Research Article Single-Machine Group Scheduling Problems with Deterioration to Minimize the Sum of Completion Times

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1 Hidawi Publishig Corporatio Mathematical Problems i Egieerig Volume 2012, Article ID , 9 pages doi:101155/2012/ Research Article Sigle-Machie Group Schedulig Problems with Deterioratio to Miimize the Sum of Completio Times Yog He 1 ad Li Su 1, 2 1 School of Ecoomics ad Maagemet, Southeast Uiversity, Najig , Chia 2 Foudatio Departmet, Huaiyi Advaced Vocatioal ad Techical Health School, Huaia , Chia Correspodece should be addressed to Yog He, heyog@126com Received 14 August 2012; Accepted 11 November 2012 Academic Editor: Gradimir Milovaovic Copyright q 2012 Y He ad L Su This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited We cosider two sigle-machie group schedulig problems with deterioratig group setup ad job processig times That is, the job processig times ad group setup times are liearly icreasig or decreasig fuctios of their startig times Jobs i each group have the same deterioratig rate The objective of schedulig problems is to miimize the sum of completio times We show that the sum of completio times miimizatio problems remais polyomially solvable uder the agreeable coditios 1 Itroductio I classical schedulig problems, the schedulig models routiely assume that job processig times are kow ad fixed throughout the period of job processig However, this assumptio may be urealistic i may situatios that the processig times of jobs may be prologed due to deterioratio or shorteed due to learig over time Schedulig with deterioratig jobs was first idepedetly itroduced by J N D Gupta ad S K Gupta 1 ad Browe ad Yechiali 2 Sice the, related models of schedulig with deterioratig jobs have bee extesively studied from a variety of perspectives Cheg et al 3 give a detailed review of schedulig problems with deterioratig jobs More recet papers which have cosidered schedulig jobs with deterioratig jobs iclude Wag 4, Voutsias ad Pappis 5, Lee et al 6, Wagetal 7, Zhu et al 8, Cheg et al 9, 10, Cheg et al 11, Su et al 12, Wei et al 13, Zhao ad Tag 14, adyietal 15, 16 O the other had, the productio efficiecy ca be icreased by groupig various parts ad products with similar desigs ad/or productio processes This pheomeo

2 2 Mathematical Problems i Egieerig is kow as group techology i the literature To the best of our kowledge, oly a few results cocerig schedulig problems with deterioratig effect uder group techology are give Wei ad Wag 17 cosider sigle-machie schedulig problems with group techology GT ad deterioratig jobs Yag 18 ivestigate group schedulig problems with simultaeous cosideratios of learig ad deterioratio effects o a sigle-machie settig The learig pheomeo is implemeted to model the setup time of groups Three models of deterioratig for the job processig time withi a group are examied Bai et al 19 cosider sigle-machie group schedulig problems with effects of learig ad deterioratio at the same time Lee ad Lu 20 discuss a sigle-machie schedulig problem of miimizig the total weighted umber of late jobs with deterioratig jobs ad setup times A brach-ad-boud with several domiace properties ad a lower boud is developed to solve the problem optimally S J Yag ad D L Yag 21 study two models of learig for the job processig time, they provide polyomial time solutios for the makespa miimizatio problems Huag et al 22 preset a sigle-machie group schedulig problems with both learig effects ad deterioratig jobs Wu ad Lee 23 cosider a situatio where both setup times ad job-processig times are legtheed as jobs wait to be processed Two sigle-machie group-schedulig problems are ivestigated where the group setup times ad the job-processig times are both icreasig fuctios of their startig times I this paper, we study the group schedulig model proposed by Lee ad Wu 24, who have cosidered the makespa problems I our study, we ivestigate the sum of completio times miimizatio problem uder the same model as Lee ad Wu 24 I additio, we suppose jobs i each group have the same deterioratig rate Hece, the problems cosidered will be oted by 1/GT p ij a ij ± b i t, s i θ i ± δ i t/ C ij Weshow that the problems remai polyomially solvable uder the agreeable coditios The remaider of this paper is orgaized i four sectios The solutio procedure for the liear icreasig fuctio problem is give i the ext sectio, ad the solutio procedure for the liear decreasig fuctio is described i Sectio 3 The, some examples are offered i Sectio 4 The coclusio is give i the last sectio 2 The 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij Problem I this sectio, we cosider the sigle-machie group schedulig problem with deterioratio to miimize the sum of completio times of all jobs I our model, we follow the otatio ad termiology used by Lee ad Wu 24 I additio, a i j is the basic ormal processig time of job J i j scheduled i the jth positio of group G i,adc i j is the completio time scheduled i the jth positio of group G i We suppose the jobs i each group have the same deterioratio rate oted by b i Theorem 21 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, the optimal schedule is obtaied by sequecig the jobs i each group i odecreasig order of a i j, that is, a i 1 a i 2 a i 3 a i 4 a i i, i 1, 2, 3,m, (the SPT rule Proof I the same group, the result ca be easily obtaied by usig iterchagig techology ad is omitted Theorem 22 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, the groups are arraged i odecreasig order of ω G i, ω G i b i 1 b i i 1 δ i 1 / 1 b i 1 b i i 1 1 δ i,

