An Approach to Develop a Dynamic Job Shop Scheduling by Fuzzy Rule Based System and Comparative Study with the Traditional Priority Rules

Transcription

1 America Joural of Egieerig ad Applied Scieces Origial Research Paper A Approach to Develop a Dyamic Job Shop Schedulig by Fuzzy Rule Based System ad Comparative Study with the Traditioal Priority Rules Rami Ahmed, Md. Muzahid Kha, Md. Hasibul Haque ad Md. Hasibur Rahma Departmet of Idustrial Egieerig ad Maagemet, Khula Uiversity of Egieerig ad Techology, Khula-9203, Bagladesh Article history Received: Revised: Accepted: Correspodig Author: Rami Ahmed Departmet of Idustrial Egieerig ad Maagemet, Khula Uiversity of Egieerig ad Techology, Khula-9203, Bagladesh ramiahmed0020@gmail.com Abstract: This paper aims at developig a dyamic job shop schedulig by establishig fuzzy rule based system ad comparig its effectiveess with the traditioal priority rules. I order to uderstad the priorities of differet jobs i a job shop, the paper ivestigates all the cotributig criteria of differet parameters. Sice the traditioal priority rules oly emphasis o a sigle parameter at a time, the authors propose a method of obtaiig a schedule which icorporates all the desired cotributio criteria of the parameters used. This method icludes establishig a Fuzzy priority rule. Fuzzy is a appropriate model whe ucertaity ad irregularities are preset. It also allows modelig of a sigificat umber of alteratives across differet parameters cosidered i a job shop. Hece, a dyamic job shop schedule ca be developed usig the established fuzzy priority rule. This research is based o a example of jobs arrivig i a regular semi-automated job shop, ot a fully automated or software based job shop. The practical implicatios of this paper is to idetify the proper parameters ad evaluate their cotributio criteria o the required coditios so that maagers ca determie the proper schedulig ad sequece of the jobs to be machied i the job shop. This study provides a determiistic method for local maagers to assess the effects of job shop schedulig ad sequece of jobs. Keywords: Fuzzy Priority Rule (FPR), Fuzzy Rule Based System (FBRS), Membership Fuctio, Parameters, Cotributio Criteria, MATLAB R203a, Fuzzy Logic, Fuzzy Decisio System, Normalized Fuctio, Geeralized Fuctio, Traditioal Priority Rules, Fuzzy Iferece System, Shortest Processig Time (SPT), Earliest Due Time (EDT), Largest Batch Size (LBS), Shortest Arrival Rate (SAR) Itroductio Schedulig is the proper allocatio of resources over a time horizo to perform a collectio of tasks i a proper sequece. The practical schedulig problem arises as the situatios ad parameters vary. The schedulig problem ca be stated as follows: N jobs to be processed by M machies, give differet parameters to be cosidered alog with its cotributig criteria i such a way that the give objectives are optimized. I developig coutries like Bagladesh, the local maufacturers do ot cosider the basics of schedulig ad sequecig. After gettig a order, they start the processig right away regardless to the ways of schedulig ad sequecig i.e., they do ot thik through that which sequece should be followed ad whether the cotributig criteria of each parameter has bee take ito cosideratio. As a result, they are suffered by the loss of time ad moey. I this study, a brief study has bee made o the above metioed icoveiece. The solutio to this icoveiece may be the use of dispatchig or priority rules. But traditioal priority rules cosider oly oe parameter at a time e.g., SPT cosiders the sequecig of jobs accordig to the icreasig processig time of the jobs. But i a dyamic job shop, all the parameters are to be cosidered alog with its fractio of importace each time. Hece combiatio of these 206 Rami Ahmed, Md. Muzahid Kha ad Md. Hasibul Haque. This ope access article is distributed uder a Creative Commos Attributio (CC-BY) 3.0 licese.

