Two Stage Interval Time Minimizing Transportation Problem

Transcription

1 wo Stage Interval me Mnmzng ransortaton Problem Sona a, Rta Malhotra b and MC Pur c Abstract A wo Stage Interval me Mnmzng ransortaton Problem, where total avalablty of a homogeneous roduct at varous sources s nown to le n a secfed nterval, s studed n the resent aer In the frst stage, the sources sh all of ther on-hand materal to the demand onts, whle a second-stage delvery covers the demand that s not fulflled n the frst shment In each stage, the obectve s to mnmze the shment tme, and the overall goal s to fnd a soluton that mnmzes the sum of the frst- and secondstage shment tmes A olynomal tme algorthm s roosed to solve the roblem to otmalty, where at varous stes of the algorthm lexcograhc otmal solutons of restrcted versons of a related standard tme mnmzng transortaton roblem are examned and fnally the global otmal soluton s determned Keywords: Combnatoral Otmzaton, Non-Convex Otmzaton, me Mnmzng ransortaton Problem, lobal Otmzaton Introducton Wde rangng lterature s avalable to study the cost mnmzng transortaton roblem (CMP If I = {,,, m} s the ndex set of m sources, = {,,, n} of n destnatons, c,( I the er unt shment cost from the source to the destnaton, a, I the avalablty of a homogeneous roduct at the source and b, the demand of the same at the destnaton, then the standard CMP s modeled as: mn c x subect to ( x = a, I x = b, I x I, (, ( where, x,( I denotes the uantty shed from the source to the destnaton he best-nown strongly olynomal tme algorthm for CMP s of order O ( mlog n( m + nlog n (Orln, 988 he tme mnmzng transortaton roblem (MP s another mortant class of transortaton roblems n terms of ts wdesread alcatons If t x, ( (, I, the shment tme from the source to the destnaton, s defned as: t ( x = t (, f x > t ( x =, f x = and shment from sources to destnatons s done n arallel, then the mathematcal model for the standard MP s: mn[ ( X = max( t ( x ] where X = x satsfes ( It may be ( th noted that shment tme from the source to the th destnaton does not deend uon the volume of the shment Clearly, t x s a concave functon ( (X s the shment tme for a feasble schedule X and s nown to be a concave functon (Bansal and Pur 98 Hence, standard MP s a concave mnmzaton roblem (CMP In lterature conventonal MP has been studed by varous authors (Ahua, 986, (Bhata, Kant Swaru and Pur 976, a Indan Insttute of echnology, Delh Inda (Emal: sona_t@redffmalcom b Kamala Nehru College, Delh nversty, Delh Inda c Indan Insttute of echnology, Delh Inda (Emal: mcur@mathstdernetn

2 (arfnel and Rao, 97, (Hammer, 969 and MPs wth mxed constrants and flow constrants have been studed by Khanna et al (Khanna, Bash and Pur 98, (Khanna and Pur 983 An otmal soluton of MP can be obtaned by fndng ts lexcograhc otmal soluton (LOS (Satya Praash, 98 A lexcograhc otmal soluton of MP s one n whch not only the shment on the longest duraton routes s mnmzed but shments on all other routes of varous duratons are also mnmzed where routes mean varous source-destnaton lns For obtanng an LOS, the set of transortaton tmes on varous routes s arttoned nto a number of dsont sets, B, =,,, s, where, + B = {( I : t = } and > =,, K, s Postve weghts, say λ, =,, K, s, are attached to these sets where, λ + >> λ =,, K, s hs yelds a standard CMP: s mnλ x where, X = ( x = ( B belongs to the transortaton olytoe over whch the orgnal MP s beng studed An otmal feasble soluton of ths CMP yelds a LOS of MP hus, a MP s also solvable n olynomal tme he ostve weghts λ, =,, K, s can be determned as descrbed by Sheral (Sheral 98 and Mazzola (Mazzola, 993 here are many more roblems where t becomes necessary to study a MP wheren the roducts are shed to the destnatons n two stages Consder for examle, the roducton of mantenancefree-sealed ndustral batteres Producton s a contnuous rocess deendng on the avalable resources However each battery has a certan shelf lfe and batteres need to be erodcally re-charged, else the whole lot becomes dead resultng n the loss of the fnshed goods Often due to lac of rechargng facltes on the roducton floor, each batch of manufactured batteres s transorted mmedately to the demand onts; ths corresonds to the frst stage In the second stage enough mantenancefree-sealed batteres from the sources are shed n order to satsfy the ndustral users' demands at the destnatons Shment s done n such a way as to mnmze the overall transortaton tme Such stuatons motvated the study of a two-stage nterval tme mnmzng transortaton roblem where the total avalablty of the roduct at the sources les n secfed ntervals In both the stages, transortaton of the roduct from the sources to the destnatons s done n arallel he current roblem may be vewed as a arametrc otmzaton roblem (al, 979 n whch avalablty at the source ( I vares n the nterval [ a, a' ] But as the resent roblem conssts of two stages, second stage beng deendent uon frst stage, the arametrc aroach wll not be of much advantage he current roblem, n ste of nterval sources' constrants, s very much dfferent from nterval lnear rogrammng because of the sum of two deendent concave functons n the obectve functon herefore, soluton strateges avalable for solvng nterval lnear rogrammng roblem (Charnes and ranot, 976, (Robers and Israel, 97 wll not only be comutatonally exensve but of no use as well Smlarly nexact otmzaton technues (Amaya and hellnc, 997, (Soyster, 973 wll not be of much use he wo Stage Interval MP s shown to be related to an ordnary nterval MP, whch s further shown to be euvalent to a standard MP Feasble solutons of the Stage-I and Stage-II roblems are derved from a feasble soluton of ths standard MP Due to the deendence of Stage-II on Stage-I, secal tyes of solutons vz lexcograhc otmal solutons (LOS are nvestgated Frst, Stage-II shment tme s beng controlled by solvng the varous restrcted versons of ths standard MP n whch some routes ertanng to Stage-II are abandoned In the Stage-I and Stage-II shment tme ars thus obtaned successvely, Stage-II shment tme strctly decreases and Stage-I shment tme strctly ncreases Smlarly on the other hand, Stage-I shment tme s controlled by solvng other restrcted versons of the same standard MP n whch some routes ertanng to Stage-I are abandoned In the ars, thus generated, Stage-I shment tme ees on decreasng and Stage-II shment tme ees on ncreasng In all the ars of Stage-I and Stage-II shment

3 tmes, the shment tme of one stage s the mnmum corresondng to the other As durng the algorthm a fnte number ( 4( s r of CMPs are to be solved, t follows that the roosed algorthm s also a olynomal tme algorthm, where s s the number of arttons of transortaton tmes on varous routes and r s the oston n the orderng of shment tmes on varous routes (arranged n the descendng order of the overall mnmum shment tme for the standard MP related to wo Stage Interval MP hs mnmum shment tme s yelded by an LOS of ths standard MP heoretcal develoment of the roblem s resented n the next secton followed by the develoment of the algorthm Concludng remars are gven towards the end after a numercal llustraton heoretcal Develoment Mathematcal Formulaton of wo Stage Interval MP Let a and a', I denote resectvely the mnmum and maxmum avalablty of a homogeneous roduct at the source and b, the demand of the same at the destnaton, where a < b < a' I I In the frst stage of the wo Stage Interval MP the uantty a ( < a' s shed from each source I and after ts comleton, enough uantty of the roduct s dsatched n the second stage so as to exactly satsfy the demand b at the destnaton, he Stage-I roblem s thus formulated as: mn [ max( t ( y ] = mn[ ( Y ] Y = ( y S ' Y S ' where, the set S ' s gven by y = a, I S' : y b, I y, ( I Corresondng to a feasble soluton Y = y of the Stage-I roblem, let ( S '( Y be the set of feasble solutons of the Stage-II roblem whch s stated as: mn [ max( t ( z ] = mn [ ( Z ] Z S '( Y Z S '( Y where, z a' a, I S' ( Y : z = b b', I z, ( I and b' = y, I hus, the wo Stage Interval me Mnmzng ransortaton Problem can be stated as mn ( Y + mn ( Z (P Y S ' Z S '( Y As shment tmes n Stage-I and Stage-II are concave functons, the wo Stage Interval MP ams at mnmzng a concave functon over a olytoe Hence (P s also a concave mnmzaton roblem As global mnmzer of a CMP over a olytoe s attanable at an extreme ont of the olytoe, t s desrable to nvestgate only ts extreme onts Closely related to the roblem (P s the nterval tme mnmzng transortaton roblem P defned as mn X S ( α [ ] [ ( X ] mn max( t ( x = P X S ( α where, a x a', I S : x = b, I x, ( I Clearly a feasble soluton of the roblem (P rovdes a feasble soluton to the roblem ( P α and conversely he standard tme mnmzng transortaton roblem ( P assocated wth ( P α s defned as mn X Sˆ where, [ ] [ ˆ( X ] mn max( tˆ ( x = P X = ( x Sˆ Iˆ ˆ (

