Two-echelon closed-loop supply chain deterministic inventory models in a batch production environment

Abstract

This paper considers a two-echelon closed-loop supply chain consisting of a manufacturer and a remanufacturer at the upper echelon and a retailer at the lower echelon. The retailer faces a constant demand from customers, which is satisfied through recovered and new products received from the remanufacturer and the manufacturer, respectively. The manufacturer produces the product with finite rate, whereas the recovery of returned product is instantaneous at the remanufacturer. We develop three models to determine the optimal production-inventory policy of the players for minimizing the joint total cost of the system. In the first model, the retailer receives the product in batches from the manufacturer and the remanufacture simultaneously, whereas in the second and third models, the batches are received alternatively. In the third model, however, the procurement of raw material at the manufacturer is also considered. Numerical illustration is presented to examine the impact of certain key parameters.

Keywords:

• Two-echelon inventory system
• remanufacturing
• reverse logistics
• closed-loop supply chain

1. Introduction

In the last decade, remanufacturing has gained considerable attention of researchers and industry practitioners to investigate the most sustainable practices such as reducing manufacturing and energy cost (Dowlatshahi 2000; Giutini and Gaudette 2003; Shi, Zhang, and Sha 2011), protection of the environment. The remanufacturing establishes the foundation of the closed-loop supply chain management. Guide and Van Wassenhove (2009) defined the closed-loop supply chain management in a business perspective as “the design, control, and operation of a system to maximize value creation over the entire life cycle of a product with dynamic recovery of value from different types and volumes of returns over time”. Presently, the total business volume of remanufacturing industries is more than US$140 billion, of which auto sector occupies 70–80% of business (Naeem et al. 2013). Other than auto sector, remanufactured products include personal computers, photocopiers, electronic products, tires, different types of batteries, printers, toner cartridges, mobile phones, televisions, bullet jackets, cameras, etc.

At one side when organizations are concerned about environmental issues, the competition on the other side compels them to minimize the cost of the system. The contemporary research on product return (Schrady 1967; Nahmias and Rivera 1979; Teunter 2001; Konstantaras and Skouri 2010; Hasanov, Jaber, and Zolfaghari 2012) has mainly contributed in the area of single-stage inventory problem. The complexity of model increases when the issue of product return is considered for multi-echelon supply chain. The key focus of this research is to determine optimal production-inventory policy of the players for minimizing the joint total cost of the system. In this context, we develop three deterministic models for inventory control problems related to two-echelon closed-loop supply chain in a batch production environment.

The reminder of this paper is organized as follows: Section 2 presents the relevant literature review. Section 3 describes the notations and assumptions used throughout this study. Section 4 presents the problem description and model formulations of all the three models. The numerical examples along with sensitivity analysis for key parameters are provided in Section 5. Finally, Section 6 concludes the paper and highlights the direction for future research.

2. Literature review

To the best of our knowledge, in the area of production-inventory decision-making problems along with the recovery of returned material, Schrady (1967) was the first reported work, where he developed a deterministic model to find EOQ for a repairable inventory system considering one manufacturing batch and at least one repairable batch. He assumed that the manufacturing and recovery (repair of returned items) rates are instantaneous and there is no shortage and disposal cost. Nahmias and Rivera (1979) extended the work of Schrady (1967) assuming finite repair rate and limited storage space. Mabini, Pintelon, and Gelders (1992) extended the model of Schrady (1967) considering backorders. Teunter (2001) extended the work of Schrady (1967) by considering different holding costs for manufactured and recovered products and allowed some quantity of the returned products to be disposed-off and also considered the multiple cycles of manufacturing and repairing which are induced alternatively. Choi, Hwang, and Koh (2007) extended the work of Teunter (2001) considering the sequence of purchase orders, manufacturing and remanufacturing setups in a production cycle as decision variables. Koh et al. (2002) generalized the work of Teunter (2001) by assuming the recovery rate to be both less and greater than the demand of the serviceable product and they developed models for the policies; one order for new product and many set-ups for recovery (1, R) and many orders for new product and one set-up for recovery (P, 1). Teunter (2004) further generalized the work of Koh et al. (2002) by deriving square root formulae for optimal lot-sizing of production and recovery batches for both the policies (1, R) and (P, 1). For applying these models in a practical environment, he proposed a simple heuristic method to modify the obtained optimal lot sizes of manufacturing and recovery batches in order to assign the integer value to P and R which provides a near-optimal solution. Dobos and Richter (2004) investigated a production-recycling model where disposal is allowed and they proved that one of the pure strategies (i.e. either produce or recycle all the products) is optimal. Konstantaras and Papachristos (2006) extended both the models of Teunter (2004) by allowing complete backordering. Konstantaras and Papachristos (2008) further improved the models of Teunter (2004) by proposing an exact solution method to find the integer values of P and R. Konstantaras and Skouri (2010) extended the models of Teunter (2004) by considering variable number of remanufacturing lots of equal size followed by a variable number of manufacturing lots of equal size, i.e. (R, P) policy. Schulz and Voigt (2014) extended the work of Choi, Hwang, and Koh (2007) by dividing the sequence of manufacturing and remanufacturing batches into subcycles and proposed a heuristic approach which allows different batch sizes of remanufacturing lot instead of equal batch sizes.

