Pooling of inventory

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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.

Fixed chargebutton.png

A cost that is some value, say LaTeX: C, regardless of the level as long as the level is positive; otherwise the fixed charge is zero. This is represented by LaTeX: Cv, where LaTeX: v is a binary variable. If LaTeX: v=0, the fixed charge is 0; if LaTeX: v=1, the fixed charge is LaTeX: C. An example is whether to open a plant LaTeX: (v=1) or not LaTeX: (v=0). To apply this fixed charge to the non-negative variable LaTeX: x, the constraint LaTeX: x \le Mv is added to the mathematical program, where LaTeX: M is a very large value, known to exceed any feasible value of LaTeX: x. Then, if LaTeX: v=0 (e.g., not opening the plant that is needed for LaTeX: x > 0), LaTeX: x=0 is forced by the upper bound constraint. If LaTeX: v=1 (e.g., plant is open), LaTeX: x \le Mv is a redundant upper bound. Fixed charge problems are mathematical programs with fixed charges. (See Myths and Counterexamples in Mathematical Programming to avoid a misconception.)

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