Abstract
In the current work, we devised a hybrid method involving a Double-Layer Neural Network (DLNN) for solving a quadratic Bi-Level Programming Problem (BLPP). For an efficient and effective solution of such problems, the proposed potential methodology includes an improved Artificial Bee Colony (ABC) algorithm, a Hopfield Network (HN), and a Boltzmann Machine (BM). The improved ABC algorithm accommodates upper-level decision problems by selecting a set of potential solutions from all combinations of solutions. However, for lower-level decision problem, HN and BM are amalgamated to manifest a DLNN that initially generates its structure by choosing a limited number of units, and will subsequently converge to an optimal solution/unit among those units and hence, constitutes an effective, efficient solution technique.We compared the accuracy, computational time and effectiveness (ability to find the true optimum) of the proposed DLNN with improved-ABC, DLNN with PSO (where PSO replaces the improved-ABC in the upper-level problem of the proposed DLNN with improved-ABC), DLNN with GA (where GAreplaces the improved-ABC in the upper-level of the proposed algorithm) and other conventional approaches and found the proposed DLNN with improved-ABC can yield high quality global optimal solutions with higher accuracy in relatively smaller time.
Highlights
The Bi-Level Programming Problem (BLPP) can be regarded as a mathematical programing problem, where one optimization problem, called the upperlevel decision problem contains another problem as a constraint
In transport network design problem (NDP) [33], the demand-performance equilibrium for a given investment is represented by the lower-level problem in the network model, whereas, the investment decision has been made by the upper-level
A number of techniques for solving BLPPs are found in literature, among them, nested evolutionary algorithms are considered as popular approaches for solving BLPPs, where lower-level problem is answered corresponding to each upper-level member [35]
Summary
The BLPP can be regarded as a mathematical programing problem, where one optimization problem, called the upperlevel decision problem contains another problem (known as lower-level decision problem) as a constraint. For the lower-level decision problem, we propose a novel DLNN (a novel neural network structure) by combining a HN (upper-layer of the DLNN) and a BM (lower-layer of the DLNN) that can delete the lower-layer units (neurons), which are not selected for the upper-layer execution and restructures the lower-layer BM using the selected units This could be considered an efficient method for solving selection problems by transforming their evaluation functions into energy function as both HN and BM converge to a minimum point of the energy function. Based on the above settings, Sherali et al [20] reported that the Stackelberg problems show BLPP-like hierarchical structures in the presence of a single leader In this case, the BLPPs and Stackelberg problems are equivalent the lower-level decision problem in the Stackelberg game is more of an equilibrium problem rather than optimization. The BLPP-complexity is briefly discussed [17]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
