Optimal Acquisition and Production Policies for Remanufacturing with Quality Grading

Abstract

Core acquisition is essential to the success of the remanufacturing business. The value of sorting and grading cores into nominal-quality classes has been certified in industry and academia. In this paper, we investigate how many unsorted cores of uncertain quality should be acquired and how many sorted cores should be remanufactured by a third-party remanufacturer (3PR) before the demand is realized. We first develop analytically tractable solutions to the acquisition and production model under deterministic demand, and then we extend it to the model under the stochastic demand by fully characterizing the structure of the optimal policy. Subsequently, we investigate the impact of core quality fraction uncertainty on the solutions. Finally, numerical analyses are conducted to further verify the proposed models. The results are as follows. First, the optimal quantity of acquisition/production and minimum expected profit increase with an increase in the selling price and decrease with an increase in the uncertainty of demand and acquisition cost. Second, the optimal production quantity does not decrease in acquisition quantity, and the rate of utilization of the recycled parts (the ratio of production quantity to acquisition quantity) increases with a decrease in the acquisition cost. Third, the growth stage is most profitable stage, so the remanufacturers should pay more attention to remanufacturing activities early in the life of products. The proposed models and solutions can not only solve the core acquisition and production problem in remanufacturing, but also solve the combinatorial optimization problem.

1. Introduction

As a mode of environmental protection and energy saving in the context of production, remanufacturing has attracted the attention of enterprises and governments all over the world [1,2,3]. An increasing number of original equipment manufacturers use remanufacturing along with manufacturing to satisfy market demands with competitive prices and environmentally friendly production, e.g., Caterpillar, ReCellular, Boeing, Deere, and Sony. However, remanufacturing not only faces the same market uncertainties as traditional manufacturing, but also encounters the variation inherent in the condition of used products. To be more precise, the supply of raw materials is relatively stable and their quality is uniform in traditional manufacturing, whereas the supply and quality of recycled materials are usually uncertain in remanufacturing [4,5,6,7,8,9].

For the above reasons, supply-side management becomes more complex in remanufacturing than in manufacturing, which is a key input to assess the economic attractiveness of reuse activities [5]. Researchers have pointed out the importance of issues related to core acquisition, and a growing number of studies have dealt with quantitative decisions in this domain [10,11,12,13]. Ferguson et al. [14] have shown that the firm does not remanufacture a unit with a lower grade of quality before one with a higher grade is out of stock; in other words, the firm always remanufactures the higher-quality grade first after the cores have been acquired and sorted. In other words, for a given acquisition quantity, the cost of production in remanufacturing is a piecewise-linear convex function.

This study is motivated by the work of Mutha, Bansal, and Guide [12], who investigated whether a third-party remanufacturer (3PR) should acquire used products or cores with uncertain levels of quality in bulk or those with known levels of quality in sorted grades, and Mutha et al. [15], who studied inter-firm transactions for buying and selling used graded products with analytically tractable models. We consider the scenario in which a 3PR acquires used products with unknown levels of quality and sorts them into different grades to satisfy the demand. We also develop models to determine the optimal quantity of products to acquire and produce. Similar to the work conducted by Mutha, Bansal, and Guide [12] and Mutha, Bansal, and Guide [15], we investigate the policy of the core acquisition of 3PRs that remanufacture products with a high time value of money, i.e., products that become obsolete quickly and whose inventory loses value quickly. To be profitable in this environment, a 3PR needs to be efficient to avoid the risk of a depreciation of the inventory while being responsive to uncertain demands [12]. Previous research on this topic [9,16,17,18,19] has not focused on developing an analytically tractable solution. We establish a model of acquisition and production under deterministic demand, generalize it to scenarios involving stochastic demand, and provide an analytically tractable solution for it.

The contributions of this paper are as follows. For core acquisition management, we establish models on the optimal acquisition/production of unsorted/sorted used products for independent remanufacturing by giving analytically tractable solutions. For inventory management, we characterize the structure of the optimal policy by using a quantity-dependent piecewise-linear convex cost. The proposed models and solutions not only solve the problems of core acquisition and production in remanufacturing, but can also be used to solve combinatorial optimization problems [20], e.g., piecewise -linear, unsplittable, multi-commodity flow problems [21], and network design and routing problems [22].

The remainder of this paper is organized as follows. We review the related literature in Section 2. Section 3 characterizes the optimal acquisition and production policies for remanufacturing under deterministic and stochastic demands. Section 4 analyzes the impact of core quality fraction uncertainty. Section 5 presents a case study to verify the validity of the proposed models and discusses their sensitivity to parametric changes. The main findings and managerial insights of this study are summarized in Section 6, as well as some highlights of possible directions for future research.

2. Literature Review

This paper is most closely related to two streams of literature: core acquisition in remanufacturing operations and inventory–production control with a piecewise-linear convex cost. For a thorough review of academic work on core acquisition in remanufacturing, we refer interested readers to Atasu et al. [23]; Wei, Tang, and Sundin [7]; Jena and Sarmah [24]; and Peng et al. [25].

In the streams of core acquisition management, Guide [4] provided the first step in research on the management of used product acquisition by providing an overview of current practices. He claimed that the manager should take actions that consistently reduce the inherent variance in a remanufacturing environment. To address the inherent variance in recycled products and uncertainty in the demand for them, previous studies have explored mitigation strategies from different perspectives, e.g., product design [26,27], channel selection [28,29,30,31,32], acquisition quantity [33,34,35], price [36,37,38,39], timing [12,40], lot sizing [41], quality information [36,42,43], and grading [14,43,44], among others. In terms of quality grading, Aras et al. (2004) investigated the issue of the stochastic nature of product returns and identified conditions under which quality-based categorization is the most cost-effective. Ferguson, Guide, Koca, and Souza [14] further exhibited the value of quality grading in remanufacturing by explicitly considering quality-related information in deciding which returned products to remanufacture or dispose of. In addition, many studies have provided a greater understanding of the effects of quality grading in remanufacturing systems from other perspectives, such as network design and configuration [45,46], and optimizing the lot size and scheduling [6,47]. Studies on acquisition according to the time of grading can be divided into three main categories: pre-purchase grading [10,15,48], post-purchase grading [9,17,49], and hybrid grading [12]. This work focuses on post-purchase grading, considers the situation when a buyer acquires used products of unknown quality and sorts them into different grades, and develops models to determine the optimal number of products to acquire and manufacture for the remanufacturer.

