IJESRT. Scientific Journal Impact Factor: (ISRA), Impact Factor: 2.114

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1 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. IJESRT ITERTIO JOR OF EGIEERIG SCIECES & RESERCH TECHOOGY BRCH D BOD TECHIQE FOR SIGE CHIE SCHEDIG PROBE SIG TYPE- TRPEZOID FZZY BERS R. Helen* R.Sumthi * PG nd Reserch Deprtment of themtics Poompuhr College (utonomous) eliyur Indi P.G. Deprtment of themtics R.V.S.College of Engg. nd Technology Krikl BSTRCT This pper dels rnch nd ound technique to solve single mchine scheduling prolem involving two processing times long with due dte using Type- Trpezoidl fuzzy numers. Our im is to otined optiml sequence of jos nd to minimize the totl trdiness. The working of the lgorithm hs een illustrted y numericl exmple. S Suject Clssifiction: 9D05 90-XX ug 00. KEYWORDS: Brnch nd Bound Technique Processing times P nd P Optiml sequence Type- Trpezoidl fuzzy numers. ITRODCTIO Jo scheduling is useful tool in decision mking prolem. The scheduling prolems re common occurrence in our dily life. The im of this technique is used to determine n optiml jo scheduling prolem nd minimizing the totl trdiness. For mny yers Scheduling prolem focused on single performnce mesure. In this pper we propose new concept in single mchine scheduling prolem. Recent development of new technology we re consider the single mchine hving doule processor to do two different works to complete jo. Ech work hving seprte processing times (ie) P nd P ddition to the due dte (dj). The most ovious ojective is to scheduling the jo nd minimizing the totl trdiness using Brnch nd Bound technique. This method is siclly stge wise serch method of optimiztion prolems whose solutions my e viewed s the result of sequence of decisions tht will help the decision mker in determining est schedule for given set of jos effectively. This method is ecome lucrtive to mke decision. In most of the rel life prolem there re elements of uncertinty in process. In prcticl sitution processing times nd due dte re not lwys deterministic. So we hve ssocited with fuzzy environment. The concept of type- fuzzy set which is n extension of the concept of n ordinry fuzzy set ws introduced y Zdeh [6]. fuzzy set is two dimensionl nd type- fuzzy set is three dimensionl type- fuzzy sets cn etter improve certin kinds of inference thn do fuzzy sets with incresing imprecision uncertinty nd fuzziness in informtion. type- fuzzy set is chrcterized y memership function (ie) the memership vlue for ech element of this set is fuzzy set in [0.] unlike n ordinry fuzzy set where the memership vlue is crisp numer in [0]. REVIEW OF ITERTRE Vrious reserchers hve done lot of work in different directions. Ishii nd Td [6] considered single mchine scheduling prolem minimizing the mximum lteness of jos with fuzzy precedence reltions. Hong et.l. [5] introduced single mchine scheduling prolem with fuzzy due dte. Itoh nd Ishii [7] proposed single mchine scheduling prolem deling with fuzzy processing times nd due dte. Gwiejnowicz et.l. [] dels with single mchine time dependent scheduling prolem. Emmons [] developed severl theorems nd dominnce rules tht cn e used to restrict the serch effort of rnch nd ound lgorithm. wler [0] pplied dynmic progrmming pproch to the single mchine totl trdiness prolem. Virktrkis nd Chung [5] proposed rnch nd ound lgorithm to minimize totl trdiness suject to minimum numer of trdy jos. zizogulu [] used rnch nd Interntionl Journl of Engineering Sciences & Reserch Technology [8]

2 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. ound method to solve the totl erliness nd totl trdiness prolem for the single mchine prolem. Rymond [] proposed rnch nd ound pproch to solve the prolem for steel plnt involving single mchine i-criteri prolem. The pper is orgnized s follows: In section dels with the preliminries. In section rithmetic opertions on type- trpezoidl fuzzy numer nd rnking function re discussed. In section we introduced rief note on Brnch nd Bound Technique. In section 5 the effectiveness of the proposed method is illustrted y mens of n exmple. PREIIRIES Definition: Fuzzy Set fuzzy set is chrcterized y memership function mpping the elements of domin spce or universe of discourse X to the unit intervl [0]. fuzzy set ~ is set of ordered pirs { ( x (x) ) / x R} where ~ ( x) : R [0 ] is upper semi continuous function ~ ( x) is clled memership function of the fuzzy set. Definition: Fuzzy umer fuzzy numer f in the rel line R is fuzzy set f: R [0] tht stisfies the following properties. (i) f is piecewise continuous. (ii) There exists n x ε R such tht f(x) =. (iii) f is convex (i.e) if x x ε R nd then λ ε [0] then f (λx + (-λ) x ) f ( x ) f ( x ). ~ Definition: Type- Fuzzy Set The type- fuzzy sets re defined y functions of the form : X ([0 ]) where ([0 ]) denotes the set of ll ordinry fuzzy sets tht cn e defined within the universl set [0]. n exmple of memership function of this type is given in fig-. Definition: Type- Fuzzy umer et ~ e type- fuzzy set defined in the universe of discourse R. If the following conditions re stisfied. (i) ~ is norml. Interntionl Journl of Engineering Sciences & Reserch Technology [9]

3 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. ~ (ii) is convex set. (iii) The support of ~ is closed nd ounded then ~ is clled type- fuzzy numer. Definition: Trpezoidl Fuzzy umer trpezoidl fuzzy numer = ( ) whose memership function is given y ( x) 0 x x x x & x x x Definition: Type- Trpezoidl Fuzzy umer type- trpezoidl fuzzy numer ~ on R is given y ~ = { x ~ ( x) ~ ( x) ~ ( x) ~ ( x)) ( x) ~ for ll ( x) ~ ( x) ~ ~ ~ x ( ) x ~ ~ ~ ~ ~ εr. (ie) where = ( ( ) ( ) ( ) ( )). x εr } nd RITHETIC OPERTIOS rithmetic Opertions on Type- Trpezoidl Fuzzy umers: et ~ ~ ~ ~ ~ = ) & B ~ ( = (( )( )( )( )) ~ ~ ~ ~ = B B B B = (( )( )( )( )) e two type- trpezoidl fuzzy numers. Then we define ddition: ~ B ~ = {( ) ( ) ( ) ( )}. Sutrction: ~ B ~ = {( ) ( - )( ) ( )}. ultipliction: Division: ~ xb ~ = {( * * * * )( * * * * )( * * * * ). ( * * * * )}. Interntionl Journl of Engineering Sciences & Reserch Technology [50]

4 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. ~ ~ B =. Rnking on Type- Trpezoidl Fuzzy umer et F( R ) e the set of ll type- norml trpezoidl fuzzy numers. One convenient pproch for solving numericl vlued prolem is sed on the concept of comprison of fuzzy numers y use of rnking function. n effective pproch for ordering the elements of F( R ) is to define liner rnking function R: F ( R ) R which mps ech fuzzy numer in to R. ~ ~ ~ ~ ~ Suppose if ( )). Then we define = ( ( ) ( ) ( ) R( ~ ) = / 6. lso we define orders on F ( R ) y R ( ~ ) R ( B ~ ) if nd only if R ( ~ ) R ( B ~ ) if nd only if R ( ~ ) = R ( B ~ ) if nd only if ~ ~ ~ B ~ B ~ = B ~ BRCH D BOD TECHIQE Brnch nd Bound: Brnching is the process of prtitioning lrge prolem into two or more suprolems nd Bounding is the process of clculting lower ound on the optiml solution of given suprolems. k Dominnce Property: While sudividing suprolem P σ into (n-k) suprolems creful nlysis would help us to crete only one suprolem insted of n-k suprolems. This is clled dominnce property. This will reduce the computtionl effort to greter extent. In suprolem P k σ if there exists jo i σ such tht d i q σ then it is k+ k sufficient to crete only one suprolem P iσ. The remining suprolems under P σ cn e ignored. In the ounding process V iσ = V σ. Trdiness: Trdiness is the lteness of jo j if it fils to meet its due dte; otherwise it is zero. It is defined s : T j = mx { oc j d j } = mx { 0 j } which mens ottions: T j = c j d j if c 0 otherwise j d n : The totl numer of independent jos. j : Represents the j th jo j =.. t j : The Processing time of the jo j. d j : The due dte of the jo j. C j : The Completion time of the jo j T j : The Trdiness of the jo j T j : umer of the trdy jos. T mn : inimum trdiness. j Interntionl Journl of Engineering Sciences & Reserch Technology [5]

