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## Appendix A. Numerical experiments

In this section, we: (i) discuss the definition of the domain of estimation used for game theory-based collaboration policies and, (ii) validate our approach for the negotiation procedure. These numerical experiments are conducted in the framework given in Section 7.1 and the obtained results are analyzed on average and discussed in Sections Appendix A.1-Appendix A.2.

### Appendix A.1. Game theory-based collaboration policies

In this section, we first discuss the quality of the estimated data within the game theory-based collaboration policies. Then, we analyze the gains of each PU applying these collaboration policies in view of baseline policies. Recall that in the conducted experiments, we consider only one unknown parameter at a time, namely: $TBO^2$, $TBO^1$, $h^1$, $h^2$, $q-b^2$, $g-b^1$, or $\hat{h}$.

Let us now place the emphasis on the impact of key characteristics of the domain of estimation of the unknown parameter {$\underline{\theta}$, ..., $\bar{\theta}$} on the problem output. It is computed based on its true value $\theta^*$ and three following features:

* **Range:** $2\Delta_{\theta}\theta^*$ is the range of the definition domain of unknown parameters, i.e., $\underline{\theta} = \theta^* - \Delta_{\theta}\theta^*$, $\bar{\theta} = \theta^* + \Delta_{\theta}\theta^*$. We fixed $\Delta_{\theta}$ in interval {0.2, 0.5, 0.8}.

* **Cardinality of the interval:** $|\Theta|$ corresponds to the number of scenarios (i.e., realizations). Let $|\Theta|$ take its values in {3, 7}.
<figure><img src="image_5.png" /><figcaption>(a) Relative gains of PU2 under PU1 Leader for unknown parameters $h^2, TBO^2, q-b^2$ against No_Co</figcaption></figure>

<caption>(b) Relative gains of PU1 under PU2 Leader for unknown parameters $h^1, TBO^1, g-b^1, \hat{h}$ against No_Co</caption>

<caption>Figure A.5: Policy $PU_i\_Leader$ ($i \in \{1, 2\}$) versus policy $No\_Co$ ($|\Theta| = 7$): Follower gains (in %)</caption>

* **Position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$:** $p_{\theta^*}$ is the position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$. Let $p_{\theta^*}$ take its values in $\{1, 2, \dots, |\{\underline{\theta}, \dots, \bar{\theta}\}|\}$. Range $\{\underline{\theta}, \dots, \bar{\theta}\}$ is shift to the right or the left depending on $p_{\theta^*}$. For example, when $|\Theta| = 3$ and $p_{\theta^*} = 2$, the range of the unknown parameter is $\{\underline{\theta}, \theta^*, \bar{\theta}\}$.

Note that, the cost functions estimating the unknown parameters are strictly increasing, thus the position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$ corresponds to the scenario number and $|\Theta| = |\{\underline{\theta}, \dots, \bar{\theta}\}|$.

Position of true value $\theta^*$ within the domain of estimation $\{\underline{\theta}, \dots, \bar{\theta}\}$: Extensive numerical experiments have shown that the economic benefit of the leader, the environmental impact, and the side payment are not impacted by the position of the true value of the unknown parameter within $\{\underline{\theta}, \dots, \bar{\theta}\}$. Figure A.5 highlights the economic benefit of the follower in collaboration policies PU1 Leader and PU2 Leader against No_Co as a function of the position of the true value within $\{\underline{\theta}, \dots, \bar{\theta}\}$ for all possible unknown parameters. Due to the high computational time needed to conduct experiments, results are provided for a subset of 733 randomly selected instances and $|\{\underline{\theta}, \dots, \bar{\theta}\}|$ is fixed to 7.

