### The single-period case

Consider an ELSR instance of *T* periods with only one period *i* fixed as positive-remanufacturing period, i.e., r_{
i
} > 0, with 1 ≤ *i* ≤ *T*, and r_{
t
} = 0 for all *t* with 1 ≤ *i* ≤ *T* and *t* ≠ *i*. The objective is to determine the optimal remanufacturing quantity *Q*
_{
i
}
^{
r
} of the period *i*, with 0 < *Q*
_{
i
}
^{
r
} ≤ *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} and *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} > 0.

First, consider the case that the number of available returns in period *i* are at most equal to the demand of the period, i.e., *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} ≤ *D*
_{
i
}. Then, by (9), the optimal remanufacturing quantity must be equal to all of the available returns, i.e., *Q*
_{
i
}
^{
r
} = *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} > 0. On the other hand, for the case that *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} > *D*
_{
i
}, we must determine the last period *j* within the planning horizon for which it is more profitable to meet at least one unit of its demand by remanufacturing in period *i*. Assume first that the number of available returns is sufficient to exactly meet the accumulative demand from the current period *i* to certain future period *k*, i.e., *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} = *D*
_{
ik
}, with 1 ≤ *i* ≤ *k* ≤ *T*. Then, the optimal remanufacturing quantity of period *i* is *Q*
_{
i
}
^{
r
} = *D*
_{
ij
}, with *j* as the last period for which ${c}_{i}^{r}{D}_{j}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}{D}_{j}}\le \phantom{\rule{0.5em}{0ex}}{K}_{j}^{p}+{c}_{j}^{p}{D}_{j}+{\displaystyle \sum _{t=i}^{T}{h}_{t}^{u}{D}_{j}}\phantom{\rule{0.5em}{0ex}}$ is fulfilled, with 1 ≤ *i* ≤ *j* ≤ *k* ≤ *T* and *D*
_{
j
} > 0. However, we note that in general the number of available returns in period *i* is sufficient to meet only a portion of the demand of a certain future period *k*, i.e., *y*
_{
i − 1}
^{
u
} + *R*
_{
i
} = *D*
_{
i(k − 1)} + *α* < *D*
_{
ik
}, with 1 ≤ *i* ≤ *k* ≤ *T* and *α* ≥ 1. Without loss of generality, let us assume that it is profitable to remanufacture in period *i* at least the needed quantity to cover the demand requirements from *i* to (*k* − 1), i.e., *D*
_{
i(k − 1)} ≤ *Q*
_{
i
}
^{
r
} < *D*
_{
ik
}. This means that at least one unit of the demand requirement of period *k* is satisfied by means of the production of new items in a certain period *t* with 1 ≤ *t* ≤ *k*. If 1 ≤ *t* ≤ *i*, then there must be that *Q*
_{
i
}
^{
r
} = *D*
_{
i(k − 1)} + *α*, since ${c}_{i}^{r}\le \phantom{\rule{0.5em}{0ex}}{c}_{t}^{p}+{\displaystyle \sum _{\tau =t}^{i-1}{h}_{\tau}^{s}}$ is true by (9). In the case of *i* < *t* ≤ *k*, we have that *Q*
_{
i
}
^{
r
} = *D*
_{
i(k−1)} + *α* only if the condition ${c}_{i}^{r}+{\displaystyle \sum _{t=i}^{t-1}{h}_{t}^{s}}\le \phantom{\rule{0.5em}{0ex}}{c}_{t}^{p}$ is fulfilled, otherwise *Q*
_{
i
}
^{
r
} = *D*
_{
i(k−1)}. This last condition can be relaxed by ${c}_{i}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}}\le \phantom{\rule{0.5em}{0ex}}{c}_{j}^{p}+{\displaystyle \sum _{t=i}^{T}{h}_{t}^{u}}\phantom{\rule{0.5em}{0ex}}$ in the case that final disposal of used items is not considered, which is supported by economic as well as ecological reasons. Teunter et al. [12] point out that disposing option ‘does not lead to a considerable cost reduction unless the remanufacturable return rate as a percentage of the demand rate is unrealistically high (above 90%) and the demand rate is very small (less than 10 per year)’. We resume the reasoning above by means of the following assumption about the profitability of maximizing the remanufacturing quantity in a certain period.