4 x = aˆ, Iˆ ˆ Sˆ : x = bˆ, ˆ Iˆ x, ( Iˆ ˆ where, Iˆ = {,, Km, m +, K,m} ˆ = { n + } aˆ aˆ bˆ bˆ tˆ tˆ tˆ tˆ n+ = a, =,, Km m+ = a' a, =,, Km = b, b = t, =,, Km, m+ n+ = m+ n+ I = t, =,, Km, a' = M, =,, Km, M =, =,, Km >> An LOS of the roblem ( P wll rovde the overall mnmum shment tme for ( P r r Let mn [ ˆ( X ] = ( = X Sˆ A feasble soluton X = ( x of the roblem ( P s sad to be an M-feasble soluton (MFS f x = ( : t M = It may be noted that f X = ( x Iˆ ˆ s an M- feasble soluton of roblem P, then Y = ( ( y and Z = ( z I are resectvely feasble solutons of Stage-I and Stage-II roblems where, y = x ( I and z = xm+ ( I Further f mn [ ( Z] = ( Zˆ, then a feasble Z S '( Y soluton of the roblem (P conssts of the feasble soluton Y for Stage-I and the feasble soluton Ẑ for Stage-II he Next two theorems establsh the euvalence between the roblem P ( α and the roblem ( P over the set of ts M- feasble solutons heorem An M-feasble soluton of the roblem ( P corresonds to a feasble soluton of the roblem ( P α and vce versa Proof Let Y = ( y be an M-feasble soluton of the roblem ( P For each I, defne x = y + ym+ It can be easly establshed that X = ( x s a feasble soluton of the roblem ( P α Conversely, let X = ( x,( I be a feasble soluton of the roblem ( P α Let x = a, I Clearly a a a' herefore, Clearly, I I x x x a = a, I = b,, ( I = b Consder the followng mbalanced MP: subect to mn ( Z = mn max( I z z z = a, I [ t ( z ] b,, ( I Let Z = ( z be an otmal feasble soluton of ths roblem and let z = b, Now, set I y y n = z,( I + =, I ( Next, consder the balanced MP defned as follows: mn ( W = mn max( [ t ( w ]

