ) (1) U Optimum cuttig speed as epressed below i Eq.(13), ca be achieved [1] by solvig differetial Eq. (3) with above tool life Eq. (1), for drillig a ew hole: U Yd 6 5 m m (3) 3.5 Zm fm Followig algorithm specific parameters for particle swarm optimisatio algorithm are obtaied through various computatioal eperimets. The effect of w, C 1 ad C o covergece for stadard umerical bechmark fuctios was provided by Bergh ad Egelbrecht [3]. The optimum selectio of operatig parameters of the algorithm like acceleratio costats C 1 ad C as well as iertia coefficiet w is essetial for the covergece of the algorithm. To esure the covergece of the PSO algorithm, the coditio specified by the followig Eq. (14) must be satisfied [3]: where: 1 ma ( 1, ) <1 (14) 1 w 1 1 w 1 1 Where, w (15) (16) 1 w 4, 1 = c 1 r 1 ad = c r. As the feasible rage for w is 0-1 ad for C 1 ad C is 0-, the selected values of w, C 1 ad C should be such that the Eq. (14) is satisfied for all possible values of radom umbers r 1 ad r i the rage 0-1. C 1=1.75; C =1.65; w=0.615; Number of variables: 8; Rage of variables: 1 to 8; Number of iteratios: 500; Number of particles: 100. Followig algorithm specific parameters for shuffled frog leapig algorithm are obtaied through various computatioal eperimets. C 1=0.95; C =1.0; w=1.0; Number of variables = 8; Rage of variables = 1 to 8; Number of frogs = 5; Number of memeplees = 5; Number of subfrogs = 5; Numbers of iteratios =15. Tab. 4 shows compariso of total processig cost of hole-makig operatios for best optimal sequece of each proposed method. Obtaied results of the eample discussed i sectio 5 usig particle swarm optimisatio, shuffled frog leapig algorithm ad miimum value of total processig cost obtaied by cosiderig all possible combiatios of sequeces are show i Tab Coclusio Optimisatio of hole-makig operatios ivolves a large umber of possible sequeces depedig upo the hole locatio ad tool sequece to be followed o part i order to miimise the total machiig cost. A applicatio eample of ijectio mould is attempted by usig particle swarm optimisatio algorithm ad shuffled frog leapig algorithm. The eact value of miimum total processig cost is obtaied by cosiderig all possible combiatios of sequeces. It is observed that the miimum value of total processig cost obtaied by usig particle swarm optimisatio ad shuffled frog leapig algorithm are almost the same as those obtaied by cosiderig all possible combiatios of sequeces. This clearly idicates the potetial of the preseted algorithms to solve the comple problem of determiig optimal sequece of hole-makig operatio. Although, the umbers of code lies for particle swarm

8 194 A. M. Dalavi, P. J Pawar, T. P. Sigh: Optimal sequece optimisatio algorithm ad shuffled frog leapig algorithm are higher tha those required for obtaiig miimum total processig cost, cosiderig all possible combiatios of sequeces approach, the computatioal time required for particle swarm optimisatio algorithm ad shuffled frog leapig algorithm is shorter tha that required for obtaiig miimum total processig cost if we cosider all possible combiatios of sequeces approach. Small error i results of PSO ad SFLA compared to miimum value of total processig cost, whe cosiderig all possible combiatios of sequeces, is due to probabilistic ature of these algorithms. Moreover, if more umbers of holes with differet diameters, depths, toleraces, surface fiish requiremets alog with tappig ad reamig operatios are cosidered i this problem, the compleity of the problem ivolved makes it harder to solve cosiderig all possible combiatios of sequeces approach. This clearly shows that particle swarm optimisatio ad shuffled frog leapig algorithm ca be effectively used i optimisatio of large scale ijectio mould hole-makig operatios. Future work for these two proposed algorithms ca be applied to a array of idustrial ijectio mould applicatios ivolvig holes with differet diameters ad depths. Table 4. Compariso of total processig cost of hole-makig operatios for best optimal sequece Method Z m, Tool switch travel cost, No-productive tool travel cost, Total processig cost, Miimum value of total processig cost by cosiderig all combiatios of sequeces PSO SFLA Table 5. Results compariso of best optimal order of hole-makig operatios Method Best possible sequece Total processig cost i Number of code lies & Eecutio time Miimum value of total processig cost by cosiderig all combiatios of sequeces {8,3,1,,6,4,7,5} &4 miutes PSO {8,3,1,,6,4,5,7} lies & 56 secods SFLA {1,3,8,5,6,,4,7} &1 mi

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