3 Mathematical Problems i Egieerig 3 if ω G i ad B i /A i have agreeable coditios, that is, ω G i ω G j if ad oly if B i /A i B j /A j, A i 1 b i 1 1 b i i 1 δ i / 1 1 b i, B i i 1 b i i k a i k 1 b i i θ i Proof Let π 1 ad π 2 be two job schedules where the differece betwee π 1 ad π 2 is a pairwise iterchage of two adjacet groups G i ad G j Thatis,π 1 S 1,G i,g j,s 2, π 2 S 1,G j,g i,s 2, where S 1 ad S 2 are partial sequeces ad S 1 ad S 2 may be empty Furthermore, we assume that t deotes the completio time of the last job i π 1 Toshowπ 1 domiates π 2,itsuffices to show that i j i C i l π 1 C j l π 1 C j l π 2 C i l π 2 21 By defiitio, the completio times of jobs i π 1 are give by C i 1 π 1 a i 1 1 b i θ i 1 b i 1 δ i t, i C i i π 1 1 b i i k a i k 1 b i i θ i 1 b i i 1 δ i t, C j 1 π 1 a j 1 ( 1 b j θj ( 1 b j ( 1 δj Ci i π 1, 22 ( j k C j j π 1 1 bj aj k ( j 1 b j θ j ( j ( 1 b j 1 δj Ci i π 1 Ad the completio times of jobs i π 2 are give by C j 1 π 2 a j 1 ( 1 b j θj ( 1 b j ( 1 δj t, ( j k C j j π 2 1 bj aj k ( j 1 b j θ j ( j ( 1 b j 1 δj t, C i 1 π 2 a i 1 1 b i θ i 1 b i 1 δ i C j j π 2, 23 i C i i π 2 1 b i i k a i k 1 b i i θ i 1 b i i 1 δ i C j j π 2

4 4 Mathematical Problems i Egieerig Therefore, we have i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 ( 1 bj ( 1 ( 1 bj j ( 1 δj t 1 ( 1 b j 1 b i ( 1 1 bi i 1 δ i C j j π b i 24 1 b i ( 1 1 b i i 1 δ i t 1 1 b i ( 1 bj ( 1 ( 1 bj j ( 1 δj Ci i π 1 1 ( 1 b j By substitutig A i 1 b i ( 1 1 b i ( ( ( i j ( 1 δ i 1 bj 1 1 bj 1 δj, A j 1 1 b i 1 (, 1 b j i B i 1 b i i k a i k 1 b i i θ i, ( j k B j 1 bj aj k ( j 1 b j θ j, 25 we ca obtai i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 ( A i (( 1 bj j ( 1 δj 1 Aj ( 1 bi i 1 δ i 1 t A i B j A j B i 26 To have i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 0, 27 whe t 0, if ad oly if A i (( 1 bj j ( 1 δj 1 Aj ( 1 bi i 1 δ i 1 0, A i B j A j B i From the result of 28, we have 1 b i i 1 δ i 1 /A i 1 b j j 1 δ j 1 /A j,thatis to say ω G i ω G j Ad from the case of 29, we ca obtai B i /A i B j /A j Therefore, if ω G i ad B i /A i have agreeable coditios, the optimal sequece betwee groups are arraged i odecreasig order of ω G i, This completes the proof From Theorems 21 ad 22,ifω G i ad B i /A i have agreeable coditio, the problem 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij ca be solved by the followig algorithm