**Definition 2.** Given two periods

*i* and

*k* of an ELSR instance of

*T* periods, with 1 ≤

*i* ≤

*k* ≤

*T* , such that

*r*
_{
i
} > 0, we say that it is profitable to maximize the remanufacturing quantity of period

*i* if the expression

${c}_{i}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}}\le \phantom{\rule{0.5em}{0ex}}{c}_{j}^{p}$

(10.1)

is fulfilled for each period

*j*, with 1 ≤

*i* ≤

*j* ≤

*k* ≤

*T*, or

${c}_{i}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}}\le \phantom{\rule{0.5em}{0ex}}{c}_{j}^{p}+{\displaystyle \sum _{t=i}^{T}{h}_{t}^{u}}\phantom{\rule{0.5em}{0ex}}$

(10.2)

in the case that the final disposal of used items is not considered.

Thus, if Definition 2 is fulfilled for any couple of periods *i* and *j* in 1,…,*T*, with *i* ≤ *j*, we can assure that the optimal remanufacturing quantity *Q*
_{
i
}
^{
r
} f a single period *i* fixed as positive-remanufacturing period is the minimum between the amount of available returns and the accumulative demand from the current period and the end of the planning horizon, i.e., to remanufacture as much as possible. We note that for a given instance, it is sufficient that Definition 2 is fulfilled between the period fixed as positive-remanufacturing period and the last one for which at least a portion of its demand is attainable by remanufacturing in the period fixed. On the other hand, if Definition 2 is not fulfilled, it is unlikely that we can determine the optimal remanufacturing quantity of a certain period without knowing the periods where production is carried out since, in the case that the available returns in period *i* are only sufficient to partially meet the accumulative demand to certain future period *k*, we need to know if the rest of the demand of period *k* is produced either in the same period or in a previous one.

Real situations where Definition 2 is fulfilled include cases where holding costs of both used and serviceable items are similar or negligible, very low remanufacturing costs, as well as instances with few periods. We also note that the problem of finding the optimal positive-remanufacturing period for an ELSR instance for which it is profitable to remanufacture as much as possible at any period can be solved in *O*(*T*^{3}) time since we must consider *T* different periods, and the corresponding optimal production and final dispose plans can be obtained in *O*(*T*^{2}) by means of a Wagner-Whitin algorithm type [16].

### The multi-period case

We now consider the problem of finding the remanufacturing quantities of a remanufacturing plan of perfect cost with at least two periods fixed as positive-remanufacturing periods. We first note that the amount to be remanufactured in a certain period depends in part of the remanufactured quantity in previous periods as well as affects the amount to be remanufactured in future periods. Then, it may not be possible to determine efficiently the optimal remanufacturing quantity for each period, even under the assumptions introduced in the previous section. In view of this difficulty, we focus on the problem of determining the total quantity of a remanufacturing plan of perfect cost. Before we tackle this problem, we provide a result about the form of the remanufacturing plan of perfect cost for a particular case.

**Proposition 1.** Consider an ELSR instance for which the number of available returns in a certain period *i* fixed as a positive-remanufacturing period is sufficient to fully cover the demand until the end of the planning horizon, i.e., *R*
_{
i
} + *y*
_{
i−1}
^{
u
} ≥ *D*
_{
iT
}, *r*
_{
i
} > 0, with 1 ≤ *i* ≤ *T*. If the optimal solution set is not empty, there is at least one optimal solution for which the total remaining demand from period *i* is satisfied only by remanufacturing from period *i* onwards, i.e., *r*
_{
iT
} = *D*
_{
iT
}, with ${r}_{\mathit{ij}}={\displaystyle \sum _{t=i}^{j}{r}_{t}},1\le i\le j\le T$.

