 # Pooling of inventory

For the EOQ case, where we want to minimize the fixed charge of ordering plus the holding cost of inventory carried, if we have two separate inventories for two demand points, each with constant demand rate $LaTeX: D,$ fixed charge of ordering $LaTeX: K,$ and holding cost/unit of inventory $LaTeX: h,$ then the minimum total cost is $LaTeX: \textstyle \sqrt{2\, D\, K\, h}$ at each demand point for a total cost of $LaTeX: \textstyle 2\, \sqrt{2\, D\, K\, h}.$ If we combine the two demand streams and carry a single inventory for them, then the minimum total cost is $LaTeX: \textstyle \sqrt{2\, (2\, D)\, K\, h} = 2\, \sqrt{D\, K\, h},$ which decreases the total cost by a factor of $LaTeX: \textstyle 1/\sqrt{2}.$
For the NV case, if we have two demands, each independently Normal distributed with mean $LaTeX: D$ and standard deviation $LaTeX: \sigma,$ a specified service level $LaTeX: \textstyle \alpha = \prob \left \{\mbox{Demand} < \mbox{inventory}\right\}$ at an inventory point, and $LaTeX: z_\alpha$ being the number of standard deviations corresponding to a left tail probability of $LaTeX: \alpha,$ then we must carry inventory $LaTeX: \textstyle D + z_\alpha \sigma$ at each inventory point for a total inventory of $LaTeX: \textstyle 2D + 2z_\alpha \sigma.$ If we can combine the two demands so we carry just one inventory, then the mean and standard deviation for the combined demands are $LaTeX: 2D$ and $LaTeX: \sqrt{2}\sigma.$ Thus, the total inventory we must carry is $LaTeX: \textstyle 2D + \sqrt{2}z_\alpha\sigma,$ and there is a square root economy of pooling in the amount of safety stock we must carry. If we say that the system service level should be $LaTeX: \alpha,$ then the benefits of pooling are even greater.