Determining the Optimal Inventory and Number of Shipments for a Two-Resource Supply Chain with Correlated Demands and Remanufacturing Products Allowing Backorder

: This study develops an integrated supplier–remanufacturer and customer (downstream manufacturer) inventory model that takes into account three-echelon system with correlated demands and remanufacturing products allowing a backorder goods condition. This paper improves the observable fact that the ﬁrst model system customer might select two sources from remanufactured products or supplier products without defective items. The second model further considers the defective items during the screening duration. The results are examined analytically and numerically to show that the policy of single shipment in large lot sizes results in less total cost than a frequent shipments policy. We also explore the impact of recovery rate on the economic beneﬁts of the inventory system. In addition, we also perform sensitivity analysis to study the impact of seven important parameters (transportation cost, recovery rate, screening rate, annual demand, defect rate, and backorder rate, holding cost,) on the optimal solution. Management insights were also discussed.

The three-echelon inventory system consists of a single supplier, a single remanufacturer and a single customer (downstream manufacturer).

2.
A single product is considered. 3.
The remanufacturing rate and demand rate are known.

4.
The screening rate is higher than the remanufacturing rate and demand rate. 5.
The lead-time and defect rate are fixed. 6.
There is an unlimited planning period. 7.
Allow supply backorder and out-of-stock inventory. 8.
Not consider quantity discounts. 9.
Not consider inventory space constraints.
The notations used in this paper are shown in Table 1.

Model Formulation
In this section, we will develop a three-echelon inventory system ( Figure 1). The downstream stage is that the customer belongs to the demand side, and the upstream stage is that the remanufacturer and supplier belong to the dual supply side. This is the total cost of the integrated supplier-remanufacturer (that is dual supply) and customer in the first model. In addition, we extend the above inventory system, adding that the downstream stage customers will completely screen out defective products, and the dual supply side remains unchanged in the second model. Symbols and diagrams for model operation contents are as follows. β Backorder ratio R q reorder of point (ROP) Q (Q = n*q)

Model Formulation
In this section, we will develop a three-echelon inventory system ( Figure 1). The downstream stage is that the customer belongs to the demand side, and the upstream stage is that the remanufacturer and supplier belong to the dual supply side. This is the total cost of the integrated supplier-remanufacturer (that is dual supply) and customer in the first model. In addition, we extend the above inventory system, adding that the downstream stage customers will completely screen out defective products, and the dual supply side remains unchanged in the second model. Symbols and diagrams for model operation contents are as follows. The system of a customer's acquirement and supplement customer demand from both external suppliers and remanufacturers can be shown in Figure 1. It is clear from Figure 1 that if q is the customer's demand quantity, the cycle length is D q . The recycle ordering amount in the remanufacturer is rq . Therefore, the order quantity with the supplier is For backorder condition model, out-of-stock and under-allocation means that the customer is allowed to make up for out-of-stock products within a certain period of time. At this time, a discounted loss of material prices may occur, or it may be to meet the allowable period by increasing the cost of overtime to rush to work. The cost of transportation or outsourcing costs can be summed up as the cost of stock-outs. Because the cost of stock-outs is estimated, customers and suppliers often have different views, so it is difficult to accurately estimate the cost.
The backorder model is shown in Figure 2. Cs is the unit cost of stock, t1 is the no out-of-stock period, and t2 is the out-of-stock period. We show derivation process of out of stock model. Average inventory level is (q -s)/2 and average lack of inventory level s/2. The geometric figure ΔABD, ΔACE and ΔEHD are similar triangles, so the following proportional relationships exist. The system of a customer's acquirement and supplement customer demand from both external suppliers and remanufacturers can be shown in Figure 1. It is clear from Figure 1 that if q is the customer's demand quantity, the cycle length is q D . The recycle ordering amount in the remanufacturer is rq. Therefore, the order quantity with the supplier is (1 − r)q. The number of orders for a supplier is n, where n (1 − r)q is an integer. For backorder condition model, out-of-stock and under-allocation means that the customer is allowed to make up for out-of-stock products within a certain period of time. At this time, a discounted loss of material prices may occur, or it may be to meet the allowable period by increasing the cost of overtime to rush to work. The cost of transportation or outsourcing costs can be summed up as the cost of stock-outs. Because the cost of stock-outs is estimated, customers and suppliers often have different views, so it is difficult to accurately estimate the cost.
The backorder model is shown in Figure 2. Cs is the unit cost of stock, t1 is the no out-of-stock period, and t2 is the out-of-stock period. We show derivation process of out of stock model. Average inventory level is (q -s)/2 and average lack of inventory level s/2. The geometric figure ∆ABD, ∆ACE and ∆EHD are similar triangles, so the following proportional relationships exist.
The total cost of the allowance model for backorders includes the purchase cost of the materials themselves, order costs per year, storage costs per year and backorder costs per year.
With the above formula differential, we get: The total cost of the allowance model for backorders includes the purchase cost of the materials themselves, order costs per year, storage costs per year and backorder costs per year.
With the above formula differential, we get:

Establishing Model 1: The Supplier and Remanufacturer Splits a Shipment into Several Small Lot Sizes for Customer (Downstream Manufacturer) without Dective Items for a Backorder Model
Based on the optimal delivery strategy, Lin (2014) [28] and Kim & Ha (2003) [29] proposed a model considering the total relevant costs for the supplier and buyer, and determined the optimal order quantity, number of deliveries/set-ups, and shipping quantities in a simple JIT single-supplier, single-remanufacturer-single-customer structure. Figure 3 shows the three-echelon inventory system diagrams for the backorder model. It can be seen from Figure 2 that customers, suppliers, and remanufacturers have the same length of supply and demand cycle. Based on customer environmental protection considerations, there are ordered both remanufactured products and new products from remanufacturers and suppliers to meet customer demand. Simultaneously we add the out-of-stock model by using heuristic derivation, importing concept of out-of-stock costs to modify Mona's 2011 formula as follows. When an out-of-stock period causes customers to wait for a long time, some customers with lower loyalty will switch to other suppliers.

Establishing Model 1: The Supplier and Remanufacturer Splits a Shipment into Several Small Lot Sizes for Customer (Downstream Manufacturer) without Dective Items for a Backorder Model
Based on the optimal delivery strategy, Lin (2014) [28] and Kim & Ha (2003) [29] proposed a model considering the total relevant costs for the supplier and buyer, and determined the optimal order quantity, number of deliveries/set-ups, and shipping quantities in a simple JIT single-supplier, single-remanufacturer-single-customer structure. Figure 3 shows the three-echelon inventory system diagrams for the backorder model. It can be seen from Figure 2 that customers, suppliers, and remanufacturers have the same length of supply and demand cycle. Based on customer environmental protection considerations, there are ordered both remanufactured products and new products from remanufacturers and suppliers to meet customer demand. Simultaneously we add the out-of-stock model by using heuristic derivation, importing concept of out-of-stock costs to modify Mona's 2011 formula as follows. When an out-of-stock period causes customers to wait for a long time, some customers with lower loyalty will switch to other suppliers. Ouyang et al. [30,31] believe that in the event of a shortage, some customers are still willing to wait out the shortage due to trust and loyalty toward the supplier. At this time, the supplier often provides backorder discounts to compensate for losses due to waiting or increase in production costs. Therefore, how to find the best relationship between the discounts of the amount owed and Ouyang et al. [30,31] believe that in the event of a shortage, some customers are still willing to wait out the shortage due to trust and loyalty toward the supplier. At this time, the supplier often provides backorder discounts to compensate for losses due to waiting or increase in production costs. Therefore, how to find the best relationship between the discounts of the amount owed and the period of backorder owing for minimizing the total cost of inventory is explored in a future study.
From Figure 3, it is apparent that the number of set-ups at customer, supplier and remanufacturer will be D q , D nq and D nq , here nq = Q, The expression for the average on-hand inventory at remanufacturer, supplier and customer can be derived as follows: The expression for the average inventory holding cost at the remanufacturer can be derived as follows: (refer to Mona 2011) [3] and Lin's research [28]: The average inventory holding cost at supplier can be derived as follows: From the customer's viewpoint, the average inventory holding cost is: According Ouyang et al. (1996) [30], we establish the allow out-of-stock mode in Equation (7); D is demand quantity for the customer and follows normal distribution with the probability density function f(D). The mean value is uL, the standard deviation is σ √ L the ROP is R q = uL + kσ √ L; the expected quantity out-of-stock each period is the Equation (7). L is the lead-time for replenishment.
The Φ and Φ(K) is the representative probability density function (pdf) and cumulative distribution function (cdf), respectively. The backorder per period is (1 − β) × B(r), such as β (0 ß 1) is the proportion of stocks understocked in stock-out period. In the out-of-stock cost function, there is a functional relationship between the out of stock discount (β) and the out-of-stock period (L). In other words, the longer the out-of-stock period, the more out-of-stock discounts may be given. We assume that the out-of-stock pattern in this case conforms to the normal distribution. Then, the cost of stock out can be expressed as follows: β is backorder ratio, π 0 is profit margin per unit of goods, π x is backorder discount per unit of goods according Ouyang et al. (1996) [30].
the formula means TC(M) the formula means TC(S) the formula means TC(Cs) There are 1 T = D Q cycles in one time period. Hence, the model average total cost is as follows: At first, we let n be fixed for finding the unique solution. Taking the Equation (10) derivative with Q will be: Let n be fixed; ETC(Q, n) is decreasing on (0 ; Q * (n)] and ETC(Q, n) is increasing on [Q * (n), ∞] . Therefore, we can obtain at one of Q * (n),ETC(Q, n) has the optimal solution. Plug Equation (11) into Equation (9), and rearranging the results will be: Since n is an integer, the optimal number of shipments from the supplier and remanufacturer to the customer must be satisfied as follows: [28] n * (n * − 1) ≤ Sθ Fϕ ≤ n * (n * + 1) For finding the overall optimal solution, to help the operations managers plan and make the decisions quickly and correctly, an algorithm is developed as follows.