The second stream of related research focuses on inventory–production control with a piecewise-linear convex cost. Research on inventory systems with a convex and variable cost of production can be traced back to the study by Karlin [50]. Recently, Lu and Song [51] characterized the optimal inventory control policy for a periodic review inventory control system with a fixed cost and a convex variable cost. Caliskan-Demirag et al. [52] and Li et al. [53] considered a case in which the cost of production was a step function of the quantity of production. Sillanpää et al. [54] proposed the optimal procurement decisions with a well-designed contract when the shortage costs of individual periods are piecewise-linear. Ardestani-Jaafari and Delage [55] examined the robust optimization of sums of piecewise-linear functions. Fortz, Gouveia, and Joyce-Moniz [21] considered multi-commodity flow problems with unsplittable flows and piecewise-linear routing costs. Lu et al. [56] conducted a periodic review-based inventory control problem with a general piecewise-linear cost. However, all the above studies assumed that the piecewise-linear convex cost of production is exogenously given, rather than a constituting decision, whereas it is determined by the quantity of acquisition in this study. We contribute to the stream of research on policies for inventory production with a quantity-dependent piecewise-linear convex cost of production, where the length of each piecewise segment depends on the quantity of products with the corresponding quality, added in order from the highest-quality grade to the worst.

The works most closely related to this one are conducted by Teunter and Flapper [9] and Mutha, Bansal, and Guide [12]. Teunter and Flapper [9] studied acquisition and remanufacturing decisions under uncertain-quality fractions of the core. With deterministic demand, they derived a simple closed-form expression for the total expected cost. Under uncertain demand, they presented optimal newsboy-type solutions for the optimal remanufacture-up-to levels and an approximate expression for the total expected cost, given the quantity of acquired cores. Mutha, Bansal, and Guide [12] investigated whether a 3PR should acquire used products or cores of uncertain quality levels in bulk or those with known quality levels in sorted grades, and whether to acquire and remanufacture cores before the demand has been realized (planned acquisition) or after it (reactive acquisition), or on both occasions (sequential acquisition). However, Teunter and Flapper [9] derived an approximate expression for the total expected cost under uncertain demand. Mutha, Bansal, and Guide [12] gave a complicated first-order functional expression that can be used to solve for the optimal quantity of acquisition but does not determine this quantity. Based on the studies above, we develop the models to derive analytically tractable solutions for models of unsorted core acquisition in this study. In short, this study makes two contributions to the literature. We establish optimal acquisition/production models of unsorted/sorted used products for core acquisition management in independent remanufacturing by proposing analytically tractable solutions. We characterize the structure of the optimal policy for managing inventory production in a system with a quantity-dependent piecewise-linear convex cost of production.

3. The Model

The remanufacturer acquires unsorted used products and remanufactures them before the demand has been realized. We develop an optimal acquisition and production model with a linear acquisition cost and a quantity-dependent piecewise-linear convex production cost, while considering uncertainty in demand. The sequence of events is as follows. First, the remanufacturer acquires $Q$ unsorted cores and classifies them into $n$ grades according to the quality levels. Second, the remanufacturer converts $z\left(z\le Q\right)$ units of sorted cores into finished products by incurring a production cost of $r_i$ for each unit of grade $i$. Finally, the demand is realized and the remanufacturer attempts to meet the demand, generating revenue $p$ for each unit sold.

The assumptions and parameters are as follows:

Assumptions:

A1. Consistent with practice in the remanufacturing industry, grades in which used products are sorted are exogenously defined [14,38].

A2. All remanufactured products have the same specifications before sale [16,17].

A3. The remanufacturer acquires unsorted used products at a fixed price per unit [12,15].

Parameters:

$i$—quality category, $i=1\left(best\right),2,\cdots ,n\left(worst\right)$;

$Q$—the acquisition quantity of cores;

${\alpha }_i$—the proportion of quality-i cores, where $i=1,2,\cdots ,n$;

$r_i$—the unit remanufacturing cost for quality-i cores, where $r_1<r_2<\cdots <r_n$;

$A$—the acquisition cost per return;

$p$—the sale price per remanufactured product;

$z$—the production quantity;

$D$—the demand quantity.

The demand $D$ for the finished products is uncertain with respect to a cumulative density function $F(·)$ and a probability density function $f(·)$, where $f\left(x\right)>0$ for all $x\ge 0$ and $f\left(x\right)=0$ otherwise. The function that aims to maximize profits for the remanufacturer can be expressed by Equation (1):

$$\underset{Q,z}{\max }\pi \left(Q,z\right)=E\left[p\min \left\{D,z\right\}\right]-R\left(Q,z\right)-AQ=p\left(z-{\displaystyle {\int }_0^{z}F\left(D\right)dD}\right)-R\left(Q,z\right)-AQ$$

(1)

The first term on the right-hand side represents the revenue raised from selling remanufactured products; the second term represents the production cost, which is jointly determined by the quantities of acquisition $Q$ and production $z$; and the third term represents the acquisition cost.

Following Ferguson, Guide, Koca, and Souza [14], we assume that the firm classifies cores based on quality, $q$, as follows: cores with $q\in \left[0,q_n\right]$ are labeled for remanufacturing and those with $q\in \left[q_n,1\right]$ are labeled as scrap for part harvesting. The range $q\in \left[0,q_n\right]$ is further divided into $n$ slices, $\left[0,q_1\right),\left[q_1,q_2\right),\cdots ,\left[q_{n-1},q_n\right]$, in order to classify the re-manufacturable cores into quality grades $i=1,\cdots ,n$.

