El Saadany et al. (2013) developed a mathematical expression that indicates the number of times a product can be remanufactured. They attempted to relax the general assumption that a product can be remanufactured for an indefinite number of times. They assumed that an item can be recovered for a limited $\left(\xi \right)$ number of times. When the system plateaus, then for any $\xi$, a fraction ${\beta }_{\xi }$ of a constant demand rate $d$ is remanufactured and ${(1-\beta }_{\xi })$ is produced, where ${\beta }_{\xi }=1-\frac{1-\beta }{1-{\beta }^{\xi +1}}$ and $\beta \left(0<\beta <1\right)$ is the proportion of used items returned for remanufacturing when an item is recovered an indefinite number of times. It is worth noting here that the mathematical expression used to derive ${\beta }_{\xi }$ focused on the returns of what was produced on previous period and ignored the rest of cumulative produced quantities that have been left or previously being remanufactured. Interested readers are referred to Table 1 in El Saadany et al. (2013). They stated that as $\xi \mathrm{}⟶\infty$, ${\beta }_{\xi }=1-\frac{1-\beta }{1-{\beta }^{\infty }}=\beta$, which is what has been suggested in the existing literature. For a pure production case, i.e., $\xi =0$, ${\beta }_{\xi }=1-\frac{1-\beta }{1-{\beta }^{0+1}}=0$, though a pure production strategy implies that $\beta =0$, i.e., there are no items returned for recovery purposes. Moreover, the mathematical expression used to derive ${\beta }_{\xi }$ assumes that no waste disposal (total repair) of the proportion $\beta$ upon recovery. Then, they modified the work of Richter (1997) and Teunter (2001) by replacing $\beta$ with ${\beta }_{\xi }$ in Richter’s and Teunter’s models.

In their model, the produced quantity $(1-{\beta }_{\xi })d$ is also disposed outside the system (e.g., Bazan et al., 2015) since they have defined $\alpha$ as the disposal rate, where $\alpha (\alpha =1-{\beta }_{\xi })$. Moreover, the role of $\beta$ in their model is somewhat ambiguous. Therefore, we can distinguish three cases: (1) As can be seen from Figure 1 in El Saadany et al. (2013), $(\alpha +{\beta }_{\xi })d$ entering the repairable stock from which ${\beta }_{\xi }d$ is remanufactured and $\alpha d=(1-{\beta }_{\xi })d$ is disposed. This implies that the return rate is $d$, however, this contradicts what the existing literature suggests, i.e., the return rate is less than demand rate; (2) ${\beta }_{\xi }=1-\frac{1-\beta }{1-{\beta }^{\xi +1}}$, which is a function of $\beta$, and therefore, the value of $\beta$ is used to compute ${\beta }_{\xi }$. In their examples, $\beta$ is defined as the collection of used items and ${\beta }_{\xi }$ is the effective proportion. In this case (case 2), one can deduce that $\beta d$ enters the repairable stock from which ${\beta }_{\xi }\beta d$ is remanufactured and $(1-{\beta }_{\xi })\beta d$ is disposed. However, $\beta$ represents the proportion of used items returned for remanufacturing when an item is recovered an indefinite number of times and only ${\beta }_{\xi }$ of $\beta$ is remanufactured; (3) The return rate is ${\beta }_{\xi }d$, which enters the repairable stock and flows in the serviceable stock to be remanufactured with no waste disposal (total repair). That is, the purpose of $\beta$, which represents (the proportion of used items returned for remanufacturing when an item is recovered an indefinite number of times) is to compute ${\beta }_{\xi }$. In this case (case 3), the system should collect ${\beta }_{\xi }$ instead of $\beta$, which is reflected in their modified version of the work of Richter (1997) and Teunter (2001). Hence, we can conclude that in all cases, $\beta$ is used to compute ${\beta }_{\xi }$, with case 2 being the most appropriate scenario. However, ${\beta }_{\xi }$ in all these cases, has no relation with an item being recovered for a limited $\left(\xi \right)$ number of times. In fact, $\xi$ is an arbitrary integer value, which implanted in ${\beta }_{\xi }$ to minimise the total cost. Therefore, considering a fraction$\gamma (0\le \gamma \le 1)$ of the return rate that meets the acceptance quality level to be remanufactured and $\left(1-\gamma \right)$ is disposed outside the system is more practicable (Dobos and Richter, 2004; El Saadany and Jaber, 2010, Alamri, 2011; Alamri, 2021).