2 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp traditioal rules or a ew rule capable of cosiderig all the parameters is to be established. As some of the decisios regardig the parameters ad their cotributig criteria to the coditios ad systems seem to be imperfectly kow due to their imprecise ad subjective ature, decisio makig becomes complex ad icosistet. Hece, fuzzy cocept has bee used. To establish a dyamic job shop schedule cosiderig the cotributio criteria of differet parameters, Fuzzy approach has bee used to establish a ew Fuzzy Priority rule. I this study, a case study has bee doe takig a raw data from a local job shop ad the calculatig the FPR ad comparig its effectiveess with the traditioal dispatchig rules. Literature Review Fuzzy logic begis with the cocept of a fuzzy set. A fuzzy set is a set without a crisp, clearly defied boudary. It cotais elemets with a partial degree of membership. To uderstad a fuzzy set, first take ito accout the defiitio of classical set. A classical set is the cotaier that fully icludes or fully excludes ay give elemet. For example, the set of days of the week uquestioably icludes Moday, Thursday ad Saturday. It just as uquestioably excludes butter, liberty ad dorsal fis ad so o. This type of set is called a classical set because it has bee aroud for a log time. It was Aristotle who first formulated the Law of the Excluded Middle, which says X must either be i set A or i set ot-a. Aother versio of this law is: Of ay subject, oe thig must be either asserted or deied. Fuzzy logic has two differet meaigs. I a arrow sese, fuzzy logic is a logical system, which is a extesio of multivalued logic. However, i a wider sese Fuzzy Logic (FL) is almost syoymous with the theory of fuzzy sets, a theory which relates to classes of objects with usharp boudaries i which membership is a matter of degree. Practically i all maufacturig systems, decisios parameters of a system are ot accurately kow because of the fact that the iformatio has imprecise ad subjective ature which makes decisio makig complicated, ucertai ad sophisticated. Fuzzy logic is a aalysis method purposefully developed to icorporate ucertaity ito a decisio model (Zadeh, 965). There are key beefits to applyig fuzzy tools. Fuzzy tools provide a simplified platform where the developmet ad aalysis of models require reduced developmet time tha other approaches. As a result, fuzzy tools are easy to implemet ad modify. Nevertheless, despite their user-friedly outlet, fuzzy tools have show to perform just as or better tha other soft approaches to decisio makig uder ucertaities (Azadega et al., 20). These characteristics have made fuzzy logic ad tools associated with its use to become quite popular i tacklig maufacturig related challeges. Fuzzy set theory is able to cope with the imprecisio ad ucertaity which is iheret i huma judgmets ad decisio makig processes by the use of liguistic terms or variables ad degrees of membership. A Liguistic Variable, the equivalet of mathematical variable, ca take words or seteces as values. For istace, a liguistic variable X with label Speed, ca assume values like Very fast, Fast, Slow, or Very slow. Fuzzy set ca classify elemets ito a cotiuous set usig the cocept of Membership Fuctios (MF) ad degree of membership (µ) to represet the gradual chages from 0 to. So i fuzzy logic, the truth of ay statemet becomes a matter of degree. Some widely used MFs are Gaussia, Geeralized bell shaped, Gaussia curves, Polyomial curves, Trapezoidal, Triagular ad Sigmoid MFs. The trapezoidal, triagular MFs which are the most widely used i literature have bee adopted for this study from the poit of view of simplicity, coveiece ad speed (Oladoku ad Okesiji, 202). May priority rules have bee proposed ad studied by researchers usig simulatio (Pierreval ad Mebarki, 997). To solve statistic ad determiistic schedulig problems a lot of aalytical ad optimizatio algorithms have bee developed (Baker, 984). If the problems are small, it is possible to geerate optimal schedules usig a brach or a boud approach (Brucker et al., 993). For larger problems, special solutio strategies based o priority rules have to be used to esure executio times of a few miutes (Peres, 995; Adams et al., 988; Subramaiam et al., 2000). There are several priority rules but there is o particular rule that is better tha the others i terms of all the parameters to be cosidered. Their efficiecy depeds o the cotributio criteria of each parameter ad their operatig coditios. Although a priority rule ca tur out to serve better result o give cotributio criterio, it may also result poorly o aother criterio. For istace, rules based o processig times ca be efficiet at reducig the time i job shop, but may ot cosider the due dates requiremets to be met (Pierreval ad Mebarki, 997). I order cope up with these icoveiece, lots of approaches have bee preseted. Amog them, steadystate simulatios have bee used to compare several priority rules applied to the resources, i various operatig coditios ad shop-floor cofiguratios. Moreover, statistical methods have bee used to obtai decisio rules, which will be used to select the suitable priority rule. Some examples of such methods are: Clusterig methods (Chu ad Portma, 99) ad machie learig (Pierreval ad Mebarki, 997). These techiques facilitate the choice of the priority rule that will be applied to each work ceter, whe the actual system is i a steady state mode. 203