According to Ferguson, Guide, Koca, and Souza [14], remanufacturing production activities always start from the item with the highest-quality grade and remanufactures until no items are left in the current quality grade. As shown in Figure 1, the production cost can be expressed by Equation (2):

$$R\left(Q,z\right)=\left[K_i\left(Q\right)+r_{i}z\right]{|}_{\left\{q_{i-1}Q<z\le q_{i}Q\right\}}+\left[K_n\left(Q\right)+r_n{q}_{n}Q\right]{|}_{\left\{z>q_{n}Q\right\}}, i=1,\cdots ,n$$

(2)

where $q_i$ denotes the cumulative ratio from quality-1 to quality-i, $q_0=0$, $q_i={\displaystyle \sum _{j=1}^i{\alpha }_j}=q_i,i=1,2,\cdots ,n$, $0=q_{0}Q<q_{1}Q<\cdots <q_{n-1}Q<q_{n}Q$, $r_0=r_1<r_2<\cdots <r_n$, $K_i\left(Q\right)=-{\displaystyle \sum _{j=1}^i\left(r_j-r_{j-1}\right)q_{j-1}Q}=-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_{j}Q}$ for all $1\le i<n$, and ${|}_{\{·\}}$ is the indicator function.

We assume that all quality levels are marginally profitable because the ratio of grades of quality is fixed, which will be relaxed in Section 4. This does not affect the final conclusion, even if a certain percentage of cores is not marginally profitable or cannot be remanufactured. Based on the above analysis of remanufacturing production costs, we first give Lemma 1.

Lemma 1.

a. 

$R\left(Q,z\right)$ is a piecewise-linear convex function and increasingly continuous in the interval $[0,q_{n}Q]$ for a given acquisition quantity $Q$.

b. 

$R\left(Q,z\right)$ is non-increasing in $Q$ for a given production quantity $z$.

The proof of Lemma 1 and the proofs of the other Lemmas and propositions are given in the Appendix A.

We now consider the model with deterministic demand and generalize it to one with stochastic demand.

3.1. Special Case: Demand Is Deterministic

The optimal production quantity is always equal to the demand when the demand is known with the assumption that all quality levels are marginally profitable, given that $z=D$. The objective function (1) can then be expressed by Equation (3):

$${\pi }_D^{}\left(Q\right)=pD-R\left(Q,D\right)-AQ$$

(3)

According to Lemma 1(b), the production cost $R\left(Q,D\right)$ is non-increasing in $Q$ for a given production quantity $D$. At the same time, the acquisition cost $AQ$ is increasing in $Q$. We can thus obtain Lemma 2.

Lemma 2.

The total cost function $C\left(Q,D\right)=R\left(Q,D\right)+AQ$ is discretely convex in $Q$ when the demand $D$ is fixed.

For the given profit function $\pi \left(Q\right)=Dp-C\left(Q,D\right)$, its first difference is $\Delta \pi \left(Q\right)=F\left(Q+1\right)-F\left(Q\right)=-\Delta C\left(Q,D\right)$. By this time, it is easy to obtain the characteristics of the profit function by Proposition 1.

Proposition 1.

The profit function ${\pi }_D^{}\left(Q\right)$ is discretely concave in $Q$ when the demand $D$ is fixed.

Based on properties of the total cost and profit functions given in Lemma 2 and Proposition 1, we aim to optimize the acquisition quantity after the demand has been determined. Suppose that the quantity demanded $D$ is in the $l-th$ interval when the purchase quantity is $Q$, and the quantity demanded $D$ is in the $m-th$ interval when the purchase quantity is $Q+1$, i.e., $Q{\displaystyle \sum _{i=1}^{l-1}{\alpha }_i}<D\le Q{\displaystyle \sum _{i=1}^l{\alpha }_i}$ and $\left(Q+1\right){\displaystyle \sum _{i=1}^{m-1}{\alpha }_i}<D\le \left(Q+1\right){\displaystyle \sum _{i=1}^m{\alpha }_i}$. At this point, we can then obtain the first difference function of the total costs below.

$$\begin{array}{l}\Delta C\left(Q,D\right)=A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i+\left(Q+1\right)\left[\left(r_l-r_m\right){\displaystyle \sum _{i=1}^{m-1}{\alpha }_i}+{\displaystyle \sum _{i=m}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\right]-\left(r_l-r_m\right)D\\ =A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i+Q\left[\left(r_l-r_m\right){\displaystyle \sum _{i=1}^{m-1}{\alpha }_i}+{\displaystyle \sum _{i=m}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\right]-\left(r_l-r_m\right)D\end{array}$$

Obviously, the larger the acquisition quantity, the smaller $j$ when the demand/production quantity is deterministic, i.e., $l\ge m$ because $Q$ is less than $Q+1$. Thus, we can derive the following two inequalities:

$$\begin{array}{l}\Delta C\left(Q,D\right)\ge A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i+\left(Q+1\right)\left[\left(r_l-r_m\right){\displaystyle \sum _{i=1}^{m-1}{\alpha }_i}+{\displaystyle \sum _{i=m}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\right]-\left(r_l-r_m\right)\left(Q+1\right){\displaystyle \sum _{i=1}^m{\alpha }_i}\\ =A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i+\left(Q+1\right){\displaystyle \sum _{i=m-1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\ge A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\end{array}$$

$$\begin{array}{l}\Delta C\left(Q,D\right)\le A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i+Q\left[\left(r_l-r_m\right){\displaystyle \sum _{i=1}^{m-1}{\alpha }_i}+{\displaystyle \sum _{i=m}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\right]-\left(r_l-r_m\right)Q{\displaystyle \sum _{i=1}^{l-1}{\alpha }_i}\\ =A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i-Q{\displaystyle \sum _{i=m}^{l-1}\left(r_i-r_m\right)}{\alpha }_i\le A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i\end{array}$$

In other words, $A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i\le \Delta C\left(Q,D\right)\le A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i$. Given any acquisition quantity $Q$, the upper and lower bounds of the first difference in the total cost $\Delta C\left(Q,D\right)$ are $\inf \left(\Delta C\right)=A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i$ and $\sup \left(\Delta C\right)=A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i$, respectively. If $A-{\displaystyle \sum _{i=1}^{m-1}\left(r_m-r_i\right)}{\alpha }_i<0$, then $\Delta C\left(Q,D\right)<0$ and $\Delta {\pi }_D^{}\left(Q\right)>0$; in other words, to increase the marginal profit, the acquisition quantity needs to be increased. If $A-{\displaystyle \sum _{i=1}^{l-1}\left(r_l-r_i\right)}{\alpha }_i>0$, then $\Delta C\left(Q,D\right)>0$ and $\Delta {\pi }_D^{}\left(Q\right)<0$; in other words, the marginal profit decreases with a reduction in acquisition quantity; otherwise, it can vary.