5 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. J : oction of i th jo on mchine k. : Totl numer of jos to e scheduled. K : chine on which i th jo is ssigned t position j. σ : The set of Scheduled jos t the end of the sequence. σ : The set of unscheduled jos. (or) complement of σ. q σ = q φ : The sum of the processing times of unscheduled jos in σ. iσ : Prtil sequence in which σ is immeditely preceded y jo i. S(σ) : The sum of the trdiness vlues of the jos K P σ : suprolem t level k in the rnching tree. In this suprolem the lst k positions in the sequence re ssigned some jos. V σ : Vlue ssocited with p k σ which is comintion of jos to totl trdiness. lgorithm The processing times of jos nd due dte re uncertin. This leds to the use of Type- trpezoidl fuzzy numers for representing these imprecise vlues. Step-: 0 Plce P φ on the ctive list; its ssocited vlues re: V φ = 0 nd q φ = n j= tj. t given stge of the lgorithm the ctive list consists of ll the terminl nodes of the prtil tree creted up to tht stge. Step- ; Remove the first suprolem P σ k from the ctive list. If k is equl to n- stop. Prefix the missing jo with σ nd tret it s the optiml sequence. Otherwise check the dominnce property for P σ k. If the property holds go to step ; otherwise go to step. Step -: et the jo j e the jo with the lrgest due dte in σ. Crete the suprolem P jσ k+ with q jσ = q σ t j V jσ = V σ j σ = Vσ. Plce P jσ k+ on the ctive list rnked y its lower ound. Return to step. Step- : Crete (n-k) suprolems one for ech i σ. For P iσ k+ let q jσ = q σ t j V iσ = V σ + mx ( 0 q σ d i ) iσ = V iσ. ow plce ech P iσ k+ on the ctive list rnked y its lower ound. Return to step -. ERIC ISTRTIO In milk producing fctory they required doule processor for production using single mchine. There re two processes done y the single mchine. (i) Crushed the soy to produce milk is the first process mde y the mchine. (ii) Tht milk will e pcked y respective quntities is the second process mde y the sme mchine. These two processes re hving seprte processing times (P P). Here we consider the two processing time nd due dte with type- trpezoidl fuzzy numers for ech jos re given in the following tle: Tle-: Jo j Processing time P Processing time P Due dte dj (560) (60) (60) (680) (68) (68) (86) (87) (88) (60) (57) (058) (-59) (-50) (68) (57) (058) (-59) (-50) (89) (68) (69) (060) (96) (580) (79) (68) Interntionl Journl of Engineering Sciences & Reserch Technology [5]

6 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. (80) (805) (806) (68) (69) (060) (-6) 5 (570) (57) (057) (-57) (79) (79) (795) (57) (058) (-59) (-50) (570) (57) (057) (-57) (69) (0650) (965) (59) (0) () () (0000) (900) (800) (700) TBE-: Jo j Processing time P Processing time P Processing time tj Due dte dj (560) (60) (60) (60) (680) (68) (68) (68) (9 8) (78) (585) (87) (57) (058) (-59) (-50) (580) (80) (805) (806) (68) (69) (060) (-6 5 (570) (57) (057) (-57) (57) (058) (-59) (-50) (79) (79) (79) (795) (57) (058) (-59) (-50) (570) (57) (057) (-57) (60) (0606) (-608) (-600) (9595) (7597) (5599) (59) (75) (77) (-79) (-7) (00) (0) (00) (-06) (86) (87) (88) (89) (68) (69) (060) (96) (68) (69) (0650) (965) (59) (0) () () (0000) (900) (800) (700) Step-: ctive list t level 0 = {P φ 0 } σ = {φ} σ = {5} V φ = 0 & q φ = Since the current level k(0) is not equl to n-(). Check the dominnce property. lso for ech of the node is. mx d i = i σ. Since this mximum is not greter thn q φ. The detils of computtions of the lower ound Interntionl Journl of Engineering Sciences & Reserch Technology [5]

