Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 0 A Siplified Analytical Approach for Efficiency Evaluation of the eaving Machines with Autoatic illing Repair DOINA CAŞCAVAL, PETRU CAŞCAVAL Dept. of Textile Engineering, Dept. of Coputer Science and Engineering Gh. Asachi Technical University of Iaşi 53B, D. Mangeron, 700050, Iaşi ROMANIA Abstract: A achine interference proble regarding a group of weaving achines with autoatic filling repair and filling break tolerance is treated in this paper. Two indicators have to be evaluated: the efficiency of the weaving achines and the work loading for the weaver. A siplified analytical ethod, based on the superposition principle and a reduced sei-markov chain, for evaluating with accuracy the two indicators previously defined is proposed. A case study in which analytical and siulation are copared deonstrates the effectiveness of this siplified analytical approach. Key ords: eaving Process, Autoatic illing Repair, Break Tolerance, Interference Tie, Markov Chains. Introduction The weaving process is a discrete event one because the warp yarns and the filling yarn break off at rando instants. In case a weaving achine (loo) is down, because a yarn has broken, a weaver (loo operator) ust reedy the broken yarn and than start up the loo again. In other words, a weaving achine is a syste with repair. The proble of allocation of loos to the weavers is a very iportant one in a large weaving ill. Two conflicting aspects ust be considered: the loo efficiency (losses caused by interference) and the work loading for the weaver. The prediction of the loo efficiency and the weaver work loading, when a group of weaving achines are allocated to one ore ore weavers is a achine interference proble. The proble of allocation of loos in weaving is widely dealt with in textile literature, both fro a theoretical and a practical point of view (for exaple, [] and [5]). In this work we focus on the achine interference proble for the loos with autoatic filling repair and filling break tolerance between packages and the prewinder. The analytical approach of achine interference proble is based on the queueing theory. The standard odel is a Markov chain for which the steady state probabilities are required. If the Markov chain has s states, s linear equations ust be solved. This ethod is siple in essence, but we ust have in view the coplexity of Markov odels [], [], [3]. or any sizable practical proble s becoes very large and the solution tie becoes very long, so that, the classical approach is difficult to apply. hen the Markov chain is very large, the two approaches available to deal with this proble are to either tolerate the largeness or avoid it. In this work, a largeness avoidance technique for an approxiate evaluating of the loo efficiency in a weaving process with autoatic filling repair is proposed. The siulation is a copleentary approach for achine interference proble [], [4]. A siulation progra based on a stochastic coloured Petri net has been used in order to validate the analytical. The reainder of this paper is organized as follows. In Section the interference proble concerning the weaving process is defined in details. In Section 3 two siple cases with one and two loos, respectively, are solved based on sei- Markov chains. In order to generalize the proble, in Section 4 a siplified analytical ethod for evaluating with accuracy the efficiency of the weaving achines is proposed. Section 5 presents a case study in which analytical and siulation are copared. The paper is closed with final rearks. Proble orulation Consider identical loos, served by one weaver, carrying out a weaving process copletely known fro a statistical point of view. Usually, a loo works with any packages for the sae filling yarn. Thus, in case the filling yarn between a package and the prewinder breaks off, an autoatic switch selects a spare package and avoids the stop of the loo. e say that the spare packages ensure a filling break
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 tolerance for the weaving process. On the other hand, when the filling yarn breaks off into the shed, the broken yarn is reoved autoatically without a weaver intervention. In this case the loo efficiency does not depend on the achine interference tie. or a stochastical odelling of a weaving process, six priary rando variables are considered: Tie to break off a warp yarn let be the warp breakage rate; Tie to break off the filling yarn into the shed let be the breakage rate into the shed; Tie to break off the filling yarn between a package and the prewinder let be the breakage rate between packages and prewinder; Tie to reedy a warp breakage let be the reedying rate of warp breakages; Tie to reedy a breakage into the shed let be the reedying rate of breakages into the shed; Tie