For the sake of brevity, let $\mathrm{\Lambda }\left(j\right)=A-{\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i$, $j=1,2,\cdots ,n$. We then have $\mathrm{\Lambda }\left(1\right)>\mathrm{\Lambda }\left(2\right)>\cdots >\mathrm{\Lambda }\left(n\right)$. $A$ is the marginal acquisition cost and ${\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i$ is the marginal savings in production costs when the production quantity $D$ is in the $j-th$ interval. In other words, $\mathrm{\Lambda }\left(j\right)$ is the marginal cost and $-\mathrm{\Lambda }\left(j\right)$ is the marginal profit when the production quantity $D$ is in the $j-th$ interval.

As described in Proposition 1, the profit function ${\pi }_D^{}\left(Q\right)$ is discretely concave in $Q$ when the demand $D$ is fixed. The optimal acquisition quantity $Q^{*}$ must satisfy the following conditions: $\Delta {\pi }_D^{}\left(Q^{*-}\right)\ge 0$ and $\Delta {\pi }_D^{}\left(Q^{*+}\right)\le 0$. Consequently, the optimal acquisition quantity is $\frac{D}{q_j}$ for $\Delta {\pi }_D^{}\left({\left(\frac{D}{q_j}\right)}^{-}\right)=-\mathrm{\Lambda }\left(j+1\right)>0$ and $\Delta {\pi }_D^{}\left({\left(\frac{D}{q_j}\right)}^{+}\right)=-\mathrm{\Lambda }\left(j\right)<0$ when $\mathrm{\Lambda }\left(j+1\right)<0$ and $\mathrm{\Lambda }\left(j\right)>0$.

Similarly, $\Delta {\pi }_D^{}\left({\frac{D}{q_j}}^{-}\right)=-\mathrm{\Lambda }\left(j+1\right)>0$, $\Delta {\pi }_D^{}\left({\frac{D}{q_j}}^{+}\right)=-\mathrm{\Lambda }\left(j\right)=0$,$\Delta {\pi }_D^{}\left({\frac{D}{q_{j-1}}}^{-}\right)=-\mathrm{\Lambda }\left(j\right)=0$, and $\Delta {\pi }_D^{}\left({\frac{D}{q_{j-1}}}^{+}\right)=-\mathrm{\Lambda }\left(j-1\right)<0$ if $\mathrm{\Lambda }\left(j\right)=0$. Consequently, the optimal acquisition quantity can be any value on the interval $\left[\frac{D}{q_j},\frac{D}{q_{j-1}}\right]$ when $\mathrm{\Lambda }\left(j\right)=A-{\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i=0$.

If $\mathrm{\Lambda }\left(n\right)\ge 0$, then $\Delta {\pi }_D^{}\left(Q\right)$ ($\Delta C\left(Q,D\right)$) is always negative (positive), which means the marginal profit is positive. With the assumption that the remanufacturing activities is always profitable, the re-manufacturable quantity equals the demand, as shown by $q_n{Q}^{*}=D$.

The optimal acquisition quantity can be obtained by Proposition 2 when the demand is deterministic.

Proposition 2.

When the demand $D$ is deterministic,

a. 

the optimal acquisition quantity is $\frac{D}{q_j}$ if $\mathrm{\Lambda }\left(j+1\right)=A-{\displaystyle \sum _{i=1}^j\left(r_{j+1}-r_i\right)}{\alpha }_i<0$ and $\mathrm{\Lambda }\left(j\right)=A-{\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i>0$;

b. 

the optimal acquisition quantity can be arbitrarily determined on the interval $\left[\frac{D}{q_j},\frac{D}{q_{j-1}}\right]$ if $\mathrm{\Lambda }\left(j\right)=A-{\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i=0$;

c. 

the optimal acquisition quantity is $Q^{*}=\frac{D}{q_n}$ if $\mathrm{\Lambda }\left(n\right)=A-{\displaystyle \sum _{i=1}^{n-1}\left(r_n-r_i\right)}{\alpha }_i>0$.

Proposition 2 reveals that the optimal acquisition quantity can be obtained at the interval where the marginal profit is equal to 0 if it exists (Proposition 2(b)) and the optimal acquisition quantity of re-manufacturable is equal to the demand when the acquisition cost is high enough, given that $A>{\displaystyle \sum _{i=1}^{n-1}\left(r_n-r_i\right)}{\alpha }_i$ (Proposition 2(c)). Otherwise, the optimal acquisition quantity can be obtained by looking for the boundary values $j$ where $\mathrm{\Lambda }\left(j+1\right)=A-{\displaystyle \sum _{i=1}^j\left(r_{j+1}-r_i\right)}{\alpha }_i<0$ and $\mathrm{\Lambda }\left(j\right)=A-{\displaystyle \sum _{i=1}^{j-1}\left(r_j-r_i\right)}{\alpha }_i>0$ (Proposition 2(a)).

3.2. Demand Is Stochastic

We will answer two questions in this section: how many unsorted cores need to be acquired and how many sorted cores need to be remanufactured into finished products, which is dependent on the acquisition quantity found in the first step. We solve for the optimal quantities of production and acquisition by backward derivation. We start by optimizing the quantity of production with a given acquisition quantity, and then further explore the optimal acquisition quantity. In the second stage, the quantity of acquisition is given, and the objective problem becomes one of inventory production with a piecewise -linear convex cost. This has been discussed by Lu and Song [51] and Lu, Song, and Yang [56].

