Bi-criteria Scheduling on Parallel Machines Under Fuzzy Processing Time

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1 22d Iteratioal Cogress o Modellig ad Simulatio, Hobart, Tasmaia, Australia, 3 to 8 December 207 mssaz.org.au/modsim207 Bi-criteria Schedulig o Parallel Machies Uder Fuzzy Processig Time Sameer Sharma a, Seemab ad Meeakshi Sharma a b P.G. Departmet of Mathematics, D A V College, Jaladhar, Pujab, Idia Research Scholar, Departmet of Mathematics, Pajab Uiversity, Chadigarh, Idia samsharma3@gmail.com Abstract: Job schedulig is cocered with the optimal allocatio of scare resources with objective of optimisig oe or several criteria. Job schedulig has bee a fruitful area of research for may decades i which schedulig resolve both allocatio of machies ad order of processig. If the s are scheduled properly, ot oly the time is saved but also efficiecy of system is icreased. The parallel machie schedulig problem is widely studied optimisatio problem i which every machie has same work fuctio ad a ca be processed by ay of available machies. Optimisig dual performace measures o parallel machies i fuzzy eviromet is fairly a ope area of research. I real life situatios, the processig times of s are ot always exact due to icomplete kowledge or a ucertai eviromet which implies the existece of various exteral sources ad types of ucertaity. Fuzzy set theory ca be used to hadle ucertaity iheret i actual schedulig problems. This paper pertais to a bi-criteria schedulig o parallel machies i fuzzy eviromet which optimises the weighted flow time ad total tardiess simultaeously. The fuzziess, vagueess or ucertaity i processig time of s is represeted by triagular fuzzy membership fuctio. The objective of the paper is to fid the optimal sequece of s processig o parallel machies so as to miimize the secodary criterio of weighted flow time without violatig the primary criterio of total tardiess. A umerical illustratio is carried out to illustrate the executio of proposed algorithm. Keywords: Fuzzy processig time, total tardiess, weighted flow time, weighted, weighted shortest processig time 382

2 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time. INTRODUCTION Job schedulig has bee a fruitful area of research for may decades i which schedulig resolve both allocatio of machies ad order of processig. If the s are scheduled properly, ot oly the time is saved but also efficiecy of system is icreased. The parallel machie schedulig problem is widely studied optimisatio problem i which every machie has same work fuctio ad a ca be processed by ay of available machies. Optimisig dual performace measures o parallel machies i fuzzy eviromet is fairly a ope area of research. A survey of literature has revealed little work reported o the bicriteria schedulig problems o parallel machie i fuzzy eviromet. Che ad Bulfi (989) have examied sigle machie schedulig problems whe all s have idetical processig time. Cea ad Tabucao (99) dealt with bicriteria schedulig with parallel idetical machie miimizig the maximum tardiess. Che ad Bulfi (994) studied schedulig o a sigle machie to miimize the maximum tardiess (maximum latess) ad umber of tardy s (late s). Parkash (997) studied the bi-criteria schedulig problems o parallel machies. Fiala (997) formulated a parallel machie schedulig problem with two criteria to miimize the makespa ad to miimize the umber of preemptio. Sari ad Harihara (2000) cosidered the bicriteria problem of schedulig s o two machies to miimize the primary criterio of maximum tardiess ad secodary criterio of umber of tardy s. Sari ad Parkash (2004) cosider the problem of schedulig s o parallel idetical machies so as to miimize primary ad secodary criteria. Aghiolfi ad Paolucci (2007) studied total tardiess schedulig problems o parallel machies. Huo et al (2007) studied bicriteria schedulig problems ivolvig umber of tardy s ad maximum weighted tardiess. Gupta et al (202) developed a algorithm for the bi-objective problem which optimises the umber of tardy s without violatig the value of maximum tardiess with ucertai processig time. Sharma et al (203) studied the bi-objective problem with total tardiess ad umber of tardy s as primary ad secodary criteria respectively for ay umber of parallel machies i fuzzy eviromet. Sharma et al (203a) developed a algorithm to schedule s o parallel idetical machies so as to miimize the secodary criterio of weighted flow time without violatig the primary criterio of maximum tardiess i fuzzy eviromet. The rest of paper is orgaized as follows: I sectio 2 problems is formulated. Sectio 3 describes the role of fuzzy i schedulig. Sectio 4 deals with the theorems derived for optimisig the bicriteria problem. Sectio 5 defies the algorithm proposed to fid the optimal sequece for bicriteria problem Weighted flow time/total tardiess. I sectio 6, umerical illustratios are carried out to test the efficiecy of the proposed algorithm. The paper is cocluded i sectio 8 followed by the refereces. 2. PROBLEM FORMULATION The followig assumptios are made before proceedig with the mathematical formulatio i developig the algorithm for bicriteria problem o parallel machies. Jobs are available at time zero Jobs are idepedet of each other No pre-emptio is allowed Machies are idetical i all respects ad are available all the time No machie ca hadle more tha oe at a time. The followig otatios will be used all the way through the preset paper. i : i th, i =, 2, 3,, : umber of s to be scheduled di : due date of the i th m : umber of machies ci : completio time of i th ta : completio time for a wi : weight of i th k : machie o which i th is assiged Ti : tardiess of the i th at the j th positio WFT : weighted flow time of s Tmax : maximum tardiess Xijk :; if i is located at positio j o j : locatio of i th o machie k, where k th machie ad 0; otherwise j =, 2, 3,,. Before formulatig the bicriteria problem, the mathematical formulatio for the sigle criterio is represeted first as give by Parkash (997). These are as discussed i sectio 2.: 2. Criterio: Total Tardiess Tardiess is give by T i = max (0, c i d i ), where c i ad d i are the completio time ad due date of i, respectively. The formulatio is as follows: 383

3 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time mi Z = j= k= i = i = Subject to costraits: X m ijk ijk i = i --- () j, k --- (2) X is biary i, j, k --- (3) ijk X i i i T T c d i alog with o-egativity costraits. --- (4) 2.2 Criterio: Weighted Flow time The formulatio to miimize the weighted flow time (WFT) is as follows: m m! mi Z = wi. X ijk j= k= Subject to: costraits set (), (2) ad (3) respectively, alog with o-egativity costraits. The formulatio of the bicriteria problems is similar to that of sigle criteria problems but with some additioal costraits requirig that the optimal value of the primary objective fuctio is ot violated. The two parts of the bicriteria problem formulatio are as follows: Primary objective fuctio Subject to: Primary problem costrait Secodary objective fuctio Subject to: a. secodary problem costrait b. primary objective fuctio value costrait c. primary problem costrait I first step, the primary costrait t a k e a s total tardiess of s is miimized ad i the secod step, the secodary costrait take as weighted flow time of s is miimized uder the objective fuctio value of primary costrait. 3. ROLE OF FUZZY Fuzzy set theory has bee used to model systems that are hard to defie precisely. Zadeh (965) stated that most of the early iterest i fuzzy set theory pertaied to represetig ucertaity i huma cogitive system. Ucertaity ca be thought of i a epistemological sese as beig the iverse of iformatio. Iformatio about a particular problem may be icomplete, imprecise, fragmetary, ureliable, vague or deficiet i some other way. Fuzzy set theory is ow applied to problems i egieerig, busiess, medical ad related health scieces ad i atural scieces. A large umber of determiistic schedulig algorithms have bee proposed i last decades to deal with schedulig problems with various objectives ad costraits. I real life situatios, decisios to be made are ofte costraied by specific requiremets. The decisio makig process gets icreasigly more complicated with icremets i the umber of costraits. The real world is complex; complexity i the world geerally arises from ucertaity. From this prospective the cocept of fuzzy eviromet is itroduced i the field of schedulig. For example, the processig times of s may be ucertai due to icomplete kowledge or ucertai eviromet which implies that there exist various exteral sources ad types of ucertaity. Fuzzy sets ad fuzzy logic ca be used to tackle ucertaity iheret i actual schedulig problems. Here, we use triagular fuzzy membership fuctio to represets the ucertaity ivolved i processig of s. Further, the system characteristics are described by membership fuctio; it preserves the fuzziess of iput iformatio. However, the desiger would prefer oe crisp value for oe of the system characteristics rather tha fuzzy set. I order to overcome this problem we defuzzify the fuzzy values of system characteristic by usig the Yager s (98) approximatio formula. For a triagular fuzzy umber = ( a a a ) ~ 3a2 + a3 a crisp(a) = Average High Rakig of A = h( A) =. 3 ~ A,, ;

4 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time 4. THEOREMS The followig theorems have bee established to optimise the bicriteria schedulig o parallel machies ivolvig maximum tardiess ad weighted flow time 4.. Theorem: A sequece S of s followig early due date (EDD) order is a optimal sequece with maximum tardiess (Tmax)..i.e. whe s are processed o ay of available parallel machies by early due date rule, the correspodig sequece of processig is optimal with respect to maximum tardiess as give by Sharma et al (203a) Theorem: A sequece S of s followig weighted smallest processig time (WSPT) rule, i which the s are placed to the earliest available locatio o the machies i WSPT order, miimizes the weighted flow time. Proof: Let, if possible sequece S obtaied by usig the WSPT rule (i.e. by arragig the s i decreasig order of their weights; break the ties (if ay) arbitrary) is ot optimal. Let there exist a better sequece of s S (say) i which adjacet s i ad j are iterchaged. Let Ci(S) ad Cj(S) be the completio times of s i ad j i schedule S respectively. Similarly, let Ci (S ) ad C j (S) be the completio times of s i ad j i schedule S. Therefore, we ca have: For sequece S: we have: Ci (S) = ta +, C j (S) = ta + 2 For sequece S : we have: Ci (S ) = ta + 2, C j (S ) = ta + Next, the WTF cotributio of these s for the sequece S is: W (S) = wi Ci (S) + w j C j (S) = wi (ta +) + w j (ta + 2) = wi ta + wi + w j ta + 2w j = (w i + w j )ta + wi + 2w j ---- (5) Similarly, the WTF cotributio of these s for the sequece S is W (S ) = wi Ci (S ) + w j C j (S )= wi (ta + 2) + w j (ta +) = wi ta + 2wi + w j ta + w j ---- (6) = (wi + w j )ta + 2wi + w j. Sice, the s i ad j are placed by WSPT rule. Therefore, we have wi wj. Hece, from results (5) ad (6), we have: W (S ) W (S).Therefore, WTF for the sequece S is more as compared to the sequece S. Hece, the sequece S followig the Weighted Shortest Processig Time (WSPT) rule miimizes the Weighted Flow Time (WTF) Theorem: A set of s iitially arraged by Early Due Date order the a late eed to be cosidered for beig exchaged oly with aother late or a havig the same due date improves the value of secodary criteria of weighted flow time give the primary criterio of miimum total tardiess. Proof: Let us pick ay two s i ad j from the EDD schedule. Let j be the late. The followig cases may arise: Case I: Job i is late ad di < dj I this case, we have either c i = c j or c i < c j. If c i = c j, the the switchig of these s will ot improve the solutio. If c i < c j, the the tardiess T ad T before ad after the exchage are Ti = max(0, ci di) + cj dj, Ti = max(0, ci dj) + cj di I case if c i > d j, the switchig i ad j will worse the primary criterio. I case if c i < d j, the switchig i ad j does ot chage the total tardiess ad weighted flow time. The oly case i which the primary criterio is ot violated ad weighted flow time improves is, if c i = d j. Case II: If i is ot late ad di < dj I this case, the total tardiess before ad after switchig i ad j ist = c d, T = c d. Here, we j j i j havet < T. Hece, the primary criterio of total tardiess is violated. Case III: If i is ot late di > dj I this case, the total tardiess before ad after switchig i ad j ist = c d, T = c d. Here, we havet < T. Hece, the primary criterio of total tardiess is agai violated. Case IV: If i is late ad di > dj I this case, we get the similar result as we get i case I, discussed above. j j i j Hece, we have show that a switchig amog ay two s will worse the EDD schedule except that made uder the exchage coditio c i = d j as stated i the algorithm. Hece, a set of s iitially arraged i EDD order, a late eeds to be cosidered for beig exchage oly with aother late or a havig the same due date to potetially improve the value of a secodary criterio, give the primary criterio of miimizig total tardiess. 385