5 subect to, I w w = a a, I = b b, Soluton Strategy for the wo Stage Interval MP he soluton strategy for the wo Stage Interval MP deends uon the followng tyes of standard MPs and CMPs w, ( I he ossblty of Stage-II shment tme If W = ( w s an otmal soluton of the less than the on hand shment tme, say above roblem, then set l, s examned by studyng the tme ym+ = w,( I mnmzng transortaton roblem l ( PL ym+ n+ = a' a, I ( derved from the roblem ( P by abandonng the routes ( m + for It can be easly seen that l Y = ( y,( Iˆ ˆ whch t as defned by ( m +, ( I e, and ( above, s an M-feasble soluton of settng tm + = M ( >>,( I for the roblem ( P whch +, l l t m If LOS of PL ( s Hence the result an MFS, then the new Stage-I shment tme s more than the on hand Stage-I shment tme and the corresondng Stage-II shment tme s less than the on hand Stage-II shment tme heorem he value of the obectve functon of the roblem ( P at an M- feasble soluton s same as the value of obectve functon of the roblem ( P α at l Suose LOS of PL the corresondng feasble soluton ( s M-feasble and let Stage-I and Stage-II shment tmes l Proof Let Y = ( y be an M-feasble corresondng to ths LOS be and resectvely o fnd the least ossble soluton of the roblem ( P wth the value Stage-II shment tme corresondng to the of ts obectve functon as Let Stage-I shment tme, the followng cost mnmzng transortaton roblem (call X = ( x be the corresondng feasble l t CPL (, s solved soluton of the roblem ( P α gvng α as l mn the value of ts obectve functon hen, cx CP Sˆ L (, I = max( tˆ ( y where, Iˆ ˆ c = M, ( I, where t > max( t ( y, =, ( I, where t ( t n y n max, + (, +, I l = max cm+ = M, ( I, where tm+ > max( tm+ ( ym+, l = λl, ( I, where tm+ = max( tm+ n+ ( ym+ n+ I = λl+, ( I, where tm+ = = max ( t y ( tm ym M M max (,max +, ( +, s = λs, ( I, where tm+ = = max{ t ( x } snce x = y + ym+ and As an M-feasble LOS of the roblem tm+ = t,( I l PL ( s a feasble soluton of = l α CPL (, and as b > a, t I l follows that otmal value n CPL (, wll be non-zero Corresondng to Stage-I l+

6 shment tme the otmal feasble l soluton of CPL (, rovdes the least Stage-II shment tme whch s less l than or eual to hus, OBFS of l CPL (, rovdes a feasble soluton of the roblem (P It may be observed that roblems l l PL ( and CPL (, for varous values of l hel n generatng the ars of the tye (( (, (: ( > ( for the Stage-I and Stage-II shment tmes Smlarly to generate the ars of the tye (( (, (: ( < (, MPs of the tye P ( and CMPs of the tye l CP (, are studed, where P ( s the MP derved from the roblem P by abandonng the routes (, I for whch t, l (, I and CP (, s defned as: l mn cx CP (, Sˆ I where, c = M, m+ c =, = M, = λ, = λ M + = λ, s ( I, where t ( I, where t ( I, where t ( I, where t,( I, where t M ( I, where t m+ m+ Procedure for wo Stage Interval MP A LOS, say X, of the roblem > > = = = + s P s obtaned Let ( r I l ( X = max tˆ ( x = r hs corresonds to ether Stage-I shment tme or Stage-II shment tme Wthout loss of generalty let t corresond to Stage-I shment tme Let Stage-II shment tme corresondng to ths LOS be r where s non-negatve nteger o fnd the mnmum Stage-II l shment tme corresondng to the Stage-I shment tme r, solve the cost mnmzng o transortaton roblem CP (, L, where = Its OBFS wll yeld the mnmum Stage-II shment tme, say r + r, corresondng to tme of Stage-I shment, where ( s a non-negatve nteger he frst recorded ar thus obtaned s (, r + yeldng r r + value + of the obectve functon of the wo Stage Interval MP (P Suose the ars thus recorded so far for Stage-I and Stage-II shment tmes be (, : > =,, K, For further generaton of such ars solve the restrcted verson P L ( of the roblem ( P wheren the routes ( m +,( I for r + whch t are abandoned If otmal feasble soluton of P ( L s an MFS, then the corresondng Stage-I shment r tme, say + r, wll be more than and Stage-II shment tme, say +, r wll be smaller than + o fnd the mnmum Stage-II shment tme corresondng to the Stage-I shment tme r +, solve ( + CP, + L Its OBFS wll yeld the ar r + r + + (,, + + for Stage-I and Stage-II shment tmes It s clamed that n these recorded ars Stage-I shment tme s also the mnmum corresondng to Stage-II shment tme If, however, LOS of P ( L s not an M- feasble soluton, then t follows that Stage-II shment tme cannot be further reduced r below + and the current best value of the obectve functon of the wo Stage Interval MP (P s r r + mn [ + ] It wll be =,, K, establshed that f the LOS, say X +, of the roblem P ( L s not an M- feasble soluton, then

