A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS

Transcription

1 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS Kumar Pardeep and Chaturved D.K. Department of Instrumentaton, Kurukshetra Unversty, Kurukshetra, , Inda e-mal: pardeep_kuk@redffmal.com, dhra_chaturved@yahoo.co.n Correspondng author Pahua G.L. Department of Electrcal Engneerng, Natonal Insttute of Technology, Kurukshetra , Inda e-mal: pahua_gl@yahoo.co.n ABSTRACT In flow networks, from the qualty and servce management pont of vew, measurement of the transmsson ablty of a network to meet the customers demand s very mportant. To meet the ultra hgh relable requrements of such networks, a heurstc method for relablty redundancy optmzaton of flow networks usng composte performance measure (CPM) ntegratng relablty and capacty has been proposed. The method s based upon the selecton of man flow paths and backup paths and then optmzng man paths on prorty bass. Thus, the reduced computaton work makes the proposed algorthm sutable for desgnng of large, relable telecommuncatons networks. Keywords: flow networks; capacty related relablty; constraned redundancy optmzaton; heurstc algorthm. 1 INTRODUCTION Constraned relablty redundancy optmzaton of networks has generally been studed wth relablty as connectvty measure. The practcal systems such as computer networks, telecommuncaton networks, transportaton systems, electrcal power transmsson networks, nternet etc. can only transport lmted amount of flow therefore these are termed as flow lmted networks. To meet the ultra hgh relablty requrements of such networks, a heurstc method for relablty redundancy optmzaton of flow networks has been proposed. Lterature s enrched wth relablty redundancy optmzaton of networks wth connectvty only as a measure of performance (Sharma & Venkateswaran 1971, Aggarwal 1976, Golden & Magnant 1977, Gopal et al. 1978, Lee 1980, Bodn et al. 1982, Xue 1985, Dnghua 1987, Fredman & Taran 1987, Km & Yum 1993, Shen 1995, Martns & Santos 1997, Schrver 1998, Ahua 1998, Kuo & Prasad 2000, Kuo et al. 2001, Park et al. 2004, Pascoal et al. 2005, Kumar et al. 2009, 2010a, b, 2011). However, relablty redundancy optmzaton of flow networks has rarely been studed. In present days context exstng methods do not fulfl the requrements of management of qualty of servce. The relablty redundancy optmzaton technques dscussed n Gopal et al. (1978), Dnghua (1987), Km & Yum (1993), Shen (1995), Park et al. (2004), Kumar et al. (2009, 2010a, b, 2011) are not suted to flow networks. Ths paper presents a technque for relablty redundancy optmzaton of flow networks usng combned performance measure named capacty related relablty (CRR). 69

