Two Uncertain Programming Models for Inverse Minimum Spanning Tree Problem

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1 Idustral Egeerg & Maageet Systes Vol, No, March 3, pp.9-5 ISSN EISSN KIIE Two Ucerta Prograg Models for Iverse Mu Spag Tree Proble Xag Zhag, Qa Wag, Ja Zhou School of Maageet, Shagha Uversty, Shagha, Cha (Receved: August 5, / Revsed: Deceber 3, / Accepted: March 5, 3 ABSTRACT A verse u spag tree proble akes the least odfcato o the edge weghts such that a predetered spag tree s a u spag tree wth respect to the ew edge weghts. I ths paper, the cocept of ucerta α-u spag tree s tated for u spag tree proble wth ucerta edge weghts. Usg dfferet decso crtera, two ucerta prograg odels are preseted to forulate a specfc verse u spag tree proble wth ucerta edge weghts volvg a su-type odel ad a a-type odel. By eas of the operatoal law of depedet ucerta varables, the two ucerta prograg odels are trasfored to ther equvalet deterstc odels whch ca be solved by classc optzato ethods. Fally, soe uercal eaples o a traffc etwork recostructo proble are put forward to llustrate the effectveess of the proposed odels. Keywords: Mu Spag Tree, Ucerta Mu Spag Tree, Iverse Optzato, Ucerta Prograg Correspodg Author, E-al: zhou_a@shu.edu.c. INTRODUCTION The verse optzato proble s a subect etesvely studed the cotet of toographc studes, sesc wave propagato, ad a wde rage of statstcal ferece wth pror probles. The verse u spag tree (IMST proble s a type of verse optzato probles. I a IMST proble, a coected graph wth edge weghts s cosdered. The obectve of IMST proble s to odfy the weghts so that a predetered spag tree s a u spag tree wth respect to the ew weghts, ad sultaeously the total odfcato of weghts s a u. The IMST proble was frst studed by Zhag et al. (996. Followg that, uch research work has bee doe othe IMST proble sce ay applcatos ca be trasfored to ths proble (Farago et al., 3; Gua ad Zhag, 7; Wag et al., 6; Yag ad Zhag, 7. Ad ay effcet algorths have bee developed for solvg the classc IMST probles ad ther dervatves (Ahua ad Orl, ; He et al., 5; Zhag et al., 6. I vew of the odeteracy of soe paraeters applcatos, soe edeavor was doe to deal wth IMST probles wth deterate forato the lteratures. For eaple, Zhag ad Zhou (6 cosdered the IMST proble whe the edge weghts were assued to be stochastc varables, ad stochastc prograg odels together wth hybrd tellget algorths were preseted for IMST probles. I practce, however, t s ot approprate to set the edge weghts as rado ubers soe cases due to a lack of observed data (Peg ad L, ; Zhou ad Peg,. Hece, we adopt the ucertaty theory, a brach of aoatc atheatcs for odelg hua ucertaty fouded by Lu (7, to hadle ths proble. I ths paper, a specfc IMST proble s dscussed uder the assupto of ucerta edge weghts. Ths paper proposes a ew cocept of ucerta α-u

2 Zhag, Wag, ad Zhou: Idustral Egeerg & Maageet Systes Vol, No, March 3, pp.9-5, 3 KIIE spag tree ad develops two ucerta prograg odels to forulate ths proble accordg to dfferet decso crtera. I ths paper, the IMST proble wth ucerta edge weghts s referred to as a ucerta verse u spag tree (UIMST proble for coveece. The rest of ths paper s orgazed as follows. Secto brefly revews the prelary cocepts of ucertaty theory. Secto 3 troduces the classc IMST proble ad the atheatcal descrpto of UIMST proble, ad the proposes a cocept of ucerta α- u spag tree. I Secto 4, two ucerta prograg odels are gve based o dfferet decso obectves. Followg that, Secto 5 presets the uercal eaples ters of the two ucerta odels. Fally, coclusos are draw Secto 6.. PRELIMINARIES Ucertaty theory provdes a ew approach to deal wth deteracy