Newsvendor Model with Random Supply

The newsvendor model is perhaps the most widely analyzed model in inventory management. In this singleperiod model, the only source of randomness is the demand during the period and one tries to determine the optimal order quantity in view of various cost factors. We consider an extention where supply is also random so that the quantity ordered is not necessarily received in full at the beginning of the period. Such models have been well-received in the literature with the assumption of independence between demand and supply. In this setting, we suppose that the random demand and supply are not necessarily independent. We focus on the resulting optimization problem and provide interesting characterizations on the optimal order quantity. Keywords. Newsvendor model, random capacity, random yield, quasi-concavity 1 Introduction The major source of randomness in inventory models is the demand. If the demand exceeds or falls short of expectations, the inventory manager will face shortage or lost sales. Moreover, the uncertainty of demand is not necessarily the only source of randomness. In fact, in recent years, there has been a lot of emphasis on models with supply uncertainty as well. The combined randomness of demand and supply enhances the level of uncertainty, thus leading to an increased complexity of the model. In this paper, we provide an example in the form of the well-known newsvendor model. Although this is a rather simple single-period model, it often forms the building block of many multi-period dynamic inventory, capacity-planning, and contract design problems. The main theme of this exposure concerns randomness in supply. This is an issue that should not be neglected or undermined. There are a lot of tragic examples concerning losses incurred due to the randomness in supply caused by uncertainties in production and transportation processes. Among many others, long machine downtimes due to unplanned maintenance, strikes, seconds and scraps in a production run, lack of raw material and rework are some reasons which leads to uncertainty during the production stage. Moreover, uncertainty during transportation is another cause for supply randomness. This is due to accidents, de…ciencies in the quality of transportation and various environmental factors. Chopra and Sodhi [3] and Serel [21] discuss some of the issues related to randomness in supply and mention a number of real cases. For example, as reported in Norrman and Jansson [15], a …re at a supplier’s plant disrupted the supply of radio-frequency chips to Ericsson in 2001 resulting in a loss of $400 million. Juttner [10] reports that in the same year, the continuity of production at Land Rover was threatened due to …nancial problems faced by the UK chassis manufacturer UPF Thompson. Kharif [12] states that Motorola failed to ship the phones promised to its major customers during the holiday season in 2003 due to component shortages. In random supply models, the quantity ordered by the inventory manager is not received in full with certainty. Instead, the manager receives a random amount that depends on the order quantity. An earlier review is provided in Yano and Lee [23] and recent advances are summarized in Grosfeld-Nir and Gerchak [7]. The earliest model of a random supply in inventory model was developed by Karlin [11]. This is followed by Shih [22], Noori and Keller [14] and Lee and Yano [13], among many others. Karlin [11] assumes that the only decision available is whether to order, and that if an order is placed, a random quantity is delivered. It is also shown that if the inventory holding and shortage cost functions are convex increasing, then there is a single critical initial on-hand inventory below which an order should be placed; otherwise, it is optimal not to order. Shih [22] assumes that inventory holding and shortage costs are linear and that the distribution of the fraction defective is invariant with the production level. The optimal production/order quantity can be found using a variant of the newsvendor model.