Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
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We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\n
Throughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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\n\t\t\t
1. Introduction
\n\t\t\t
With the widespread concept of partnership in supply chain management, the traditional replenishment process is fast giving way to the Quick Response (QR) and Continuous Replenishment Programme (CRP). QR is a movement in industries to shorten the replenishment lead time which is critical to reduce inventory level and improve the levels of customer service. Wal-Mart, Seven-Eleven Japan, and many other retailers apply tremendous pressure on their suppliers to reduce the replenishment lead time (Chopra and Mendil 2007). CRP is an efficient replenishment initiative which focuses on removing excess inventory throughout the pipeline and synchronizes demand with production (EAN International 2000). In CRP, the inventory of retailer is planned, monitored, and replenished by the supplier on behalf of the consumers. To enable CRP, sales data and inventory level of the retailer must be provided to the supplier via Electronic Data Interchange (EDI) or other electronic means. Thus, to successfully implement CRP requires the supplier and the retailer to work in a cooperative manner based on mutual trust and joint gains.
\n\t\t\t
The lead time is needed for several operations among trading parties, such as ordering, manufacturing, delivering, and handling. In practice, lead time can be shortened with extra crashing costs. Many researchers have studied the inventory decision incorporating lead time reduction under various assumptions. For example, Liao and Shyu (1991) first developed an inventory model in which lead time was the unique decision variable. Ben-Daya and Raouf (1994) extended the previous model by including both lead time and order quantity as decision variables. Pan and Yang (2002) and Pan and Hsiao (2005) considered lead time crashing cost as a function of both the order quantity and the reduced lead time. Since the reduction of lead time may involve with the vendor and the buyer to improve the related operations, a dyadic vendor-buyer viewpoint for shortening lead time is often suggested. For instance, Iyer and Bergen (1997) compare the profits of a single vendor and a single buyer supply chain before and after QR based on a newsboy inventory model. Ben-Daya and Hariga (2004), Ouyang et al. (2004), Chang et al. (2006), and Ouyang et al. (2007) investigated the decision of lead time reduction in a context of a single-vendor-single-retailer integrated inventory system, in which the possibility of stockout of the supplier was ignored.
\n\t\t\t
The study of inventory decision of CRP based inventory systems was initiated by Goyal (1976) by presenting a joint economic lot-size model for a supplier and a retailer. Subsequently, many researchers investigated this issue under various assumptions. For example, Banerjee (1986) generalized the model of Goyal (1976) by incorporating a finite replenishment rate for the supplier. Goyal (1988) extended Banerjee’s work by relaxing the lot-for-lot policy for the supplier and assumed that the supplier’s lot size is an integral multiple of retailer’s order quantity. Goyal and Srinivasan (1992) further extended the model by relaxing the assumption that the supplier can supply to the retailer only after completing the entire production lot. At the same time, Banerjee and Banerjee (1992) extended the study of integrated inventory control to a multiple-retailer case by considering that the supplier delivers items to several buyers at a coordinated common cycle time. Banerjee and Banerjee (1994) generalized their previous model by dealing with the normally distributed demand case and assuming a fixed shortage cost attributable to each stockout incident.
\n\t\t\t
To understand the effects of QR on the CRP based supply chains, we build a formal model of lead time reduction and replenishment decisions for a supply chain consisting of a manufacturer and multiple retailers in which the inventory throughout the supply chain is managed in a CRP context. The model extends the work of Banerjee and Banerjee (1994) by including the ordering cost and adopting QR with the expenditure regarded as a function of the reduced lead time. In stead of explicitly estimating the shortage cost, a service level constraint (SLC) approach is applied to the problem for which may skirt the difficult practical issue of explicitly determining the shortage cost (Chen and Krass 2001). According to the definition in Ouyang and Chuang (2000), the service level in this study is measured by the expected fraction of demand met from inventory on hand in an inventory cycle. The objective of the model is to determine the common shipment cycle time and the replenishment lead time for the retailers, the manufacturer’s production cycle time, and the target levels of replenishment for the manufacturer and each retailer so that the expected total system cost can be minimized under the service level constraint.
\n\t\t
\n\t\t
\n\t\t\t
2. Notations and assumptions
\n\t\t\t
To develop the proposed models, we adopt the following notation which is principally the same as that in Banerjee and Banerjee (1994):
\n\t\t\t
\n\t\t\t\tn= total number of retailers;
\n\t\t\t
\n\t\t\t\td\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= demand rate of retailer i, which follows a probability density function (p.d.f.) f\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t(d\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t) with mean D\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t and variance\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, i =1, 2,…, n;
\n\t\t\t
\n\t\t\t\th\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= carrying cost per unit per year for retailer i ($/unit/year);
\n\t\t\t
\n\t\t\t\th\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t\n\t\t\t\t= carrying cost per unit per year for the manufacturer ($/unit/year);
\n\t\t\t
\n\t\t\t\tl= length of lead time for retailers (year), a decision variable;
\n\t\t\t
\n\t\t\t\tx\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= demand during the protection period (T+l) on retailer i, which has a p.d.f. \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twith the mean \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the variance \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t(T+l);
\n\t\t\t
\n\t\t\t\ty= demand during the production cycle time of KT on manufacturer, which has a p.d.f. g(y) with the mean \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the variance\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t;
\n\t\t\t
\n\t\t\t\tz\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= safety factor for retailer i with z\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t ≥ 0;
\n\t\t\t
\n\t\t\t\tz\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t= safety factor for the manufacturer with z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t ≥ 0;
\n\t\t\t
\n\t\t\t\tα\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= the threshold of retailer i’s service level, 0 ≤ α\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t ≤1;
\n\t\t\t
\n\t\t\t\tα\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t= the threshold of manufacturer’s service level, 0 ≤ α\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t ≤1;
\n\t\t\t
\n\t\t\t\tA= setup cost for the manufacturer ($/per setup);
\n\t\t\t
\n\t\t\t\tC= common ordering cost shared by all retailers ($/per order);
\n\t\t\t
\n\t\t\t\tC\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= individual ordering cost for retailer i ($/per order);
\n\t\t\t
\n\t\t\t\tC\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t(l)= lead-time crashing cost per order;
\n\t\t\t
\n\t\t\t\tP= production rate of manufacturer, which is a known constant;
\n\t\t\t
\n\t\t\t\tD=\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, total demand from all the retailers on the manufacturer per year, which has mean \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and variance\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Note that, of necessity, P>\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t;
\n\t\t\t
\n\t\t\t\tK= integral number of shipments to retailers per manufacturer’s production cycle, which is a decision variable with K ≥ 1;
\n\t\t\t
\n\t\t\t\tS\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= replenish-up-to level after placing a new order for retailer i, a decision variable;
\n\t\t\t
\n\t\t\t\tS\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t= produce-up-to level after placing a new manufacture order for the manufacturer, a decision variable;
\n\t\t\t
\n\t\t\t\tT= common shipment cycle time to all retailers (year), a decision variable;
\n\t\t\t
\n\t\t\t\tEC\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t= the expected annual inventory cost for retailer i;
\n\t\t\t
\n\t\t\t\tEC\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t= the expected annual inventory cost for the manufacturer;
\n\t\t\t
\n\t\t\t\tETC= the expected annual total cost for the system.
\n\t\t\t
The major assumptions made in this study are the following:
\n\t\t\t
In the CRP context, the manufacturer plans, monitors, and replenishes the inventory of retailers based on the information of sales data and inventory level provided by all the retailers to minimize the expected annual total cost of the supply chain.
The demand rate of each retailer is independently distributed.