factors whe there s a lack of observed data (Lu, 7,. Nowadays, ucertaty theory has becoe a brach of aoatc atheatcs for odelg hua ucertaty, wdely appled ay research areas (Che, ; L ad Peg, ; Sheg ad Yao, ; Xu ad Zhu,. Ths secto s teded to revew soe basc cocepts ucertaty theory whch wll be used to establsh ucerta prograg odels for the UIMST proble. where Λ k are arbtrarly chose evets fro L k for k =,, L, respectvely. A ucerta varable ξ s essetally a easurable fucto fro a ucertaty space to the set of real ubers. I order to descrbe a ucerta varable practce, Lu (7 defed a cocept of ucertaty dstrbuto as follows. Defto (Lu, 7. Let ξ be a ucerta varable. The, ts ucertaty dstrbuto s defed by ( M{ ξ } Φ = ( for ay real uber. Furtherore, Peg ad Iwaura ( showed that a fucto M: R [, ] s a ucertaty dstrbuto f ad oly f t s a ootoe creasg fucto ecept Φ( ad Φ(. For stace, a ucerta varable ξ s called lear f t has a lear ucertaty dstrbuto (Fgure,, f a Φ ( = ( a/( b a f a b, f b deoted by ξ : Lab (,, where a ad b are real ubers wth a < b. Defto (Lu, 7. Let L be a σ -algebra o a oepty set Γ. A set fucto M: L [, ] s called a ucerta easure f t satsfes the followg aos: Ao (Noralty Ao. M{ L } = for the uversal set Γ ; c Ao (Dualty Ao. M{ Λ } + M{ Λ }= for ay evet Λ ; Ao 3 (Subaddtvty Ao. For every coutable sequece of evets Λ, Λ, L, we have M { U Λ} M{ Λ}. = The trplet ( Γ, L, M s called a ucertaty space. Besdes, the product ucerta easure o the product σ -algebra was defed by Lu (9 va the followg product ao: Ao 4 (Product Ao. Let ( Γ k, Lk, M k be ucertaty spaces for k =,, L. The product ucerta easure M s a ucerta easure satsfyg M = Λ = M Λ { } k k k k =, Fgure. Lear ucertaty dstrbuto. A ucertaty dstrbuto Φ s sad to be regular f ts verse fucto Φ ( α ests ad s uque for each α (,. It s clear that a lear ucertaty dstrbuto La (, b s regular, ad ts verse ucertaty dstrbuto s Φ ( α = ( α a + ab. ( The verse ucertaty dstrbuto plays a portat role the operatos of depedet ucerta varables. Defto 3 (Lu, 9. The ucerta varables ξ, ξ L, ξ are sad to be depedet f,

3 Two Ucerta Prograg Models for Iverse Mu Spag Tree Proble Vol, No, March 3, pp.9-5, 3 KIIE for ay Borel sets M B M B = = I ( ξ = { ξ } (3 B, B, L, B of real ubers. Theore (Lu,. Let ξ, ξ, L, ξ be depedet ucerta varables wth regular ucertaty dstrbutos Φ, Φ, L, Φ, respectvely. If the fucto f (,,, L s strctly creasg wth respect to,, L, k ad strctly decreasg wth respect to k+, k+, L,, the ξ = f ( ξ, ξ, L, ξ s a ucerta varable wth verse ucertaty dstrbuto Ψ ( α = f ( Φ ( α, L, Φ ( α, Φ ( α, L, Φ ( α. k k+ 3. UNCERTAIN INVERSE MINIMUM SPANNING TREE PROBLEM I ths secto, a classc cocept of u spag tree as well as a path optalty codto s revewed brefly, ad the a UIMST proble s talzed by troducg ts applcato backgrouds ad atheatcal descrpto. Fally, a ew cocept of ucerta α -u spag tree s preseted. 3. Classc IMST Proble Defto 4 (Mu Spag Tree. Gve a coected graph G = ( V, E wth edge weghts, E{,, L, }, a spag tree T s sad to be a u spag tree f (5 T T holds for ay spag tree T. I a classc IMST proble, a predetered spag tree T s gve. The obectve of IMST proble s to fd soe ew edge weghts such that T s a u spag tree wth respect to the ew edge weghts ad accordgly the odfcato of edge weghts s a u. Fgure. A eaple of verse u spag tree proble. I order to provde the atheatcal descrpto of IMST proble, soe otos are proposed as follows. Frstly, we refer to the edges the gve spag tree T as tree edges, ad the edges ot T as o-tree edges. Hece the set of all the o-tree edges s E \ T. I the spag tree T, there s a uque path betwee the two vertces of ay o-tree edge, referred to as tree path of edge ad