The difference between $\beta$ and ${\beta }_{\xi }$ represents about 50% when $\xi =1$ and $\beta =0.9$ (see Figure 1). Moreover, for a fixed value of $\beta$, this difference decreases with $\xi$ (see Figure 2 in El Saadany et al. (2013) page 600 and Figure 1 in this paper). This seems logical in their expression, since as $\xi \mathrm{}⟶\infty$, ${\beta }_{\xi }⟶\beta$ because they assumed that all returned items have been remanufactured $\xi$ number of times. On the contrary, however, this difference should increase as a returned item with a greater number of remanufacturing times recovers with inferior quality. Furthermore, implementing ${\beta }_{\xi }$ as suggested by El Saadany et al. (2013) would result in a large disposal quantity especially for products that are associated with relatively small number of recovery times since ${\beta }_{\xi }$increases with $\xi$ (see Figure 2). Finally, incorporating such ${\beta }_{\xi }$ in the mathematical formulation entails that all returned items have been remanufactured $\xi$ number of times. However, fact remains that returned items may arrive out of sequence, i.e., with different number of remanufacturing times assuming also that previous number of remanufacturing times is labelled. It is true to say that considering such classification of returned items in the mathematical expression is not an easy task, however, this limitation will be discussed in the next section.

The comprehensive discussion in the previous section is necessary to position our study in the existing literature and highlights its research contribution. This paper aims to enhance this line of research by developing a new mathematical expression that models the percentage of returns as a function of the number of times an item is recovered, the corresponding quality for the recovery item and the expected number of times an item can be remanufactured on its life cycle. In this paper, we assume that returned items are collected at a rate of $c_j\left(t\right)$ (decision variable) where $j$ denotes the cycle index. Note that a pure production strategy occurs when $c_j\left(t\right)=0$.

Only a fraction ${\gamma }_{\xi j}$ of these retuned items can be remanufactured. Namely, ${\gamma }_{\xi j}=e^{\frac{-\xi q_{\xi }}{\tau }}$, where $q_{\xi j}\mathrm{}(0<q_{\xi j}<1)$ denotes the quality level of an item that has been recovered $\xi$ number of times. We assume that $q_{\xi j}=e^{\frac{-\xi }{\tau }}$, where $q_{0j}=1$, i.e., it refers to the quality of a newly manufactured item. In this paper, $\xi$ refers to the maximum number of times an item can be technologically (or optimally) remanufactured and $\tau \ge \xi$ denotes the expected number of times an item can be remanufactured on its life cycle. Note that $q_{\xi j}$ decreases as $\xi$ increseas and it attains a minimum value as $\xi \mathrm{}⟶\tau ,$ in this case, $q_{\xi j}=q_{\tau }=q_{min}=e^{-1}$ (Table 1). Accordingly, a recovery item with a quality less than the minimum acceptance quality level $q_{min}$ is considered defective and incurs a disposal cost. Note that ${{\gamma }_{\xi j}}^{\text{'}}<0$ and ${{\gamma }_{\xi j}}^{\text{'}\text{'}}>0$ $\forall \xi >0$, i.e., ${\gamma }_{\xi j}$ is a monotonically decreasing function over $\xi$ and, as $\tau ⟶\infty ,{\gamma }_{\xi j}⟶1$. The same arguments hold true for $q_{\xi j}$. This implies that ${\gamma }_{\xi j}$ is modelled as a function of the expected number of times an item can be remanufactured on its life cycle and the number of times an item can be technologically remanufactured based on its quality upon recovery. Moreover, ${\gamma }_{\xi j}$ and $q_{\xi j}$ are free from any judgmental measurements.

