The integral decision on production/remanufacturing technology and investment time in product recovery

Abstract

This paper examines the financial impact of technological decisions firms face when introducing a new product which, as firms increasingly assign responsibility for the recovery of their used products, must be expanded to include their influence on available options on how to deal with used products. The integrated choice between a low (disposal) or high (remanufacturing) level of product recovery and, in the second case, at which time to start remanufacturing activities and whether the firm would dispose of initial product returns or store them until start time result in three generic options: (a) design for single use, (b) design for reuse, and (c) design for reuse with stock-keeping. After introducing a basic dynamic framework consisting of a product life cycle for the demand and a similar development for an external return stream, the Net Present Values of the relevant payments connected with each option are given and optimal policies for options (b) and (c) are derived. An extensive numerical study is used to examine the potential benefits from using the inventory and the applicability of simple heuristic rules for stock-keeping and for determining when to start remanufacturing.

Appendices

Appendix 1‐ proofs

Properties of NPV _b

When inserting u(t)=0 for t<Δ and replacing the max/min operators in Eq. 2 the objective is (omitting time indices) given by

$${\text{NPV}}_{b} = \left\{ {\begin{array}{*{20}l} {{K^{r}_{p} + {\int\limits_0^\Delta {e^{{ - \alpha t}} } }c^{r}_{p} d\,\operatorname{d} t + e^{{ - \alpha t_{r} }} K_{r} } \hfill} \\ {{ + {\int\limits_\Delta ^{t_{I} } {e^{{ - \alpha t}} } }{\left[ {c^{r}_{p} {\left( {d - u} \right)} + c_{r} u} \right]}\operatorname{d} t + {\int\limits_{t_{I} }^\infty {e^{{ - \alpha t}} } }{\left[ {c_{r} d + c_{w} {\left( {u - d} \right)}} \right]}\operatorname{d} t} \hfill} \\ {{K^{r}_{p} + {\int\limits_0^\Delta {e^{{ - \alpha t}} } }c^{r}_{p} d\,\operatorname{d} t + {\int\limits_\Delta ^{t_{r} } {e^{{ - \alpha t}} } }{\left[ {c^{r}_{p} d + c_{w} u} \right]}\operatorname{d} t + e^{{ - \alpha t_{r} }} K_{r} } \hfill} \\ {{ + {\int\limits_{t_{r} }^{t_{I} } {e^{{ - \alpha t}} } }{\left[ {c^{r}_{p} {\left( {d - u} \right)} + c_{r} u} \right]}\operatorname{d} t + {\int\limits_{t_{I} }^\infty {e^{{ - \alpha t}} } }{\left[ {c_{r} d + c_{w} {\left( {u - d} \right)}} \right]}\operatorname{d} t} \hfill} \\ {{K^{r}_{p} + {\int\limits_0^\Delta {e^{{ - \alpha t}} } }c^{r}_{p} d\,\operatorname{d} t + {\int\limits_\Delta ^{t_{r} } {e^{{ - \alpha t}} } }{\left[ {c^{r}_{p} d + c_{w} u} \right]}\operatorname{d} t + e^{{ - \alpha t_{r} }} K_{r} } \hfill} \\ {{ + {\int\limits_{t_{r} }^\infty {e^{{ - \alpha t}} } }{\left[ {c_{r} d + c_{w} {\left( {u - d} \right)}} \right]}\operatorname{d} t} \hfill} \\\end{array} \begin{array}{*{20}l} {{} \hfill} \\ {{t_{r} <\Delta } \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ {{\Delta \leqslant t_{r} < t_{I} } \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ {{} \hfill} \\ {{t_{I} \leqslant t_{r} } \hfill} \\ {{} \hfill} \\\end{array} } \right.$$

(26)

