Alice Yalaoui

254 IEEE TRANSACTIONS ON RELIABILITY, VOL. 54, NO. 2, JUNE 2005 A New Dynamic Programming Method for Reliability & Redundancy Allocation in a Parallel-Series System Alice Yalaoui, Eric Châtelet, and Chengbin Chu , Member, IEEE Abstract—Reliab ility & redun dancy alloca tion is one of the most frequ ently encounter ed prob lems in system design. This probl em is subject to constraints related to the design, such as required structural, physical, and technical characteristics; and the components available in the market. This last constraint implies that system components, and their reliability, must belong to a finite set. For a parallel-series system, we show that the problem can be modeled as an integer linear program, and solved by a decomposi- tion approach. The pr oblem is deco mpo sed int o as many sub-pr ob- lems as subsystems, one sub-problem for each subsystem. The subproblem for a given subsystem consists of determining the number of components of each type in order to reach a given reliability target with a minimum cost. The global problem consists of determining the reliability target of subsystems. We show that the sub-problems are equivalent to one-dimensional knapsack problems whi ch can be solv ed in pseu dopolynomial time wit h a dynami c programming approach. We show that the global problem can also be sol ved by a dyn ami c pro grammi ng tec hni que . We also show that the obtained method YCC converges toward an optimal solution. Index T erms— Allocat ion, dynamic progr amming , parall el-series system, redundancy, reliability. NOTATIONS number of subsystems number of different component types available for subsystem , lower , and upper bounds on the number of re - dunda nt compo nents in subsy stem minimum reliability required for the system , reliability, and cost of component of type for subsystem , re liability, and cost of subsystem number of components of type in subsystem numbe r of compo nents of ty pe used in the solution of the subproblem : maximum number of free places leaving after having chosen the number of com- pone nt s of type to ta ke i n the subsys te m in the al loca tio n pr ob lem. is als o the number o f ite ms of t ype tak en in the knapsack number of it ems of ty pe used in th e solution of the subproble m maximum volume for the knapsack Manuscript receive d March 25, 2003; revised May 2, 2004 and June 2, 2004. Associate Editor: K. Kanoun. The authors are with ISTIT, Université de Technologie de Troyes, 10010 Troyes, France (e-mail: yalaouia@utt .fr; [email protected]; [email protected]). Digital Object Identifier 10.1109/TR.2005.847 270 the volume of items of type integer parameter of the method so that minimum reliability for subsystem max imu m vol ume of the kna psa ck ben efit bro ugh t by the vo lume in kn apsack number of different solutions obtained for subsystem feasible reliability for the subsystem , op ti m al re l ia bi l it y and cost of subsys te m for a given , opti mal re li ab il it y , and cost of the sy st em fo r a given optimal volume of the knapsack for a given V ol ume of the item of type in the globa l problem optima l quan tity o f items of t ype for an y va lue of pro fit brou ght by items c hos en between the it ems of ty pe so th at th e vol ume us ed is exactly opt imal number o f ite ms to ta ke in order to obtain the maximal profit maximal total benefit obtained by choosing one item of each types so that the total volume used is exactly V olume of the i tem of t ype chose n amon g ite ms av ail abl e pro vid ing the max ima l tot al ben - efit I. INTRODUCTION R ELIABILITY engineering has attracted many researchers since 1960 due to reliability’s critical importance in a va- riety of systems. As explained by Tzafestas in 1980 [ 1], one of the undeniable steps in the design of multi-component systems is the problem of using the available resources in the most ef- fective way so as to maximize the overall system reliability, or so as to minimize the consumption of resources while achieving spe cific rel iab ili ty goa ls. The re exi st seve ral met hod s to imp rov e the system reliability. The most known are: • reduction of the system complexity, • allocation of components’ reliability, • allocation of redundancy alone or combined with reliability allocation, or 0018-9529/$20.00 © 2005 IEEE

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Alice Yalaoui