1. Introduction
An effective inventory management policy is meant to enhance the efficiency of inventory control and minimize inventory costs. Conventional economic order quantity (EOQ) models assume that the items provided by suppliers are in perfect condition; however, variability in manufacturing and accidents during shipping can result in retailers receiving defective items. The subsequent sale of defective products can lead to complaints, returns, and damage to the retailer’s reputation. Retailers should perform careful inspections of all items prior to sale; however, rigorous assessments are not always possible. Thus, inventory models must account for sub-lot inspections and the possibility that defective units could end up in the inventory.
Inventory models have been developed to meet the ever-evolving challenges of real-world applications. There has been considerable research into the issue of defective units and sub-lot inspection in inventory models. Paknejad et al. [
1] proposed a modified EOQ model in which the number of defective goods is treated as a random variable. Wu and Ouyang [
2] presented a continuous review inventory model combining backorders and lost sales, under the assumption that each lot contained a random number of defective units. Ouyang et al. [
3] presented an iterative algorithm aimed at deriving the optimal vendor–buyer strategy for three situations involving defective items, respectively using the crisp defect rate approach, a triangular fuzzy number approach, and a statistical fuzzy approach. Sarkar et al. [
4] proposed an integrated inventory model that deals with defective items using an inspection policy such as those decision variables that are the lead time and the ordering quantity. Sarkar et al. [
5] provided a distribution-free approach to generalize inventory systems where the quality improvement and setup cost reduction are examined. Khan et al. [
6] studied a vendor-managed inventory of consignment stock for a supply chain with defective items. An economic production quantity (EPQ) inventory system with a reworking of all defective items and multiple shipments was developed by Taleizadeh et al. [
7]. Kang et al. [
8] incorporated the effects of random defects on lot size and expected total cost function to assess a work-in-process-based inventory assuming that manufacturing defects follow a random distribution. Manna et al. [
9] presented an imperfect production inventory model in which the defect follows the production rate as dependent. An EOQ system with a returned policy of deteriorated products and sample quantity inspection was constructed by Cheikhrouhou et al. [
10]. An integrated inventory system to consider ordering quantity, lead time, number of shipments, and safety factor was provided by Kim et al. [
11]. A continuously reviewed inventory system with multiple items was provided by Malik and Sarkar [
12] to consider limited storage space, fuzzy demand, and stochastic lead time demand. A coordinated supply chain with a reliable seller and an unreliable seller was constructed by Malik and Sarkar [
13] to reduce the transportation lead time. A continuously reviewed inventory system was examined by Bhuiya et al. [
14] for a partially known distribution of lead time demand where backlogs and the lead time-dependent lost sales are considered as decision variables. A supply chain to minimize the total cost by setup cost investment and crashable lead time was developed by Ganguly et al. [
15]. A traditional EPQ model with one vendor and one buyer was examined by Malik and Kim [
16] to obtain the optimal solution. A sustainable EPQ system with cap-based production and the carbon tax was considered by Mishra et al. [
17] to control the carbon emission rate under several shortage environments to invest in green technology. A transportation model was developed by Hota et al. [
18] to show that unequal lot size is the best policy to transfer items from the manufacturer to the retailer.
Wu and Ouyang [
2] tried to show the existence and uniqueness of the interior optimal solution; the minimum boundary for this solution is the goal of the proposed approach. Wu et al. [
19] tried to generalize Wu and Ouyang [
2] by considering such that their first model has a mixed normal probability density function and their second model has a mixed cumulative distribution where the first moment and the second moment for two distributions are known. Wu et al. [
19] claimed that the optimal solution can be derived by an iterative method for the system of first partial derivatives. Hence, Wu et al. [
19] claimed that their optimal solution must be an interior solution. If we assume their two distributions are identical, then the second model of Wu et al. [
19] will degenerate to the distribution-free model of Wu and Ouyang [
2]. In this study, we will show that sometimes the optimal solution will occur on the boundary such that the iterative method of Wu et al. [
19] will contain questionable results. In the current study, this proposed model questions the assertion provided by Wu and Ouyang [
2]. A detailed derivation is provided to obtain the criteria to decide the location of the optimal solution. This study then develops a theorem to show the condition for the boundary minimum solution. There are two numerical examples such that the first one is demonstrated for the interior optimal solution, in which our result is superior to that of Wu and Ouyang [
2]. The second one is a boundary optimal solution to indicate the questionable assertion proposed by Wu and Ouyang [
2]. In the past, for inventory models under the distribution-free approach, researchers always believed that the two decision variables: ordering quantity and the safety factor are both interior solutions and they can be derived by the iterative method through the first partial derivative system of the objective function. In this study, conditions are provided to help decision-makers to search the optimal solution on the boundary.