The first derivative of Eq. 26 differs for each of the three regions as defined in Section 2.1 and is given by

$$\frac{{\partial \text{NPV}_b }} {{\partial t_r }} = \left\{ {\begin{array}{*{20}c} { - \alpha e^{ - \alpha t_r } K_r } \hfill & {\text{for}\,\text{t}_\text{r} < \Delta } \hfill & {\left( {\text{Region}\,\text{1}} \right)} \hfill \\ {\text{undefined}} \hfill & {\text{for}\,\text{t}_\text{r} = \Delta } \hfill & {\left( {\text{Region}\,2} \right)} \hfill \\ {e^{\text{ - }\alpha t_r } \left[ {\left( {c_p^r + c_w - c_r } \right)u\left( {t_r } \right) - \alpha K_r } \right]} \hfill & {\text{for}\,\Delta < \text{t}_\text{r} < t_I } \hfill & {} \hfill \\ {e^{\text{ - }\alpha t_r } \left[ {\left( {c_p^r + c_w - c_r } \right)d\left( {t_r } \right) - \alpha K_r } \right]} \hfill & {\text{for}\,\text{t}_\text{I} \leqslant t_r } \hfill & {\left( {\text{Region}\,3} \right).} \hfill \\ \end{array} } \right.$$

(27)

Note that except for t _r =Δ, Eq. 27 is continuous. It has negative sign in Region 1 and from assumption A.1 there exists a time t>t _I in Region 3 for which it holds $\forall t_{r} > t:d{\left( {t_{r} } \right)} <\frac{{\alpha K_{r} }} {{c_{p} + c_{w} - c_{r} }}$ and thus the objective finally must decrease. A similar argumentation holds for u(t _r ) if t _I =∞. Let ${\mathop x\limits^ \bullet }$ denote the first derivative of x with respect to time. The second derivative of Eq. 26 is given by

$$\frac{{\partial ^2 \text{NPV}_b }} {{\partial t_r^2 }} = \left\{ {\begin{array}{*{20}c} {\alpha ^2 e^{ - \alpha t_r } K_r } \hfill & {\text{for}\,t_r < \Delta } \hfill \\ {\text{undefined}} \hfill & {\text{for}\,t_r = \Delta } \hfill \\ {e^{ - \alpha t_r } \left[ {\left( {c_p^r + c_w - c_r } \right)\left( {\mathop u\limits^ \bullet \left( {t_r } \right) - \alpha u\left( {t_r } \right)} \right) + \alpha ^2 K_r } \right]} \hfill & {\text{for}\,\Delta < t_r < t_I } \hfill \\ {\text{undefined}} \hfill & {\text{for}\,t_r = t_I } \hfill \\ {e^{ - \alpha t_r } \left[ {\left( {c_p^r + c_w - c_r } \right)\left( {\mathop d\limits^ \bullet \left( {t_r } \right) - \alpha d\left( {t_r } \right)} \right) + \alpha ^2 K_r } \right]} \hfill & {\text{for}\,t_I < t_r } \hfill \\ \end{array} } \right.$$

(28)

It shows a positive sign in Region 1 and as time reaches infinity where d and ${\mathop d\limits^ \bullet }$ approach zero. Hence if there exists a local minimum, it must be followed by a local maximum.

Proof of Proposition 1

Candidates for t* _r are given by time points where Eq. 27 changes its sign from negative to positive. This is not possible in Region 1, where Eq. 27 is negative. Time point t _r,1=Δ is a candidate for t* _r if ${\mathop {\lim }\limits_{t_{r} \to \Delta + 0} }\frac{{\partial {\text{NPV}}_{b} }} {{\partial t_{r} }} \geqslant 0$. This yields

$$u{\left( \Delta \right)}{\left( {c^{r}_{p} + c_{w} - c_{r} } \right)} \geqslant \alpha K_{r} .$$

(29)

Further candidates, t _r,2 and t _r,3, are given by setting the first derivative in the two remaining regions to zero, which gives the following conditions

$$u{\left( {t_{{r,2}} } \right)}{\left( {c^{r}_{p} + c_{w} - c_{r} } \right)} = \alpha K_{r} \,{\text{for}}\,\Delta {\text{ $<$ }}t_{{r,2}} < t_{I} \,\,{\text{or}}$$