Each production lot of the manufacturer will be delivered in an integral number, K, of shipments to all retailers periodically.
The target level of replenishment of S\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t units for retailer i is equal to the sum of retailer i’s expected demand during the protection period, D\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t(T+l), and the safety stock, where safety stock = z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\t\tσ\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
The target level of production of S\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t units for the manufacturer is equal to the sum of manufacturer’s expected demand during the production cycle, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, and safety stock, where safety stock = z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
For each order, a common ordering cost C is incurred to and shared by all retailers and an individual ordering cost C\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t is incurred to the retailer i.
The manufacturer incurs a setup cost A for each production run.
The service level measures, i.e., fraction of demand met per cycle, for the retailer i and for the manufacturer are defined as \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, respectively.
The expenditure for implementing QR is modeled as the lead-time crashing cost per order, C\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tr\n\t\t\t\t\t\t(l), and is assumed to be a non-increasing stairstep function of l, such as \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t⋮\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t⋮\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where l\n\t\t\t\t\t\t0 and l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tb\n\t\t\t\t\t\t represent the existing and the minimal length of lead times, respectively. This cost could be expenditure on equipment improvement, order expediting, or special shipping and handling.
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3. Model development and solution
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\n\t\t\t\tFigure 1 depicting the inventory time plots for a manufacturer and n retailers, illustrates the CRP based inventory control system proposed in this work. For the retailer i, there is a target level of replenishment of S\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t units. For the manufacturer, there is a target level of production of S\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t units. Given a common shipment cycle time of T years, the manufacturer produces a batch of item every KT years. After the Kth shipment cycle since the beginning of the last production cycle, the manufacturer begins the production of another lot DT/P years prior to the next scheduled shipment time, as shown in Figure 1, which is illustrated the case of K =3. In order to cope with a stochastic demand, all of the trading parties carry safety stocks.
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The demand during the protection period for retailer i is assumed to be normally distributed with mean \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and standard deviation\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Then, the total demand from all retailers on the manufacturer during a production cycle of KT is normally distributed with the mean \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the standard deviation\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. For the retailer i, with a cycle of T, a replenishment lead time of l, and a target level of S\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t, the expected net inventory level just before receipt of an order is given by S\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t–D\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t\n\t\t\t\tl–D\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t\n\t\t\t\tT and the expected net inventory level immediately after the successive order is\n\t\t\t\tS\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t–D\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t\n\t\t\t\tl. Then, the average inventory over the cycle is approximated by\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. According to the fourth assumption, the target levels of replenishments, S\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t, is set as
The annual individual ordering cost incurred to retailer i is C\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t/T. Then, the expected annual inventory cost for all retailers, consisting of common ordering cost, individual ordering cost, and carrying cost, is given by
where the first term represents the average annual setup cost and the second term is the average carrying cost.
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Based on the service level measure defined in Ouyang and Chuang (2000), the following service level constraints are specified:
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service level constraint on retailer i =\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t≥ α\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t, i =1, 2,…, n,
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service level constraint on the manufacturer =\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t≥ α\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t.
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As the demand is normally distributed, (3) and (4) can be rewritten as (see Appendix for proof):
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service level constraint on retailer i =\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, i =1, 2,…, n,
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service level constraint on the manufacturer =\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tΦ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t,
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where (z) and Φ(z) represent the standard normal distribution and the complementary cumulative normal distribution function, respectively. The expected annual total cost for the system, comprising the expected annual inventory costs for all parties and the lead time crashing cost, is given by
where the vector z ={z\n\t\t\t\t1, z\n\t\t\t\t2,…, z\n\t\t\t\t\n\t\t\t\t\tn\n\t\t\t\t}. Then, the decision is to minimize ETC(K, T, l, z, z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t) under the constraints (5) and (6).
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In order to solve the nonlinear programming problem, the following propositions are needed.
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\n\t\t\t\tProposition 1. For any given K, T, z, and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t, the minimal ETC will occur at the end points of the interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t] when C\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t(l) is a stairstep function.
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Proof:
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As C\n\t\t\t\t\n\t\t\t\t\tr\n\t\t\t\t(l) is a stairstep function, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tfor any l in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t]. The second derivative of (7) with respect to l is equal to
for any l in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t]. Therefore, the proposition holds.
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\n\t\t\t\tProposition 2. For any fixed K, l, z, and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t, there is a unique optimal T to ETC in (7).
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Proof:
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For any given K, l, z, and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t], a necessary condition for a solution of T to be optimal for (7) is
That is, the second derivative of ETC at each T\n\t\t\t\t\n\t\t\t\t\tf\n\t\t\t\t is positive. The result implies that, for given K, l, z, and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t, an only solution for T, corresponding to the minimum of ETC, can be derived from (8). This completes the proof.
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\n\t\t\t\tProposition 3. For any given K, T, and l in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t], the boundary solution for z and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t derived from (5) and (6) is the optimal solution to this nonlinear programming problem.
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\n\t\t\t\tProof:\n\t\t\t
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By using the method of Lagrange multipliers, the problem can be transformed into
where λ is a vector of Lagrange multipliers with λ={λ\n\t\t\t\t1, λ\n\t\t\t\t2,…, λ\n\t\t\t\t\n\t\t\t\t\tn\n\t\t\t\t, λ\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t}≥ 0, and s is a vector of slack variables with s = {s\n\t\t\t\t1\n\t\t\t\t2, s\n\t\t\t\t2\n\t\t\t\t2,…, s\n\t\t\t\t\n\t\t\t\t\tn\n\t\t\t\t\n\t\t\t\t2, s\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t\n\t\t\t\t2}. According to Kuhn-Tucker theorem (Taha, 2002), for any given l in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t], it can be shown that s = {0,0,…,0} is a necessary condition for a solution in (9) to be optimal. Then, for any given l in a continuous interval [l\n\t\t\t\t\n\t\t\t\t\tj+1, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t], the necessary conditions for (9) to be minimized are
Therefore, constraints (5) and (6) are binding. In addition, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t = \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t>0 (i=1, 2,…, n, m), then the z and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t derived from (12) and (13) (i.e., the boundary solutions to (5) and (6)) are optimal to (9) with given K, T, and l.
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Based on the above propositions, the optimal solution for K, T, l, S, and S\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t to this nonlinear programming problem can be obtained by the following algorithm:
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\n\t\t\t\tAlgorithm\n\t\t\t
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Set K=1.
Find ETC*(K,lj/j=0,1,…,b)
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2.1Let j =0
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2.2Set l = l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t. z\n\t\t\t\t\n\t\t\t\t\ti\n\t\t\t\t (i =1, 2,…, n) = 0, and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t = 0.
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2.3Compute T\n\t\t\t\t* by substituting K, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t, z and z\n\t\t\t\t\n\t\t\t\t\tv\n\t\t\t\t into (8). Set T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t = T\n\t\t\t\t\n\t\t\t\t\t*\n\t\t\t\t.
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2.4Compute z\n\t\t\t\t* by substituting l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t and T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t into (12) and compute \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t by substituting K and T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t into (13).
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2.5Compute T\n\t\t\t\t* by substituting K, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t, z\n\t\t\t\t*, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t into (8).
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2.6If T\n\t\t\t\t* = T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t, go to step 2.7; otherwise, set T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t = T\n\t\t\t\t* and go to step 2.4.
