An application of interval differential equation on a production inventory model with interval‐valued demand via center‐radius optimization technique and particle swarm optimization

Abstract

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.