7 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. Piσ Viσ = Vσ + mx (0 qσ- di) iσ = viσ P P P P P5 0 + mx 0 + mx 0 + mx 0 + mx 0 + mx ctive list = { {P P 5 P P P } Step-: Check the dominnce property σ= {} σ = {5} q σ = mx d i= i σ = Since this mximum vlue is not greter thn Interntionl Journl of Engineering Sciences & Reserch Technology [5]

8 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. q σ = The detils of computtions of the lower ound for ech of the node is Piσ Viσ = Vσ + mx (0 qσ- di) iσ = viσ P P P P mx mx mx mx ctive list = { P 5 P P P P 5 P P P }. Proceeding in this wy we get Step- 7: This suprolem occurs t level which is not equl to {}. Hence check the dominnce property σ = {5} σ = {} q 5 = q t 5 = Interntionl Journl of Engineering Sciences & Reserch Technology [55]

9 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. = q σ mx i σ d i= Since the mximum vlue is equl to Step-8: Jo hs n element in σ which hs the highest due dte. Hence sed on the dominnce property the suprolem P 5 is further prtitioned with single rnch P 5.σ= {5} & j = q σ = = + mx V jσ = V σ + mx (0 q σ d j ) = TREE DIGR Interntionl Journl of Engineering Sciences & Reserch Technology [56]

10 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. The minimum totl trdiness vlue is The minimum totl trdiness vlue is 59. The required optiml sequence is 5. COCSIO We considered single mchine scheduling prolem (SSP) with fuzzy processing time nd fuzzy due dte to minimize the totl trdiness. This method is very esy to understnd ech stge tht will help the decision mker in determining est schedule for given set of jos effectively. This method hs significnt use of prcticl results in industries. REFERECES [] zizoglu Kondcki Sun nd omer. Interntionl Journl of Production Economics.99. [] Emmons H One mchine sequencing to minimize certin functions of Trdiness Opertions Reserch Interntionl Journl of Engineering Sciences & Reserch Technology [57]

11 [Helen. (): pril 05] ISS: Scientific Journl Impct Fctor:.9 (ISR) Impct Fctor:. [] Gwiejnowicz S. et.l. nlysis of time dependent scheduling prolem y signtures of Deteriortion rte sequences Discrete pplied themtics vol [] Henri P. sing fuzzy set theory in scheduling prolem: cse study. Fuzzy sets nd Systems vol [5] Hong T.P. nd Chung T.. new tringulr fuzzy Johnson lgorithm Computers nd ppliction Industril Engineering vol [6] Ishii H nd Td. Single mchine scheduling prolem with fuzzy precedence reltion Europen Journl of Opertionl Reserch vol [7] Itoh T. nd Ishii H. Fuzzy due-dte scheduling prolem with fuzzy processing time Interntionl Trnsctions in Opertionl Reserch vol [8] John R.J. Type- fuzzy sets nd pproch of theory nd pplictions Interntionl Journl Fuzziness Knowledge Bsed Systems 6(6) [9] Kurod nd Wng Z Fuzzy jo shop scheduling Interntionl Journl of Production Economics vol [0] wler E. Pseudopolynomil lgorithm for sequencing jos to minimize totl trdiness nnls of Discrete themtics [].izumoto nd K.Tnk Some properties of fuzzy sets of type- Informtion nd Control vol [] J endl Fiu nd D.Zhi α-plne representtion for type- fuzzy sets: theory nd pplictions IEEE Trns. Fuzzy Systems vol [] Q. endoz P.elin nd Gice hyrid pproch for imge recognition comining type- fuzzy logic modulr neurl networks nd the sugeno integrl Informtion Sciences vol [] Rymond B. Europen Journl of Opertionl Reserch 6(-). 99. [5] Virklrkis G.. nd YEE ee Chung IIE Trunction 7() 995. [6] Zdeh I.. The fuzzy concept of inguistic vrile nd its pplictions to pproximte Resoning Interntionl Journl Inform Science Interntionl Journl of Engineering Sciences & Reserch Technology [58]