2 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS A path s a sequence of arcs and nodes connectng a source to a snk. All the arcs and nodes of network have ts own attrbutes lke delay, relablty and capacty etc.. From the qualty and servce management pont of vew, measurement of the transmsson ablty of a network to meet the customers demand s very mportant (Ln 2006). When a gven amount of flow s requred to be transmtted through a flow network, t s desrable to optmze the network relablty to carry the desred flow. In such cases, the system relablty s the measure of qualty of the system capablty to transmt desred flow. The capacty of each arc (the maxmum flow passng the arc per unt tme) has two levels, 0 and/or a postve nteger. The system relablty s the probablty that the maxmum flow through the network between the source and the snk s not less than the demand (Golden & Magnant 1977, Lee 1980, Bodn et al. 1982, Fredman & Taran 1987, Ahua 1998, Ln 2003, 2004, Pahua 2004, Ln 2006, 2007a, b). For determnng the relablty t s generally assumed that network s capable of transmttng any requred amount of flow between source (s) and termnal (t) nodes of the network. Ths presumpton s nether vald nor ustfable for real lfe systems as lnks and nodes can carry only lmted amount of flow. In 1980 Lee dd the poneer work of ntegrated both capacty and relablty & named t combned performance measure as capacty related relablty (CRR) and also termed such networks as flow networks. Max-Flow- Mn-Cut theorem has been used to determne the capacty of the network (Sharma & Venkateswaran 1971, Xue 1985, Shen 1995, Martns & Santos 1997, Schrver 1998, Ln 2003, 2004, Park et al. 2004, Ln 2006, 2007a, b). 2 COMPOSITE PERFORMANCE MEASURE Relablty under flow constrant s a more realstc performance measure for flow networks. The concept of weghted relablty ntroduced by Aggarwal (1976) requres that all the successful states qualfyng connectvty measure of the network be enumerated. The probablty of each success state s evaluated and s multpled by the normalzed weght (Aggarwal 1985). Rushd (1988) evaluated the same performance ndex as evaluated by Aggarwal 1985 usng decomposton approach. Methods gven by Aggarwal (1985), Rushd (1988), and Shakt (1995) generate both cancellng (faled) and non-cancellng (success) terms. The followng secton presents a heurstc algorthm for relablty redundancy optmzaton of flow networks usng composte performance measure (CPM) and the method has been utlzed to determne the capacty related relablty performance ndex. 2.1 Notaton a l (X) Senstvty factor of l th mnmal path set b (x ) Subsystem selecton factor for th subsystem wth x components C Total amount of resource avalable g c ( x ) / ( x ) Amount of resources- consumed n subsystem- wth x components / Cost of subsystem for th constrant. h(.) Functon yeldng system relablty; dependent on number of subsystems (n) and confguraton of subsystems k Number of constrants, = 1, 2,, k L (x) ( L, L,..., L ), Lower lmt of each subsystem, x1 x2 xn m Number of man mnmal path sets, l = 1, 2,., m n Number of subsystems, = 1, 2,, n P l th mnmal path set of the system l Ps (l1,l 2,,l mn ): prorty vector s.t. l 1 and l mn are the number of mnmal path sets arranged 70

3 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS n decreasng order of path selecton parameter a l (X). Q ( ) Unrelablty of subsystem wth x components. x r Relablty of a component at subsystem. ( x R ) Relablty of subsystem wth x components. R r Resdual resources [total resource avalable (C ) - resources consumed ( g x )] R s (X ) System relablty S ( x) Set of varables that have been used as key-elements n a gven decomposed expressons U (x) ( U, U,..., U ), Upper lmt of each of subsystem, x1 x2 xn x Optmal soluton x Number of components n subsystem ; = 1,2,.n X A vector (x 1, x n ) Increment n th stage relablty when a unt s added n parallel to the th stage R 2.2 Assumptons Followng are the assumptons for the rest of the sectons: 1. The system and all ts subsystems are coherent. 2. Subsystem structures (other than coherence) are not restrcted. 3. The networks are modelled wth the help of graphs, the paths (ordered par of arcs and the members of the ordered par are relablty and capacty respectvely) where n are assgned as the weght of each lnk. 4. Each lnk can have only two stages up and down. 5. The network nodes are perfect. If the nodes are not perfect, the method needs to be modfed to deal wth nodes falures. 6. All component states are mutually and statstcally ndependent. 7. All constrants are separable and addtve among components. 8. Each constrant s an ncreasng functon of x for each subsystem. 9. Redundant components cannot cross subsystem boundares. 2.3 Composte Performance Measure (CPM) The weghted relablty measure.e. composte performance measure (CPM), ntegratng both capacty and relablty may be stated as by [27, 28]: CPM t R (1) S( x) Where ωt s the normalzed weght and s defned as: t = Cap / Cap max.e. the rato of capacty n the th state to the maxmum capacty (Cap max ) of the system and R probablty of the system beng n state S and s computed as: R P r S p k / 1 k / S 0 S k q (2) 71

