An Efficient Heuristic Algorithm for m- Machine No-Wait Flow Shops

Transcription

1 An Effcent Algorthm for m- Machne No-Wat Flow Shops Dpak Laha and Sagar U. Sapkal Abstract We propose a constructve heurstc for the well known NP-hard of no-wat flow shop schedulng. It s based on the assumpton that the prorty of a job n the sequence s gven by the sum of ts processng tmes on the bottleneck machne(s) for selectng the ntal sequence of jobs. The computatonal expermentatons show that there s a sgnfcant mprovement n soluton qualty over the exstng heurstc, especally for large szes whle not affectng ts tme complexty. Statstcal tests are used to substantate the sgnfcance of the results by the proposed method. Index Terms No-wat flow shop schedulng, heurstcs, total flow tme, combnatoral optmzaton A I. INTRODUCTION no-wat flow shop s a manufacturng system where each job s processed untl completon wthout nterrupton ether on or between any two consecutve machnes; that s, once a job s started on the frst machne, t has to be contnuously processed through machnes wthout nterrupton. In addton, we assume that each machne can handle no more than one job at a tme and each job has to vst each machne exactly once wthout preempton. Therefore, when needed, the start of a job on the frst machne must be delayed n order to meet the no-wat requrement. A detaled survey of the methods and applcatons of these schedulng s s gven by Hall and Srskandarajah [1]. The no-wat flow shop schedulng s wth more than two machnes belong to the class of NP-hard [2], [3]. Varous researchers have developed the constructve heurstcs as well as metaheurstcs for solvng these s for dfferent crteron such as makespan and total flow tme. For no-wat flow shop schedulng s, noteworthy constructve heurstcs wth the total flow tme objectve have been studed by Rajendran and Chaudhur [4], Bertolss [5], Aldowasan and Allahverd [6], and Framnan, Nagano, and Moccelln [7]. Rajendran and Chaudhur [4] proposed two constructve heurstcs consderng two heurstc preference relatons as the bass for selectng the seed sequence of jobs. The seed sequence of jobs thus generated s then mproved further by Manuscrpt receved January 03, 11. Dpak Laha, Reader, Mechancal Engneerng Department, Jadavpur Unversty, Kolkata , Inda. (e-mal: dpaklaha_jume@yahoo.com, dlaha@mech.jdvu.ac.n ). Sagar U. Sapkal, Research Scholar, Mechancal Engneerng Department, Jadavpur Unversty, Kolkata , Inda. (correspondng author, phone: ; fax: ; e-mal: sagar_us@ndatmes.com). usng the job nserton method of Nawaz-Enscore-Ham (NEH) [8] n the remanng parts of the heurstcs. Ther heurstcs, especally heurstc 1 perform sgnfcantly better than those of Bonney and Gundry [9], and Kng and Spachs []. Bertolss [5] presented a heurstc based on calculatng the mnmum apparent flow tme of each par of jobs and then fndng the number of tmes (here, marks) of the startng jobs of the pars as a bass for selectng the seed sequence of jobs. The seed sequence thus generated s then mproved further usng the job nserton algorthm n the same manner as done by Rajendran and Chaudhur [4]. The computatonal results reveal that the heurstc of Bertolss performs better than those gven by the heurstc of Rajendran and Chaudhur [4], and Bonney and Gundry [9]. The heurstcs of Aldowasan and Allahverd [6] and Framnan, Nagano, and Moccelln [7] have shown superor performance compared to the exstng heurstcs. However, these heurstcs take much larger computatonal tmes compared to those gven by the exstng heurstcs. Therefore, these heurstcs are not consdered here for comparson of the heurstcs for the total flow tme mnmzaton. In ths paper, we present a constructve heurstc for mnmzng the total flow tme whch s based on the assumpton that the prorty of a job n the sequence s gven by the sum of ts processng tmes on the bottleneck machne(s). We show, through computatonal expermentaton that the proposed heurstc performs sgnfcantly well compared to the Bertolss heurstc [5] whch has shown better than the heurstcs of Rajendran and Chaudhur [4], and Bonney and Gundry [9]. Also, the mean CPU tme used by the proposed method s found to be less compared to the Bertolss heurstc. II. THE PROPOSED HEURISTIC