3 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Methodology Basically, fuzzy logic is a multi-valued logic which allows itermediate values to be defied betwee traditioal evaluatios like true/false, yes/o, black/white, etc. Notios like warm or pretty cold ca be desiged ad formulated mathematically ad algorithmically. I this way a approach is made to apply a more huma-like way of thikig i the programmig of computers. Fuzzy logic addresses the imprecise iput ad output variables by defiig fuzzy umbers ad fuzzy sets expressed i liguistic variables (e.g., low, medium ad high) ( Fuzzy rule-based approach is based o verbally formulated rules combied with ad cosiderig the cotributio criteria throughout the parameter space. Numerical iterpolatio ca be used to hadle complex o-liear relatioships ( I a broader sese, a FRBS is a Rule-Based System i which Fuzzy Logic ca be used as a tool to represet differet forms of techiques ad kowledge about the problem solvig, as well as for modelig the relatioships ad the iteractios existig betwee its variables ( The implemetatio of the proposed methodology ivolves goig through seve mai stages, as described i Fig.. As it ca be see withi the algorithm, the first step i the proposal of the parameters which ofte may have more or less elemets as per give coditios. It follows the to determie the ormalized ad geeralized fuctio of the data with respect to the selected parameters. Stage 3 cosists of determiig the fuzzy rules accordig to the cotributio criteria of each parameter. I stage 4 takes place the applicatio ad simulatio of the algorithm i MATLAB 203a. Here the rules are first set i order ad the either the ormalized or geeralized fuctios are put to obtai specific outcome. Hece a sequece of jobs is obtaied. I the phase 5 takes place calculatio of the average flow, average tardiess ad average umber of jobs at work ceter accordig to the obtaied sequece. Phase 6 implies calculatio of the average flow, average tardiess ad average umber of jobs at work ceter accordig to the traditioal rules take ito accout i.e., i this case SPT, EDT, LBS ad LAR. I the last stage, takes place measurig the effectiveess of fuzzy priority rule by comparig them with those of the traditioal rules. Proposal of the Parameters The proposed method assigs priorities to the part types takig four parameters ito accout. I this case, a coditio has bee assumed where batch size, due time, processig time ad arrival rate are take as the iput parameters as show i Fig. 2. The reasos for selectig these parameters are as follows: Processig time Batch size Arrival rate Due time The geerated output is a percetage of priority i relatio with all the parameters take ito accout basig o a specific cotributio criteria of the parameters. Cosiderig these four parameters, Table depicts the iput data parameters to be cosidered alog with its cotributig criteria i such a way that the give objectives are optimized. As arrival rate preseted i previous table follows Poisso distributio, Table 2 gives the revised arrival rate. Determiatio of the Normalized ad Geeralized Fuctio Normalized Fuctio Batch size Due time d [ b ( b ) ] [( b ) ( b ) ] i i mi ( b ) = i i max i mi [ d ( d ) ] [( d ) ( d ) ] i i mi ( ) = i i max i mi [ ti ( ti) ] Processig time i [( t ) ( t ) ] Arrival rate () (2) mi ( t ) = (3) i max i i mi ( a ) = i i max i mi i mi [ a ( a ) ] [( a ) ( a ) ] (4) Normalized cotributio fuctio of each parameter of part type is defied as the ratio of the differece of the particular value ad its miimum value to the differece of the maximum ad miimum value. Each data is coverted i terms of ormalized fuctios ad preseted i Table 3. Geeralized Fuctio Geeralized fuctio refers to the value of each part take for each parameter ad plotted withi the rage of miimum ad maximum respective values i MATLAB 203a. Determiatio of the Fuzzy Rules for a Give Coditio of Cotributio Criteria I this case, four parameters have bee cosidered. All the four iput variables are associated with idetical liguistic values: Low (L), medium (M) ad high (H). The output to be geerated should be a priority percetage such as 0, 20, 30, 40, 50, 60, 70, 80 ad 90%. There are = 8 rules for the system. The membership fuctios for ormalized batch size, ormalized due time, ormalized processig time ad ormalized arrival rate are similar as show i Fig. 3 (Bilkay et al., 2004). 204

4 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Fig.. The proposed methodology Fig. 2. Parameter of fuzzy job shop schedulig Triagular membership fuctios have bee used because it is the most commo type of membership fuctio. Iitially, differet membership fuctios have bee studied but triagular membership fuctios depicted the most promisig performace i both computig time ad performace. 205

