Specially structred flow shop scheduling with fuzzy processing time to minimimze the rental cost

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1 vailable olie at dvaces i pplied Sciece Research, 0, 3 (4): ISSN: CODEN (US): SRFC Specially structred flow shop schedulig with fuzzy processig time to miimimze the retal cost Deepak Gupta, Sameer Sharma* ad Shefali ggarwal Departmet of Mathematics, M. M. Uiversity, Mullaa, mbala, Haryaa, Idia Departmet of Mathematics, D..V. College, Jaladhar City, Pujab, Idia BSTRCT Schedulig is a edurig process where the existece of real time iformatio frequetly forces the review ad modificatio of pre established schedules. 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 theory of schedulig. This paper pertai to a specially structured -jobs, -machies flow shop schedulig i which processig times are described by triagular fuzzy umbers. Further the average high rakigs of fuzzy processig time are ot radom but bear a well defied relatioship to oe aother. The preset work is a attempt to develop a ew heuristic algorithm, a alterative to the traditioal algorithm as proposed by Johso s (954) to fid the optimal sequece to miimize the utilizatio time of machies ad hece, their retal cost uder specified retal policy. Keywords: Specially structured flow shop, processig time, fuzzy schedule, average high rakig, retal policy. INTRODUCTION I a real world, fial deadlies deped upo types of productio priority of jobs / customers. For example, exports are to be completed rigidly before shippig. But i some cases slight delay is allowed. I the literature dealig with a flowshop schedulig problems, processig times ad relevat data are usually assumed to be kow exactly. Yet this is seldom the case i most situatios. s i case of real life decisio makig situatios, there are may vaguely formulated relatios ad imprecisely quatified data values i real world descriptio sice precise details are simply ot available i advace. s a result, the decisio makig is much easier i providig approximate duratio ad to specify most ad least possible values tha to give exact ad precise values. s the fuzzy approach seems much more atural, we ivestigate its potetial i solvig the flow shop problem i real-life situatios. Moreover, the fuzzy approach seems a atural extesio of its crisp couterpart so that we eed to kow how the fuzziess of processig times affects the job sequece itself. flow shop schedulig problems has bee oe of the classical problems i productio schedulig sice Johso [8] proposed the well kow Johso s rule i the two ad three stage flow shop schedulig problem. MacCaho ad Lee [9] discussed the job sequecig with fuzzy processig time. Ishibuchi ad Lee [6] addressed the formulatio of fuzzy flowshop schedulig problem with fuzzy processig time. Hog ad Chuag [5] developed a ew triagular Johso algorithm. Mari ad Roberto [0] developed fuzzy schedulig with applicatio to real time systems. Some of the oteworthy approaches are due to Yager [5], McCaho [9], Shukla ad Che [], Yao ad Li [7], Sigh ad Gupta [], Sauja ad Sog [3], Sigh, Suita ad llawalia [4]. Gupta, D., Sharma, S. ad Shashi [4] studied specially structured two stage flow shop schedulig to miimize the retal cost. I the preset paper we have itroduced the cocept of fuzzy processig time for a specially structured two stage flowshop schedulig i which processig times are described by triagular fuzzy umbers. The proposed algorithm is more efficiet ad less time cosumig as compared to the algorithm proposed by Johso s [8] to miimize the utilizatio time of machies ad hece their retal cost for specially structured flow shop schedulig. 986