ALGORITHM Gven the processng tme p(, j) of job on machne j n the no-wat flow shop schedulng, each of n jobs s processed on m machnes n the same technologcal order wthout preempton and nterrupton on or between any two consecutve machnes. The s to determne a sequence of n jobs that mnmzes the total flow tme crteron. Let σ = {σ 1, σ 2,, σ n } represent the sequence of n jobs to be processed on m machnes, and d(, k) the mnmum delay on the frst machne between the start of job and the start of job k (requred because of the no-wat restrcton). Also, let p(σ, j) represent the processng tme on machne j of the job n the th poston of a gven sequence, and let d(σ -1, σ ) denote the mnmum delay on the frst machne between the start of two consecutve jobs found n

2 the (-1)th and th poston of the sequence. The total flow tme (TFT) of the sequence of n jobs n the no-wat flow shop schedulng s gven by, n n m TFT = ( n + 1 ) d ( σ, σ ) + p(, j) 1 = 2 = 1 j = 1 where, the delay matrx D, of the d(, k) values are calculated as n Fnk and Voß [11]. It may be noted that snce the release tmes of the jobs are all zeros, the total completon tme crteron s equvalent to the TFT crteron. The shortest processng tme (SPT) rule optmzes the mean flow tme of a set of jobs processng on sngle machne [12], [13] and also t has been shown to be effectve n m- machne flow shop schedulng [14]. Based on ths dea, we make a smlar attempt to obtan a sequence of jobs consderng bottleneck machne(s) [], [16], whch s used as a startng sequence of the proposed method. The proposed method s based on the assumpton that the prorty of a job n the sequence s gven by the sum of ts processng tmes on the bottleneck machnes. The seed sequence of jobs thus obtaned s then appled to the remanng parts of the heurstc followng a job nserton algorthm as descrbed n the work of Rajendran and Chaudhur [4]. The proposed heurstc algorthm s gven as follows: Step 1: Set z = 1. Consder sngle bottleneck machne among m machnes wth the largest sum of the processng tmes of n jobs and determne the sequence of jobs by arrangng them n ascendng order of ther processng tmes on t. Step 2: For z = 2 to m, consder z adjacent machnes as bottlenecks wth largest sum of the processng tmes of jobs on these machnes and obtan the sequence of jobs n ascendng order of the sum of the processng tmes of the ndvdual jobs on these machnes. Step 3: A total of m sequences of jobs are generated from steps 1 2 and the best one s selected as the ntal sequence of jobs for the rest part of the heurstc. Step 4: Set k = 1. Select the frst job n the ntal sequence and nsert t n the frst poston of the partal sequence σ and call ths partal sequence as the current partal sequence σ. Step 5: Update k = k+1. Select the k-th job from the ntal sequence and nsert t n the r-th possble poston of the current partal sequence σ where r s the nteger varyng k/2 r k to produce r sequences. Among these r sequences, select the one as the new partal sequence σ wth the mnmum k value of the expresson: ( k + 1 ) d ( σ, ) 2 1 σ = and make t the current partal sequence σ. Step 6: If k = n, go to Step 7, else go to step 5. Step 7: The sequence σ s the fnal sequence. The algorthmc complexty of the proposed heurstc s as follows: Step 1 fnds a sum of n terms for each of m machnes wth complexty O(nm), and then gves sortng of n tems usng the bubble sort technque [] wth an average case complexty of O(n 2 ). The complexty of a schedule of n jobs on m machnes s O(nm). Thus, the total complexty of step 1 s O(2nm + n 2 ) or O(n 2 ). Step 2 actually dctates the complexty of the proposed method. In the frst part of step 2, the total number of comparsons (varyng the number of adjacent machnes from 2 to m) s equal to m(m 1)/2 and ts complexty s O(m 2 ). In the next part of step 2, (m 1) schedules of n jobs on m machnes are generated, thereby resultng n total complexty of O(m 2 + nm). Hence, the overall complexty of the proposed method s O(2nm + 2n + m 2 + nm) or O(n 2 + 3nm). III. BERTOLISSI HEURISTIC [5] The man dea of the Bertolss heurstc [5] s to obtan an ntal schedule as a startng pont whch s used for generatng the job seed and defnng the order n whch jobs wll be selected for