#### Abstract

This paper allows the backorder rate as a control variable to widen applications of a continuous review inventory model. Moreover, we also consider the backorder rate that is proposed by combining Ouyang and Chuang (2001) (or Lee (2005) with Pan and Hsiao (2001) to present a new form. Thus, the backorder rate is dependent on the amount of shortages and backorder price discounts. Besides, we also treat the ordering cost as a decision variable. Hence, we develop an algorithmic procedure to find the optimal inventory policy by minimax criterion. Finally, a numerical example is also given to illustrate the results.

#### 1. Introduction

In most of literatures dealing with inventory problems, either in deterministic or probabilistic model, lead time is viewed as a prescribed constant or a stochastic variable, and is not subject to control (see, e.g., Naddor [1], Liberatore [2], Magson [3], Kim and Park [4], Silver and Peterson [5], Foote et al. [6], Azoury and Brill [7] and Chiu [8]). However, as pointed out by Tersine [9], lead time usually comprises several components, such as setup time, process time, wait time, move time, and queue time. In many practical situations, lead time can be reduced using an added crash cost; in other words, it is controllable. By shortening the lead time, we can lower the safety stock, reduce the stock-out loss and improve the service level to the customer so as to increase the competitive edge in business. Firstly, Liao and Shyu [10] presented a probabilistic model in which the order quantity is predetermined and the lead time is the only decision variable. Secondly, Ben-Daya and Raouf [11] have also extended the model of Liao and Shyu [10] by considering both the lead time and the order quantity as decision variables and the situation of shortage is neglected. Subsequently, Ouyang et al. [12] considered an inventory model with a mixture of backorders and lost sales to generalized Ben-Daya and Raouf's [11] model, where the backorder rate is fixed.

In this study, we consider to allow the backorder rate as a control variable. Under most market behaviors, we can often observe that many products of famous brands or fashionable commodities may lead to a situation in which customers prefer their demands to be backordered while shortages occur. Certainly, if the quantity of shortages is accumulated to a degree that exceeds the waiting patience of customers, some may refuse the backorder case. However, the supplier can offer a price discount on the stock-out item in order to secure more backorders. In the real market as unsatisfied demands occur, the longer the length of lead time is, the larger the amount of shortages is, the smaller the proportion of customers can wait, and hence the smaller the backorder rate would be. But, the larger the backorder discount is, the larger the backorder rate would be. Thus, the backorder rate is dependent on the amount of shortages and backorder price discounts. Therefore, we also consider the backorder rate that is proposed by combining Ouyang and Chuang [13] (or Lee [14]) with Pan and Hsiao [15] to present a new form. In addition, there are many authors that (Porteus [16–18], Billington [19], Nasri et al. [20], Kim et al. [21], Paknejad et al. [22], Sarker and Coates [23], Ouyang et al. [24, 25], Moon and Choi [26], Chuang et al. [27], Lin and Hou [28], and Chang et al. [29]) have investigated the effects of investing in reducing ordering cost. Hence, we treat the ordering cost as a decision variable in this study.

Because the demand of different customers is not identical in the lead time, we cannot only use a single distribution (such as [13]) to describe the demand of the lead time. It is more reasonable that mixture distribution is applied to describe the lead time demand than single distribution is used. Besides, in many practical situations, the probability distributional information of lead time demand is often quite limited. Since Lee et al. [30] consider that the lead time demand follows a mixture of normal distributions, we relax the assumption about the form of the mixture of distribution functions of the lead time demand. Therefore, we consider that any mixture of distribution functions (d.f.s); say , of the lead time demand has only known finite first and second moments (and hence, mean and standard deviations are also known and finite) but we make no assumption on the distribution form of . That is, and of belong to the class of all single d.f.s' with finite mean and standard deviation. Our goal is to solve a mixture inventory model by using the minimax criterion. This is, the minimax criterion (such as Wu et al. [31]) for our model is to find the most unfavorable d.f.s and in for each decision variable and then to minimize over the decision variables. Finally, one numerical example is also given to illustrate that when or 1, the model considers only one kind of customers' demand; when , the model considers two kinds of customers' demand for the fixed backorder parameters and . It implies that the minimum expected total annual costs of two kinds of customers' demand are larger than the minimum expected total annual cost of one kind of customers' demand. Thus, the minimum expected total annual cost increases as the distance between and 0 (or 1) increases for the fixed backorder parameters and . Hence, if the true distribution of the lead time demand is a mixture of normal distributions, we use a single distribution (such as [13]) to substitute the true distribution of the lead time demand then the minimum expected total annual cost will be underestimated.

