Abstract The objective of this article is to introduce several new methods or techniques for solving simultaneous linear and nonlinear system of equations with the help of a new hybrid algorithm based on advanced quantum behaved particle swarm optimization and the concept of binary tournamenting process. Depending on different options of binary tournamenting, six different variants of hybrid algorithms are proposed. To examine the effectiveness of the proposed hybrid algorithms five well known benchmark bound‐constrained optimization problems are solved. Among the six different variants of hybrid algorithms, the best algorithm is selected on the basis of their performances in these problems. This best algorithm is then applied in solving simultaneous linear and nonlinear system of equations transforming these equations into optimization problems. In case of linear system, just two systems are solved while in case of nonlinear system seven complicated problems are solved and finally a comparison of best found solutions are estimated with the same of existing algorithms provided in the literature.

Due to the fluctuation of market economy and uncertainty of customers' demand, it is quite difficult to develop an appropriate inventory model under uncertain situations. To overcome this difficulty, a food production model with preservation technology and credit-linked demand under default risk of capital in uncertain environment is developed with the help of parametric approach and interval mathematics. In this proposed model, all the inventory parameters, including demand rate, production rate and deterioration rate are considered as interval-valued. Because of the consideration of demand rate, production rate as well as deterioration rate as interval-valued, to represent the proposed model mathematically, the interval differential equations have been used. Solving these differential equations by using parametric approach, all the cost components and the corresponding average profit are obtained in the form of intervals. Therefore, the optimization problem of this model becomes interval-optimization problem. Then, to solve the interval-optimization problem, the center-radius optimization technique is established with the help of interval order relations. With the help of this technique, the interval-optimization problem is converted into crisp problem and then it is solved numerically by using different variants (Gaussian Quantum-behaved Particle Swarm Optimization, Weighted Quantum-behaved Particle Swarm Optimization, and Adaptive Quantum-behaved Particle Swarm Optimization) of Quantum-behaved Particle Swarm Optimization technique. Some real-life problems are considered and solved to justify the validity of the proposed model. The same model is also analyzed in crisp environment to verify the result of interval environment. Finally, the sensitivity analyses of both crisp and interval environments are performed separately with respect to the different system parameters.