5 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time 5. ALGORITHM The followig algorithm is proposed to fid the optimal sequece for bi-criteria problem WFT/Total Tardiess: Step : Fid the crisp values of the fuzzy processig time of various s o differet machies usig Yager (98) approximatio formula. Step 2: Arrage all the s o the available parallel machies i a early due date (EDD) order. If there is a tie the use Weightage Shortest Processig Time (WSPT) to break the tie. Step 3: Let C be a set of s that caot be switched ad L be a set of all late s. Iitially, C = { }. Step 4: Cosider first late i C. If oe exit the go to the step 6; else go step 5. Step 5: Check all the late s after i. If there exist j L, j i such that c i d j ad w i < w j, So we exchage the i with j which has maximum weight amogst all s satisfyig these coditios; update L, if ecessary else set C = C + {i} ad go to Step 4. Step 6: Cosider the first o late i C. If oe exist the exit else go to step 7. Step 7: Check all the early s after i. If there exist a o late j after i for which c i d j ad c j d i ad w i < w j ; exchage the i with j which has maximum weight amogst all s satisfyig the above coditios else set C = C + {i} ad go to Step NUMERICAL ILLUSTRATION The followig umerical illustratios are carried out to test the efficiecy of algorithm proposed to optimise the bicriteria WFT/Total Tardiess o parallel machies i fuzzy eviromet. 6.. Cosider a example of s with processig time i fuzzy eviromet, due date ad s Weight give i table to be scheduled o three parallel machies i a flowshop. The objective is to obtai a Sequece of the s processig optimisig the bicriteria take as WTF/Total Tardiess. Table. Jobs with fuzzy processig time Jobs Processig Time Due Date Weight (w i ) (8,9,0) 20/3 4 2 (5,6,7) 29/3 6 3 (9,0,) 32/3 3 4 (7,8,9) 26/3 5 5 (5,6,7) 25/3 6 (0,,2) 35/3 2 Solutio: The crisp values for fuzzy processig time of give s is as follow Table 2. Jobs with crisp values for processig time Jobs Processig Time Due Date Weight(w i ) 29/3 20/ /3 29/ /3 32/ /3 26/ /3 25/3 6 35/3 35/3 2 O arragig the s i EDD order o the parallel available machies M, M 2 ad M 3, We get Table 3. Jobs schedulig followig EDD order Jobs M M 2 M 3 w i d i T i 0 29/3 4 20/3 9/ /3 25/ /3 5 26/3-2 20/3 40/3 6 29/3 /3 3 26/3-58/3 3 32/3 26/3 6 29/3 64/3 2 35/3 29/ Therefore, Total Tardiess = = uits ad weighted flow time 3 3 wc i i Weighted Flow Time = = w = 5.34 uits. i 386

6 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time Therefore, set of late s = L= ad set of s that caot be switched C = O cosiderig st late L ad C there is a late j = 2 L, j i after i such that c i d j ad w i < w j. Therefore o iterchagig the i with j, the schedule becomes Table 4. Reduced Job schedulig table Jobs M M 2 M 3 w i d i T i /3 6 29/ /3 25/ /3 5 26/3-20/3 49/3 4 20/3 29/3 3 20/3 52/3 3 32/3 20/3 6 26/3-6/3 2 35/3 26/3 Therefore, Total Tardiess = 75/3 uits ad weighted flow time wc i i WFT = = = w i =.8 uits. Therefore, set of late s = L= ad set of s that caot be switched C = O cosiderig st late i = L ad C, there is o late after i satisfyig c i d j ad w i < w j, therefore C = C + =. Now, o cosiderig the ext late i = 3 L ad 3 C, there is o late after i satisfyig c i d j ad w i < w j, therefore C = C + =. Now, o cosiderig the ext late 6 L ad 6 C, there is o late after i. So set C= C + =. Now, there is o late i C, so we pick the first o late i =2 C. Next, we observe that there is o early j after i i the schedule for which c i d j ad c j d i ad w i < w j ; so set C = C + = Now, o cosider the ext o late i =5 C, there is o early j after i i the schedule for which c i d j ad c j d i ad w i < w j, so set C = C + = Similarly, o cosider the ext o late i =4, there is o early j after i i the schedule. Therefore C = C + = Hece, the optimal sequece of s processig is with miimum WFT as.8 uits ad Total tardiess as 75/3 uits. 7. DISCUSSION 7.. The proposed algorithm optimises the bi-criteria problem of miimizig the weighted flow time for a miimum value of total tardiess. Proof: The result ca be verified by cosiderig the followig two cases: Case : Whe s i ad j are late s,.i.e. i ad j L This case correspods to step 5 of the algorithm.we kow that if s are iitially arraged i early due date order, a late eed to be cosidered for beig exchaged oly with aother late i order to potetially