7 ( X + ( + mn[ + X + =,,, whch n turn would mean that there can not exst any other ar ( (, (: ( ( ( yeldng value of the sum of Stage-I and Stage-II shment tmes less than r r + mn [ + ] It may be observed =,,, that for these recorded ars > and <, =,, K, Next, f ossble, the ars ( (, (: ( ( ( < n whch Stage-I shment tme s less than the Stage-II shment tme are generated Frst, the restrcted verson, call t P (, of the roblem ( P s constructed by abandonng the routes r + (, I for whch t e, settng t = M ( I, for whch r + t If LOS of P ( s not an M-feasble soluton, then t s clamed that there does not exst a ar (( (, (: ( < ( such that the corresondng value of the obectve functon of the wo Stage Interval MP (P s better than mn ( + =,,, r X + On the other hand, f LOS, say, of the roblem P ( s an M-feasble soluton, then the corresondng Stage-I r shment tme, say +, would be less r than + Let Stage-II shment tme at ths M-feasble LOS of the roblem P r ( + r be o obtan the mnmum Stage-I shment tme r corresondng to the tme of Stage-II shment, the cost mnmzng transortaton roblem ( CP, s solved Its OBFS yelds the mnmum Stage-I shment r tme, say +, corresondng to the r Stage-II shment tme Suose the r + r r + r ars (, : <,,, K, = have been generated so far Exstence of next such ar s examned by ] solvng the restrcted verson P ( derved from P by settng t = M >>,( I t ( for whch If LOS, say P ( note r + + X, of s an M-feasble soluton, then r + + ( + r + X r + + ( + r X and o fnd the mnmum Stage-I shment tme corresondng to the Stage-II shment tme +, the cost mnmzng transortaton roblem + ( + CP, s solved Its OBFS yelds the mnmum Stage-I shment tme (call t + corresondng to the Stage II shment tme, + hus, OBFS of + ( + CP, rovdes the ar + + (, + X of P r ( + On the other hand, f the LOS s not an MFS, then Stage-I shment tme can not be reduced below and t s establshed that ( r ( mn [ X + X + ] =,, K, whch n turn means that no more ars (( (, (: ( < ( can be obtaned yeldng value of the sum of Stage- I and Stage-II shment tmes less than r + r mn [ + ] It s clamed that n =,, K, r the ars (, : <,,, K, = thus generated, Stage-II shment tme s also the mnmum corresondng to Stage-I shment tme Hence these ars also corresond to feasble solutons of the wo Stage Interval me Mnmzng ransortaton Problem hus, the global mnmum value of the obectve functon of the wo Stage Interval MP s mn mn + + ( + ( + r r r,mn o gve the above stated rocedure a sound mathematcal foundaton, the varous clams are establshed n the followng theorems