the job nserton as descrbed by Rajendran and Chaudhur [4]. The prncple for the selecton of the seed job and the order of nserton of jobs n the avalable partal schedule s based on the work of Chan and Bedworth [18]. The heurstc conssts of the followng steps: Step 1: Compute the flow tmes for each par of jobs, k by usng the equaton (, ) 2 (,1) m F k = p + p (, j ) + R, where R j= 2 m( k ) m( k ) s recursvely computed as m Rm ( k ) = p( k, m) + max( Rm 1( k ), p(, r)), and r= 2 R1( k ) = p( k,1). Step 2: Compare each par of flow tmes (F(,k) and F(k,)) and select the smallest one, and mark the startng job of the par. Step 3: perform step 2 for all the pars of flow tmes. Step 4: Count the number of marks of each job and order the jobs n decreasng number of marks and use ths orderng as the ntal sequence of jobs. Step 5: Set k = 1. Select frst job from ths ntal sequence and nsert t n the frst poston of the partal sequence σ. Call t as current sequence σ. Step 6: Increment k, k = k+1. Select the k-th job from the sorted array of step 1 and nsert t n the r possble postons of the current sequence σ, where k/2 r k. Select the best one among r sequences wth the mnmum value of k ( k + 1 ) d ( σ, ) 2 1 σ and set t as current = sequence σ. Step 7: If k = n, go to step 8, else go to step 6. Step 8: The sequence σ s the fnal soluton of the heurstc. The algorthmc complexty of Bertolss heurstc s as follows: Step 1 nvolves the generaton of n(n 1)/2 schedules of 2 jobs. Also, the complexty of each flow tme calculaton for each schedule of a par of jobs on m machnes s O(2m). Thus, the complexty of step 1 s O(n(n 1)2m) or O(n 2 m). In step 2, the number of comparson between two flow tmes s n(n 1)/2 and ts complexty s O(n(n 1)/2) or O(n 2 ). Step 4 gves sortng n tems and usng bubble sort technque [], an average case complexty s O(n 2 ). Hence, the overall complexty of steps 1 4 of the Bertolss heurstc s O(n 2 m), whch s more than that of the proposed method wth the complexty of O(n 2 + 3nm). It has been shown, through the exhaustve computatonal expermentaton, that the Bertolss heurstc produces near optmal solutons, whch generally gves better results than

3 the ones provded by Rajendran and Chaudhur [4] heurstc and Bonney and Gundry [9] for both small and large szes. However, Bertolss heurstc requres more computatonal tmes compared to those gven by the exstng heurstcs. Based on ths evaluaton, the method proposed by Bertolss s consdered as the best exstng heurstc for the mnmzaton of total flow tme n no-wat flow shop schedulng. IV. COMPUTATIONAL EXPERIENCE The proposed heurstc, and the heurstc of Bertolss [5] were coded n C and run on an Intel Core 2 Duo, 2 GB RAM, 2.93 GHz PC. To compare the proposed heurstc wth the exstng heurstc, we carred out the expermentaton n two phases. In the frst phase, we consdered small szes wth number of jobs (n) = 5, 6, 7, 8, and 9 and number of machnes (m) = 5,,,, and. The second phase was formed takng the large szes wth n =,,,,,, and 70 and m = 5,,,, and. Thrty ndependent nstances were consdered for each sze. Each nstance corresponds to a new processng tme matrx where each processng tme was generated from a unform random dscrete u(1, 99) dstrbuton, commonly used by researchers [19], []. The followng two performance measures, popular n the schedulng lterature [4], [6], [7] are used n the present expermentaton: average relatve percentage devaton (ARPD), and percent of optmal solutons (for small szes) or percent of best heurstc solutons (for large szes). ARPD for small number of jobs s s gven by, 0 k Optmal ARPD = k = 1 Optmal ARPD for large number of jobs s s gven by, 0 k Best ARPD = k = 1 Best where, denotes the objectve functon value obtaned for -th nstance by a heurstc, Optmal s the optmal soluton value obtaned for that nstance, Best s the best soluton value obtaned for that nstance, and k s the number of nstances for a sze. Table I dsplays comparatve evaluaton of the proposed method, and the heurstc of Bertolss [5] based on ARPD, and the percent of optmal solutons for the small szes (n = 5, 6, 7, 8, and 9). The results of Table I show that the overall performance of the proposed heurstc wth respect to ARPD, and percent