to reedy a breakage between a package and the prewinder let be the reedying rate of this type of breakages. or the loos with autoatic filling repair (Dornier AS, for exaple), the broken yarn is reoved autoatically fro the shed in a constant interval of tie, noted by RTS. In this case, =. RTS Assue that all paraeters,,,,, and RTS are known. The proble of prediction the loo efficiency (E) and the weaver work loading (L), depending on the nuber of loos allocated to the weaver, is treated in this paper. A siilar proble is presented in [] and [], where weaving achines with filling break tolerance are considered. In this work we focus especially on the weaving achine with autoatic filling repair, but the filling break tolerance is also considered. Assutions: The weaving process is in a steady state condition. All rando variables are exponentially distributed. A weaving achine is either up or down, with no partial or interediate states. All break events are stochastically independent. 3 Exaples Two exaples regarding the weaving achines with autoatic filling repair are presented in this section. In the weaving process, two identical packages are used to feed the shed with filling yarn. 3. The first exaple Take the ost siple case in which the weaver serves a single weaving achine (=). To estiate the loo efficiency, two different approaches based on Markov chains are presented: first, a classical approach and than, a siplified one. a) A classical approach The weaving process is odeled by a sei-markov chain with the following states: S The loo is running on and no yarn breaks exist (the initial state); S hile the loo is running on, the weaver works to reedy a broken yarn between a package and the prewinder; S 3 The loo is down and the weaver works to reedy a warp breakage; S 4 The loo is down because of a yarn breakage into the shed; the broken yarn is reoved autoatically fro the shed; S 5 The loo is down and the weaver works to reedy a warp breakage. A yarn between a package and the prewinder is also broken, but this reedying is teporary suspended. S 6 The loo is down because both packages are unavailable. The weaver works to reedy a breakage between a package and the prewinder. S 7 The loo is down because of a breakage into the shed; while the broken yarn is reoved autoatically fro the shed, the weaver works to reedy a breakage between a package and the prewinder. The state diagra of the sei-markov chain describing this weaving process is given in ig.. S 3 S 5 µ µ µ S S S 6 µ µ µ S 4 S 7 µ ig. Markov chain for a weaving achine. The atrix M, illustrated in ig., presents the transition rates between states. The location (i, j) in atrix M, i j, coprises the transition rate fro state j to state i. The value of location (i, i) is equal to
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 the su, taken with inus, of the transition rates in colun i. ( + + ) 0 0 0 ( + + + ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( + ) ig. Transition atrix M of Markov chain presented in ig.. Let p be the steady state probability of state S i, i i {,,., 7}. To deterine these probabilities, the set of linear equations () ust be solved, in which P=[ p, p,..., p7 ] T, and Z=[0, 0,..., 0] T. S 3 µ µ S µ M P = Z p + p + + p7 = () This set of equations leads to the probabilities, p = + + + ( + RTS + + + ) + p = p, () + where =, = RTS, = and =. + The loo efficiency E and the work loading for the weaver L are equal to E = p + p, L = p. (3) b) A siplified approach In order to reduce the Markov chain, the rule of superposition is applied, so that, the loo efficiency is deterined in two steps: in the first step, the warp breakages and the breakages between packages and the prewinder are considered, whereas, in the second one, the attention is oved to the filling breakages into the shed. Step. If the filling breakages into the shed are ignored, only the states S, S, S 3, S 5 and S 6 are possible. The Markov chain describing this weaving process is presented in ig. 3. or this case, the steady state probabilities p and p are p =, p = p. (4) + + + + The first estiation for the loo efficiency is S 5 S S 6 µ ig. 3 Markov chain for a weaving achine when the filling breaks into the shed are ignored. E + =. (5) + + + + Step. Now, let us consider only the filling breakages into the shed. Reeber that the rate of filling breaks is, and the reedying tie for a breakage is a constant noted by RTS. In this case, the loo efficiency ( E ) is equal to E =. (6) + RTS hen all types of yarn breakages are considered, we can use the superposition rule to obtain the loo efficiency, so that E = E E. (7) To deonstrate the effectiveness of this siplified ethod, a nuerical evaluation is presented. Take a weaving process with filling break tolerance and autoatic filling repair, described by the following paraeters: = 4.77 warp breakages/h; =.05 filling breakages into the shed/h; =.37 yarn breakages between packages and prewinder/h; = 58.88 warp reedies/h; = 0.0 filling reedies/h (RTS=0.004545 h); = 43.0 yarn reedies between packages and prewinder/h. 3