Figure A.5 shows that both collaboration policies for one-sided asymmetric information sharing do not react in the same way to the position of $\theta^*$ within set $\{\underline{\theta}, \dots, \bar{\theta}\}$. Hence, let us discuss the results under the lens of the estimated parameter:

* **Inventory holding cost of the main product ($h^2, h^1$):** The more the leader underestimates $h^2$, the more the gain of the follower decreases. On the contrary, when the leader overestimates the gains, she/he leads to an increase of the gain of the follower, up to 20% when $\theta^* = \underline{\theta}$. The behavior of the gain of the follower is symmetric when the unknown is $h^1$ and when PU2 Leader.
<figure><img src="image_7.png" /><figcaption>(a) Relative gains of PU2 under PU1 Leader for unknown parameters $h^2, TBO^2, q-b^2$ against No_Co</figcaption></figure>

<caption>(b) Relative gains of PU1 under PU2 Leader for unknown parameters $h^1, TBO^1, g-b^1, \hat{h}$ against No_Co</caption>

<caption>Figure A.5: Policy $PU_i\_Leader$ ($i \in \{1, 2\}$) versus policy $No\_Co$ ($|\Theta| = 7$): Follower gains (in %)</caption>

* **Position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$:** $p_{\theta^*}$ is the position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$. Let $p_{\theta^*}$ take its values in $\{1, 2, \dots, |\{\underline{\theta}, \dots, \bar{\theta}\}|\}$. Range $\{\underline{\theta}, \dots, \bar{\theta}\}$ is shift to the right or the left depending on $p_{\theta^*}$. For example, when $|\Theta| = 3$ and $p_{\theta^*} = 2$, the range of the unknown parameter is $\{\underline{\theta}, \theta^*, \bar{\theta}\}$.

Note that, the cost functions estimating the unknown parameters are strictly increasing, thus the position of $\theta^*$ in set $\{\underline{\theta}, \dots, \bar{\theta}\}$ corresponds to the scenario number and $|\Theta| = |\{\underline{\theta}, \dots, \bar{\theta}\}|$.

*Position of true value $\theta^*$ within the domain of estimation $\{\underline{\theta}, \dots, \bar{\theta}\}$:* Extensive numerical experiments have shown that the economic benefit of the leader, the environmental impact, and the side payment are not impacted by the position of the true value of the unknown parameter within $\{\underline{\theta}, \dots, \bar{\theta}\}$. Figure A.5 highlights the economic benefit of the follower in collaboration policies *PU1 Leader* and *PU2 Leader* against *No_Co* as a function of the position of the true value within $\{\underline{\theta}, \dots, \bar{\theta}\}$ for all possible unknown parameters. Due to the high computational time needed to conduct experiments, results are provided for a subset of 733 randomly selected instances and $|\{\underline{\theta}, \dots, \bar{\theta}\}|$ is fixed to 7.

Figure A.5 shows that both collaboration policies for one-sided asymmetric information sharing do not react in the same way to the position of $\theta^*$ within set $\{\underline{\theta}, \dots, \bar{\theta}\}$. Hence, let us discuss the results under the lens of the estimated parameter:

* **Inventory holding cost of the main product ($h^2, h^1$):** The more the leader underestimates $h^2$, the more the gain of the follower decreases. On the contrary, when the leader overestimates the gains, she/he leads to an increase of the gain of the follower, up to 20% when $\theta^* = \underline{\theta}$. The behavior of the gain of the follower is symmetric when the unknown is $h^1$ and when *PU2.Leader*.
* **Time between orders (TBO¹, TBO²):** When PU1 Leader, the position of θ* in set Θ has no impact on the gain of the follower. It is distributed between 0% and 20%. Under PU2 Leader, the distribution of the gain of the follower is almost the same except when closing θ. When θ* = θ, the average gain of PU1 is slightly higher than 0.

* **Gain of reusing the by-product ($q - b^2$, $g - b^1$):** The more the leader underestimates $q - b^2$ and $g - b^1$, the more the gain increases, up to almost 20%. Note that, in case of underestimation, the gain of reusing the by-product is almost always strictly greater than 0. On the contrary, when the leader overestimates the gains, she/he leads to a decrease in the gain of the follower, which is always between 0% and 1% when $θ^* = \bar{θ}$.

