Alchemical Theory in v6 (Equivalent Exchange)

Equivalent Exchange version 6, much like its predecessors, is very much a mod focused on exponential gains. The Collector and Relay are capable of producing EMC which can be reinvested in further Collectors and Relays indefinitely. However, there are rules about how they function, and the question about how best to expand one's machine over time is fairly subtle. In this article, I hope to give the basic concepts required to begin evaluating machine designs, and show how some common designs compare with each other.
To begin with, let's look at the machines in isolation.
A fully lit Collector is simple and produces 4*k EMC/s where k is 1, 3, or 6 depending on the level of the Collector (Mk 1, 2 or 3).
The following table summarises the production rate and cost of each collector, along with a very important value, the cost of the device divided by its production rate, which is a quantity in seconds, the amount of time it takes the device to produce itself. We obviously want this to be as low as possible in our designs.
Collector
Production (EMC/s)
Cost (EMC)
Cost/Production (s)
Mark 1
4
82953
20738
Mark 2
12
232969
19414
Mark 3
24
448521
18688
Note that we'll be assuming that the Collectors all have light level 15 on their top face, as is provided by a glowstone block (or an interdiction torch, though those are a good deal more expensive). Mark 3 Collectors and Relays also give off this much light, and so once one can afford them, it becomes sensible to put Collectors beneath other machines. A collector which isn't receiving full light will have its production rate cut proportionally to the light level. It is almost never a good plan to build in this fashion, and the issue disappears as soon as we can afford better equipment, so we'll ignore it, and assume that somehow the Collectors are well-lit. Glowstone blocks cost 54 times less than even a Mark 1 collector, so we can usually fairly safely ignore their cost in calculations without affecting the numbers much.
From the table, we can immediately see that the higher level collectors are more cost effective as well, as the rate at which they produce EMC slightly outpaces the increase in cost. So, upgrading a collector is rarely a bad idea, however, there will be cases where upgrading a Relay instead will be better, so let's have a look at those.
A Relay produces n*k EMC/s where k is again 1, 3, or 6 according to the level of the Relay, and n is the number of adjacent passive Collectors which are sending it EMC to store (that is, not Collectors which are working on upgrading fuels, or charging a Klein Star). This fact, that a Relay's production rate depends on the surrounding machines, is what makes machine layout important to consider.
Because it is possible to surround a Relay on 5 sides while leaving one face still conveniently open for access (for example, to put a Klein Star in to collect the generated EMC), we'll call a Relay with passive Collectors on 5 sides "saturated". It is possible to surround Relays on 6 sides with Collectors, and while this produces more EMC/s, it means that we have to break down our machine in order to collect the EMC to afford more machines. Constantly breaking down and rebuilding the machine is not only tedious, but has a heavy cost in terms of time that the machine is out of operation, and thus EMC. Perhaps once RedPower 2 has a bit more functionality, devices will become possible that automatically break down a machine, collect the Klein Stars from it, and quickly rebuild it again with fresh storage, but for now, this is an unreasonable thing to do for more than one or two Relays.
Let's look at the costs and production for saturated relays:
Saturated Relay
Production (EMC/s)
Cost (EMC)
Cost/Production (s)
Mark 1
5
83402
16680
Mark 2
15
223114
14874
Mark 3
30
428362
14279
One thing which should be immediately pointed out is that even a Mark 1 relay, once saturated, is more cost effective than a Mark 3 Collector, and the cost effectiveness of a saturated Mark 3 Relay is the lowest out of any machine that we can build.
This suggests that we want as many Relays in our machines as possible. However, we need the Collectors in order to keep the Relays saturated and producing 5 EMC/s. Just for the sake of familiarity, let's look at the numbers for Relays with Collectors on only 4 sides, which produce the same amount of EMC/s as a Collector:
Unsaturated (4 side) Relay
Production (EMC/s)
Cost (EMC)
Cost/Production (s)
Mark 1
4
83402
20850
Mark 2
12
223114
18593
Mark 3
24
428362
17848
If we ever have, say, a Relay Mark 3 with Collectors on only 4 sides, then the cost effectiveness of adding one more collector is rather good. The cost of a Mark 3 Collector is
448521 EMC, and in addition to the 24 EMC/s it will produce, we will obtain an additional 6 EMC/s from the Relay, for a total of 30 EMC/s.
Thus the time cost of this is (448521 EMC) / (30 EMC/s) = 14951 s. This will almost always make it a good idea (decrease the overall price for performance of the machine) to fully saturate a Relay that only has 4 of its sides covered by Collectors.
If we ever have a Relay which is only surrounded on 3 (or fewer) sides, the picture begins to look quite grim:
Unsaturated (3 side) Relay
Production (EMC/s)
Cost (EMC)
Cost/Production (s)
Mark 1
3
83402
27801
Mark 2
9
223114
24790
Mark 3
18
428362
23798
These numbers are all worse than the numbers for a single Collector. Any completed design which involves a Relay that has collectors on only 3 or fewer sides would thus be better off having that Relay simply replaced by a Collector.
So this rule that we should have every Relay surrounded on 5 sides by Collectors is actually a very good idea. One of the earliest popular designs, which I'll call the 17/5 Flower takes some advantage of this:
Yellow means Collector, grey means Relay, and the green box in the middle is either a Condenser producing items, or another Collector charging a Klein Star (and providing an additional 24 EMC/s, but no extra Relay bonuses). This will be accessed from beneath. The numbers in the diagram reflect the Condenser design, because it seems to be popular to build it that way. We're assuming that all the devices are upgraded to Mark 3 because it is always cost effective to do so before building additional machines.
One thing which this design suffers from is a very low number of Relays per Collector, which means that it tends toward the higher Collector numbers for self-reproduction time. If we could fit in more Relays per Collector in our design, we could do better. If in some ideal case, we had one saturated Relay for every Collector (which is impossible with a finite machine), the Cost/Production would become the average of that of a Mark 3 Collector and a Mark 3 Relay, which is (18688 + 14279)/2 or about 16484 EMC/s. This number is the best that we can expect to do, because any design which consists of more than 50% Relays will have to start putting Relays next to other Relays, which will leave them not fully saturated, and/or inaccessible. So the 17/5 Flower takes an extra 17 minutes to build itself than this ideal machine. Can we do better?
Well sure! But we will have to give up on having a single collection point, and we may need to build a few more machines to do it...
(to be continued)