Choosing an Optimal Set of Libraries

Transcription

1 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 2, 1996 JUNE Choosig a Optimal Set of Libraries Oded Berma Uiversity of Toroto, Toroto Michal Cutler, Member IEEE State Uiversity of New York, Bighamto Keywords - Software reliability, Software library, Software cost, Optimizatio, Heuristics, Brach & boud Abstract - This paper presets optimizatio models for selectig a subset of software libraries, viz, collectios of programs, residig o floppy disks or compact disks, available o the market. Each library cotais a variety of programs whose reliabilities are assumed to be kow. The objective is to maximize the reliability of the computer system subject to a budget costrait o the total cost of the libraries selected. The paper icludes six models, each of which applies to a differet software structure ad assumptios. A detailed brach & boud algorithm for solvig oe of the six models is described; it cotais a simple greedy-procedure for geeratig a iitial solutio. For solvig the rest of the models, see Berma & Cutler (1995). 1. INTRODUCTION The relatioship betwee reliability & cost is importat for software developmet The reliability of the software is very ofte a measure of its quality [6] ad the cost of software programs must be cosidered whe the user has limited budget for ivestig i software. Ref 121 preseted two optimizatio models for selectig a set of available commercial programs to maximize the system reliability, subject to a budget costrait. I model 1, redudacy of various versios of the programs is ot allowed, whereas i model 2 such redudacy is permitted. Ref 131 exteded the work i 121 by cosiderig software systems where each program cosists of sequece of modules which, upo executio, perform a fuctio. Ref 141 cosiders the tradeoff betwee reliability & cost whe usig N-versio programmig i the framework of 12, 31. This paper cosiders the tradeoff betwee reliability & cost whe the problem is to choose a set of software libraries from may that are available i the market. Each oe of the available libraries cotais a variety of programs. The user has a list of programs that are required for the computer system, ad wishes to esure that each oe of the programs required is icluded i at least oe of the libraries chose so as to maximize the system reliability, subject to a budget costrait. This paper presets optimizatio models to derive the optimal selectio of commercial libraries available i the market. Sectio 2 itroduces the models. Sectio 3 describes the solutio method. Sectio 4 discusses limitatios of the models., N [umber, set] of programs required by the user N = {1,2,..., ) m umber of commercially available libraries cosidered Ll,..., L, m partitios (librairies) of L; U FZl Li = N Ci cost of library Li Ri,j, Rj reliability of program j i EL,, the software system] - implies the complemet; eg, R= 1 -R 4(.) idicator fuctio: S(True) = 1, S(Fa1se) =O K S(1ibrary i is selected) Xij S(1ibrary i AND program j are selected) 4 frequecy of use of program j B budget available sk set of programs that are required to be used i a sequece Fk frequecy of use of sk V umber of Sk Other, stadard otatio is give i Iformatio for Readers & Authors at the rear of each issue. 2.1 Geeral Models 2. THEMODELS Three models are developed separately for each of 2 cases (total of 6 models). The 2 cases are: Case 1. No redudacy. The reliability of a particular program is the largest amog selected libraries that cotai it. Case 2. Redudacy. All versios of the selected programs cotribute to the reliability. 4 The 3 models (pertaiig to 3 system structures) are: a. Each program is stad-aloe. b. The set of required programs is used to costruct 1 complex user program (system). A11 programs i the user s list are executed i sequece (geeratig 1 stad-aloe system). c. The set of required programs is used to costruct several complex user programs (systems): SI,..., S,. These complex programs (systems) are executed i sequece. 4 Thus there are 6 situatios (Cases 1 & 2, ad Models a,b,c): ClMa, ClMb, ClMc; C2Ma, C2Mb, C2Mc. Assumptios 1. Each library has a kow cost. 2. Each program cotaied i each oe of the libraries has a kow reliability /96/$ IEEE