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2.7Set T\n\t\t\t\t\n\t\t\t\t\t*\n\t\t\t\t(K,l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t)=T\n\t\t\t\t\n\t\t\t\t\tkj\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t=z\n\t\t\t\t\n\t\t\t\t\t*\n\t\t\t\t, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t=\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Compute \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t by substituting K, l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t, T\n\t\t\t\t\n\t\t\t\t\t*\n\t\t\t\t(K,l\n\t\t\t\t\n\t\t\t\t\tj\n\t\t\t\t), \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t into (7).
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2.8Let j = j+1 and go back to step 2.2 until j = b.
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Set ETC(K,l) =Min[ETC(K,l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t)|j = 0,1,…,b].
If K = 1, set ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t = ETC(K,l), K = K+1, z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t (i =1, 2,…, n) = 0, z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t = 0 and go back to step 2; otherwise go to step 5.
If ETC(K,l) < ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t, let ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t = ETC(K,l), K = K+1, z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t (i =1, 2,…, n) = 0, z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t = 0, and go back to step 2; otherwise let ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t= ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t along with the corresponding values of K and l being the value of K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t and l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, respectively.
Find T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t from (8) with the obtained K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t and l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t. Find z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t from (12) and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t from (13) with the obtained K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, and T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t. Find S\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t from (1) and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t from (2) with the obtained K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, z\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, and\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Output the values of K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, S\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t and Stop.
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4. Computer implementation
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The algorithm proposed in Section 3 has been implemented as a decision support system on a personal computer in which Intel Pentium D 2.8 GHz CPU with 1024 MB RAM inside it. Visual Basic 2005 is utilized as the software platform to develop the decision support system. Figure 2 shows the window to input the numbers of retailers, e.g., n =3 in this case, and steps of the lead-time crashing cost function, e.g., three steps in this case, involved in the replenishment and lead time reduction decisions. Once the decision maker inputs the numbers, a window to specify the relevant parameters will be displayed as shown in Figure 2. If all of the relevant parameters are input for the system, the decision maker then clicks the label “Find Solution” to select the lead time policy and to set the options of output. Figure 3 illustrates the result that the system derives the solution in which K\n\t\t\t\t* = 2, l\n\t\t\t\t* = 0.005 years, T\n\t\t\t\t* = 0.0709 years, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t=708, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t= 760, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t= 1276, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t= 3574, and ETC\n\t\t\t\t* = $19455.5 for the case of implementing QR (representing by the controllable lead time option) after the
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Figure 2.
The system window to input parameters
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Figure 3.
The replenishment and lead time reduction decisions under CRP provided by the decision support system
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Figure 4.
The coordinated replenishment decision under CRP provided by the decision support system
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Figure 5.
The “what-if analysis” function of the decision support system
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decision maker clicks “Perform”. From the value of “Saving”, the benefit derived from implementing QR can be learned. As shown in Figure 4, the system also provides the inventory decision before QR (represented by the constant lead time option). One advantage of the system is that the decision maker can learn the sensitivity of replenishment and lead time decisions and the expected annual total cost of the chain by specifying the parameter with the variation range in the Window of “What-if Analysis”. Figure 5 illustrates the output of the what-if analysis of K\n\t\t\t\t*, l\n\t\t\t\t*, T\n\t\t\t\t*, S\n\t\t\t\t*, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, ETC\n\t\t\t\t*, and the cost saving after QR against the variation of standard deviation of demand.
a: K\n\t\t\t\t\t\t0 and T\n\t\t\t\t\t\t0 represent the optimal K and T, respectively, before QR is implemented.b: ΔETC(%)=(1-ETC\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t/ETC\n\t\t\t\t\t\t0)×100%, where ETC\n\t\t\t\t\t\t0 represents the optimal ETC before QR is implemented.Sensitivity analysis for parameters
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The sensitivity of the optimal solution has been further examined by conducting several numerical experiments. The results are illustrated by a base case and are shown in Table 1. Some findings are summarized as follows:
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The amount of ΔETC(%) measures the improvement rate of expected annual total system cost after implementing QR. The behavior of ΔETC(%) reveals that the supply chain may benefit from shortening the replenishment lead time.
The value of ΔETC(%) is especially sensitive to the variation of σ\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t or h\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t. As σ\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t or h\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t increases, the value of ΔETC(%) increases. The results mean that the return of investment in QR is especially significant for a system with high uncertainty in demand or with high carrying costs incurred to the retailers.
The common shipment cycle time is more sensitive to the variance of demand, the variation of manufacturer’s production rate, the individual ordering cost, and the retailer’s carrying cost.
The shipment cycle time, T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, will be reduced after implementing QR in which the replenishment lead time for retailers has been shortened. Moreover, the target levels of replenishments for the manufacturer and the retailers may become lower after bringing QR into practice.
The number of shipments per production cycle, K\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, after implementing QR is always no less than that before implementing QR. When the values of K for the two situations are equal, the optimal shipment cycle time, T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t as well as the protection period, T\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t+l\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t*\n\t\t\t\t\t\t, after QR is always no large than those before QR. The result implies that under a fixed number of shipments during a production cycle, the protection period for the retailer will be reduced after implementing QR.
The increase of production rate will result in a longer shipment cycle time and higher target levels of replenishments for the retailers. In contrast, as the value of P increases, the number of shipment per production cycle and the target level of production for the manufacturer will become smaller.
When the ordering cost increases, the shipment cycle time and the target levels of replenishments for each party will increase.
The value of α\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t specifies the minimal fraction of demand met per cycle for the retailer and directly relates with the length of protection period (T+l). It can be found that the amount of ΔETC(%) decreases as the retailer’s maximal fraction of demand unfilled per cycle increases. The result implies that the benefit from implementing QR is significantly related to the retailer’s service level threshold and the benefit is substantial for a supply chain requesting a high customer service level.
The numerical example shows that as 1-α\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t increases from 1% to 2%, the length of common shipment cycle time increases by 2.54% (from 0.0709 to 0.0727) but the lead time is unaffected. The result implies that the effect of manufacturer’s service level threshold is more significant on the common shipment cycle time than on the reduction of lead time.
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5. Conclusions
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Speed, service, and supply chain management have been core capabilities for business competition. The reduction of overall system response time with a satisfied service level has received a great deal of attention from researchers and practitioners. In this study, we investigated the effect of investing in QR on a CRP based supply chain where a manufacturer produces and delivers items to multiple retailers at a coordinated common cycle time with the minimized expected total system cost and satisfied service levels. Extending the work of Banerjee and Banerjee (1994) by involving ordering costs and the reducible replenishment lead time, a model and an algorithm are proposed to simultaneously determine the optimal shipment cycle time, target levels of replenishments, lead time, and number of shipments pre production cycle under service level constraints for the supply chain.
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A numerical experiment along with sensitivity analysis was performed and the results explain the effect of QR on the replenishment decisions and the total system cost. The results provide the following findings about our model:
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The system can reduce the stocks throughout the pipeline and remain service levels of retailers via investing in QR initiative.
The benefit from implementing QR is especially significant for a supply chain with high uncertainty in demand or the retailers requesting high service levels or incurring high carrying costs.
The shipment cycle time will decrease after QR implemented. Additionally, the shipment cycle time is especially sensitive to the variation of manufacturer’s production rate, the individual ordering cost, the variance of demand, and the retailer’s carrying cost.
The decision of adopting QR is mainly influenced by the variance of demand and the retailer’s service level threshold. The higher the demand uncertainty or the higher the retailer’s service level threshold, the more beneficial to implement QR in supply chains.
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Acknowledgments
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This research was partially supported by the National Science Council, Taiwan ROC (Plan No. NSC 98-2410-H-415 -007).