5 Mathematical Problems i Egieerig 5 Algorithm 23 Step 1 Jobs i each group are scheduled i odecreasig order of the basic processig time a i j,thatis,a i 1 a i 2 a i 3 a i 4 a i i, i 1, 2, 3,m, the SPT rule Step 2 Calculate ω G i b i 1 b i i 1 δ i 1 / 1 b i 1 b i i 1 1 δ i Step 3 Groups are scheduled i odecreasig order of ω G i,thatis,ω G 1 ω G 2 ω G 3 ω G m Obviously, It is easy to show that the total time for Algorithm 23 is O log 3 The 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij Problem I order to discuss the problem 1/GT, p ij suppose 0 b i 1ad0 θ i 1 a ij b i t, s i θ i δ i t/ C ij coveietly, we Theorem 31 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, the optimal schedule is obtaied by sequecig the jobs i each group i odecreasig order of basic processig time a i j, that is, a i 1 a i 2 a i 3 a i 4 a i i, i 1, 2, 3,m, (the SPT rule Proof I the same group, the result ca be easily obtaied by usig iterchagig techology ad is omitted Theorem 32 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, the groups are arraged i odecreasig order of ω G i, ω G i b i 1 b i i 1 δ i 1 / 1 b i 1 1 b i i 1 δ i, if ω G i ad B i /A i have agreeable coditios, that is, B i /A i B j /A j if ad oly if ω G i ω G j, A i 1 b i 1 1 b i i 1 δ i / 1 1 b i, ad B i i 1 b i i k a i k 1 b i i θ i Proof Here, we still use the same otatios as that i Theorem 22Toshowπ 1 domiates π 2, it suffices to show that i j i C i l π 1 C j l π 1 C j l π 2 C i l π 2 31 By defiitio, the completio times of jobs i π 1 are give by C i 1 π 1 a i 1 1 b i θ i 1 b i 1 δ i t, i C i i π 1 1 b i i k a i k 1 b i i θ i 1 b i i 1 δ i t, C j 1 π 1 a j 1 ( 1 b j θj ( 1 b j ( 1 δj Ci i π 1, 32 ( j k C j j π 1 1 bj aj k ( j 1 b j θ j ( j ( 1 b j 1 δj Ci i π 1

6 6 Mathematical Problems i Egieerig Ad the completio times of jobs i π 2 are give by C j 1 π 2 a j 1 ( 1 b j θj ( 1 b j ( 1 δj t, ( j k C j j π 2 1 bj aj k ( j 1 b j θ j ( j ( 1 b j 1 δj t, C i 1 π 2 a i 1 1 b i θ i 1 b i 1 δ i C j j π 2, 33 Therefore, we have i C i i π 2 1 b i i k a i k 1 b i i θ i 1 b i i 1 δ i C j j π 2 i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 ( 1 bj ( 1 ( 1 bj j ( 1 δj t 1 ( 1 b j 1 b i ( 1 1 bi i 1 δ i C j j π b i 34 1 b i ( 1 1 b i i 1 δ i t 1 1 b i ( 1 bj ( 1 ( 1 bj j ( 1 δj Ci i π 1 1 ( 1 b j By substitutig A i 1 b i ( 1 1 b i ( ( ( i j ( 1 δ i 1 bj 1 1 bj 1 δj, A j 1 1 b i 1 (, 1 b j i B i 1 b i i k a i k 1 b i i θ i, ( j k B j 1 bj aj k ( j 1 b j θ j, 35 we ca obtai i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 ( A i (( 1 bj j ( 1 δj 1 Aj ( 1 bi i 1 δ i 1 t A i B j A j B i 36 To have i i C j l π 2 C i l π 2 C i l π 1 C j l π 1 0, 37