improve the value of secodary criterio of weighted flow time, give the primary criterio of miimum total tardiess. If the coditios c i d j ad w i < w j for i, j L, the j appearig after i i the schedule violate the primary criterio of miimum total tardiess. Hece, s i ad j must be exchaged i order to optimise the secodary criterio of weighted flow time for a give miimum value of primary criterio of total tardiess. Case 2: Whe s are early s This case correspods to step 7 of the algorithm. Sice, early s remai early s eve whe they are exchaged. Therefore, if there exist a o late j after a early i for which c i d j ad c j d i ad w i < w j the iterchage the i with j which has maximum weight amogst all s satisfyig the above coditios otherwise, the secodary criterio of weighted flow time will remai optimised with miimum total tardiess If the problem of sigle criteria, Total Tardiess, is NP-hard, the schedulig problem o parallel machies optimisig the bi-objective fuctio WFT / Total Tardiess will also be NP-hard. Solutio: We shall prove the result by the method of cotradictio: Let if possible the bi-objective fuctio WFT / Total Tardiess is ot NP-hard. Therefore, there must exists a polyomial algorithm which ca solve the problem of optimisig the bi-objective fuctio WFT / Total Tardiess o parallel processig machies. This implies that sigle criteria of Total Tardiess ca be optimised i polyomial time;.i.e. Total Tardiess is ot NP-hard. This is a cotradictio as Total Tardiess is NP-hard. 387

7 Sharma et al., Bi-criteria Schedulig o Parallel Machies uder Fuzzy Processig Time Hece, the schedulig problem optimisig the bi-objective fuctio WFT / Total Tardiess o parallel processig machies will also be NP-hard. 8. CONCLUSION I this paper a heuristic algorithm to optimise the bicriteria schedulig problem ivolvig total tardiess ad weighted flow time o parallel machies is developed. I real life situatio the processig time of s may vary due to lack of complete kowledge, ucertaity ad vagueess. The cocept of fuzzy processig time is itroduced i processig of s to hadle uder these ucertaities. The optimal sequece of s processig for the problem disused above is with miimum WFT as.8 uits ad Total tardiess as 75/3 uits. The study may further be exteded by usig trapezoidal fuzzy umbers, by geeralizig the umber of machies, by itroducig the cocepts of o availability costraits ad machies processig the s with differet speeds. REFERENCES Aghiolfi, D. ad Paolucci, M. (2007). Parallel machie total tardiess schedulig problem, Computers ad Operatio Research, 34() Cea, A.A. ad Tabucao, M.T. (99). Schedulig problem i a shop with parallel processor, Iteratioal Joural of Productio Ecoomics, 25(-3) Che, C.L. ad Bulfi, R.L. (989). Schedulig uit processig time s o a sigle machie with multiple criteria, Computers ad Operatios Research, 7-7. Che, C.L. ad Bulfi, R.L. (994). Schedulig a sigle machie to miimize two criteria: maximum tardiess ad umber of tardy s, IIE Trasactio, Falia, P. (997). Heuristic solvig a bicriteria parallel machie schedulig problem, Kyberetika, Gupta, D., Sharma, S. ad Aggarwal, S. (202). Bi-objective schedulig o parallel machies with ucertai processig time, Advaces i Applied Sciece Research, 3(2) Huo, Y., Leug, J.Y.T. ad Zhao, H. (2007). Bicriteria schedulig problems: umber of tardy s ad maximum weighted tardiess, Europea Joural of Operatioal Research, 77() Parkash, D. (997). Bi-criteria Schedulig problems o parallel machies Ph.D. Thesis, Uiversity of Birekshurg, Virgiia. Sari, S.C. ad Harihara, R. (2000). A two machie bicriteria schedulig problem, Iteratioal Joural of Productio Ecoomics, 65(2) Sari, S.C. ad Parkash, D. (2004). Equal processig time bicriteria schedulig o parallel machies, Joural of Combiatorial Optimisatio, 8, Sharma, S., Gupta, D. ad Seema (203). Bicriteria schedulig o parallel machies: total tardiess ad weighted flowtime i fuzzy eviromet, Iteratioal Joural of Mathematics i Operatioal Research, 5(4) Sharma, S., Gupta, D. ad Seema (203). Bi-Objective schedulig o parallel machies i fuzzy eviromet, Advaces i Itelliget System ad Computig, Yager, R.R. (98). A procedure for orderig fuzzy subsets of the uit iterval, Iformatio Scieces, 24, Zadeh, L.A. (965). Fuzzy Sets, Iformatio ad Cotrol, 8,

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ. 2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For