4 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS 2.4 Capacty Functons of Networks The capacty functon of dfferent arcs connected n parallel s [25]: C X Par Cap x and the capacty functon of dfferent arcs connected n seres s: C X mn (4) Ser Cap The rules for connectng seres and parallel arcs to ntegrate capacty and relablty to gve composte performance measure are expressed as: n (5) CR X mn Cap r CR Ser X Par x 1 n n Cap. r 1 1 CPM for seres and parallel networks can be defned as: CPM Par = CR(X) Par / Cap max (7) and CPM Ser = CR(X) Ser / Cap max (8) 3 PROBLEM FORMULATION AND HEURISTIC METHOD 3.1 Problem Formulaton The general constraned redundancy optmzaton problem n complex systems can be reduced to the followng nteger programmng problem (Kuo et al. 2001): Maxmse Rs ( X ) h( R1 ( x1),..., Rn ( xn )), (9) n (10) subect to g ( x ) C, = 1, 2,, k and 1 U x 3.2 Proposed Heurstc Method 1 x, = 1, 2,, n. In real lfe systems all the arcs are not smultaneously connected to carry flow from source to snk. Hence a flow path set s the arcs and nodes that actually carry traffc. In these practcal systems all the path sets are not utlzed for transfer of nformaton (Hayash & Abe 2008). The flow s transmtted through the man path(s) and n case of falure of ths path(s), a backup path completes the task of man path. The backup path(s) come n operaton only when the man paths fal thus, enhancng the relablty of the network, t s presumed that the man path(s) and backup path(s) for the network are known. The proposed algorthm frst optmzes the man path(s) and then back up path(s) usng redundancy optmzaton technque. Ths leads to more effcent use of resources whch are generally lmted. Unlke exstng heurstcs, a stoppng crteron has been appled to swtch from relablty redundancy optmzaton of man path(s) to relablty redundancy optmzaton of backup paths. The algorthm consders senstvty factor as the crtera for selectng the man path from the lst of gven man paths and then a subsystem for applyng redundancy wthn the chosen flow path. 3.3 Steps of the Proposed Method Step1: Frstly select the man path sets m and back up flow paths of the flow Network. (3) (6) 72

5 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS Step2: Let x = 1 for all ; = 1, 2,, n. Step3: Calculate a l (X ) a l ( ) = max [ (X ) X a l ] and, l = 1, 2,, m for each mnmal path set and fnd l such that a l ( X l 1,2,3...m. ) P l k 1 P ( g l R ( x ( x ) ) / k C ), Step4: For the chosen mnmal path set l fnd such that ( x ) b = max [ b x ) ( ] and b ( x ) R P, k ( g ( x ) / k C ) for each l 1 2 where R 1 Q ( x ) R ( x ) Step5: Check, by addng one redundant subsystem to unsaturated subsystem : ) f no constrants are volated, add one redundant subsystem to unsaturated subsystem by replacng x wth x 1, and go to step 3. ) f at least one constrant s exactly satsfed and others are not volated, then add one redundant subsystem to unsaturated subsystem by replacng x wth x 1. The x = X s the optmal soluton. Go to step 6. ) f at least one constrant s volated, then remove mnmal path set l from further consderaton and consder the next path havng maxmum a l ( ) value and go to step 4. v) f all mnmal path sets are now excluded from further consderaton, then x = X s the optmal soluton; else go to step 3 Step6: In case any resources are stll avalable optmze the backup paths as dscussed n Step2. Step4: Evaluate the composte performance measure (CPM) for each subsystem of the network. Step5: Evaluate the system relablty usng the CPM of the each subsystem. 4 COMPUTATION AND RESULTS To llustrate the performance of the proposed algorthm a network havng sx arcs {x 1, x 2, x 3, x 4, x 5, x 6 } and fve mnmal path sets {y 1, y 2, y 3, y 4, y 5 } as shown n the Fgure 1 s consdered and solved for capacty related redundancy relablty optmzaton usng CPM (7 and 8). System relablty s determned usng Bayes method. The network shown n Fgure 1 s a bench mark problem, consdered by Hayash & Abe (2008). X Usng Baye s method, the Relablty of the above system can be expressed as: 73

6 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS R s (X) = R 3 [1- Q 6 {1-(1-Q 1 Q 2 )(1-Q 4 Q 5 )}] + Q 3 [1-(1-R 2 R 5 )(1-R 1 R 4 )]Q6 (11) y 5 y 2 y 1 x 2 y 4 x 1 x 3 x 5 x 6 x 4 y 3 Fgure 1 Illustraton Network The smple mnmal path sets of the Network are P1 = {1, 3, 5}, P2 = {2, 3, 4}, P3 = {1, 4}, P4 = {6}, P5 = {2, 5} The problem s solved for data gven n Table 1. Usng ths ntal data the general problem of constraned relablty redundancy allocaton has been solved usng the steps dscussed n Secton 3.3 above. The problem s solved by consderng the flow path sets P1, P2, P3 and P4 as man path sets and P5 as backup path. The proposed algorthm gves the optmal soluton (3, 1, 1, 2, 1, 4) wth system relablty R s = , the optmzed subsystem relablty probablty R and unrelablty probabltes Q are shown n Table 2. Table 1 Data for Fg r g 1 / c C 1 30 Table 2 Optmzed subsystem relablty/unrelablty for Fg. 1 x 1 x 2 x 3 x 4 x 5 x 6 X R Q The capacty of each subsystem of the flow path s taken as 100 and the capacty of flow paths of the network s determned usng proposed approach as: 74

7 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS P1 =mn {3100, 100, 100} =100 P2 =mn {100, 100, 100} =100 P3 =mn {3100, 2100} =200 P4 =mn {4100} =400 (12) Next the composte performance measure CPM expresson (13) s derved usng (7 and 8) and the value for CPM for an assumed flow of 200 s suppose to pass through the flow path and t comes out to be CPM P1 mn Cap Cap max [ R1R3 R5 ] = (100/200) x x 0.8 x 0.7 = CPM P2 mn Cap [ R2 R3R4 ] Cap max = (100/200) x 0.75 x 0.8 x = mn Cap CPM P3 [ R1R 4 ] Cap max = (200/200) x x = CPM P4 mn Cap [ R6 ] Cap max = (400/200) x = (2) x as 0 (mn Cap / Cap max ) 1 so = 1x = (13) (14) (15) (16) Composte performance measure ntegratng the relablty wth capacty s calculated as: CPM Network = 1- (1-CPM P1 ) (1-CPM P2 ) (1-CPM P3 ) (1-CPM P4 ) = 1- ( ) x ( ) x ( ) x ( ) = The above result shows that proposed method s capable of optmzng the flow network to transport the desred capacty through the network wth hghest relablty. However, the selecton of man paths and backup paths wll affect the qualty of composte performance measure. Hence the proper choce of these paths may be done usng cardnalty crtera (Kumar et al. 2010b) or any other herarchcal measures of mportance. 5 CONCLUSIONS Ths paper has presented a new model for desgnng relable flow networks capable of transmttng requred flow. The proposed algorthm utlzes the concept of man and backup flow paths. The choce of backup and flow paths s applcaton specfc and paths wth mnmum cardnalty may be selected as man path and dsont paths can be the backup paths. The numercal example demonstrates that the proposed algorthm s fast for desgnng large, relable telecommuncatons networks because the task of optmzaton s reduced, as only few paths are selected as man paths. 75