(30)

$$d{\left( {t_{{r,3}} } \right)}{\left( {c^{r}_{p} + c_{w} - c_{r} } \right)} = \alpha K_{r} \,{\text{for}}\,t_{I} \leqslant t_{{r,3}} .$$

(31)

Inserting Eqs. 30 and 31 for αK _r in the respective part of Eq. 28, conditions for a local minimum can be derived. t _r,2 is a local minimum if

$$\left. {\frac{{\partial ^{2} {\text{NPV}}_{b} }}{{\partial t^{2}_{r} }}} \right|_{{t_{r} = t_{{r,2}} }} = e^{{ - \alpha t_{r} }} {\left( {c^{r}_{p} + c_{w} - c_{r} } \right)}{\mathop u\limits^ \bullet }{\left( {t_{{r,2}} } \right)} > 0 \Leftrightarrow {\mathop u\limits^ \bullet }{\left( {t_{{r,2}} } \right)} > 0$$

(32)

and thus, $t_{{r,2}} < t^{{\max }}_{u}$. Analogously, t _r,3 is a local minimum if ${\mathop d\limits^ \bullet }{\left( {t_{{r,3}} } \right)} > 0$. Since demand must decrease for any time point t _r,3≥t _I (by definition of t _I ), candidate t _r,3 fails the second order necessary conditions. From our assumptions about the return function (unimodal) it follows that if inequality (29) holds, i.e. t _r,1=Δ is a candidate for an optimal solution, there will be no candidate t _r,2 and vice versa. Hence, there exists at most a single finite solution, being located in a half open interval [Δ, min {t _u ^max, t _I }).

Proof of Proposition 2

Since NPV _b decreases for sufficient high t _r solution candidate $t^{\infty }_{r} = \infty$ (invest never) has to be considered. In order to find the best alternative, the Net Present Value of the payment stream arising by assuming a relevant finite candidate $\widetilde{t}_{r} \in \left[ {\Delta ,\min {\left\{ {t^{{max}}_{u} ,t_{I} } \right\}}} \right)$, i.e. ${\text{NPV}}_{b} {\left( {\widetilde{t}_{r} } \right)}$, has to be compared with ${\text{NPV}}_{b} {\left( {t^{\infty }_{r} } \right)}$, which is given by

$${\text{NPV}}_{b} {\left( {t^{\infty }_{r} } \right)} = K^{r}_{p} + {\int_0^\infty {e^{{ - \alpha t}} } }{\left[ {c^{r}_{p} d{\left( t \right)} + c_{w} u{\left( t \right)}} \right]}\operatorname{d} t.$$

(33)

This gives an expression of the total discounted advantage of remanufacturing A ^b _r

$$\begin{array}{*{20}c} {A^{b}_{r} = {\text{NPV}}_{b} {\left( {t^{\infty }_{r} } \right)} - {\text{NPV}}_{b} {\left( {\widetilde{t}_{r} } \right)}} \\ { = - e^{{ - \alpha \widetilde{t}_{r} }} K_{r} + {\int_{\widetilde{t}}^\infty {e^{{ - \alpha t}} {\left[ {{\left( {c^{r}_{p} + c_{w} - c_{r} } \right)}\min {\left\{ {d{\left( t \right)},u{\left( t \right)}} \right\}}} \right]}{\text{d}}t} }} \\ \end{array} $$

(34)

Therefore, $t^{ * }_{r} = \widetilde{t}_{r} \,{\text{if}}\,A^{b}_{r} > 0$, i.e.

$$K_{r} \leqslant {\int_{\widetilde{t}_{r} }^\infty {e^{{ - \alpha {\left( {t - t^{r}_{r} } \right)}}} {\left[ {c^{r}_{p} + c_{w} - c_{r} } \right]}\min {\left\{ {d{\left( t \right)},u{\left( t \right)}} \right\}}dt.} }$$

(35)

Otherwise t _r *=∞.

