A Scenario-Based Objective Function for an M/M/K Queuing Model with Priority (A Case Study in the Gear Box Production Factory)

Transcription

1 Proceedngs of the World Congress on Engneerng 20 Vol I WCE 20, July 6-8, 20, London, U.K. A Scenaro-Based Objectve Functon for an M/M/K Queung Model wth Prorty (A Case Study n the Gear Box Producton Factory) H. R. Rezae, A. Asgar, H. Mogoue, Amrhossen Amr Abstract In most optmzaton problems of an M/M/K queung model wth prorty, the scenaros are defned and optmzed separately. However, sometmes, the scenaros are dependent and should be optmzed smultaneously. In ths paper, an effcent approach s developed for ths purpose. The performance of the proposed approach s evaluated through a real case study n a ware house of a Gear Box producton factory. Key words Queung model, Optmzaton, M/M/K wth prorty T I. INTRODUCTION HE case study nvestgated n ths paper, s related to Gear Box producton company whch has a hgh market share of Gear Box n Iran. In ths company, the queung system for the fnal product warehouse s studed to determne the optmum number of docks. Now, there exsts three docks avalable and each dock requres two operators for loadng trucks. Ths company has two types of customers. The man customer makes up 40% of the cash flow and s consdered as the most mportant customer among the others. The other 60% cash flow s dstrbuted among four less prorty customers. The cash flow shares and the prorty of all customers are summarzed n TABLE. If the customers are wated n the queues, the company should pay a penalty cost to customers. The top manager of the company has decded to decrease the watng tme as much as possble. As faster as the company can send products to the customer, the cash flow would ncrease. Hence, less wat n queue tme s requred even f the top Manuscrpt receved March 7, 20; revsed Aprl 09, 20 H. R. Rezae s wth Industral Engneerng Department, Shahed Unversty, Tehran, Iran (Correspondng author; phone: (+982) ; e-mal:hamdreza.rezae2@gmal.com) A. Asgar s wth Industral Engneerng Department, Shahed Unversty, Tehran, Iran (e-mal: alasgar_987@yahoo.com). H. Mogoue s wth Industral Engneerng Department, Shahed Unversty, Tehran, Iran (e-mal: mogoue@ shahed.ac.r) Amrhossen Amr s wth Industral Engneerng Department, Shahed Unversty, Tehran, Iran (e-mal: amr@shahed.ac.r) manager has to omt a low prorty customer n future contracts. Costumer No. TABLE Customers Informaton Demand Share Prorty 40% 2 2% 2 3 6% 3 4 2% 4 5 % 5 As mentoned above, top manager polcy wll accept to elmnate one of the customers among customers number 2 to 5 from future contract lst. The way that queung theory models have been used n practce s categorzed n two major groups. The frst group contans the cases where queung theory models have been used to obtan the value of queung system evaluaton ndces such as and for a few decson scenaros and choosng the best optmal soluton. In more advanced cases an objectve functon s consdered n whch decson varables are or and the value of ths functon s calculated under any of scenaros l S q and S w q and fnally, the best alternatve among avalable scenaros are selected. The applcaton of the frst category was optmzaton of Berth Crane operatons n a martme termnal []. Also a few other researches have used ths approach n varous cases to mprove queung systems. Ths approach has shown a good effcency n optmzaton of complex queung networks [2]. By combnng queung theores wth schedulng knowledge, optmzng the loadng and unloadng operatons n contaner termnal s consdered n [3]. Another example s the usage of ths approach for selectng the best confguraton for a manufacturng system n whch queung networks exsted [4]. The other category that can be called optmzaton based applcaton of queung theory models focuses on defnng an objectve functon n whch the decson varables are or and the cost or beneft functon are computed based on or. For the second category, an optmzaton procedure for fndng the optmal value of N n an N polcy M/ / queung system wth a removable servce staton s consdered n [5]. A recursve method to control an M/G/queung system s proposed by [6]. The optmal polcy of an M/ E k / whle the servce staton was removable s gven n [7]. An M/G/ queung model was studed n [8] to determne the optmal control of the queung model whle the capacty of the queue was defnte. ISBN: WCE 20

