Journal of Applied Science and Agriculture

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1 AENSI Journals Journal of Appled Scence and Agrculture Journal home page: Developng a New Mult-Obectve Model for Locaton-Routng Problem along wth Travelng Tmes and Fuzzy Due dates Mad Noavan, 2 Reza Tavaol-Moghaddam, Al Kazem Yousef Department of Industral Engneerng, Islamc Azad Unversty, South Tehran Branch, Tehran Iran, 2 Department of Industral Engneerng, College of Engneerng, Unversty of Tehran, Iran ARTICLE INFO Artcle hstory: Receved 7 October 23 Receved n revsed form 23 November 23 Accepted 28November 23 Avalable onlne 25 February 24 Keywords: Locaton-Routng Problem Facltes Locaton Fuzzy Plannng Mult-Obectve Optmzaton ABSTRACT Bac ground: In present study, a locaton-routng problem along wth trp tmes and Fuzzy delvery tmes are taen nto account for whch one two-obectve model s proposed. Dstrbuton of goods has consderable mportance n supply chan management. Dstrbuton networ desgn problem ams to specfy the best way to transfer goods from suppler ponts for customers so that customers requests are met and dstrbuton cost s mnmzed. Obectve: The study ams to mnmze dstrbuton networ costs and total weghted tardness. Dstrbuton networ costs nclude the ependtures for depots nstallaton and shppng. About the latter obectve, a delvery deadlne s consdered for each customer and mnmum total weghted tardness s taen as the second obectve nto account. Results: Approprate meta-heurstc algorthms are proposed to resolve ths problem. Calculaton results ndcate that for all small, bg and medum problems, Smulated Annealng Algorthm has better performance and the obtaned boundary by ths algorthm overcome answers reached by genetc algorthm. Ths ssue s appeared n all performance measurements, ecept epanson one. Concluson: Therefore, mult-obectve Smulated Annealng Algorthm s recommended to resolve ths problem. 23 AENSI Publsher All rghts reserved. To Cte Ths Artcle: Mad Noavan, Reza Tavaol-Moghaddam, Al Kazem Yousef., Developng a New Mult-Obectve Model for Locaton-Routng Problem along wth Travelng Tmes and Fuzzy Due dates. J. Appl. Sc. & Agrc., 8(7: 72-78, 23 INTRODUCTION Dstrbuton of goods has consderable mportance n supply chan management. Dstrbuton networ desgn problem ams to specfy the best way to transfer goods from suppler ponts for customers so that customers requests are met and dstrbuton cost s mnmzed. Locaton-routng problem s of a specfc type of routng problems where depots quantty and locaton as well as vehcles route from depots for customers should be specfed to meet customers needs. The locaton-routng problem s a NP-hard one as t consoldates two other NP-hard problems (facltes locaton and vehcles routng. Here, one model s developed consderng trp tmes and Fuzzy delvery tmes. The locaton-routng problem encompasses selectng some places to nstall depots from a potental set of places and specfyng vehcles route to delver servce for customers, mnmzng ether the total taen route or networ costs.watson-gandy and Dohrn (973 employed an nventve algorthm based on consecutve methods wth route length estmaton to resolve the locaton-routng problem. Rand (976 studed on researchers and authors who have wored practcally on the locaton-routng problems. Tuzun and Bure (999 presented some calculatve comparsons about performance benchmarng of the locaton-routng problem solutons. Chan et al (2 eamned stochastc locaton-routng problem by an nventve algorthm based on groupng method. Labbe et al (24 conducted an nvestgaton on Plant Cycle Locaton problem where smultaneously both locatng the rado statons and desgnng the felds whch connect rado antennas to the statons, are performed. Gunnarsson et al (26 studed the locaton-routng problem wth heterogeneous fleet, eamnng for a Cycle Medan-le problem. The transportaton fleet ncludes shp, tran and truc. Ambrosno et al (26 presented an nventve model wth mult-echange technques for the locaton-nventory-routng problem. Jafar and Golozar (2 also consdered the locaton-routng problem along wth fuzzy trp tmes and L-R fuzzy Numbers. They proposed a mathematcal plannng model to resolve the problem for whch then they suggested an nventve resolvng method. Belenguer et al (2 also desgned a branch and cut method for ths problem. Baldacc et al (2 proposed a branch and prce algorthm based on dvson of set formulaton. Nguyen et al (22 presented a numercal model and an nventve algorthm for solvng the two-echelon locaton routng problem by smultaneous learnng process and path relnng. Furthermore, Karaoglan et al (22 offered a Correspondng Author: Al Kazem Yousef, Department of Industral Engneerng, Islamc Azad Unversty, South Tehran Branch, Iran, Tehran. E-mal: al.a987@yahoo.com

2 73 Mad Noavan et al., 23 meta-heurstc algorthm accompaned by consderng smultaneous pcup and delvery. Tng and Chen (23 too the capactated locaton routng problem nto account and presented an optmzaton algorthm based on multple ant colones. Methodology: Mathematcal Model: In ths secton, n order to specfy actve depots, to select vehcles n both levels of networ as well as to determne servce delvery routes for customers, a lnear plannng two-obectve model ned wth ntegers s developed. The resulted values for customers delvery prorty n former secton wll be consdered as nput parameter and used n target functon of customer satsfacton. Then, the usng notaton n proposed model wll be qute descrbed. Inde for potental stes depots (, 2,..., m Inde for customers (, m+, m+ 2,..., m+ n Inde for vehcles n layer (,2,..., p Inde for vehcles n layer 2 ( p+, p+ 2,..., p+ q Customer demand Customer s desred delvery tme The dstance between customers and The dstance between customer and depot The dstance between depot and the factory Capacty of the vehcle Capacty of the vehcle Worng tme lmts for each of the second vehcles Transportaton tme between customer and Transportaton tme between customer and depot Servce tme at customer ste Servce tme at depot depot Fed cost of usng vehcle Fed cost of usng vehcle Fed cost of settng up depot Average cost of vehcle transportaton for one unt of dstance Average cost of vehcle transportaton for one unt of dstance Arbtrary large number An enough-long perod of tme The weght of customer whch s obtaned by TOPSIS method, d p ( p, p, p b b c c c W t ( t, t, t t ( t, t, t g ( g, g, g g ( g, g, g r r r h h M T tw :

3 74 Mad Noavan et al., 23 In case of usng depot, the value s, otherwse t s. If customer s suppled by vehcle through depot, ths customer wll be l th customer at vehcle route and get value of. Otherwse, t wll be. l,2,..., n+ If second layer vehcle s assgned to depot, t wll be otherwse t wll be. If second layer vehcle s assgned to depot, t wll be otherwse t wll be. If customer s the last customer vsted by vehcle at depot, t wll be, otherwse t s. Customer s vsted at depot by vehcle mmedately before customer and wthout recharge, t wll be otherwse t wll be. If customer s vsted at depot by vehcle mmedately before customer and after recharge, t wll be otherwse t wll be. If customer s vsted by a vehcle (wthout recharge mmedately before customer, t wll be otherwse t wll be. If customer s vsted by a vehcle (wth recharge mmedately before customer, t wll be otherwse t wll be. z l y y w u v Dr Ir Gven the above shown parameters and varables, nteger lnear mathematcal model of our desred model s modeled as follows: Z y r + y r + rz + h b y mn + hb + hbw + hb u ( + hb + hb + hb v Mn Z 2 tw D + n ( l t ( (, My l l ( (,, l l ( l l ( d c l l d c y d c ( l l y l zm l ( ( ( (,,, l,..., n ( w l l u + V ( (,,,, l,..., n ( l l + (8 (9 ( ( (2 (3 (4 (5 (6 (7 (8 (9