Model 2: The Supplier and Remanufacturer Splits a Shipment into Several Small Lot Sizes for Demand (Downstream Manufacturer) with Defective Items
In this model, we extend the previous model, and we assume that the customer 100% checks and screens out defects that are immediately removed from the original inventory. Figure 4 displays the behavior of the inventory level for this model. Suppliers and remanufacturers arrive at customers with replenishment; the cycle length remains the same, but the cycles in one-time period at customer, supplier, and remanufacturer will be D (1−p)q , D n(1−p)q and D n(1−p)q , respectively. From the remanufacturer's viewpoint, the total cost of a product should include the set-up cost, holding cost, product delivery cost, and screening cost. S m is the set-up cost (including maintenance); ] is the remanufacturer's holding costs for transporting batches; nF m is the shipping freight cost.
The remanufacturer's holding cost is derived as follows (refer to Mona 2011 [3] and Lin's research [28]): The total cost for the remanufacturer is: and screens out defects that are immediately removed from the original inventory. Figure 4 displays the behavior of the inventory level for this model. Suppliers and remanufacturers arrive at customers with replenishment; the cycle length remains the same, but the cycles in one-time period at customer, supplier, and remanufacturer will be The total cost for the remanufacturer is: The supplier's holding cost is derived as follows: The total cost for the supplier is: From the customer's viewpoint, the average on-hand inventory cost is: The total cost for the customer is: The cost of stock out is: The combined remanufacturer, supplier and customer and backorder total costs is  There are 1 T = D (1 − p)Q cycles in one time period. Hence, the model average total cost, as follows: Since n is an integer, the optimal number of shipments from the vendor to the customer must be satisfied as follows:

Numerical Analysis
This analysis considers two kinds of materials, the resources of the supplier and the remanufacturer. The customer (downstream manufacturer) will consider two kinds models of inventory without defective items when considering the backorder condition with defective items. We further calculate the total relevant customer, remanufacturer and supplier cost by determining the optimal order quantity cycle lot size, a number of deliveries, and dual supplier shipment quantity, and Table 2 is a numerical example using Kim and Ha (2003) as the parameters needed for the analysis model in this paper [29].
Some values in Table 2 are based on Salameh & Jaber's (2000) [32] values for fixed costs, screening rate, expected defect rate, E[p] = 0.02, and another definition of demand, the remanufacturing rate. Considering the cost structure and using the algorithm in Section 3, we can get the optimal order quantity, the best shipment times and the lowest annual total cost as follows: According to the Figure 5 as shown below, assume supplier and remanufacturer have same condition (such as quality, cost, etc.). Then, we integrate the supplier and the remanufacturer, and the recovery rate dropped slightly by 50.3%; therefore, we can obtain the quantity of remanufactured r * q = 274, the quantity of suppliers (1 − r * )q = 252, Q * * = 2088 units ETC(2088, 4) = $8391 per year.
However, if the single delivery policy (n = 1) were used, we also compare Model 1 in terms of the effects on the optimal strategies and the performance of a shipment split into several small lot sizes (compared with a single shipment). The values for M, D, S m , Ss, S b , H m , H s , H b , x, and [P] are the same as those given in Table 2. We assume F s = 25, and F m = 10, 25, 100. Also, we assume r takes the six values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6. and β = 0.05. Substituting these values into the derived formulas, we obtain the results summarized in Table 3. By comparing a shipment of several lot sizes with a single shipment strategy, we will obtain cost savings (CS), as follows: [28] CS = ETC * * n=1 − ETC * * ETC * * × 100%. However, if the single delivery policy (n = 1) were used, we also compare Model 1 in terms of the effects on the optimal strategies and the performance of a shipment split into several small lot sizes (compared with a single shipment). The values for M, D, Sm , Ss , Sb, Hm , Hs ,Hb, x, and [P] are the same as those given in Table 2. We assume Fs = 25, and Fm= 10, 25, 100. Also, we assume r takes the six values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6. and β=0.05. Substituting these values into the derived formulas, we obtain the results summarized in Table 3. By comparing a shipment of several lot sizes with a single shipment strategy, we will obtain cost savings (CS), as follows: [28] CS = % 100 The results is shown in Table 3 and the sensitivity analysis is shown in Table 4. Table 3. Comparison of performance of a shipment split into several small lot sizes and a single shipment under backorder condition. Figure 5. Remanufacturer and supplier-ordering system for the customer economic curve.

1.
When the screening rate (x), demand rate (D) and defect rate (p) increase, then Q ** and ETC ** also increase. The screening rate (x) increases, that is, the more defective products are selected, the more customers will increase the number of orders and reduce the cost of holding, thus Q ** and ETC ** increase. This result corresponds to the works of Maddah and Jaber (2008) As Demand rate (D) increases, Q ** and ETC ** increase. Because customers will generate a large number of orders in order to meet the demand. When the defect rate(p) increases, Q ** and ETC ** increases. Since the higher the defect rate, the higher the number of orders and the higher the number of deliveries, the total annual cost will increase as well.

3.
When backorder rate (β) is higher, Q ** and ETC ** increase. Because the higher the out-of-stock rate and the higher the number of deliveries, the total cost will increase

Summary and Conclusions
As inventory management issues increase, businesses face a greater need to improve their financial performance by cutting (shipping) on inventory holding cost, and integrating the supply chain to allow all members to share the minimum joint total cost. Although Mitra (2012) [27] can solve the problem of defective returned products, in real life, defective products are returned to upstream manufacturers. In addition, environmental awareness is rising. To ensure an image of corporate social responsibility, this study specifically proposes to purchase a certain proportion of remanufacturing products and mixed new products. We developed an algorithm with the ability to make the decisions quickly and correctly to find the overall best solution. In order to better match the inventory situation in the real world, this article considers an inventory model with two states-nondefective and defective items-and incorporates different recovery rates for each model. From these models, we derive closed-form formulas and derive the optimal ordering and shipping strategies.
In reality, shortage cost refers to the cost incurred when inventory is in short supply, which can be further divided into backorder cost or the cost of a loss in sales. Losses caused by stock shortages often have different cost valuation methods due to different positions of buyers and sellers. Therefore shortage costs are the hardest to estimate inventory costs. The level of this cost and the number of stock-outs are related to the unit of shortage cost. When performing inventory model analysis, it can choose different inventory costs according to different needs. It must be specifically stated that the total cost include the ordering cost, holding cost and backorder cost.
From the parameters analysis and the illustrative numerical example, we assume that storage is neglected. Thus, we find that: 1.
In the two-stage remanufacturer and supplier supportable stocks, it shows the best ordering recovered rate r* (r* q) for the customer. Therefore, the small batch delivery method can save costs more than a single shipment, so it is better to adopt the upstream and downstream integration strategy of the supply chain than the single strategy.

2.
The replacement rate and the backorder rate will increase expected total cost. After considering the backorder rate, under a specific number of shipments, replacement rate, and screening rate, the expected total cost will increase as the backorder rate increases. In the sensitivity analysis, it is also found that the increase of the screening rate, demand rate, defect rate and backorder rate will increase the total cost. Therefore, the ordering strategy adopted by the management personnel needs to decentralize the source of supply. Procurement should avoid concentrating on a few resettlement or supplier sources to ensure the company's best inventory policy.
It is suggested that future research should consider more realistic conditions and more complex inventory models to coordinate conflicts in the supply chain system and achieve a win-win policy for all parties, such as the time value of money, customers willing to acquire defective products at low prices, and multiple suppliers, including multi-customer cooperation and restrictions between each other.
Lin (2014) [28] does not allow the backorder in the supply chain system. However, in real case, there are often unexpected situations (sudden increase in demand, mechanical equipment factors such as repairs required for damage or careless operation of personnel) lead to backorder. Therefore, this study considers allowing backorders for shortages, and finds the optimal number of deliveries (n), the backorder rate (β) and the recovery rate (r), which will affect the total cost. Subsequent research can explore the interaction effects of these three variables to achieve the lowest total cost model.