Principle (Variability Pooling): The variability of combined independent random processes is less than the sum of the variability of the individual processes.
Motivation
Variability pooling is a fundamental concept from the field of statistics. It is the reason that confidence intervals on sample means grow tighter as the sample size is increased. For instance, consider IQ scores, which are normally distributed with a mean of 100 and a standard deviation of 15. By the nature of the normal distribution, this implies that roughly 95% of the population have IQ's between 70 and 130 (i.e., within 2 standard deviations of the mean). Suppose we take a random sample of individuals and compute the mean IQ of the group. If we took samples of 10 people, roughly 95% of the sample means would lie between 90 and 110. If we took samples of 100 people, roughly 95% of the sample means would lie between 97 and 103. Larger samples are unlikely to have extreme means because high scores tend to cancel out low scores. This is what is meant by pooling variability. There are many important manufacturing applications of variability pooling, a few of which are listed below.
Examples
Multi-item Inventory Policy: Consider an inventory system that stocks repair parts. Each part is needed for the repair of a different machine type and therefore we assume that the demands of the different parts are independent. A common way to control inventories of such parts is the base stock policy, under which we set a target stock for each item as:
Target Stock = Cycle Stock + Safety Stock
where Cycle Stock equals the expected demand during the replenishment
lead time and Safety Stock equals the additional inventory carried to prevent
stock-outs due to variability in demand. Therefore the total safety stock
is the sum of the safety stocks for the individual parts.
Now suppose that instead of stocking individual parts for each machine, it is possible to stock a single generic part that would fit all different types of machines. Although this would not affect total cycle stock, it would make it possible to achieve the same service level with less safety stock because it pools the variability of the demands. The reason is that high demand tends to average out low demand and hence generic safety stock is more effective than specialized safety stock. The plot below shows the ratio of the total safety stock required to achieve a fixed service level with and without pooling as a function of the number of number of part types. Click here for the mathematical derivation of this relationship.
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Notice that while the total safety stock decreases in the number of
part types (i.e., the amount of pooling), it does so at a decreasing rate.
This is also an illustration of the principle
of diminishing return.
Parallel Capacity: All other things being equivalent, a station with multiple machines in parallel will experience less WIP buildup and queuing delay than a station with a single (fast) machine. By "equivalent" we mean that the two stations have the same capacity and that the variability of individual machines (measured by the coefficient of variability of process times) is the same. The reason for this is that the parallel machine station pools the variability of the individual machines and thus mitigates its effect. For instance, when one machine in the parallel machine station experiences an unusually long process time (e.g., due to a breakdown), the other machines can continue working, so that the station rate decreases but does not fall to zero. In contrast, when a single machine station experiences a long process time, the whole station comes to a stop, allowing WIP to build up and delays to occur. Thus, if cost, capacity, and reliability are the same for a single fast machine or two half-speed machines, it is preferable to equip a station with the two half-speed machines.
Closely related to parallel machine stations are systems making
use of
queue sharing. An everyday life example of queue
sharing is a bank. Instead of having individual waiting lines for
each teller, most banks have a single line feeding all of the tellers.
This ensures that if a teller experiences a long service time (e.g., someone
wants to deposit $250 in pennies) people waiting in line do not experience
an inordinately long delay. In contrast, grocery stores generally
use separate queues for cashiers, which means that a customer that gets
stuck behind a slow transaction experiences a long delay (or tries to jump
queues, which makes the system operate closer to the queue sharing behavior
of the bank queue). In production systems, queue sharing can be facilitated
through the use of flexible equipment, which enables machines to switch
to handle multiple types of jobs instead of being dedicated to a single
job type.
Postponement (Late Customization): If a set of products can be designed so that they remain generic for a significant portion of the production process and are only customized at the end, then it may be attractive to build the generic portion of the product to stock. Since demands for end items will be pooled with respect to their requirement of the generic parts, efficient levels of safety stock will be possible. Furthermore, the portion of the product that is built to stock will have its lead time invisible to the customer. Hewlett-Packard used this approach for their DeskJet printers. By manufacturing printers without power supplies and documentation (the two features that differentiate products destined for different European countries), they were able to postpone customization until the printers reached the distribution center in Europe. By pooling demand across countries, this policy made more efficient use of safety stock and reduced obsolescence costs.
Cross Training: An increasingly common practice in modern manufacturing organization is the use of cross-trained operators who are capable of staffing multiple functions. Since these workers can "float" to places in the system in need of labor, they serve to pool the variability in workloads at different stations. The result is reduced staffing requirements relative to the same system staffed only by specialized workers.