A multi-item inventory system with expected shortage level-dependent backorder rate with working capital and space restrictions

Article history: Received 1 June 2011 Received in revised form July, 29, 2011 Accepted 30 July 2011 Available online 3 August 2011 In this paper, a new multi-item inventory system is considered with random demand and random lead time including working capital and space constraints with three decision variables: order quantity, safety factor and backorder rate. The demand rate during lead time is stochastic with unknown distribution function and known mean and variance. Random constraints are transformed to crisp constraints with using the chance-constrained method. The Minimax distribution free procedure has been used to lead proposed model to the optimal solution. The shortage is allowed and the backlogging rate is dependent on the expected shortage quantity at the end of cycle. Two numerical examples are presented to illustrate the proposed solution method. © 2012 Growing Science Ltd. All rights reserved


Introduction
The shortage cost calculation is an important problem in estimating the inventory systems costs including purchasing, set up, holding, stock out costs and, etc. Shortage cost is divided to the backorder and lost sale. Furthermore, Lead time management is a significant issue in production and operation management. As stated in Tersine (1994), lead time usually comprises several components, such as set up time, waiting time, move time and queue time. In many practical situations, lead time can be reduced using an added crashing cost. In other words, lead time is controllable. Liao and Shyu (1991), Ben Daya and Rauf (1994), , Park (2007) considered lead time as a variable and controlled it by paying extra crashing cost. They assumed that the lead time could be decomposed into n mutually independent components where each component has fixed crashing cost. Besides Callego and Moon (1993) assumed unfavorable lead time demand distribution and solved both the continuous and periodic review models with a mixture of backorder and lost sale using Minimax distribution free method.
There are some multi-product inventory models with shortage including restrictions on inventory investment, space or reorder work load. Brown and Gerson (1967) proposed some models for multi-item stochastic inventory system with the limitation on total inventory investment. Schrady and Choe (1971) proposed a model with the total time weighted shortages with limitations on inventory investment and reorder work load. Gardner (1983) developed models for minimizing expected approximate backordered sales with the restrictions on aggregate investment and replenishment work load. Schroeder (1974) presented a model constrained by total expected annual ordering with an objective of minimizing the expected number of unit's backordered per year.
In many real-world situations, during a shortage period, the longer the waiting time is, the smaller the backlogging rate is. For instance, for fashionable commodities and high-tech products with a product life cycle, the willingness for a customer to wait for backlogging is diminishing with the length of the waiting time. In this way, the researcher used β as the function of τ, the time remaining until the next replenishment. Montgomery et al. (1973) proposed linear function for β τ . Abad (1996) introduced exponential β τ originally, but Papachristors and Skouri (2000) referred it as exponential. Abad (1996) proposed rational β τ for the first time and then San Jose et al. (2005) and Silica et al. (2007) used this form and the first to use this form to it. Silica et al. (2009) proposed mixed exponential β τ in their study. Some other authors considered β as a function of expected shortage quantity at the end of cycle. Their studies are based upon this assumption, which the larger amount of the expected shortage at the end of cycle, the smaller amount of customer can wait and hence the smaller backorder rate would be. Ouyang and Chaung (2001) were first to introduce this assumption in their model and some other authors generalized this assumption in their models for backorder rate (Lee, 2005;Lee et al., 2007;Lee et al. , 2006).
Many models for continuous inventory system with stochastic demand and allowable stock out such as Hadley and Whithin (1963), Parker (1964), Tinareli (1983 and Yano (1976) and some models in the stochastic demand and stochastic lead time environment such as Ord and Bagchi (2006), Burgin (2007) have been studied to find optimal solution on order quantity and backorder rate, which depends on expected shortage quantity and reorder point, which is replaced by safety factor. Ouyang and Chaung (2001) observed that many products of well-known brand and modish goods like certain brand gum shoes and clothes may lead to a state in which clients prefer their demands to be backordered, whereas shortage happened. Doubtlessly, if the amount of shortage exceeds the waiting patience of client, some clients avoid the backorder case. This phenomenon reveals that as shortage occurs, in the stochastic demand and deterministic lead time area, the longer the length of lead time is the larger amount of shortage is, the smaller proportion of customers can wait and hence the smaller backorder rate would be. However, in our new suggested model, in the stochastic lead time and stochastic demand environment, we consider safety factor, order quantity and backorder rate as the decision variables and assume that the backorder rate is dependent on the expected shortage quantity at the end of cycle. Thus, the larger amount of the safety factor is, the larger amount of safety stock is, the larger holding cost is, the smaller amount of expected shortage quantity through the stochastic lead time is, the larger back order rate is and therefore, the smaller stock out cost would be. Therefore, our suggested model balances holding cost and stock out cost to minimize the total expected cost (TECU).
In this paper, first we consider multi product inventory system with three variables (order quantity, backorder rate and safety factor) with space and working capital constraints. We assume that the probability distribution of demand during lead time is unknown, but mean and variance of demand are known. Then, we utilize Minimax distribution free method to minimize the total expected cost per unit time. We transform random working capital and random space constraints to crisp constraints with using the chance-constrained method and then, solve the problem with Lagrange's multiplier method. At the end of paper, we present two numerical examples to illustrate our solution procedure.

