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We study inventory systems with product recovery. Recovered items are as-good-as-new and satisfy the same demands as new items. The demand rate and return fraction are deterministic. The relevant costs are those for ordering recovery lots, for ordering production lots, for holding recoverable items in stock, and for holding new/recovered items in stock. We derive simple formulae that determine the optimal lot sizes for the production/procurement of new items and for the recovery of returned items. These formulae are valid for finite and infinite production rates as well as finite and infinite recovery rates, and therefore more general than those in the literature. Moreover, the method of derivation is easy and insightful.

Product recovery (repair, refurbishing, remanufacturing) is receiving increasing attention. In the past, engagement in recovery activities was often driven by legislation or by associated environmentally friendly image. But nowadays, the main reason for companies to become involved with product recovery is economical. Being active in product recovery reduces the need for virgin materials and thus leads to reduced costs. Our attention is focused on Original Equipment Manufacturers that are involved with product recovery. These differ from specialized recovery companies in that they also produce/procure new items. Moreover, we assume that recovered items are as-good-as-new and sold on the same market as new items. See Fig. 1. We address the problem of determining optimal lot sizes for production/procurement and recovery. This problem was first studied by Schrady (1967). He analyzes the problem in the traditional Economic Order Quantity (EOQ) setting: deterministic and continuous demand and return, infinite production and recovery rates. The objective is to minimize the total cost per time unit for placing orders and for holding inventory (different holding cost rates for recoverable items and serviceable items). He considers policies that alternate one production lot with a fixed number R of recovery lots ((1,R) policies for short), and derives a pair of simple EOQ formulae. Mabini, Pintelon, and Gelders (1992) discuss an extension of Schrady's model to a multi-item case. Teunter (2001) generalizes the EOQ formulae for (1,R) policies from Schrady (1967) by including a disposal option for non-serviceable items, and by using different holding cost rates for produced and recovered serviceable items. Furthermore, he also derives EOQ formulae for policies that alternate a fixed number P of production lots and one recovery lot ((P,1) policies for short). Richter, 1996a and Richter, 1996b also includes a disposal option (though a sub-optimal constant disposal rate is required) and studies both (1,R) and (P,1) policies. However, his model differs from that of Schrady, 1967 and Teunter, 2001 in the existence of a collection point, where used items are collected and from there returned in batches. Richter derives a formula for the total average cost, but simple formulae for the optimal lot sizes are not obtained. Nahmias and Rivera (1979) study an EPQ variant of Schrady's model with a finite recovery rate. The production rate is still infinite. They assume that the recovery rate is larger than the demand rate, and derive lot-sizing formulae for (1,R) policies. Koh, Hwang, Sohn, and Ko (2002) also assume that the production rate is infinite and that the recovery rate is finite. Their study is more general that that of Nahmias and Rivera (1979), since they allow the recovery rate to be both smaller and larger than the demand rate, and they consider both (P,1) and (1,R) policies. For all four combinations, they derive a closed-form expression for the average total cost which can be used to determine the optimal lot sizes numerically. We remark that the (1,R) policies proposed by Koh et al. (2002) differ from those proposed by Nahmias and Rivera (1979) (which are generalizations of Schrady's policies) in the timing of the recovery lots. As is explained in Appendix A, the (1,R) policies in Nahmias and Rivera (1979) are better if the holding cost rate for serviceable items is larger than that for recoverable items, which is usually the case since recovery adds value to an item. To summarize the above mentioned findings, lot-sizing formulae have been derived for both (P,1) and (1,R) policies, for infinite and finite recovery rates, but only for infinite production rates. In this paper, we derive more general lot-sizing formulae for both policies. (The (1,R) policies are generalizations from those in Nahmias and Rivera (1979).) These formulae are also valid for finite production rates. Furthermore, the formulae are derived in an insightful graphical way, avoiding the tedious calculations in previous work. The remainder of this paper is organized as follows. The model and the assumptions are described, and notations are introduced in Section 2. Lot-sizing formulae for (P,1) and (1,R) policies are derived in 3 and 4, respectively. Their use is illustrated in Section 5. We end with conclusions and discuss limitations in Section 6.

We derived square-root formulae that determine the optimal production and recovery lot-sizes for two classes of policies: (1,R) and (P,1). Moreover, we proposed simple modifications of resulting lot sizes that ensure discreteness of R or P. These results are quite general, since they are valid for finite and infinite production rates as well as finite and infinite recovery rates. Furthermore, they are easy to apply in practice due to their simple form and due to the simple structure of the policies. A limitation of the considered policies is that they seem too rigid for cases with high variations in the demand rate and in the return fraction. In such cases, it is better to use a policy that allows variations in the number of production or recovery lots per cycle. One reasonable policy is as follows: if the stock drops to 0, place a recovery order of size Qr if enough recoverable items are available, and place a production order of size Qp otherwise. For such a policy, the formulae that we presented can still be used to as approximations for the optimal lot-sizes. See also Teunter (2002).