7 Mathematical Problems i Egieerig 7 whe t 0 if ad oly if A i (( 1 bj j ( 1 δj 1 Aj ( 1 bi i 1 δ i 1 0, A i B j A j B i From the result of 38, we have 1 b i i 1 δ i 1 A i ( 1 bj j ( 1 δj 1 A j, 310 that is to say ω G i ω G j Ad from the case of 39, we ca obtai B i /A i B j /A j Therefore, if ω G i ad B i /A i have agreeable coditios, the optimal sequece betwee groups are arraged i odecreasig order of ω G i, This completes the proof From Theorems 31 ad 32,ifω G i ad B i /A i have agreeable coditio, the problem 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij ca be solved by the followig algorithm Algorithm 33 Step 1 Jobs i each group are scheduled i odecreasig order of the basic processig time a i j,thatis,a i 1 a i 2 a i 3 a i 4 a i i, i 1, 2, 3,m, the SPT rule Step 2 Calculate ω G i b i 1 b i i 1 δ i 1 / 1 b i 1 1 b i i 1 δ i Step 3 Groups are scheduled i odecreasig order of ω G i,thatis,ω G 1 ω G 2 ω G 3 ω G m Obviously, It is easy to show that the total time for Algorithm 33 is O log 4 Examples Example 41 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, we cosider six jobs are divided ito three groups The basic job ad group processig times, the job ad group deterioratio rates for each group are give, respectively, as follows: p 11 5, p 12 8, p 21 4, p 22 10, p 23 6, p 31 7, b 1 004, b 2 002, b 3 001, θ 1 5, θ 2 4, θ 3 6, 41 δ 1 001, δ 2 005, δ Solutio 1 Accordig to Algorithm 23, we solve the example as follows Step 1 I group G 1, the optimal job sequece is J 11 J 12 I group G 2, the optimal job sequece is J 21 J 23 J 22 I group G 3, the optimal job sequece is J 31 Step 2 calculate: B 1 /A , B 2 /A , B 3 /A , ad ω G , ω G , ω G It is easy to say B 2 /A 2 <B 1 /A 1 <B 3 /A 3 ad ω G 2 <ω G 1 <ω G 3

8 8 Mathematical Problems i Egieerig Therefore, the optimal group sequece is G 2 G 1 G 3 ad the optimal schedule is J 21 J 23 J 22 J 11 J 12 J 31 The completio times of jobs are C , C , C , C , C , ad C , ad the sum of completio times is Cij Example 42 For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, we cosider six jobs are divided ito three groups as Example 41, the job ad group deterioratio rates for each group are give, respectively, as follows: p 11 9, p 12 7, p 13 5, p 21 10, p 31 8, p 32 6, b 1 003, b 2 005, b 3 002, θ 1 4, θ 2 8, θ 3 5, 42 δ 1 015, δ 2 001, δ 3 01 Solutio 2 Accordig to Algorithm 23, we solve the example as follows Step 1 I group G 1, the optimal job sequece is J 13 J 12 J 11 I group G 2, the optimal job sequece is J 21 I group G 3, the optimal job sequece is J 32 J 31 Step 2 calculate: B 1 /A , B 2 /A , B 3 /A , ad ω G , ω G , ω G It is easy to say B 1 /A 1 <B 3 /A 3 <B 2 /A 2 ad ω G 1 < ω G 3 <ω G 2 Therefore, the optimal group sequece is G 1 G 3 G 2 ad the optimal schedule is J 13 J 12 J 11 J 32 J 31 J 21 The completio times of jobs are C , C , C , C , C , C , ad the sum of completio times is C ij Coclusios I this paper, we ivestigate the sigle-machie group schedulig problems with deterioratio which is studied by Lee ad Wu 24 We cosider the sum of completio times miimizatio problem, ad we show the problem to miimize total completio time of all jobs is polyomially solvable uder agreeable coditios i our study Ackowledgmets The authors thak the valuable commets of the referees for a earlier versio of this paper Their commets have sigificatly improved the paper This work is supported by the Natioal Natural Sciece Foudatio of Chia o Also, this research is partly supported by the Program for New Cetury Excellet Talets i Uiversity o NCET ad the Miistry of Educatio of Chia: Grat-i-aid for Humaity ad Social Sciece Research o 11YJCZH139