*Proof*. Let us consider an optimal solution of the ELSR with *r*
_{
i
} > 0, *R*
_{
i
} + *y*
_{
i−1}
^{
u
} ≥ *D*
_{
iT
} , and *r*
_{
iT
} < *D*
_{
iT
}. Then, the quantity (*D*
_{
iT
} − *r*
_{
iT
}) > 0 is satisfied by means of the production of new items. We can determine a new solution with *r*
_{
iT
} = *D*
_{
iT
} from the current solution as follows: First, for each period *t* with *i* ≤ *t* ≤ *T* and *p*
_{
t
} > 0, we replace the entire production in *t* by remanufacturing, i.e., *r*
_{
t
} ← *p*
_{
t
}, *p*
_{
t
} ← 0. Note that the replacement operation is possible as we are assuming the returns are sufficient. Second, while *r*
_{
iT
} < *D*
_{
iT
}, take the last period *t* with *p*
_{
t
} > 0 and 1 ≤ *t* < *i*, and transfer units of the production of period *t* to the remanufacturing of period *i*, until *r*
_{
iT
} = *D*
_{
iT
} or *p*
_{
t
} = 0. By (9), the cost of the new solution is at most equal to the cost of the original. Therefore, there must be an optimal solution of the ELSR for which *r*
_{
iT
} = *D*
_{
iT
}, if *r*
_{
i
} > 0 and *R*
_{
i
} + *y*
_{
i−1}
^{
u
} ≥ *D*
_{
iT
} is are complied.

Proposition 1 helps us to identify the form of a remanufacturing plan of perfect cost for the ELSR in the particular case that the number of available returns in a period fixed as positive-remanufacturing period is sufficient to meet all the remaining demand until the end of the planning horizon. We must note that if the amount of available returns in a certain period is sufficient to meet all the remaining demand but the period is not fixed as a positive-remanufacturing period, we cannot ensure the result above unless the period under consideration is the first one (see [10]).

We consider now the problem in general sense, i.e., no kind of relationship is assumed between the returns and the demand values. First, we provide the following definitions about the costs and the quantities of remanufacturing.

**Definition 3.** We say that the remanufacturing costs are non-speculative with respect to the transfer when they satisfy the following expressions:

${K}_{i}^{r}+{c}_{i}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}}-{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{u}}\phantom{\rule{1em}{0ex}}\ge \phantom{\rule{0.5em}{0ex}}{K}_{j}^{r}+{c}_{j}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{u}}\phantom{\rule{1em}{0ex}}$

(11.1)

${c}_{i}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{s}}-{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{u}}\phantom{\rule{1em}{0ex}}\ge \phantom{\rule{1em}{0ex}}{c}_{j}^{r}+{\displaystyle \sum _{t=i}^{j-1}{h}_{t}^{u}}$

(11.2)

for any couple of period *i* and *j* in 1,…,*T*.

Expression (11.1) states that it is profitable to transfer the entire remanufacturing quantity from a certain period to other future period that was inactive, while (11.2) states that it is profitable to transfer forward at least one unit between two periods with positive remanufacturing. We note that the expressions given in (11) are fulfilled in different settings of practical interest, e.g., when all the costs involved are stationary or they do not increase over time.

**Definition 4.** Given an ELSR instance with a set of periods fixed as positive remanufacturing periods and a feasible remanufacturing plan *r*, we define the *upper bound of remanufacturing* of a certain period *i* to the quantity *u*
_{
i
} = 0 if *r*
_{
i
} = 0 and *u*
_{
i
} = min (*R*
_{
i
} + *y*
_{
i
}
^{
u
}, *D*
_{
i(j−1)}) if *r*
_{
i
} > 0, where *j* is either the next positive-remanufacturing period within the planning horizon, or (*T* + 1) if *i* is the last positive-remanufacturing period, i.e., r_{
t
} = 0 for all periods *t* in (*i* + 1), …, *T*.

**Proposition 2**. Given an ELSR instance, there is at least one optimal solution for which the remanufacturing quantity of each period is at most equal to its upper bound of remanufacturing, i.e., 0 ≤ *r*
_{
t
} ≤ *u*
_{
t
}, for all periods *t* = 1, …, *T*.

*Proof*. Without loss of generality, consider an optimal solution of an ELSR instance with only one period *i* for which *r*
_{
i
} > *u*
_{
i
} = min (*R*
_{
i
} + *y*
_{
i
}
^{
u
}, *D*
_{
i(j−1)}) and *r*
_{
j
} > 0 with 1 ≤ *i* ≤ *j* ≤ *T*. First, we note that the case *u*
_{
i
} = *R*
_{
i
} + *y*
_{
i
}
^{
u
} is not feasible since the remanufacturing quantity is greater than the amount of available returns. Now, consider the case that *u*
_{
i
} = *D*
_{
i(j−1)}. Then, by (11), we can obtain a new solution with at most the same cost than the original by transferring remanufactured units from period *i* to the consecutive period *j* with *r*
_{
j
} > 0, until *r*
_{
i
} = *D*
_{
i(j−1)} in the new solution. Therefore, an optimal solution for the same ELSR instance for which *r*
_{
t
} ≤ *u*
_{
t
} can be obtained, for all periods *t* = 1, …, *T*.

Proposition 2 states that the remanufacturing quantity of a certain period is upper-bounded by the minimum between the number of available returns and the accumulative demand until the period preceding the next period with positive remanufacturing. We note that the upper bound value of certain period depends on the remanufacturing quantities of the previous periods. In addition, it may not be possible to determine how close or how far to its upper bound is the remanufacturing quantity of a certain period in an optimal solution of the ELSR. Despite these facts, the upper bound of remanufacturing allows us to determine the total remanufacturing quantity of a remanufacturing plan of perfect cost, as we show in the following proposition:

**Proposition 3.** Consider an ELSR instance with a set of periods *F* fixed as positive-remanufacturing periods such as for any pair of consecutive periods *i* and *j* of *F*, the Definition 2 is fulfilled for any pair of meaningful periods, i.e., pairs (*i*, *t*) with *i* ∈ *F* and *t* the last period before *j* for which at least a portion of its demand is attainable by remanufacturing in i with *i* ≤ *t* < *j*. Then, consider the remanufacturing plan $\overline{r}$ obtained by remanufacturing in each period the amount given by the upper bound of remanufacturing applied in ascending order, i.e., $\overline{{r}_{t}}={u}_{t}$, assuming that $\overline{{r}_{1}}={u}_{1},\overline{{r}_{2}}={u}_{2},\dots ,\overline{{r}_{\left(t-1\right)}}={u}_{\left(t-1\right)}$, for all periods *t* = 1, …, *T*. Then, there is an optimal solution with a remanufacturing plan *r*^{*} for which ${r}_{1T}^{*}=\overline{{r}_{1T}}$ where ${r}_{\mathit{ij}}^{*}={\displaystyle \sum _{t=i}^{j}{r}_{t}^{*}}$ and $\overline{{r}_{\mathit{ij}}}={\displaystyle \sum _{t=i}^{j}\overline{{r}_{t}}}$, with 1 ≤ *i* ≤ *j* ≤ *T*.

*Proof*. We note that by Proposition 2 and Definition 4, there must be that ${r}_{1T}^{*}\le \overline{{r}_{1T}}$. Without loss of generality, let us assume that ${r}_{1T}^{*}=\overline{{r}_{1T}}-1$. Then, there exists a period *i*, with 1≤*i* ≤ *T*, for which $0<{r}_{i}^{*}=\overline{{r}_{i}}-1$, ${r}_{1\left(i-1\right)}^{*}=\overline{{r}_{1\left(i-1\right)}}$, and ${y}_{\left(i-1\right)}^{*u}=\overline{{y}_{\left(i-1\right)}^{u}}$. This means that the upper bound of remanufacturing of period *i* is the same for both remanufacturing plans under consideration, with $0<{r}_{i}^{*}<{u}_{i}=\overline{{r}_{i}}$. We also note that *y*
_{
t
}
^{*u
} ≥ 1 is fulfilled for all periods *t* = i, …, *T*. Therefore, we can obtain a new feasible solution for the same ELSR instance with at most the same cost by increasing the remanufacturing in period *i* in one unit, i.e., ${r}_{i}^{*}\leftarrow {r}_{i}^{*}+1=\overline{{r}_{i}}$, without affecting the remanufacturing of the future periods and in the meantime by reducing the production of a certain period *j* in 1, …, *T*. This new solution fulfills that ${r}_{1T}^{*}=\overline{{r}_{1T}}$, and this cost is at most the same than the cost of the original optimal solution as we are assuming that maximizing the remanufacturing quantity of the periods with positive remanufacturing is profitable according to Definition 2.

Proposition 3 states that in order to determine a remanufacturing plan of perfect cost for an ELSR instance with certain periods fixed as positive-remanufacturing periods, we only need to explore those remanufacturing plans for which the total remanufacturing quantity is equal to the sum of the upper bounds of remanufacturing. These values can be determined efficiently (linear time) by applying Definition 4 period by period, beginning with the first period fixed as positive-remanufacturing period. We show the usefulness of Proposition 3 through the following numeric example.