Proof of Propositions 3 to 8

Propositions 3 to 8 are results of the following optimization approach. Since both the objective function NPV _c and constraint (14) are in general not convex, in accordance with Sydsæter and Hammond (1995), p 608 the following solution method is used:

• (1) Determination of the partial derivatives of the objective function.

• (2) Identification of possible solution candidates (Steps 1 and 2 in Sydsæter and Hammond 1995) using standard methods of Nonlinear Programming. This proofs Proposition 3. Exploring a joint property of all valid cases proofs Proposition 4 while individual properties confirm results stated in Propositions 5–8.

• (3) Comparison of values of NPV _c at candidate points against each other (Step 3) and with the Net Present Value of investing never ${\text{NPV}}_{b} {\left( {t^{\infty }_{r} } \right)}$ as given in (33). Smallest value is the (global) minimal value of NPV _c (Step 5). As this requires actual data, we will omit this part.

(1) Partial derivatives of the objective function

Subsequently, the partials of t _x with respect to t _e and t _r will be needed. This is applied by using implicit differentiation rules leading to

$$\frac{{\partial t_{x} }} {{\partial t_{e} }} = - \frac{{\frac{{\partial f}} {{\partial t_{e} }}}} {{\frac{{\partial f}} {{\partial t_{x} }}}} = - \frac{{u{\left( {t_{e} } \right)}}} {{d{\left( {t_{x} } \right)} - u{\left( {t_{x} } \right)}}},$$

(36)

$$\frac{{\partial t_{x} }} {{\partial t_{r} }} = - \frac{{\frac{{\partial f}} {{\partial t_{r} }}}} {{\frac{{\partial f}} {{\partial t_{x} }}}} = \frac{{d{\left( {t_{r} } \right)}}} {{d{\left( {t_{x} } \right)} - u{\left( {t_{x} } \right)}}}.$$

(37)

The first partial derivative of NPV _c (t _e , t _r ) with respect to t _e is (after collecting terms and inserting y _u (t _e )=y _u (t _x )=0) given by

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} = e^{{ - \alpha t_{e} }} c_{w} u{\left( {t_{e} } \right)} + {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }h_{u} \frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{e} }}\operatorname{d} t + {\left( { - e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} } \right]}{\left( {d{\left( {t_{x} } \right)} - u{\left( {t_{x} } \right)}} \right)} + {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} h_{u} } }\frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{x} }}\operatorname{d} t} \right)}\frac{{\partial t_{x} }} {{\partial t_{e} }}.$$

Since the partial of y _u (t) with respect to t _e equals

$$\frac{{\partial y_{u} {\left( t \right)}}}{{\partial t_{e} }} = \left\{ {\begin{array}{*{20}l} {{{\text{undefined}}} \hfill} & {{{\text{for}}\,t = t_{e} } \hfill} \\ {{{\text{ - }}u{\left( {t_{e} } \right)}} \hfill} & {{{\text{for}}\,t \in {\left( {t_{e} ,t_{x} } \right)}} \hfill} \\ {{{\text{undefined}}} \hfill} & {{{\text{for}}\,t = t_{x} } \hfill} \\ {{\text{0}} \hfill} & {{{\text{otherwise}}} \hfill} \\ \end{array} } \right.,$$

(38)

and by additionally replacing $\frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{x} }} = 0$ and $\frac{{\partial t_{x} }} {{\partial t_{e} }}$ by Eq. (36) it follows

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} = {\left( {e^{{ - \alpha t_{e} }} c_{w} - h_{u} {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }\operatorname{d} t + e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} } \right]}} \right)}u{\left( {t_{e} } \right)}.$$

(39)

Equation (39) can be interpreted as follows. If t _e increases, a marginal return u(t _e ) arriving at this time is no longer stored but disposed of, leading to additional unit costs of $e^{{ - \alpha t_{e} }} c_{w} u{\left( {t_{e} } \right)}$. Since this (marginal) return is not available for later remanufacturing, t _x decreases and thus, costs for producing a (marginal) new product at t _x are caused, given by $e^{{ - \alpha t_{x} }} {\left( {c^{r}_{p} - c_{r} } \right)}u{\left( {t_{e} } \right)}$. On the other hand, storing less returns reduces inventory holding costs per unit by $h_{u} {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }\operatorname{d} t\,u{\left( {t_{e} } \right)}$.

Evaluating the integral in Eq. 39 finally yields

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} = {\left( {e^{{ - \alpha t_{e} }} {\left[ {c_{w} - \frac{{h_{u} }} {\alpha }} \right]} + e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} + \frac{{h_{u} }} {\alpha }} \right]}} \right)}u{\left( {t_{e} } \right)}.$$

(40)

The first partial derivative of NPV _c (t _e , t _r ) with respect to t _r is (after collecting terms and inserting y _u (t _x )=0) given by

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{r} }} = e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} + {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }h_{u} \frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{r} }}\operatorname{d} t + {\left( { - e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} } \right]}{\left( {d{\left( {t_{x} } \right)} - u{\left( {t_{x} } \right)}} \right)} + {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }h_{u} \frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{x} }}\operatorname{d} t} \right)}\frac{{\partial t_{x} }} {{\partial t_{r} }}.$$

Replacing

$$\frac{{\partial y_{u} {\left( t \right)}}} {{\partial t_{r} }} = \left\{ {\begin{array}{*{20}l} {{{\text{undefined}}} \hfill} & {{{\text{for}}\,t = t_{r} } \hfill} \\ {{d{\left( {t_{r} } \right)}} \hfill} & {{{\text{for}}\,t \in {\left( {t_{r} ,t_{x} } \right)}} \hfill} \\ {{{\text{undefined}}} \hfill} & {{{\text{for}}\,t = t_{x} } \hfill} \\ {{\text{0}} \hfill} & {{{\text{otherwise}}} \hfill} \\\end{array} } \right.$$

(41)

and $\frac{{\partial t_{x} }} {{\partial t_{r} }}$ by (37) yields

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{r} }} = {\left( {{\left( {e^{{ - \alpha t_{r} }} - e^{{ - \alpha t_{x} }} } \right)}{\left[ {c^{r}_{p} - c_{r} } \right]} + h_{u} {\int_{t_{r} }^{t_{x} } {e^{{ - \alpha t}} } }\operatorname{d} t} \right)}d{\left( {t_{r} } \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} .$$

(42)

A later investment time t _r decreases the Net Present Value of the investment expenses by $\alpha e^{{ - \alpha t_{r} }} K_{r}$. A (marginal) demand d(t _r ) is no longer satisfied by remanufacturing returns at t _r , which instead are stored for a later use at t _x . Therefore, a cost reduction at t _x by remanufacturing instead of producing faces an increase in costs at t _r . Additional holding costs are caused by storing the (marginal) return.

Continuing in the same manner as above gives

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{r} }} = {\left( {e^{{ - \alpha t_{r} }} - e^{{ - \alpha t_{x} }} } \right)}{\left[ {c^{r}_{p} - c_{r} + \frac{{h_{u} }} {\alpha }} \right]}d{\left( {t_{r} } \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} .$$

(43)

(2) Identification of solution candidates

By introducing Lagrange multipliers μ _i , i=1, 2, 3 which are associated with constraints (12)–(13) the Lagrangian $\mathcal{L}{\left( {t_{e} ,t_{r} ,\mu _{1} ,\mu _{2} ,\mu _{3} } \right)}$ is defined as

$$\mathcal{L}{\left( {t_{e} ,t_{r} ,\mu _{1} ,\mu _{2} ,\mu _{3} } \right)} = {\text{NPV}}_{c} {\left( {t_{e} ,t_{r} } \right)} - \mu _{1} {\left( {t_{e} - \Delta } \right)} - \mu _{2} {\left( {t_{r} - t_{e} } \right)} - \mu _{3} {\left( {{\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}\operatorname{d} s} }} \right)}.$$

(44)

The partials of $\mathcal{L}{\left( {t_{e} ,t_{r} ,\mu _{1} ,\mu _{2} ,\mu _{3} } \right)}$ have to equal zero

$$\begin{array}{*{20}c} {\frac{{\partial \mathcal{L}}} {{\partial t_{e} }} = \frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} - \mu _{1} + \mu _{2} - \mu _{3} u{\left( {t_{e} } \right)} = 0,} \\{} \\\end{array} $$

(45)

$$\frac{{\partial \mathcal{L}}} {{\partial t_{r} }} = \frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{r} }} - \mu _{2} + \mu _{3} d{\left( {t_{r} } \right)} = 0.$$

(46)

The complementary slackness conditions are given by

$$\frac{{\partial \mathcal{L}}} {{\partial \mu _{1} }} = \Delta - t_{e} \leqslant 0,\mu _{1} \geqslant 0,\mu _{1} {\left( {\Delta - t_{e} } \right)} = 0,$$

(47)

$$\frac{{\partial \mathcal{L}}} {{\partial \mu _{2} }} = t_{e} - t_{r} \leqslant 0,\mu _{2} \geqslant 0,\mu _{2} {\left( {t_{e} - t_{r} } \right)} = 0,$$

(48)

$$\frac{{\partial \mathcal{L}}} {{\partial \mu _{3} }} = - {\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}} }\operatorname{d} s \leqslant 0,\mu _{3} \geqslant 0,\mu _{3} {\left( {{\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}} }\operatorname{d} s} \right)} = 0.$$

(49)

Having three constraints, each either being active or inactive, in total eight cases have to be distinguished. It can be shown that all Cases where stockkeeping does not occur (i.e. where t _e =t _r ) can be excluded (see Kleber 2004, for the straightforward proof). Thus, for the optimal solution (t* _e , t* _r ) it holds t* _e <t* _r . The remaining cases (1)–(4) constitute the different types of solution candidates as stated in Proposition 3. This completes the proof to Proposition 3.

For all remaining cases (1)–(4) it holds t* _e <t* _r which requires from Eq. 48 μ ₂=0. Inserting this value into Eq. 45 (reconsidering non-negativity of μ ₁, μ ₃, and u(t* _e ) necessitates $\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} \geqslant 0$, yielding

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} = {\left( {e^{{ - \alpha t^{ * }_{e} }} {\left[ {c_{w} - \frac{{h_{u} }} {\alpha }} \right]} + e^{{ - \alpha t^{ * }_{x} }} {\left[ {c^{r}_{p} - c_{r} + \frac{{h_{u} }} {\alpha }} \right]}} \right)}u{\left( {t^{ * }_{e} } \right)} \geqslant 0$$

(50)

Because of u(t)>0 ∀t≥Δ this is equivalent to

$$e^{{ - \alpha t^{ * }_{e} }} {\left[ {c_{w} - \frac{{h_{u} }} {\alpha }} \right]} + e^{{ - \alpha t^{ * }_{x} }} {\left[ {c^{r}_{p} - c_{r} + \frac{{h_{u} }} {\alpha }} \right]} \geqslant 0$$

and solving for t* _x −t* _e finally yields

$$t^{ * }_{x} - t^{ * }_{e} \leqslant \frac{1} {\alpha }\ln {\left( {\frac{{\alpha {\left( {c_{p} - c_{r} } \right)} + h_{u} }} {{ - \alpha c_{w} + h_{u} }}} \right)} = :\tau .$$

(51)

This completes the proof to Proposition 4.

Case (1)$t_e < t_r ,\Delta < t_e ,\int_{t_x }^{t_I } {\left( {d\left( s \right) - u\left( s \right)} \right)} \,ds > 0 \Leftrightarrow \Delta < t_e < t_x < t_I$. None of the conditions is active. Thus, μ₁=0, μ₂=0 as well as μ₃=0. Inserted into Eq. 45, this yields

$$\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }} = {\left( {e^{{ - \alpha t_{e} }} c_{w} - h_{u} {\int_{t_{e} }^{t_{x} } {e^{{ - \alpha t}} } }\operatorname{d} t + e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} } \right]}} \right)}u{\left( {t_{e} } \right)} = 0.$$

Proceeding as above leads to

$$t_{x} - t_{e} = \tau .$$

(52)

Further, inserting Lagrange Multipliers into Eq. 46 requires

$$- e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} = h_{u} {\int_{t_{r} }^{t_{x} } {e^{{ - \alpha t}} } }\operatorname{d} t\,d{\left( {t_{r} } \right)} - e^{{ - \alpha t_{x} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} .$$

(53)

This completes the proof to Proposition 5.

Case (2)$t_{e} < t_{r} ,\Delta = t_{e} ,{\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}} }\operatorname{d} s = 0 \Leftrightarrow \Delta = t_{e} < t_{r} < t_{x} = t_{I}$. Inserting μ₂=0 into (46) yields

$${\begin{array}{*{20}c} {\mu _{3} = \left. { - \frac{{\partial {\text{NPV}}_{c} }}{{\partial t_{r} }}} \right|_{{t_{x} = t_{I} }} \frac{1}{{d{\left( {t_{r} } \right)}}}} \\ { = - {\left( {e^{{ - \alpha t_{r} }} - e^{{ - \alpha t_{I} }} } \right)}{\left[ {c^{r}_{p} - c_{r} } \right]} - h_{u} {\int_{t_{r} }^{t_{I} } {e^{{ - \alpha t}} \operatorname{d} t} } + \alpha e^{{ - \alpha t_{r} }} K_{r} \frac{1}{{d{\left( {t_{r} } \right)}}}.} \\ \end{array} }$$

μ₃≥0 requires

$$- e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} \geqslant h_{u} {\int\limits_{t_{r} }^{t_{I} } {e^{{ - \alpha t}} } }\operatorname{d} t\,d{\left( {t_{r} } \right)} - e^{{ - \alpha t_{I} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} .$$

(54)

Further, inserting μ ₃ into Eq. (45) leads to

$$\begin{array}{*{20}c} {\mu _{1} = \frac{{\partial {\text{NPV}}_{c} }}{{\partial t_{e} }}\left| {_{{t_{e} = \Delta ,\,t_{x} = t_{I} }} } \right. + \frac{{\partial {\text{NPV}}_{c} }}{{\partial t_{r} }}\left| {_{{t_{x} = t_{I} }} } \right.\frac{{u{\left( \Delta \right)}}}{{d{\left( {t_{r} } \right)}}}} \\ { = {\left( {e^{{ - \alpha \Delta }} c_{w} - h_{u} {\int_\Delta ^{t_{r} } {e^{{ - \alpha t}} } }\operatorname{d} t + e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}} \right)}u{\left( \Delta \right)} - \alpha e^{{ - \alpha t_{r} }} K_{r} \frac{{u{\left( \Delta \right)}}}{{d{\left( {t_{r} } \right)}}}.} \\ \end{array} $$

(55)

Since μ₁≥0, Eq. 55 implies

$$e^{{ - \alpha \Delta }} c_{w} d{\left( {t_{r} } \right)} \geqslant h_{u} {\int_\Delta ^{t_{r} } {e^{{ - \alpha t}} } }\operatorname{d} t\,d{\left( {t_{r} } \right)} - e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} + \alpha e^{{ - \alpha t_{r} }} K_{r} .$$

(56)

This completes the proof to Proposition 6.

Case (3)$t_{e} < t_{r} ,\Delta = t_{e} ,{\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}} }\operatorname{d} s > 0 \Leftrightarrow \Delta = t_{e} < t_{r} < t_{x} < t_{I}$. Both second and third conditions are inactive. Then, μ₂=0 and μ₃=0 from Eqs. 48 and 49, respectively. Inserting both values into Eq. 46 again yields Eq. 53. This completes the proof to Proposition 7.

Case (4)$t_{e} < t_{r} ,\Delta < t_{e} ,{\int_{t_{x} }^{t_{I} } {{\left( {d{\left( s \right)} - u{\left( s \right)}} \right)}} }\operatorname{d} s = 0 \Leftrightarrow \Delta < t_{e} < t_{x} = t_{I}$. Both, first and second conditions are inactive. Then, μ₁=0 and μ₂=0 from Eqs. 47 and 48, respectively. Both values inserted into Eq. 45 yields the value for μ₃

$$\mu _{3} = \frac{1}{{u{\left( {t_{e} } \right)}}}\frac{{\partial {\text{NPV}}_{c} }}{{\partial t_{e} }}\left| {_{{t_{x} = t_{l} }} = e^{{ - \alpha t_{e} }} c_{w} - h_{u} {\int_{t_{e} }^{t_{l} } {e^{{ - \alpha t}} } }dt + e^{{ - \alpha t_{l} }} {\left[ {c^{r}_{p} - c_{r} } \right]}} \right.$$

Inserting μ ₃ into Eq. 46 gives

$$\begin{array}{*{20}c} {{\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{r} }}\left| {_{{t_{x} = t_{I} }} } \right. + \frac{{d{\left( {t_{r} } \right)}}} {{u{\left( {t_{e} } \right)}}}\frac{{\partial {\text{NPV}}_{c} }} {{\partial t_{e} }}\left| {_{{t_{x} = t_{I} }} } \right. = 0}} \\ {{ \Leftrightarrow e^{{ - \alpha t_{e} }} c_{w} d{\left( {t_{r} } \right)} = h_{u} {\int_{t_{e} }^{t_{r} } {e^{{ - \alpha t}} } }\operatorname{d} t\,d{\left( {t_{r} } \right)} - e^{{ - \alpha t_{r} }} {\left[ {c^{r}_{p} - c_{r} } \right]}d{\left( {t_{r} } \right)} + \alpha e^{{ - \alpha t_{r} }} K_{r} .}} \\\end{array} $$

(57)

This completes the proof to Proposition 8.

Appendix 2 — list of symbols

Generic dynamic environment
t	time index
(.)(t)	time dependence
.*	optimal values
d(t)	demand rate at time t
[[f]]t^{{\max }}_{d}[[/f]]	time point of maximum demand rate
u(t)	return rate at time t
Δ	time delay of returns
[[f]]t^{{\max }}_{u}[[/f]]	time point of maximum return rate
t I	intersection point of demand and return functions
Processes (states and variables)
y u (t)	physical stock recoverables
p(t)	production rate
r(t)	remanufacturing rate
w(t)	disposal rate
Cash flow parameters
α	discount rate or continuous interest rate
c s p	variable unit production cost at single use production
c r p	variable unit production cost at reuse production
c r	variable unit remanufacturing cost
c w	variable unit disposal cost or negative salvage revenue
h u	holding cost rate recoverables
K s p	initial investment expenditures for single use production
K r p	initial investment expenditures for reuse production
K r	investment expenditures for remanufacturing facility
Policy parameters and optimization oriented notation
t r	time of remanufacturing investment
t e	start time of storing collected returns
t x	time where all stored returns are used up
u crit	critical return rate in investment project (b)
A b r	total discounted net advantage of remanufacturing in investment project (b)
D b p	increase of total discounted expenditures for production in investment project (b)
τ	maximal holding time of returns in investment project (c)
Parameters used in numerical investigation
M	number of potential adopters (parameter in Bass model)
P	coefficient of innovation (parameter in Bass model)
Q	coefficient of immitation (parameter in Bass model)
F	fraction of demanded products being available for remanufacturing