In real life settings, returned items may recover out of sequence. This can be attributed to random number of times these items have been remanufactured. Let us define the returned amount for cycle $j$ as $R_j$, where this amount undergoes a 100 per cent inspection. Assuming an automated remanufacturing system, the observation of $R_j$ seems realistic since all returned items are inspected. Therefore, returned items that are subjected to a 100 per cent screening upon recovery to the repairable stock would imply that $\underset{\_}{R_j}=\left(r_{\xi j},r_{\xi -1j},\dots ,r_{0j}\right)$. That is, $r_{kj}$ is the collected used/returned items with $k(k=\xi ,\xi -1,\dots ,0)$ is the number of times these items have been previously remanufactured. Here, $r_{0j}$ denotes returned items that have not yet being remanufactured, and $r_{\xi j}$ refers to defective (disposed) returned items that have been remanufactured $\xi$ number of times or items that do not meet the minimum acceptance quality level, $q_{min}$.

It is worth noting here, that the above-mentioned classification seams realistic because the system can deal with items based on such classification upon recovery. Therefore, as the number of times an item can be recovered increases, its corresponding use proportion decreases. This finding, however, contradicts that of El Saadany et al. (2013). To justify this, suppose that among the returned quantity that can be remanufactured say, 5 times there exists a sub-quantity arrived at the repairable stock with its first-time recovery. In this case, implementing ${\gamma }_{\xi j}$ for this sub-quantity would result in disposing an equal fraction as that of items with a greater recover time, though these items have not yet been remanufactured. Table 2 depicts the corresponding use proportion where $\underset{\_}{{\gamma }_{ij}}=\left({\gamma }_{1j},{\gamma }_{2j},\dots {\gamma }_{\xi j}\right)$. For instance, if a quantity of returned items that can be remanufactured say, 5 times, then each sub-quantity is associated with its corresponding accepted fraction, i.e., ${\gamma }_{5j}=0.692,{\gamma }_{4j}=0.698,{\gamma }_{3j}=0.719,{\gamma }_{2j}=0.765and{\gamma }_{1j}=0.849$. That is, ${\gamma }_{ij}$ represents the use proportion of the sub-quantity of the retuned items that can be remanufactured for its $i^{th}$ remanufacturing time. However, considering the above-mentioned classification in the mathematical formulation emerges as a challenge that affects the tractability problems in modelling. Therefore, to tackle this issue, we suggest that ${\gamma }_j=\frac{\sum _{i=1}^{\xi }{\gamma }_{ij}}{\xi }$ which constitutes an approximation of the average fraction (cumulative average up to $\xi$) that can be remanufactured in cycle $j$ (Figure 3).

As can be seen from Table 2, assuming ${\gamma }_{\xi j}$ to be remanufactured in cycle $j$ implies that the proportion $\left(1-{\gamma }_{\xi j}\right)c_j\left(t\right)$ is disposed outside the system. Conversely, $\left(1-{\gamma }_j\right)c_j\left(t\right)\le \left(1-{\gamma }_{\xi j}\right)c_j\left(t\right)$, i.e., ${\gamma }_j$ considers the cumulative average up to time $\xi$ of returned items (Table 4). This is a key consideration because it governs the behaviour of returned items and ensures reducing the disposal of unnecessary amount. In addition, returned items are coupled with distinct purchasing price $c_{prj}=c_{pm}{e}^{\frac{-1}{q_j}}$, where $c_{pm}$ denotes unit purchasing price for new items.

This paper develops a new mathematical expression that specifies the number of times a product can be remanufactured. In particular, the proposed expression is modelled as a function of the expected number of times an item can be remanufactured on its life cycle and the number of times an item can be technologically (or optimally) remanufactured based on its quality upon recovery.