8 heorem 3 In a ar (, corresondng to the OBFS of the roblem CPL (,, s the mnmum Stage-I shment tme corresondng to r tme of Stage-II shment, where r + r + and are the Stage-I and Stage-II shment tmes resectvely corresondng to the M-feasble LOS of P ( L Proof o rove that the Stage-I shment r tme s the mnmum corresondng r to the Stage-II shment tme +, r assume the contrary Suose that s not the mnmum shment tme for Stage-I r corresondng to tme + of Stage-II shment hs mles that there exsts a soluton, say Xˆ, of the roblem ( P such that r r ( Xˆ ˆ r = < and Xˆ + ( = r Clearly Xˆ r ( < as ( = s the mnmum shment tme for Stage-I yelded by the LOS of the roblem ( P M- r feasble LOS of P ( + yelds the L r Stage-I shment tme, whch s also the overall shment tme for ths MP r Also by defnton of P ( + t follows that Xˆ s ts feasble soluton By assumton Xˆ yelds Stage-I shment r tme ˆ r ( < and Stage-II shment r tme + hus Xˆ yelds overall shment tme for the tme mnmzng r transortaton roblem P ( + L smaller than the one yelded by ts LOS, r whch cannot be true Hence s the mnmum Stage-I shment tme corresondng to the Stage-II shment tme r + heorem 4 In a ar (, corresondng to the OBFS of the roblem r + r r CP (,, s the mnmum Stage-II shment tme corresondng to tme of Stage-I shment, where r + and are the Stage-I and L Stage-II shment tmes corresondng to the M-feasble LOS of P ( Proof he roof s smlar to the roof of the theorem 3 heorem 5 If LOS, say X, of the tme mnmzng transortaton roblem r + P L ( s not an MFS, then ( X r ( + mn { + X + =,,, where, each of roblems P L ( =,,, has M-feasble LOS Proof As LOS of P ( L s not an M- feasble soluton; the Stage-II shment tme r can not be further reduced below + Currently best value of the sum of the shment tmes n Stage-I and Stage-II r s [ ] r + mn + =,,, At non M-feasble LOS of P ( L ether ( the Stage-I shment tme s one of the frst ( + recorded tmes:,,, K,, n whch case the corresondng mnmum Stage-II shment tme s already nown or ( the Stage-I shment tme s none of the frst ( + recorded tmes but t les n the nterval [, ], n whch case one of the recorded Stage-I and Stage-II shment tmes would yeld a smaller value of the sum of the shment tmes or ( the r Stage-I shment tme s more than, r n whch case [ ] r + + mn wll be =,,, smaller than the sum of the Stage-I and Stage-II shment tmes at the current non M-feasble LOS of P ( L Hence the sum of the Stage-I and Stage-II shment tmes corresondng to a non M-feasble LOS of the roblem P ( L s not less than the current best value of the sum hat s, ( X ( X mn =,,, } [ + ]

9 Remar If LOS of the roblem P L ( s not an MFS, then no further restrcted verson of the roblem P, namely ( P L ( can rovde a soluton of the roblem (P yeldng value better r r + mn + than ( =,,, Remar Let ars n hand of Stage-I and r r + Stage-II shment tmes be (,, =,,, Let LOS of the roblem r + P L ( be an MFS hen, Stage-I shment tme corresondng to ths M- r feasble LOS s more than snce for a gven Stage-I shment tme less than or r eual to the Stage-II shment tme r can not be less than + (Recall that M- r feasble soluton LOS of P L ( + yelds r Stage-II shment tme less than + heorem 6 If LOS of the roblem r P (, say X +, s not an MFS, then ( X + ( X mn{ + } r X + Proof As s not an MFS, we have r + x > for some (, I for whch r + t, e, ( r X + Also we have ( r + r r X = (snce = herefore, ( X + ( X Hence the result mn{ + + Remar If LOS of the roblem P ( s not an MFS, then the restrcted versons P ( of the roblem P can not rovde an otmal soluton of the roblem (P hs also mles that there does not exst a feasble soluton of the roblem (P havng Stage-I tme less r than + } + + r heorem 7 If LOS, sayx, of P ( ( X s not an MFS, then r ( mn { X + =,, K, P ( where, each of the roblem =,, K has an M-feasble LOS Proof he roof s smlar to the roof of the theorem 5 Remar If LOS of the roblem P ( s not an MFS, then no further restrcted r + verson of ( P, namely P (, can rovde a soluton of the roblem (P yeldng value better than r + r mn + =,, K, ( Remar If LOS of the roblem ( s an MFS for all =,, K,, then as r + r X ( + < r + r X ( > P +, t follows that + snce f ( X + then corresondng mnmum Stage-I tme wll be greater than or eual to he next theorem roves that the roosed soluton methodology ndeed obtans the global otmal soluton of the wo Stage Interval MP (P heorem 8 If the generated ars of Stage-I and Stage-II shment tmes are r r + (,, and (,, then, the otmal value of the obectve functon of the roblem (P s mn mn + + ( + ( + r r r,mn Proof If the theorem s not to be true, then there must exst a feasble soluton, say X, of the roblem (P such that the corresondng Stage-I and Stage-II shment tmes (call them ( X and ( X resectvely are such that }