5 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Fig. 3. Triagular membership fuctio Table. Iput data for priority calculatio Uit Arrival Part Batch Operatio Due date processig rate per o size umbers i mi time i mi hour Sice four parameters each with three associates are used, all 8 combiatios are obtaied by the followig JAVA programmig laguage. Table 2. Revised arrival rate Part type Arrival rate Table 3. Iput data with ormalized fuctios Normalized form Batch Arrival Due Processig Part size rate date time I this study, the cotributio criteria cosidered for the parameters are: Batch size-50% Arrival Rate-25% Due date-75% Processig time-50% Basig o these priorities, the set rules for fuzzy decisio system are as follows: If batch size is LOW AND arrival rate is LOW AND due time is LOW AND processig time is LOW THEN decisio is

6 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp If batch size is LOW AND arrival rate is LOW AND due time is MEDIUM AND processig time is MEDIUM THEN decisio is 0.65 If batch size is LOW AND arrival rate is HIGH AND due time is MEDIUM AND processig time is LOW THEN decisio is 0.65 If batch size is HIGH AND arrival rate is MEDIUM AND due time is LOW AND processig time is MEDIUM THEN decisio is 0.55 If batch size is HIGH AND arrival rate is HIGH AND due time is HIGH AND processig time is HIGH THEN decisio is 0. Implemetatio of the Rules ad Obtaiig the Results from MATLAB For Normalized Fuctio, Fig. 4 ad 5 refers to the settig of the Rage ad Membership Fuctios of the Parameters i MATLAB Plottig the Normalized ad Geeralized Values of the Data i MATLAB For Normalized Fuctios I the view form of the rules, the ormalized fuctios of the parameters of each part type are plotted as show by Fig. 6 ad 7. The the output shows the priority percetage i the form of result. As stated earlier, ormalized fuctios may show some icoveieces such as showig same priority for two or more part types. Hece geeralized fuctio has bee take ito accout ad the outcome, after plottig all part types, has bee depicted i Table 4. Calculatio the Average Flow, Average Tardiess ad Average Number of Jobs Fuzzy Rule Based System The obtaied sequece of Table 4 go through further calculatios. That is, the average flow, average tardiess ad average umber of jobs at work ceters are calculated basig o the data i Table 5. Average flow time, F = = Fj Mi Average Tardiess, T = Tj = Mi Traditioal Priority Rules This set of calculatio is to be doe basig o the traditioal priority rules. I this case, the rules cosidered are: Largest Batch Size (LBS) Shortest Arrival Rate (SAR) Earliest Due Time (EDT) Shortest Processig Time (SPT) Calculatio for LBS Table 6 shows the raw data for LBS estimatio: Average flow time, F = = Fj Mi Average Tardiess, T = Tj = 72.9 Mi Average Number of Jobs, N = δ ( Tj) Calculatio for SAR T = 6.5 Table 7 shows the raw data for SAR estimatio: Average flow time, F = = Fj Mi Average Tardiess, T = Tj = Mi Average Number of Jobs, N = δ ( Tj) Calculatio for EDT T = 5.53 Table 8 shows the raw data for EDT estimatio: Average flow time, F = = Fj Mi Average Tardiess, T = Tj = Mi Average Number of Jobs, N = δ ( Tj) T = 4.72 Average Number of Jobs, N = δ ( Tj) T =

7 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Calculatio for SPT Table 9 shows the raw data for SPT estimatio: Average flow time, F = Fj = Mi Average Tardiess, T = Tj = Mi Average Number of Jobs, N = δ ( Tj) T = 4.20 Fig. 4. Normalized membership fuctio for batch size Fig. 5. Normalized membership fuctio for due time 208

8 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Fig. 6. Plottig the data of part type Table 4. Part sequece ad fuzzy part priority Part sequece Table 5. Calculatio for fuzzy rule based system Fuzzy part priority Flow Due Days Processig time time tardy Part time (Mi) (Mi) (Mi) (Mi) Total Total Total processig flow days time (Mi) time (Mi) tardy (Mi) =445 =68338 =46364 Table 6. Calculatio for largest batch size Processig Flow Due Days Part time (Mi) time (Mi) time (Mi) tardy (Mi) Total Total Total processig flow Days time (Mi) time(mi) tardy (Mi) = 445 = = 729 Table 7. Calculatio for shortest arrival rate Processig Flow Due Days Part time (Mi) time (Mi) time (Mi) tardy (Mi) Total Total Total processig flow Days time (Mi) time(mi) tardy (Mi) = 445 = =