In this paper, we present a general reverse logistics inventory model with a single manufacturing cycle and a single remanufacturing cycle. Demand, deterioration, manufacturing, remanufacturing and return rates are arbitrary functions of time. Therefore, a diverse range of time-varying forms can be disseminated from the general model. The mathematical formulation consists of serviceable and reparable stocks. The serviceable stock is for new and remanufactured items and the reparable stock is for collecting returned items to be remanufactured in the serviceable stock as good as new. Therefore, different holding costs and deterioration rates are considered for manufactured, remanufactured and returned items (e.g., Alamri, 2011; Alamri et al., 2016; Jaber and El Saadany, 2009; Teunter, 2001).

Only a proportion of the returned items that specifies the number of times a product can be remanufactured flows in the serviceable stock. In the first remanufacturing cycle, the initial inventory of retuned items is zero since there are no returned items to be remanufactured. Therefore, the accumulated amount of returned items (during the time gap of non-production and non-remanufacturing processes) represents the initial inventory of returns for the second cycle. This amount, indeed, should differ from that accumulated for subsequent cycles. This is key in our formulation, and therefore, ensures that all optimal values vary for each cycle before the system plateaus. The proposed model accounts for setup changeover costs when switching from manufacturing phase to remanufacturing phase. The proposed model also considers an investment cost associated with the number of times a product is recovered. We assume that returned, manufactured and remanufactured items deteriorate while they are effectively in stock. The return rate of the returned items is a decision variable, which is a function of the demand rate. The purchasing price of returned items is a function of the purchasing price of new items and the quality of items upon recovery. All functions and input parameters can be adjusted for subsequent cycles.

The remainder of the paper is organised as follows: In Section 4, we present our joint manufacturing and remanufacturing model and the solution procedure. Illustrative examples, and special cases are offered in Section 5. Managerial insights and concluding remarks are provided in Section 6 and Section 7 respectively.

Our model is developed under the following notations:

$j$          The cycle index;

$z$          $(z=gm,gr,r)$  $gm$

  is for manufactured items, $gr$   is for remanufactured items and $r$ is for returned items;

$I_{zj}\left(t\right)$          The inventory level at time $t$ ;

$P_{mj}\left(t\right)$          The manufactured rate per unit time for new items;

$P_{rj}\left(t\right)$                   The remanufactured rate per unit time for returned items;

$D_j\left(t\right)$         The demand rate per unit time;

$c_j\left(t\right)$          The return rate per unit time for returned items (decision variable), where $c_j\left(t\right)={\varnothing }_j{D}_j\left(t\right),$ and $0\le {\varnothing }_j<1$ ;

${\delta }_{zj}\left(t\right)$                  The deterioration rate per unit time;

$d_{zj}$         The deteriorated quantity for cycle $j$;

$Q_{mj}$         The manufactured quantity for cycle $j$;

$Q_{rj}$         The remanufactured quantity for cycle $j$;

$R_j$         The returned quantity for cycle $j$;

${∆}_j$          The accumulated quantity of returned items (during the time gap of non-production and non-remanufacturing processes);

$\xi$         The maximum number of times an item can be remanufactured;

$\tau$         The expected number of times an item can be remanufactured on its life cycle, where $\tau \ge \xi$ ;

$q_{\xi j}$            The actual quality level of an item that has been recovered $\xi$   number of times in cycle $j$ , where $q_{\xi j}=e^{\frac{-\xi }{\tau }}$  ( Table 1 );

$q_j$            The average fraction (cumulative average up to $\xi$ ) of the quality level of items that have been recovered for their $i^{th}$   time in cycle $j$ , where $q_j=\frac{\sum _{i=1}^{\xi }{q}_{ij}}{\xi }$  ( Table 3 );