* **Inventory holding cost of the by-product when PU2 Leader ($\hat{h}$):** When PU2 underestimates $\hat{h}$ (especially when $θ^* = \bar{θ}$), the average gain of the follower is negligible and this last one is legitimatized to questions the well-founded of the collaboration for him/her. On the contrary, when PU2 overestimates the by-product inventory holding cost, the economic benefit increases for PU1 but it never exceeds 5%.

In view of these findings and to avoid any bias, let us position $θ^*$ in the middle of the range of estimation in what follows.

Range and cardinality of estimation domain {$\underline{\theta}, \dots, \bar{\theta}$}: Table A.7 summarizes the gaps between PU1 Leader (resp. PU2 Leader) and collaboration policy No_Co for all unknown parameters {$TBO^2, TBO^1, h^1, h^2, q - b^2, g - b^1, \hat{h}$}, and for different ranges and cardinalities of estimation domains. This gap is calculated as follows: $100 \times (c^1 + c^2 - c_p^1 - c_p^2)/(c^1 + c^2)$, where $p \in \{\text{PU1 Leader, PU2 Leader}\}$ and $c^i$ is the nominal cost of PU $i \in \{1, 2\}$.

Table A.7 shows a number of empirical pieces of evidence of the impact of the range and cardinality of estimation domains on game theory-based collaboration policies. When the unknown parameter is the gain of reusing the by-product (i.e., $g - b^1$ or $q - b^2$), the gap to No_Co is the highest. We can also notice that the number of values in the estimation domain has a low impact on the global economic gain of the industrial symbiosis network. On the contrary, a large range leads to a significant difference in terms of gain. The more the range is tight, the lower the economic gain is.

<table><caption>Table A.7: Gaps between game theory based collaboration policies and No_Co (in %) for unknown parameters {$TBO^2, TBO^1, h^1, h^2, q - b^2, g - b^1, \hat{h}$} ($\theta^*$ in the middle of the estimation range)</caption><thead><tr><th></th><th>θ</th><th colspan="3">h<sup>2</sup></th><th colspan="3">TBO<sup>2</sup></th><th colspan="3">q - b<sup>2</sup></th></tr></thead><tbody><tr><th>PU1 Leader</th><th>Δ<sub>θ</sub></th><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td></tr><tr><th></th><th>|Θ|</th><td>3</td><td>4.2</td><td>4.6</td><td>4.9</td><td>4.2</td><td>4.6</td><td>4.4</td><td>4.3</td><td>5.1</td><td>5.8</td></tr><tr><th></th><th></th><td>7</td><td>4.2</td><td>4.6</td><td>4.9</td><td>4.1</td><td>5.0</td><td>5.1</td><td>4.3</td><td>5.1</td><td>5.8</td></tr><tr><th></th><th>θ</th><th colspan="3">h<sup>1</sup></th><th colspan="3">TBO<sup>1</sup></th><th colspan="3">g - b<sup>1</sup></th><th>ĥ</th></tr><tr><th>PU2 Leader</th><th>Δ<sub>θ</sub></th><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td></tr><tr><th></th><th>|Θ|</th><td>3</td><td>3.5</td><td>3.9</td><td>4.2</td><td>3.4</td><td>3.8</td><td>3.5</td><td>3.8</td><td>4.8</td><td>5.7</td><td>3.2</td><td>3.3</td><td>3.4</td></tr><tr><th></th><th></th><td>7</td><td>3.5</td><td>4.0</td><td>4.2</td><td>3.5</td><td>4.2</td><td>4.2</td><td>3.8</td><td>4.8</td><td>5.7</td><td>3.2</td><td>3.3</td><td>3.4</td></tr></tbody></table>
* **Time between orders (TBO¹, TBO²):** When **PU1 Leader**, the position of $\theta^*$ in set $\Theta$ has no impact on the gain of the follower. It is distributed between 0% and 20%. Under **PU2 Leader**, the distribution of the gain of the follower is almost the same except when closing $\underline{\theta}$. When $\theta^* = \underline{\theta}$, the average gain of PU1 is slightly higher than 0.