10 + ( X mn < mn mn ( +, ( + ( X As X s a feasble soluton of the roblem (P, t follows that ( X s the mnmum Stage-II shment tme corresondng to the Stage-I shment tme ( X herefore, X s an M-feasble soluton of the roblem P ( he above neualty mles that ( X + ( X < + Wthout loss of generalty, assume that ( X ( X As ( = r s the otmal transortaton tme for the tme mnmzng transortaton roblem ( P, t follows that ( X and hence ( X < herefore, there exsts an ndex, say d (a ostve nteger not less than, such that d d < ( X < < hs mles that X s an M-feasble r soluton of the roblem P ( + d L r M-feasble LOS of P ( + d L yelds r d Stage-I shment tme By hyothess, d d ( X + ( X < + r + d d As ( X >, we have ( X < hs mles that X s a soluton better r than the M-feasble LOS of P ( + d L, whch s not true Hence there does not exst any feasble soluton of (P yeldng sum of Stage-I and Stage-II shment tmes less than mn mn + + ( + ( + r r r,mn he formal algorthm for the wo Stage Interval MP s gven below Algorthm Ste Obtan an LOS of the roblem ( P Note the corresondng Stage-I tme as r = r and Stage-II tme as + Solve cost mnmzng transortaton o roblem CP (, L to fnd the mnmum Stage-II shment tme, say r + r, of Stage-II corresondng to the tme of Stage-I shment Record ths ar as (, r + If r = s or =, then sto and go to ste 3 Else, go to ste Ste ( Construct the roblem r + P ( L and fnd ts LOS If t s not an MFS, then go to ste 3 Else, solve the cost mnmzng transortaton roblem CP (, L to fnd the mnmum r Stage-II shment tme + r corresondng to the tme of Stage-I shment Record the ar (, s If = or =, then sto and go to ste 3 Else, execute ste for next hgher value of Ste 3 Construct the roblem P r ( + and obtan ts LOS If t s not an M-feasble soluton, then go to ste 5 Else, note the Stage-I shment tme as r + r and Stage-II tme as o fnd r the mnmum Stage-I shment tme, +, r corresondng to tme of Stage-II shment, solve the cost mnmzng transortaton roblem ( CP, Record the ar (, s If = or r =, then sto and go to ste 5 Else, go to ste 4 Ste 4 ( Construct the roblem P ( and fnd ts LOS If t s not an M-feasble soluton, then go to ste 5 Else, r note the Stage-I shment tme as + r and Stage-II shment tme as

11 S Solve cost mnmzng transortaton roblem CP (, to fnd the mnmum Stage-I shment tme corresondng to Stage-II shment tme, r Record the ar (, If, s = or =, then sto and go to ste 5 Else, reeat ths ste for next hgher value of Ste 5 Fnd + + ( + ( + r r r,mn mn mn hs wll be the otmal value of the obectve functon of the roblem (P Numercal Illustraton Consder the 3 6 wo Stage Interval MP gven below n able D D D 3 D D 4 5 D6 6 t = a a' S S b able where S s the th source, =,, 3 and th D s the destnaton, =,, K, 6 he artton of transortaton tmes on varous routes s: ( = 59 > 5 ( = 6 > ( = 48 > 6 ( = 3 > 3 ( = 4 > 7 ( = > 4 ( = 38 > 8 ( = 9 s = 8 = 9 and therefore, s = 8 he corresondng ( P roblem s gven below n able D D D 3 D D 4 5 D 6 D 7 a S M 6 S M 5 S M 3 S S S b able LOS of the roblem ( P yelds the Stage-I shment tme as 6 and Stage-II shment tme as 38 herefore, r = 4 = 38 and hence r = 4 o obtan the mnmum Stage-I shment tme corresondng to Stage-II shment tme 38, solve the cost mnmzng transortaton roblem CP (6,38 Its otmal soluton yelds the same ar (6,38 Hence the frst recorded ar of the Stage-I and Stage-II shment tmes s (6,38 o obtan a new ar, the tme mnmzng transortaton roblem P (6 s solved Its LOS s M-feasble and yelds the ar (3,4 of the Stage-I and Stage-II shment tmes he otmal soluton of the cost mnmzng transortaton roblem CP (3,4 gves bac the ar (3,4 Hence the second recorded ar of the Stage-I and Stage-II shment tmes s (3,4 Next the tme mnmzng transortaton roblem P (3 s solved Its LOS s not an M-feasble soluton Hence the tme mnmzng transortaton roblem P L (6 s constructed whose LOS s M-feasble and yelds the ar (38,3 hen, we solve the cost mnmzng transortaton roblem CP L (38,3 to obtan the mnmum Stage-II shment tme corresondng to shment tme 38 of Stage-I Otmal soluton of CP L (38,3 yelds the Stage-II tme as

12 S S S 3 Hence the thrd recorded ar of the Stage-I and Stage-II shment tmes s (38, Next, LOS of P L ( yelds the ar (4,9 he cost mnmzng transortaton roblem CP L (4,9 gves bac the ar (4,9 Snce Stage-II tme has reached s = 9 we sto here he fourth recorded ar of the Stage-I and Stage-II shment tmes s (4,9 Now, mn{6+38, 3+4, 38+, 4+9}=58 Hence for the otmal soluton of the roblem (P the Stage-I shment tme s 38 and Stage-II shment tme s yeldng the sum of Stage-I and Stage-II shments as 58 Stage-I and Stage-II shment schedules for ths otmal soluton are gven n able 3 D D D 3 D D 4 5 D6 D M M M a 6 5 hence no comaratve study could be carred out As the roblem under study s a non-convex otmzaton roblem, some sort of enumeraton of feasble solutons has to be resorted to o mae enumeraton a vable oton very udcous enumeraton s roosed b In the develoed algorthm not more than ( 4( s r number of CMPs are solved to generate the dfferent ars of Stage-I and Stage-II shment tmes It s nown that a CMP s solvable n olynomal runnng tme O ( mlog n( m + nlog n (Orln, 988 Hence the roosed algorthm s also a olynomal tme algorthm c he algorthm roosed for wo Stage Interval MP has been coded n C++ and verfed successfully wth the hel of a lot of test roblems of varous szes Recordngs of some of these examles are lsted n able 4 S 4 S 5 S b able 3 Note that only 7 CMPs ( < 4( s r = 4 are to be solved Concludng Remars a wo Stage Interval MP has been ntroduced for the frst tme and as such we are not aware of any soluton strategy for the same and

13 Sze of Longest Shortest No of No of No of Otmal Otmal the Duraton Duraton arttons MPs CMPs ars ars(s value roblem me me solved solved obtaned (4, (3, (7, (7, (8, (3,6, (5, (, (4, (7, (9, (6,6, (9, (8, (4, (9, (5,6 8 4 (4, (3,,(, (6, (4, (6,5, (7,4, (4, (9, (6, (,7 9 able-4 Acnowledgement We are ndebted to Prof S N Kabad nversty of New Brunswc-Fredercton, Canada, wth whom we had very useful dscussons We are also thanful to Mr Avneet Bansal who greatly heled us n mlementaton of the roosed algorthm References [] R K Ahua, Algorthms for the Mnmax ransortaton Problem, Naval Research Logstcs Quarterly, vol 33, , (986 [] Amaya and De hellnc, Dualty for Inexact Lnear Programmng Problems, Otmzaton, vol 39, 37-5, 997 [3] S Bansal and M C Pur A Mn-max Problem, ZOR, vol 4, 9-, 98 [4] H L Bhata, Kant Swaru and M C Pur me-cost rade-off n a ransortaton Problem, Osearch, vol 3 (3-4, 9-4, 976 [5] A Charnes and F A ranot, Modfed Algorthm for Solvng Interval Lnear Programmng Problems, Caher, vol 8(3, , 976 [6] al, Post Otmal Analyss, Parametrc Programmng and Related ocs, Mcraw-Hll, New Yor, 979 [7] R S arfnel and M R Rao, Bottlenec ransortaton Problem, Naval Research Logstcs Quarterly, vol 8, , 97

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