# Pooling of inventory

When there are two or more demand points for a product, it may possible to save money if the separate inventories for these demand points can be combined or "pooled." There is a "square root economy of scale" from pooling for two of the simplest inventory models: a) the Economic Order Quantity (EOQ) model, and b) the Newsboy (NV) model.

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.

There is an interesting anomaly wherein pooling can increase inventory. Specifically, for the NV problem, if instead of there being a constraint on service level, the objective is simply to minimize the expected cost of unsatisfied demand and the cost of surplus inventory, and the demand distributions at each of two demand points (with associated inventory) are right skewed, e.g., the mean is greater than the median, then pooling the two inventories may in fact result in a higher total optimal inventory. For example, if the cost/unit short is 5, the cost/unit of inventory left over is 4, the demands at each of two demand points are independent Poisson distributed with mean 3.3, then the optimal amount to stock at each location is 3. If, however, the demands are pooled and a single inventory is carried so the demand is Poisson with mean 6.6, then the optimal amount to stock is 7. So pooling increased the optimal inventory. See Myths and Counter Examples for an additional example.

# Newsboy problem

A newspaper is concerned with controlling the number of papers to be distributed to newstands. The cost of a paper varies (i.e., Sunday vs. daily), and the demand is a random variable, $LaTeX: q,$ with probability function $LaTeX: P(q).$ Unsold papers are returned, with no salvage value the next day, losing millions of dollars in the production cost. The demand for newspapers is a random variable, with probability function $LaTeX: P(q)$ = probability that demand equals $LaTeX: q.$ It is possible, however, for a newstand to order more papers the same day. There are holding and shortage (penalty) costs. The problem is to determine a reorder policy so as to minimize total expected cost. This problem was used to consider a reorder policy with a 2-parameter decision rule:

1. $LaTeX: s$ = inventory level at which an order is placed;
2. $LaTeX: S$ = inventory level to which to order.

Then, the decision rule to be employed is the following policy:

Order nothing if the inventory of papers is $LaTeX: \ge s;$
Order $LaTeX: S-s$ if there are s papers on hand and $LaTeX: s < S.$

The significance of this problem is that it was used to introduce the notion (and optimality) of an $LaTeX: (s, S)$ policy in inventory theory.