A Single-Machine Deteriorating Job Scheduling Problem of Minimizing the Makespan with Release Times

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1 A Sngle-Machne Deteroratng Job Schedulng Problem of Mnmzng the Makespan wth Release Tmes Wen-Chung Lee, Chn-Cha Wu, and Yu-Hsang Chung Abstract In ths paper, we study a sngle-machne deteroratng ob schedulng problem wth ob release tmes where the obectve s to mnmze the makespan. The problem s known to be strongly NP-complete. Therefore, we frst develop a branch-and-bound algorthm ncorporatng wth several domnance propertes and a lower bound to derve the optmal soluton. Then, we propose two heurstc algorthms for the near-optmal solutons. Fnally, we conduct some computatonal experments to evaluate the performance of the proposed algorthms. Keywords: sngle machne; deteroratng obs; makespan; release tme 1. Introducton For many years, ob processng tmes are assumed to be known and fxed from the frst ob to be processed untl the last ob to be completed. However, there are many stuatons n whch a ob that s processed later consumes more tme than the same ob when processed earler. Examples can be found n the fre fghtng, emergency surgery, machne mantenance, and cleanng assgnment where any delay n processng a ob s penalzed and often mples addtonal tme for accomplshng the ob. Schedulng n ths settng s known as deteroratng ob schedulng problems and has receved ncreasng attenton n recent years. Gupta and Gupta [1] and Browne and Yechal [2] were the poneers that ntroduced the deteroratng ob schedulng problems. They constructed models where the processng tme of a ob s a functon of ts startng tme. Snce then, many researchers have devoted to ths vvd area. For nstance, Mosheov [3] consdered the smple lnear deteroraton model. He showed that the problems of mnmzng the makespan, total flow tme, sum of weghted completon tmes, total lateness, number of tardy obs, maxmum lateness, and maxmum tardness are polynomally solvable. Cheng and Dng [4] examned the sngle machne makespan schedulng problems nvolvng startng tme Manuscrpt receved November 20, Ths paper s support by NSC under grant number NSC E Wen-Chung Lee s wth Feng Cha Unversty, Tachung, Tawan, ROC. (phone: x4016; e-mal: wclee@fcu.edu.tw). Chn-Cha Wu s wth Feng Cha Unversty, Tachung, Tawan, ROC (e-mal: cchwu@fcu.edu.tw). Yu- Hsang Chung s wth Feng Cha Unversty, Tachung, Tawan, ROC. dependent tasks wth release tmes and lnearly ncreasng/decreasng processng rates. They solved a seres of models n the lterature, and further gave a sharp boundary delneatng the complexty of the problems. Bachman and Janak [5] proved that the maxmum lateness problem s NP-hard f the actual processng tme s a lnear functon of ts startng tme. Under the same model, Bachman et al. [6] proved that mnmzng the total weghted completon tme s NP-hard. Under the decreasng lnear deteroraton model, Wang and Xa [7] provded optmal solutons for sngle-machne problems of mnmzng the makespan, maxmum lateness, maxmum cost and number of tardy obs. Wu et al. [8] studed the total weghted completon tme problem on a sngle machne. In addton, Janak and Kovalyov [9] studed an mportant problem of schedulng obs executed by human resources n a contamnated area. The specfcty of ths problem s that the dynamcs of the harmful factor should be taken nto account as well as the norms of organsm recovery n rest perods. They also constructed two polynomal tme algorthms for the both cases-- wth and wthout precedence constrants. Lee et al. [10] studed the two-machne makespan problem whle Wu and Lee [11] consdered the two-machne total completon tme problem. On the other hand, there are some stuatons n whch the ob wll requre extra tme for successful accomplshment f certan mantenance procedures fal to be completed pror to a pre-specfed deadlne. Kubak and van de Velde [12] consdered a sngle machne problem of schedulng ndependent obs to mnmze the makespan, and they showed the problem s NP-complete n the ordnary sense. Cheng and Dng [13] assumed that the actual processng tme p s a step functon of ts startng tme and showed the makespan problem s NP-complete n the ordnary sense, and the total completon tme problem s NP-complete. They further dentfed a varety of solvable cases for some related problems. Researchers have formulated the deteroraton phenomenon nto dfferent models and solved dfferent problems for varous crtera. Aldaee and Womer [14] surveyed the rapdly growng lterature, whle Cheng et al. [15] gave a detaled revew of schedulng problems wth deteroratng obs. However, most of the works assume that the obs are avalable at all tmes. To the best of our knowledge, Cheng and Dng [4] were the only authors that consdered deteroratng ob schedulng problems wth release tmes. They consdered a famly of schedulng problems for a set of startng tme dependent tasks wth release tmes and lnearly ncreasng or decreasng processng rates on a sngle

2 machne to mnmze the makespan. In partcular, they showed that the makespan problem wth dentcal basc processng tme, arbtrary release tmes and ncreasng processng rates s strongly NP-complete. However, no reports on developng the exact soluton or provdng the approxmately solutons are found. In ths paper, we wll nvestgate the model where each ob has a common basc processng tme a, a deteroraton rate b, and a release tme r. We wll develop a branch-and-bound and two heurstc algorthms to obtan the optmal and near-optmal solutons, respectvely. The remander of ths paper s organzed as follows. The problem formulaton s gven n the next secton. The elmnaton rules and a lower bound for the problem are presented n Secton 3. In Secton 4 we develop two heurstc algorthms to solve the proposed problem. In Secton 5, we provde the results of the computatonal experment. The concluson s gven n the last secton. 2. Problem formulaton The problem descrpton s gven as follows. Let N ={ 1,..., n } be the set of obs to be scheduled. Each ob has a basc processng tme a, a deteroraton rate b, and a release tme r. The actual processng tme of J s p = a+ bt, where t s the startng tme for J. Moreover, let C ( S ) denote the completon tme for J under schedule S. The obectve of ths paper s to fnd an optmal schedule S such that max{c 1 ( S ), C 2 ( S ),, C n ( S )} max{c 1 (S), C 2 (S),, C n (S)} for any schedule S. 3. Elmnaton rules and lower bound Cheng and Dng [4] showed that the problem under consderaton s strongly NP-complete. Therefore, the branch-and-bound method s a good way to derve the optmal soluton. To facltate the search process, some elmnaton rules are needed. Suppose that S and S are two ob schedules where the dfference between S and S s a parwse nterchange of two adacent obs and. That s, S = ( π π ) and S = ( π π ), where π and π denote the partal sequences. Furthermore, let A denote the completon tme of the last ob n π. To show that S domnates S, t suffces to show that C ( S) C ( S ). Property 1. If r A and (1 + b) A+ a r, then S domnates S. Proof: From A r, t mples that C( S) = (1 + b) A+ a. Snce (1 + b ) A+ a r, we have C ( S) = (1 + b ) r + a. From r Thus, A and (1 + b) A+ a r, t mples that r r. C( S ) = (1 + b) r + a, and C ( S ) = (1 + b )(1 + b ) r + (1 + b ) a+ a. Snce b > 0, we have C ( S) < C ( S ). Therefore, S domnates S. The proofs of Propertes 2 to 6 are omtted snce they are smlar to that of Property 1. Property 2. If r A r, (1 + b) A+ a r, and b < b, then S domnates S. Property 3. If A max{ r, r} and b > b, then S domnates S. Property 4. If r A r, (1 + b ) A+ a r, and ab ( b) < (1 + b)(1 + b )( A r), then S domnates S. Property 5. If A r r, b > b, and (1 + b) r + a > r, then S domnates S. Property 6. If A r and (1 + b ) r + a r, then S domnates S. To further facltate the searchng process, a theorem s used n the branch-and-bound algorthm to determne the c order of unscheduled obs. Let ( π, π ) be a sequence of obs where π s the scheduled part conssted of k obs and c π s the unscheduled part. In addton, let ( π, π ) be a sequence n whch the unscheduled obs n π are arranged n non-ncreasng order of deteroraton rates b. For smplcty, we use the symbol [ ] to sgnfy the order of obs n a sequence. Theorem 1. If C [ k ] > max N{ r }, then sequence ( π, π ) c domnates sequences of any other types of ( π, π ). 3.1 Lower bound In ths subsecton, we establsh a lower bound to curtal the branchng tree. Suppose that PS s a partal schedule n whch the order of the frst k obs s determned and S s a complete schedule obtaned from PS. By defnton, the completon tme for the (k+1)th ob s C = (1 + b ) max{ C, r } + a [ k+ 1] [ k+ 1] [ k] [ k+ 1] (1 + b[ k+ 1] ) C[ k] + a. Smlarly, the completon tme for the (k+r)th poston ob s C = (1 + b ) max{ C, r } + a [ k+ r] [ k+ r] [ k+ r 1] [ k+ r] r r 1 r, (1) (1 + b ) C + (1 + b ) a+ a [ k+ ] [ k] [ k+ ] = 1 = 1 = r + 1 where k+ r = n. In equaton (1), the frst term s constant, and the second term s an ncreasng functon of. Thus, t s mnmzed by arrangng the deteroraton rates n a non-ncreasng order for the unscheduled obs. Therefore, the lower bound can be obtaned as n the followng way. r r 1 LB = (1 + b ) C + (1 + b ) a + a [ k+ ] [ k] ( k+ ) = 1 = 1 = 1, (2)

3 where b( k+ 1) b( k+ 2) b( k+ r) rates n the unscheduled part. K denotes the deteroraton 4. The heurstc algorthms An alternatve approach to solve an NP-hard problem s to provde the heurstc algorthm. In ths secton, two heurstc algorthms are presented. The man dea of the frst algorthm s to choose the ob wth the smallest release tme. However, the schedule mght be affected by the deteroraton rates, whch n turn yelds a larger value of makespan. To overcome ths problem, the second heurstc algorthm s proposed. Instead of only choosng the ob wth the smallest release tme among all the unscheduled obs, we choose the smallest value of release tme and deteroraton rates from the obs already released. The steps of the proposed algorthms are descrbed as follows. Heurstc Algorthm 1 ( HA 1 ) Step 1: Let N={ 1,..., n } and k=1. Step 2: Choose ob wth the smallest release tme from N and place the ob on the kth poston. Compute the completon tme for the kth ob, say C [ k]. Delete ob from N. Step 3: If C[ ] > max { r }, then go Step 5, otherwse, k N go to Step 4. Step 4: Let k=k+1. If k n, go to Step 2. Step 5: Arrange the unscheduled obs n a non-ncreasng order of the deteroraton rates and go to Step 6. Step 6: Output the fnal soluton. Heurstc Algorthm 2 ( HA 2 ) Step 1: Let N={ 1,..., n }, C [0] = 0, and k=1. Step 2: Choose ob whch has the smallest value of (1 + b ) max{ C [ 1], r } a k + n N and place t on the kth poston. Compute the completon tme for the kth ob, say C [ k]. Delete ob from N. Step 3: Let k=k+1. If k n, then go to Step 2. Otherwse, go to Step 4. Step 4: Output the fnal soluton. To further refne the qualty of the proposed heurstcs, some neghborhood search movements are necessary. In ths paper, we use the parwse nterchange, extracton and backward-shfted, extracton and forward-shfted movement n our neghborhood search snce they mprove the qualty of the soluton sgnfcantly. 5. Computatonal experment A computatonal experment s conducted to evaluate the performance of the branch-and-bound and the heurstc algorthms. The programs are coded n Fortran 90 and run on a Pentum 4 personal computer. The basc processng tmes take the values of 5 and 10, and the release tmes are generated from a unform dstrbuton between 0 to 50.5nλ, where n s the ob number and λ s a factor of the range of the release tmes. The computatonal experment conssts of two parts. In the frst part, the branch-and-bound and the heurstc algorthms are conducted wth three dfferent ob numbers (n= 20, 30 and 40) assocated wth ten dfferent ranges of λ ( λ =0.2, 0.4, 0.6, 0.8, 1.0, 1.25, 1.5, 1.75, 2.0, and 3.0). The deteroraton rates are generated from a unform dstrbuton between 0 and These tests are desgned to study the effects of the domnance rules and the lower bound, and the accuracy of the heurstc algorthms. The average and maxmum number of nodes and the average and maxmum executon tme (n seconds) are reported for the branch-and-bound algorthms. For the heurstc algorthms, the average and maxmum error percentages are recorded. The error percentage of the soluton produced by the heurstc algorthm s calculated as ( V V ) / V 100%, where V s the value of the makespan generated from the heurstc method and V s the optmal value of the makespan obtaned from the branch-and-bound method. In addton, the performance of the heurstc BH = mn{ HA1, HA2} s also reported. As a consequence, 60 expermental condtons are examned, and 100 replcatons are randomly generated for each condton. The results are summarzed n Table 1. It s observed that the number of nodes and the executon tme grow exponentally as the ob sze ncreases. The most tme-consumng case takes about 20 mnutes and the maxmal number of nodes s 6 over when n= 40. As λ ncreases, the executon tme and the number of nodes ncrease for a fxed number of obs. In addton, the problems are easer to solve as the values of the basc processng tmes ncrease. Ths s due to the fact that t s easer to surpass the maxmal release tme, and Theorem 1 s more powerful n that case. Both heurstc algorthms perform well n terms of the average error percentages. Overall, the performance of HA 1 heurstc s better than that of HA 2, especally for small values of λ. Moreover, the average error percentages of both heurstcs tend to decrease as λ ncreases when the ob number s fxed. In addton, the performance of BH heurstc s much superor to those of HA 1 and HA 2, whch means there s no clear domnance relaton between the performances of HA 1 and HA 2. Thus, BH s recommended for small ob-szed problems.