Some of the studies in this area considered return rate and waste disposal rate of the returned product as a decision variables (Richter and Dobos 1999; Dobos and Richter 2004), whereas in the previous models number of manufacturing and remanufacturing cycles are the decision variables along with the lot size of manufactured and remanufactured products. Some researchers considered minimizing the discounted total cost as an objective function (Teunter and Van Der Laan 2002; Corbacıoglu and van der Laan 2007), while some of them (Nikolaidis 2009; Jena and Sarmah 2014) addressed the issue of acquisition management and pricing policies of the returned product. Few studies considered the quality of the recovered products different from the manufactured product (Jaber and El Saadany 2009; Hasanov, Jaber, and Zolfaghari 2012), whereas Tsai (2012) studied the learning effects on remanufacturing and manufacturing process.

Further, few exhaustive literature reviews also have been carried out in this area focused on the literature related to quantitative models in reverse logistics system design (Fleischmann et al. 1997), production planning and control in remanufacturing (Guide, Jayaraman, and Srivastava 1999), inventory control and production planning models related to closed-loop supply chain (Akcalı & Cetinkaya 2011), issues in reverse supply chain (Sasikumar and Kannan, 2008a, 2008b, 2009).

In the case of two-echelon closed-loop supply chain, Mitra (2009) suggested a generalized inventory management problem considering one depot and one distributor for a single retailer. He developed deterministic as well stochastic models considering independent demand and return rate for optimal lot sizing and shipment policy, whereas Mitra (2012) developed similar models but considered correlated demand and return rate. Teng et al. (2011) extended the work of Mitra (2009) by optimizing the partial backordering inventory model with product returns and excess stocks.

In the case of multi-echelon closed-loop supply chain, Chung, Wee, and Yang (2008) developed a joint profit maximization model for the system consisting of multiple players such as supplier, manufacturer for single retailer and a third party who collects used products. In this system, manufacturing and remanufacturing operations are carried out alternatively in the common production line at manufacturer. Yuan and Gao (2010) and Yang et al. (2013) extended the work of Chung, Wee, and Yang (2008) by developing profit maximization models for (1, R) and (P, 1) policies.

As mentioned above, the majority of the models developed dealing inventory management problem in a closed-loop supply chain belongs to single-echelon and considered only one player in the system. However, limited studies (Mitra 2009; Teng et al. 2011; Mitra 2012) are available for those belonging to two-echelon closed-loop supply chain and considered more than one player in the system and all these models assumed procurement of the finished product from the outside vendor, rather than producing at manufacturer (or warehouse) and supply to retailer(s)/customer(s). The proposed study belongs to two-echelon closed-loop supply consisting of three players in the system; manufacturer, remanufacturer and retailer. All the three models of the proposed study consider the production of the new product at a finite manufacturing rate (i.e. batch production environment) at manufacturer instead of procurement from outside vendor. Further, in the third model, the cost associated with the raw material of the manufacturer is also included, which none of the above studies included in the case of two-echelon closed-loop supply chain system. The generic problem description under study is explained in brief in subsequent paragraph of this section, however the detailed description of each model is provided in Section 4.

In this paper, an integration issue of two-echelon closed-loop supply chain consisting of manufacturing and remanufacturing player for a single retailer has been studied to determine the optimal lot sizing and shipment policy for the minimum joint total cost of the integrated system in a batch production environment. The system under study is shown in Figure 1, which consists of a manufacturer, a remanufacturer and a retailer. The retailer faces a constant demand and simultaneously the remanufacturer receives a fraction of demand as returned material for recovery purpose. The retailer receives the supplies from manufacturer and remanufacture for satisfying the demand of the customers. The remanufacturer accumulates the returned material till the end of its cycle and then remanufacture the returned material at the end its each cycle. The remanufacturer dispatches to the retailer the recovered material immediately after the recovery of the returned material. The remanufacturing time is assumed negligible. The remaining requirement of the retailer is satisfied by the manufacturer. The manufacturer produces product in a batch and dispatches in batches to retailer. The manufacturer used fresh raw material which is purchased in batches from outside vendor.

Figure 1. Inventory system set-up.