\n\t\t
\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/18794.pdf",chapterXML:"https://mts.intechopen.com/source/xml/18794.xml",downloadPdfUrl:"/chapter/pdf-download/18794",previewPdfUrl:"/chapter/pdf-preview/18794",totalDownloads:3160,totalViews:425,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"November 7th 2010",dateReviewed:"February 28th 2011",datePrePublished:null,datePublished:"September 6th 2011",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/18794",risUrl:"/chapter/ris/18794",book:{slug:"efficient-decision-support-systems-practice-and-challenges-in-multidisciplinary-domains"},signatures:"Shu-Lu Hsu and Chih-Ming Lee",authors:[{id:"37729",title:"Prof.",name:"Shu-Lu",middleName:null,surname:"Hsu",fullName:"Shu-Lu Hsu",slug:"shu-lu-hsu",email:"slhsu@mail.ncyu.edu.tw",position:null,institution:null},{id:"37730",title:"Prof.",name:"Chih-Ming",middleName:null,surname:"Lee",fullName:"Chih-Ming Lee",slug:"chih-ming-lee",email:"cmlee@scu.edu.tw",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Notations and assumptions",level:"1"},{id:"sec_3",title:"3. Model development and solution",level:"1"},{id:"sec_4",title:"4. Computer implementation",level:"1"},{id:"sec_5",title:"5. Conclusions",level:"1"},{id:"sec_6",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBanerjee\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1986A joint economic lot size model for purchaser and vendor. Decis Sci 17\n\t\t\t\t\t3\n\t\t\t\t\t92\n\t\t\t\t\t311\n\t\t\t\t\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBanerjee\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBanerjee\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1992Coordinated, orderless inventory replenishment for a single supplier and multiple buyers through electronic data interchange. Int J Technol Manage 79(4-5):328-336.\n\t\t\t'},{id:"B3",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBanerjee\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBanerjee\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994A coordinated order-up-to inventory control policy for a single supplier and multiple buyers using electronic data interchange. Int J Prod Econ 35(1-3):85 EOF\n\t\t\t\t\t91 EOF\n\t\t\t\t\n\t\t\t'},{id:"B4",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBen-Daya\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHariga\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2004Integrated single vendor single buyer model with stochastic demand and variable lead time. Int J Prod Econ 92\n\t\t\t\t\t1\n\t\t\t\t\t75\n\t\t\t\t\t80\n\t\t\t\t\n\t\t\t'},{id:"B5",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBen-Daya\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRaouf\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994Inventory models involving lead time as a decision variable. J Oper Res Soc 45\n\t\t\t\t\t5\n\t\t\t\t\t579\n\t\t\t\t\t582\n\t\t\t\t\n\t\t\t'},{id:"B6",body:'\n\t\t\t\t\n\t\t\t\t\tChang HC, Ouyang LY, Wu KS, Ho CH\n\t\t\t\t\t2006Integrated vendor-buyer cooperative inventory models with controllable lead time and ordering cost reduction. Eur J Oper Res 170\n\t\t\t\t\t2\n\t\t\t\t\t481\n\t\t\t\t\t495\n\t\t\t\t\n\t\t\t'},{id:"B7",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChen\n\t\t\t\t\t\t\tF. Y.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKrass\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2001Inventory models with minimal service level constraints. Eur J Oper Res 134\n\t\t\t\t\t1\n\t\t\t\t\t120\n\t\t\t\t\t140\n\t\t\t\t\n\t\t\t'},{id:"B8",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChopra\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMeindl\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007Supply chain management. Pearson Education Inc., New Jersey.\n\t\t\t'},{id:"B9",body:'\n\t\t\t\t\n\t\t\t\t\tEAN International\n\t\t\t\t\t2000Continuous replenishment: how to use the EAN.UCC standard. http://www.gs1.org/docs/EDI0002.pdf.\n\t\t\t'},{id:"B10",body:'\n\t\t\t\t\n\t\t\t\t\tGoyal SK\n\t\t\t\t\t1976An integrated inventory model for a single-supplier-single- customer problem. Int J Prod Res 15\n\t\t\t\t\t107\n\t\t\t\t\t111\n\t\t\t\t\n\t\t\t'},{id:"B11",body:'\n\t\t\t\t\n\t\t\t\t\tGoyal SK\n\t\t\t\t\t1988A joint economic-lot-size model for purchaser and vendor: a comment. Decis Sci 19\n\t\t\t\t\t1\n\t\t\t\t\t236\n\t\t\t\t\t241\n\t\t\t\t\n\t\t\t'},{id:"B12",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoyal\n\t\t\t\t\t\t\tS. K.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSrinvasan\n\t\t\t\t\t\t\tG.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1992The individually responsible and rational comment. Decis Sci 23\n\t\t\t\t\t3\n\t\t\t\t\t777\n\t\t\t\t\t784\n\t\t\t\t\n\t\t\t'},{id:"B13",body:'\n\t\t\t\t\n\t\t\t\t\tLiao CJ, Shyu CH\n\t\t\t\t\t1991An analytical determination of lead time with normal demand. Int J Oper Prod Manage 7\n\t\t\t\t\t4\n\t\t\t\t\t115\n\t\t\t\t\t124\n\t\t\t\t\n\t\t\t'},{id:"B14",body:'\n\t\t\t\t\n\t\t\t\t\tIyer AV, Bergen ME\n\t\t\t\t\t1997Quick response in manufacturer-retailer channels. Manage Sci 43\n\t\t\t\t\t4\n\t\t\t\t\t559\n\t\t\t\t\t570\n\t\t\t\t\n\t\t\t'},{id:"B15",body:'\n\t\t\t\t\n\t\t\t\t\tOuyang LY, Chuang BR\n\t\t\t\t\t2000Stochastic inventory model involving variable lead time with a service level. Yugosl J Oper Res 10\n\t\t\t\t\t1\n\t\t\t\t\t81\n\t\t\t\t\t98\n\t\t\t\t\n\t\t\t'},{id:"B16",body:'\n\t\t\t\t\n\t\t\t\t\tOuyang LY, Wu KS, Ho CH\n\t\t\t\t\t2004Integrated vendor-buyer cooperative models with stochastic demand in controllable lead time. Int J Prod Econ 92\n\t\t\t\t\t3\n\t\t\t\t\t255\n\t\t\t\t\t266\n\t\t\t\t\n\t\t\t'},{id:"B17",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tOuyang\n\t\t\t\t\t\t\tL. Y.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWu\n\t\t\t\t\t\t\tK. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHo\n\t\t\t\t\t\t\tC. H.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007An integrated vendor-buyer inventory model with quality improvement and lead time reduction. Int J Prod Econ 108(1-2):349 EOF\n\t\t\t\t\t358 EOF\n\t\t\t\t\n\t\t\t'},{id:"B18",body:'\n\t\t\t\t\n\t\t\t\t\tPan JC, Hsiao YC\n\t\t\t\t\t2005Integrated inventory models with controllable lead time and backorder discount consolidations. Int J Prod Econ 93-94:387-397.\n\t\t\t'},{id:"B19",body:'\n\t\t\t\t\n\t\t\t\t\tPan JC, Yang JS\n\t\t\t\t\t2002A study of an integrated inventory with controllable lead time. Int J Prod Res 40\n\t\t\t\t\t5\n\t\t\t\t\t1263\n\t\t\t\t\t1273\n\t\t\t\t\n\t\t\t'},{id:"B20",body:'\n\t\t\t\t\n\t\t\t\t\tTaha HA\n\t\t\t\t\t2002Operations research: an introduction. Prentice Hall, Upper Sadle River, New Jersey.\n\t\t\t'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Shu-Lu Hsu",address:"",affiliation:'
Department of Management Information Systems, National Chiayi University, Taiwan
As per immunculus concept, natural antibodies (NAbs) are formed in response to gut microflora and environment in addition to self-antigens through feedback network to maintain homeostasis [1, 2, 3] bridging innate and adaptive immune response. Thus, any chronic inflammation combined with compromised central tolerance can culminate into autoimmune disease [4]. However, autoimmune diseases have gender bias with prevalence in females owing to ‘autoimmune X chromosome’ and autoimmune infertility (AI) is no exception [5]. Concomitantly, reproductive autoimmune failure could result from an activated immune system or by anti-ovarian antibodies (AOA) alone as described in endometriosis patients [6]. Other reproductive disorders such as POI, polycystic ovary syndrome (PCOS), unexplained infertility, and repeatedly unsuccessful IVF attempts may be responsible for the pathophysiology of preeclampsia or spontaneous abortions and may also have presence of multiple autoantibodies (AAbs) [7, 8, 9, 10, 11].
