Square root law of logistics
An MRP controller puzzle: “How high must the parts inventory be if two decentralized warehouses of the same type are merged into one central warehouse , there are no other relevant changes and the parts inventory in the decentralized warehouses before centralization is 1000 pieces each?” – I often ask this question at the beginning of my training courses for dispatchers.
Take your time and try to solve the problem first using your intuition or “common sense”!
Intuition and common sense tend towards linear solutions and think they have found the solution in 2000 pieces. Others are sure that their intuition is misguided, but are still unable to solve the puzzle and others simply guess. It is extremely rare for a dispatcher – even one with many years of experience – to solve the puzzle straight away.
The first solution to the puzzle – optimal lot-sizing calculation
One possible solution to the MRP controller puzzle is the famous economic order quantity (EOQ) formula by Andler and Wilson/Harris. According to this formula, the optimal lot size is simply the result of minimizing the sum of linear inventory costs and inversely exponentially distributed fixed setup costs (e.g. transport costs, procurement costs). These two cost components together have a minimum at one point, the mathematical derivation of which is the famous EOQ. If, for example, the period requirement (e.g. annual consumption) is doubled, then the EOQ does not increase linearly, i.e. it does not double, but only increases by the root mean square factor 2 (= root of the factor 2), which is approx. 1.41; therefore the correct answer to the MRP controller puzzle is: The parts stock in the new centralized warehouse must comprise approx. 1410 pieces. If, for example, we were to triple the period requirement, i.e. 3 warehouses were merged, then the period requirement would increase by the factor of the square root of 3, i.e. to approx. 1730 pieces. This is why in logistics we speak of the square root law of logistics(Timm Gudehus).
We can see that merging decentralized warehouses into a central warehouse (under otherwise unchanged conditions) results in a cost-effective inventory effect. Of course, there is still a lot to consider in practice. For example, merging warehouses will increase other costs, such as transportation costs. We cannot avoid taking a comprehensive look at each individual case, but without knowledge of the interrelationships between statistics, logistics and supply chain management, managers will not get very far.
The second solution to the puzzle – the law of large numbers
A second possible way of solving the dispatcher puzzle is through analytical statistics. This states that the sum of the scatter of two decentralized locations is always greater than the scatter of the sum of the decentralized locations. And – in model terms – always by the square root of the factor. This is based on the famous statistical “law of nature”, the law of large numbers. Based on this statistical law, it is advantageous, for example, to aggregate the orders received locally for items sold nationwide in a central order collection point or to bring them together virtually in a central computer, analyze them there and schedule further processing. The advantages of such centralized scheduling increase with the size of an item’s sales territories.
For more information on goods supply, see The cost of unused opportunities in supply chain management.
Image source: Wikimedia Commons, Averater, License:(CC0 1.0)