${\gamma }_{\xi j}$           The actual proportion of returned items that can be remanufactured in cycle $j$ , where ${\gamma }_{\xi j}=e^{\frac{-\xi q_{\xi }}{\tau }}$   ( Table 2 );

${\gamma }_j$           The average fraction (cumulative average up to $\xi$ ) of returned items that can be remanufactured for their $i^{th}$   time in cycle $j$ , where ${\gamma }_j=\frac{\sum _{i=1}^{\xi }{\gamma }_{ij}}{\xi }$  ( Table 4 );

$c_{pm}$            The unit purchasing cost for new items;

$c_{prj}$            The unit purchasing price for retuned items in cycle $j$ , where $c_{prj}=c_{pm}{e}^{\frac{-1}{q_j}}$ ;

$c_{inv}$            The remanufacturing investment cost in the design process of an item to, technologically, be able to remanufacture it $\tau$   number of times ;

$c_{invj}$            The remanufacturing investment cost in cycle $j$  in the design process of an item to, technologically, be able to remanufacture it $\xi$   number of times, where $c_{invj}=c_{inv}\left(1-e^{\frac{-\xi }{q_j}}\right)$ ;

$c_m$            The unit manufacturing cost;

$c_r$            The unit remanufacturing cost;

$c_s$           The unit screening cost;

$c_w$           The unit disposal cost for deteriorated and scrap items;

$h_z$            The holding cost per unit per unit time;

$S_z$           The set-up/order cost per cycle;

$w_m$            The switching cost from remanufacturing phase to manufacturing phase;

$w_r$            The switching cost from manufacturing phase to remanufacturing phase;

Below is a list of all assumptions used in the paper:

In our model, demand in the first cycle is satisfies from production only (see Figure 5), as the inventory of returned items in the first cycle is zero (there are no returned items to be remanufactured). The process is repeated until inventory of product returns can be technologically attainable. Then, at the beginning of each cycle$j$, the system starts the production prosses until time $T_{1j}$, by which point in time $Q_{mj}$ units have been produced and stored in the serviceable stock. At time $T_{2j}$, the inventory level of new items becomes zero and $d_{gmj}$ units have deteriorated, which refers to the difference between the satisfied demand during production cycle and $Q_{mj}$ units that have been manufactured in cycle$j$. The remanufacturing process starts at time $T_{2j}$ until time $T_{3j}$, by which point in time the remanufactured quantity $Q_{rj}$ units have been accumulated and stored in the serviceable stock. The returned items are collected throughout the time interval at a rate $c_j\left(t\right)$, in which a fraction${\gamma }_j{c}_j\left(t\right)$ has been remanufactured and the remaining quantity $\left(1-{\gamma }_j\right)c_j\left(t\right)$ is disposed as waste outside the system. The remaining quantity $\left(1-{\gamma }_j\right)c_j\left(t\right)$ refers to returned items that have been remanufactured $\xi$ number of times or items that do not meet the minimum acceptance quality level, $q_{min}$. The inventory level of remanufacturing items becomes zero by time $T_{4j}$ and $d_{grj}$ units have deteriorated, which refers to the difference between the satisfied demand during remanufacturing cycle and $Q_{rj}$ units that have been remanufactured in cycle $j$. At time $T_{4j}$ (the end of cycle $j$), ${∆}_j$ units have been accumulated and stored in the repairable stock, which constitutes the initial inventory of returned items for the next cycle. In our model, the term ${∆}_{j-1}$ governs the behaviour of each cycle and at the beginning of the first remanufacturing cycle, ${∆}_{j-1}={∆}_0=0$. That is, the initial inventory of returned items in the first remanufacturing cycle is set equal to zero. The deteriorated quantity in the repairable stock is $d_{rj}$, which denotes the difference between the returned quantity that have been accepted to be remanufactured and $Q_{rj}$ units that have been remanufactured in cycle$j$. The process is repeated. Figure 4 depicts a general framework of production and remanufacturing unified system, and Figure 5 depicts the behaviour of such a unified system.