deoted by P. A eaple of classc IMST proble wth 6 vertces ad edges s show Fgure, where c ad deote the orgal ad ew weghts o edge, ad the sold le represets a gve spag tree T. The set of o-tree edges s E \ T = { 6, 7,8,9, }, ad the tree path of o-tree edge BD s AB-AE-DE,.e., P 9 = {, 3, 5 }. Moreover, Ahua et al. (993 proved a equvalet codto of u spag tree, called a path optalty codto as follows. Theore (Ahua et al., 993. T s a u spag tree wth respect to the edge weghts f ad oly f E T P (6, \, where tree path of edge. Accordg to Theore, the classc IMST proble ca be forulated as the followg odel, E \ T s the set of o-tree edges, ad -c = subect to : E T P, \, P s the (7 where c ad are the orgal ad ew weghts of edge, E, respectvely. Note that the obectve fucto c = ca be replaced wth soe other obectve fuctos f ecessary (Sokkalga et al., Applcato Backgrouds May recostructo probles practce ca be trasfored to ucerta probles. Let us cosder a LAN recostructo proble as follows. Much research work shows that the spag tree structure s the best topology for telecoucato etwork desgs (Kershebau, 993, especally coputer etwork systes. LANs are cooly used as a coucato frastructure that eets the deads of users a local evroet. These coputer etworks typcally cosst of several LAN segets coected va brdges. Suppose that there s a old LAN, whch several servce ceters are tercoected va brdges. Because of the treedous etwork cogesto, the badwdths o brdges ust be odfed. The decso-aker hopes that a predetered spag tree becoes a u

4 Zhag, Wag, ad Zhou: Idustral Egeerg & Maageet Systes Vol, No, March 3, pp.9-5, 3 KIIE spag tree wth respect to the travelg te (whch eas hgh et-speed betwee the a servce ceters. Also the total badwdth odfcato should be zed so as to dsh the cost of recostructo. Sce the travelg tes as well as the et speeds are related to badwdths, t s atural to descrbe the travelg te o a brdge as a ucerta uber stead of a deterstc oe wth respect to badwdths of brdges whe there are o forer statstcal data. Ths s a typcal verse spag tree proble wth ucerta weghts,.e., a UIST proble. 3.3 Notatos ad Proble Descrpto I ths paper, a specfc IMST proble wth ucerta edge weghts s vestgated. I order to provde a atheatcal descrpto for ths proble, the followg otatos are used: G = ( V, E : a coected graph wth set of vertces V ad edge set E = {,, L, } ; T : a predetered spag tree of G; c : the orgal edge weghts, E; : decso varables represetg the ew edge weghts, E; ξ ( : the ucerta edge weghts wth respect to, E. For our purpose, we assue that c, E, whch s practcal ay stuatos. For stace, a traffc syste recostructo proble, the roads are ofte requred to be broadeed stead of beg arrowed order for accoodatg the creasg traffc flow. Hece the obectve of UIMST proble here s to fd a ew edge weght vector to ze the odfcato (, c = ad sultaeously T s a u spag tree wth respect to the ucerta edge weghts ξ (, E. 3.4 Ucerta α-u Spag Tree I a UIMST proble, Defto 4 becoes powerless due to the ucertaty of edge weghts ξ. Therefore, before odelg the UIMST proble, a u spag tree wth respect to ucerta weghts ust be defed frst. I ths secto, by usg the ucertaty easure (see Secto, a ew cocept of ucerta α - u spag tree s recoeded as follows. Defto 5 (Ucerta α -Mu Spag Tree. Gve a coected graph G = ( V, E wth ucerta edge weghts ξ, E, ad a gve cofdece level α, a spag tree T s sad to be a ucerta α -u spag tree f M ξ ξ holds for ay spag tree T. (8 T T Defto 5 ples that a ucerta α -u spag tree has a chace ot less tha α of ot havg a ucerta weght larger tha every other spag tree, whch s tutvely reasoable. As troduced Secto 3., Theore s a ecessary ad suffcet codto