Multicommodity Distribution System Design

Transcription

1 Mummodt strbun stem esgn Consder the probem where ommodtes are produed and shpped from pants through potenta dstrbun enters C usmers. Pants C Cosmers

2 Mummodt strbun stem esgn Consder the probem where ommodtes are produed and shpped from pants through potenta dstrbun enters C usmers. The probem s determned quantt of ommodt shpped from pant usmer through C. whh C s used serve usmer { f C s servng usmer otherwse f C s operatng { f C s operatng otherwse

3 v f or or ubet mn Probem formuan

4 . one s served through eat Eah usmer C v f or or ubet mn Probem formuan

5 C pant at ommodt of the supp shoud not eeed through a usmers a pant shpped from ommodt quantt of Tota pant at ommodt of supp v f or or ubet mn Probem formuan

6 . a for that foows t Furthermore. through pants shpped from a quantt the ta must be equa ommodt for ustumer of Hene the demand. other a for and t foows that. other for a and then s served through uppose that usmer C C v f or or ubet mn Probem formuan

7 Probem formuan mn ubet or or f v For a usmer served through C Hene gong through C f ndates that the throughput must be smaer than or equa than an upper bound f C C s operatng then s not operatng then s equa the ta throughput. fores the throughput be equa. equa a ower bound. and rean and arger than or and. rean

8 Probem formuan mn ubet or or f v s ommodt from pant usmer. the unt ost of shppng ost. produng and shppng through C s the ta produn and

9 v f or or ubet mn Probem formuan operatng. s the the ost f s. the fed ost for operatng s C f C f

10 v f or or ubet mn Probem formuan. the throughput at the ost for handng the ta s. at the unt ost for handng throughput C v C v

11 Probem formuan mn ubet or or f v enote X YZ Proen on the spae of the annong varabes mn f v mn YZ X Consder the varabes and as the annong varabes. Mae a proen on the spae of these varabes

12 Proen on the spae of the annong varabes Consder the nteror mnman probem n X YZ v f mn mn ubet mn ubet mn

13 ubet mn µ π Mutpers spef the dua ubet ma µ π µ π µ

14 mn ubet Assumng that.e. ta supp ta demand for eah ommodt The one C n the Godman resoun of the feasbe doman of the dua redues the orgn ma ubet µ µ π µ π

15 ma ubet µ µ π µ π enote the set of etreme ponts of whh s a poedra onve set : { 2 2 H µ π µ π µ H π }. the feasbe doman of the dua

16 mn YZ f v mn X ma ubet µ µ π µ π enote the set of etreme ponts of whh s a poedra onve set : 2 2 H { µ π µ π µ H π }. the feasbe doman of the dua ne the one C n the Godman resoun of the feasbe doman redues the orgn then the equvaent probem taes the form mn ubet f v µ h YZ π h h H

17 A sequene of reaed probems of the equvaent probem are soved. Consder a reaed probem of the equvaent probem spefed wth a subset of the etreme ponts of the feasbe doman of the dua R { 2 H} mn ubet f v µ h YZ π h h R enote b an optma soun of ths reaed probem.

18 Mehansm verf optmat ubet mn probem mnman nteror the Consder probem beomes mnman nteror the Then. a for that foows t Furthermore. other a for and t foows that. usmer the servng we now the probem. the equvaent an reaed probem of of For a soun C ubet mn

19 ubet mn Ths probem s separabe n transportan probems assoated wth the dfferent ommodtes. ubet mn the transportan probem s as foows For ommodt enote b the optma soun of the transportan probem assoated wth the ommodt.

20 f then the optma soun of the reaed probem s aso feasbe and hene optma for the equvaent probem and the proedure sps. Otherwse usng the optma souns we an determne a orrespondng etreme pont µ π of doman of the dua probem generate a µ π be added the probem n order generate a new reaed probem wth an addna onstrant. of the transportan probems new onstrant the feasbe

21 ovng the reaan appromate n ther paper Geoffron and Graves uses a soun strateg where the reaon of the equvaent probem s not soved optmat. The rather sp the resoun whenever the vaue of the obetve funn reahes some spefed bound B. Ths strateg an be ustfed as foows. Frst note that the feasbe doman of the reaan appromates and nudes that of the equvaent probem. As the number of terans nreases the approman beomes better. But durng the frst terans the approman s not ose enough n genera ustf the omputana effort of sovng the reaan optmat.

22 As the number of terans nreases the approman beomes better. But durng the frst terans the approman s not ose enough n genera ustf the omputana effort sove the reaan optmat. Hene t seems more effent redue the omputana effort of sovng the reaan n order use ths effort for ompetng a arger number of terans generatng n dong so a arger number of onstrants more rapd. Thus the approman mproves more rapd. To mpement ther strateg the authors termnates the resoun of the reaan whenever the obetve funn reahes the bound B UB ρ. Of ourse ρ > and t must be adusted redued as the number of terans nreases.

23 optma vaue of reaan vaue of soun UB téran

24 rawbas: uboptma vaue not neessar non dereasng uboptma vaue not neessar ower bound ρ ρ ρ ρ 2 ρ optma vaue of reaan vaue of soun UB suboptma vaue optma vaue téran

25 Referene A.M.GeoffronG.W.Graves "Mummodt strbun stem esgn b Benders eomposn" Management ene