2 304 leee TRANSAaIONS ON RELIABILITY, VOL. 45, NO. 2, 1996 JUNE 3. The collectios of library programs are s-idepedet. 4. Failure of differet versios of the same program are kl = X,,J.R,,J. s-idepedet. { E I E Lr} 5. Usage frequecy for each program is kow, viz, provided by the user. 2.3 Case 2: Redudacy 6. Usage frequecy for each required sk is kow, viz, provided by the user. The ureliability of program j E N is: 7. Each program has 2 states (it either performs well, or - fails). R. I = IT Ri,j, (9) 8. Complex systems ca be geerated directly by combi- {i: Y,=l, j E Lr} ig existig software. 4 Cosider the 3 models (idexed as i sectio 2.1) for this 2.2 Case 1: No Redudacy case 2: Oly 1 program, j, is used by the system - the oe with maximum reliability. The reliability of program j E N is: C2Ma, C2Mb, C2Mc. The set of costraits (2) - (5) is modified as show i (10) - R. = max {Rjj). (1) (12). Sice redudacy is allowed, the Xi,j= 1 U (y= 1) J {i: Y,=I, j E LJ Cosider the 3 models (idexed as i sectio 2.1) for this case 1: AND (j E Li). Therefore, the Xi,j are ot eeded ad the aalogous costrait for (3) is omitted, ad the oes for (2) & (5) are chaged appreciably. The costraits, (10) - (12), are the same for each of the 3 models: ClMa, ClMb, ClMc. The costraits, (2) - (5), are the same for each of the 3 models: y 2 1, forj = 17 2,...7 ; (10) {i: j E L} 0 E L,l = 1, forj = I, 2,..., ; i= 1-1 ~.yi ~ I 1 0, for i = 1,..., m; (3) Y, = 0, 1,fori = 1,..., m. (12) Each model has its ow objective fuctio (ObF) to be m (4) maximized, subject to the 3 costraits, (10) - (12). i= 1 ObF(C2Ma) = F,.k2:] Xjj = 0, 1, for Y, = 0, 1, fori = 1,..., m, adj E L,. (13) (5) ObF(C2Mb) = q2,. * Costraits (2) guaratee that exactly 1 program, j, is selected. j= 1 0 Costraits (3) esure that, if = 0 the o programj for j E Li ca be selected; else, at most, all programs i L, ca be selected. (15) Costrait (4) is required so that the budget is ot exceeded. Each model has its ow objective fuctio (ObF) to be maximized, subject to the costraits, (2) - (5). ObF(C1Ma) = J=1 F,.\kl, ObF(C1Mb) = PI] ObF(C1Mc) = I= 1 V k=l Fk- j E sk (7) J=1 k2:j = 1 - q. 3. SOLVING PROBLEM ClMa Problem ClMa is a iteger program (ad ca be solved as oe) while the remaiig 5 problems have o-liear objective fuctios. All 6 problems ca be solved usig brach & boud algorithms where the brach is o the Xi,? The mai ideas of

3 BERMANKUTLER: CHOOSING AN OPTIMAL SET OF LIBRARIES 305 the algorithms are explaied via problem ClMa; see [l] for details o all the problems. The example i the remaider of this sectio illustrates the algorithm. 3.1 Statemet of Example There are 5 programs ad 4 libraries. N = (11, 2, 3, 4, 5}, h_ = (2, 3, 51, L3 = (3, 47 51, Y,=2 for ay library i ot yet cosidered i the partial solutio reliability of the chose versio of program j i the R1OJ partial solutio: Y,. = 1 ad X,OJ = 1, for ay j E J Rt*, highest reliability that ca be achieved for program j : Rt*,] = maxi, WO} {RtJj for ay j +Z J LB reliability of the best feasible solutio UP upper boud for a partial solutio for ClMa C, C* [lower, upper] boud o cost for the solutio correspodig to [a give partial solutio i, UP] I* cotais the idexes of