A fuzzy approach to reliability based design of storm water drain network

Abstract

This paper proposes an approach to estimate reliability of a storm water drain (SWD) network in fuzzy framework. It involves: (i) use of proposed fuzzy Monte-Carlo simulation (FMCS) methodology to estimate fuzzy reliability of conduits in the network, (ii) construction of a reliability block diagram (RBD) for the network (system) using suggested guidelines, and (iii) use of the RBD and reliability estimates of the conduits in the network to compute system reliability based on a proposed procedure. In addition, a system reliability based methodology is proposed for design/retrofitting of SWD network by optimization of its conduit dimensions. Conventionally used reliability analysis approaches assume that the cumulative distribution function (CDF) of performance function (marginal safety) of conduits follows Gaussian distribution, which cannot be ensured in the real world scenario. The proposed approach alleviates the need for making such assumptions and can account for linguistic ambiguity in variables defining the performance function. Effectiveness of the proposed approach is demonstrated on a hypothetical SWD network and a real network in Bangalore, India. Comparison of the results obtained from the proposed approach with those from conventional Monte-Carlo simulation (MCS) reliability assessment approach indicated that the estimate of system reliability and conduit reliability are higher with FMCS approach. Consequently, conduit dimensions required to attain required system (network) reliability could be expected to be lower when FMCS approach is used for designing or retrofitting a system.

Appendix

This section provides finer details of the analysis performed to arrive at estimates of minimum reliability (R ^min _i , i = 1,…, 10) required for the ten conduits in the hypothetical SWD network to achieve a higher system reliability (R _s  = 0.99).

Let R _C1, …, R _C10 denote reliability of conduits numbered 1 to 10, respectively. Further, let R _Ca denote reliability of the network within the dotted lines in Fig. 7a. If the system reliability R _S  = 0.99, then R _S  = R _Ca  × R _C8 × R _C9 × R _C10 = 0.99 [from Eq. (9)]. Assuming R _Ca , R _C8, R _C9 and R _C10 to be equal, R _Ca  = R _C8 = R _C9 = R _C10 = 0.997. The required minimum reliability for conduits 8, 9 and 10 ($R_8^{min },R_9^{min },R_{10}^{min }$) is thus considered to be 0.997.

Fig. 7

Illustration of the analysis performed to estimate reliability values of the individual conduits in the hypothetical network to achieve a higher system reliability (R _S  = 0.99)

If R _Cb denotes reliability of the network within the dotted lines in Fig. 7b, R _Ca  = 0.997 = 1 − (1 − R _Cb )(1 − R _C6)(1 − R _C7) [from Eq. (10)]. This yields R _Cb  = R _C6 = R _C7 = 0.856, when R _Cb , R _C6 and R _C7 are assumed to be equal. The required minimum reliability for conduits 6 and 7 (R ^min₆ , R ^min₇ ) is therefore 0.856.

If R _Ce denotes reliability of the network within the dotted lines in Fig. 7c, then, R _Cb  = R _Ce  × R _C4 × R _C5 = 0.856 [from Eq. (9)]. Assuming R _Ce , R _C4 and R _C5 to be equal, R _Ce  = R _C4 = R _C5 = 0.949. Hence, the required minimum reliability for conduits 4 and 5 (R ^min₄ , R ^min₅ ) is 0.949.

Similarly, R _C1, R _C2, andR _C3 can be estimated using Fig. 7d, as R _Ce  = 0.949 = 1 − (1 − R _C1)(1 − R _C2)(1 − R _C3) [from Eq. (10)]. This yields R _C1 = R _C2 = R _C3 = 0.630, when reliability of all the three conduits are assumed to be equal. Thus, the required minimum reliability for conduits 1, 2 and 3 (R ^min₁ , R ^min₂ , R ^min₃ ) is 0.630.