\n
Immune dysregulation is the cause of unexplained or idiopathic infertility in 20–30% of infertile couples [12]. AI is diagnosed when spontaneously synthesized antibodies bind or react with sperm/oocyte to prevent any one or several events: fertilization, acrosome reaction, capacitation or embryo implantation. Despite much research into organ specific biomarkers, no specific and sensitive biomarkers have been identified making detection of AI elusive. Organ-specific autoimmune disease gets treated using established protocols without sufficient consideration for fertility of women. Detection of AAbs mandates management of endometriosis, POI and other idiopathic infertility as an autoimmune disease with the treatment having adverse effects. This chapter will focus on AI, briefly in males but mainly in females, to include:
autoantigenic targets identified in female infertility with special emphasis on endometriosis and POI,
current understanding of effect of autoantibodies using animal models of disease,
including (AAbs) as diagnostic tools: current practices and
future research.
\n
\n
\n
2. Male autoimmune infertility: anti-sperm antibodies (ASA)
\n
Sperm are specialized haploid cells with autoantigenic and isoantigenic potential. Thus, ASA can be present in blood, semen, follicular fluid and cervicovaginal secretions affecting sperm movement, capacitation, fertilization and embryo implantation [13, 14]. ASA are far more frequent than oocyte antibodies.
\n
In testis, the Sertoli cells through tight junctions form the impervious blood-testis barrier of two compartments: basal and adluminal. Basal compartment, which houses spermatogonia and young spermatocytes, is connected to vasculature through phagocytic Sertoli cells, which in turn act as antigen presenting cells to induce tolerance. The adluminal surface housing sperm undergoing meiosis and spermiogenesis is segregated from vasculature. Thus leakage of autoantigens from basal compartment can potentially generate ASAs. However, the exact mechanism of ASA generation is still unclear [13]. In some cases, Human Leucocyte Antigen system is associated with ASA and AI [15]. In 0.9–4% of normal fertile adult males as well as pre-pubertal boys, ASA are found in blood serum, seminal plasma, or directly attached to sperm surface indicating these to be NAbs generating confusion on their role in human infertility [16, 17].
\n
Very few ASA are sperm specific [18] and never directed to multiple organs (except in animals). These can appear more frequently due to testicular failures: cryptorchidism, undescended testes, mobile testes and orchitis (especially due to infectious diseases such as mumps). Additionally, varicocele increases the risk of ASA production by two-fold [19]. The reduced testosterone levels due to altered Leydig cell function in undescended testes could theoretically result in reduced T regulatory cells and compromised central tolerance, however, exact mechanism is unclear. Elevated ASA could lead to low sperm count or low progressive motility. Hence, surgery at an early age, followed by steroid therapy to suppress immune reaction is recommended to prevent future infertility in cases with testicular failure.
\n
ASA could be against carbohydrate moieties and sperm antigens example integral membrane proteins (exposed due to undescended testes) mainly through molecular mimicry. Natural ASA are reported in rodents due to sperm antigenic ‘leak’ to ensure immune tolerance. ASA are generally associated with genital tract infections. Vasectomy induces AAbs to antigens of mature human sperm [20, 21] with HLABw22 and A28 having increased predisposition post vasectomy [22]. Incidence reported is 61% pre- and 73–80% post-vasectomy. Antigens could be of either testicular or epididymal origin (epididymal maturation) with Abs directed to acrosome, equatorial and postacrosomal regions, tail midpiece and sperm nucleus. This could be due to sperm leakage in either the vas or cauda epididymis [21]. AutoAbs to FA-1 antigen (44%) and protamine (28%) seen post vasectomy in sera (none in seminal plasma) with prevalence of reduced fertilization rate in vitro. These were either of IgG, M or A subclass [23]. Post vasectomy ASA are seen only in serum while in seminal plasma and ejaculate post vasovasostomy. Fertile men with no ASA before vasovasostomy will show ASA that can affect sperm count [24, 25]. Further, there is no overlap of ASA between infertile men, post vasectomy [26] and post vasovasostomy. However, there are conflicting reports on their influence on pregnancy rate [27, 28]. Table 1 enlists ASA in men with autoimmune infertility.
\n
\n
\n
\n
\n
\n\n
\n
Autoantigen
\n
Body fluid compartment
\n
Function
\n
Reference
\n
\n\n\n
\n
Nuclear autoantigenic sperm protein (NASP) histone binding
List of autoantigens in men with autoimmune infertility.
\n
High titers of IgA-ASA found in seminal plasma of infertile men bind sperm head and impair fertilizing ability, the IgG elicit opsonization, and IgM from vaginal washings of vaginitis cases reduce fertilization by 44% [13]. ASAs directed to surface antigens are clinically relevant since they affect semen quality (not morphology or count) by any one of: premature acrosome reaction making the sperm moribund, sperm agglutination leading to impairment in cervical mucus penetration, opsonization through female genital tract via complement pathway.
\n
ASA may aid sperm capacitation with no adverse effects on sperm-oocyte fusion. However, ASA binding outer acrosomal membrane proteins are washed away during procedure and do not affect IVF-intracytoplasmic sperm injection (IVF-ICSI) outcomes unlike those in females which are reported to reduce cleavage rate [47, 48, 49], with multiple autoantigenic targets necessary for AI [50].
\n
Typically in women, the mucosal immunity protects entire reproductive tract up to Fallopian tubes against incoming sperm or any microbes. Thus vaginal and cervical secretions may contain ASA due to multiple semen exposures causing autoantigenicity to seminal fluid proteins. In rare cases of Human Seminal Plasma Allergy, first exposure can elicit antibodies [51] though it is not always associated with infertility [52, 53].
\n
ASA in females are of IgG, IgA and IgE subtypes in blood, lymph and cervical-vaginal mucus [50]. IgA antibodies in the cervical secretions can bind and agglutinate sperm with eventual clearance by circulating macrophages while the predominant IgG [54] can lead to opsonization and local clearance of antibody-antigen complexes. The uterus and Fallopian tubes are also protected by circulating macrophages and NK cells that clear the incoming sperm. Thus sperm coated with IgA-ASA are unaffected unlike those by IgG which are opsonized and cleared via macrophages. Both subtypes in the mucus individually affect fertilization alone while a combination significantly affects fertilization rate [55, 56, 57, 58].