* **Gain of reusing the by-product ($q-b^2, g-b^1$):** The more the leader underestimates $q-b^2$ and $g-b^1$, the more the gain increases, up to almost 20%. Note that, in case of underestimation, the gain of reusing the by-product is almost always strictly greater than 0. On the contrary, when the leader overestimates the gains, she/he leads to a decrease in the gain of the follower, which is always between 0% and 1% when $\theta^* = \bar{\theta}$.

* **Inventory holding cost of the by-product when PU2 Leader ($\hat{h}$):** When PU2 underestimates $\hat{h}$ (especially when $\theta^* = \bar{\theta}$), the average gain of the follower is negligible and this last one is legitimatized to questions the well-founded of the collaboration for him/her. On the contrary, when PU2 overestimates the by-product inventory holding cost, the economic benefit increases for PU1 but it never exceeds 5%.

In view of these findings and to avoid any bias, let us position $\theta^*$ in the middle of the range of estimation in what follows.

Range and cardinality of estimation domain {$\underline{\theta}, \dots, \bar{\theta}$}: Table A.7 summarizes the gaps between **PU1 Leader** (resp. **PU2 Leader**) and collaboration policy **No-Co** for all unknown parameters {$TBO^2, TBO^1, h^1, h^2, q-b^2, g-b^1, \hat{h}$}, and for different ranges and cardinalities of estimation domains. This gap is calculated as follows: $100 \times (c^1 + c^2 - c_p^1 - c_p^2)/(c^1 + c^2)$, where $p \in \{\text{PU1 Leader, PU2 Leader}\}$ and $c^i$ is the nominal cost of PU $i \in \{1, 2\}$.

Table A.7 shows a number of empirical pieces of evidence of the impact of the range and cardinality of estimation domains on game theory-based collaboration policies. When the unknown parameter is the gain of reusing the by-product (i.e., $g-b^1$ or $q-b^2$), the gap to **No-Co** is the highest. We can also notice that the number of values in the estimation domain has a low impact on the global economic gain of the industrial symbiosis network. On the contrary, a large range leads to a significant difference in terms of gain. The more the range is tight, the lower the economic gain is.

<table><caption>Table A.7: Gaps between game theory based collaboration policies and No-Co (in %) for unknown parameters {$TBO^2, TBO^1, h^1, h^2, q-b^2, g-b^1, \hat{h}$} ($\theta^*$ in the middle of the estimation range)</caption><thead><tr><th></th><th>θ</th><th colspan="3">h<sup>2</sup></th><th colspan="3">TBO<sup>2</sup></th><th colspan="3">q-b<sup>2</sup></th></tr></thead><tbody><tr><th rowspan="3" scope="rowgroup">PU1 Leader</th><th scope="row">Δ<sub>θ</sub></th><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td></tr><tr><th scope="row">|Θ|</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th scope="row">3</th><td>4.2</td><td>4.6</td><td>4.9</td><td>4.2</td><td>4.6</td><td>4.4</td><td>4.3</td><td>5.1</td><td>5.8</td></tr><tr><th scope="row"></th><th scope="row">7</th><td>4.2</td><td>4.6</td><td>4.9</td><td>4.1</td><td>5.0</td><td>5.1</td><td>4.3</td><td>5.1</td><td>5.8</td></tr><tr><th rowspan="3" scope="rowgroup">PU2 Leader</th><th scope="row">θ</th><th colspan="3">h<sup>1</sup></th><th colspan="3">TBO<sup>1</sup></th><th colspan="3">g-b<sup>1</sup></th></tr><tr><th scope="row">Δ<sub>θ</sub></th><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td><td>0.2</td><td>0.5</td><td>0.8</td></tr><tr><th scope="row">|Θ|</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th scope="row"></th><th scope="row">3</th><td>3.5</td><td>3.9</td><td>4.2</td><td>3.4</td><td>3.8</td><td>3.5</td><td>3.8</td><td>4.8</td><td>5.7</td></tr><tr><th scope="row"></th><th scope="row">7</th><td>3.5</td><td>4.0</td><td>4.2</td><td>3.5</td><td>4.2</td><td>4.2</td><td>3.8</td><td>4.8</td><td>5.7</td></tr></tbody></table>
To summarize, even if the position of $\theta^*$ within $\{\theta, \dots, \bar{\theta}\}$ does not have an impact on the cost of the leader, gains of the follower are more or less high when $\theta^*$ is an endpoint of the estimation domain depending on the unknown parameter. The leader has to be careful when defining the domain of estimation because additional values can have a negative global effect.

