Determining Replenishment Lot Size and Shipment Policy for an EPQ Inventory Model with Delivery and Rework

The determination of production-shipment policies for a vendor-buyer system is dealt within this paper. The main objective is to derive the optimal replenishment lot size and shipment policy for an EPQ inventory model with multiple deliveries and rework. This inventory model contains two decision variables: the replenishment lot size and the number of deliveries. Previous researches solve this inventory model considering both variables to be continuous. However, the number of deliveries must be considered as a discrete variable. In this direction, this paper solves the inventory model considering two cases: Case 1: the replenishment lot size as a continuous variable and the number of shipments as a discrete variable and Case 2: the replenishment lot size and the number of shipments as discrete variables. The final results are two simple and easy-to-apply solution procedures to find the optimal values for the replenishment lot size and the number of deliveries for each case.


Introduction
A key challenge in the inventory management in any organization is to answer the following simple question: How many products to order?This question is responded to by the traditional economic order quantity (EOQ) inventory model.It is well known that the EOQ inventory model was derived by Harris [1] in 1913.Later, the economic production quantity (EPQ) inventory model was proposed by Taft [2] in 1918.It is worthy to mention that since then, several extensions to both inventory models have been derived by several scholars, that is, Taleizadeh et al. [3], Chen [4], and Pal et al. [5], just to name a few recent researches.In Cárdenas-Barrón et al. [6] Ford Whitman Harris is named as the founding father of the inventory theory.
Recently, Chiu et al. [7] and Chiu et al. [8] determine the optimal replenishment lot size and shipment policy for an EPQ inventory model with multiple deliveries and rework.We have read the papers by Chiu et al. [7] and Chiu et al. [8] with substantial interest and we have found that their inventory model contains two decision variables: the replenishment lot size and the number of deliveries.The works of Chiu et al. [7] and Chiu et al. [8] solve the inventory model considering both variables to be continuous.However, the number of deliveries must be considered to be a discrete variable.We think that the researchers that have been attracted by works of Chiu et al. [7] and Chiu et al. [8] may be interested in knowing the solution procedure that gives the optimal solution to the decision variables according to its nature.In this direction, this paper solves their inventory model considering two cases: Case 1: the replenishment lot size as a continuous variable and the number of shipments as a discrete variable and Case 2: the replenishment lot size and number of shipments as discrete variables.To solve both cases we use the algebraic method of completing perfect squares and the result of García-Laguna et al. [9].The algebraic method of completing perfect squares has been used amply by many scholars since 1996 (Grubbström [10], Omar et al. [11], Wee et al. [12], and Yang and Wee [13]).Conversely, Chiu et al. [7] consider both decision variables to be continuous and then use the classical optimization technique (differential calculus) to determine the replenishment lot size and shipment policy for an extended EPQ model with delivery and quality assurance issues.It is important to point out that Chiu et al. [8] derive the same inventory problem of Chiu et al. [7] using the algebraic method of completing perfect squares and they also considered both variables to be continuous.
Recently, Treviño-Garza et al. [14] solve optimally a family of inventory models that deal with an EPQ for an integrated vendor-buyer system considering that the production system creates defective products.Taleizadeh et al. [15] propose EPQ inventory model with rework of defective items when multishipment policy is used.Their inventory model determines optimally the selling price, the lot size, and the number of shipments.Sana [16] develops an EOQ inventory model for conforming and nonconforming quality products in which the nonconforming products are sold at a reduced price.Pal et al. [17] derive an EPQ inventory model to determine the optimal buffer for a stochastic demand considering preventive maintenance.Pal et al. [18] propose an EPQ inventory model for an imperfect production system that takes into account that defective items are reworked after the regular production time.Das Roy et al. [19] develop an economic production lot size model for a manufacturing system that produces defective items.These defective items are accumulated and then reworked.This inventory model also permits shortages with partial and full backordering.

Optimizing the Replenishment Lot Size and the Number of Shipments
This section presents the optimizing procedure of the replenishment lot size and the number of shipments for the inventory problem given in Chiu et al. [7] and Chiu et al. [8].

The Replenishment Lot Size and Shipment Policy for an
Extended EPQ Model with Delivery and Quality Assurance Issues (Chiu et al. [7] and Chiu et al. [8]).Chiu et al. [7] and Chiu et al. [8] derived the following long-run average costs [()]: where the notation is as follows.Variables : the replenishment lot size (units), : the number of shipments.

Parameters
The detailed derivation of each term in [()] can be found in Chiu et al. [7].Note that they consider only one variable ().Conversely, this paper considers the long-run average costs with two decision variables:  and , where Q is a continuous variable and  is a discrete variable.Hence [(, )] is rewritten as follows: where the constants  1 ,  2 , and  3 are given by It is important to remark that the total cost [(, )] is a mixed integer nonlinear optimization problem when the lot size () is a continuous variable and the number of shipments () is a discrete variable.On the contrary, the total cost [(, )] is an integer nonlinear optimization problem when both variables ( and ) are considered to be discrete variables.
To optimize the total cost [(, )] a sequential optimization procedure of two stages is applied.First stage optimizes the replenishment lot size () by the algebraic method of completing perfect squares (see, e.g., Cárdenas-Barrón [20]).Second stage optimizes the number of shipments () using the result of García-Laguna et al. [9].

Stage I (Optimizing the Replenishment Lot Size (𝑄)).
In the research work of Cárdenas-Barrón [20] it was demonstrated by the algebraic method of completing perfect squares that a function of type  1  +  2 / is always minimized for  = √ 2 / 1 , which attains the minimum at () = 2 √  1  2 where  is a continuous variable and  1 and  2 are both greater than zero.Consequently, the replenishment lot size And the minimal total cost is where It is worthy to mention that the function   has the same mathematical form  1 + 2 / but in this function the decision variable  is discrete, and obviously  must be greater or equal than one.In the work of García-Laguna et al. [9] it was showed that when  is discrete and  1 and  2 are greater than zero, then function of type  1  +  2 / attains its minimum when  is given as follows: or It is important to remember that ⌈⌉ and ⌊⌋ are the smallest integer greater than or equal to  and the largest integer less than or equal to , respectively.Clearly, it is easy to see that ⌈⌉ = ⌊ + 1⌋ if and only if  is not a discrete value.In this circumstance the minimization problem has a single solution for  which is  * =  (given by either of the two expressions in ( 7) and ( 8)).Otherwise, the minimization problem has two solutions for  that are  * =  and  * =  + 1.
Considering the result of ( 7) and ( 8), then the solution to the discrete variable () is as follows: or It is obvious that  4 > 0 and therefore  4 =  1  4 is always greater than zero.However,  5 =  5 can be positive, zero, or negative because the following term ] can be positive, zero, or negative.When  5 takes positive values then the optimal solution for  is given by ( 9) or (10).On the contrary, for zero and negative values of  5 , it is easy to see that  1  +  2 / attains its global minimum value at  = 1.
In many situations of the real life the replenishment lot size () could be an integral value too.If we constrain the replenishment lot size to be a discrete variable then applying the previous result (( 7) and ( 8)) the replenishment lot size () is given by or A lower bound for the total cost [()] can be obtained straightforwardly.The lower bound is derived just considering both variables ( and ) to be continuous variables.Therefore, the lower bound is given by ) +  3 . ( The above lower bound is usable only when ] is negative then the lower bound is just the [()] given by (2).It is important to mention that this lower bound ( 13) can be attained just for the case when  is a continuous variable and  is a discrete variable if and only if  = √ 5 / 1  4 is an integer value.On the contrary, for the case when both  and  are considered to be discrete variables then this lower bound ( 13) can be attained when the optimal solutions for both variables,  and , are discrete values, in other words, when  given by ( 4) is a discrete value and at the same time  given by  = √ 5 / 1  4 is a discrete value too.It is important to mention that the lower bound for the total cost obtained by ( 13) or ( 2) is just for Case 1. Obviously, the lower bound for the total cost for Case 2 is the total cost for Case 1. Step 2. Compute the integral value for .
Step 3. Given the discrete value of  then compute the continuous value for the lot size  using (4).
It is important to remark that if there exist two solutions for  then there are two optimal solutions for the inventory problem.For each solution of  do Steps 3 and 4 and report the two optimal solutions.A flow diagram for solution procedure of Case 1 is given in Figure 1.Step 1.
Step 2. Compute the integral value for .
Step 3. Given the discrete value of  then compute the discrete value for the lot size  using (11) or (12).
Here, it is important to notice that if two solutions for  exist then there are two solutions.For each solution of  do Steps 3 and 4 and choose the solution with the minimal total cost.The flow diagram of the solution procedure for Case 2 is given in Figure 2.
The example of Chiu et al. [7] and Chiu et al. [8] is solved with the proposed solution procedures.The data for the example is shown in Table 1 and the solutions are given in Table 2. Now, we illustrate the situation when (ℎ 2 −ℎ)[(1−())− (1/ + ()(1 − )/ 1 )] is lower and equal to zero.Consider that the value of ℎ is 98 and the other values for the parameters are the same as given in Table 1.The value of (ℎ 2 − ℎ)[(1 − ()) − (1/ + ()(1 − )/ 1 )] is negative and it is equal to −12.53271429.Then by applying the proposed solution procedures we obtain the following.
For Case 1: the replenishment lot size () being continuous and the number of shipments () being discrete, the optimal solution is  = 1110.748506; = 1 and [] = 519292.08791321.
For n 1 and n 2 compute Q 1 and Q 2 with ( 4) an integer?For Case 2: the replenishment lot size () being discrete and the number of shipments () being discrete, the optimal solution is  = 1111;  = 1 and [] = 548596.14413939.

Discussion
When applying the solution procedure for Case 1, we get the same solution as Chiu et al. [7] and Chiu et al. [8] but in a simple manner.On the other hand, the solution procedure of Chiu et al. [7] rounds the value of the number of shipments ().This action could provide us with a nonoptimal value for the number of shipments ().Although Chiu et al. [8] fix the previous mentioned shortcoming, their solution procedure requires to evaluate the total cost twice (one for each ) in order to compute the number of shipments ().Moreover, it is important to mention that Chiu et al. [7] and Chiu et al. [8] do not consider the situation when (ℎ 2 − ℎ)[(1 − ()) − (1/ + ()(1 − )/ 1 )] ≤ 0. Also, they do not solve the inventory problem when both variables are discrete.Furthermore, both solutions procedures proposed in this paper are easy to implement in a spreadsheet and do not require any computational effort.These are some important advantages that our paper has with respect to Chiu et al. [7] and Chiu et al. [8].9) or (10) Compute n with ( 9) or (10) Compute with ( 2) For n 1 and n 2 compute Q 1 and Q 2 with (11) or ( 12) Is an integer?(given ( 13)) LB = 425862.39472for Case 2 Note that Chiu et al. [7] reported for the special case that the replenishment lot size is 2018 units and that the number of shipments is 3 with a total cost of 427938.The reader can notice that the solution for the special case reported by Chiu et al. [7] is erroneous.

Conclusions
The main and new contribution of this paper is to present two easy-to-apply solutions procedures to determine jointly both the optimal replenishment lot size and the optimal number of shipments for the inventory model proposed by Chiu et al. [7] and Chiu et al. [8].The solution procedures are developed for solving two cases: Case 1: the replenishment lot size () being continuous and the number of shipments () being discrete and Case 2: the replenishment lot size () being discrete and the number of shipments () being discrete.The proposed solution procedures are simple and require no tedious computational effort.Furthermore, the proposed solution procedures discriminate between the situation in which there is a single solution and when there are two solutions for each discrete variable.Chiu et al. [7] considered the decisions variables ( and ) to be continuous and then round up the decision variable number of shipments ().This could give us a nonoptimal value for .Our paper improves and complements Chiu et al. [7] and Chiu et al. [8] research works since it treats both variables according to their nature.The readers who are interested in this paper may also refer to the research works of Cárdenas-Barrón et al. [21][22][23].Finally, this paper can be extended in several ways.example,

Case 2 :
The Replenishment Lot Size () Being Discrete and the Number of Shipments () Being Discrete Solution Procedure for Case 2

Table 1 :
Data for the example.
*Remember that for a uniform distribution with range (a, b) the expected value is defined as () = ( + )/2.

Table 2 :
Results for the example.