2 Sameer Sharma et al dv. ppl. Sci. Res., 0, 3(4): Practical Situatio Fuzzy set theory is applicable to problems i egieerig, busiess, medical ad related health scieces, ad the atural scieces. Various practical situatios occur i real life whe oe has got the assigmets but does ot have oe s ow machie or does ot have eough moey or does ot wat to take risk of ivestig huge amout of moey to purchase machie. Uder such circumstaces, the machie has to be take o ret i order to complete the assigmets. I his startig career, we fid a medical practitioer does ot buy expesive machies say X-ray machie, the Ultra Soud Machie, Rotatig Triple Head Sigle Positro Emissio Computed Tomography Scaer, Patiet Moitorig Equipmet, ad Laboratory Equipmet etc., but istead takes o ret. Retal of medical equipmet is a affordable ad quick solutio for hospitals, ursig homes, physicias, which are presetly costraied by the availability of limited fuds due to the recet global ecoomic recessio. Retig eables savig workig capital, gives optio for havig the equipmet, ad allows upgradatio to ew techology. Fuzzy Membership Fuctio ll iformatio cotaied i a fuzzy set described by its membership fuctio. The triagular membership fuctios are used to represet fuzzy processig times i our algorithm. Figure shows the triagular membership fuctio of a fuzzy set ~ P, ~ P =(a, b, c). The membership value reaches the highest poit at b, while a ad c deote the lower µ + boud ad upper boud of the set P ~ respectively. The membership value of the x deoted by x, x R calculated accordig to the followig formula. 0 ; x a x a ; a x b b a µ ( x ) = c x ; b < x < c c b o ; x c, ca be µ x P a b c Figure x.. verage High Rakig <.H.R.> To fid the optimal sequece, the processig times of the jobs are calculated by usig Yager s (98) average high 3b + c a rakig formula (HR) = h( ) =. 3.. Fuzzy rithmetic Operatios If = ( m, α, β ) ad = ( m, α, β ) be the two triagular fuzzy umbers, the + = ( m, α, β ) + ( m, α, β ) = ( m + m, α + α, β + β ) = ( m, α, β ) ( m, α, β ) = ( m m, α α, β β ) if the followig coditio is ~ ~ satisfied DP DP, where ~ β m ~ β m DP( ) = ad DP( ) =. Here DP deotes differece poit of a Triagular fuzzy umber. k = k( m, α, β ) = ( km, kα, kβ ) ; if k>0. k = k( m, α, β ) = ( kβ, kα, km ) ; if k<0. 987

3 Sameer Sharma et al dv. ppl. Sci. Res., 0, 3(4): Notatios S : Sequece of jobs,, 3,., S k : Sequece obtaied by applyig Johso s procedure, k =,, 3, M j : Machie j, j=, M : Miimum makespa a ij : Fuzzy processig time of i th job o machie M j i=,,3,...,; j=, ij : HR of processig time of i th job o machie M j t ij (S k ) : Completio time of i th job of sequece S k o machie M j I ij (S k ) : Idle time of machie M j for job i i the sequece S k U j (S k ) : Utilizatio time for which machie M j is required R(S k ) : Total retal cost for the sequece S k of all machie C i : Retal cost of i th machie. CT(S i ) : Total completio time of the jobs for sequece S i.3. Defiitio Completio time of i th job o machie M j is deoted by t ij ad is defied as: t ij = max (t i-,j, t i,j- ) + i,j, where i,j =HR of processig time of i th job o j th machie..4. Retal Policy(P) The machies will be take o ret as ad whe they are required ad are retured as ad whe they are o loger required. i.e. the first machie will be take o ret i the startig of the processig the jobs, d machie will be take o ret at time whe st job is completed o the st machie. Problem Formulatio Let some job i (i =,, 3,..., ) is to be processed o two machies M ad M i the order M M such that o passig is allowed. Let a ij be the processig time of i th job o j th machie i fuzzy eviromet. Let ij ; i=,,3,...; j=, be the average high rakig (HR) of the processig times o two machies M & M such that either