Supply Chain Model for Expiring Items Following Ramp-Type Demand With Stochastic Lead Time Under Crisp and Fuzzy Environment

1. Introduction

In the existing literature, most of the inventory/supply chain model was developed considering three types of demand rates: (i) constant demand rate; (ii) linearly positive/negative demand rate; and (iii) exponentially increasing/decreasing demand rate. However, demand for a commodity cannot increase/decrease continuously over time. It is observed that demand for seasonal products like fruits and the fashionable products over the entire time horizon is three folded. At the beginning of the season demand increases more rapidly, as the time passes, it becomes steady in the middle of the season and decreases more rapidly towards the end of the season and becomes asymptotic. The term “ramp type” is used to represent such kind of demand pattern. In the existing inventory and supply chain models, after the development of classical economic ordering quantity model by Wilson (1934) under the assumption of constant demand rate, researchers extensively studied several aspects of inventory and supply chain modeling by assuming time-dependent demand rate. The assumption of constant demand rate is usually valid in the mature stage of a product’s life cycle. In the growth and/or end-stage life cycle, demand rate may well be approximated by a time-dependent function. Resh et al. (1976) were the first who developed a model with linearly time-varying demand. Hill (1995) first resolved the indiscipline of time-dependent demand pattern by considering the demand as the combination of two different types of disciplined demand in two successive time periods over the entire time horizon and developed an inventory model with increasing demand (general power of time) followed by a constant demand and term it as ramp-type time-dependent demand pattern. Wu and Ouyang (2000) studied inventory models under two different replenishment policies: (i) those starting with no shortages; and (b) those starting with shortages. Chen et al. (2006) derived optimal ordering quantity model with ramp type demand rate and time-dependent deterioration rate. Teng et al. (2011) developed an inventory model with ramp type demand rate assuming that shortages are partially backlogged for the items following Weibull deterioration rate. Giri and Bardhan (2012) derived supply chain coordination for a deteriorating item with stock and price-dependent demand under a revenue sharing contract. Tripathy and Mishra (2011) discussed an EOQ model with time-dependent Weibull deterioration and ramp type demand. Dhagat et al. (2012) studied an EOQ inventory model with ramp type demand for the items having generalized Weibull distribution deterioration to deal with shortages. Garg et al. (2012) developed an economic production lot size model with price discounting for non-instantaneous deteriorating items with ramp-type production and demand rates. Chakraborty et al. (2015) derived a vendor-managed inventory scheme as a supply chain coordination mechanism. Sharma et al. (2018) analyzed a replenishment model of deteriorating items with ramp-type demand and trade credit under the learning effects.