The variations in the inventory levels depicted in Figure 5 are given by the following differential equations:

with the boundary conditions:

Considering the boundary conditions, the solutions of the above differential equations are: respectively, where

From Equations (1)-(14), we note that each function is solely modelled and, therefore, functions may or may not be related to each other.

The per cycle total cost components for the underlying inventory model are given as:

Purchase price for returned items ($c_{prj}$) + Inspection cost ($c_s$) + Disposal cost for waste and deteriorated items ($c_w$) + Material cost for new items ($c_{pm}$) + Manufacturing cost ($c_m$) + Remanufacturing cost ($c_r$) + Holding costs ($h_z)$+ Switching cost for manufacturing ($w_m$) + Switching costs for remanufacturing ($w_r$) + Investment cost ($c_{invj}$) + Set-up and order costs ($S_z)$=

Now, denote $K=w_m+w_r+S_{pm}+S_{pr}+S_r$, and $d_j=d_{gmj}+d_{grj}+d_{rj}$.

As can be seen, the returned, manufacturing and remanufacturing costs cover and include deteriorated items. This is very well recognised in the literature because items deteriorate while they are effectively in stock (e.g., Inderfurth et al. 2005; Jaggi et al. 2015; Alamri et al. 2016; Polotski et al. 2019).

From Equations (8)-(14), the holding costs are as follows:

Holding costs for produced and remanufactured items at the serviceable stock:

Holding cost for returned items at the repairable stock:

Therefore, the per unit time total cost function of the unified inventory model during the cycle $\left[{0,T}_{4j}\right]$, as a function of $T_{1j},T_{2j},T_{3j}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{T}_{4j}$ denoted by $L\left(T_{1j},T_{2j},T_{3j},T_{4j}\right)$ is given by

where

Note that Equation (16) is a modified version of that of Alamri (2021). Therefore, and to avoid repetition, the existence, uniqueness and global optimality of the solution can be obtained by a quite similar way. Interested readers are referred to (Alamri, 2011; 2021).

The variables $T_{ij},i(i=\mathrm{1,2},3,4)$ that minimise $L\left(T_{ij}\right)$ given by Equation (16) are governed by the following relations:

For example, relations 19 and 20 guarantee that the inventory levels for the production and the remanufacturing phases have equal values for $t=T_1$ and for $t=T_3$. Note that the term${∆}_{j-1}$ is modelled as a deterministic value, i.e., it impacts the behaviour of each cycle until the system plateaus. This is a key in the mathematical formulation and, consequently, it ensures that the model is a viable solution for each cycle, whether the input parameters change their values or remain static (Alamri (2021)).

It can be seen from $\mathrm{E}\mathrm{q}.\left(19\right)$ that $P_{mj}\left(t\right)>D_j\left(t\right)⟹\mathrm{E}\mathrm{q}.\left(19\right)\iff T_{1j}<T_{2j}$. Also, from Equation (19), $T_{1j}=0⟹T_{2j}=0⟹$ a pure strategy of no manufacturing option. In this case, $P_{rj}\left(t\right)>D_j\left(t\right)⟹\mathrm{E}\mathrm{q}.\left(21\right)\iff T_{3j}<T_{4j}$. Conversely, $T_{3j}=0⟹T_{4j}=0$. Thus, from Equations (21) and (22), $T_{2j}=T_{3j}⟹T_{3j}=T_{4j}⟹T_{1j}<T_{2j}⟹$ a pure strategy of no remanufacturing option. Conversely, $T_{1j}=0⟹T_{2j}=0⟹T_{3j}=0⟹T_{4j}=0$. Thus, $T_{1j}>0⟹T_{1j}<T_{2j}$ and $T_{2j}<T_{3j}⟹<T_{3j}<T_{4j}$. Hence, Equations (19)-(22) implies constraint (18), and, therefore, constraint (18) can be ignored. Thus, our goal is to solve the following objective function: