Using Lead Time Data from Discontinuous Processes

The teaching of lean concepts is typically tuned to continuous processes: Day in, day out, value flows continuously from suppliers until the final product reaches the customer. The concepts of lead time (the time it takes individual “flow-units” to travel through a process), Takt time (the available time divided by customer demand), and many other lean concepts are based on the assumption of continuous processing. The concept of “available time” helps expand that to interruptions in the process, such as work being done in one or two shifts only.

However, this extension is only valid if how a shift starts and how a shift ends can be neglected in the face of what is going on during the shift. Work in progress must also be virtually “frozen” between shift end and shift start. The less that is true, the more analysis tools must be specifically tuned to discontinuous processes.

The logistics industry is heavily built on discontinuous processing. Start and end cannot be neglected. This can readily be seen from the cycle of a typical distribution center: Until the late afternoon it is virtually empty. During the course of the evening, aircraft and line-haul trucks bring in shipments, and the center fills up. These shipments are sorted onto other aircraft and line hauls, which depart during the course of the late night and early morning. At the beginning of office hours, the center is again empty.

This sort process is highly repeatable—but it is discontinuous. Between the two sort periods there is also no “work in progress” left in the process. As a consequence, Takt time is highly dependent on the time of the night. It just cannot be calculated from the number of shipments and whatever the “available time” might be.

A manager rather is confronted with the following dilemma: how to ramp up and down process capacity so that both work in progress and idle capacity are held at a minimum?

To help local management and also to “democratize” problem solving within the framework of a continuous improvement program, we have looked for tools that help teams better understand discontinuous processes. To be broadly applicable, these tools must be easy to:
• Set up using a standard piece of software
• Teach and use
• Interpret by a large community of nonspecialist practitioners

In other words, graphical analysis tools that can be built into Excel spreadsheets.

Lead times from measurements have a simple data structure

Let us first understand the structure of the raw data used to measure lead time (table 1). What is most obvious is that “lead time” as such is not measured. What is measured are time stamps for processing start and processing end for a flow unit (or in the logistics industry, a shipment). The lead time is then calculated from the difference between these two time stamps.

Shipment ID	Process start time	Process end time
A	TA, in	TA, out
B	TB, in	TB, out
C	TC, in	TC, out
…	…	…

Table 1: All information needed for the analysis is represented in the above table.

 

Graphical analysis tool No. 1:
The histogram

Building a histogram from continuous data is a commonly used technique. Standard spreadsheet programs like Excel have add-ins for that purpose. From data such as those seen in table 1, histograms can be built for the start times, the end times, and the lead times. In our example these three histograms answer the following questions:

• How are arrivals distributed over time? This helps plan capacity for handling arrivals (Figure 1a)
• What does “end of processing” over time look like? In our example this helps to plan capacity for handling sorted shipments and match that with the departure schedule of line hauls and aircraft (figure 1b)
• What is the flow units’ lead time distribution? This helps identify “slow movers” and understand the process capability (Figure 1e).

Because they are so simple to obtain, we recommend always using all three histograms.

 

 Figure 1a: The “analysis six pack” for discontinuous processes
Histogram of arrivals of shipments into the process (a) and of end of sort (b). Shipments arrive in batches. The readiness for loading on local trucks follows a unique structure.

 Figure 1b: End of sort

Figure 1c: Individual value plots for end of processing vs. arrival time. The red dots display the ideal situation of end of processing equals arrival time.

 Figure 1d: Lead time vs. arrival time. Each dot corresponds to one shipment.

 

 Figure 1e: Histogram of the lead time

  Figure 1f: Cumulated throughput diagram.

 

Graphical analysis tool No. 2
Individual value plots

For any given flow unit, the time it gets out of the process (Y) may depend on the time it got into the process (x). Data analysis helps reveal the key drivers in Y = f(x). In principle, the relation between x and Y can be visualized with three individual value plots:
1. Time-in vs. time-out (figure 1c)
2. Lead time vs. time in (figure 1d)
3. Lead time vs. time out

We have found that the information contained in the latter graph is the one that can be extracted from the first two. For that reason we have not included this last graph in our “analysis six pack” in figure 1.

Even though these graphs are easy to build and easy to interpret, they are not commonly employed. Because they often give surprising insight, we recommend using them for just about any situation.

These graphs answer questions such as the following:
• How is end of processing linked to start of processing?
• What patterns are there between start of processing and the processing time?

This reveals the time-dependency of lead time.

Graphical analysis tool No. 3
The cumulative throughput diagram

The cumulative throughput diagram displays over time (i.e., cumulatively) how flow units get into and then come out of a process (figure 1f). Graphically, the vertical distance between these two curves allows estimating the amount of work in process over time (i.e., how many flow units got into the process but not yet out).

As the appendix explains, the work in progress can more accurately be calculated from the raw data. For small numbers of items in process, the before estimation cannot be used. We have used these calculated values to display work in progress (abbreviated as “WIP”). The horizontal distance between the two curves can also be interpreted: It is the average lead time of shipments to get through the process at any given moment in time. Whenever the cumulative curves for incoming and outgoing shipments are not parallel, then the average lead time is time-dependent.

How to interpret the “analysis six pack” for discontinuous processes

For discontinuous processes—and not only with the example we use here—we realize that the histogram for lead times can only be interpreted together with the other graphs. We will now interpret the graphical data analysis in Figure 1.