\n
IgA alloantibodies to FSH are seen in some normal fertile women and can be produced during tolerance to partner antigens (sperm proteins and shared maternal antigens) through semen [59, 60]. Patients with increased intestinal permeability in bowel inflammatory disease show higher production of ASA through molecular mimicry or epitope sharing between intestinal microbes and spermatozoa [61]. An upregulated normal mucosal immune response could lead to the elevated levels of anti-FSH IgA antibodies in IVF patients. Another possible explanation could be a deficit in producing antibodies that neutralize anti-FSH immunoglobulins, which has been noted in patients who produce ASA [62]. These results together suggest that the elevated values of anti-FSH IgA in IVF patients could represent a failure in mucosal tolerance in the genital tract, which could be genetically determined [12] (Table 2). Enlists ASA detected in sera of women.
List of autoantigenic targets against sera of women with ASA.
\n
\n
2.1. Diagnostic approaches and treatment modalities for couples with ASA
\n
Presence of ASA in serum of seminal fluid binding to sperm outer membrane antigens and thereby altering fertilization rate are relevant, is inversely correlated with pregnancy and not a good indicator of pregnancy outcome. Testing for ASA is indicated for men with genitourinary infections (e.g., Chlamydia) or acquired genital tract obstructions. Nevertheless, these ASA may not always hinder pregnancy.
\n
Sexually active homosexual individuals who have also undergone pelvic surgery should be advised to test for ASA [69]. Routine semen samples can be tested for sperm bound antibodies by IgG-mixed antiglobulin reaction (IgG-MAR [70]), immunobead test (IBT) [71] or sperm-MAR test [72]. However, none of the available diagnostic tests quantitate, are neither effective nor specific [73, 74]. Hence, instead of ineffective generalized immunosuppressive therapy IVF-ICSI should be considered [75, 76, 77, 78, 79].
\n
Post vasovasostomy couples are advised IVF for pregnancy depending on body mass index and age which affect serum testosterone levels as well as ASA in men. In these cases, IVF may be beneficial only after testing for hypogonadism and serum testosterone levels [80]. ASA post vasovasostomy can cause necrospermia and deteriorate sperm count hence IVF-ICSI using testicular sperm is an option [81].
\n
\n
\n
\n
3. Female autoimmune infertility
\n
Women are prone to autoimmune diseases due to hormonally dictated cytokine and chemokine milieu [82] often leading to other autoimmune dysfunctions [83] including reproductive autoimmune failure. Gleicher and co-workers [6] postulated that endometriosis could be an autoimmune disease and studies from our lab show 30% prevalence [84]. Commonly seen serum AAbs are anti-phospholipid, anti-nuclear, anti-thyroid, anti-annexin V, anti-prothrombin, anti-laminin, anti-ZP (Table 3 for entire list), with the high level of NK cells as the risk factors but not as those pathognomonic [85]. However, none of the AAb biomarkers tested were effective [86]. A recent study reported better sensitivity of 6 new biomarkers [87]. With detection of AAbs to steroid producing cells and thyroglobulin in cases with concomitant adrenal or thyroid disease in PCOS, it is now considered an autoimmune disease. However, anti-ovarian antibodies were reported in only one study [7, 88] with no clarity on their role in PCOS pathogenesis [89]. Organ-specific AAbs such as ovary, adrenal and thyroid (endocrine autoimmune) disease are reported to cause infertility due to premature ovarian insufficiency (POI) [90].
List of autoantigenic targets against sera of women with reproductive infertility.
\n
Both PCOS and endometriosis are also causative factors of POI. 40–60% women with endometriosis possess anti-ovarian Abs in addition to anti-endometrial Abs [103]. Several AAbs to non-organ specific targets are seen in women with unexplained infertility [104]. Further, 22% of patients with SLE show anti-corpus luteum antibodies and elevated FSH levels typical of POI [57] and 60% POI cases are of autoimmune origin [105, 106]. POI is typically detected late with both non-organ and organ-specific antibodies in conjunction with an autoimmune disease thus evading a specific and accurate biomarker for diagnosis and prognosis [107, 108]. Whether AAbs are causative of or a by-product of underlying disease is unclear.
\n
Nevertheless, elaborate animal models of the disease as well as case studies have provided relevant data. Day three neonatal thymectomy mouse model showed that multi-organ autoimmune disease prevails. Immunization with a single antigen causes oophoritis alone while those to multiple antigens completely compromises ovarian function. Additionally, concomitant presence of the autoantigens was mandatory [109].
\n
Efforts to identify target autoantigens based on discovery of an ovary specific autoantigen by ELISA, immunofluorescence or immunohistochemistry approach were unfruitful. This interference was due to non-specific reactivity of natural albumin antibodies [110]. Attempts to identify target autoantigens using sera and proteomics approach were fruitful enough to identify several somatic proteins: alpha actin, alpha actinin-4, heat shock proteins 70 and 90β in 30% of POI and 26% of IVF-ET failure cases [100, 111, 112]. Of these, 47% cases showed presence of AAbs to HSP90β. Reactivity of these antibodies was seen against several follicular components (Table 4). Note, besides oocyte the corpus luteum seems to be a major cellular target while HSP90β the molecular target contributing to early POI (bold and italics in Table 4) [111]. AAbs to MATER led to assuming it to be an ovary specific target [113] however, these AAbs were also seen in idiopathic hypoparathyroidism cases only in context of autoimmune polyendocrinopathy-candidiasis-ectodermal dystrophy syndrome [114].
\n
\n
\n
\n
\n
\n\n
\n
Condition
\n
Age at detection
\n
Cellular target
\n
Molecular target
\n
\n\n\n
\n
POI
\n
22
\n
Oocyte, theca, corpus luteum
\n
90
\n
\n
\n
33
\n
Oocyte, corpus luteum
\n
30
\n
\n
\n
38
\n
45, 90
\n
\n
\n
24
\n
90, 97
\n
\n
\n
33
\n
Oocyte, theca, corpus luteum
\n
97
\n
\n
\n
90
\n
\n
\n
39
\n
Oocyte, theca
\n
\n
\n
33
\n
Theca
\n
120
\n
\n
\n
38
\n
Oocyte
\n
90
\n
\n
\n
33
\n
Ooplasm and nucleus of oocyte, theca
\n
\n
\n
35
\n
Oocyte
\n
55
\n
\n
\n
36
\n
97
\n
\n
\n
35
\n
Oocyte of primordial follicle
\n
70, 75
\n
\n
\n
32
\n
Oocyte
\n
70
\n
\n
\n
35
\n
Granulosa, corpus luteum
\n
30, 45
\n
\n
\n
IVF-ET
\n
29
\n
Oocyte, corpus luteum
\n
97
\n
\n
\n
28
\n
120
\n
\n
\n
39
\n
30, 90
\n
\n
\n
32
\n
Oocyte
\n
120
\n
\n
\n
34
\n
50, 75, 90
\n
\n
\n
30
\n
Oocyte, granulosa
\n
80, 97
\n
\n
\n
28
\n
Oocyte
\n
120
\n
\n
\n
31
\n
Theca
\n
45, 97
\n
\n
\n
29
\n
Oocyte
\n
90, 120
\n
\n
\n
32
\n
30, 50, 90
\n
\n
\n
90
\n
\n
\n
33
\n
Oocyte
\n
90, 97
\n
\n
\n
30
\n
Zona pellucida
\n
45
\n
\n\n
Table 4.
List of antigens and cellular targets detected using sera of women with premature ovarian insufficiency (POI) and in vitro fertilization-embryo transfer (IVF-ET); compiled from [97].
\n
Though a 75–90% accuracy was observed in ELISA assays using immunodominant epitopes from the identified targets, the AAbs were also present in normal population, highlighting the fact that these were NAbs. These were also validated to induce aPOI in a mouse model. The immunodominant epitopes tested were able to induce POI and alter ovarian cytoarchitecture. Folliculogenesis was severely affected at each developmental stage with gross lack of mature Graafian follicles and a persistent corpus luteum [101].