optmal soluton s comparable to that of Bertolss heurstc for small job sze s. The proposed method performs better than the Bertolss heurstc for cases wth respect to the ARPD, and cases wth respect to the percent tmes optmal soluton found out of each cases for small szes. Table II presents comparatve results based on ARPD, and the percent of best heurstc solutons for the large szes (n =,,,,,, and 70). Table II ndcates that the proposed method also performs sgnfcantly better than the Bertolss heurstc for large job szes. The proposed method performs better than the Bertolss heurstc for 26 cases wth respect to the ARPD, and 23 cases wth respect to TABLE I ARPD, AND PERCENT TIMES OPTIMAL SOLUTION OBTAINED FOR SMALL PROBLEM SIZES ARPD Percent optmal nstances Bertolss Bertolss Average TABLE II ARPD, AND PERCENT TIMES BEST SOLUTION OBTAINED FOR LARGE nstances PROBLEM SIZES ARPD Bertolss Bertolss Percent best Average

4 the percent tmes best soluton found out of each 35 cases for large szes. Next, we show the statstcal sgnfcance [21] of the results obtaned by the proposed method over those produced by the Bertolss heurstc. The number of nstances for each sze s taken as. We test the null hypothess, H 0 : µ = 0 aganst alternatve hypothess, H 1 : µ > 0;.e., f H 0 holds true, then, statstcally, the dfference between the two methods s not sgnfcant. At 5 % level of sgnfcance, the crtcal value, t 0.05,ν s obtaned from the relaton Probablty (t t 0.05,ν ) = α = Usng the standard tables of t-dstrbuton, we obtan, t 0.05,ν = for ν = N-1 = 29 degrees of freedom. We also compare the level of sgnfcance wth the p-value. The results are presented n Table III. nstances TABLE III RESULTS OF STATISTICAL TEST Bertolss heurstc versus proposed heurstc TFT dfference t p-value Mean Std. dev The average computatonal tmes (n seconds) requred for solvng each nstance by the heurstcs are gven n Table IV. The results show that the proposed method takes less computatonal tme than that of the Bertolss heurstc. Overall, the results of the proposed method are better by 38% n ARPD and 42% n percent tmes best found especally for large sze nstances whle takng about % less CPU tme than requred by the Bertolss heurstc. V. CONCLUSION In ths paper, we have presented a constructve heurstc for the no-wat flow shop schedulng wth the objectve of mnmzng total flow tme crteron. The method s based on the prncple of the sum of processng tmes of ndvdual jobs on the bottleneck machnes to determne the ntal sequence of jobs. Based on the computatonal expermentaton, the proposed method gves comparable performance as that of the Bertolss heurstc for small szes, whereas, there s sgnfcant mprovement n soluton qualty for large szes. Also, t has been shown that the CPU tme requred by the proposed method s less. TABLE IV MEAN CPU TIME (IN SECONDS) REQUIRED BY THE BERTOLISSI HEURISTIC AND THE PROPOSED HEURISTIC n m Bertolss Average REFERENCES [1] N. G. Hall and C. Srskandarajah, A survey of machne schedulng s wth blockng and no-wat n process, Opeatons Research, vol. 44, pp. 5-5, [2] C. H. Papadmtrou and P. C. Kanellaks, Flowshop schedulng wth lmted temporary storage, Journal of Assocate Computer Machnery, vol. 31, pp. 3-3, [3] H. R ock, The three-machne no-wat flow shop s NP-complete, Journal of Assocate Computer Machnery, vol. 31, pp , [4] C. Rajendran and D. Chaudhur, algorthms for contnuous flow-shop, Naval Research Logstc Quarterly, vol., pp , [5] E. Bertolss, algorthm for schedulng n the no-wat flowshop, Journal of Materals Processng Technology, vol. 7, pp , 00. [6] T. Aldowasan and A. Allahverd, New heurstcs for m-machne nowat flowshop to mnmze total completon tme, Omega, vol. 32, pp , 04. [7] J. M. Framnan, M. S. Nagano, and J. V. Moccelln, An effcent heurstc for total flowtme mnmzaton n no-wat flowshops, Internatonal Journal of Advanced Manufacturng Technology, vol. 46, pp. 49-,. [8] M. Nawaz, E. E. Jr. Enscore, and I. Ham, A heurstc algorthm for the m machne, n job flowshop sequencng, Omega, vol. 11, pp , 19. [9] M. C. Bonney and S. W. Gundry, Solutons to the constranted flowshop sequencng, Operatonal Research Quarterly, vol., pp , 1976.

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