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 3 Based on the result of classical approach (Eqs. (3) and (4)), the loo efficiency is E=0.96, and by using the ethod in two steps, E=0.959. 3. The second exaple Consider two weaving achines with autoatic filling repair and filling break tolerance, served by one weaver. In this case, the classical approach for evaluating the loo efficiency is difficult to apply because the Markov chain is coposed of 46 states. or this reason, the ethod in two steps based on the rule of superposition is ore appropriate. Step. If the filling breaks into the shed are ignored, the Markov chain that odels the weaving process is coposed of 6 states, as illustrated in Table. The following notations are used to denote the states of a weaving achine: the weaving achine is running on, no break exists; the weaving achine is down because of a warp breakage; the weaving achine is running on but a yarn breakage between a package and prewinder has occurred; the weaving achine is down because both packages are unavailable. Regarding the weaver, the notation R denotes a warp break reedying, whereas, R reflects a filling break reedying between a package and the prewinder. The transition atrix is M=[a i,j ], i, j {,,, 6}, where a i,j, i j, denotes the rate transition fro state j to state i, and a i,i is the su of transitions in colun i, taken with inus. To obtain the steady-state probabilities, MATLAB progra can be used, and then, the loo efficiency E can be calculated by applying Eq. (8). E = p + p3 + p + p 8 + p9 + 0.5( p + p5 + p6 + p + p + p + p + p + p 4 6 Step. If only the filling breakages into the shed are considered, the loo efficiency does not depend on the achine interference tie. Consequently, the loo efficiency ( E ) is given by Eq. (6). 7 8 7 0 + ). (8) Table The states and the transition rates of Markov chain (the second exaple). S i The states specification The transition rates a i,j, i j. S {},{} a, = ; a,3 = S {} R,{} a, = ; a,4 = ; a,6 = ; a,7 = S 3 {} R,{} a 3, = ; a 3,5 = ; a 3,8 = ; a 3,9 = ; a 3, = S 4 {} R,{} a 4, = ; a 4,3 = S 5 {} R,{} a 5, = ; a 5,4 = ; a 5,5 = ; a 5,9 = S 6 {} R,{} a 6,3 = S 7 {} R,{} a 7,3 = S 8 {} R,{} a 8,3 = ; a 8, = ; a 8,8 = S 9 {} R,{} a 9,3 = ; a 9,6 = ; a 9,0 = S 0 {} R,{} a 0,5 = ; a 0, = S {},{} R a,0 = ; a,7 = S {} R,{} a,5 = ; a, = S 3 {} R,{} a 3,6 = ; a 3,7 = S 4 {} R,{} a 4,6 = ; a 4,9 = S 5 {} R,{} a 5,7 = ; a 5,8 = S 6 {} R,{} a 6,8 = ; a 6,9 = ; a 6,4 = ; a 6,6 = S 7 {} R,{} a 7,9 = S 8 {} R,{} a 8,9 = S 9 {},{} R a 9, = S 0 {},{} R a 0, = ; a 0,3 = ; a 0,5 = S {} R,{} a,4 = ; a,7 = ; S {} R,{} a,4 = ; a,8 = ; S 3 {} R,{} a 3,6 = ; a 3,7 = ; S 4 {} R,{} a 4,6 = ; a 4,8 = ; S 5 {},{} R a 5,0 = ; 4
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 4 S 6 {},{} R a 6,0 = ; inally, Eq. (7) is used to obtain the loo efficiency. ith the paraeters presented in section 3., the loo efficiency is equal to E=0.905. A closed result has been obtained by siulation, naely, E=0.9053. As a conclusion, the nuber of states of Markov chain increases draatically when the weaver serves ore weaving achines. Usually, one weaver serves up to ten weaving achines, when the Markov chain has hundreds of states. Taking into account the coplexity of Markov chains, the classical approach for exact evaluation of loo efficiency is difficult to apply. or this reason, an approxiate analytical ethod is proposed in the following section. 4 Siplified analytical approach for achine interference proble or the general case in which one weaver serves weaving achines, the Markov chain is too large if all rando variables presented in section are considered. or this reason, we focus on an approxiate ethod and propose a reduced Markov chain able to predict with accuracy the efficiency of the weaving achines. To siplify the analytical odel, two points were having in view, as follows. ) Regarding the autoatic filling repair The down tie for reedying a breakage into the shed does not depend on the interference tie, so that the loo efficiency can be evaluated in two steps, as presented in Section 3. ) Regarding the filling break tolerance To reduce the Markov chain when any packages are used for the sae filling yarn, soe serial and parallel transforations can be applied []. In this way, an approxiite odel with only two rando variables describing the weaving process can be obtained. These two rando variables are: the tie to stop the weaving process because of a breakage (a warp or a filling breakage), and the tie to reedy a yarn breakage. Let and be the stop and the reedying