We develop three models and considered only set-up/ordering and inventory holding costs of retailer, manufacturer and remanufacturer as we have considered the cost structure only for inventory point of view. In Model-I, the retailer receives the lots of manufactured and remanufactured product simultaneously from the manufacturer and remanufacturer at the beginning of each cycle, whereas in Model-II and Model-III, the retailer receives lots of manufactured and remanufactured product alternatively just after the consumption of the previous lot. In Model-III, we have considered the costs associated with raw material also in addition to the costs considered in Model-I and Model-II.

The models presented in this study can be applicable to the lead-acid battery manufacturing industries. In the case of lead-acid battery, the residual value of the used battery is very high, as almost all the parts of the battery is reusable without spending much amount on remanufacturing operation. Further, lead being a hazardous material, it is the obligatory responsibilities of the lead-acid battery manufacturing industries to recover and reuse the maximum possible quantity of the lead from the used product for minimizing environmental hazardous.

3. Notations and assumptions

To establish the mathematical models, the following notations and assumptions are used.

Notations
μ	=	demand per unit time on the retailer
r	=	fraction of demand returned per unit time, 0 < r < 1
α	=	conversion factor of the returned product to the remanufactured product, 0 < α ≤ 1
rα	=	fraction of demand, which is remanufactured per unit time
f	=	conversion factor of raw material to finished product, where f ≤ 1
P	=	production rate of the manufacturer, P > μ(1 − αr)
Q	=	lot size of the product received in one replenishment cycle at the retailer from the manufacturer and the remanufacturer (decision variable)
Qfi	=	procurement lot size of raw material of the manufacturer for case i (i = 1, 2) of Model-III
A1	=	cost per order of the retailer ($/order)
A2	=	cost per production setup of the manufacturer ($/setup)
A3	=	cost per production setup of the remanufacturer ($/setup)
A4	=	cost per order for raw material of the manufacturer ($/order)
h1	=	inventory holding cost per unit per unit time of the retailer ($/unit/time)
h2	=	inventory holding cost per unit per unit time for the finished product of the manufacturer ($/unit/time)
h3	=	inventory holding cost per unit per unit time for the returned material of the remanufacturer ($/unit/time)
h4	=	inventory holding cost per unit per unit time for the raw material of the manufacturer ($/unit/time)
mj	=	number of shipments made to the retailer during one production cycle of the manufacturer in Model-j (j = 1, 2, 3), where mj is a positive integer (decision variable)
ki	=	number of shipments of raw material received in one production cycle at the manufacturer in case i (i = 1, 2) of Model-III. Here ki = {1, 1/2, 1/3, … ,1/n1}∪{1, 2, 3, … ,n2},where n1 and n2 are positive integers (decision variables)

Assumptions

(1)	Remanufactured product is as good as the new product and used by the retailer for satisfying the demand of customers. Consumable items needed for recovery are not included in the models.
(2)	Demand and return rates are deterministic, stationary and uniform over infinite time horizon.
(3)	Returned product is remanufactured in batches and remanufacturing time of a batch is negligible.
(4)	Rejected material during the remanufacturing operations is immediately discarded.

4. Problem description and model formulation

This paper considers a two-echelon closed-loop supply chain consisting of a manufacturer and a remanufacturer at the upper echelon and a retailer at the lower echelon. The retailer faces a constant demand from customers, which is satisfied through recovered and new products received from the remanufacturer and the manufacturer, respectively. Simultaneously, a fraction of the demand is received by the remanufacturer as returned product for recovery purpose. The returned products are accumulated at the remanufacturer till the end of collection cycle of the remanufacturer, and then accumulated returned products are remanufactured and immediately shipped to the retailer. The retailer receives a lot of quantity (1 − αr)Q of the new product from the manufacturer and a quantity of αrQ from the remanufacturer to satisfy the demand of quantity Q in each cycle of the retailer. The manufacturer produces the product in a batch of size m_j(1−αr)Q for model-j at the rate of P units per unit time in each production cycle and ships the manufactured product to the retailer in m_j lots each of size (1 − αr)Q at the interval of time Q/μ. This description is common for all the models, whereas the specific characteristics of the models are discussed while explaining the individual model. The objectives of the models are to determine the optimal production-inventory policy of the players for minimizing the joint total cost of the system.

4.1. Model-I (case of simultaneous replenishment policy at the retailer)

In this model, the lots from the manufacturer and the remanufacturer are received simultaneously at the retailer. Figure 2 shows the inventory variation pattern for the manufacturer, the remanufacturer and the retailer. The retailer receives the product in a lot of quantity (1 − αr)Q from the manufacturer and quantity αrQ from the remanufacturer simultaneously at the beginning of each cycle of the retailer. Therefore, the retailer will always have a lot of quantity Q at the beginning of each cycle and thereafter inventory declines at the rate of μ quantity per unit time and becomes zero at the end of each cycle of the retailer. The remanufacturer has zero inventory of the returned product at the beginning of each cycle and it accumulates at the rate of rμ quantity per unit time and the inventory level of returned product becomes rQ during the period Q/μ. Then, the accumulated returned products are remanufactured instantaneously (remanufacturing time is negligible) and the recovered product of quantity αrQ is shipped to the retailer as a single lot at the end of each cycle of the remanufacturer.