libraries with Y, = 2 that iclude the most reliable versios for programs ot selected yet. L4 = (1, 2, 5). UP = E $-Rio,; + $*R'i:;; ; J ;ej R1,1 = 0.90, R1,2 0.80, RI,4 = 0.70; c* = ci + ci; R2,2 = 0.85, R2,3 = 0.90, R2,5 = 0.95; {i: x=1} i E I* R3,3 = 0.70, R3,4 = 0.90, R3,5 = 0.80; I* = {i*: there exists j sf J with Ri:j R4,1 = 0.90, R4,2 = 0.80, R4,5 zz = {i: max Y,=2} (Ri,;) > {i,~a211 {Ri,j>) C1 = 10, Cz = 15, C3 = 20, C4 = 17. B = 44. FJ = 0.2 for all j E N. 3.2 Observatios About a Partial Solutio Four observatios about a partial solutio idetified i the brach & boud process are: 1. If (Xi,J = 1) i a partial solutio, the Y, = If (Xi,j = 1) AND (x = 1) i a partial solutio, the Yk = 0 for all libraries k such that Rk,; > Rjj. 3. If for a give partial solutio, there is: a. a program t which is ot part of the partial solutio, ad b. oly 1 library Lk (that is ot part of the partial solutio) that icludes program t; the set Yk = 1 ad xk,t = If for a give partial solutio: a. XiJ = 1, ad b. there is a program t, ot selected yet i L,, with reliability strictly greater tha that of program t i all libraries ot selected yet; the set = Geeral Procedure J set of all programs cosidered so far i a partial solutio If, from (17), C* 5 B, the the solutio correspodig to UP is feasible ad the curret ode o the brach & boud tree should ot be further brached out. Whe J = 0, viz, before ay program is selected, j=l ie, for every program j, take the miimum cost of program j if each program were sold separately, ad if all the programs of a sigle library had the same cost. Sice libraries are sold as a whole, 'actual cost for program j' 2 mi { Ci/ I Li I }. Let the partial solutio have J # 0; the the lower boud o the cost of this partial solutio is: c, = CA + CB. Q CA CB CBI U y,= I Li: set of all programs i selected libraries {i: y,=l} cm E (j: j E Q Ci: actual cost of selected libraries max(cb1, Cm) mi(ci: yi = 2, Li Q: Q} E mi ~~, K=21 [-,LjN- e,

4 306 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 2, 1996 JUNE Whe C* > B, the partial solutio should ot be further bra- Yl = 1. ched out sice the partial solutio caot be exteded to a feasible solutio. Iteratio 2 A iitial feasible solutio ad LB is ot easy to fid. Eve the miimum cost problem which disregards reliability is NP- i2 = 2 (sice mi[15/2, 20/2, 17/11 = 71/21, complete'. However the greedy heuristic for the Set Coverig problem [6] ca be used to fid a low cost solutio that satisfies N3 = 0, costraits (2), (3), (5). If this solutio also satisfies costrait (4) the it ca serve as a iitial solutio ad its reliability is L3 = {3, 41, LB. Moreover, this solutio ca be easily improved. Y, = Greedy Heuristic This heuristic fids a cheap solutio. Iitialize: N' = N, L1 = (1, 2,..., }, Y, = 0 for i E L'. Iteratio t: Select Library it such that, Choose, Yl = Y, =1, Y3 = Y4 = 0, so that x,, = XI4 = x,, = x,, = x25 = 1, C = 25; which is feasible for the example. The versio of program 2 i 4 is chose sice R22 > RI2. The reliability of this solutio is This procedure might improve the solutio obtaied by the greedy-heuristic from sectio 3.4. qt = 1. S set of libraries i the solutio If N"' # 0, set t - ts 1 ad repeat iteratio t. R(SU {i}) reliability of SU {i}. STOP Use this heuristic o the example. N' = (1, 2, 3, 4, 5}, 4 Step 1 L' = (1, 2, 3, 43, Step 2 Y = Iteratio 1 (0, 0, 0, 0). P = {i: Yi = 0 AND C' + Ci 5 B) If P # 0, fid, j S R = max{r(su{i}: i E P} = R(SU{i*}) i' = 1 (sice mi[l0/3, 15/3, 2013, 17/31 = 3%), N2 = (3, 5}, L2 = (2, 3, 4}, 'Show by reducig the miimum cover problem 141 (kow to be NP-complete) to it. If R > R(S), the set, s = SU{i*}, yi* = 1; GoTo step 1. (Ed of step 2) STOP Use this procedure o the example.

5 BERMANKUTLER: CHOOSING AN OPTIMAL SET OF LIBRARIES 307 Step 1 P = {4}, (S = {1,2}). Step 2 R(SU(4)) = = 0.86 = R(S). Thus S = { 1,2} remais as the iitial solutio ad LB = Brach & Boud Algorithm Apply the brach & boud algorithm [7]; the result is the optimal solutio: Y1 = Y2 = 1, Y, = Y4 = 0; This is also the iitial solutio. 4. LIMITATIONS Assumptios 2, 4, 8 limit the usefuless of the models ad idicate the eed for further research. Their limitatios are discussed here. 2. I reality these umbers are ot available. Moreover, eve whe serious attempts are made to evaluate the reliability of systems, the results are ot completely accurate. A possible remedy for this problem might be to assume fuzzy reliabilities such as highly-reliable i our models. 4. I reality the failure behavior of differet versios is s-correlated because they are subject to the same iput [ 11, 121. Although s-correlatio is a importat issue i N-versio programmig, it is less severe i our models sice we demad oly that oe versio must be reliable for a give iput. 8. Experiece shows that some code must be added to cotrol the executio of the system, ad to deal with the compatibility of the programs i the system. 4 Despite these limitatios, we believe that costructig reliable systems that use commercially available libraries uder budgetary costrait is importat ad the models i this paper ca be used as buildig blocks for more pragmatic models. We pla to cotiue our research effort i this directio, icludig the aalysis of the sesitivity of the results to the assumptios. REFERENCES [l] 0. Berma, M. Cutler, Choosig a optimal set of libraries, Workig Paper, 1995; Faculty of Maagemet, Uiv. of Toroto. N. Ashrafi, 0. Berma, Optimizatio models for selectio of programs, Cosiderig cost & reliability, IEEE Tras. Reliability, vol 41, 1992 Ju, pp , Berma, N. Ashrafi, Optimizatio models for reliability of modular software systems, IEEE Tras. Software Eg g, vol 19, 1993 Dec, pp N. Ashrafi, 0. Berma, M. Cutler, Optimal desig of large software systems usig -versio programmig, IEEE Tras. Reliability, vol43, 1994 Ju, pp M.R. Garey, D.S. Johso, Computers ad Itractability: A Guide to the Theory of NP-Completeess, 1979; Freema. D.J. Musa, Software quality ad reliability basics, Proc Fall Joit Computer Co5 1987, pp ; IEEE Explorig Techology Today & Tomorrow, 1987 Oct. G.L. Nemhauser, L.A. Wolsey, Iteger ad Combiatorial Optimizatios, 1988; Joh Wiley & Sos. R. M. Reiss, A predictio experiece with three software reliability models, Workshop o Quatitative Sofhyare Models for Reliability, Complexiry, Cost : A Assessmet of the State of Art, 1979 October, pp ; IEEE Cat- CH F. Zahedi, N. Ashrafi, Software reliability allocatio based o structure, utility, price, ad cost, IEEE Tras. Sofhyare Eg g, vol 17, 1991 Apr, pp B.W. Boehm, Software Egieerig Ecoomics, 1981 ; Pretice-Hall. V.F. Nicola, A. Goyal, Modellig of correlated failures ad commuity error recovery i multiversio software, IEEE Tras. Sojiware Eg g, vol 16, 1990 Mar, pp D.E. Eckhardt, L.D. Lee, A theoretical basis for the aalysis of multiversio software subject to coicidet errors. IEEE Tras. Software Eg g, vol 11, 1985 Dec, pp AUTHORS Dr. Oded Berma; Faculty of Maagemet,; Uiv. of Toroto; Toroto, Otario M5S 3E6 CANADA. Iteret ( ) : berma@ fmgmt. mgrt. utoroto. ca Oded Berma is a Full Professor ad Associate Dea of Programs at the Faculty of Maagemet at the Uiv. of Toroto. He received his PhD (1978) i Operatios Research from the Massachusetts Istitute of Techology. He has bee with the Electroic Systems Lab at MIT, the Uiv. of Calgary, ad the Uiv. of Massachusetts at Bosto, where he was also the Chair of the Dept. of Maagemet Scieces. He has published over 80 articles ad has cotributed to several books i his field. His mai research iterests iclude operatios maagemet i the service idustry, locatio theory, etwork models, ad software reliability. He is the area editor of public services ad military for Operatios Research, a member of the editorial board for Computers ad Operatios Research, ad Associate Editor of Maagemet Sciece ad Trasportatio Sciece. Dr. Michal Cutler; Dept. of Computer Sciece; State Uiversity of New York; Bighamto, New York USA. Iteret ( ): cutler@bigsuy. cc. bighamto. edu Michal Cutler (M 86) received a BA i Applied Mathematics from Tel- Aviv Uiversity, ad MA & PhD i Computer Sciece from the Weizma Ist. of Sciece, Rehovot. She is a Associate Professor of Computer Sciece at Bighamto Uiversity. Her iterests are fault-tolerat computig, iformatio retrieval, ad artificial itelligece. She is a member of the IEEE Computer Society. Mauscript received 1995 November 4 Publisher Item Idetifier S (96) TRb