## Appendix A.2. Negotiation procedure managed by a mediator (Algorithm 1)

In this section, we perform several experiments to analyze the contracts negotiated by a mediator in the case of bilateral asymmetric information sharing. The goal of this section is to evaluate the impact of the critical parameters and to show the industrial soundness of the proposed approach. To do this, we carry out the comparison between the following approaches:

* **TBO_Estim**: Negotiation procedure (Algorithm 1).

* **Monte_Carlo**: Negotiation procedure (Algorithm 1) without Phase 1, i.e., based only on the Monte Carlo sampling procedure. It is a crude approach, which consists in randomly generating a large number of contracts to converge towards efficient and attractive contracts by virtue of the law of large numbers.

* **Full_Co**: Centralized collaboration policy (i.e., Model (2)-(14)) of the ULS-IS problem.

By definition, **Full_Co** provides the best contract that the mediator can propose in terms of economic benefits. Given that, to discuss the competitiveness of **TBO_Estim** against **Monte_Carlo**, we compute for each contract its gap to **Full_Co** using the following formula: $(c_p^1 + c_p^2 - c) \times 100/c$, where $c_p^1$ (resp. $c_p^2$) is the cost of PU1 (resp. PU2) for approach $p \in \{\text{TBO\_Estim}, \text{Monte\_Carlo}\}$, and $c$ is the global cost associated with **Full_Co**. For each problem instance, only the minimal gap to **Full_Co** is kept.

Note that the computational time needed to solve both versions of the negotiation scheme depends linearly on the number of proposed contracts. For instance, the computational time of **TBO_Estim** and **Monte_Carlo** for 50 contracts is about 6 seconds on average.

<figure><img src="image_6.png" /><figcaption>Figure A.6: Average gap to Full_Co (in %)</figcaption></figure>
To summarize, even if the position of $\theta^*$ within $\{\theta, \dots, \bar{\theta}\}$ does not have an impact on the cost of the leader, gains of the follower are more or less high when $\theta^*$ is an endpoint of the estimation domain depending on the unknown parameter. The leader has to be careful when defining the domain of estimation because additional values can have a negative global effect.

## Appendix A.2. Negotiation procedure managed by a mediator (Algorithm 1)

In this section, we perform several experiments to analyze the contracts negotiated by a mediator in the case of bilateral asymmetric information sharing. The goal of this section is to evaluate the impact of the critical parameters and to show the industrial soundness of the proposed approach. To do this, we carry out the comparison between the following approaches:

* **TBO_Estim**: Negotiation procedure (Algorithm 1).

* **Monte_Carlo**: Negotiation procedure (Algorithm 1) without Phase 1, i.e., based only on the Monte Carlo sampling procedure. It is a crude approach, which consists in randomly generating a large number of contracts to converge towards efficient and attractive contracts by virtue of the law of large numbers.

