What we discussed in my last posting is what happens in a balanced line, but what about an unbalanced line? What happens with C/T, TH, and WIP? Consider the four-step process below. In this process we see that Step A has a processing time of 1 minute, Step B’s P/T equals 2 minutes, Step C’s P/T equals 3 minutes and Step D’s P/T equals 1 minute. Clearly this is an unbalanced line because the processing times are not all equal.

The assumptions we make here are that no variation exists and only one machine exists at each process step. Here we see that the capacity of each process step is obtained by dividing the processing time in minutes/part into 60 minutes/hour. The capacity of the line is dictated by the bottleneck step which in this process is Step C at 20 parts per hour. Applying Little's law to this process, results in a critical WIP level needed to maximize throughput while minimizing cycle time as follows:

Critical WIP = TH x C/T

or

W10 = rbTo

Where: W0 is the critical WIP, rb is maximum throughput, and T0 is minimum cycle time

W0 = 20 parts/hour x 7/60 hour = 20 parts/hr x 0.116667 hr = 2.3333 parts

This extension of Little’s Law tells us that if we want to achieve maximum throughput at minimum cycle time, then our critical WIP is 2.3333 parts, or 3 parts. Any number of parts above 3 will lengthen the cycle time and any number of parts below 3 will negatively impact throughput. Since we are interested in maximizing revenue and on-time delivery, Little’s Law will help us achieve this. The table below is a summary of what we just discussed.

As we have just seen, the most efficient form of manufacturing from a flow perspective is single piece flow. But having said this, there are times when one piece flow is not appropriate or not even possible, so common sense must play a role in the decision to use it. For example, suppose the next step in a process is bead blasting or heat treatment of parts. Would it make sense to run the bead blaster or a heat treat oven with a single part or with a small batch? From a manufacturing efficiency perspective, a small batch probably makes more sense.

We always want to minimize the non-value-added activities in a manufacturing process which includes travel time. If one piece flow is not possible and transfer batches are needed, then one way to keep the transfer batch size small is through the use of cellular manufacturing. Cellular manufacturing positions all work stations needed to produce a family of parts in close physical proximity with each other. Because material handling is minimized, it is much easier to move parts between stations in small batches.

Since some processes don’t lend themselves to one piece flow and are better served by producing in batches or lots then how do we know what that batch size should be?. Once again we turn to Hopp and Spearman1. In 1913 Harris developed a mathematical model to compute the optimal manufacturing batch size. His model, the Economic Order Quantity (EOQ) is considered the foundation of research on inventory management. In order to develop this model Harris made six assumptions as follows:

1.

*Instantaneous production.*There is no capacity constraint, so the entire lot is produced simultaneously.

2.

*Immediate delivery.*There is no lag time between production and availability to satisfy demand.

3.

*Demand is deterministic*. There is no uncertainty about the quantity or timing of demand.

4.

*Demand is constant over time*. Demand is linear meaning that if annual demand was 365 units, it translates into a daily demand of one unit/day.

5.

*A production run incurs a fixed setup cost*. Regardless of the size of the lot or status of the factory, the setup cost is the same.

6.

*Products can be analyzed individually*. Either there is only a single product or there are no interactions (e.g. shared equipment) between products.

As Harris developed his model he assumed constant, deterministic demand, ordering Q units whenever the inventory reached zero with an average inventory of Q/2 (i.e. maximum + minimum divided by 2). Harris also presented the holding cost of the inventory as hQ/2 per year, with h being the holding cost in dollars/unit/year. Continuing, Harris next added the setup cost, A, per order to his equation to have AD/Q per year with D being equal to demand, since we must place D/Q orders/year to satisfy the demand. Harris then included the production cost/unit, c, or cD/year to complete the equation for cost. The final equation for cost, then, is as follows:

Y(Q) = hQ/2 + AD/Q + cD

EOQ = √2AD/h

The inference we can take away from this formula is that the optimal order quantity increases with the square root of the setup cost or the demand rate and decreases with the square root of the holding costs. What this all boils down to is, there is a tradeoff between lot size and inventory. Increasing the lot size will increase the average amount of inventory in the factory, but also reduces the frequency of ordering and by using a setup cost to penalize frequent replenishments Harris was able to articulate this tradeoff in concise financial terms.

There is another law, the law of move batching, presented by Hopp and Spearman1 that suggests one of the easiest ways to reduce cycle times in some manufacturing systems is to reduce transfer batch sizes. This law states that cycle times over a segment of a routing are roughly proportional to the transfer batch sizes used over that segment, providing there is no waiting for the conveyance device. The bottom line here is that by holding the transfer batch size to its optimal level, cycle time will also be optimal. So, if one piece flow isn’t ideal for your process, then at least calculate the optimal transfer batch size.

1

*Factory Physics*by Hopp & Spearman

Bob Sproull

ras8202@live.com

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