Tutorial:Quality upcycling math - Revision history

707090: Show that the matrix L is raised to the 100th power in the equation for the last matrix

2026-01-22T22:56:59Z

Show that the matrix L is raised to the 100th power in the equation for the last matrix

← Older revision		Revision as of 22:56, 22 January 2026
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	<math>		<math>
	\begin{align}		\begin{align}
	\textbf{L}		\textbf{L}^{100}
	&=		&=
	\begin{bmatrix}		\begin{bmatrix}

Cardinal: Adopted the normative term.（common → normal）

2026-01-14T02:47:47Z

Adopted the normative term.（common → normal）

← Older revision		Revision as of 02:47, 14 January 2026
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	=== Quality Matrix ===		=== Quality Matrix ===
	When any machine with a quality chance operates on its ingredients it will have a chance to increase its quality with the probability mentioned above. We can model this quality increase chance as a transformation matrix. ~~Common~~ outputs only have one term because they are only created from ~~common~~ inputs which fail to increase in quality since Uncommon items and above can only increase in quality. Uncommon outputs are created from uncommon inputs which failed to increase in quality, as well as all ~~common~~ inputs which got a single rarity upgrade. This equation repeats for the higher tiers, summing up the chances of a jump from lower tiers as well inputs which failed to upgrade from its own teir. Additionally, the number of outputs can be increased with machine productivity. The quantity of outputs from each tier machine can be multiplied by the productivity of the machine.		When any machine with a quality chance operates on its ingredients it will have a chance to increase its quality with the probability mentioned above. We can model this quality increase chance as a transformation matrix. Normal outputs only have one term because they are only created from normal inputs which fail to increase in quality since Uncommon items and above can only increase in quality. Uncommon outputs are created from uncommon inputs which failed to increase in quality, as well as all normal inputs which got a single rarity upgrade. This equation repeats for the higher tiers, summing up the chances of a jump from lower tiers as well inputs which failed to upgrade from its own teir. Additionally, the number of outputs can be increased with machine productivity. The quantity of outputs from each tier machine can be multiplied by the productivity of the machine.

	Inputs starting at each tier have different potential upgrade upside since quality is capped at legendary. This means that it is optimal to vary the ratio of productivity to quality modules used in each tier of machine (<math>p_i</math> will not be the same for all tiers).		Inputs starting at each tier have different potential upgrade upside since quality is capped at legendary. This means that it is optimal to vary the ratio of productivity to quality modules used in each tier of machine (<math>p_i</math> will not be the same for all tiers).
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	\end{bmatrix}</math>		\end{bmatrix}</math>

	At this point in the tutorial I want to review what this all means for those of you who are new or rusty on your matrix algebra. This transformation matrix represents the ratio of outputs of each rarity that will be produced for every input. Each column represents the input rarity. Remember in our equations that we derived this matrix from, ~~common~~ items were always the first term. Each row represents the output rarity of the items after the machine has processed it. So for example the at <math>0.005</math> at <math>A_{3,1}</math> means that for every ~~common~~ input, we will get 0.005 rare (because it is in row 3) outputs. The beautiful thing about this transformation matrix is that it captures every transition chance from any rarity to any other rarity all at once when something is processed by the machine (a recycler with 4 legendary quality module 3s in this case).		At this point in the tutorial I want to review what this all means for those of you who are new or rusty on your matrix algebra. This transformation matrix represents the ratio of outputs of each rarity that will be produced for every input. Each column represents the input rarity. Remember in our equations that we derived this matrix from, normal items were always the first term. Each row represents the output rarity of the items after the machine has processed it. So for example the at <math>0.005</math> at <math>A_{3,1}</math> means that for every normal input, we will get 0.005 rare (because it is in row 3) outputs. The beautiful thing about this transformation matrix is that it captures every transition chance from any rarity to any other rarity all at once when something is processed by the machine (a recycler with 4 legendary quality module 3s in this case).

	=== Combined model ===		=== Combined model ===
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	</math>		</math>

	Because products can only get higher quality, or be destroyed by the recycler's negative productivity, all products will eventually either become legendary or be destroyed by the recycler. We can see what the ratio of our inputs become legendary be assessing the ratio of items in different tiers after many cycles though the loop. If we choose a sufficiently high iteration <math>k = 100</math> the ratio of items left in the ~~common~~-epic rows will be nearly zero.		Because products can only get higher quality, or be destroyed by the recycler's negative productivity, all products will eventually either become legendary or be destroyed by the recycler. We can see what the ratio of our inputs become legendary be assessing the ratio of items in different tiers after many cycles though the loop. If we choose a sufficiently high iteration <math>k = 100</math> the ratio of items left in the normal-epic rows will be nearly zero.

