A survey on fuzzy transportation problems

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1 IOP Coferece Series: Materials Sciece ad Egieerig PAPER OPEN ACCESS A survey o fuzzy trasportatio probles To cite this article: D Auradha ad V E Sobaa 2017 IOP Cof. Ser.: Mater. Sci. Eg View the article olie for updates ad ehaceets. Related cotet - Solvig fully fuzzy trasportatio proble usig petagoal fuzzy ubers P Ua Maheswari ad K Gaesa - Optial solutio of full fuzzy trasportatio probles usig total itegral rakig M Sa a, Farikhi, S ariyato et al. - Fuzzy ulti obective trasportatio proble evolutioary algorith approach T Karthy ad K Gaesa Recet citatios - P. Sethil Kuar This cotet was dowloaded fro IP address o 07/07/2018 at 04:57

2 A survey o fuzzy trasportatio probles D Auradha ad V E Sobaa Departet of Matheatics, School of Advaced Scieces, VIT Uiversity, Vellore ,Idia E-ail: auradhadhaapal1981@gail.co Abstract. A ucertai trasportatio proble is a trasportatio proble i which the paraeters are fuzzy ubers. This paper presets a survey o sigle obective fuzzy trasportatio proble (SOFTP) ad ulti-obective fuzzy trasportatio proble (MOFTP) with its atheatical odels. 1. Itroductio The trasportatio proble (TP) is a iiu-cost plaig proble for trasportig a coodity fro factories to warehouses with the shippig cost fro oe poit to aother poit. The basic TP was origially developed by itchcock [36]. TP ca be a sigle obective proble or ulti obective proble. The ai of the TP is to fid the trasportatio schedule that iiizes the total trasportatio cost. The uit costs of trasportatio, supply ad dead quatities are the paraeters of the TP. This paper is orgaized as follows: Sectio 2 proects the ebership fuctios. Sectio 3 presets SOFTP with its atheatical forulatio ad MOFTP with its atheatical forulatio i sectio 4 ad the last sectio presets the coclusio. 2. Mebership fuctios ad rakig techique Oe of the ai assuptios i solvig fuzzy atheatical prograig probles ivolves the use of o-liear (expoetial ad hyperbolic) ebership fuctios ad liear ebership fuctio for all fuzzy sets ivolved i a decisio akig process. 2.1.Triagular Mebership Fuctio Triagular ebership fuctio ca be stated as follows. 0, y<b1 y b1 b1 yb2 b2 b1 μ B (y) = b3 y b2 yb3 b3 b2 0 y b3 (1) 2.2.Trapezoidal Mebership Fuctio Trapezoidal ebership fuctio ca be defied as follows. Cotet fro this work ay be used uder the ters of the Creative Coos Attributio 3.0 licece. Ay further distributio of this work ust aitai attributio to the author(s) ad the title of the work, oural citatio ad DOI. Published uder licece by Ltd 1

3 0 for y b (y-b ) 1 for b1 yb2 (b2-b 1) 1 for b B (y) = (b -y) 1 yb for b3 yb4 (b4-b 3) 0 for y b 4 (2) 2.3.Liear Mebership Fuctio A liear ebership fuctio ca be defied as follows. 1 if z L, z- L μ (z (y)) = 1 if L z U, (3) U- L 0 if z U. Where L is the aspiratio level of achieveet ad U is the highest acceptable level of achieveet for the th obective fuctio. 2.4.Expoetial Mebership Fuctio A expoetial ebership fuctio ca be defied as follows. 1 if z L, s (y) s E e e ( z (y)) if L z U, 1 e s (4) 0 if z U. where (y) (z L ) / (U L ) 1,2,...,M ad s is a o-zero paraeter recoeded by the decisio aker yperbolic Mebership Fuctio A hyperbolic ebership fuctio ca be defied as follows. 1 if z L, U L z U L z e e μ (z (y)) = if L z U, 2 2 U L z U L z 2 2 e e 0 if z U. (5) 6 where. U -L yperbolic ebership fuctio has the followig properties. (1) μ (z (y)) is strictly ootoously decreasig fuctio with respect to Z (y); 2

