Heuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transportation Problems

Transcription

1 Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: Volume Issue 6 (July 04) Heurstc Algorm for Fndng Senstvty Analyss n Interval Sold Transportaton Problems D.Anuradha Department of Maematcs, School of Advanced Scences, VIT Unversty, Vellore-63 04, Inda K.Kava Department of Maematcs, School of Advanced Scences, VIT Unversty, Vellore-63 04, Inda Abstract-Ths paper develops a heurstc algorm for fndng e ranges of cost n e nterval sold transportaton problem such at optmal bass s nvarant. The procedure of e proposed approach s llustrated by numercal example. Keywords- Interval sold transportaton problem (ISTP), Cost senstvty analyss I. INTRODUCTION The Sold transportaton problem (STP) s an mportant augmentaton of e transportaton problem (TP). The STP arses when bounds are gven on ree tems namely, supply, demand and conveyance. As a generalzaton of TP, e STP was ntroduced by Haley [7]. Pandan and Anuradha [4] have dscussed new soluton procedure for solvng a STP. Senstvty analyss (SA) s one of e most nterestng and preoccupyng areas n optmzaton. SA s to analyze e effect of e changes of e parameters n e optmzaton problems on e optmal value of e obectve functon as well as e valdty ranges of ese effects. SA for a lnear programmng problem were categorzed and summarzed by Kolta and Terlay [] and Hadgheh and Terlay [5,6]. Doustdarghol et al. [3] dscussed a new SA approach for RHS parameter n a TP. Ma and Wen [8] studed e cost coeffcents SA of e degenerate TP. An algorm for fndng e SA of costs n a TP was presented by Luca Cabulea []. Kava and Pandan [0] have ntroduced an algorm for e cost SA n e STP. Jen et al. [] dscussed SA of e degenerate TP usng labelng algorm. ITP can arse when uncertanty exsts n data problem and decson maers are more comfortable expressng t as ntervals. Pandan and Anuradha [5] dscussed e soluton approach for ISTP. Badya et al. [] presented mult teterval valued STP w safety factor. Kava and Pandan [9] proposed an algorm for solvng SA of costs n ITP. In s paper, a heurstc algorm for fndng e SA of costs n an ISTP s proposed and e same s llustrated w e help of numercal example. The SA of costs n an ISTP by e proposed algorm can help e decson maers to determne what level of accuracy s necessary for a parameter to mae e model suffcently useful and vald when ey are handlng dstrbuton problem havng ree constrants. II. PRELIMINARIES Let D denote e set of all closed bounded ntervals on e real lne R. That s, D= [ a, b], a b and a and b are n R. We need e followng defntons of e basc armetc operators and partal orderng on closed bounded ntervals whch can be found n [4,3]. Defnton : Let A = [a, b] and B = [c, d] be n D. Then, () A B [ a c, b d]; () A B [ a d, b c]; () A = [a, b] f s a + ve real number; (v) A = [b, a] f s a ve real number and (v) A B [ p, q] where p = mn {ac, ad, bc, bd} and q = max {ac, ad, bc, bd} Defnton : Let A = [a, b] and B = [c, d] be n D. Then, () A B f a c and b d ; () A B f a c and b d ; () A B f B A, at s a c and b d and (v) A B f A B and B A, at s, a = c and b = d. Consder e followng ISTP: Mnmze [ z, z ] [ c, d ] [ x, y ] subect to III. INTERVAL SOLID TRANSPORTATION PROBLEM 04, IJIRAE- All Rghts Reserved Page -

