Reliability evaluation of multi-state systems under cost consideration

Assessing the Reliability of a Multistate Logistics Network under the Transportation Cost Constraint

“…The reliability index ( , ) of logistics network is defined as the probability that at least units of flow demand can be successfully transmitted from the source to the destination with the total transportation cost less than or equal to c [3][4][5][6][7][8][9]. One of the general algorithms for computing 2 Discrete Dynamics in Nature and Society ( , ) is using ( , )-minimal paths (( , )-MPs) [4][5][6][7][8][9]. A ( , )-MP, x, is a minimal state vector meeting the demand and the cost constraint c, which means that, for any < , does not meet the demand or the cost constraint [6].…”

“…A ( , )-MP, x, is a minimal state vector meeting the demand and the cost constraint c, which means that, for any < , does not meet the demand or the cost constraint [6]. If all ( , )-MPs are known, the well-known Inclusion-Exclusion rule is available to calculate ( , ) [7][8][9]. So, the most important work is how to efficiently determine all ( , )-MPs.…”

“…And yet, the algorithm by Yeh [8] is an exhaustive enumeration method in itself, so its searching efficiency is low. By adding some constraints to the minimal capacities of arcs, Niu and Xu [9] proposed an algorithm to improve the searching efficiency of solving ( , )-MPs.…”

“…Furthermore, the time complexity of the MPs based algorithms [5][6][7] is directly proportional to the number of MPs which grows exponentially with the scale of the network; thus, the computational efficiency of these algorithms in [5][6][7] is far from satisfactory. The methods in [8,9] require no MPs information and thus are an improvement to the ones in [5][6][7]. But, one remarkable limitation of Yeh's method [8] is that a huge number of state vectors need to be enumerated, which results in the lower searching efficiency.…”

“…But, one remarkable limitation of Yeh's method [8] is that a huge number of state vectors need to be enumerated, which results in the lower searching efficiency. While Niu and Xu [9] have made some efforts to advance the searching efficiency, the method by Niu and Xu is even inferior to the one by Yeh when the demand level is low, and thus its searching efficiency is also not satisfactory. Hence, the purpose of this paper is to propose a new efficient algorithm for solving all ( , )-MPs.…”