\n
AAbs to a single immunodominant epitope (EP6) HSP90β led to 9% dissociated oocyte-cumulus complexes, granulosa cells undergoing apoptosis, 48% empty follicles, and 12% degenerated follicles. These animals demonstrated significant pre- and post-implantation loss with concomitant decrease in fertility index along with an increased polymorphonuclear cell infiltration of the ovarian follicles. The infiltration may have contributed to generation of antibodies against the EP6 peptide [115, 116].
\n
In normal physiological inflammatory processes like ovulation, follicular atresia, corpus luteum regression and tissue remodeling, the ovarian leukocytes like T cells and macrophages play an important role [117, 118]. Interestingly, NAbs especially, IgM play a role in clearing apoptotic cells, maintaining B cell homeostasis, inflammation, atherosclerosis and autoimmunity. Any drop in IgM levels is associated with ineffective clearance of apoptotic cells culminating into autoimmune disease. Alternatively, strong and persistent recognition of apoptotic cells by such NAbs may overactivate the immune system and cause chronic inflammation [3]. Corticosteroid treatment resolves the ensuing infertility [119]. However, there are no randomized controlled trials (RCT) to date. Our animal studies showed high dose corticosteroid was better able to rescue fertility in mice immunized with immunodominant epitopes of HSPA5 (Table 5). An interesting finding was the epitope spreading observed: AAbs to HSPA5 cross-react with immunodominant epitope (EP6) of HSP90β at high titer [120]. Thus, autoreactivity to HSP90β could have diagnostic and prognostic value.
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
\n
AutoAb target
\n
Cellular target
\n
Effect on estrus cycle
\n
Delay in vaginal plug
\n
Preimplantation loss
\n
Fertility reduction
\n
Effect of corticosteroid treatment
\n
\n\n\n
\n
Alpha actinin-4
\n
Ooplasm, theca and corpus luteum
\n
Not determined
\n
30%
\n
24%
\n
32%
\n
44%
\n
\n
\n
HSPA5
\n
Ooplasm, granulosa, theca and corpus luteum
\n
–
\n
44%
\n
\n
\n
Alpha actin
\n
Ooplasm, granulosa and theca,
\n
30%
\n
36.4%
\n
\n
\n
HSP90-beta (EP6)
\n
Granulosa cells, developing embryo
\n
Not significant
\n
Not determined
\n
\n
\n
\n
\n
\n
MATER/NALP5 (parathyroid autoantigen)
\n
oocytes of later-stage small follicles
\n
Not determined
\n
\n
\n
\n
\n\n
Table 5.
Effect of autoantibodies on fertility and extent of rescue with corticosteroid therapy.
\n
Thyroid autoimmunity is commonly found with other systemic autoimmune diseases [121, 122] and is associated with anti-phospholipid syndrome (APS) due to anti-phospholipid antibodies [123] which in turn mediate recurrent miscarriages common to APS [124]. Thus women with thyroid autoimmunity and APS have greater risk of recurrent miscarriages mandating screening for anti-phospholipid antibodies. AAbs to ANA (12%), ANCA (20%), AECA (24%), ACLA (8%), anti-dsDNA (0%), β2 microglobulin (14%), and anti-HLA antibodies (10%) have been reported among Indian RSA patients [125]. This indicates that women with thyroiditis, endometriosis, SLE, APS also run the risk of repeated miscarriages.
\n
At least 20–30% of POI cases have an additional autoimmune disorder [126] including several endocrinopathies, thyroid diseases, Addison’s disease, rheumatoid arthritis and polyglandular syndrome with greater prevalence of thyroid autoimmunity (14–27% at initial diagnosis) and thyroid peroxidase AAbs [127, 128]. At least 10% women with Addison’s disease manifest AAbs to 21- or 17-hydroxylase and autoimmune oophoritis [129]. Thyroid peroxidase antibodies (TPO Abs) are also prevalent in PCOS cases. Thus, these along with HSP90β could be included in an antibody detection panel.
\n
In women with endometriosis, use of biomarkers including CA-125 for diagnosis of endometriosis was prohibited [130, 131]. However as per recent guidelines, use of biomarkers has been recommended for both diagnosis and disease monitoring [132] and is still a researchable area. Anti-endometrial antibodies exist but their sensitivity and accuracy varies from 0 to 100% [131, 133, 134].
\n
\n
3.1. Treatment modalities and management of autoimmune infertility
\n
Endometriosis management guidelines are valid for women with mild to moderate disease and do not recommend hormonal therapy for managing ovulation to improve fertility rate [135]. Despite reduction in ovarian function, one time laparoscopic operation to remove endometriosis and improve pregnancy rates is often recommended [136, 137]. Adjunctive hormonal therapy is prohibited pre- or post-surgery to improve pregnancy rates [138]. Intra uterine insemination along with controlled ovarian stimulation is recommended 6 months post-surgery since it shows similar pregnancy rates as that of women with unexplained infertility [139]. ART can also be recommended especially in cases of tubal factor or male factor infertility as controlled ovarian stimulation does not increase chances of recurrence of endometriosis after IVF/ICSI [140, 141, 142, 143] however, it may not always be effective [144, 145].
\n
POI seems to be an end-stage disease in women with an autoimmune disorder since it is detected at a late stage when the ovary has been substantially ravaged with little scope for fertility management. Thus treatment options for fertility management of women with POI are limited. Counseling for early marriage and pregnancy to complete the family is applicable only in case of early diagnosis or known familial origin. Other options include egg donation and IVF-ICSI or surrogacy. The women are administered corticosteroids in case of known autoimmune disease diagnosis and advised IVF-ICSI when AAb titers fall. However, this is not an option since it entails risk of osteoporosis and iatrogenic Cushing’s syndrome [119]. In most cases, adoption is the only option along with psychological counseling and cardiovascular and bone health management of hypoestrogenism effects [146].
\n
Additionally, there should be efforts to increase awareness among reproductive endocrinologists to recommend testing for undiagnosed autoimmune disease to couples on a case basis before embarking on ART-IVF [147].
\n
\n
\n
\n
4. Future research
\n
Presence of AAbs is hallmark of autoimmune disease with no clarity on their role in disease pathogenesis and ensuing AI. With few exceptions these are not organ-specific indicating them to be NAbs [148, 149, 150, 151]. Obtaining clarity on role of AAbs will guide further treatment modalities for patients with AI [93, 101, 152]. Global high dose immunosuppressive therapy seems to be the only effective option for autoimmune reproductive failure despite its shortcomings [153, 154].
\n
Targeted interventional therapy by inducing antigen-specific tolerance is another option [155, 156]. Till such a time as a definitive therapy is available, pan autoimmune disease diagnostic panels can be designed using autoantigenic targets (recombinant proteins or peptides) such as β2-glycoprotein I and HSP90β (EP6) [151, 157, 158, 159] followed by management with corticosteroid therapy. A loss of reactivity to key autoantigens (predetermined to affect ovarian function) would serve as biomarkers to better manage immunosuppressant therapy.
\n
\n
\n
5. Conclusion
\n
The very lack of any organ-specific biomarker till date along with the preponderance of NAbs indicates that warped self-tolerance would lead to AI. AAbs in females alone appear to be significant in AI. Fertility studies need to be undertaken to gauge effect of such AAbs identified thus far and immunodominant epitopes gleaned could prove useful to design a pan autoimmune disease diagnostic peptide array to manage AI. Global immunosuppressant therapy and IVF-ICSI are the only current hope for such couples.