2 Proceedngs of the World Congress on Engneerng 20 Vol I WCE 20, July 6-8, 20, London, U.K. Accordngly, there are some researches whch mproved the second approach for problems wth fuzzy nput parameters. As an example, an M/M/K queung model wth fuzzy nput parameters was studed n [9]. In ths paper the number of servers s optmzed based on the degree of customer's satsfacton. The research for computng queue length n a MAP/G//N model [0] as well as an N polcy G/M/queung system can be notable examples [] n ths category. Furthermore, examples of queung networks wth fnte capacty queues can be seen n [2]. Defnng the servce rate whle parameters of the queung model are fuzzy s another example of ths category [3]. The rest of the paper s organzed as follows: In secton II, proposed approach s explaned. In secton III, the problem s modeled and the queung system s analyzed. Moreover, the objectve functon s defned. In secton IV, the optmzaton procedure s presented and the results of optmzaton are llustrated. In the fnal secton concluson s dscussed. II. PROPOSED APPROACH A. Modelng the Problem Modelng a queung system ncludes estmatng and, mathematcal defnton of queue polces as well as recognton the type of queue n the case that t matches any of avalable models. For estmatng and, a samplng procedure should be done to calculate the average tme between nterval and the average tme of servng. The dstrbuton tme of arrval and servce tme n queung systems should be determned n advance. Statstcal methods such as goodness-of-ft test or graphcal methods such as probablty plots can be used for ths purpose. B. Evaluaton of the Current System In ths step by knowng the type of queung model the performance measures of the current system are calculated. III. ILLUSTRATIVE EXAPLE Ths paper nvestgates the applcaton of an M/M/K wth prorty model to optmze an objectve functon for the daly benefts for a warehouse of Gear Box producton company. The full descrpton of the case s gven n the ntroducton secton. A. Modelng the Problem To model the problem, estmatons of are requred. For ths purpose, a dataset of a 5 workng days perod s gathered. The dataset for one of these days s shown n TABLE 2. For determnng the dstrbuton of tme between arrvals, plottng the hstogram of the dataset n TABLE 2 can be helpful. The hstogram of the dataset n TABLE 2 s shown n Fg.. TABLE 2 Data Set For The Tme Between Intervals Demand No Tme of Tme between Arrval arrvals(mnute) 08:6-2 08: : : : : : : : :30 38 :0 3 2 : : : : : :55 24 C. Cost and Beneft Terms Before defnng the objectve functon, an ntal step s to defne the cost terms affectng the objectve whch are n accordance wth or. The benefts of servng a customer wth a lower wat n queue are defned through a proporton of cash flow per customer. D. Objectve Functon Usng the terms of costs and benefts, objectve functon s easly defned by subtractng costs from benefts. E. Optmzaton Procedure To optmze the objectve functon, an teratve algorthm s proposed n whch, n every step by changng decson varables, the value of objectve functon s obtaned untl the tme no more mprovement of the objectve functon s acheved. Fg.. Hstogram of Tme between arrvals As shown n the hstogram of Fg., the dataset follow an exponental dstrbuton, but for an accurate statstcal judgment, a goodness-of-ft test approves the assumpton of exponental dstrbuton for the dataset. Mntab results for ths test are llustrated n Fg. 2. F. Senstvty Analyss Fnally a senstvty analyss can be helpful for evaluatng possble changes of nput parameters of the model. ISBN: WCE 20