#### 2. Model Formulation

To establish the mathematical model, the notation and assumptions employed throughout the paper are as follows:

: odering cost per order,: average demand per year,: inventory holding cost per item per year,: length of lead time,: order quantity,: reorder point,: lead time demand with the mixtures of distribution function,: fraction of the demand backordered during the stock-out period, gross marginal profit per unit,: back-order price discount offered by the supplier per unit, : back-order parameters, : the weight of the component distributions, : maximum value of and 0, that is, : maximum value of and 0, that is, : : the expected shortage quantity at the end of cycle,: the allowable stock-out probability during : the safety factor which satisfies : the mean of lead time demand with the mixture of distributions,: the standard deviation of lead time demand with the mixture of distributions,: original ordering cost,: capital investment required to achieve ordering cost : fractional opportunity cost of capital per unit time,: percentage decrease in ordering cost per dollar increase in investmentThe assumptions of the model are exactly the same as those in Ouyang and Chuang [13] who expect the following assumptions: the reorder point = expected demand during the lead time + safety stock (SS), and SS = (standard deviation of lead time demand), that is, , where , , , (it means that ), and is the safety factor. Moreover, the mixtures of distribution functions are unimodal for all if (or ) (see [32]). Besides, the reorder point must satisfy the following equation which implies a service level constraint , where represents the allowable stock-out probability during .

In addition, we assume that the capital investment, , in reducing ordering cost is a logarithmic function of the ordering cost . That is,

This function is consistent with the Japanese experience as reported in Hall [33], and has been utilized in many articles (see [16, 17, 20–24, 34], etc.).

In this study, we relax the restriction about the form of the mixtures of d.f. of lead time demand, that is, we assume here that the lead time demand *X* has the mixtures of d.f. , where has finite mean and standard deviation and has finite mean and standard deviation , , . Then the expected shortage at the end of the cycle is defined by . Thus, the expected number of backorders per cycle is and the expected lost sales per cycle is . Hence, the expected annual stock-out cost is .

The expected net inventory level just before the order arrives is

and the expected net inventory level at the beginning of the cycle is

Therefore, the expected annual holding cost is

Finally, the total expected annual cost (EAC) can be expressed as follows:

In practical situations, as shortage occurs, the longer the length of lead time is, the larger the amount of shortage is, the smaller the proportion of customers can wait, and hence the smaller the backorder rate would be; in addition, the larger backorder price discount is, hence the larger the backorder rate would be. Therefore, we also consider the backorder rate that is proposed by combining Ouyang and Chuang [13] (or Lee [14]) with Pan and Hsiao [15] at the same time. Thus, we define , where

Hence, the total expected annual cost (2.6) reduces to Besides, we also consider that the ordering cost can be reduced through capital investment and the ordering cost as a decision variable. Hence, we seek to minimize the sum of capital investment cost of reducing ordering cost and the inventory costs (as expressed in (2.8) by optimizing over , , and constrained on . That is, the objective of our problem is to minimize the following total expected annual cost: subject to

Now, we attempt to use a minimax criterion to solve this problem. If we let be the class of all single c.d.f. (included and ) with finite mean and standard deviation, then the minimax criterion for our problem is to find the most unfavorable c.d.f.s and in for each decision variable and then to minimize over the decision variables; that is, our problem is to solve

In addition, we also need the following Proposition which was asserted by Gallego and Moon [35] to solve the above problem.

Proposition 2.1. *For any ,
**
Moreover, the upper bound (2.11) is tight. In other words, we can always find a distribution in which the above bound is satisfied with equality for every .*

Using the inequality (2.11) for and , we obtain

where

Then, the problem (2.10) is equivalent to minimize where subject to

In order to solve this nonlinear programming problem, we first ignore the restriction and take the first partial derivatives of with respect to and , respectively. We can obtain where , , and is expressed as (2.15).

Since is the expected shortage quantity at the end of cycle, we know that if shortages occur; otherwise. It is clear that is positive. By examining the second-order sufficient conditions (SOSCs), it can be easily verified that is not a convex function of . However, for fixed , and , is concave in because

Therefore, for fixed , and , the minimum total EAC will occur at the end points of the interval . On the other hand, for a given value of , by setting (2.16) equal to zero, we obtain where and is expressed as (2.15).

Theoretically, for fixed , from (2.19)–(2.21), we can get the values of , and (we denote these values by , and ).

Moreover, it can be shown that the SOSCs are satisfied since the Hessian matrix is positive definite at point (see the appendix for the proof). Hence, for a fixed , the point is the local optimal solution such that the total expected annual cost has minimum value.

We now consider the constraint From (2.20), we note that is positive. Also, if , then is an interior optimal solution for given . However, if , then it is unrealistic to invest in changing the current ordering cost level. For this special case, the optimal ordering cost is the original ordering cost, that is, , and our model reduces to (2.8) (i.e., the model of Lee et al. [36] with any mixture of distribution functions, not just mixture of normal distributions).

Substituting (2.20) and (2.21) into (2.19), we get

where and is expressed as (2.15).

Theoretically, for fixed , , , , , , , , , , , and each the optimal pair given can be obtained by solving (2.22) iteratively until convergence. The convergence of the procedure can be shown. Furthermore, using (2.9), we can obtain the corresponding total expected annual cost . Hence, the minimum total expected annual cost is . However, in practice, since the p.d.f. of the lead time demand is unknown, even if the value of is given, we cannot get the exact value of . Thus, in order to find the value of , we need the following proposition.

Proposition 2.2. *Let be a random variable which has a p.d.f. with finite mean and standard deviation , then for any real number ,
*

So, by using , the recorder point and Proposition 2.2, we get
Further, it is assumed that the allowable stock-out probability *q* during lead time is given, that is, , then from (2.24), we get . It is easy to verify that has a smooth curve for . Hence, we can establish the following algorithm to obtain the suitable *k* and hence the optimal , , and .

*Algorithm 2.3. **Step 1. *Input the values of , , , , , , , , , , , , , and , .*Step 2. *For a given *q*, we divide the interval into equal subintervals, where is large enough. And we let , and , *Step 3. *Use the , , to compute , *Step 4. *For each , , compute by using (2.22) for given , . Then, compute and by using (2.20) and (2.21).*Step 5. *Compare and . If , then take into Step 6; if , then take into Step 6. Compare and . If , then take into Step 6; if , then take into Step 6.*Step 6. *For each pair and , compute the corresponding total expected annual cost , .*Step 7. *Find . If , then find . If , then is the optimal solution; the value of such that exists is the optimal safety factor, and we denote it by .*Step 8. *Stop.

#### 3. A Numerical Example

In order to illustrate the above solution procedure, let us consider an inventory system with the following data: units/year, per order, , , units/week, units/week, and the lead time has three components with data shown in Table 1. Besides, for ordering cost reduction, we take per dollar per year and

We assume here that the lead time demand follows a mixture of distribution functions and want to solve the case when , , , and

If we knew the form of the c.d.f.s and , we could solve the problem optimally for that particular distribution. For example, if is c.d.f. of mixture of normal distributions, then the total expected annual cost is

We can obtain the optimal by the standard procedure and incur an expected cost (see Lee et al. [30]). For fixed , if we use instead of the optimal for mixture of normal distributions, then we can get an expected cost . Hence, as the c.d.f. is mixture cumulative of normal distributions, the added cost by using the minimax mixture of distributions free procedure instead of the standard procedure is . This is the largest amount that we would be willing to pay for the knowledge of d.f. . This quantity can be regarded as the expected value of additional information (EVPI). The results of the solution procedure are solved by using the subroutine ZREAL of IMSL from the computer software Compaq Visual Fortran V6.0 (Inclusive of IMSL) [37] and summarized in Table 2. From Table 2, we note that (i) the order quantity , the ordering cost , the backorder price discount and the minimum total expected annual costs increase as increases (i.e., the back-order rate decreases) for and the fixed ; (ii) the minimum total expected annual cost increase and then decrease as *p* increases for the fixed and , thus for and the fixed , when or 1, the model considers only one kind of customers' demand; when , the model considers two kinds of customers' demand. It implies that of two kinds of customers' demand are larger than of one kind of customers' demand, thus if the true distribution of the lead time demand is a mixture of normal distributions, we use a single distribution (such as [13]) to substitute the true distribution of the lead time demand then the minimum expected total annual cost will be underestimated; (iii) the order quantity , the ordering cost , the backorder price discount and the minimum total expected annual cost decrease as increases for the fixed and *p*; (iv) no matter what values of *p*, the optimal lead time is approached to a certain value (3 weeks) for and (v) while for the fixed , EVAI increases and then decreases as increases (i.e., the backorder rate decreases) for and fixed ; (vi) EVAI increases and then decreases as *p* increases except , when ; (vii) the cost penalty / of using the distribution free operating policy instead of the optimal one is increasing and then decreasing as increases (i.e., the backorder rate decreases) for the fixed except and , , when . Conveniently, we organize the above (i)–(vii) in Table 3.

#### 4. Concluding Remarks

In this article, we consider that the backorder rate is dependent on the amount of the shortages and the backorder price discount by using the idea of Ouyang and Chuang [13] and Pan and Hsiao [15]. Hence, we regard the backorder rate as controllable variable. In addition, the ordering cost can be controlled and reduced through various efforts such as worker training, procedural changes, and specialized equipment acquisition. So, we also treat the ordering cost as a decision variable. Moreover, we make no assumption about the form of the mixtures of distribution functions of the lead time demand and apply the minimax criterion to solve the problem. We also develop an algorithmic procedure to find the optimal inventory policy.

In this study, we consider the service level constraint to satisfy the equation as . But, the reorder point *r* is a controllable variable. Hence, it would be interesting to treat the reorder point as a decision variable in future research.

#### Appendix

For a given value of , we first obtain the Hessian matrix as follows:

where

Then we proceed by evaluating the principal minor of at point . The first principal minor of is The second principal minor of is (note that from (2.16) ) The third principal minor of is where , .

Therefore, from (A.3)–(A.5), it is clearly seen that the Hessian matrix *H* is positive definite at point .

#### Acknowledgments

The authors wish to thank the referees for valuable suggestions which led to the improvement of this paper. This research was partially supported by the National Science Council, Taiwan (Plan no. NSC 96-2221-E-309-001 and NSC 97-2221-E-309-008).