2 Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: Volume Issue 6 (July 04) [ x, y ]=[ a, a ],,,..., [ x, y ] [ b, b ] [ x, y ] [ e, e ],,,..., n,,,..., l m () () (3) x 0, y 0 for all, and (4) where c, d, a,, a, b, b e and e are postve real numbers for all, and. A set {[ x, y ], for all,,..., m,,,..., n and,,..., l } s sad to be a feasble soluton of e ISTP f ey satsfy e equatons (), (), (3)and (4). A feasble soluton of (ISTP) whch mnmzes e total shppng cost, at s, [ c, d ] [ x, y ] s called an optmal soluton (OS) to (ISTP). We consder e followng two problems as an upper bound (UB) problem and a lower bound (LB) problem of e gven problem (ISTP): (UB) Problem Mnmze Mnmze z d y (LB) Problem z c x subect to y a,,,..., y = b,,,..., y = e,,,..., y 0 for all, and m. (5) n. (6) l. (7) subect to x a,,,..., x = b,,,..., x = e,,,..., m. (8) x 0 for all, and n. (9) l. (0) In [0] Pandan and Kava proved e followng results whch s used n e proposed algorm. Theorem [0]: Let (,, ) cell be a non-basc cell correspondng to an OS of e STP w c u v w ( 0). If c s e perturbed cost of c, en e range of [, ). Theorem [0]: Let (,, ) cell be basc cell correspondng to an OS of e STP w c u v w ( 0). If c s e perturbed value of c and U s e mnmum value of for all non-basc cells n e orgn, s e mnmum value of for all non-basc cells n e destnaton and W s e mnmum value of for all non-basc cells n e conveyance, en e range of (, M ] where M e maxmum { U, V, W }. V 04, IJIRAE- All Rghts Reserved Page - 3

3 Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: Volume Issue 6 (July 04) IV. HEURISTIC ALGORITHM A heurstc algorm for fndng e SA of ISTP s proposed below: Step : Construct two ndvdual problems of e gven ISTP namely, (UB) problem and (LB) problem. Step : Compute an OS to (UB) problem by e proposed meod n [4]. Step 3: Create e MODI ndex matrx for e soluton obtaned n Step: (a) For all basc cells, use e relaton ( u v w ) c, and startng w any two MODI ndces values zero, compute e remanng MODI ndces. (b) For all non-basc cells, compute c u v w. Step 4: Compute e cost ranges of all non-basc cells usng e Theorem [0] and en, compute e cost ranges of all basc cells usng e Theorem [0] to e (UB) problem. Step 5: Repeat e steps from to 4 for e (LB) problem w e upper bound constrants x y, for all, and. V. NUMERICAL EXAMPLE The proposed meod s llustrated by e followng example. Example : Consder e followng ISTP: O O Capacty E E E [6,33] E E E [3,8] E3 E3 E3 [4,7] D D D3 Supply f f f3 f f f3 f 3 f3 f [9,30] 33 f f f3 f f f3 f 3 f3 f [8,] 33 where O3 f 3 f3 f33 f3 f 3 f33 f 33 f33 f [6,6] 333 Demand [9,7] [4,9] [30,3] [53,68] f =[37,4]; f =[65,7]; f 3 =[80,84]; f =[68,73]; f =[93,97]; f 3 =[84,87]; f 3=[,6]; f 3 =[3,7]; f 33 =[5,0]; f =[8,84]; f =[38,4]; f 3 =[44,46]; f =[67,7]; f =[49,53]; f 3 =[83,88]; f 3=[78,84]; f 3 =[38,4]; f 33 =[9,95]; f 3=[5,8]; f 3 =[8,]; f 33=[0,34]; f 3=[35,49]; f 3 =[56,70]; f 33=[,3]; f 33=[48,50]; f 33 =[3,6]; f 333=[47,49]. Now, e (UB) problem of e gven problem (ISTP) s gven below: Capacty E E E 33 E E E 8 E3 E3 E3 7 D D D3 Supply O O O Demand , IJIRAE- All Rghts Reserved Page - 4