Figure 2. Model I and II – On-hand Inventory pattern for manufacturer, remanufacturer and retailer.

4.1.1. The joint total cost of the system

It can be seen from Figure 2, if lot size of the retailer is Q, the cycle length of the retailer and the remanufacture is Q/μ and the production cycle length of the manufacture is m₁Q/μ. Accordingly, the number of orders placed per unit time by the retailer is μ/Q and the number of production set-up per unit time of the manufactures is μ/(m₁Q). The relevant total cost of each player is derived as follows.

4.1.1.1. Average inventory of the retailer

The area of triangle due to one cycle of the retailer is Q²/2μ; hence, the average inventory per unit time at the retailer is Q/2.

4.1.1.2. Average inventory of the manufacturer

The manufacturer’s average inventory per unit time is calculated by taking the difference of the manufacturer’s accumulated inventory and the retailer’s accumulated inventory due to the manufacturer’s supply per unit time. The expression for the manufacturer’s average inventory per unit time is derived as

and on simplification, it can be expressed as .

4.1.1.3. Average inventory of the remanufacturer

Area of the triangle due to one cycle of the remanufacturer is rQ²/2μ, therefore, the average inventory per unit time will be equal to rQ/2.The number of remanufacturing production set-up per unit time will be equal to μ/Q.

Therefore, the joint total cost per unit time of the closed-loop supply chain of Model-I is the sum of the ordering and the inventory holding costs per unit time of the retailer, production set-up and inventory holding costs per unit time of the manufacturer and the remanufacturer. This can be expressed as(1)

To find the minimum value of JTC(Q, m₁) and the optimal value of Q, we take the first partial derivative of JTC(Q, m₁) with respect to Q for fixed value of m₁ and obtain

Further to proof the convexity, we take the second derivative of JTC(Q, m₁) with respect to Q for fixed value of m₁ and obtain

, which is always a positive quantity. Hence, JTC(Q, m₁) is convex in Q for fixed value of m₁.

Thus, to find the optimal value of Q for fixed value of m₁, the first partial derivative is set to zero and the optimal value of Q can be given by(2)

Now, after substituting the value of Q * (m₁) from (2) in (1), the joint total cost can be expressed as(3)

Let , therefore

To obtain the optimal value of m₁, m₁ is temporarily considered as a continuous variable. The function F(m₁) is differentiated with respect to m₁ and set to zero. The optimal value of m₁is given by(4)

Since the function F(m₁) is convex in m₁ (proof is provided in Appendix 1) and m₁ takes positive integer value, so the optimal value of m₁ can be found according to the following rule. If the value of is obtained as an integer from (4), then is the optimal value of m₁. Otherwise, evaluate JTC() and JTC() using (3), where and are the two integers surrounding , and the optimal value of m₁ is selected as  =  if JTC() < JTC() else  = .

4.2. Model-II (case of alternate replenishment policy at the retailer)

The inventory variation pattern for all the players of the closed-loop supply chain of Model-II is shown in Figure 2.In this model, the only difference from Model-I is that the retailer receives the product in a lot of quantity (1 − αr)Q from the manufacturer at the beginning of each cycle of the retailer. The inventory of this lot declines at the rate of μ quantity per unit time and becomes zero after the time (1 − αr)Q/μ and then immediately the retailer receives another lot of quantity αrQ from the remanufacturer, which satisfy the demand of the customer for the remaining period of the same cycle of the retailer.

4.2.1. The joint total cost of the system

Similar to Model-I, the cycle length of the retailer and remanufacturer is Q/μ and the production cycle length of the manufacturer is m₂Q/μ. Hence, the number of order placed per unit time by the retailer is μ/Q and the number of production set-up of manufacturer is μ/(m₂Q).

The average inventory per unit time of the retailer, the manufacturer and the remanufacturer is calculated for Model-II (see Figure2) similar to Model-I and which can be expressed as

, , and , respectively.

Therefore, the joint total cost per unit time of the closed-loop supply chain of Model-II can be expressed as(5)

To find the minimum value of JTC(Q, m₂) and the optimal value of Q, we take the first partial derivative of JTC(Q, m₂) with respect to Q for fixed value of m₂, and obtain

Further to proof the convexity, we take the second derivative of JTC(Q, m₂) with respect to Q for fixed value of m₂, and obtain

, which is always a positive quantity. Hence, JT(Q, m₂) is convex in Q for fixed value of m₂.