of u spag tree, whch provdes a useful approach for odelg a IMST proble. I the UIMST proble, a slar result ca be obtaed for ucerta α -u spag tree by adoptg oly a ty chage as follows. Theore 3. T s a ucerta α -u spag tree wth respect to the ucerta edge weghts f ad oly f { ξ ξ } α M ( (, E \ T, P, (9 where E \ T s the set of o-tree edges, ad tree path of edge. P s the Proof. It follows drectly fro Theore ad Defto UNCERTAIN PROGRAMMING MODELS Based o the cocept of ucerta α -u spag tree ad Theore 3, two ucerta prograg odels are bult up for the UIMST proble ths secto cludg a ucerta su-type odel ad a ucerta a-type odel. Furtherore, the operatoal law of depedet ucerta varables,.e., Theore Secto, s used to derve two equvalet deterstc odels. 4. Ucerta Su-Type Model Let us cosder a traffc etwork recostructo proble, where soe roads should be broadeed for soe reasos. The decso-aker hopes that the predetered spag tree becoes a ucerta α -u spag tree wth respect to ucerta travelg tes betwee soe a traffc hubs, where α s provded as a approprate safety arg by the decso-aker. Ad the total odfcato of road wdths s also requred to be zed whch eas decreasg the cost of recostructo. I ths case, a so-called ucerta sutype odel ca be obtaed fro Theore 3 as follows, ( c = subect to : M { ξ ξ } α E T P c, =,, L,. ( (, \, ( where α s a predetered cofdece level. Ths odel eas to ake the least odfcato o the deterstc edge weghts, E, such that a gve spa-

5 Two Ucerta Prograg Models for Iverse Mu Spag Tree Proble Vol, No, March 3, pp.9-5, 3 KIIE 3 g tree becoes a ucerta α -u spag tree wth respect to the ucerta edge weghts ξ, E. By the use of Theore, t s easy to covert odel ( to the followg crsp equvalet odel, ( c = subect to : Φ (, α Φ (, α, E\ T, P c, =,, L, ( where Φ represet the verse dstrbutos of ucerta weghts ξ, =,, L,, whch are related to the ew edge weghts. 4. Ucerta Ma-Type Model Soetes, the decso-aker hopes to ze the aal oe aog all the odfcatos so as to keep a balace of cost durg the syste recostructo. I order to eet ths obectve, a ucerta atype odel ca be establshed as follows, a ( c = subect to : M { ξ ξ } α E T P c, =,, L,. ( (, \, ( The obectve fucto of odel (3 s to ze the largest odfcato all the edges provded that a gve spag tree T becoes a ucerta α - u spag tree regardg the ucerta edge weghts. Utl ow, two ucerta odels together wth ther equvalet deterstc odels are preseted for the UIMST proble accordg to dfferet decso obectves. We ca see that odels ( ad (3 have o dfferece wth classcal atheatcal prograg odels whe the verse ucertaty dstrbutos are kow. Thus, we ay solve the by classcal optzato ethods or tellget algorths. 5. COMPUTATIONAL EXAMPLES I order to llustrate the effectveess of the above two ucerta prograg odels, ths secto, a traffc etwork recostructo proble wth 6 traffc hubs ad roads s cosdered (Fgure 3, wth the sold le represetg the predetered spag tree T. There are three weghts o each road, where c ad are the orgal ad ew wdths of road, respectvely, ad ξ deote the ucerta travelg tes o road, whch are assued to be lear ucerta varables oly wth respect to,.e., ξ = ξ( = L(, + a (Table. Two ucerta odels are gve to forulate ths proble ad the solved by MATLAB (MathWorks, Natck, MA, USA. Slarly, usg the verse ucertaty dstrbutos Φ of edge weghts, E, a equvalet deterstc odel ca be obtaed as follows, a ( c = subect to : Φ (, α Φ (, α, E\ T, P c, =,, L,. (3 Fgure 3. Ucerta verse u spag tree proble for coputatoal eaples. Table. Edge weghts for UIMST proble Fgure 3 Edge Orgal weght Paraeter weght Ucerta weght c a ξ = L(, + a 5 (-, (-, (- 3, (- 4, (- 5, (- 6, (- 7, (- 8, (- 9, (-, 3- UMIST: ucerta verse u spag tree.

6 Zhag, Wag, ad Zhou: Idustral Egeerg & Maageet Systes Vol, No, March 3, pp.9-5, 3 KIIE 4 Eaple. Frstly, whe the su-type odel ( s used, the obectve s to ze the total odfcato of road wdths wth a gve cofdece level α =.8 so as to ze the total cost of recostructo, the the followg ucerta prograg odel s obtaed, ( c = subect to : Φ (,.8 Φ (,., E\ T, P c, =,, L, (4 where the o-tree edge set E \ T = { 6,7,8,9, }, P s the tree path of o-tree edge, ad Φ are verse ucertaty dstrbutos of ξ, =,, L,. Sce Φ (, = + Φ = + accord-.8.8 a, (,..a g to (, t follows that Φ (,.8 Φ (,. = + +.8a. a. As a result, a equvalet forulato of odel (4 ca be gaed as follows, ( c = subect to : + + a a E T P c, =,, L,.8., \, (5 whch s a lear prograg odel. Hece, we solve t by MATLAB 7. ad get the optal soluto = (, 68, 7, 9, 5, 6, 6, 4, 7, 3 ad the u total odfcato o road wdths s. Eaple. Secodly, by the use of the a-type IMST odel (3, we have the correspodg forulato for ths proble lke: a ( c subect to : + + a a E T P c, =,, L,..8., \, (6 Slarly, wth the help of MATLAB 7., odel (6 s solved easly wth the fal soluto = (, 68, 7,, 57, 6, 6, 4, 7, 65 as well as the optal obectve value 4. By akg a further vestgato o the eperetal results, we have ( c ( c a = a = 4, ( c ( c =, = 38, = = whch ply that both ad are optal solutos of odel (6, whle oly s the optal soluto of odel (5. We ca coclude fro the coputatoal results that odel (6 has ore tha oe optal soluto cludg that of odel (5. 6. CONCLUSION I ths paper, the cocept of ucerta α -u spag tree s tated for ucerta u spag tree based o the ucertaty easure. After that, two ucerta prograg odels are preseted to forulate the verse u spag tree proble wth ucerta edge weghts ad the trasfored to crsp equvalet odels. Fally, the uercal eaples o a traffc syste recostructo proble are put forward to llustrate the effectveess of odels proposed. The results of the coputatoal eaples led us to coclude that the ucerta a-type odel has ore tha oe optal soluto, ad the optal soluto of ucerta su-type odel s also a optal soluto of ucerta a-type odel. The ultple optal solutos Eaple dcate that there are ore tha oe pla for the decso-aker to choose for the recostructo of the traffc etwork. As a copleetary of ths paper, a further study o the aalyss of the solutos s eeded. Soe effcet algorths to deal wth the UIST proble should be also volved the future work. ACKNOWLEDGMENTS Ths work was supported part by a grat fro the Mstry of Educato Fuded Proect for Huates ad Socal Sceces Research (No. JDXF5, ad the Shagha Phlosophy ad Socal Scece Plag Proect (No. BGL6 ad Iovato Progra of Shagha Mucpal Educato Cosso (No. 3ZS65. REFERENCES Ahua, R. K., Magat, T. L., ad Orl, J. B. (993, Network Flows: Theory, Algorths, ad Applcatos, Pretce Hall, Eglewood Clffs, NJ. Ahua, R. K. ad Orl, J. B. (, A faster algorth for the verse spag tree proble, Joural of Algorths, 34(, Che, X. (, Aerca opto prcg forula for ucerta facal arket, Iteratoal Joural of