8 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS REFERENCES Sharma J. and Venkateswaran K. V. 1971, A drect method for maxmzng the system relablty, IEEE Trans. Relablty, vol. R-20, no. 1, pp Aggarwal K. K. 1976, Redundancy optmzaton n general system IEEE Trans. Relablty, vol R-25, pp Golden B. L. and Magnant T. L. 1977, Determnstc network optmzaton: A bblography, Networks, vol. 7, no. 2, pp Gopal K., Aggarwal K. K., and Gupta J. S. 1978, An mproved algorthm for relablty optmzaton, IEEE Trans. Relablty, vol. R-27, no. 5, pp Lee S. H. 1980, Relablty Evaluaton of Flow Networks, IEEE Tr. Rel. Vol. R-29, pp Bodn L. D., Golden B. L., Assad A. A., and Ball M. O. 1982, Routng and schedulng of vehcles and crews: the state of the art, Computers and Operatons Research, vol. 10, no. 2, pp Aggarwal K. K. 1985, Integraton of Relablty and Capacty n Performance Measure of Telecommuncaton Network, IEEE Trans. Relablty, vol. R-34, pp Xue J. 1985, On multstate system analyss, IEEE Trans. Relablty, vol. 34, no. 4, pp Dnghua S. 1987, A new heurstc algorthm for constraned redundancy-optmzaton n complex systems, IEEE Trans. Relablty, vol. R-36, no.5, pp Fredman M. L. and Taran R. E. 1987, Fbonacc heaps and ther uses n mproved network optmzaton algorthms, J. ACM, vol. 34, no. 3, pp Rushd M. 1988, Performance Indexes of Telecommuncaton Networks, IEEE Trans. Relablty, vol. R-37, pp Ra S., and Soh S. 1991, A computer Approach for Relablty Evaluaton of Telecommuncaton networks wth heterogeneous Lnk capactes, IEEE trans. On Relablty, Vol. 40, pp Km J. H. and Yum B. J. 1993, A heurstc method for solvng redundancy optmzaton problems n complex systems, IEEE Trans. Relablty, vol. 42, no. 4, pp Varshney, Josh, and Chang A. R. 1994, Relablty modelng and performance evaluaton of varable lnk capacty networks, IEEE Trans. Relablty, vol. 43(3), pp Shen Y. 1995, A new smple algorthm for enumeratng all mnmal paths and cuts of a graph, Mcroelectroncs and Relablty, vol. 35, no. 6, pp Shakt K. 1995, Some Aspects of Relablty Related Performance Evaluatons of Computer Communcaton Networks, Ph.D. Thess, Kurukshetra Unversty, Kurukshetra. Martns E. D. Q. V. and Santos J. L. E. D. 1997, An algorthm for the quckest path problem, Operatons Research Letters, vol. 20, no. 4, pp Schrver 1998, Theory of Lnear and Integer Programmng. NY: John Wley & sons, pp Ahua R. K. 1998, Mnmum cost-relablty rato problem, Computers and Operatons Research, vol. 16, no. 1, pp Kuo W. and Prasad V. R. 2000, An Annotated Overvew of System-Relablty Optmzaton, IEEE Trans. Relablty, vol. 49(2), pp Kuo, W. Prasad, R., Tllman, F.A. and Hwang, C.L Optmal Relablty Desgn Fundamentals and Applcatons, Cambrdge Unversty Press. Ln Y. K. 2003, Extend the quckest path problem to the system relablty evaluaton for a stochastc-flow network, Computers and Operatons Research, vol. 30, no. 4, pp Park C. K., Lee S., and Park S. 2004, A label-settng algorthm for fndng a quckest path, Computers and Operatons Research, vol. 31, no. 14, pp Ln Y. K. 2004, Relablty of a stochastc-flow network wth unrelable branches & nodes 76

9 Kumar Pardeep and Chaturved D.K., Pahua G.L. - A HEURISTIC METHOD FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS under budget constrants, IEEE Trans. Relablty, vol. 53, no. 3, pp Pahua G. L. 2004, Relablty Evaluaton and Optmzaton Recent and New Approaches, Ph.D. thess, Kurukshetra Unversty, Kurukshetra. Pascoal M. M. B., Captvo M. E. V., and Clímaco J. C. N. 2005, An algorthm for rankng quckest smple paths, Computers and Operatons Research, vol. 32, no. 3, pp Ln Y. K. 2006, Relablty of a computer network n case capacty weght varyng wth arcs, nodes and types of commodty, Relablty Engneerng and System Safety, vol. 2(5), pp Ln Y. K. 2007, a. Relablty evaluaton for an nformaton network wth node falure under cost constrant, IEEE Trans. Systems, Man and Cybernetcs-Part A: Systems and Humans, vol. 37, no. 2, pp b. On a mult-commodty stochastc-flow network wth unrelable nodes subect to budget constrant, European Journal of Operatonal Research, vol. 176, no. 1, pp Hayash M. and Abe T. 2008, Evaluatng Relablty of Telecommuncatons Networks Usng Traffc Path Informaton, IEEE Trans. Relablty, vol. 57(2), pp , pp Kumar P., Chaturved D. K. and Pahua G. L. 2009, Heurstc algorthm for constraned redundancy relablty optmzaton and performance evaluaton. Proc. IMechE, Part O: J. Rsk and Relablty, vol. 223 (4), Kumar P., Chaturved D. K. and Pahua G. L. 2010, a. Heurstc Methods for Solvng Redundancy Allocaton n Complex Systems, Int. J. of Relablty and Safety, vol. 4, No. 2/3, pp b. An effcent heurstc algorthm for determnng optmal redundancy allocaton of complex networks, Relablty Theory and Applcatons, vol. 3(18), pp Kumar P., Chaturved D. K. and Pahua G. L. 2011, Constraned Relablty Redundancy Optmzaton of Complex Systems usng Genetc Algorthm, MIT Int. J of Electrcal and Instrumentaton Engneerng, vol. 1, No. 1, pp