Queueing-inventory system with return of purchased items and customer feedback

¹ Department of Mathematics, T K Madhava Memorial College, Nangiarkulangara, Alappuzha 690513, Kerala, India
² Department of Mathematics, Sanatana Dharma College, Alappuzha 688003, Kerala, India
³ Baku Engineering University, Khirdalan, Hasan Aliyev Str., 120 AZ 0101, Azerbaijan

^* Corresponding author: dhanya.shajin@gmail.com

Received: 29 June 2024
Accepted: 7 April 2025

Abstract

In this paper, a model of single server queueing-inventory system (QIS) with Markovian Arrival Process (MAP) and phase-type distribution (PH-distribution) of the service time of consumer customers (c-customers) is considered. After completing the service of c-customer, he (she) can make one of the following decisions: (1) eventually leave the system with probability (w.p.) σ_ℓ; (2) after a random “thinking” time returns the purchased item w.p. σ_r; (3) after a random “thinking” he (she) feedback to buy a new item w.p. σ_f . It is assumed that σ_ℓ+σ_r +σ_f = 1. If upon arrival of the c-customer the system main warehouse (SMW) is empty, then the incoming customer, according to the Bernoulli scheme, is either joins the infinite queue or leaves the system. A virtual finite orbit can be considered as a waiting room for feedback customers (f-customers). Returned items are considered new and are sent directly to SMW if there is at least one free space; otherwise, this item is sent to a special warehouse for returned items (WRI). After completing the service of each customer, one item is instantly sent from the WRI (if any) to the SMW. In SMW, the (s, S) replenishment policy is used and it is assumed that the lead time follows exponential distribution with finite parameter. When the stock level reaches its maximum value due to items returns, the system immediately cancels the regular order. Along with classical performance measures of QIS new specific measures are defined and numerical method for their calculation as well as maximization of the revenue function are developed. Results of numerical examples to illustrate the effect of different parameters on the system’s performance measures are provided and analyzed. We also provide a detailed analysis of an important special case of the Poisson process/exponential service time model.