Thus, to find the optimal value of Q for fixed value of m₂, the first partial derivative is set to zero and the optimal value of Q can be given by(6)

Now, after substituting the value of Q * (m₂) from (6) in (5), the joint total cost can be(7)

Similar to Model-I, let , therefore

Like Model-I, to obtain the optimal value of m₂, m₂ is temporarily considered as continuous variable and the function F(m₂) is differentiated with respect to m₂ and set to zero. The optimal value of m₂ is given by(8)

The convexity of function F(m₂) can be proved like Appendix 1 and the optimal positive integer value of m₂ can be found like m₁ of Model-I.

The joint total cost of Model-II will be always less than the joint total cost of Model-I when m₁ = m₂.Since in the case of Model-II, the shipment from remanufacturer is deployed at the instant when the shipped product from manufacturer is completely consumed. Thus, the average inventory level at the retailer is always less in Model-II compared to Model-I. As in Model-I, the shipments from both manufacturer and remanufacturer are consolidated.

4.3. Model-III (Model-II is extended by incorporating raw material of manufacturer)

Model-III is an extension of Model-II by considering the procurement of raw material by the manufacturer to produce the new product, it is also in line with the Lee (2005) model for single-manufacturer single-buyer in a forward supply chain problem. The inventory variation pattern of the manufacturer, the remanufacturer and the retailer is presented in Figure 3. The model is formulated to include the ordering and holding costs of the raw material of the manufacturer in addition to the cost components considered in Model-II. To derive the expression for the costs associated with the procurement lot size of the raw material by the manufacture is explained in the following paragraphs.

Figure 3. Model-III – On-hand Inventory pattern for manufacturer, remanufacturer and retailer.

There can be two situations for procurement lot size of the raw material.

Case-1: In this situation, each procurement lot size of the raw material will be used for n₁ (k_i = 1/n₁) number of production cycle of the manufacturer, where n₁ is a positive integer.

Case-2: each production lot size of the manufacturer will use n₂ (k_i = n₂) number of lots of the raw material, where n₂ is a positive integer.

Now onwards for explaining the Model-III, we will use 1 as suffix for case-1 and 2 as suffix for case-2 for decision variables Q, m₃ and n.

It can be seen from Figure 3 that the number of orders per unit time for the raw material in case-1 and case-2 are and , respectively. Therefore, the raw material ordering cost per unit time of the manufacturer will be and for case-1 and case-2, respectively. The average inventory per unit time of the raw material at the manufacturer for case-1 and case-2 are given by and , respectively. Therefore, the raw material holding cost per unit time is and for case-1 and case-2, respectively.

4.3.1. Computing the joint total cost

The joint total cost for case-1 and case-2 are computed as follows:

4.3.1.1. Case-1

The joint total cost per unit time of the system in case-1 is the sum of the ordering and the inventory holding costs per unit time of the retailer, the ordering and inventory holding cost per unit time of raw material of the manufacturer, production set-up and inventory holding costs per unit time of finished products of the manufacturer and the production set-up and inventory holding costs per unit time of returned products of the remanufacturer. This can be expressed as(9) (10)

To solve the above problem, we temporarily relax the integer requirements of m₃₁ and n₁. In order to find the minimum value of JTC₁(Q₁, m₃₁, n₁), we take the first and second partial derivatives of JTC(Q₁, m₃₁, n₁) with respect Q₁ for fixed m₃₁ and n₁and obtain(11) (12)

Since for fixed m₃₁ and n₁, the second derivative of the function JTC₁(Q₁, m₃₁, n₁) with respect to Q₁ is positive, hence JTC₁(Q₁, m₃₁, n₁) is convex in Q₁.

Similarly, we take the first and second partial derivatives of JTC₁(Q₁, m₃₁, n₁) with respect to m₃₁ for fixed value of Q₁ and n₁ and with respect to n₁ for fixed Q₁ and m₃₁ and obtain the following expressions:(13) (14) (15) (16)

Since (14) and (16) are positive, JTC₁(Q₁, m₃₁, n₁) is convex in m₃₁ for fixed Q₁ and n₁ and in n₁ for fixed Q₁ and m₃₁.