* **Full_Co**: Centralized collaboration policy (i.e., Model (2)-(14)) of the ULS-IS problem.

By definition, Full_Co provides the best contract that the mediator can propose in terms of economic benefits. Given that, to discuss the competitiveness of TBO_Estim against Monte_Carlo, we compute for each contract its gap to Full_Co using the following formula: $(c_p^1 + c_p^2 - c) \times 100/c$, where $c_p^1$ (resp. $c_p^2$) is the cost of PU1 (resp. PU2) for approach $p \in \{\text{TBO\_Estim}, \text{Monte\_Carlo}\}$, and $c$ is the global cost associated with Full_Co. For each problem instance, only the minimal gap to Full_Co is kept.

Note that the computational time needed to solve both versions of the negotiation scheme depends linearly on the number of proposed contracts. For instance, the computational time of TBO_Estim and Monte_Carlo for 50 contracts is about 6 seconds on average.

<figure><img src="image_8.png" /><figcaption>Figure A.6: Average gap to Full_Co (in %)</figcaption></figure>
**TBO_Estim versus Monte_Carlo.** Figure A.6 provides the average gap between Full_Co and both versions of the negotiation scheme (TBO_Estim and Monte_Carlo) for different numbers of proposed contracts for two cases: when the by-product is unstorable ($B = 0$), and when it is storable with a limited capacity ($B/\bar{d}^1 > 0$).

When the by-product is unstorable, the gap of TBO_Estim to Full_Co converges quickly to very small values (below 0.001%). When the by-product is storable with a limited capacity, the gap of TBO_Estim to Full_Co starts from 0.019%, and converges to values below 0.003% after 200 contracts. The gap of TBO_Estim at the first iteration is better than the results obtained with Monte_Carlo after 500 contracts, which reaches 0.011%. To summarize, TBO_Estim allows us to reduce the number of proposed contracts, and consequently the computational time. TBO_Estim provides better results than Monte_Carlo, whatever the storability of the by-product is. Due to the stabilization of the results for TBO_Estim after 200 contracts, the number of contracts is fixed at 200 in the rest of the experiments.

<figure><img src="image_7.png" /><figcaption>Figure A.7: Average gap to Full_Co (in %) depending on $B/\bar{d}^1$, $TBO^1 - TBO^2$ and $\bar{d}^1 - \bar{d}^2$ for 200 contracts</figcaption></figure>

**Focus on the critical parameters.** As illustrated in Figure A.6, the negotiation procedure described in Algorithm 1 is very competitive (the average gap to Full_Co is very close to 0%). Figure A.7 provides the boxplots representing the distribution of the minimal gap of TBO_Estim and Monte_Carlo to Full_Co of each instance, depending on the critical parameters of instances $\{B/\bar{d}^1, TBO^1 - TBO^2, \bar{d}^1 - \bar{d}^2\}$. Note that the results are not discussed with respect to $\Delta$ since its impact on the results is very low, as shown in (Suzanne et al., 2021).

Let us mention, first, that all the gaps to Full_Co are below 0.9% for TBO_Estim and 1.5% for Monte_Carlo. Figure A.7 shows that TBO_Estim is persistently better than Monte_Carlo, and the gap of at least 75% of instances is at 0% for both TBO_Estim and
**TBO_Estim versus Monte_Carlo.** Figure A.6 provides the average gap between Full_Co and both versions of the negotiation scheme (TBO_Estim and Monte_Carlo) for different numbers of proposed contracts for two cases: when the by-product is unstorable ($B = 0$), and when it is storable with a limited capacity ($B/\bar{d}^1 > 0$).