Let $g\left(z\right)=pE\left[\min \left\{D,z\right\}\right]-az$, and define $S\left(a\right)=\min \arg \underset{z}{\max }g\left(z\right)$. Then, $S\left(a\right)$ decreases in $a$. This implies that $S\left(r_1\right)\ge S\left(r_2\right)\cdots \ge S\left(r_n\right)$ because $R\left(Q,r_i\right)$ is discretely convex. $S\left(r_i\right)$ can be interpreted as the optimal production level if the unit cost of production is $r_i$, $i=0,1,\cdots ,n$. It is then easy to demonstrate that $\frac{d^{2}g\left(z\right)}{d{z}^2}<0$. Thus, $S\left(r_i\right)$ can be obtained by letting $\frac{dg\left(z\right)}{dz}=0$; thus, $S\left(r_i\right)={\overline{F}}^{-1}\left(\frac{r_i}{p}\right)$. Let $g^i\left(z\right)=p\min \left\{D,z\right\}-r_{i}z$; it monotonically increases in $\left(0,S\left(r_i\right)\right]$ and monotonically decreases in $\left(S\left(r_i\right),+\infty \right)$. For simplicity but not generality, let $S\left(r_{n+1}\right)=0$.

Based on the method given in Lu and Song [51], we can obtain the optimal production quantity in the second stage by Proposition 3.

Proposition 3.

For a given acquisition quantity, the optimal production quantity is $S\left(r_i\right)$ if $q_{{}_{i-1}}Q<S\left(r_i\right)\le q_{{}_i}Q$, for $1\le i\le n$, and the optimal production quantity is $q_{{}_i}Q$ if $S\left(r_{i+1}\right)<q_{{}_i}Q\le S\left(r_i\right)$ for $1\le i\le n$; thus, $z\left(Q\right)=\{\begin{array}{l}S\left(r_i\right),\begin{array}{c}\end{array}{q}_{{}_{i-1}}Q<S\left(r_i\right)\le q_{{}_i}Q\\ q_{{}_i}Q,\begin{array}{c}\end{array}S\left(r_{i+1}\right)\le q_{{}_i}Q<S\left(r_i\right)\end{array}$.

For any given acquisition quantity, we can find the optimal production quantity according to Proposition 3, and this is a one-to-one relationship. For $S\left(r_n\right)\le S\left(r_{n-1}\right)\cdots \le S\left(r_1\right)$ and $q_{{}_1}\left(Q\right)\le q_{{}_2}\left(Q\right)\cdots \le q_{{}_n}\left(Q\right)$, we can obtain Lemma 3 by using the result given by Proposition 3.

Lemma 3.

The optimal production quantity $z$ does not decrease in acquisition quantity $Q$.

We analyze the optimization of the acquisition quantity in the first stage after analyzing the characteristics of the production of the second stage. To obtain the optimal quantity of acquisition, we first analyze the properties of the first difference in the profit function $\pi \left(Q,z\right)$ by Lemma 4.

Lemma 4.

The first difference in $\pi \left(Q,z\right)$ does not increase in $Q$; thus,

$$\frac{d\pi \left(Q,z\right)}{dQ}=\{\begin{array}{l}p{q}_i\overline{F}\left(q_{i}Q\right)-r_i{q}_i+{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}-A,\begin{array}{c}\end{array}\frac{S\left(r_{i+1}\right)}{q_i}<Q<\frac{S\left(r_i\right)}{q_i},i=1,\cdots ,n\\ {\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}-A, \begin{array}{c}\frac{S\left(r_i\right)}{q_i}<Q<\frac{S\left(r_i\right)}{q_{i-1}},i=2,\cdots ,n\end{array}\\ p{q}_1\overline{F}\left(q_{1}Q\right)-r_1{q}_1-A, Q>\frac{S\left(r_1\right)}{q_1}\end{array}$$

According to Lemma 4, the profit function $\pi \left(Q,z\right)$ is piecewise-differentiable and does not increase. We can thus obtain Proposition 4.

Proposition 4.

The profit function $\pi \left(Q,z\right)$ is discretely concave with respect to the acquisition quantity $Q$.

According to Proposition 4, the profit function $\pi \left(Q,z\right)$ is discretely concave in the acquisition quantity $Q$, so the optimal acquisition quantity is the one (those) which satisfies the following conditions: $\Delta \pi \left(Q^{*-}\right)\ge 0$ and $\Delta \pi \left(Q^{*+}\right)\le 0$.

If $A-{\displaystyle \sum _{j=1}^{n-1}\left(r_n-r_j\right){\alpha }_j}<0$, then $\frac{d\pi \left(Q,z\right)}{dQ}{|}_{\frac{S\left(r_n\right)}{q_n}<Q<\frac{S\left(r_n\right)}{q_{n-1}}}={\displaystyle \sum _{j=1}^{n-1}\left(r_n-r_j\right){\alpha }_j}-A<0$, according to Lemma 4 and Proposition 4, and the optimal acquisition quantity is in the first interval $\left[0,\frac{S\left(r_n\right)}{q_n}\right)$. Consequently, we can obtain the optimal acquisition quantity by letting $\frac{d\pi \left(Q,z\right)}{dQ}{|}_{0<Q<\frac{S\left(r_n\right)}{q_n}}=0$, which is $Q^{*}={\overline{F}}^{-1}\left(\frac{{\displaystyle \sum _{j=1}^n{r}_j{\alpha }_j}+A}{p{q}_n}\right)$.