4 Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: Volume Issue 6 (July 04) Now, usng e proposed meod n [4], e optmal soluton to e (UB) problem s y 3 4, y 3 6, y 8, y, y 3, y 3 9 and y Usng Step 3, e MODI ndex matrx correspondng to e above soluton s gven below: E E E w 5 E E E w 4 E3 E3 E3 w3 35 D D D3 O u 35 O u 0 O u3 43 v 0 v v3 0 x and Usng Step 4, e range of e (UB) problem are gven below: Now, e (LB) problem of e gven problem (ISTP) w e upper bound constrants s gven below. y,,,..., m ;,,..., n and,,..., l. E E E E E E E3 E3 E3 D D D3 O [ 5, ) [ 64, ) [ 84, ) [ 46, ) [ 79, ) [ 76, ) (,0] (,0] [ 0, ) O [ 33, ) (,3] [, ) [ 9, ) (,3] [ 4, ) [ 33, ) (,0] [ 60, ) O3 (,3] [ 3, ) [ 4, ) [ 30, ) [ 60, ) (,0] [ 4, ) [ 7, ) [ 57, ) Capacty E E E 6 E E E 3 E3 E3 E3 4 D D D3 Supply O O O Demand Usng e procedure followed as n e soluton of (UB) problem, e range of e (LB) problem s obtaned as E E E E E E E3 E3 E3 D D D3 O [ 6, ) [ 6, ) [ 84, ) [ 46, ) [ 79, ) [ 77, ) (,9] (,9] [ 9, ) O [ 35, ) (,] [ 3, ) [ 0, ) (,] [ 4, ) [ 3, ) (,9] [ 6, ) O3 (,] [, ) [ 30, ) [ 9, ) [ 48, ) (,3] [ 43, ) [ 6, ) [ 57, ) 04, IJIRAE- All Rghts Reserved Page - 5

5 Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: Volume Issue 6 (July 04) VI. CONCLUSION In s paper, we obtaned e perturbaton range of costs SA of ISTP. The necessty of consderng cost SA of e ISTP arses when heterogeneous conveyances are avalable for shpment of products n publc dstrbuton systems. The proposed meod can help e decson maers to determne what level of accuracy s necessary for a parameter to mae e model suffcently useful and vald when ey are handlng dstrbuton problem havng ree constrants. REFERENCES [] A.Badya, U.K.Bera, M.Mat, Mult teterval valued sold transportaton problem w safety measure under fuzzy stochastc envronment, J. Transp Secur, Vol.6, pp. 5-74, 03. [] Ch-Jen, Ln; Kang-Tng, Ma; Ue-Pyng, Wen, Type II senstvty analyss of cost coeffcents n e degenerate transportaton problem, European Journal of Operatonal Research, Vol.7, pp , 03. [3] S.Doustdarghol, D. Derahshan Asl and V. Abasgholpour, Senstvty analyss of rght hand-sde parameter n transportaton problems, Appled Maematcal Scences, Vol.3, pp. 50-5, 009. [4] George, J.Klr and Bo Yuan, Fuzzy sets and fuzzy logc: Theory and Applcatons. Prentce-Hall, 008. [5] A.G. Hadgheh and T. Terlay, Senstvty analyss near optmzaton: nvarant Support set ntervals, Eur. J. of Ope. Research, Vol.69, pp.58-75, 006. [6] A.G.Hadgheh, T.Terlay, Actve constrant set nvarancy senstvty analyss near optmzaton, J. of Opt. Theory and App., Vol.33, pp , 007. [7] K.B. Haley, The sold transportaton problem, Oper. Res. Vol., pp ,96. [8] Kang-Tng Ma, Ue-Pyng Wen and Ch-Jen Ln, Support set nvarant senstvty analyss n degenerate transportaton problem, The 4 Asa Pacfc Industral Engneerng and Management Systems conference, Melaa, 00. [9] K. Kava and P. Pandan, Senstvty analyss of costs n nterval transportaton problems, Appled Maematcal Scences, Vol.6, pp , 0. [0] K. Kava and P. Pandan, Senstvty analyss n sold transportaton problem, Appled Maematcal Scences, Vol.6, pp , 0. [] T.Kolta and T. Terlay, The Dfference Between e Manageral and Maematcal Interpretaton of Senstvty Analyss Results n Lnear Programmng, Internatonal J. of Producton Economcs, Vol.65, pp , 000. [] Luca Cabulea, Senstvty analyss of costs n a transportaton problem, ICTAMI, Alba Iula, Romana, Vol., pp.39-46, 006. [3] Moore, R.E, Meod and applcaton of nterval analyss, SLAM. Phladelpha, PA, 979. [4] P.Pandan and D. Anuradha, A new approach for solvng sold transportaton problems, Appled Maematcal Scences, Vol.4, pp , 00. [5] P.Pandan and D. Anuradha, A new optmal soluton procedure for fully nterval sold transportaton problems, Int. J. of Ma. Sc. and Engg. Appls., Vol.5, pp , 0. 04, IJIRAE- All Rghts Reserved Page - 6