4 75 Mad Noavan et al., 23 ( (,, l,..., n ( l ( + u u u u u + + ( ( u u u l l l t + w t + g W u u a t a t a (, m m l l ( (, ( (, ( t U t t t g V t u u u u a a + g + t M ( Dr m m m m a a + g + t M ( Dr l l l l a a + g + t M ( Dr ( ( ( u u u u u u u a a + g + t + t + t + g V M ( Ir m m m m m m m a a + g + t + t + t + g V M ( Ir l l l l l l l a a + g + t + t + t + g V M ( Ir U MDr u l pu nu m m pm nm l u pl nl pu pm pl ( D 4 D D a p D D a p D D a p D D V MIr + + ( D + 6 ( (,,,, l, y, y, w, u, v {,} l (2 (2 (22 (23 (24 (25 (26 (27 (28 At above model, Equaton (8 shows costs mnmzaton so that costs for layer 2 vehcles settng up, layer vehcles settng up, depot settng up, transportaton n layer and transportaton n layer 2 are modeled n 4 relatons, respectvely. Second target functon (9 shows total weghted tardness. The lmt n ( ndcates that the customer should be assgned to one depot, one vehcle and n one place on route of that vehcle. The lmt n ( specfes that f layer 2 vehcle K s not used at depot ( y, no customer can be assgned to. The lmt (2 ndcates that one customer can be at most assgned to each locaton l from each vehcle at depot. The lmt (3 shows that the customers should be assgned only as much as capacty of the depot. The lmt (4 determnes that for each depot, total capacty of layer vehcles assgned to depot should be at least equal to customers demands assgned to the depot. The lmt (5 addresses that for each vehcle n layer 2 (, total assgned customers demands should not be more than capacty of the vehcle. The lmt (6 mentons that each vehcle n layer ( should at most assgned to one depot. The lmt (7 addresses that f depot s not used, no customer wll be able to be assgned to that depot. The lmt (8 s made to specfy the last customer vsted by vehcle n depot.

5 76 Mad Noavan et al., 23 Equaton (9 addresses that f customer s vsted by vehcle (no matter f recharged or not eactly before customer, ether u or v wll be. The lmt (2 depcts that the routes related to each vehcle n layer 2 ( should be flled n ascendng order by customers. Equaton (2 shows worng hours lmtaton of each vehcle n layer 2 (. The lmt (22 shows f customer s placed n frst locaton of vehcle, that customer s vst tme s at least equal to the tme between depot and customer. Inequalty (23 addresses that f customer s vsted by vehcle mmedately after customer (wthout recharge, vst tme of customer wll be at least equal to vst of customer plus departure tme from customer to customer and servce tme n place of customer. Equaton (23 addresses that f customer s vsted by vehcle mmedately after customer after recharge, vst tme of customer wll be at least equal to vst tme of customer plus servce tme at place of customer n addton to departure tme from customer to depot plus servce tme at place of depot n addton to departure tme from depot to customer. Also, Equaton (24 determnes values of Dr, so that f customer s vsted by vehcle mmedately after customer (wthout recharge, Dr wll be equal to. Also, Equaton (25 specfes values of Ir so that f customer s vsted mmedately after customer by vehcle after recharge, Ir wll be. Usng lmt (26, hgh, low and mdlevels of non-determnstc tardness amount are calculated accordng to the dfference between calculated delvery tme on commtted model and tme for each customer. In lmt (27, determnstc tardness tme for each customer s calculated by weghted average method. As t can be seen, the numbers.6, 4.6 and.6 are consdered for weghtng pessmstc, epected and optmstc values. 3. Results: 3.. Proposed algorthm: Here, two NSGA-II and SA algorthms are used whose nput parameters are done through DOE method. Algorthms results for dfferent answer szes are as follows: Table : Calculaton results for mean eamples. Eample propertes Customers Depots Quantty Convergence Quantty NSGA II Epanson Soluton algorthm Rato Convergence 3.69E-5 7.9E-5 9.9E-5.74E-5.94E-5 4.3E E-5 MOSA Epanson Rato Obtaned Pareto Boundary s as follows: Fg. : Obtaned Pareto Boundary by both algorthms.

