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the final cut by oliver johnson logging the world logging the world subscribe sign in the final cut baby i know the first cut is the deepest oliver johnson dec 01 2024 412 37 43 share there are some maths topics which always do well on twitter patterns with numbers bayes theorem exponential growth wald s aeroplane but that s just the start there s a mathematical theorem which is so simple that i can explain it to you here but which wasn t written down until 1956 once you get your head around it you ll see it everywhere like the glasses in cult classic they live it can genuinely change the way you look at the world by seeing how it helps explain nhs congestion the ukraine war energy policy the vaccine rollout and a lot else besides you ll be ready to chew bubblegum and kick ass metaphorically speaking at least and the slogan i want to give you is to focus on your cuts let me explain subscribe max flow min cut suppose you own a widget manufacturing factory you want to get your widgets to the port there is a network of one way rail lines looking like this the capacity of each line in trains per hour say is marked in a circle at what rate can we manufacture widgets and get them to the port we ll call this the flow as with any maths problem i recommend that you play around with it yourself first have a go at thinking how many widgets you might send down each line hopefully if you do that you will convince yourself that the answer is a flow of 6 trainloads per hour this answer comes via the work of ford and fulkerson and is often referred to as max flow min cut the key idea is that of a cut in the picture below it s the red line drawn across the network with the factory on one side and the port on the other in fact we can think about a cut as defining a set the set of places on the same side of the line as the factory if you add up the ringed numbers on the edges crossing the red cut you get 6 and we ll say 6 is the value of the cut that tells us that we can t get a flow of more than 6 widgets hour through the network as a whole because every strategy must involve widgets crossing this cut line and so this cut limits the rate of the whole system using this argument we know that no flow can be bigger than the value of any cut and so the maximum possible flow can t be bigger than the minimum value of the cuts in fact it turns out you can t find a cut with a lower value than 6 feel free to try so this is the minimum value in this case then we can t get a flow bigger than 6 what s maybe less obvious is that we can achieve a flow of 6 i ve marked one way of doing this in green and you ll notice that all the edges crossing the red cut are at capacity there are other ways to route the widgets through the network to achieve a flow of 6 but in each case the edges across the red cut will be at capacity in fact whatever the network looks like whatever the numbers on the edges we can always find a flow which hits this target the maximum possible flow is equal to the minimum value of the cuts the argument is simple but i ll put it in a footnote for people who don t care about details like this 1 the implications of this are profound we can see that it is pointless thinking only about supply and demand without considering the flow through the network and the constraints that this critical red cut puts on it there would be no point in boosting widget making capacity or the number of ships in the port because the network couldn t cope with that widgets would pile up in the factory or ships would sit idle we need to focus on our cuts in lots of ways this is all obvious but ford and fulkerson only formalised this intuition in 1956 it s a great example of how maths research is an ongoing thing and that all the results that matter weren t invented by the greeks or fermat or whoever centuries ago warfare so far i ve described a scenario where we have to do our best with a network that s given to us but in the real world we aren t just passive observers we can affect the network itself and the joy of the max flow min cut theorem is that it tells you how to do this i ll talk later about some examples where we might try to build capacity to improve the value of the flow by adding more links across the critical cut however we can also think of the situation the other way around where we seek to lower the flow by reducing the value of the critical cut i recently wrote about 617 squadron and their attempts to destroy strategic infrastructure in the second world war you can think of that in precisely these terms by targetting bottlenecks in the transport network you can degrade the whole system by bombing the bielefeld viaduct or the saumur tunnel the flow of german troops and supplies to the front was seriously disrupted and the minimal cut degraded you can see it even more starkly in the ukraine war where the cut can be already on the ground in the form of a river by identifying that geographical constraint and destroying the bridges across it ukraine managed to seriously inhibit flows of russian supplies into the territory they seized in kursk in the summer and are so far managing to hold indeed many forms of warfare whether open or covert can be seen in these mathematical terms by cutting submarine internet cables blowing up the nordstream pipeline or attacking the kerch bridge each side has identified critical cuts and understood that targetting them will have a much greater effect on the flow of resources than indiscriminately bombing in general building the grid we can also think in terms of max flow and min cut when understanding the challenges of decarbonising the electricity grid here the situation is more complicated because there isn t a single source of energy trying to supply a single place where it is consumed instead we need a grid robust enough to link multiple power stations wind and solar farms to homes factories and shops bearing in mind that supply and demand will vary by the minute the day and the season and that links may sometimes go down for maintenance or because of bad weather however many of the max flow min cut principles do still apply uk national grid map taken from https www nationalgrid com electricity transmission network and infrastructure network route maps fundamentally there s no point in building too much wind and hydro generating capacity in scotland if the majority of the uk s demand is in england and there aren t high enough capacity links between the two indeed you can see this problem very clearly on the uk national grid map there s a pretty clear cut more or less along the line of the england scotland border and so the constraints on the two cables crossing that border govern how much electricity can flow from north to south of course the national grid are aware of this and are looking to rectify the problem there are plans afoot to boost the flow across this cut by building a new undersea cable but that s not going to happen overnight and in the meantime the lack of realistic storage capacity means that very often scottish windfarms are being paid for energy which can t be usefully used with the problem only getting worse as more generation capacity comes online nhs capacity we can think about the operation of systems like the nhs in a similar way the previous examples involved some physical network of train tracks roads or wires existing in space however we can abstract this into a more mathematical idea of a virtual network one that doesn t necessarily have a physical manifestation but that we can try to optimise in the same way for example we can think about patients flowing through a network where each edge represents some form of intervention booking an appointment seeing a doctor getting treated we d like to make sure that as many people as possible receive a successful outcome and efficiently navigate through this network this is a bewildering problem but the same principles apply imagine that for a particular form of cancer there is a flow from screening to needing a preliminary biopsy to processing that in the lab to performing an operation if necessary we need to know where the bottleneck is in order to focus on the right cut you could easily imagine a situation where politicians announce boosts in funding for screening or operations which would have no effect on the overall efficiency of the system if the critical constraint is the amount of lab capacity for example in some ways it s encouraging if we can identify the real constraints then we just need to optimise those rather than having to increase funding across the board tim leunig has been thinking about some of these on his substack lately do we need trained doctors to explain procedures and ask patients to get ready or can someone else do it can we have a standby waiting list to balance out missed appointments it would be nice to think that this kind of system level thinking could be put place right across the board project management indeed it feels to me that everyone in managerial or leadership roles should understand their organisation well enough to know where the critical cut is that s preventing it from being more productive which members of the team are most under pressure which resources form the overall constraint by knowing this and targetting improvements in the right place by focusing on the cut the efficiency of the whole system can be boosted in the most telling way reading the kate bingham book it s striking that much of the success of the vaccine taskforce came from thinking in this way by investing in a range of possible vaccines there were multiple routes through the uncertain clinical trials process which could deliver success but bingham and her team were thinking about much more than simply the vaccines themselves she describes confirming we have got 150 million vials stoppers and overseals do we have enough tubular glass to make the vials and do we have enough of the borate silicate sand to make tubular glass every step of the process was considered and optimised from rehearsing carrying vaccines around using pizza boxes to investing in a new type of syringe and training vaccinators even before the vaccines were ready by thinking ahead by identifying each of these possible constraints and thinking how they could be overcome a huge range of possible cuts were improved allowing for the successful flow of patients through the vaccination system in due course if any one of them had gone wrong or not even been considered the whole system could have run far less smoothly of course most of us will never have to manage a system as complicated as the vaccine task force did but nonetheless the same mathematical principles apply even in everyday life we should all be focusing on our cuts and looking for ways to do better thanks for reading logging the world subscribe for free to receive new posts and support my work subscribe 1 suppose we have the best possible flow think about the set of places in the network to which we can increase the flow of widgets a all the edges leaving that set must be at capacity or the set would be bigger than it is b the factory is in this set the port isn t or the flow wouldn t be optimal so this choice of set gives us a cut putting a and b together we can see that this flow is equal to the value of the corresponding cut that tells us that this must be the minimum cut value there can t be a better cut elsewhere because it would be lower than the value of the flow and we are done 412 37 43 share discussion about this post comments restacks tim coote dec 2 2024 liked by oliver johnson am i correct in thinking that the process network has to be a dag any loops would make it impossible to draw a cut in turn this would have an impact on process analysis for situations like nhs where a the process for a presenting individual cannot be defined a priori b some node types e g imaging lab analysis could appear many times on any given route c capacity could have a large variation depending on individuals situations i think it s still a useful framework for such situations but the limitations and changes to approach need to be understood reply share 2 replies by oliver johnson and others andy kelly dec 1 2024 liked by oliver johnson regarding your bit on the nhs it s worth reading the evaluation of a programme that sought to do just what you described taking a process and breaking it down to its components to reduce steps improve efficiency etc there s an online copy at https www elft nhs uk sites default files import news re engineering_leicester_royal_infirmary pdf reply share 1 reply by oliver johnson 35 more comments top latest discussions no posts ready for more subscribe 2026 oliver johnson privacy terms collection notice start your substack get the app substack is the home for great culture this site requires javascript to run correctly please turn on javascript or unblock scripts