	==== Example ====		==== Example ====
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	</math>		</math>

	When we raise this loop matrix to the 100th power to see what would happen to each ~~common~~ item after putting the outputs back into the upcycler 100 times, we get this:		When we raise this loop matrix to the 100th power to see what would happen to each normal item after putting the outputs back into the upcycler 100 times, we get this:

	<math>		<math>
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	</math>		</math>

	This this shows us what would happen to inputs of each rarity after 100 cycles. The number in the bottom left corner (0.013 in our example), which comes from the far left column (~~common~~ input) and bottom row (legendary output), shows how many legendary outputs will be created for every ~~common~~ input. We can use an program to try out different ratios of productivity to quality for each stage to try to maximize this value.		This this shows us what would happen to inputs of each rarity after 100 cycles. The number in the bottom left corner (0.013 in our example), which comes from the far left column (normal input) and bottom row (legendary output), shows how many legendary outputs will be created for every normal input. We can use an program to try out different ratios of productivity to quality for each stage to try to maximize this value.

707090: The original article is very high level and low on explanation for the way the math works. I have attempted to make the article easier to follow along for those who are rusty on their matrix algebra

2026-01-13T01:08:16Z

The original article is very high level and low on explanation for the way the math works. I have attempted to make the article easier to follow along for those who are rusty on their matrix algebra

Show changes

707090: Update all the icons to represent level 3 modules (as that is what the article is talking about)

2026-01-12T19:48:05Z

Update all the icons to represent level 3 modules (as that is what the article is talking about)

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Cardinal: /* Mathematical model */ Adopted the normative term.

2025-12-08T00:51:14Z

Mathematical model: Adopted the normative term.

← Older revision		Revision as of 00:51, 8 December 2025
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	The mathematical model is time discrete. As opposed to dealing with derivatives in respect to time, the next state is a direct function of the previous state.		The mathematical model is time discrete. As opposed to dealing with derivatives in respect to time, the next state is a direct function of the previous state.

	* <math>m_{i,k}</math> ... Number of materials of tier <math>i</math> (whereas 1 is ~~common~~, 2 is uncommon, 3 is rare, 4 is epic, and 5 is legendary) after the <math>k</math>-th iteration before being crafted. (this doesn't mean that an item only needs one type of ingredient, but that "1 materials" can be crafted into 1 item from them)		* <math>m_{i,k}</math> ... Number of materials of tier <math>i</math> (whereas 1 is normal, 2 is uncommon, 3 is rare, 4 is epic, and 5 is legendary) after the <math>k</math>-th iteration before being crafted. (this doesn't mean that an item only needs one type of ingredient, but that "1 materials" can be crafted into 1 item from them)
	* <math>n_{i,k}</math> ... Number of items of tier <math>i</math> after the <math>k</math>-th iteration, after being crafted together.		* <math>n_{i,k}</math> ... Number of items of tier <math>i</math> after the <math>k</math>-th iteration, after being crafted together.
	* <math>p_0</math> ... the crafting machines inherent productivity bonus		* <math>p_0</math> ... the crafting machines inherent productivity bonus
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	<math>		<math>
	\begin{align}		\begin{align}
	\text{~~common~~ material:}\quad m_{1,k+1} &= \frac{n_{1,k}}{4} \cdot (1 - q_r)\\		\text{normal material:}\quad m_{1,k+1} &= \frac{n_{1,k}}{4} \cdot (1 - q_r)\\
	\text{uncommon material:}\quad m_{2,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{2,k}}{4} \cdot (1 - q_r)\\		\text{uncommon material:}\quad m_{2,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{2,k}}{4} \cdot (1 - q_r)\\
	\text{rare material:}\quad m_{3,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.09 &+& \frac{n_{2,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{3,k}}{4} \cdot (1 - q_r)\\		\text{rare material:}\quad m_{3,k+1} &= \frac{n_{1,k}}{4} \cdot q_r \cdot 0.09 &+& \frac{n_{2,k}}{4} \cdot q_r \cdot 0.9 &+& \frac{n_{3,k}}{4} \cdot (1 - q_r)\\
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	<math>		<math>
	\begin{align}		\begin{align}
	\text{~~common~~ item:}\quad n_{1,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot (1 - q_1)\\		\text{normal item:}\quad n_{1,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot (1 - q_1)\\
	\text{uncommon items:}\quad n_{2,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.9 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot (1 - q_2)\\		\text{uncommon items:}\quad n_{2,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.9 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot (1 - q_2)\\
	\text{rare items:}\quad n_{3,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.09 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot q_2 \cdot 0.9 &+& m_{3,k+1} \cdot ( 1 + p_3 ) \cdot (1 - q_3) \\		\text{rare items:}\quad n_{3,k+1} &= m_{1,k+1} \cdot ( 1 + p_1 ) \cdot q_1 \cdot 0.09 &+& m_{2,k+1} \cdot ( 1 + p_2 ) \cdot q_2 \cdot 0.9 &+& m_{3,k+1} \cdot ( 1 + p_3 ) \cdot (1 - q_3) \\