Now, to get the optimal value of Q₁, m₃₁ and n₁, when all the three decision variables are considered continuous, Equations (11), (13) and (15) are set to zero, and we obtain(17) (18) (19)

After solving the three simultaneous Equations (17), (18) and (19), we get the optimal value of the decision variables as follows:(20) (21)

and(22)

For representing the expressions in a simpler way, let(23)

and(24)

Thus, (20), (21) and (22) can be written as(25) (26)

and(27)

Now for fixed m₃₁ and n₁, the optimal value of Q₁ can be obtained from (17) and the corresponding minimum joint total cost from (10). Thus, the expression for the minimum value of joint total cost for fixed m₃₁ and n₁ will be given by(28)

4.3.2. Computing the joint total when m₃₁ and n₁ are discrete variables

Minimization of JTC₁(m₃₁, n₁) in (28) is equivalent to minimization of [JTC₁(m₃₁, n₁)]². Therefore, or(29)

Let ψ₁(m₃₁, n₁) denote the function of [JTC₁(m₃₁, n₁)]² after subtracting the terms which are independent of m₃₁ and n₁. Hence, the minimization problem of (29) becomes simply minimizing ψ₁(m₃₁, n₁). Thus(30)

Since function (28) is convex for fixed m₃₁ and n₁, the same applies to function (30).

Now to minimize the ψ₁(, ) for the fixed , an associated discrete optimal value of will have to satisfy both the inequalities (31) and (32) as given below:(31) (32)

Considering inequality (31) and ignoring all the terms which are independent of , we obtained

and after simplifying the above expression, we will obtain , and similarly from (32) we get . Hence, by combining both the inequalities, we obtain(33)

As the minimum joint total cost of the model may exist either in case-1 or case-2 (except the situation when  = , where the minimum joint total cost is same in both the cases), therefore, it is required to derive all the functions and relations of case-2 similar to case-1.

4.3.2.1. Case-2

Similar to case-1of Model-III, the joint cost per unit time of the system in case-2 can be expressed as(34) (35)

Now like Case-1, the function JTC₂(Q₂, m₃₂, n₂) can be proved convex in Q₂ for fixed m₃₂ and n₂, in m₃₂ for fixed Q₂ and n₂ and in n₂ for fixed Q₂ and m₃₂.

Further, by using the same procedure as case-1 to get the optimal value of Q₂, m₃₂ and n₂, when considering all the three decision variables as continuous, we obtain the following equations:(36) (37) (38)

After solving the three simultaneous Equations (36), (37) and (38), we get the optimal value of the decision variable as follows:(39) (40) (41)

Further, when m₃₂ and n₂ are discrete variables; then for fixed n₂, the optimal discrete value of m₃₂ can be obtained by the inequality given below:(42)

4.3.3. Procedure for finding the global optimal solution

An iterative method is used to obtain the global optimal solution when m_3i and n_i are discrete variables. Hence, an algorithm is developed to find the optimal solution for Model-III. For this we use ( denotes the largest integer less than equal to ) from (22) and ( is the largest integer less than equal to ) from (38) as the starting points in the iterative methods while finding the optimal solutions in case-1 and case-2, respectively.

The optimal solutions are obtained through the following algorithm:

4.3.3.1. Algorithm

Step-1: Obtain the value of from (22) for case-1 ( from (38) in case-2);

Step-2: Let n₁ =  for case-1 (n₂ =  for case-2, here and denotes the largest integer less than or equal to and , respectively), go to Subroutine.Let  = JTC₁ for case-1( = JTC₂ for case-2);

Step-3: Let n₁ =  + 1 for case-1(n₂ =  + 1 for case-2), go to Subroutine. Let JTC₁(+)^* = JTC₁ for case-1 (JTC₂(+)^* = JTC₂ for case-2);

Step-4: Let n₁ =  − 1 for case-1 (n₂ =  for case-2), go to Subroutine. Let JTC₁(−)^* = JTC₁ for case-1 (JTC₂(−)^* = JTC₂ for case-2);

Step-5: If JTC₁(−)^* ≥  ≤ JTC₁(+)^*(JTC₂(−)^* ≥  ≤ JTC₂(+)^* for case-2), then ( for case-2) is optimal, stop;

If JTC₁(−)^* ≥  ≥ JTC₁(+)^* (JTC₂(−)^* ≥  ≥ JTC₂(+)^* for case-2), let  = JTC₁(+)^* ( = JTC₂(+)^* for case-2), n₁ =  + 1 (n₂ =  + 1 for case-2), go to step 6;

If JTC₁(−)^* ≤  ≤ JTC₁(+)^*(JTC₂(−)^* ≤  ≤ JTC₂(+)^* for case-2), let  = JTC₁(−)^* ( = JTC₂(−)^* for case-2), n₁ =  (n₂ =  for case-2), go to step 8;

Step-6: Let n₁ = n₁ + 1 (n₂ = n₂ + 1 for case-2), go to Subroutine;

Step-7: If  ≥ JTC₁ ( ≥ JTC₂ for case-2), Let  = JTC₁ ( = JTC₂ for case-2), go to Step-6;

otherwise, the optimal solution was found in the previous run, JTC₁(, m₃₁, n₁) (JTC₂(, m₃₂, n₂) for case-2); stop.