7 Two Ucerta Prograg Models for Iverse Mu Spag Tree Proble Vol, No, March 3, pp.9-5, 3 KIIE 5 Operatos Research, 8(, 7-3. Farago, A., Szetes, A., ad Szvatovszk, B. (3, Iverse optzato hgh-speed etworks, Dscrete Appled Matheatcs, 9(, Gua, X. ad Zhag, J. (7, Iverse costraed bottleeck probles uder weghted l or, Coputers ad Operatos Research, 34(, He, Y., Zhag, B., ad Yao, E. (5, Weghted verse u spag tree probles uder Hag dstace, Joural of Cobatoral Optzato, 9(, 9-. Kershebau, A. (993, Telecoucato Network Desg Algorths, McGraw-Hll, New York, NY. L, S. ad Peg, J. (, A ew approach to rsk coparso va ucerta easure, Idustral Egeerg & Maageet Systes, (, Lu, B. (7, Ucertaty Theory (d ed., Sprger- Verlag, Berl. Lu, B. (9, Soe research probles ucertaty theory, Joural of Ucerta Systes, 3(, 3-. Lu, B. (, Ucertaty Theory: A Brach of Matheatcs for Modelg Hua Ucertaty, Sprger- Verlag, Berl. Peg, J. ad L, S. (, Spag tree proble of ucerta etwork, Proceedgs of the 3rd Iteratoal Coferece o Coputer Desg ad Applcatos, X a, Shaa, Cha. Peg, Z. ad Iwaura, K. (, A suffcet ad ecessary codto of ucertaty dstrbuto, Joural of Iterdscplary Matheatcs, 3(3, Sheg, Y. ad Yao K. (, Fed charge trasportato proble ad ts ucerta prograg odel, Idustral Egeerg ad Maageet Systes, (, Sokkalga, P. T., Ahua, R. K., ad Orl, J. B. (999, Solvg verse spag tree probles through etwork flow techques, Operatos Research, 47(, Wag, Q., Yag, X., ad Zhag, J. (6, A class of verse doat probles uder weghted l or ad a proved coplety boud for Radzk s algorth, Joural of Global Optzato, 34(4, Xu, X. ad Zhu, Y. (, Ucerta bag-bag cotrol for cotuous te odel, Cyberetcs ad Systes, 43(6, Yag, X. ad Zhag, J. (7, Soe verse -a etwork probles uder weghted l ad l ors wth boud costrats o chages, Joural of Cobatoral Optzato, 3(, Zhag, B., Zhag, J., ad He, Y. (6, Costraed verse u spag tree probles uder the bottleeck-type Hag dstace, Joural of Global Optzato, 34(3, Zhag, J., Lu. Z., ad Ma, Z. (996, O the verse proble of u spag tree wth partto costrats, Matheatcal Methods of Operatos Research, 44(, Zhag, J. ad Zhou, J. (6, Models ad hybrd algorths for verse u spag tree proble wth stochastc edge weghts, World Joural of Modellg ad Sulato, (5, Zhou, C. ad Peg, J. (, Models ad algorth of au flow proble ucerta etwork, Proceedgs of the 3rd Iteratoal Coferece o- Artfcal Itellgece ad Coputatoal Itellgece, Tayua, Sha, Cha, -9.