Bathtub System Dynamics

A simple bathtub problem “If it takes 10 minutes to fill a 200 litre bathtub and 12 minutes to empty it, how long would it take to fill if there was no drain plug in place?”

The answer requires some logic and a bit of arithmetic, but not too much as this is part of the curriculum for 10 year olds. However, it may still help to soak in a hot bathtub to think it through. What is required is to work out the inflow and outflow rates and to calculate how long it will take to fill the bath at the difference between these – the net flow rate. As shown in the Appendix to this chapter the answer is 60 minutes. As a child the world seemed puzzlingly obsessed with bathtub, wall building and hole digging problems, but no explanation was ever given for why they might be important – other than for passing examinations. Certainly no mention was made of the counter intuitive nature of the answers.

For example, who would have thought that filling the bath would take so long! There were also many unspoken assumptions in these problems. An astute child, for example, might have commented that the outflow rate would not be constant, since as the volume of water in the bath decreased, then so would the outflow rate. A really questioning child might have aspired to challenge who in their right minds would ever be so silly and wasteful as to leave the drain plug out of a bath at all and what a theoretical and unreasonable problem this was. However, having spent most of our lives solving management problems we now realise that such bathtub issues are indeed everywhere in the world and would argue strongly for even more time to be spent on them in school. In particular, it is our experience that not understanding bathtubs and flow rates and, in particular, not recognising or not taking account of draining rates, or deliberately ignoring them, are the ways in which wrong and evasive answers are given to many important questions in business and government and can be the source of many costly mistakes.

Distinguishing between bathtub levels and rates is vital. For example, much political mileage can be made by suggesting that current investment in resources such as police recruitment rates are the highest ever, whilst not mentioning that the number of police in post is actually declining, because the retirement rate is increasing faster than the recruitment rate. In other words the ‘net’ recruitment rate is negative and the level of the ‘bath tub’ of police is actually falling. Advertisers also tend to choose whether to use bathtub levels or rates when promoting products. The fastest growing mobile phone network might sound good even though the bathtub level of users is quite small. Turnover (revenue generation rate) is a well accepted measure of the size and activity of a company, but large turnovers are no guide to profitability. Such an inflow rate to the bathtub of cash means little without knowing the cash outflow rate (costs of the company). Just as bad is not taking into account drainage rates, and our astute child might be comforted to know that leaks are a common feature of all bathtubs. For example, calculating the university intake rate of doctors for training needed to meet future targets for hospital consultants without taking into account later drop-out rates can result in vast undershoot of targets. Ignoring the draining rate means the target level of the bathtub of doctors is never reached. In ordering new parts for assembly lines, production can be seriously lost if topping up of inventories is based on stock shortfalls only, rather than this figure plus the average usage (drainage) rate. A number of major motor companies have found this out the hard way when using the wrong algorithms in their expensive computer automated reorder systems. A similar misconception occurs in admitting people to hospital, where spare beds can be underestimated if based solely on the gap between bed capacity and occupancy without taking into account the discharge rate. In water distribution companies having to live with leaking bathtubs is a way of life as leakage from Victorian pipe systems in the UK are typically over 30%.

In business, perhaps the best known, but least addressed, problem is the waste resulting from allowing talent and knowledge, particularly female talent, to drain away. This often happens in management restructuring, only to find later that the talent is really actually needed. The question in all these cases is not so much how long would a bath with a leak take to fill, but more how to recognise leaks and how to plug leaks or how to regulate inflow rates to compensate for leaks that cannot be repaired. Sometimes we focus entirely on bathtub inflow rates and imagine we will never have to think about reducing accumulations in bathtubs. Most of us are hoarders to some degree and will collect books and clothes until book shelves and wardrobes are full – and then buy more storage (bigger bathtubs). As societies, our approach to waste management through landfill and our filling of near earth orbits with satellites are other examples. There is much to be learned from nature’s balanced approach to replacement and the evolution of materials which degrade. In climate change, because we are increasing the rates of emission of greenhouse gases into the atmosphere far faster than the rate at which the atmosphere can deal with them, the bath tub level (volume) of such gases is increasing rapidly. It is both interesting and depressing to realise that even if we stopped all green house emissions today the gases already in the bathtub of the atmosphere would take hundreds of years to dissipate and for the planet to repair itself. In other cases it is not the inflow to bathtubs which we need to address, but the bathtub outflow. In manufacturing, production rates, which draw down the bathtub of orders (order backlog), need to be based on new order rates to keep order backlog bathtubs at a steady level. Focussing entirely on outflows of course means depleting bathtubs, as we do in closing down sales and as we do in depleting natural resources. Often we are able to influence both inflows and outflows. In management either can be supply or demands and we often have to combine demand management with supply provision.

All these examples emphasise the true meaning of sustainability of bathtub levels, which is to keep bathtub inflows and outflows in balance. An interesting thought is what governs the point in any situation at which we become aware of imbalance and try to do something about it. This is usually when resources dwindle and costs rise and we are forced to account for items on our profit and loss sheets that were previously considered free. However, waiting for market forces to solve our problems does not take into account delays in perception and action and is usually much too late. Bathtub thinking is about spotting bathtubs and their levels and adjusting flows at a stage early enough to prevent unnecessary wastage of time and resources. It is also about spotting the unintended consequences of actions and designing adjustments to flow rates which trade off their benefits and costs.