Step-8: Let n₁ = n₁–1 (n₂ = n₂ − 1 for case-2), go to Subroutine;

Step-9: If  ≥ JTC₁ ( ≥ JTC₂ for case-2), Let  = JTC₁ ( = JTC₂ for case-2), go to Step-8; otherwise, the optimal solution was found in the previous run, JTC₁(, m₃₁, n₁) (JTC₂(, m₃₂, n₂) for case-2); stop.

Subroutine:

(1)	Obtain the associated discrete value of m31 from (33) for case-1 (m31 from (42) for case-2);
(2)	Put the value of n1 and m31 in (17) and obtain the value of Q1 for case-1 (n2 and m32 in (36) and obtain Q2 for case-2);
(3)	Put the value of Q1, m31 and n1 in (9) and obtain the value of JTC1for case-1(Q2, m32 and n2 in (34) and obtain JTC2 for case-2); return.

Note that, since the cost functions are convex, hence it is not necessary to compute joint total cost for both the cases simultaneously. One can apply the algorithm in case-1 only, as long as the value of is more than one as in this situation the optimal solution will definitely fall into case-1. While in the situation where is less than or equal to one, then one should use the algorithm for case-2 also and find out which case will provide the lowest joint total cost. However, in the situation when  =  = 1, the optimal solution will be same in both the cases.

5. Numerical illustration and sensitivity analysis

The minimum joint total cost of Model-I and Model-II are computed from the Equations (3), (4) and (7), (8) respectively. Whereas in Model-III, algorithm is used to obtain the minimum joint total cost. The calculation results of Model-III for the parameters: μ = 10,000, p = 15,000, A₁ = 100, A₂ = 400, A₃ = 200, A₄ = 100 and 6000, h₁ = 40, h₂ = 20, h₃ = 10, h₄ = 12, r = 0.25, α = 0.9 and f = 0.8, are shown in Table 1.

Table 1. Calculation results of Model-III for two values of A_4.

ki	n1	n2	Qf1	Qf2	m3i	Qi	JTC(Qi, m3i, ni)
When A4 = 100
1/3	3	–	1372.0	–	–	472.1	31069.0
1/2	2	–	1019.4	–	–	526.1	28510.0
1	1	1	–	858.2	2	443.0	24833.0
2	–	2*	–	474.3*	2*	489.62*	24509.0*
3	–	3	–	417.1	3	430.5	24776.0
When A4 = 6000
1/4	4	–	3454.7	–	2	445.8	56083.0
1/3	3	–	3464.8	–	3	397.4	55361.0
1/2	2*	–	3265.4*	–	4*	421.3*	54588.0*
1	1	1	–	3006.5	7	443.4	54777.0
2	–	2	–	2368.1	11	444.5	64227.0
3	–	3	–	2018.3	14	446.5	72316.0

* Optimal value Q_f1 and Q_f2 are the lot sizes of raw material in case-1 and case-2, respectively.

From the results of Model-III, we can observe that when the cost per order of raw material is low (A₄ = 100), the optimal solution exists in case-2, whereas when it is high (A₄ = 6000), the optimal solution exists in case-1. Further, it is also observed that the optimal lot size of raw material is changed from 474.32 units to 3265.37 units when we increased per order cost of raw material from 100 to 6000. Hence, we conclude that ordering cost of raw material is a significant factor for deciding the ordering policy and lot size of the raw material.

5.1. Sensitivity analysis

Sensitivity analysis is carried out to examine the impact of ‘rα’ and P and also all the three models are compared with respect to the minimum joint total cost (MJTC). The value of the parameters for the purpose of analysis is taken as follows:

μ = 10,000, p = 15,000, A₁ = 100, A₂ = 400, A₃ = 200, A₄ = 250, h₁ = 40, h₂ = 20, h₃ = 10, h₄ = 12, r = 0.25, α = 0.9 and f = 0.8.

The computational results of the sensitivity analysis are presented in Figures 4 and 5 and the observations of the sensitivity analysis are discussed in subsequent sections.

Figure 4. Variation of the joint total cost with ‘rα’.

Figure 5. Variation of the joint total cost with ‘P’.

5.1.1. Impact of ‘rα’

To examine the impact of ‘rα’, r is varied while keeping α constant. It is evident from the data taken for the sensitivity analysis that the cost of remanufacturing is lower than the manufacturing. We observe from Figure 4 that MJTC of Model-I decreases with the increase of ‘rα’, whereas in Model-II, the MJTC is lowest when the value of ‘rα’ is close to 0.5. This is because the value of the term of joint total cost function of Model-II is lowest when ‘rα’ is equal to 0.5. In Model-III, the MJTC decreases quickly with the increase of ‘rα’ as in this case the cost component of raw material of the manufacturer is also reduced with the increases of ‘rα’.

