An Integrated Vendor-Buyer Model with Uncertain Lead Time, Life Time under Inflation and Variable Holding Cost

Introduction

Besides, in almost every business, supplier suffers from the problem of uncertain lead time at his end as well as the lead time is inversely proportional to the managing cost of the supplier. More the supplier is ready to spend as the managing expenses; smaller will be the lead time and vice-versa. However, there is a constraint upon the cost; the supplier can afford to spend. Due to his goodwill supplier is also willing to owe the extra cost incurred on retailer in terms of shortages cost and lost sales cost due to uncertain lead time. Due to the uncertainty of the arrival of replenishment unsatisfied demand is fulfilled at an imprecisely partially backlogging rate.

Liao and Shyu (1991) showed that lead time can be controlled through crashing. Later on, Ouyang et al. (1999) investigated uncertain lead time and studied the effect of cost reduction in a continuous review inventory model. Even then, this is a realm which has not been sufficiently explored by researchers.

Another area which is comparatively untouched is the concept of capital constraint for the supplier. However, as is very much evident from the face of facts, this constraint is very common. Khouja and Mehraz (1996) explored a constrained multi product newsboy problem with progressive multi discount. Lau and Lau (1996) in the same year studied the newsstand problem for a capacitated multi product single period inventory system. Other researchers related to this area such as Khouja (1999) and Pasternack (2001).

The supply chain models in inventory are a comparatively new foray for researchers. There has been especially very limited research for a supply chain in inflationary environment. The idea of joint total cost of the supplier and the customer was first introduced by Goyal (1976). Later, Cohen and Lee (1988) determined material requirement for all materials at every stage in a supply chain. Subsequent contributions in this direction came from researchers like Pake and Cohen (1993), Gyana and Bhabha (1999), Sarker et al. (2000), Chein and Lin (2004) and Ahmed et al. (2007).

Singh et al. (2009) developed an Economic Order Quantity models for retail supply chain with time dependent demand.

In recent years, inventory problems for deteriorating items have been widely studied after Ghare and Schrader (1963). They presented an EOQ (Economic Order Quantity) model for an exponentially decaying item with constant demand. Later, Covert and Philip (1973) used a variable deterioration rate of two-parameter Weibull distribution with constant demand rate and no shortages. Philip (1974) then developed the inventory model with a three-parameter Weibull distribution rate and no shortages. Shah (1977) extended Philip’s (1974) model and considered that shortage was allowed. Goyal and Giri (2001) provided an excellent and detailed review of deteriorating inventory literatures since the early 1990s. This work was extended by Goyal and Giri (2003), Ghosh and Chaudhari (2004) and Dye (2007). Singh et al. (2009) developed a supply chain model for ameliorating and deteriorating products.

The holding cost is explicitly assumed to be varying over time in only few inventory functions. This is particularly true in the storage of deteriorating and perishable items such as food products. Goh (1992) has taken holding cost variation over time as a continuous nonlinear function. Shao et al. (2000) determined a manufacturing process with variable holding costs. Beltran and Krass (2002) considered deterministic time varying demands and concave holding costs. Singh et al. (2009) assumed an inventory model with power demand, partial backlogging and incremental holding cost under inflation.