First, shipments with a long lead times typically arrived early, and shipments with a short lead times typically arrived late in the process. Therefore, a comparison of the lead time histogram to any common distribution cannot be used as a basis of interpretation: Early and late shipments come from different populations. We have added a best-fit gamma distribution curve to show how easily one might be misguided when treating these discontinuous data as if they came from a continuous process.

We use the start of processing and end of process plots in figure 1c as our first step in the analysis. A perfect first in-first out (FIFO) process would display a narrow line of dots, separated vertically from the time in = time out reference line by the lead time. The triangular shape of the data here displays the absence of FIFO. Shipments coming into the process participate in a kind of a “lottery”: whatever is in may “win” the way out immediately or at any time later until the end of processing. The individual value plots also reveal batching far better than the histograms do: Figures 1c and 1d display a nonstochastic distribution of dots that appear vertically aligned. These are shipments that came in at about the same time (i.e., in the same batch). However, the members of these batches got out of the process randomly.

A rate of arriving shipments that is higher than the rate at which shipments reach end of processing also leads to a steady buildup of work in process, which is only stopped by the fade-off of new arrivals, that is, by the very discontinuity of the process. When FIFO is lacking in the process, in a logistics distribution center, this spells a continuous buildup of chaos in the process over the course of the night, a fact that can be observed and documented with tools such as “waste walks.” To estimate the rate of arrivals (1.5 per second) and of shipments reaching end of processing (1.2 per second), we use the times when 10 percent and 90 percent of shipments have arrived or been processed.

For the special case of continuous and steady processing, the cumulative throughput diagram would display the incoming and outgoing flows as more or less straight and parallel lines. In this special case, the average lead time has no time dependency, and all shipments can be treated as coming from the same population.

Using the analysis six pack for the design of discontinuous processes

In the sort process we aimed to improve, we improved FIFO through a kanban WIP-control system: Once a container of shipments is empty, the next container is pulled from the upstream process step. We also accounted for the fact that container size, the rate of shipments going through the process, and their lead times are interdependent.

It may also be of interest how we explained the concept to workers. We used a “rhythm and pint” analogy: you only order the next pint (of beer) after you have finished the first. It also does not make sense to hand over a pint to the next station before they are ready to “drink.” So the entire process needs to align to a given rhythm.

We have set up such a WIP-control system experimentally over the course of several nights. This system also allows workers to adapt to varying workloads (e.g., small volume during summer holidays, high volume at the end of the year). We have further installed visual management screens to display the arrival rate, the rate of shipments having reached end of processing, and the work in progress at any given moment in time. The underlying data are extracted from a data center. Such visual control methods help workers allocate resources where they are most needed in the process.

Improvement was also made on the sort process capacity. As can be seen from figure 1f, sort capacity should be increased from 1.2 shipments per second to 1.5 shipments per second or even more in order to match the rate of incoming shipments. Any increase in capacity means a process owner is faced with the concern of having idle capacity at a given moment in time. However, figure 1f shows that shipments come in from 00:45 onwards. Setting up the full process capacity of 1.5 shipments per second at 01:00—and no earlier—will keep all resources busy until about 02:15 to 02:30.

Breaks should also be planned throughout the process and in small groups rather than a major part of the team abandoning the process at its peak time (see the dip in figure 1b).

With all these improvements, realized over a series of rapid-improvement kaizen events, the sort process has not only become much faster, it is also far less obstructed because work in progress at any given time has reduced to a fraction of what it initially was.

Appendix: how to calculate WIP from time in, time out data

One can estimate the amount of WIP by simply calculating the vertical difference between the cumulative arrivals and cumulative end-of-processing curves in the cumulative throughput diagram. Depending on the resolution of the time axis, there will be errors, though. One way to address this is with the following process (see also the attached Excel sheet):
1. Take the original data and sort start of processing and flow unit by start of processing. Add a column with the rank obtained in this way (rank 1 is the earliest flow unit).

2. Take the original data and sort end of processing and flow unit by end of processing. Add a column with the rank. This will produce a new set of data columns given in Table 2.

3. For each end of processing, calculate the largest start of processing time that is still smaller. In Excel, this can be realized with the following formula (column Q in the CTD sheet):
= SUM PRODUCT (MAX ((I$6:I$10097 < N6) * I$6:I$10097)).

Here the start of processing data are stored in the column I6:I10097. The end of processing time is stored in cell N6. The formula then provides the largest value in column I that is still smaller than the value in cell N6.

4. Using the search function in Excel, find the rank of that largest start of processing which is still smaller than the given end of processing. The difference between the ranks is the number of flow units in process: They have already gotten into the process but not yet gotten out.

5. If within the time resolution of the data two or more arrival times are the same, then take the highest rank for this arrival time.

 

This procedure builds a new table for arrival times and end-of-processing times vs. work in process.

Flow unit	Time in (hh-mm-ss)	Order	Flow unit	Time out (hh-mm-ss)	Order
A	TA, in	1	…	…	…
B	TB, in	2	X	TX, out	N-2
C	TC, in	3	Y	TY, out	N-1
…	…	…	Z	TZ, out	N

Table 2: Sorted time in and time out tables. The link between the two is broken during this sort.

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