\n
\n
Acknowledgments
\n
The work was co-funded by Indian Council of Medical Research and Dept. of Biotechnology, Govt. of India.
\n
Conflict of interest
None.
\n',keywords:"autoantibodies, premature ovarian insufficiency, endometriosis, autoimmune infertility",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/59246.pdf",chapterXML:"https://mts.intechopen.com/source/xml/59246.xml",downloadPdfUrl:"/chapter/pdf-download/59246",previewPdfUrl:"/chapter/pdf-preview/59246",totalDownloads:599,totalViews:316,totalCrossrefCites:0,dateSubmitted:"July 13th 2017",dateReviewed:"January 12th 2018",datePrePublished:"March 2nd 2018",datePublished:null,dateFinished:null,readingETA:"0",abstract:"Autoimmune diseases have gender bias with predominance in females, autoimmune infertility (AI) being no exception. This chapter will focus on AI in females with brief reference to the same in males. Autoimmune diseases have established protocols for detection and management of ensuing infertility, however similar protocols for unexplained infertility [tubal blockage, endometriosis, premature ovarian insufficiency (POI), undiagnosed underlying autoimmune disease (Sjögren’s syndrome, IBS, celiac disease) and tubal blockage] are not established. Endometriosis and POI, in particular, have autoimmune etiology yet lack specific and sensitive biomarkers for accurate diagnosis. If autoantibodies are indeed diagnosed, then treatment regimen focuses on AI which has known adverse effects. The detection of natural antibodies as autoantibodies presents a viable alternative to organ specific biomarker panel for better management of AI.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/59246",risUrl:"/chapter/ris/59246",signatures:"Kaushiki M. Kadam, Purvi Mande and Asmita Choudhury",book:{id:"6243",title:"Autoantibodies and Cytokines",subtitle:null,fullTitle:"Autoantibodies and Cytokines",slug:"autoantibodies-and-cytokines",publishedDate:"April 24th 2019",bookSignature:"Wahid Ali Khan",coverURL:"https://cdn.intechopen.com/books/images_new/6243.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"64042",title:"Dr.",name:"Wahid Ali",middleName:null,surname:"Khan",slug:"wahid-ali-khan",fullName:"Wahid Ali Khan"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Male autoimmune infertility: anti-sperm antibodies (ASA)",level:"1"},{id:"sec_2_2",title:"2.1. Diagnostic approaches and treatment modalities for couples with ASA",level:"2"},{id:"sec_4",title:"3. Female autoimmune infertility",level:"1"},{id:"sec_4_2",title:"3.1. Treatment modalities and management of autoimmune infertility",level:"2"},{id:"sec_6",title:"4. Future research",level:"1"},{id:"sec_7",title:"5. Conclusion",level:"1"},{id:"sec_8",title:"Acknowledgments",level:"1"},{id:"sec_11",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Poletaev A, Boura P. The immune system, natural autoantibodies and general homeostasis in health and disease. Hippokratia. 2011;15(4):295-298. PMCID: PMC3876841\n'},{id:"B2",body:'Siloşi I, Siloşi CA, Boldeanu MV, Cojocaru M, Biciuşcă V, Avrămescu CS, Cojocaru IM, Bogdan M, Folcuţi RM. The role of autoantibodies in health and disease. Romanian Journal of Morphology and Embryology. 2016;57(2):633-638. PMID: 27833954\n'},{id:"B3",body:'Panda S, Ding JL. Natural antibodies bridge innate and adaptive immunity. Journal of Immunology. 2015;194:13-20. DOI: https://doi.org/10.4049/jimmunol.1400844. PMID: 25527792\n'},{id:"B4",body:'Weiss G, Goldsmith LT, Taylor RN, Bellet D, Taylor HS. 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High frequency of natural autoantibodies mice in normal Newborn. Journal of Immunology. 1985;134(2):765-771. PMID: 4038410\n'},{id:"B149",body:'Dighiero G, Guilbert B, Fermand JP, Lymberi P, Danon F, Avrameas S. Thirty-six human monoclonal immunoglobulins with antibody activity against cytoskeleton proteins, thyroglobulin, and native DNA: Immunologic studies and clinical correlations. Blood. 1983;62(2):264-270. PMID: 6409187\n'},{id:"B150",body:'Savitskaya YA, Duarte C, Marín N, Téllez R, Alfaro A, Ibarra C. Identification of circulating natural antibodies against endogenous mediators in the peripheral blood sera of patients with osteoarthritis of the knee: A new diagnostic frontier. Journal of Molecular Biomarkers & Diagnosis. 2012;3:135. DOI: 10.4172/2155-9929.1000135\n'},{id:"B151",body:'Zhen X, Qiao J, Li R, Wang L, Ping L. Serologic autoimmunologic parameters in women with primary ovarian insufficiency. BMC Immunology. 2014;15:11. PMID: 24606591. DOI: 10.1186/1471-2172-15-11\n'},{id:"B152",body:'Cline AM, Kutteh WH. Is there a role of autoimmunity in implantation failure after in-vitro fertilization? Current Opinion in Obstetrics & Gynecology. 2009;21(3):291-295. PMID: 19469047\n'},{id:"B153",body:'Boumpas DT, Chrousos GP, Wilder RL, Cupps TR, Balow JE. Glucocorticoid therapy for immune mediated diseases: Basic and clinical correlates. Annals of Internal Medicine. 1993;119:1198-1208. PMID: 8239251\n'},{id:"B154",body:'Boumpas DT, Austin HA 3rd, Vaughan EM, Yarboro CH, Klippel JH, Balow JE. Risk for sustained amenorrhea in patients with systemic lupus erythematosus receiving intermittent pulse cyclophosphamide therapy. Annals of Internal Medicine. 1993;119(5):366-369. PMID: 8338289\n'},{id:"B155",body:'Otsuka N, Tong Z-B, Vanevski K, Tu W, Cheng MH, Nelson LM. Autoimmune oophoritis with multiple molecular targets mitigated by transgenic expression of mater. Endocrinology. 2011;152(6):2465-2473. PMID: 21447630. DOI: 10.1210/en.2011-0022\n'},{id:"B156",body:'Zhang D, Tu E, Kasagi S, Zanvit P, Chen Q, Chen W. Manipulating regulatory T cells: A promising strategy to treat autoimmunity. Immunotherapy. 2015;7:1201, 11. PMID: 26568117-1211. DOI: 10.2217/imt.15.79\n'},{id:"B157",body:'Vojdani A. Antibodies as predictors of complex autoimmune diseases. International Journal of Immunopathology and Pharmacology. 2008;21:267-278. PMID: 18547471. DOI: 10.1177/039463200802100203\n'},{id:"B158",body:'Busnelli A, Paffoni A, Fedele L, Somigliana E. The impact of thyroid autoimmunity on IVF/ICSI outcome: A systematic review and meta-analysis. Human Reproduction Update. 2016;22(6):775-790. PMID: 27323769. DOI: 10.1093/humupd/dmw019\n'},{id:"B159",body:'Unuane D, Velkeniers B, Anckaert E, Schiettecatte J, Tournaye H, Haentjens P, Poppe K. Thyroglobulin autoantibodies: Is there any added value in the detection of thyroid autoimmunity in women consulting for fertility treatment? Thyroid. 2013;23(8):1022-1028. PMID: 23405888. DOI: 10.1089/thy.2012.0562\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Kaushiki M. Kadam",address:"kaushikikadam@gmail.com",affiliation:'
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