In production rate. Goyal (1988) extended Banerjee’s work by

In this paper, we develop an integrated bi-objective model of two stage supply chain composed of a vendor and a buyer under an imperfect production process and stochastic inflationary condition wherein the first objective is minimizing the expected costs of the proposed supply chain model and the second objective is minimizing buyer’s shortage variance. We assume lead time and ordering cost are controllable parameters and lead time crashing cost is considered as a function of both order quantity and reduced lead time. An effective solution procedure is developed to determine optimal policy of the proposed model. Finally, a numerical example and sensitivity analysis is proposed to show the performance of the model.   

Keywords:  Supply Chain; Integrated Vendor-Buyer Inventory Model; Lead Time; Inflation; Stochastic; Multi-Objective Programming.

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1. Introduction

For both classical economic order quantity (EOQ) and economic production quantity (EPQ) models the vendor and buyer minimized their own inventory costs independently. This independent decision making for inventory problem usually cannot guarantee the optimal policy for both vendor and buyer. However, the recent inventory control literature shown that by integrating EOQ model of buyer and EPQ model of vendor will result in lower inventory total costs for the supply chain system rather than obtaining optimal polices of chain’s members individually.         

The study of an integrated inventory model for vendor and buyer was pioneered by Goyal (1976). Subsequently, many researchers have investigated this issue under various assumptions. For instance, Banerjee (1986) expanded the model of Goyal (1976) with considering a finite vendor’s production rate. Goyal (1988) extended Banerjee’s work by relaxing the lot-for-lot production policy for the vendor and assumed that the vendor’s production quantity can be true multiple of the buyer’s order quantity. Ha and Kim (1997) further extended Goyal’s (1988) model and proposed an integrated lot-slitting model of facilitating multiple shipment in small lots. Yang and Wee (2000) proposed an integrated vendor-buyer inventory model for deteriorating items. Lee (2004) developed a single-vendor single-buyer supply chain model where the vendor orders raw materials from its supplier, then using its manufacturing processes converts the raw materials to finished goods, finally delivers the finished goods to its customer.   Hans et al. (2006) developed a new methodology to obtain the joint economic lot size in distribution system with multiple shipment policy. Hariga and Al-Ahmari (2013) proposed a supplier-retailer supply chain system with hybrid vendor-managed inventory and consignment stock policy and stock dependent demand at retailer’s shelf space. Previously mentioned literature on integrated inventory model of vendor and buyer most concentrated on deterministic demand for the system. However, when demand is considered as a stochastic variable, lead time becomes an imprtant issue and shortening it leads to reduce safety stock, reduce the stock-out loss and improve the customer service level so as to gain competative advantages in business. If it is assumed that lead time can be decomposed into several components, such as setup time, process time, and queue time, it can be assumed that each component may be reduced at a crashing cost. One of first papers dealing with a variable lead time in an inventory model was due to Lia and Shyu (1991). The authors assumed that lead time can be decomposed to its elements and these elements can be reduced to their given minimum duration. Under the assumption that the lot size is predetermined and demand is normally distributed, they calculated an optimal lead time and show that reducing lead time may result in lower expected total cost. Pan and Yang (2002) assumed lead time to be a decision variable and obtained a lower joint total expected cost and shorter lead time of integrated supply chain. Ouyang et al., (2004) extended Pan and Yang (2002) model considering reorder point as one of decision variable and shortages allowed. Pan and Hsiao (2005) proposed an integrated inventory model with backorder price discount and controllable lead time wherein lead time crashing cost is a function of both ordering quantity as well as reduced lead time. Yedes et al. (2012) presented an integrated vendor-buyer inventory model wherein the production unit is assumed randomly shift form an in-control to an out-of-control situation. They considered simultaneously production, inventory and maintenance policies. On the other hand, almost all of previously stated papers on integrated vendor- buyer inventory systems didn’t study on ordering/setup cost reduction. However, studies regarding ordering/setup cost reduction have shown that investment in ordering/setup cost can be reduced system’s cost meaningfully. Porteus (1985) was pioneer in presenting the concept of investing in reducing setup cost on the classical EOQ model without backorders which contains a set up as a function of capital expenditure. Lin (2009) presented an integrated vendor-buyer model with backorder price discount and ordering cost reduction. Glock (2012) considered a single-vendor single-buyer inventory model with lot-size-dependent lead time and lead time reduction. He assumed vendor’s set up cost can be reduced with an extra crash cost with a piece-wise linear function. Giri and Sharma (2017) proposed an integrated vendor-buyer inventory model with imperfect production process wherein ordering cost can be reduced with extra investment.