4 1 1 (2) μ (z (y)) z (y) ( U +L ); 2 2 (3) μ 1 (z (y)) strictly covex for z (y) ( U +L ) ad 2 strictly cocave for 1 z (y) ( U +L ); 2 (4) μ (z (y)) satisfies 0<μ (z (y)) 1 for L z (y) U ad approaches asyptotically μ (z (y)) 0 ad μ (z (y)) 1 as Z (y) ad -, respectively. 2.6.Rakig Techique Rakig of fuzzy data is a essetial part of the decisio process i various applicatios. Fuzzy data ust be raked before a actio is take by a decisio aker. I [41,42], Jai itroduced a ethod usig the otio of axiizig set to order the fuzzy ubers. I [96], Yager preseted a robust s rakig ethod to covert the FTP ito TP. L U Robust srakig Techique.The Robust s rakig is defied as R(c) = 0.5(c,c ) d, where 1 0 L U ( c, c ) is the α-level cut of the fuzzy uber c.robust s rakig approach satisfies the property of copesatio, liearity ad additive. 3. Sigle obective fuzzy trasportatio proble (SOFTP) I ay real life situatios, trasportatio probles are odelled ad solved as sigle obective probles. The occurrece of radoess ad iprecisio i the real life is uavoidable due to soe uexpected circustaces. To deal quatitatively with ucertai data i akig decisios, Bella ad Zadeh [3] ad Zadeh [97] proposed the cocept of fuzziess. I practice, due to soe ucotrollable factors cost coefficiets, the supply ad dead aout of a TP ay be iprecise. Such TP is kow as FTP. The obective of the FTP is to fid the trasportatio schedule that iiizes the total fuzzy trasportatio cost. O heigeartaigh [35] proposed a heuristic approach for the solutio of TP where the availability ad requireets are fuzzy ubers with liear triagular ebership fuctios. Usig the paraetric prograig approach i ters of the Bella-Zadeh criterio, Chaas et al. [11] discussed TP with fuzzy paraeters. Their ethod solved the solutio which siultaeously satisfied the costraits ad the obective to a axiu level. Lio ad wag [61] developed trasportatio odel which solved the proble whe supply ad dead are fuzzy ad costs are crisp. Chaas et al. [10] forulated the FTP i three differet cases ad preseted a algorith for solvig the forulated FTP. Chaas ad Kuchta [12] discussed the cocept of the solutio of the LPP with the obective fuctios as iterval coefficiets. Chaas ad Kuchta [13] discussed a algorith for fidig a optial solutio to fuzzy iteger TP by cosiderig the paraeters that are availability ad requireet as L R type fuzzy ubers. Parra et al. [75] preseted a ew procedure for solvig FTP ad the to obtai the possibility distributio of the obective value of the TP. For fidig a optial solutio to iterval TP, Segupta ad Pal [81] applied a techique of fuzzy prograig ad they also cosidered the idpoit ad width of the iterval i the obective fuctio. Nagoor Gai ad Abdul Razak [69] solved the TP by cosiderig the costraits as ucertai paraeters ad they discussed the utilizatio of Kuh-Tucker coditios related to the paraetric proble. For fidig a optial solutio to a TP uder fuzziess, Oar et al. [73] applied a paraetric techique. Based o the cocept of extesio priciple, Liu ad Kao [62] discussed a ew procedure for solvig the FTP by cosiderig the obective value ad paraeters as fuzzy ubers. 3

5 Jersha Chiag [43] discussed the optial solutio of the FTP, i which requireet ad product are fuzzy. NagoorGai ad Abdul Razak [70] discussed a two-stage cost iiizig FTP i which deads ad supplies are trapezoidal fuzzy ubers. Nagoor Gai ad Abdul Razak [71] have solved TP with availability ad dead as fuzzy values ad also with a itegratio coditio iposed o the solutio usig a procedure. For fidig the fuzzy iitial basic feasible solutio of FTP, Das ad Baruah [15] applied a ethod of VAM. Li et al. [55] discussed a ew procedure for solvig FTP with fuzzy costs ad their idea was based o the cocept of goal prograig. Che et al. [14] discussed the ethods for solvig TP o a fuzzy etwork. Li [60] proposed a geetic algorith for solvig TP by cosiderig the coefficiets as fuzzy. Diagar ad Palaivel [20] itroduced a fuzzy MODI ethod for fidig the optial solutio to the FTP, i which paraeters are trapezoidal fuzzy ubers. I [21], they applied trapezoidal ebership fuctios for solvig FTP. Padia ad Nataraa [74] itroduced a ew algorith aely, fuzzy zero poit ethod for fidig the optial solutio to the FTP i which paraeters are trapezoidal fuzzy ubers. GuzelNura [32] ivestigated the FTP at two stages, i first stage they calculated the level of satisfactio betwee ucertai supply ad dead ad i secod stage by cosiderig the uit costs of TP fro zero to axiu satisfactio level. Kaur ad Kuar [45] discussed a heuristic approach for solvig FTP by cosiderig that a decisio aker is iprecise about the precise values of the uit cost of trasportatio, availability ad requireet of the quatity. For solvig FTP, Gai et al. [29] discussed a siplex type procedure. Edward Sauel ad Vekatachalapathy [22] discussed VAM for solvig FTP. Sobha [83] foud the axiu profit cost of soe products through a capacity etwork, whe the availability ad requireet of odes ad the cost ad capacity of odes are cosidered as triagular fuzzy ubers. A ew dual based procedure for the ubalaced FTP was discussed by Edward Sauel ad Vekatachalapathy [23]. Kaur ad Kuar [46] proposed a heuristic ethod for fidig the optial solutio to the FTP, i which paraeters are geeralized trapezoidal fuzzy ubers. I [24], Edward Sauel ad Vekatachalapathy ivestigated a heuristic algorith for solvig geeralized trapezoidal FTP. Mohaaselvi ad Gaesa [66] preseted a ew algorith to fid the iitial fuzzy feasible solutio for the FTP ad they applied fuzzy versio of MODI ethod to fid the fuzzy optial solutio. Shugai Pooa et al. [82] solved a FTP where the paraeters are trapezoidal fuzzy ubers. Robust s rakig fuctio is used to trasfor the FTP ito crisp TP. Fegade et al. [28] solved fuzzy trasportatio proble usig zero suffix ad robust rakig ethod. Maiekalai et al. [65] proposed a algorith for solvig FTP with iiu cost usig robust rakig ethod. Narayaaoorthy et al. [72] proposed fuzzy Russell s ethod to fid the iitial basic feasible solutio of fuzzy trasportatio proble. They applied Yager s rakig ethod to trasfor fuzzy trasportatio proble to crisp trasportatio proble. Edward Sauel ad Vekatachalapathy [26] proposed a ew ethod for solvig a special type of fuzzy trasportatio proble o the assuptio of the ucertaity of the decisio aker about the precise values of trasportatio cost. They [25] also ivestigated the iproved zero poit ethod for solvig fuzzy trasportatio probles usig rakig fuctios. Solaiappa ad Jeyaraa [84] proposed a algorith for solvig a fuzzy trasportatio proble usig zero teriatio ethod with trapezoidal fuzzy ubers. Sriivasa ad Geetharaai [86] discussed a iovative ethod for solvig fuzzy trasportatio proble. Reovig soe variables fro equatios by the eliiatio ethod ad applyig the Fourier eliiatio ethod, Pooa Shugai et al. [77] have foud the best coproise solutio for the fuzzy trasportatio proble. The sae authors [78] itroduced dual siplex ethod to solve trasportatio proble with fuzzy obective fuctios. Edward Sauel ad Vekatachalapathy [27] discussed iproved zero poit ethod for solvig the ubalaced fuzzy trasportatio probles. Dayi e et al. [18] trasfored the fuzzy trasportatio proble ito four types of crisp liear prograig probles by the paraetric ethod usig possibility theory i fractile ad odality approach. Sriivas ad Gaesha [85] used robust rakig idices to covert the fuzzy trasportatio proble ito crisp trasportatio proble ad steppig stoe ethod to fid a optial solutio to the fuzzy trasportatio proble. Thaaraiselvi ad Sathi [89] discussed a fuzzy trasportatio proble i which the values of trasportatio costs are represeted by idiscriiate hexagoal fuzzy ubers. Gaurav Shara et al. 4

6 [31] proposed a algorith for fidig the fuzzy optial solutio for a fuzzy trasportatio proble i which the paraeters are trapezoidal fuzzy ubers. Viala et al. [93] proposed oalisha's approxiatio ethod for solvig the fuzzy trasportatio proble. Isail Mohidee et al. [40] applied octago fuzzy ubers with α-cut ad rakig techique for solvig fuzzy trasportatio proble.murugaada ad Sriivasa [67] foud the least trasportatio cost of soe coodities through a capacitated etwork whe the origi ad destiatio of odes ad the capacity ad cost of edges are represeted as fuzzy ubers. Krisha Prabha ad Viala [49] discussed a ew algorith for axiizig the profit of the trasportatio proble i fuzzy eviroet. By usig the ethod of agitude rakig, fuzzy quatities were trasfored ito crisp quatities. Krisha Prabha ad Viala [48] proposed a ew techique for fidig the axiu profit cost for fuzzy trasportatio Proble. The circuceter of cetroids rakig techique was used by the [50] for axiizig the profit for a ubalaced fuzzy trasportatio proble. Kalpaapriya ad Auradha [44] proposed a algorith for the two vehicle cost varyig balaced trasportatio proble ad ubalaced trasportatio proble with ucertai data. Daruee uwisai ad PooKua [17] used Robust s rakig techique for trasforig ucertai data to precise data. They used allocatio table ethod for fidig a iitial basic feasible solutio to the fuzzy trasportatio proble. 3.1.Matheatical Stateet of SOFTP Now, the atheatical odel of a SOFTP is give as follows Miiize z = c subect to =1 i=1 i i=1 =1 x =a, i=1,2,..., x =b, =1,2,..., x 0, forall i ad x where c is the fuzzy cost of trasportig oe uit fro source i to the destiatio ad x is the fuzzy uber of uits trasported fro source i to destiatio. a is the aout of aterial available at i source poit i ad b is the aout of the aterial required at destiatio poit. 4.Multi obective fuzzy trasportatio proble (MOFTP) I geeral, real-life trasportatio probles ay be odelled ore profitably with the cocurret cosideratio of ulti criteria because a trasportatio syste decisio-aker geerally pursues ultiple goals. For solvig LPP ivolvig costraits of equality ad obectives coflictig with each other, we use MOTP. Lee ad Moore [54] suggested the applicatio of goal prograig approach for solvig the MOTP. Ziera [100] discussed the fuzzy prograig techique for solvig ulti criteria probles. Isera [38] proposed a ew approach for fidig all the o-doiated solutios for a liear MOTP. Leberlig [52] used hyperbolic ebership techique for solvig MOLPP ad foud that the solutios obtaied usig this ethod is always efficiet. Luhadula [64] discussed the usage of copesatory operators i fuzzy liear prograig with ultiple obectives. Ziera s [101] developed fuzzy approach to solve TP ad MOLPP. Riguest ad Riks [79] proposed the usage of iteractive algoriths to obtai ore tha k o-doiated ad doiated solutios for liear MOTP. Biswal [4] solved ulti obective geoetric prograig probles by usig fuzzy prograig techique.bit et al. [7] discussed MOTP by usig the fuzzy prograig approach with liear ebership fuctio, ad arrived at efficiet solutios as well as a optial coproise solutio for MOTP. I [8], usig additive fuzzy prograig odel, they cosidered (6) 5

7 weights ad priorities for all o-equivalet obectives for the TP. Lee ad Li [53] ivestigated the possibility of fuzzy ultiple obective prograig ad coproised prograig with paretoo ptiu for solvig MOFTP. Bit et al. [9] discussed fuzzy prograig techique to chace costraied MOTP. Biswal ad Siha [6] solved MONLPP usig fuzzy prograig approach. Biswal [5] solved ulti-obective FLPP usig proective ad scalig algorith. Vera et al. [92] discussed a fuzzy prograig techique for solvig MOTP with soe oliear ebership fuctios. Vera et al. [92] tried solvig MOTP with soe oliear ebership fuctios after discussig a fuzzy prograig approach. ussei [37] ivestigated the coplete solutios of MOTP with possibilistic coefficiets ofthe obective fuctios. Das et al. [16]discussed the procedure of derivig solutio to the MOTP i which all the paraeters have bee cosidered as itervals. Li ad Lai [56] obtaied a o-doiated coproise solutio to the MOTP usig a fuzzy coproise prograig ethod. Waiel ad Wahed [95] dealt with MOTP uder fuzziess. Wahed ad Sia [94] applied a fuzzy techique to fid the optial coproise solutio of a MOTP ad calculated the closeess degree of the coproise solutio to the ideal solutio usig the cocept of distace fuctios. Gao ad Liu [30] used a two-phase fuzzy algorith for solvig MOTP with oliear ad liear ebership fuctios. Aar ad Youess [1] discussed the efficiecy of the solutios ad the stability of MOTP i which paraeters are fuzzy. Wahed ad Lee [2] obtaied a coproise solutio to the MOTP usig a iteractive fuzzy goal prograig techique. Liag [57] dealt with distributio of plaig decisios by applyig iteractive fuzzy MOTP. Isla ad Roy [39] solved the ulti obective etropy TP with a additioal delivery tie costrait where its shippig costs were i the for of geeralized trapezoidal fuzzy ubers. Tie-Fu Liag [59] optiized trasportatio plaig decisio usig a iteractive fuzzy MOTP techique. Zagiabadi ad Maleki [98] preseted a fuzzy goal prograig techique with hyperbolic ebership fuctio to obtai a optial coproise solutio for the MOTP. Surapati ad Roy [88] discussed a priority based fuzzy goal prograig techique with ebership fuctio for fidig a coproise solutio of a MOTP with fuzzy coefficiets. They coverted the ebership fuctios ito ebership goals, by prioritisig the highest degree of a ebership fuctio as a level of aspiratio ad by itroducig deviatioal variables to each of the. For solvig TP usig fuzzy LPP, Liag [58] proposed a iteractive ulti obective techique. Lau et al. [51] solved the MOTP usig fuzzy logic guided o-doiated sortig geetic algorith. Deshabrata Roy Mahapatra et al. [19] apprehesive about the usage of fuzzy prograig approach to the obective fuctio, the stochastic techique was used for the radoess of supply ad dead paraeters i iequality type of costraits of ulti obective stochastic ubalaced TP. Lohgaokar ad Baa [63] applied fuzzy prograig approach with ebership fuctio(liear, hyperbolic ad expoetial) to obtai the optial coproise solutio of a ulti obective capacitated trasportatio proble. ale GoceKocke ad Mehet Ahlatcioglu [33] ivestigated fuzziess i the obective fuctios. They preseted a copesatory techique to solve the MOLTP with cost coefficiets as fuzzy. Their ethod geerated both copesatory ad Pareto optial coproise solutios. Vekatasubbaiah et al. [91] used fuzzy goal deviatio fuctio to idetify a coproise solutio for the MOTP ad