Similarly, if $A-{\displaystyle \sum _{j=1}^k\left(r_{k+1}-r_j\right){\alpha }_j}<0$ and $A-{\displaystyle \sum _{j=1}^{k-1}\left(r_k-r_j\right){\alpha }_j}>0$, the optimal acquisition quantity is in the interval $\left(\frac{S\left(r_{i+1}\right)}{q_i},\frac{S\left(r_i\right)}{q_i}\right)$, so the optimal acquisition quantity can be obtained by letting $\frac{d\pi \left(Q,z\right)}{dQ}{|}_{\frac{S\left(r_{i+1}\right)}{q_i}<Q<\frac{S\left(r_i\right)}{q_i}}=0$, which is $Q^{*}={\overline{F}}^{-1}\left(\frac{{\displaystyle \sum _{j=1}^k{r}_j{\alpha }_j}+A}{p{q}_k}\right)/q_k$. If $A-{\displaystyle \sum _{j=1}^{k-1}\left(r_k-r_j\right){\alpha }_j}=0$, the optimal acquisition quantity can be any value in the interval $\left[\frac{S\left(r_k\right)}{q_k},\frac{S\left(r_k\right)}{q_{k-1}}\right)$.

Consequently, we can derive the optimal acquisition quantity using Proposition 5.

Proposition 5.

When the demand $D$ is stochastic,

a. 

the optimal acquisition quantity is ${\overline{F}}^{-1}\left(\frac{{\displaystyle \sum _{j=1}^n{r}_j{\alpha }_j}+A}{p{q}_n}\right)$ if $A-{\displaystyle \sum _{j=1}^{n-1}\left(r_n-r_j\right){\alpha }_j}>0$;

b. 

the optimal acquisition quantity is ${\overline{F}}^{-1}\left(\frac{{\displaystyle \sum _{j=1}^k{r}_j{\alpha }_j}+A}{p{q}_k}\right)/q_k$ , where $k$ satisfies the following two conditions: $A-{\displaystyle \sum _{j=1}^k\left(r_{k+1}-r_j\right){\alpha }_j}<0$ and $A-{\displaystyle \sum _{j=1}^{k-1}\left(r_k-r_j\right){\alpha }_j}>0$ ;

c. 

the optimal acquisition quantity is $Q^{*}=\frac{S\left(r_k\right)}{q_k}$ if $A-{\displaystyle \sum _{j=1}^{k-1}\left(r_k-r_j\right){\alpha }_j}=0$.

Proposition 5 analytically gives tractable solutions to obtain the optimal acquisition quantity as long as we know the sum of $A$ and $-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}$, where $-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}$ is the first difference in the fixed production cost of quality $i$, given that $K_i\left(Q\right)$. At the same time, Proposition 3 describes a simple method that can be used to obtain the optimal production quantity for the acquisition quantity given in the first stage, which only depends on $S\left(r_i\right)$ and $q_{{}_i}Q$, $i=1,2,\cdots ,n$.

We analytically provide tractable solutions to the problems of core acquisition and production by considering stochastic demands. Next, we will further explore the impact of core quality fraction uncertainty 0 in Section 4.

4. The Impact of Core Quality Fraction Uncertainty

In this section, we explicitly consider the cases when the quality of acquired cores is uncertain. We only know the probability that an acquired core is of quality-i is ${\alpha }_i$. Consider a collection lot of $Q$ cores. Let $X_{\left(1\right)},X_{\left(2\right)},\cdots ,X_{\left(Q\right)}$ denote the order statistics of the $Q$ cores, ordered by condition. According to Evans et al. [57], the probability density function of $X_{\left(r\right)}$ can be expressed by Equation (4)

$$P_{X_{\left(r\right)}}\left(i\right)=\{\begin{array}{l}{\displaystyle \sum _{\omega =0}^{Q-r}\left(\begin{array}{l}Q\\ \omega \end{array}\right){\left({\alpha }_1\right)}^{Q-\omega }{\left(1-{\alpha }_1\right)}^{\omega }, i=1}\\ {\displaystyle \sum _{u=0}^{r-1}{\displaystyle \sum _{\omega =0}^{Q-r}\left(\begin{array}{l}\begin{array}{ccc}& & Q\end{array}\\ u,Q-u-\omega ,\omega \end{array}\right){\left({\displaystyle \sum _{j=1}^{i-1}{\alpha }_j}\right)}^u{\left({\alpha }_i\right)}^{Q-u-\omega }{\left({\displaystyle \sum _{j=i+1}^Q{\alpha }_j}\right)}^{\omega },i=2,\cdots ,Q-1}}\\ {\displaystyle \sum _{u=0}^{r-1}\left(\begin{array}{l}Q\\ u\end{array}\right){\left(1-{\alpha }_n\right)}^u{\left({\alpha }_n\right)}^{Q-u} i=}Q\end{array}$$

(4)

The formula for the probability density function of $X_{\left(r\right)}$ is a direct result of the following observation. For the $r-th$ order statistic to equal $i$, there must be $u$ values less than or equal to $i-1$ and $\omega$ values greater than or equal to $i+1$, where $u=0,1,\cdots ,r-1$ and $\omega =0,1,\cdots ,Q-r$. The other $Q-u-\omega$ values must be equal to $i$. The cumulative density function procedure is used to determine the probability of obtaining a value less than or equal to $i-1$, the survivor function procedure is used to determine the probability of obtaining a value greater than or equal to $i+1$, and the probability density function procedure is used to determine the probability of obtaining the value $i$. The multinomial coefficient expresses the number of combinations resulting from a specific choice of $u$ and $\omega$. Consequently, the order statistical probability of obtaining the quality-i cores is $P_{X_{\left(r\right)}}\left(i\right)$.

So far, we can use the methods given in Proposition 5 to optimize acquisition decisions with substituting order statistical probability $P_{X_{\left(r\right)}}\left(i\right)$ for the fixed proportion ${\alpha }_i$. Subsequently, the optimal production quantity can be achieved according to Proposition 3 after the core quality fractions are realized.