Cardinal: Improved layout&wording.

2025-12-06T09:16:34Z

Improved layout&wording.

Show changes

Bilka: Don't use tabber

2025-03-05T16:19:43Z

Don't use tabber

← Older revision		Revision as of 16:19, 5 March 2025
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	The table below shows the best ratio for [[Quality module 3\|quality]] to [[Productivity module 3\|productivity modules]] in the crafting machines, while the [[Recycler\|recyclers]] always take only quality modules. The values are not given in whole numbers because often it is not just a single crafting machine per tier that will be used, then the ratios can change between different crafting machines in the same tier. e.g. "3.67 quality / 1.33 productivity" could have 4 machines where 3 have a ratio 4 to 1, and one a ratio 3 to 2.		The table below shows the best ratio for [[Quality module 3\|quality]] to [[Productivity module 3\|productivity modules]] in the crafting machines, while the [[Recycler\|recyclers]] always take only quality modules. The values are not given in whole numbers because often it is not just a single crafting machine per tier that will be used, then the ratios can change between different crafting machines in the same tier. e.g. "3.67 quality / 1.33 productivity" could have 4 machines where 3 have a ratio 4 to 1, and one a ratio 3 to 2.

	~~<tabber>~~
	~~common modules =~~
	The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.		The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
	{\| class="wikitable"		{\| class="wikitable"
			\|+ common modules
	! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*		! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*
	\|-		\|-
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	\| Cryogenic plant \|\| 6 quality <br />2 productivity \|\| 6 quality <br />2 productivity \|\| 6 quality <br />2 productivity \|\| 6.5 quality <br />1.5 productivity \|\| 0 quality <br />8 productivity \|\| 0.119134%\|\| 840		\| Cryogenic plant \|\| 6 quality <br />2 productivity \|\| 6 quality <br />2 productivity \|\| 6 quality <br />2 productivity \|\| 6.5 quality <br />1.5 productivity \|\| 0 quality <br />8 productivity \|\| 0.119134%\|\| 840
	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	~~\|-\|~~
	~~uncommon modules =~~
	~~The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ uncommon modules
	! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*		! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*
	\|-		\|-
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	\| Cryogenic plant \|\| 4.6 quality <br />3.4 productivity \|\| 4.6 quality <br />3.4 productivity \|\| 4.67 quality <br />3.33 productivity \|\| 5 quality <br />3 productivity \|\| 0 quality <br />8 productivity \|\| 0.257621% \|\| 389		\| Cryogenic plant \|\| 4.6 quality <br />3.4 productivity \|\| 4.6 quality <br />3.4 productivity \|\| 4.67 quality <br />3.33 productivity \|\| 5 quality <br />3 productivity \|\| 0 quality <br />8 productivity \|\| 0.257621% \|\| 389
	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	~~\|-\|~~
	~~rare modules =~~
	~~The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ rare modules
	! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*		! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*
	\|-		\|-
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	\| Cryogenic plant \|\| 3.6 quality <br />4.4 productivity \|\| 3.6 quality <br />4.4 productivity \|\| 3.6 quality <br />4.4 productivity \|\| 3.9 quality <br />4.1 productivity \|\| 0 quality <br />8 productivity \|\| 0.565030% \|\| 177		\| Cryogenic plant \|\| 3.6 quality <br />4.4 productivity \|\| 3.6 quality <br />4.4 productivity \|\| 3.6 quality <br />4.4 productivity \|\| 3.9 quality <br />4.1 productivity \|\| 0 quality <br />8 productivity \|\| 0.565030% \|\| 177
	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	~~\|-\|~~
	~~epic modules =~~
	~~The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ epic modules
	! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*		! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*
	\|-		\|-
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	\| Cryogenic plant \|\| 2.6 quality <br />5.4 productivity \|\| 2.6 quality <br />5.4 productivity \|\| 2.6 quality <br />4.4 productivity \|\| 2.8 quality <br />4.2 productivity \|\| 0 quality <br />8 productivity \|\| 1.122444% \|\| 90		\| Cryogenic plant \|\| 2.6 quality <br />5.4 productivity \|\| 2.6 quality <br />5.4 productivity \|\| 2.6 quality <br />4.4 productivity \|\| 2.8 quality <br />4.2 productivity \|\| 0 quality <br />8 productivity \|\| 1.122444% \|\| 90
	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	~~\|-\|~~
	~~legendary modules=~~
	~~The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ legendary modules
	! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*		! Crafting machine \|\| Modules for common items \|\| Modules for uncommon items \|\| Modules for rare items \|\| Modules for epic \|\| Modules for legendary \|\| Percentage yield \|\| Items recycled*
	\|-		\|-
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	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.		<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	~~</tabber>~~