3 Proceedngs of the World Congress on Engneerng 20 Vol I WCE 20, July 6-8, 20, London, U.K. Percent Probablty Plot of tme between arrvals Exponental - 95% CI 0 tme between arrvals Fg. 2. A-D test for tme between arrvals 00 Mean N 6 AD 0.49 P-Value 0.49 Some descrptve statstcs and the P-value of Anderson- Darlng(AD) Test for the dataset n TABLE 2 are summarzed n TABLE 3. N * N 6 0 Medan 24 Goodness of Ft Test: Dstrbuton Exponental TABLE 3 Descrptve Statstcs Mean Mnmum 2 AD 0.49 St Dev Maxmum 79 P 0.49 Based on the number of the orders for a four months perod gven n TABLE 4, λ, the arrval rate, s estmated by usng Equaton (). ˆ n n x TABLE 4 Data Set Needed For Estmatng Month Number of receved orders Average Number of orders n a day ( x ) March Aprl May June The number of orders n a day summarzed n TABLE 4 s based on the assumpton that the number of workng days n month s 26. The estmaton of λ s computed as () In Table 5, the needed tme for servng the customer at each dock are shown. The A-D test results whch approves exponentally dstrbuton for servce rate per any servers (docks) s gven n TABLE 6. Also the results of goodnessof-ft test for dataset n TABLE 5 are llustrated n TABLE 6. TABLE 5 Data Set Needed For Estmatng Servce Tme Dock (Mn) Servce Tme Dock2 (Mn) Servce Tme Dock3 (Mn) TABLE 6 Goodness of Ft Test for Exponental Dstrbuton of servce tme Dock AD P The procedure for estmaton of whch represents the rate of servng each customer (loadng trucks) s roughly smlar to estmatng λ. For a more accurate estmaton of, the servce tme for ten days was recorded and fnally the average of servce tmes was ftted for μ mnute Results n μ 6.46 orders per day Hence, the servce rate for all three servers (docks) s equal to 6.46 orders per day. Based on the calculatons of ths secton, the tme between arrvals and servce tme for all docks follow exponental dstrbuton wth 26 and 6.46 orders per day, respectvely. There are three servers, and each customer has a specfc prorty, an M/M/K model wth consderng prortes of customers s sutable for ths case. Note that preempton of servng s not allowed n our case..e., when servng a customer starts, t cannot be nterrupted n any ways. ˆ orders par day 4 ISBN: WCE 20

4 Proceedngs of the World Congress on Engneerng 20 Vol I WCE 20, July 6-8, 20, London, U.K. B. Calculaton of queung system evaluaton ndces To calculate evaluaton ndces, frst s calculated by equatons (2), (3), and (4). Then, s obtaned by equaton (5). Fnally, usng equatons (6), (7), w and l are calculated, respectvely. The notatons are as follows: l q : The average number of customers watng n queue per day. w q : The average tme of watng n a queue per customer. w: The average tme that a customer should spend n a system. (Average tme of watng n the queue plus the average tme of beng served) l: The average number of customers watng n the system A m! ( m ) m m j j j j! j m (2) B0 and B, =,2,3,4,5 (3) m where s the number of sever. wq,,2,3,4,5 (4) AB B lq w q (5) w w q (6) l w (7) The results of evaluaton ndces by usng equatons (4), (5), (6) and (7) for each customer n current status are shown n TABLE 7. TABLE 7 Evaluaton Indexes For Dfferent Customers Customer w q w l l q C. Objectve functon To model the objectve functon, note that buldng a new dock mposes cost of constructon as well as cost of hrng two operators for each new dock. The notatons of the objectve functon are defned as followng: customer th s reman W = 0 customer th s elmnated C: A set of customers c: Number of customers m 0 : Number of exstng docks TS : Average of daly cash flow wth th customer PR : The percentage of proft for dealng wth th customer W q : The expected tme of watng n queue for th customer LR : The penalty percentage for lateness n servng th customer TCD: The daly costs for constructng a new dock (cost n ths case s consdered for ten years lfe and hrng 2 operator for each dock) The objectve functon s as follows: TS PR W TS LR m m TCD c Z W q 0 (8) Equaton (8) consders total proft, lateness penalty and the cost of new dock, respectvely. All of terms are determned based on a day. m and w are decson varables. m 0 and c n ths case are 3 and 5, respectvely. Also, note that, for =( C ), W equals one and t s not a decson varable. IV. OPTIMIZATION PROCEDURE Accordng to W q n TABLE 7, the ntal total proft s equal to Z 0 = Note that currency of Iran s Ral. Snce each term n the objectve functon affects on each other, teratve optmzaton s suggested for ths problem. For ths purpose, MATLAB software s appled to fnd the best soluton. Note that m and W 's are decson varables and should be reported to top manager, n addton to the value of the objectve functon. The best decson varables for company are shown n TABLE 8. TABLE 8 Result of Optmzaton Best customer for elmnatng Number of requrement servers (docks) Intal Total proft per day(rals) Optmzed total proft per day (Rals) Customer 5 (w 5 =0) 3,(m=3) e e+006 In Fg. 3 and Fg. 4, neghborhood solutons are presented for selectng the best decson varables. ISBN: WCE 20

5 Proceedngs of the World Congress on Engneerng 20 Vol I WCE 20, July 6-8, 20, London, U.K. REFERENCES [] P. Canonaco, P. Legato, R. M. Mazza, R.Musmanno, A Queung Network Model for the Management of Berth crane operatons, Computers & Operatons Research, vol. 35, no. 8, pp , [2] D. R. Bngham, Y. C. Hung, G. Mchalds, Developng effcent smulaton methodology for complex queueng networks, n Proceedngs of the wnter smulaton conference, pp , [3] E. K. Bsh, A Multple-Crane-constraned Schedulng problem n a Contaner Termnal, European Journal of Operatonal Research, vol. 44, no., pp , Fg. 3. Proft chart accordng to customer elmnatng and number of docks(m=3 or m=4) m=3,w 5 =0 [4] P. Canonaco, P. Legato, F. Vocaturo, Steady state Smulaton of Queung Network Models of Manufacturng Systems for Purposes of rankng and selecton of the best Confguraton, n Proceedngs of the AIRO XXXV annual conference, [5] W. L. Pearn, Y. C. Chang, Optmal Management Of the N polcy M/ / Queung system wth a Removable Servce Staton a senstvty Investgaton, Computers and Operatons Research, vol. 3, no.7 pp , [6] K. H. Wang, J. C. Ke, A recursve method to the optmal control of an M/G/ queung system wth Defnte capacty and ndefnte capacty, Appled Mathematcal Modelng, vol. 24, no.2, pp , [7] K. H. Wang, H. M. Huang, Optmal control of an M/E k / queung system wth a removable servce staton. Mcroelectroncs and Relablty, vol. 35, no. 7, pp , 995. Fg. 4. Iteraton plot for selectng the best soluton As shown n Fg. 4., the company does not need to ncrease the number of docks. In addton, customer 5 s not proftable for company. Hence, f the customer 5 elmnated, beneft would be maxmum. V. CONCLUSION In ths paper, we analyzed a queung system n Gear Box Producton Company. The man problem of ths company was busy docks for servng customers whch had a penalty cost for management. To analyze the problem, we proposed an approach to compare between ncreasng docks or elmnatng the secondary customers smultaneously. Ths approach determned the best number of docks n addton to canddate customer for elmnatng as decson varables and led to proft for company. [8] Jr. J. Teghem, Optmal control of a removable server n an M/G/ queue wth Defnte capacty, European Journal of Operatonal Research, vol. 3, no. 3, pp , 987. [9] M. J. Pardo, D. Fuente, Desgn of a fuzzy fnte capacty queung model based on the degree of customer satsfacton: Analyss and fuzzy optmzaton, Fuzzy Sets and Systems, vol. 59, no.24, pp , [0]U. C. Gupta, K. Skdar, Computng Queue length dstrbutons n MAP/G//N queue under sngle and multple vacaton, Appled Mathematcs and Computatons, vol. 7, no. 2, pp , [] J.C. Ke, K.H. Wang, A recursve method for the N polcy G/M/ queung system wth fnte capacty, European Journal of Operatonal Research, vol. 42, no. 3, pp , [2] S. Balsamo, V. D. Persone, P. Inverard, A revew on queung network models wth fnte capacty queues for software archtectures performance predcton, Performance Evaluaton, vol. 5, no. 2, pp , [3] M. J. Pardo, D. de la Fuente, Optmal selecton of the servce rate for a fnte nput source fuzzy queung system, Fuzzy Sets and Systems, vol. 59, no. 3, pp , ACKNOWLEDGMENT The authors gratefully acknowledge the top managers of the Gear Box Producton Company for provdng the dataset used n ths paper. ISBN: WCE 20