i i or i i for all values of i. Our aim is to fid the sequece {S k } of the jobs which miimize the retal cost of the machies. Mathematically, the problem is stated as: Miimum R( Sk ) = i C + U j ( Sk ) C i= Subject to costrait: Retal Policy (P) Our objective is to miimize retal cost of machies while miimizig the utilizatio time. Theorem 6.. Theorem: If i i for all i, j, i j, the k, k.k is a mootoically decreasig sequece, where K =. i i i= i= Solutio: Let i j for all i, j, i j i.e., max i mi j for all i, j, i j Let K = i i i= i= Therefore, we have k = lso k = + = + ( ) (Q ).. k k Now, k 3 = = + + ( 3 ) = k + ( 3 ) k (Q 3 ) Therefore, k 3 k k or k k k 3. Cotiuig i this way, we ca have K k k 3. k, a mootoically decreasig sequece. 988

4 Sameer Sharma et al dv. ppl. Sci. Res., 0, 3(4): Corollary: The total retal cost of machies is same for all the sequeces. Proof: The total elapsed time T ( S) = i + k = i + i= i= = Costat Therefore total elapsed time ad hece total retal cost of machies is same for all the sequeces. 6.. Theorem: If i j for all i, j, i j, the k, k. k is a mootoically icreasig sequece, where K = Proof: Let. i i i= i= K = i i i= i= Let i j for all i, j, i j i.e., mi i max j for all i, j, i j Here k = k = + = + ( ) k (Q j ) Therefore, k k. lso, k 3 = = + + ( 3 ) = k + ( 3 ) k (Q 3 ) Hece, k 3 k k. Cotiuig i this way, we ca have k k k 3. k, a mootoically icreasig sequece. Corollary: The total retal cost of machies is same for all the possible sequeces. Proof: The total elapsed time T ( S) = + k = + = + = + i= i= i= i= i= i= i= i= i i i i i i i i.= Costat It implies that uder retal policy P the utilizatio time of machie M is same. Therefore total retal cost of machies is same for all the sequeces. lgorithm The followig algorithm is proposed to miimize the retal cost for a specially structured flow shop schedulig, the processig times are uder fuzzy eviromet ad represeted by triagular fuzzy umber. Step : Fid the average high rakig (HR) ij ;i=,,3,...,; j=, of the processig times for all the jobs o two machies M & M. Step : Obtai the job J (say) havig maximum processig time o st machie. Step 3: Obtai the job J (say) havig miimum processig time o d machie. Step 4: If J J the put J o the first positio ad J as the last positio & go to step 7, Otherwise go to step 5. Step 5: Take the differece of processig time of job J o M from job J (say) havig ext maximum processig time o M. Call this differece as G.lso, Take the differece of processig time of job J o M from job J - (say) havig ext miimum processig time o M. Call the differece as G. Step 6: If G G put J o the last positio ad J o the first positio otherwise put J o st positio ad J - o the last positio. Step 7: rrage the remaiig (-) jobs betwee st job & last job i ay order, thereby we get the sequeces S, S S r. Step 8: Compute the total completio time CT(S k ) k=, r. Step 9: Calculate utilizatio time U of d machie U = CT(S k ) (S k ); k=,,. r. Step 0: Fid retal cost R( Sk ) = i ( Sk ) C + U C, where C & C are the retal cost per uit time of st & d machie respectively. i= Numerical Illustratio Cosider 6 jobs ad machie problem to miimize the retal cost i which the processig times are represeted by triagular fuzzy umbers. The retal costs per uit time for machies M ad M are 6 uits ad 5 uits respectively. The objective is to obtai a optimal sequece of job schedulig with miimum retal cost. 989

5 Sameer Sharma et al dv. ppl. Sci. Res., 0, 3(4): Table : The machies with fuzzy processig time Jobs Machie M Machie M i a i a i (7,8,9) (6,7,8) (,3,4) (5,6,7) 3 (8,0,) (4,5,6) 4 (0,,) (5,6,7) 5 (9,0,) (5,6,8) 6 (8,0,) (3,4,5) Solutio The HR of the processig time of the job is as follows: Table : verage High Rakig of Processig time Jobs Machie M Machie M i i i 6/3 3/3 4/3 0/3 3 34/3 7/3 4 35/3 0/3 5 3/3 /3 6 34/3 4/3 Here each i i for all i. lso, Max i = 4/3 which is for job.i.e. J =. Mi i = 4/3 which is for job 6.i.e. J = 6.i.e. J J, therefore J = will be o st positio ad J =6 will be o the last positio. Therefore, the optimal sequeces are: S = , S = , S 3 = , The total elapsed time is same for all these possible 4 sequeces S, S, S 3... S 4. The I Out table for ay of these 4 sequeces say S = is Table 3: The I Out table for the optimal sequece Jobs Machie M Machie M i I - Out I - Out (0,0,0)--(,3,4) (,3,4)--(7,9,) (,3,4)--(9,,3) (9,,3)--(5,8,3) 3 (9,,3)--(7,3,35) (7,35,3)--(3,36,4) 4 (7,3,35)--(37,4,47) (37,4,47)--(4,48,54) 5 (37,4,47)--(46,5,58) (46,5,58)--(5,58,66) 6 (46,5,58)--(54,6,70) (54,6,70)--(57,66,75) The total elapsed time, CT(S ) = (57,66,75) Utilizatio time for M, U (S ) = (57,66,75) (7,9,) = (40, 47, 54) Therefore, total retal cost for each of sequece R(S k ) = 6(54, 6, 70) + 5(40, 47, 54) = (54, 607, 690) uits. The.H.R. or retal cost = uits. Remarks If we solve the above problem by Johso s rule [8], we get the optimal sequece as S = The I-Out flow table for the sequece S is Table 4: The I Out table flow table The total elapsed time, CT(S ) = (57,66,75) Jobs Machie M Machie M i I - Out I - Out (0,0,0)--(7,8,9) (7,8,9)--(3,5,7) 5 (7,8,9)--(6,8,0) (6,8,0)--(,4,8) 4 (6,8,0)--(6,9,3) (6,9,3)--(3,35,39) (6,9,3)--(38,4,46) (38,4,46)--(43,48,58) 3 (38,4,46)--(46,5,58) (46,5,58)--(50,57,64) 6 (46,5,58)--(54,6,70) (54,6,70)--(57,66,75) 990

6 Sameer Sharma et al dv. ppl. Sci. Res., 0, 3(4): Utilizatio time for M, U (S ) = (57,66,75) (7,8,9) = (50, 58, 64) Therefore, total retal cost for each of sequece R(S k ) = 6(54, 6, 70) + 5(50, 58, 64) = (574, 65, 750) uits. The.H.R. or retal cost = uits which is much more as compared to the retal cost of the machies by proposed algorithm although the total elapsed time remais same. CONCLUSION The algorithm proposed i this paper for specially structured two stage flowshop schedulig problem is less time cosumig ad more efficiet as compared to the algorithm proposed by Johso s (954) to fid a optimal sequece miimizig the utilizatio time of the machies ad hece their retal cost. Due to our retal policy, the utilizatio time of secod machie is always miimum ad hece, thereby retal cost will also be miimum. REFERENCES [] Barker, K. R., (974). Itroductio to sequecig ad schedulig. New York: Joh Willy & Sos Ic. [] Cythia, S. ad MacCaho, (99). Europea joural of Operatioal Research, 6(3), [3] Cowlig, P. L. ad Johaso, M.,(00). Europea Joural of Operatioal Research, 39(), [4] Gupta, D., Sharma, S. ad Bala, S., (0). Iteratioal Joural of Emergig treds i Egieerig ad Developmet, (), [5] Hog, T. ad Chuag, T., (999). Computer ad Idustrial egieerig, 36(), [6] Ishibuchi, H. ad Lee, K. H.,(996). Proceedig of IEEE iteratioal coferece o Fuzzy system, [7] Jig-Shig Yao ad Freg Tsc Li,(00). Iteratioal Joural of ppropriate Reasoig, 9(3), [8] Johso, S. M., (954). Naval Research Logistics Quarterly, (), [9] McCaho, S. ad Lee, E. S.,(990). Computer ad mathematics with applicatios, 9(7), 3-4. [0] Marti, L. ad Roberto, T., (00). Fuzzy sets ad Systems, (3), [] Shukla, C. S. ad Che, F. F.,(996). Joural of itelliget Maufacturig, 4, [] Sigh, T. P. ad Gupta, D, (005). Miimizig retal cost i two stage flow shop, the processig time associated with probabilies icludig job block. Reflectios de ER, (), [3] Sauja, P. ad Xueya, S.,(006). Optimizatio & Egieerig, 7(3), [4] Sigh, T. P., Suita ad llawalia, P., (009). rya Bhatta Joural of mathematics & Iformatics, (-), [5] Yager, R. R., (98). Iformatio Scieces, 4,