rate, respectively. Assuing that both rando variables are exponentially distributed, the weaving process with weaving achines can be odeled by a reduced Markov chain with + states, as presented in ig. 4. The steady state probabilities are given by the Eqs. (9) and (0), where =. p =. (9) i + ( k) k = 0 i i pi = p ( k), i = k = 0, 3,...,. (0) The following notations are introduced: d the ean nuber of achines down in a certain tie; the ean stop rate in the group of weaving achines; tr the ean reedying tie for a yarn breakage; td the ean down tie of a weaving achine because of a breakage; ti the ean interference tie. The following equations can be written: d + = ( i ) p i tr =, td d, = ( i + ) () p i = (Little forula) () td = ti + tr. (3) ( i ) pi It follows that, ti =. (4) ( i + ) p + Let c v be the coefficient of variation for tie to reedy a weaving achine, obtained by siulation. As shown in [5, pp.70], the estiation of ti can be iproved by applying a correction factor, so that * + cv ti = ti. (5) It follows that, d * = ( ti * + ), and finally, * d E =. (6) Sybol * is used to denote the estiation when the correction factor is considered. i S S S 3 S i S S+ (-) (-) (-i+) (-i) 5
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 5 ig. 4 Reduced Markov chain for weaving achines served by one weaver. The work loading for the weaver is equal to ([], [4]). Note the good accordance between the + analytical and the siulation. L = p i. (7) Table Anaytical and siulation (expressed as %). 5 Case Study In this section, analytical and siulation are copared in order to verify the effectiveness of the reduced odel presented in this paper. Take the weaving process with filling break tolerance and autoatic filling repair as presented in section 3.. Reeber the paraeters of the weaving process: =4.77, =.05, =.37 breakages/h and =58.88, =0.0, = 43.0 reedies/h. Note that, in this case study, the weaver does not suspend the reedying process of a broken yarn between a spare package and the prewinder, when other breakage occours, as considered in section 3.. In order to evaluate the loo efficiency when loos are allocated to the weaver, the ethod in two steps will be applied. Step. In this stage, the filling breakages into the shed are ignored. A reduced Markov odel with only two rando variables is obtained by applying a parallel and a serial transforation, as proposed in []. Thus, for a weaving achine, the breakage rate and the reedying rate are given by Eq. (8) and Eq. (9), respectively. = +. (8) + eaving achines allocated to the weaver Machine efficiency (E) analytical siulation Percentage of working tie (L) analytical siulation M= 9.37 9.39.4.4 M= 90.55 90.58.5.4 M=3 89.70 89.7 3.8 3.8 M=4 88.7 88.70 43.7 43.8 =5 87.55 87.56 53.4 53.4 =6 86.5 86.7 63.9 63.8 =7 84.77 84.79 7.48 7.47 =8 83. 83. 8.0 8. 6 inal Reark An approxiate analytical ethod able to predict with accuracy the efficiency of the weaving achine with autoatic filling repair and filling break tolerance is proposed. This work iproves the result presented in [], where a siilar proble is treated. In this paper, all rando variables describing the weaving process are exponentially distributed. But, as shown in [5, pp.69], in any cases it is necessary to consider a noral or gaa distribution for the reedying tie. This point will be the subject for upcoing papers. + + =. (9) + ( + ) ith these values of and, the loo efficiency E and the work loading L can be estiated by using Eqs. () (7). Step. Only the filling breakages into the shed are considered. The loo efficiency E is given by Eq. (6), where RTS =. hen all types of yarn breakages are considered, the loo efficiency is E = E E. Nuerical when one weaver serves up to eight weaving achines are presented in Table. or siulation, a odel of stochastic coloured Petri net has been used References: [] Caşcaval, D., Caşcaval, P., Analytical and Siulation Approach for Efficiency Evaluation of the eaving Machines with illing Break Tolerance, SEAS Trans. on Inforation Science & Applications, Vol.,, 005, pp. 43 5. [] Caşcaval, P., Craus, M., Caşcaval, D., A Syplified Approach for Machine Interference Proble, Recent Advances in Circuits, Systes and Signal Processing, SEAS Press, Athens, 00, pp. 63 68. [3] Caşcaval, D., Ciocoiu, M., An Analytical Approach to Perforance Evaluation of a eaving Process, ibres & Textiles in Eastern Europe, Vol. 8, No. 3 (30), 000, pp. 47 49. [4] Caşcaval, D., Ciocoiu, M., A Siulation Approach Based on the Petri Net Model to Evaluate the Global Perforance of a eaving 6
Proceedings of the 6th SEAS International Conference on Siulation, Modelling and Optiization, Lisbon, Portugal, Septeber -4, 006 6 Process, ibres & Textiles in Eastern Europe, Vol. 8, No. 3 (30), 000, pp. 44 47. [5] Bona, M., Statistical Methods for the Textile Industry, Texilia, Torino, 993. 7