Notation and assumption
The following notations have been used in this paper:

Parameters:
Penalty cost per unit for i-th item Marginal profit per unit for i-th item Purchasing cost per unit for i-th item Space used per unit for i-th item Ordering cost per order for i-th item Backorder parameter ( 0) for i-th item Maximum inventory investment for all items Maximum available space for all items

Decision variables:
Order quantity for i-th item Reorder point (which is replace by safety factor) for i-th item Safety factor for i-th item The fraction of demand which is backordered during stockout period for i-th item

Random variables:
Demand rate per unit time for i-th item Length of lead time for i-th item Demand during lead time for i-th Maximum value of x and 0 · Mathematical expectation The developed model is based on these assumptions:  Shortage is allowed and partially backlogged.  Demand rate is a random variable with mean and standard deviation .  Lead time is randomly distributed with mean and standard deviation .  Demand during lead time is convolution of the demand rate and lead time. If demand rate, , and lead time, , be independent to each other the mean and variance of is (Tersine (1994) . (2)  The reorder point is the expected demand during lead time plus safety stock (ss) and (standard deviation of lead time) i.e. where is safety factor satisfying , represents the standard normal random variable and represents the allowable stockout probability during lead time .
 Inventory is continuously reviewed. The replenishments are made whenever the inventory level falls to the reorder point .  The purchasing cost for i-th item is paid at the time of order received.

Model formulation
The system manager places an order of amount for i-th item, when the inventory level reaches to reorder level. The expected demand during shortage at the end of cycle is: (3) The expected net inventory level at the end of cycle is calculated by (see Fig. 1): (4) So, the expected number of backorder at the end of cycle is and the expected number of lost sale at the end of cycle is 1 . The expected net inventory level just before the order arrives is + 1 and the expected net inventory level at the beginning of the cycle is + 1 and holding cost per cycle is calculated as follow (first we do not consider backorder rate): In the above equations, for calculating holding cost per cycle, we don't consider the fraction demand during stockout period which will be backordered (β). It means that all of the demands are backordered. If we consider (β), we have lost sales and holding cost per cycle will be changed as follow (see Appendix A): Thus, the mathematical model of the expected cost per cycle can be expressed by: Therefore, the total expected cost per unit time (TECU) is simply calculated by multiplying Eq. (8) in the expected number of cycle and model is transformed as follow: , , ordering cost + holding cost + stockout cost , , 2 1 1 We consider backorder rate as a variable, which is dependent on the expected shortage quantity at the end of cycle. It means that when shortage occurs, the larger amount of shortage is, the smaller ratio of client can wait and therefore, the smaller backorder rate would be. Thus, backorder will be function of the expected shortage quantity, which can be expressed as follow: Backorder parameter ( is a positive constant, which exhibits the importance of shortage in calculating backorder rate ( . Our objective is to minimize total expected cost per unit time with two restrictions, working capital and space and our model is formulated as follow: 1 . According to Tersine (1994), the distribution of demand is normal at the factory level; the Poisson at the retail level; and the exponential at wholesale and retail level. In addition, the distribution of lead time may be gamma, exponential geometric and normal. Bagchi et al. (1986) discussed elaborately on these topics. We relaxed the assumption on the distribution of demand rate lead time and demand during lead time with the following assumptions: 1. The distribution function of belong to the class of distribution function with finite mean and standard deviation .
2. The distribution function of belong to the class of distribution function with finite mean and standard deviation .
3. The distribution function of belong to the class of distribution function with finite mean and standard deviation .
Lemma 1: Callego and Moon (1993) where is overcapacity and is random variable with mean and standard deviation . Using the above lemma for any , it can be deduced that (see Appendix B): Then, with considering the definition of and the above inequality, Eq. (13), we have: If the purchasing cost of i-th item is paid at the time of receiving order then the problem can be formulated by objective function with the random working capital and random space constraints, which are given below (see Appendix B): This problem can be solved with using several methods. We use chance-constrained programming technique in this paper, which is explained in proposition 1: Perposition1: (chance-constrained) as the name indicates, the chance-constrained programming technique can be used to solve problems involving chance constrained, i.e. constraints having finite probability of being violated, this technique originally developed by Charnes and Cooper (1959).if and are the probabilities of non-violation of the constraints then the constraints can be written as: First, we use chance-constrained technique for space constraint: We can solve this model with Lagrange multiplier method. Therefore, the Lagrange function will be: , , , , To minimize the above unconstrained function, the Kuhn-Tucker condition for the minimization of a function subject to two inequality constraints are invoked as follow: , , , 0 1 … , , , 0 1 … is obtained as follow: The solution procedure is as follow: Step1 obtain from partial derivative of Eq. (27), in Eq. (31). Put 1 … in Eq. (28), Eq. (29) and Eq. (30) .

Example1:
To illustrate the developed model, consider the numerical data, which has been stated in Table 1 (two items have been considered). Table 2 shows the distribution functions of monthly demand and lead time in days. These two distribution functions are independent. The maximum inventory investment is 15000 $ and the total space is 13000 . We consider 0.9 0.9.  In the Table below, we consider different values for backorder parameter, which is shown the importance of shortage for i-th item. When importance of shortage for i-th item is increased, the safety factor for i-th item will be increased, consequently. It means that the safety stock and holding cost is increased and optimal ordering quantity is decreased by Eq. (31). Table 3 shows that the larger amount of backorder rate is, the smaller amount of total expected cost (TECU) would be.   Table 5. According to Table 5, the bigger total space and total inventory investment is, the smaller TECU (total expected cost per unit time) would be.

Conclusion
The purpose of this study was to extend multi product inventory system by adding two limitations (working capital and space) and considering backorder rate as a decision variable which is dependent on the expected demand during shortage. It means that when shortage happen, the larger amount of shortage is, the smaller ratio of client can wait and therefore, the smaller backorder would be. We supposed that distribution of demand during lead time was unknown, but the mean and variance of demand during lead time was known. With this assumption, we utilized minimax distribution free method to minimize our model. With chance programming method, we transformed our random constraints to the crisp constraints. Then, we applied Lagrange multiplier method to solve our model. At the end of paper, we prepared two numerical examples and compared them together. We assume that 1 Therefore, our model is reduced to: