• Solution and stability analysis of non-homogeneous difference equation followed by real life application in fuzzy environment

The study fuzzy difference equation becomes very important as huge numbers of real-life problems in the field of engineering; ecology social science, etc. can be mathematically represented in the form of difference equation where impreciseness is inherently involved. In this paper, we have focused on the solution techniques of non-homogeneous fuzzy linear difference equation with different cases involving fuzzy initial conditions, fuzzy forcing function and fuzzy coefficient. The idea of fuzzy equilibrium point is introduced and its stability analysis has been performed. The whole theoretical work is followed by real-life applications which show the impact of fuzzy concepts in mathematical modelling for better understanding the behaviour of the system in an elegant manner.

• Synergetic study of inventory management problem in uncertain environment based on memory and learning effects

Due to the involvement of human intelligence in the inventory planning procedures, memory and learning from repeated tasks in the planning horizon are two important facts that make great impressions on the decision taken in reality. However, the concepts of learning and memory related to the inventory theory are rarely illustrated in literature and till date we have not noticed any work where the effects of memory and system learning have been explored simultaneously. Making an attempt to close the gap, the present paper extends an economic order quantity (EOQ) model into a memory and learning sensitive set-up. The primary structure of the model is established on the assumption that the demand of the EOQ model is constant in inventory run period and the same is a decreasing function of time in the shortage period. Introducing fractional calculus as a replacement of integer one, the notion of memory is included in the proposed theory. Finally, using Zadeh’s extension principle, the fuzzzification of the fractional deterministic model is executed and ultimately the senseof learning based decision making is incorporated letting the demand to be a triangular dense fuzzy number. Here, considering different underlying scenarios, four different models have been illustrated and solved numerically. The a-cut method of defuzzification is used for the numerical simulation of two fuzzy models. It is worth mentioning that the joint impact of learning and memory creates positive results on the cost reduction objective of the proposed lot-sizing problem.

• Designing a multi-depot multi-period vehicle routing problem with time window: hybridization of tabu search and variable neighbourhood search algorithm

This research formulates a real-life multi-depot and multi-period vehicle routing problem (MDMPVRP) by imposing time window (TW) and many other constraints. A set of customers, spread in different locations, are to be served by a fleet of heterogeneous vehicles over a finite number of periods. Eachcustomer is associated with combinations of routes and vehicles over the period. A customer must be served in one of the allowable combinations. The objective of this MDMPVRP-TW is to minimize the total distance traversed by the fleet over the planning horizon. The proposed MDMPVRP-TW is an extension of vehiclerouting problem (VRP), and is hence an NP-hard problem. In order to optimize it, we propose a hybrid metaheuristic approach by combining tabu search (TS) and variable neighbourhood search (VNS) algorithms. Furthermore, to provide richer insights, the efficacy of the proposed method and mathematical formulation isdemonstrated through numerical experiments for a number of instances varying from small to large scale.