itroduced a fuzzy ax-i operator, a auxiliary variable, the equivalet fuzzy iteractive goal prograig techique was forulated to axiize λ. Peidro ad Vasat [76] used odified S-curve o-liear ebership fuctio to deterie a coproise solutio for the MOTP. Zagiabadi ad Maleki [99] proposed a fuzzy goal prograig techique with special type of oliear ebership fuctio to fid a optial coproise solutio for the liear MOTP. Thorai ad Ravi Shakar [90] discussed a algorith for aalyzig a FMOTP by applyig a liear prograig odel based o a heuristic approach for rakig geeralized LR fuzzy ubers. Subhra ad Goswai et al. [87] proposed two-vehicle cost varyig MOTP. They trasfored the cost varyig MOTP to MOTP by NWCR ethod ad the used MODI ethod for fidig the optial solutio. Kha ad Das [47] proposed a review of the coectio betwee oder era approaches ad fuzzy ulti obective optiizatio to deal with its shortcoig ad fuzzy ulti obective optiizatio used i TP. ale GoceKocke [34] proposed a copesatory fuzzy techique to the MOLTP i which uit cost supply ad dead quatities are 6

8 triagular fuzzy ubers. This techique had three stages. I the first stage, by usig Ziera s i operator, the ucertai availability ad requireet aout was reoved, that is, the precise availability ad requireet aout were obtaied fro ucertai aout to satisfy the balace coditio. I the secod stage, breakig poits ad cost-satisfactio iterval sets were fixed for each obective. I the third stage, cosiderig cost-satisfactio iterval sets of all obectives, a overall costsatisfactio iterval set was idetified. Murugaada ad Sriivasa [68] preseted a heuristic approach for fidig the optial solutio to the two stage cost iiizig FTP with ulti obective costraits. For MOTP, SaruKuari ad Priyavada Sigh [80] applied fuzzy efficiet iteractive goal prograig techique. 4.1.Matheatical Stateet of MOFTP Now, the atheatical odel of a MOFTP is give as follows =1 i=1 r i=1 =1 Miiize z ( x) = c x, r 1,2,..., K subect to x r x a, i 1,2,..., i x b, 1,2,..., 0, for all i ad where a i is the aout of the aterial available at the ith source ad b is the aout of the aterial required at th destiatio. r is the uber of the obective fuctio of MOFTP. (7) r c is the ucertai uit trasportatio cost fro source i to destiatio for the obective r ad x is the ucertai uber of uits shipped fro source i to destiatio. 4.2.Fuzzy Goal Prograig Approach with Liear ad No-liear Mebership fuctios Usig the liear ebership fuctio as defied i (3) tha a equivalet liear odel for the odel (7) ca be forulated as: i : subect to z- L +d + d =1, U - L d, =1,2,...,k, d d =0, i 1 i1 i1 x a, i=1,2,...,; x b, =1,2,...,, d d 0, 1, 0, x 0, for all i,, where the equlibriu coditio a i 1 b 7

9 Usig the expoetial ebership fuctio as stated i (4) the a equivalet liear odel for the odel (7) ca be forulated as: i : subect to k (y) e e k +d + d =1, 1 e k d, =1,2,...,M, d d =0, x a, i=1,2,...,; x b, =1,2,...,, 1 i i 1 d d 0, 1, 0, x 0, for all i,, where the equlibriu coditio a i1 i b 1 Usig the yperbolic ebership fuctio as stated i (5) tha a equivalet liear odel for the odel (7) ca be forulated as: i : subect to U L z U L z e e +d + d =1, 2 2 U L z U L z 2 2 r e e d, =1,2,...,M, d d =0, x a i, i=1,2,...,; x b, =1,2,...,, i 1 1 d d 0, 1, 0, x 0, for all i,, where the equlibriu coditio a b 1 i i 1 5. Coclusio This paper provides the survey o sigle obective FTP ad ulti-obective FTP with its atheatical odels. The survey also studies the approaches used to solve such probles. Refereces 8

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