When the by-product is unstorable, the gap of TBO_Estim to Full_Co converges quickly to very small values (below 0.001%). When the by-product is storable with a limited capacity, the gap of TBO_Estim to Full_Co starts from 0.019%, and converges to values below 0.003% after 200 contracts. The gap of TBO_Estim at the first iteration is better than the results obtained with Monte_Carlo after 500 contracts, which reaches 0.011%. To summarize, TBO_Estim allows us to reduce the number of proposed contracts, and consequently the computational time. TBO_Estim provides better results than Monte_Carlo, whatever the storability of the by-product is. Due to the stabilization of the results for TBO_Estim after 200 contracts, the number of contracts is fixed at 200 in the rest of the experiments.

<figure><img src="image_9.png" /><figcaption>Figure A.7: Average gap to Full_Co (in %) depending on $B/\bar{d}^1$, $TBO^1 - TBO^2$ and $\bar{d}^1 - \bar{d}^2$ for 200 contracts</figcaption></figure>

*Focus on the critical parameters.* As illustrated in Figure A.6, the negotiation procedure described in Algorithm 1 is very competitive (the average gap to Full_Co is very close to 0%). Figure A.7 provides the boxplots representing the distribution of the minimal gap of TBO_Estim and Monte_Carlo to Full_Co of each instance, depending on the critical parameters of instances $\{B/\bar{d}^1, TBO^1 - TBO^2, \bar{d}^1 - \bar{d}^2\}$. Note that the results are not discussed with respect to $\Delta$ since its impact on the results is very low, as shown in (Suzanne et al., 2021).

Let us mention, first, that all the gaps to Full_Co are below 0.9% for TBO_Estim and 1.5% for Monte_Carlo. Figure A.7 shows that TBO_Estim is persistently better than Monte_Carlo, and the gap of at least 75% of instances is at 0% for both TBO_Estim and
**Monte_Carlo.** In the following, we focus on the distribution of gaps depending on the critical parameters:

* **By-product inventory capacity:** The more the by-product inventory capacity is close to the average demand of PU1 ($\bar{d}^1$), the higher the gap of TBO_Estim to Full_Co is. For the particular case, where the by-product is unstorable, the gap is below 0.05% for all instances. Note that in this configuration, the difference between the gap of Monte_Carlo and the gap of TBO_Estim is the highest.

* **Setup cost-holding cost ratios TBO¹ and TBO²:** The more TBO¹ and TBO² are different, the higher the gaps to Full_Co can be. We can also notice when TBO¹ > TBO² the gaps are more significant than when TBO¹ < TBO².

* **Average demands $\bar{d}^1$ and $\bar{d}^2$:** When $\bar{d}^1 > \bar{d}^2$, the gap of TBO_Estim to Full_Co is very close to 0% for all instances. When $\bar{d}^1 < \bar{d}^2$, gaps are below 0.3%. Finally, for instances having $\bar{d}^1 = \bar{d}^2$, TBO_Estim provides the worst results even if the gaps of TBO_Estim are better than those provided by Monte_Carlo.
**Monte_Carlo.** In the following, we focus on the distribution of gaps depending on the critical parameters:

* **By-product inventory capacity:** The more the by-product inventory capacity is close to the average demand of PU1 ($\bar{d}^1$), the higher the gap of TBO_Estim to Full_Co is. For the particular case, where the by-product is unstorable, the gap is below 0.05% for all instances. Note that in this configuration, the difference between the gap of Monte_Carlo and the gap of TBO_Estim is the highest.

* **Setup cost-holding cost ratios TBO¹ and TBO²:** The more TBO¹ and TBO² are different, the higher the gaps to Full_Co can be. We can also notice when TBO¹ > TBO² the gaps are more significant than when TBO¹ < TBO².

* **Average demands $\bar{d}^1$ and $\bar{d}^2$:** When $\bar{d}^1 > \bar{d}^2$, the gap of TBO_Estim to Full_Co is very close to 0% for all instances. When $\bar{d}^1 < \bar{d}^2$, gaps are below 0.3%. Finally, for instances having $\bar{d}^1 = \bar{d}^2$, TBO_Estim provides the worst results even if the gaps of TBO_Estim are better than those provided by Monte_Carlo.