	=== Number of crafting machines ===		=== Number of crafting machines ===
	If we assume a constant input stream of uncommon items, which will always fill back up, we can additionally figure out what ratio of items will be inside the system at once, and with that we can figure out how many crafting machines we need per tier of quality. This is done by setting <math>m_{1,k+1} = 100 \, \% - m_{2,k+1} - m_{3,k+1} - m_{4,k+1}</math> after each iteration and further adjusting for the crafting machines change in speed dependant on the modules (assuming the machines which only house productivity modules are not additionally boosted by [[Speed module 3\|speed moduled]] [[beacon]]s). See the calculations further below for a full explanation of the calculations.		If we assume a constant input stream of uncommon items, which will always fill back up, we can additionally figure out what ratio of items will be inside the system at once, and with that we can figure out how many crafting machines we need per tier of quality. This is done by setting <math>m_{1,k+1} = 100 \, \% - m_{2,k+1} - m_{3,k+1} - m_{4,k+1}</math> after each iteration and further adjusting for the crafting machines change in speed dependant on the modules (assuming the machines which only house productivity modules are not additionally boosted by [[Speed module 3\|speed moduled]] [[beacon]]s). See the calculations further below for a full explanation of the calculations.

	~~<tabber>~~
	~~per recycler=~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ per recycler
	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
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	\| Cryogenic plant \|\| 1 \|\| 5.23556 \|\| 0.21384 \|\| 2.53374 \|\| 0.1034 \|\| 0.594		\| Cryogenic plant \|\| 1 \|\| 5.23556 \|\| 0.21384 \|\| 2.53374 \|\| 0.1034 \|\| 0.594
	\|}		\|}
	~~\|-\|~~
	~~per legendary crafter (exact) =~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ per legendary crafter (exact)
	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
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	\| Cryogenic plant \|\| 10.8718 \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1		\| Cryogenic plant \|\| 10.8718 \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1
	\|}		\|}
	~~\|-\|~~
	~~per legendary crafter (conservative)=~~
	{\| class="wikitable"		{\| class="wikitable"
			\|+ per legendary crafter (conservative)
	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
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	\| Cryogenic plant \|\| 9 \|\| 41 \|\| 16 \|\| 8 \|\| 4 \|\| 1		\| Cryogenic plant \|\| 9 \|\| 41 \|\| 16 \|\| 8 \|\| 4 \|\| 1
	\|}		\|}
	~~</tabber>~~

	== The crafting machines ==		== The crafting machines ==

Rorshan: fixed broken tabber tabs

2025-02-23T21:46:22Z

fixed broken tabber tabs

← Older revision		Revision as of 21:46, 23 February 2025
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	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.		<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	\-\		\|-\|
	uncommon modules =		uncommon modules =
	The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.		The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
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	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.		<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	\-\		\|-\|
	rare modules =		rare modules =
	The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.		The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
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	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.		<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	\-\		\|-\|
	epic modules =		epic modules =
	The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.		The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
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	\|}		\|}
	<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.		<nowiki>*</nowiki> "items recycled" quantifies how many items need to be crafted and put into the entire upcycler to yield a single legendary item, it doesn't account for those which get recycled multiple times.
	\-\		\|-\|
	legendary modules=		legendary modules=
	The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.		The columns "Modules for X items" denote which modules need to go into the machines that are set to produce items of X quality.
Line 112:		Line 112:
	\| Cryogenic plant \|\| 1 \|\| 5.23556 \|\| 0.21384 \|\| 2.53374 \|\| 0.1034 \|\| 0.594		\| Cryogenic plant \|\| 1 \|\| 5.23556 \|\| 0.21384 \|\| 2.53374 \|\| 0.1034 \|\| 0.594
	\|}		\|}
	\-\		\|-\|
	per legendary crafter (exact) =		per legendary crafter (exact) =
	{\| class="wikitable"		{\| class="wikitable"
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	\| Cryogenic plant \|\| 10.8718 \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1		\| Cryogenic plant \|\| 10.8718 \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1
	\|}		\|}
	\-\		\|-\|
	per legendary crafter (conservative)=		per legendary crafter (conservative)=
	{\| class="wikitable"		{\| class="wikitable"

Cardinal: Added language suffix.

2025-02-22T17:03:59Z

Added language suffix.

← Older revision		Revision as of 17:03, 22 February 2025
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			{{Languages}}
	How do we get the most amount of [[Quality\|legendary]] items out of an upcycling plant?		How do we get the most amount of [[Quality\|legendary]] items out of an upcycling plant?

Untitled7: /* Number of crafting machines */

2024-11-24T11:54:48Z

Number of crafting machines

← Older revision		Revision as of 11:54, 24 November 2024
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	=== Number of crafting machines ===		=== Number of crafting machines ===
	If we assume a constant input stream of uncommon items which will always fill back up, we can additionally figure out what ratio of items will be inside the system at once, and with that we can figure out how many crafting machines we need per tier. This is done by setting <math>m_{1,k+1} = 100 \, \% - m_{2,k+1} - m_{3,k+1} - m_{4,k+1}</math> after each iteration. See the calculations further below for a full explanation of the calculations.		If we assume a constant input stream of uncommon items, which will always fill back up, we can additionally figure out what ratio of items will be inside the system at once, and with that we can figure out how many crafting machines we need per tier of quality. This is done by setting <math>m_{1,k+1} = 100 \, \% - m_{2,k+1} - m_{3,k+1} - m_{4,k+1}</math> after each iteration and further adjusting for the crafting machines change in speed dependant on the modules (assuming the machines which only house productivity modules are not additionally boosted by [[Speed module 3\|speed moduled]] [[beacon]]s). See the calculations further below for a full explanation of the calculations.

	<tabber>		<tabber>
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	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
	\| Chemical plant \|\| 1 \|\| 3.~~4427~~ \|\| 0.~~0290~~ \|\| 1.~~0086~~ \|\| 0.~~0077~~ \|\| 0.~~2979~~		\| Chemical plant \|\| 1 \|\| 4.418130519 \|\| 0.037216657 \|\| 1.294369664 \|\| 0.009779 \|\| 0.431955
	\|-		\|-
	\| Assembling machine 3 \|\| 1 \|\| 3.~~3027~~ \|\| 0.~~0365~~ \|\| 1.~~0657~~ \|\| 0.~~0108~~ \|\| 0.~~3286~~		\| Assembling machine 3 \|\| 1 \|\| 4.733868899 \|\| 0.052316655 \|\| 1.527502978 \|\| 0.015336 \|\| 0.52576
	\|-		\|-
	\| Foundry \|\| 1 \|\| 2.~~9551~~ \|\| 0.~~0563~~ \|\| 1.~~1528~~ \|\| 0.~~0208~~ \|\| 0.~~3653~~		\| Foundry \|\| 1 \|\| 4.314446 \|\| 0.082198 \|\| 1.683088 \|\| 0.03016 \|\| 0.58448
	\|-		\|-
	\| Electromagnetic plant \|\| 1 \|\| 2.~~7239~~ \|\| 0.~~0709~~ \|\| 1.~~1726~~ \|\| 0.~~0301~~ \|\| 0.~~3474~~		\| Electromagnetic plant \|\| 1 \|\| 4.494435 \|\| 0.116985 \|\| 1.93479 \|\| 0.049665 \|\| 0.60795
	\|-		\|-
	\| Cryogenic plant \|\| 1 \|\| 2.~~3798~~ \|\| 0.~~0972~~ \|\| 1.~~1517~~ \|\| 0.~~0470~~ \|\| 0.~~2700~~		\| Cryogenic plant \|\| 1 \|\| 5.23556 \|\| 0.21384 \|\| 2.53374 \|\| 0.1034 \|\| 0.594

	\|}		\|}
	\-\		\-\
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	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
	\| Chemical plant \|\| ~~108~~.~~8978~~ \|\| ~~374~~.~~9134~~ \|\| 43.~~5367~~ \|\| 12.~~7555~~ \|\| 3.~~3856~~ \|\| 1		\| Chemical plant \|\| 90.1223 \|\| 331.8198 \|\| 38.5325 \|\| 11.2893 \|\| 2.9653 \|\| 1
	\|-		\|-
	\| Assembling machine 3 \|\| 70.~~4576~~ \|\| ~~232~~.~~7032~~ \|\| 33.~~9340~~ \|\| 10.~~9500~~ \|\| 3.~~2432~~ \|\| 1		\| Assembling machine 3 \|\| 52.8432 \|\| 208.4632 \|\| 30.3992 \|\| 9.8094 \|\| 2.8783 \|\| 1
	\|-		\|-
	\| Foundry \|\| 33.~~0224~~ \|\| 97.~~5872~~ \|\| 21.~~8308~~ \|\| 8.~~5164~~ \|\| 3.~~1550~~ \|\| 1		\| Foundry \|\| 24.7668 \|\| 89.0483 \|\| 19.9206 \|\| 7.7712 \|\| 2.8592 \|\| 1
	\|-		\|-
	\| Electromagnetic plant \|\| 24.~~1178~~ \|\| 65.~~6959~~ \|\| 18.~~4556~~ \|\| 7.~~9448~~ \|\| 3.~~3747~~ \|\| 1		\| Electromagnetic plant \|\| 16.5379 \|\| 61.9418 \|\| 17.4010 \|\| 7.4908 \|\| 3.1822 \|\| 1
	\|-		\|-
	\| Cryogenic plant \|\| 19.~~9316~~ \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1		\| Cryogenic plant \|\| 10.8718 \|\| 47.4350 \|\| 18.2124 \|\| 8.8138 \|\| 4.2654 \|\| 1
	\|}		\|}
	\-\		\-\
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	! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items		! Crafting machine \|\| Recyclers \|\| Machines producing common items \|\| Machines producing uncommon items \|\| Machines producing rare items \|\| Machines producing epic items \|\| Machines producing legendary items
	\|-		\|-
	\| Chemical plant \|\| 92 \|\| ~~318~~ \|\| 37 \|\| 11 \|\| 3 \|\| 1		\| Chemical plant \|\| 53 \|\| 198 \|\| 23 \|\| 7 \|\| 2 \|\| 1
	\|-		\|-
	\| Assembling machine 3 \|\| 62 \|\| ~~205~~ \|\| 30 \|\| 10 \|\| 3 \|\| 1		\| Assembling machine 3 \|\| 31 \|\| 123 \|\| 18 \|\| 6 \|\| 2 \|\| 1
	\|-		\|-
	\| Foundry \|\| 30 \|\| 89 \|\| 20 \|\| 8 \|\| 3 \|\| 1		\| Foundry \|\| 14 \|\| 53 \|\| 12 \|\| 5 \|\| 2 \|\| 1
	\|-		\|-
	\| Electromagnetic plant \|\| 20 \|\| 56 \|\| 16 \|\| 7 \|\| 3 \|\| 1		\| Electromagnetic plant \|\| 14 \|\| 56 \|\| 16 \|\| 7 \|\| 3 \|\| 1
	\|-		\|-
	\| Cryogenic plant \|\| 17 \|\| 41 \|\| 16 \|\| 8 \|\| 4 \|\| 1		\| Cryogenic plant \|\| 9 \|\| 41 \|\| 16 \|\| 8 \|\| 4 \|\| 1
	\|}		\|}
	</tabber>		</tabber>



	== The crafting machines ==		== The crafting machines ==