This study compared with other inventory models from the above-mentioned literature review have been summarized as
Table 1.
3. Review of Distribution-Free Model
For the distribution-free model, this study recalls the minimum problem that is proposed by Wu and Ouyang [
2], for
,
and
, the expected total annual cost (EAC) is denoted as
where
A is the ordering cost for one replenishment.
is the holding cost for one replenishment.
is the stockout cost for one replenishment.
is the inspecting cost for one replenishment.
is the uninspected defective penalty cost for one replenishment.
is the lead time crashing cost for one replenishment. The duration time for one replenishment is
.
Wu and Ouyang [
2] took the sum of (a) the ordering cost, (b) the holding cost, (c) the stockout cost, (d) the inspecting cost, (e) the uninspected defective penalty cost, and (f) the lead time crashing cost for one replenishment to find the total cost for one replenishment. Wu and Ouyang [
2] divided the total cost for one replenishment by the duration time to derive the average cost per unit of time, denoted as
.
They proved that is concave up with respect to and then they implied that the objective function will reduce to or . To simplify the expression, this study still uses to represent for or .
The system of the first partial derivative is derived by Wu and Ouyang [
2] as follows
with
, and
Wu and Ouyang [
2] mentioned that explicit general solutions for
and
are not possible because the evaluation of Equations (2) and (3) requires knowledge of the value of the other. Thus, an iterative algorithm is established to find the optimal value of
and
. Hence, Wu and Ouyang [
2] assumed that
and inserted it into Equation (2) to find a value of
, denoted as
, the value of
is then used in Equation (3) to obtain a value of
, denoted as
. Iteratively,
is plugged into Equation (2) to derive
,
is then plugged into Equation (3) to find the next one,
. They derive two sequences:
and
, until no change occurs in the values of
and
.
The procedure of Wu and Ouyang [
2] contains three questionable results. First, why will the two sequences
and
converge? Second, given two different initial values of
as
and
, if the results of the two sequences, (a) based on
,
and
, and (b) based on
,
and
, do indeed converge then they may have different limit points. That is, the limits may be dependent on the initial point. Third, if the two sequences converge, why are the limit points the optimal solution? Sometimes the iterative sequence approach may not derive the desired result, an example is provided in the following to illustrate this point. To solve
for positive solutions, there are two solutions
and
. Our goal is to illustrate that by using the iterative method, it is almost impossible to construct a sequence that converges to 10. If the researchers try to find
by the iterative method, implicitly motivated by
, the researchers use the following iterative formula
From Equation (4), the researchers know that if and only if or , which means that for the starting point, say , there are the following five cases:
- (a)
, the generated sequence will increase and converge to its limit, 2. For example, , , , , , and , which indicates that ;
- (b)
, the researchers have , a constant sequence. It converges to 2;
- (c)
, owing to , the generated sequence will decrease and converge to its limit, 2. For example, if researchers select , then , , , , , , and , which indicates that the derived sequence will converge to its limit, ;
- (d)
, the researchers have , a constant sequence. It converges to 10;
- (e)
, according to , the generated sequence will increase and diverge to . For example, if the researchers select , then , , , , , , and , which indicates that the derived sequence will diverge to .
Based on the above example, the starting point will influence the result. Moreover, most of the time will not be selected. Therefore, there is little chance to construct a convergent sequence which will converge to our goal, with its limit of . Our example demonstrates that sometimes the iterative approach may not be as robust as previously thought. The result of the above discussion provides us with enough motivation to apply an analytical approach to prove the existence and uniqueness of the interior optimal solution and find the conditions for the boundary minimum.
In the next section, this study provides an analytical approach to prove that with three reasonable conditions, there exists a unique pair of
and
that is the optimal solution. This study will show that the three reasonable conditions for the optimal order quantity are as one lower bound, Equation (10) and two upper bounds, and Equations (6) and (14). The first partial derivative system of Wu and Ouyang [
2] consists of Equations (2) and (3). This study combines them into one equation by deleting the variable
such that in our findings, Equation (15) only contains one variable,
. Owing to that, the order quantity must be positive and the safety factor must be non-negative, so during our derivation (Equations (5)–(14)), this proposed model derived that there are three necessary conditions to ensure the existence of the optimal solution.
4. Improved Mathematical Analysis
This study rewrites Equation (3) as
where
with
. Owing to
, this study finds
so that
to derive an upper bound of
as
This study rewrites Equation (5) as
This study squares Equation (7) and cross multiplies to derive
in a function of
, then
This study rewrites Equation (2) as
where
, and
.
From Equation (9), owing to
, it yields a lower bound for
such that
From Equation (9), it yields that
If the researchers take the sum and the difference of Equations (9) and (11), then it implies that
and
Based on Equation (13) with
and Equation (10), this study obtains an upper of
as
If this study plugs Equations (12) and (13) into Equation (7), then it yields a fifth-degree polynomial as
The goal is to find the condition that guarantees Equation (15) will have a unique positive solution. From Equation (6), it follows that
to assure the well-defined of Equation (15).
However, the restriction in Equation (16) is unnecessary, owing to Equation (6), as this study already has
. Based on Equations (6) and(14), this study knows that
In the following, this study will prove that under the conditions of Equations (10) and (17),
, there is a unique solution for Equation (15). This study assumes an auxiliary function, say
, where
From Equation (6), it yields that
. On the other hand, owing to
, this study knows that
such that
is a convex function. According to
,
and
From Equation (21), depending on the sign of , this study will divide the solution procedure into the following three cases: (i) , (ii) and , and (iii) and .
For case (i), this study knows that
and then
Due to the convexity of
, there is therefore a unique point, say
, with
that satisfies Equation (15). Using Equation (13), it implies that
such that
and
are the interior optimal solution for the distribution-free inventory model.
For case (ii), owing to and , this study still has . Hence, the rest of the derivation is the same for case (i).
For case (iii), owing to
and
, this study implies that
. From the convexity property of
, this study derives that there is no point with
which satisfies Equation (15). Consequently, the interior minimum solution does not exist. Though subsequently, by using our approach, this study finds the conditions
to locate the boundary minimum.
Next, this study considers the minimums that are located on the boundary with
or
. Though in the paper of Wu and Ouyang [
2], there is no discussion about the domain for their distribution free inventory model. This study knows that
is the safety factor with
and
is the order quantity with
.
During their derivation, Wu and Ouyang [
2] adopted
to simplify their inventory model. That means that they assumed that
would be a very large number. The assumption of
is supported from their numerical examples, the range of
is derived to be from 145 to 184. Based on the above discussion, this study will first prove that the local minimum will not happen along the line with
. This study derives that
and
From Equation (27), , this study knows that is convex in when and are fixed, where or .
Moreover,
and
there is a point, say
satisfying
, such that
is the minimum along the ray
. Hence, there is no need to discuss the boundary along the line of
.
On the other hand, along the boundary of
, the objective function of Equation (1) is reduced to
Based on Equation (29), this study assumes that.
and
with
,
, and
being positive.
Hence, based on the expression of Equations (30)–(32), this study can simplify Equation (29) as
From Equation (33), this study derives that
to the boundary minimum point
and minimum value
.
In the next theorem, the results will be summarized as follows.
Theorem 1. If or , there is an interior that satisfies Equation (15) and which satisfies Equation (13).
If, then the boundary minimum will happen atand.
Lead time is an important factor for inventory models. This study presents a detailed examination between the crashable lead time and annual total cost. Hence, the decision-makers can decide how many investments should be provided to balance the cost and efficiency.
5. Numerical Examples
In order to demonstrate our approach, this study considers the same numerical examples that are proposed by Wu and Ouyang [
2], with the following data:
units/year,
per order,
,
,
,
,
,
,
units/week, the lead time has three components such that
,
,
L2 = 4, and
with the crashing cost,
,
,
, and
, the defective rate
has a Beta distribution with parameters
and
to imply the probability density function is
for
to imply that
and
, the backordered rate,
, is assumed as 0, 0.5, 0.8, and 1. This study derives
from Equation (15) and then uses Equation (13) to find
, to plug them into Equation (1) to find the minimum value. This study lists our findings in
Table 2.In
Table 3 shown below, this study compares our findings with that of Wu and Ouyang [
2].
Based on the comparison of the minimum solution and the minimum value, both approaches are comparatively very similar, under the parameters set in Wu and Ouyang [
2]. In the following, this study constructed a counter-example to demonstrate that sometimes the iterative method cannot work. This study assumes that
,
with
,
and
to imply that
which satisfies the condition of Equation (25). By our approach, according to the derivation of Equation (34), this study finds that
and
with
. On the other hand, by using the iterative method proposed by Wu and Ouyang [
2] with
, by Equation (2), it yields that
. If the researchers follow the suggestion of Wu and Ouyang [
2], this study then faces the following problem
such that this study cannot derive a non-negative
from Equation (36). The above counter-example demonstrates that in our Theorem 1, if the condition
is satisfied, then the iterative approach proposed by Wu and Ouyang [
2] can not generate two sequences that converge to the optimal solution.