6 77 Mad Noavan et al., 23 Table 2: Calculaton results for mean eamples. The eample propertes Customers Depots Quantty Convergence Quantty NSGA II Epanson Soluton algorthm Rato Convergence 3.69E-5 7.9E-5 9.9E-5.74E-5.94E-5 4.3E E-5 MOSA Epanson Rato And, Obtaned Pareto Ponts are as follows: Fg. 2: Obtaned Pareto Boundary by Genetc Algorthm and Smulated Annealng Algorthm. Table 3: Calculaton results for great eamples. The eample propertes Customers Depots Quantty Convergenc Quantty e NSGA II Epanson Soluton algorthm Rato Convergence.4E E-4.48E E E E-5.E+ MOSA Epanson Rato Pareto results are also as follows: Fg. 3: Obtaned Pareto Boundary by Genetc Algorthm and Smulated Annealng Algorthm.

7 78 Mad Noavan et al., Dscusson: Calculaton results ndcate that for all small, bg and medum problems, Smulated Annealng Algorthm has better performance and the obtaned boundary by ths algorthm overcome answers reached by genetc algorthm. Ths ssue s appeared n all performance measurements, ecept epanson one. Regardng epanson measurement, two algorthms appromately have a same performance. Therefore, mult-obectve Smulated Annealng Algorthm s recommended to resolve ths problem. 5. Concluson and future research study: Therefore, mult-obectve Smulated Annealng Algorthm s recommended to resolve ths problem. Presentng other meta-heurstc algorthms such as Ants Colony Algorthm and Partcle Swarm Optmzaton Algorthm as well as consderng other obectves for ths problem such as servce delvery tme to the last customer are proposed as the solutons for future researches. REFERENCES Ambrosno, D., A. Scomachen, M.G. Scutellà, 29. A heurstc based on mult-echange technques for a regonal fleet assgnment locaton-routng problem. Computers & Operatons Research, 36: Baldacc, R., A. Mngozz, R. Wolfler Calvo, 2. An eact method for the capactated locaton routng problem. Operatons Research, 59(5: Belenguer, J.M., E. Benavent, C. Prns, C. Prodhon, R.W. Calvo, 2. A Branch-and-Cut method for the Capactated Locaton-Routng Problem. Computers & Operatons Research, 38: Chan, Y., W.B. Carter, M.D. Burnes, 2. A multple-depot, multple-vehcle, locaton-routng problem wth stochastcally processed demands. Computers and Operatons Research, 28: Gunnarsson, H., M. Ronnqvst, D. Carlsson, 26. A combned termnal locaton and shp routng problem. Journal of operatonal research socety, 57: Jafar, A., F. Golozar, 2. Applcaton of Ranng Functon to Solve Fuzzy Locaton-Routng Problem wth L-R fuzzy Numbers. 2nd IEEE Internatonal Conference on Informaton and Fnancal Engneerng (ICIFE, Karaoglan, I., F. Altparma, I. Kara, B. Dengz, 22. The locaton-routng problem wth smultaneous pcup and delvery: Formulatons and a heurstc approach. Omega, 4: Labbe, M., G. Laporte, 24. Mamzng user convenence and postal servce effcency n post bo locaton. Belgan Journal of Operatonal Research, Statstcs and Computer Scence, 26: Nguyen, V.P., P. Prns, C. Prodhon, 22. Solvng the two-echelon locaton routng problem by a GRASP renforced by a learnng process and path relnng. European Journal of Operatonal Research, 26: 26 Rand, G.K., 976. Methodologcal choces n depot locaton studes. Operatonal Research Quarterly, 27: Tng, C.J., C.H. Chen, 23. A multple ant colony optmzaton algorthm for the capactated locaton routng problem. Internatonal Journal of Producton Economcs, 4: Tuzun, D., L.I. Bure, 999. A two-phase tabu search approach to the locaton routng problem. European Journal of Operatonal Research, 6: Watson-Gandy, C.D.T., P.J. Dohrn, 973. Depot locaton wth van salesmen a practcal approach. Omega :