5. Numerical Study

In this section, we perform a numerical study to verify the analytical results above and intuitively illustrate them based on data from ReCellular reported by Mutha, Bansal, and Guide [12]. The selling price is $p=61.41$, the cost of acquisition is $A=11.58$, and the uncertainty in demand is normal with $\mu =1000$ and $\sigma =250$. The cores are sorted into four grades; the costs of remanufacturing are $r_1=5$, $r_2=20$, $r_3=30$, and $r_4=40$; and their proportions are ${\alpha }_1=47.05\%$, ${\alpha }_2=18.55\%$, ${\alpha }_3=15.05\%$, and ${\alpha }_4=19.35\%$, respectively.

5.1. Results and Discussion

Table 1. Numerical analyses.

i	1	2	3	4
$S\left(r_i\right)$	1348.90	1112.97	1007.20	902.75
$\frac{S\left(r_i\right)}{q_i}$	2866.94	1696.60	1248.85	902.75
$\frac{S\left(r_i\right)}{q_{i-1}}$	/	2365.50	1535.36	1119.35
$A-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}$	11.58	4.52	−2.04	−10.10

shows the values of the functions directly related to the solution: $S\left(r_i\right)$, $\frac{S\left(r_i\right)}{q_i}$,$\frac{S\left(r_i\right)}{q_{i-1}}$, and $A-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}$. According to Proposition 5, the optimal acquisition quantity is $Q^{*}={\overline{F}}^{-1}\left(\left({\displaystyle \sum _{j=1}^2{r}_j{\alpha }_j}+A\right)/p{q}_2\right)/q_2=1583.91$ because $A-{\displaystyle \sum _{j=1}^2\left(r_3-r_j\right){\alpha }_j}=-2.04<0$ and $A-\left(r_2-r_1\right){\alpha }_1=4.53>0$.

As shown in

Table 2. The value of $S\left(r_i\right)$ and $q_{{}_i}Q$ when the acquisition quantity is $Q^{*}=1583.91$.

i	1	2	3	4
$S\left(r_i\right)$	1348.90	1112.97	1007.20	902.75
$q_{{}_i}Q$	745.23	1039.05	1277.423	1583.91

, $S\left(r_3\right)\le q_2{Q}^{*}<S\left(r_2\right)$; thus, according to Proposition 3, the optimal production quantity is $z=q_{2}Q=1039.05$. Consequently, the expected profit is $28465.55$.

To clarify the characteristics of the problem, we show the trends of change in the optimal production quantity and expected profits in Figure 2. The blue area represents the trend of the optimal production quantity and the orange line represents the maximum expected profit for a given acquisition quantity. We also mark the position of $S\left(r_i\right)$, $\frac{S\left(r_i\right)}{q_i}$, and $\frac{S\left(r_i\right)}{q_{i-1}}$ in blue, red, and yellow in Figure 2, which can help readers understand the characteristics of the optimized problem more intuitively. As demonstrated in Proposition 3, the optimal production quantity is $S\left(r_i\right)$ if $q_{{}_{i-1}}Q<S\left(r_i\right)\le q_{{}_i}Q$ and the optimal production quantity is $q_{{}_i}Q$ if $S\left(r_{i+1}\right)<q_{{}_i}Q\le S\left(r_i\right)$. As demonstrated in Lemma 3 and Proposition 4, the optimal production quantity $z$ is piecewise-non-decreasing with respect to acquisition quantity $Q$, and the profit function $\pi \left(Q,z\right)$ is discretely concave with respect to the acquisition quantity $Q$. Specifically, the optimal production quantity is equal to the acquisition quantity when the acquisition quantity is not greater than $S\left(r_4\right)=902.75$. The optimal production quantity does not change whereas the expected profit continues to decrease when the acquisition quantity exceeds $\frac{S\left(r_1\right)}{q_1}=2866.94$.

5.2. Sensitivity Analysis

We also analyze the sensitivity of the parameters to explore the effect of the unit acquisition cost $A$ on the results. We change it from 0 to 23.37 in steps of 2.895. The results are shown in

Table 3. The result for various unit acquisition costs $A$.

$\mathit{A}$		$\mathit{Q}$	$\mathit{z}$	$\mathit{z}\mathbf{/}\mathit{Q}$	$\mathit{\pi }\left(\mathit{Q},\mathit{z}\right)$
−100%	0.00	2866.94	1348.90	0.4705	54,097.18
−75%	2.895	2608.51	1227.31	0.4705	46,205.95
−50%	5.790	2432.24	1144.37	0.4705	38,919.91
−25%	8.685	1654.82	1085.56	0.656	33,152.82
0	11.58	1583.91	1039.05	0.656	28,465.55
+25%	14.475	1235.39	996.34	0.8065	24,222.77
+50%	17.370	1189.76	959.54	0.8065	20,712.22
+75%	20.265	1143.00	921.83	0.8065	17,335.15
+100%	23.160	1093.91	882.24	0.8065	14,096.49

and Figure 3. The optimal quantity of acquisition/production and the minimum expected profit decrease as the unit acquisition cost $A$ increases. Note that the optimal quantity of production is always the product of the optimal quantity of acquisition and $q_i$ because $A-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}\ne 0$, which can be deduced from Proposition 3. As $A$ increases, the rate of utilization of cores $z/Q$ increases from $q_1$ to $q_3$, which means that a lower cost of acquisition can improve the expected profit and quantity of the acquisition of remanufacturers while improving the rate of utilization of the recycled parts.

Similarly, we analyze the effects of the selling price $p$ and the standard deviation of the demand ${\sigma }_D$ on the results, as shown in

Table 4. The results for various selling prices $p$.

$\mathit{p}$		$\mathit{Q}$	$\mathit{z}$	$\mathit{z}/\mathit{Q}$	$\mathit{\pi }\left(\mathit{Q},\mathit{z}\right)$
0	61.41	1583.91	1039.05	0.656	28,465.55
+20%	73.692	1655.97	1086.31	0.656	39,873.61
+40%	85.974	1710.32	1121.97	0.656	51,467.56
+60%	98.256	1753.67	1150.40	0.656	63,184.75
+80%	110.538	1789.53	1173.93	0.656	74,989.36
+100%	122.82	1820.00	1193.92	0.656	86,859.06

and

Table 5. The results for the various standard deviations of demand ${\sigma }_D$.