5.1.2. Impact of P

From Figure 5, it is observed that initially, with the increase of P in Model-I and model-II, MJTC first increases and then becomes constant, but in Model-II, further it starts decreasing. However, in both the models, the MJTC is lowest, when the P is just above the demand rate of retailer from the manufacturer. From the Equations (1) and (5) it is observed that as long as the value of m_j (j = 1, 2) is equal to 2, the MJTC will be constant even with the increase of P. However, in the case of Model-III, the MJTC decreases continuously when P increases even when m_3i is equal to 2. This is because the raw material holding cost per unit time of manufacturer decreases when P increases. Hence, the difference of MJTC between the Model-II and Model-III decreases when P increases.

5.2. Comparison between traditional forward supply chain model and closed-loop supply chain models

Model-I, Model-II and Model-III of closed-loop supply chain is compared with the corresponding traditional forward supply chain models (by putting r = 0, A₃ = 0 and h₃ = 0) for different set of values of the parameters r, h₃ and A₃, while keeping the value of the other parameters same as mentioned earlier in Section 5.1.

We observe from the Table 2 that the minimum joint total cost of the closed-loop supply chain models are lower than the forward supply chain models when the set-up and inventory holding costs of the remanufacturer are significantly lower as compared to the corresponding ordering/set-up and inventory holding costs of the retailer and the manufacturer, and also when the fraction of demand recovered material (rα) is significantly high.

Table 2. Comparison of joint total cost as changing the value of r, A₃ and h_3.

rα	A3	h3	Joint total cost
			Closed-loop supply chain			Forward supply chain (r, A3 & h3 = 0)
			Model-I	Model-II	Model-III	Model-I & Model-II	Model-III
0.09	50	2	19068	18189	24797	18974	24152
		5	19107	18230	24841
		10	19173	18298	24915
	150	2	22419	21269	27975
		5	22470	21323	28025
		10	22555	21413	28108
	250	2	25174	23882	30390
		5	25232	23944	30456
		10	25327	24044	30567
0.45	50	2	18157	14898	18809
		5	18362	15180	19099
		10	18699	15640	19573
	150	2	21262	17024	20973
		5	21531	17416	21380
		10	21971	18050	22043
	250	2	23875	18820	22724
		5	24177	19254	23165
		10	24671	19955	23883
0.81	50	2	15973	14104	15867
		5	16335	14538	16301
		10	16921	15233	16999
	150	2	19122	16718	18474
		5	19608	17274	19050
		10	20352	18163	19927
	250	2	21683	18811	20592
		5	22236	19493	21253
		10	23129	20578	22312

6. Conclusion and scope for future research

In this paper, an integration issue of the two-echelon closed-loop supply chain consisting of manufacturing and remanufacturing player for a single retailer has been studied to determine the optimal lot sizing and shipment policy for minimum joint total cost. Models were developed for an infinite time horizon at constant demand and return rate, considering set-up/ordering and inventory holding costs of all the players in the system. Sensitivity analysis is carried out to examine the impact of ‘rα’ and P, and from the sensitivity analysis it is found that the minimum joint total cost of the closed-loop supply chain system decreases when ‘rα’ and ‘P’ increases. In addition to the sensitivity analysis, the closed-loop supply chain models were also compared with the corresponding forward supply chain (by putting r = 0, A₃ = 0 and h₃ = 0) models. The findings of the comparison revealed that when the product recovery cost is significantly low and the ‘rα’ is significantly high, then only the closed-loop supply chain is economically sustainable.

The models presented in this paper can be applicable for the lead-acid manufacturing industries, as in the case of lead-acid battery manufacturing industries, the fraction of demand return is very high and the cost of remanufacturing is low, hence the closed-loop supply chain is economically sustainable and also it is the obligatory responsibility of the lead-acid battery manufacturing to implement the closed-loop supply chain for preventing environmental hazardous.

In this research, we have modelled for constant demand situation. Modelling for the dynamic demand, uncertain demand and modelling for multi-echelon can be a direction for future research. In this study, we have assumed that the conversion factor of returned material to recovered product is constant and the remanufacturing time is insignificant, relaxing these assumptions can also be a direction for future research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix 1

The function F(m₁) can be expressed as

Now for proving the convexity of the function F(m₁)with respect to m₁, we find the second derivative of F(m₁) with respect to m₁ and obtain the following expression:(A-1)

Therefore, the function F(m₁) will be convex with respect to m₁ if in (A-1) is positive.

Let

.

Since P > μ(1 − αr) (assumptions made in Section 3), i.e. , therefore

, which is always true for 0 < r < 1 and 0 < α ≤ 1, as all other parameters are positive. Thus, the function F(m₁) is convex with respect to m_1.