9 Mathematical Problems i Egieerig 9 Refereces 1 J N D Gupta ad S K Gupta, Sigle facility schedulig with oliear processig times, Computers ad Idustrial Egieerig, vol 14, o 4, pp , S Browe ad U Yechiali, Schedulig deterioratig jobs o a sigle processor, Operatios Research, vol 38, o 3, pp , T C E Cheg, Q Dig, ad B M T Li, A cocise survey of schedulig with time-depedet processig times, Europea Joural of Operatioal Research, vol 152, o 1, pp 1 13, J-B Wag, Sigle machie schedulig with decreasig liear deterioratio uder precedece costraits, Computers & Mathematics with Applicatios, vol 58, o 1, pp , T G Voutsias ad C P Pappis, A brach ad boud algorithm for sigle machie schedulig with deterioratig values of jobs, Mathematical ad Computer Modellig, vol 52, o 1-2, pp 55 61, W-C Lee, P-J Lai, ad C-C Wu, Some sigle-machie ad flowshop schedulig problems with a o-liear deterioratio fuctio, Computers & Mathematics with Applicatios, vol 62, o 6, pp , J-B Wag, L-H Su, ad L-Y Su, Sigle-machie total completio time schedulig with a timedepedet deterioratio, Applied Mathematical Modellig, vol 35, o 3, pp , Z Zhu, L Su, F Chu, ad M Liu, Due-widow assigmet ad schedulig with multiple rate-modifyig activities uder the effects of deterioratio ad learig, Mathematical Problems i Egieerig, vol 2011, Article ID , 19 pages, T C E Cheg, W-C Lee, ad C-C Wu, Sigle-machie schedulig with deterioratig jobs ad past-sequece-depedet setup times, Applied Mathematical Modellig Simulatio ad Computatio for Egieerig ad Evirometal Systems, vol 35, o 4, pp , T C E Cheg, S J Yag, ad D L Yag, Commo due-widow assigmet ad schedulig of liear time-depedet deterioratig jobs ad a deterioratig maiteace activity, Iteratioal Joural of Productio Ecoomics, vol 135, pp , W Cheg, P Guo, Z Zhag, M Zeg, ad J Liag, Variable eighborhood search for parallel machies schedulig problem with step deterioratig jobs, Mathematical Problems i Egieerig, vol 2012, Article ID , 20 pages, L-H Su, L-Y Su, M-Z Wag, ad J-B Wag, Flow shop makespa miimizatio schedulig with deterioratig jobs uder domiatig machies, Iteratioal Joural of Productio Ecoomics, vol 138, o 1, pp , C-M Wei, J-B Wag, ad P Ji, Sigle-machie schedulig with time-ad-resource-depedet processig times, Applied Mathematical Modellig, vol 36, o 2, pp , C Zhao ad H Tag, Two-machie owshop schedulig with deterioratig jobs ad chai precedece costraits, Iteratioal Joural of Productio Ecoomics, pp , Y Yi, S-R Cheg, ad C-C Wu, Schedulig problems with two agets ad a liear o-icreasig deterioratio to miimize earliess pealties, Iformatio Scieces, vol 189, pp , Y Yi, D Xu, S-R Cheg, ad C-C Wu, A geeralisatio model of learig ad deterioratig effects o a sigle-machie schedulig with past-sequece-depedet setup times, Iteratioal Joural of Computer Itegrated Maufacturig, vol 25, o 9, pp , C-M Wei ad J-B Wag, Sigle machie quadratic pealty fuctio schedulig with deterioratig jobs ad group techology, Applied Mathematical Modellig, vol 34, o 11, pp , S-J Yag, Group schedulig problems with simultaeous cosideratios of learig ad deterioratio effects o a sigle-machie, Applied Mathematical Modellig, vol 35, o 8, pp , J Bai, Z-R Li, ad X Huag, Sigle-machie group schedulig with geeral deterioratio ad learig effects, Applied Mathematical Modellig, vol 36, o 3, pp , W-C Lee ad Z-S Lu, Group schedulig with deterioratig jobs to miimize the total weighted umber of late jobs, Applied Mathematics ad Computatio, vol 218, o 17, pp , S J Yag ad D L Yag, Sigle-machie group schedulig problems uder the effects of deterioratio ad learig, Computers ad Idustrial Egieerig, vol 58, o 4, pp , X Huag, M Z Wag, ad J B Wag, Sigle-machie group schedulig with both learig effects ad deterioratig jobs, Computers ad Idustrial Egieerig, vol 60, o 4, pp , C C Wu ad W C Lee, Sigle-machie group-schedulig problems with deterioratig setup times ad job-processig times, Iteratioal Joural of Productio Ecoomics, vol 115, o 1, pp , W C Lee ad C C Wu, A ote o optimal policies for two group schedulig problems with deterioratig setup ad processig times, Computers ad Idustrial Egieerig, vol 58, o 4, pp , 2010

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