4 Table 1. The Performance of Branch-and-Bound and Heurstc Algorthms for b ~ U (0,0.25). Error Percentage Number of Nodes CPU Tmes HA 1 HA 2 BH n a λ mean max mean max mean max mean max mean max

5 Table 2. The Performance of Branch-and-Bound and Heurstc Algorthms for b ~ U (0,0.50) Error Percentage Number of Nodes CPU Tmes HA 1 HA 2 BH n a λ mean max mean max mean max mean max mean max

6 In order to study the performance of HA 1 and HA 2 algorthms for large ob-szed problems, three dfferent ob numbers (n= 60, 80, and 100) combned wth ten dfferent values of λ are tested n the second part of the experment. The deteroraton rates are generated from a unform dstrbuton between 0 and 0.5. As a consequence, 60 expermental condtons are examned, and 100 replcatons are randomly generated for each condton. The results are summarzed n Table 2. The same phenomenon s also observed from Table 2. Both heurstc algorthms perform well n terms of the average error percentages. Overall, the performance of HA 1 heurstc s better than that of HA 2, especally for small values of λ. Moreover, the average error percentages of both heurstcs tend to decrease as λ ncreases when the ob number s fxed. 6. Conclusons Ths paper consders a sngle-machne makespan problem where each ob has a basc processng tme, a deteroraton rate, and a release tme. Ths problem s known to be NP-complete. Thus, a branch-and-bound algorthm wth several domnance propertes and a lower bound s proposed to derve the optmal soluton. In addton, two effectve heurstc algorthms are also provded. The computatonal results show that the branch-and-bound algorthm performs well n terms of the number of nodes and the executon tme. Moreover, the computatonal experments also show the proposed heurstc algorthms perform well. The extenson of arbtrary basc processng tmes mght be an nterestng ssue for future research. Acknowledgements. The authors are grateful to the edtor and the referees, whose constructve comments have led to a substantal mprovement n the presentaton of the paper. [8] C.C. Wu, W.C. Lee, and Y.R. Shau, Mnmzng the total weghted completon tme on a sngle machne under lnear deteroraton. Internatonal Journal of Advanced Manufacturng Technology, 33, 2007, pp [9] A. Janak, and M.Y. Kovalyov, Schedulng n a contamnated area: A model and polynomal algorthms. European Journal of Operatonal Research, 173, 2006, pp [10] W.C. Lee, C.C. Wu, C.C. Wen, and Y.H. Chung A two-machne flowshop makespan schedulng problem wth deteroratng obs. Computers & Industral Engneerng, (DOI /.ce ) [11] C.C. Wu and W.C. Lee Two-machne flowshop schedulng to mnmze mean flow tme under lnear deteroraton. Internatonal Journal of Producton Economcs, 103, 2006, pp [12] W. Kubak, and S. van de Velde, Schedulng deteroratng obs to mnmze makespan. Naval Research Logstcs, 45, 1998, pp [13] T.C.E. Cheng, and Q. Dng, Sngle-machne schedulng wth step-deteroratng processng tmes. European Journal of Operatonal Research, 134, 2001, pp [14] B. Aldaee, and N.K. Womer, Schedulng wth tme dependent processng tmes: Revew and extensons. Journal of the Operatonal Research Socety, 50, 1999, pp [15] T.C.E. Cheng, Q. Dng, and B.M.T. Ln, A concse survey of schedulng wth tme-dependent processng tmes. European Journal of Operatonal Research, 152, 2004, pp References [1] J.N.D. Gupta, and S.K. Gupta, Sngle faclty schedulng wth nonlnear processng tmes. Computers Industral Engneerng, 14, 1988, pp [2] S. Browne, and U. Yechal, "Schedulng deteroratng obs on a sngle processor. Operatons Research, 38, 1990, pp [3] G. Mosheov, Schedulng obs under smple lnear deteroraton. Computers and Operatons Research, 21(6), 1994, pp [4] T.C.E. Cheng, and Q. Dng, The complexty of schedulng startng tme dependent tasks wth release tmes. Informaton Processng Letters, 65(2), 1988, pp [5] A. Bachman, and A. Janak, Mnmzng maxmum lateness under lnear deteroraton. European Journal of Operatonal Research, 126, 2000, pp [6] A. Bachman, T.C.E. Cheng, A. Janak, and C.T. Ng, Schedulng start tme dependent obs to mnmze the total weghted completon tme. Journal of the Operatonal Research Socety, 53(6), 2002, pp [7] J.B. Wang, and Z.Q. Xa, Schedulng obs under decreasng lnear deteroraton. Informaton Processng Letters, 94, 2005, pp