${\mathit{\sigma }}_{\mathit{D}}$		$\mathit{Q}$	$\mathit{z}$	$\mathit{z}/\mathit{Q}$	$\mathit{\pi }\left(\mathit{Q},\mathit{z}\right)$
0	100	1548.20	1015.62	0.656	32,095.74
−40%	150	1560.10	1023.43	0.656	30,885.64
−20%	200	1572.01	1031.24	0.656	29,675.54
0	250	1583.91	1039.05	0.656	28,465.55
+20%	300	1595.82	1046.86	0.656	27,257.40
+40%	350	1607.72	1054.67	0.656	26,058.72
+60%	400	1619.63	1062.48	0.656	24,884.36
+80%	450	1631.53	1070.29	0.656	23,751.78
+100%	500	1643.44	1078.09	0.656	22,675.64

and Figure 4 and Figure 5. The optimal quantity of acquisition/production and the minimum expected profit increase (decreased) as the selling price $p$ (the standard deviation of demand ${\sigma }_D$) increases, whereas the rate of utilization of cores $z/Q$ remains constant because it depends only on $A-{\displaystyle \sum _{j=1}^{i-1}\left(r_i-r_j\right){\alpha }_j}$, which is determined by the unit cost of acquisition $A$, the unit cost of production $r_i$ and the proportion of cores ${\alpha }_j$.

5.3. Analysis of Changes along Product Lifecycle

A remanufactured product follows the standard lifecycle pattern of a new product, with five stages: introduction, growth, maturity, decline, and end-of-life. However, these stages lag compared with their corresponding stages for a new product in time [58]. In the remainder of this section, we refer to the stages of the lifecycle of a remanufactured product. The data shown in

Table 6. The data from ReCellular along the lifecycle in Mutha, Bansal, and Guide [12].

Parameter	Introduction	Growth	Maturity	Decline	End
A	35.62	23.15	17.81	11.58	5.34
p	85	85	72.25	61.41	44.37
$\alpha$	20%, 0, 0, 80%	47.05%, 18.55%, 15.05%, 19.35%
${\mu }_D$	1000	1000	1000	1000	1000
${\sigma }_D$	50	150	200	250	100

are from the survey of ReCellular in Mutha, Bansal, and Guide [12].

As shown in

Table 7. The results along the lifecycle.

	$\mathit{Q}$	$\mathit{z}$	$\mathit{z}/\mathit{Q}$	$\mathit{\pi }\left(\mathit{Q},\mathit{z}\right)$
Introduction	956.60	956.60	1	15,216.66
Growth	1004.57	1004.57	1	38,448.34
Maturity	1247.90	1006.43	0.8065	31,289.88
Decline	1583.91	1039.047	0.656	28,465.55
End	2196.78	1033.59	0.4705	26,347.34

and Figure 6, the expected profit for the remanufacturer increases from the introduction to the growth stage, and decreases from the growth stage to the end-of-life. On the one hand, profits in the introduction stage are the lowest because of the low quality of cores, where the lowest rating is as high as 80%. This is mainly due to defective products among the new products. The profit during the growth stage is the highest, mainly because the quality of cores is better than in the introduction stage, the selling price is higher, and the rate of change in demand is lower than in the mature, decline, and end-of-life stages. The decline in profits from the growth stage to the decline stage occurs because the selling price decreases and the standard deviation of demand increases.

On the other hand, the quantity of acquisition increases with the lifecycle, whereas the quantity of production remains stable. The rate of utilization of cores $z/Q$ decreases and the cores are fully utilized in the introduction and growth stages at a high selling price.

The most profitable stage is the growth stage, mainly because the product is well recognized by customers in the growth stage and its price is relatively high. Therefore, remanufacturers should seize business opportunities in the early stage rather than the late stage of the product’s lifecycle.

6. Conclusions and Proposals for Further Research

In this study, we considered the optimal acquisition of products with a high monetary value, e.g., electronic products, by a third-party remanufacturer. The 3PRs acquire used products of unknown quality and sort them into different grades. We developed models to determine the acquisition and quantity of production for the remanufacturer. The optimal models of acquisition and production under deterministic and stochastic demands were established, and the corresponding analytically tractable solutions to them were given. The proposed models were further verified and visualized through numerical work. We also analyzed the trend of changes in the remanufacturer’s profit, the quantity of acquisition, and rate of utilization of cores from the perspective of the entire lifecycle. The results show that the optimal production quantity does not decrease in acquisition quantity, the rate of utilization of the recycled parts increases when the acquisition cost decreases, and the optimal quantity of acquisition/production and the minimum expected profit increases when the selling price increases and decreases when the uncertainty of demand and acquisition cost increases. The practical management implications of this study are as follows. First, we provided analytically tractable solutions for the acquisition and production problem when the used products purchased are unsorted, which will give accurate quantitative guidance to the acquisition and production decisions of remanufacturing enterprises. Second, we analyzed the influence of relevant factors on the acquisition quantity, remanufacturing quantity, and enterprises profits, which will provide qualitative guidance for the remanufacturing enterprises to weigh various factors. Third, the results noted that remanufacturers should pay more attention to remanufacturing activities early in the life of products.

Some interesting avenues for future research emerge from this work, as the same methods and similar models can be used to explore other combinatorial optimization problems—for example, piecewise-linear, unsplittable, and multi-commodity flow problems, and network design and routing problems. This study also offers several interesting directions for future research. First, we built a single-period model to optimize the acquisition of products with a short lifecycle. A multi-period model for optimizing the acquisition of products with a long lifecycle should be built in future research. Second, we ignored the cost structure and simply considered a linearly variable cost of acquisition. Verifying this assumption is an interesting challenge for future research [28]. Third, we explored the policy of the acquisition of unsorted cores and the quantity-dependent piecewise costs of production from the perspective of third-party remanufacturers. This can be extended from the perspective of the supply chain in future work.