9 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Table 8. Calculatio for earliest due time Flow Due Days Processig time time tardy Part time (Mi) (Mi) (Mi) (Mi) Total Total Total processig flow Days time (Mi) time(mi) tardy (Mi) = 445 = = Table 9. Calculatio for shortest processig time Flow Due Days Processig time time tardy Part time (Mi) (Mi) (Mi) (Mi) Total Total Total processig flow Days time (Mi) time(mi) tardy (Mi) = 445 = = Compare the Results Obtaied from Fuzzy Priority Rule from those of the Traditioal Rules I this sectio, a comparative study has bee doe betwee the data obtaied i sectio 3..5 ad sectio For doig so, the cotributio criteria of each parameter viz. batch size-50%, arrival rate-25%, due time-75%, processig time-50% came ito use. The values obtaied from each property LBS are multiplied by 0.5, those obtaied from SAR are multiplied by 0.25, those obtaied from EDT are multiplied by 0.75 ad those obtaied from SPT are multiplied by 0.5. These are the added ad divided by the total weight age: Average flow time= 2 = mi Average Tardiess= 2 = Average Number of Jobs= 2 = 5.25 Results Proper schedulig ad sequecig of jobs should be performed for all job shops i order to miimize the loss of time ad moey. Based o the data i terms of differet parameters alog with their cotributio criteria, several rules have bee established. By the aalytical aid of MATLAB, output priority percetage of differet part types have bee obtaied. Beside fuzzy calculatios, all the combiatios of parameters with respect to the give coditios have bee illustrated graphically MATLAB. Eve the surface view of differet parameters with respect to each other has bee show where Fig. 8 ad 9 suggest two combiatios. Fially, the obtaied compariso betwee Fuzzy priority rule ad combied traditioal priority rule alog the horizos (i.e., average flow time, average tardiess ad average umber of jobs at work ceter) is show graphically usig MATLAB i Fig. 0 ad umerically i Table 0. Discussio Huma behavior is all about fidig easier ad shortest way of doig work. I developig coutries like Bagladesh, this sese of behavior has ot yet developed by far meas especially i the maufacturig ad schedulig regio. After gettig a order, they start the processig right away regardless to the cosideratio of the importace of differet criteria. I this case, job shop priority or dispatchig rules may play the role as a savior. These traditioal priority rules cosider oly oe parameter or criteria at a time i.e., i this case traditioal priority rules will cosider ay of the LBS, SAR, SPT or EDT. So i this study, a rule-based fuzzy priority system i.e., Fuzzy Priority Rules (FPR) has bee itroduced ad the cotributio criteria of each parameter has bee icorporated. For differet percetage of each parameter importace, every elemet of the 8 rules eeds to be assiged with their proper values. By doig so, a output is obtaied as show i Table for first sample ru. This output formulates a sequece of jobs ad accordig to this sequece, the average flow time, average tardiess ad average umber of jobs at work ceter have bee determied. 20

10 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp Fig. 9. Surface view betwee due time ad batch size Fig. 0. Compariso betwee fuzzy priority rules ad combied traditioal priority rules Table 0. Compariso of fuzzy priority rule with traditioal priority rule Average Average Average flow tardiess umber of jobs Fuzzy priority rule Combied traditioal priority rule The the average flow time, average tardiess ad average umber of jobs of those traditioal rules have bee determied ad each of the particular rule is measured agaist their respective cotributio criteria to icorporate to a combied form for the satisfactio of the coditios. The obtaied data from FRBS ad traditioal priority rules have bee compared alog several costraits where it is observed that Fuzzy Priority Rule proves to be more efficiet tha the traditioal priority rules takig every possible demads ad cocers ito accout. I this study, the relatios betwee differet parameters i terms of their parametric ad surface structure have also bee show. Fially, a graph has bee show which illustrates the comparative study betwee Fuzzy Rulebased system ad traditioal priority rule. Coclusio Fuzzy set theory allows the complexity of real life issues to be icluded withi the cofies ad rigors of the mathematical model. The proposed approach is 2

11 Rami Ahmed et al. / America Joural of Egieerig ad Applied Scieces 206, 9 (2): DOI: /ajeassp used to solve the problems related to schedulig ad sequecig of jobs i a job shop i order to reduce average flow time, average tardiess ad average umber of jobs at work ceter. The proposed methodology ca be successfully applied for the purposes of pursuig a course of actio i terms of developig a dyamic job shop through proper schedulig ad sequecig of jobs. Here, the cotributio criteria of each parameter has a greater or lesser ifluece o the output as take by the user. I this study, the cotributio criteria have bee assumed but the method, of utilizig ay set of cotributio as oe ca demad, has bee show. Ackowledgemet The authors wat to express the gratefuless ad humbleess to Almighty Creator, who is the most merciful ad the most gracious for the successful completio of this thesis work. They also would like to express their sicere thaks ad heartfelt gratitude to their supervisor Assistat Professor, Md. Fashiar Rahma, Departmet of Idustrial Egieerig ad Maagemet, Khula Uiversity of Egieerig ad Techology (KUET), Khula, for his proper guidace, utirig geerous help ad advice ad the systematic approach of the supervisor i aalyzig the problem, useful commets ad ufailig ethusiasm ad overall presece will derive ispiratio i all of their future activities. They are also idebted to all other teachers of Idustrial Egieerig ad Maagemet departmet of KUET, Khula for their direct ad idirect assistace at differet times. Fially, they express their deepest gratitude to their family members ad to all their well-wishers. Fudig Iformatio This paper was a thesis work of the authors. All the cost icurred i order to perform the related operatios we bore by the authors themselves. Author s Cotributios This research work had bee cotributed by all the authors. Rami Ahmed: Proposed the hypothesis, performed mathematical aalysis ad wrote the paper. Md. Muzahid Kha: Collected raw data from a local job shop, assisted i 8 rule formulatio ad eve i mathematical aalysis. Md. Hasibul Haque: Formatted for this mauscript ad provided useful suggestios. Md. Hasibur Rahma: Made figures, tables ad provided useful suggestios. Ethics This article is origial ad cotais upublished material. The correspodig author cofirms that all of the other authors have read ad approved the mauscript ad o ethical issues ivolved. Refereces Adams, J., E. Balas ad D. Zawack, 988. The shiftig bottleeck procedure for job shop schedulig. Maage. Sci., 34: DOI: 0.287/msc Azadega, A., L. Porobic, S. Ghazioory, P. Samouei ad A.S. Kheirkhah, 20. Fuzzy logic i maufacturig: A review of literature ad a specialized applicatio. Iter. J. Product. Ecoom., 32: DOI: 0.06/j.ijpe Baker, K.R., 984. Sequecig rules ad due-date assigmet i a job shop. Maage. Sci., 30: DOI: 0.287/msc Bilkay, O., O. Alaga ad S.E. Kilic, Job shop schedulig usig fuzzy logic. It. J. Adv. Maufactur. Techol., 23: DOI: 0.007/s Brucker, P., B. Jurisch ad M. Jurisch, 993. Ope shop problems with uit time operatios. Zeitschrift für Operatios Res., 37: DOI: 0.007/BF Chu, C. ad M. Portma, 99. Some ew efficiet methods to solve the //ri/ϵti schedulig problem. Europea J. Operat. Res., 58: DOI: 0.06/ (92)9007-G Oladoku, V.O. ad O.O. Okesiji, 202. Applicatio of fuzzy logic to the optimal locatio of public utilities: A case study of pedestria bridges. Proceedigs of the Iteratioal Coferece o Microelectroic Test Structures, (MTS 2), IEEE Xplore Press, Phoeix, AZ, USA, pp: Pierreval, H. ad N. Mebarki, 997. Dyamic schedulig selectio of dispatchig rules for maufacturig system. It. J. Product. Res., 35: DOI: 0.080/ Peres, D., 995. Miimizig late jobs i the geeral oe machie schedulig problem. Europea J. Operatioal Res., 8: DOI: 0.06/ (94)006-T Subramaiam, V., T. Ramesh, G.K. Lee, Y.S. Wog ad G.S. Hog, Job shop schedulig with dyamic fuzzy selectio of dispatchig rules. It. J. Adv. Maufactur. Techol., 6: DOI: 0.007/s Zadeh, L.A., 965. Fuzzy sets. Iform. Cotrol, 8: