Many of the production systems in which industrial engineers practice can be described as follows: products or customers enter the system, and the system uses resources to perform operations that add value to the product or customer. For example, in a typical factory both machines and people constitute resources. Some machines are fixed in location, such as metalworking machines, ovens, and conveyors, while some machines move around, such as cranes, lift trucks, and all sorts of hand tools. People also can move between locations and processes. The processes that are performed using these resources, e.g., might take sheet metal and transform it into enclosures for avionics systems, or take plastic pellets and transform them into enclosures for cable modems. A hospital also is a production system, in which patients enter, various procedures are performed, and - hopefully - patients leave in a better condition than when they arrived. In both the factory and the hospital, value has been added. A coal-fired electricity generating plant also is a production system–coal enters the system, operations are performed, and energy leaves the system through the electrical grid to which it is attached.

Merriam Webster defines *logistics* as: the things that must be done to plan and organize a complicated activity or event that involves many people. Clearly, production systems are logistics systems. We are interested in a subset of all production systems, namely those where flows are in discrete units, e.g., individual product units or components of product units, and where processes have well-defined start and end events, e.g., the start of a machining or heat-treating operation, and the completion of same. These systems are quite different from continuous flow systems, such as energy flow through an electrical grid, water flow through a municipal water system, or air flow around an airfoil. Our focus will be on production systems that may be described as “discrete event logistics systems” or DELS.

The concepts of DELS extend far beyond factories. A warehouse also is a DELS, albeit one with much simpler resources and processes. Similarly, a supply chain is a DELS, but one in which the “facility” is not a building, but the geographical organization of factories, warehouses, and transportation resources.

Clearly, many seemingly quite different systems fall into the category of DELS. What we will see is that there are generic ways of thinking about, analyzing, and designing DELS that can be applied across this broad spectrum. The reason this is possible is because we can abstract the essential aspects of DELS into a relatively small set of concepts, which will support very powerful generic analyses.

There is a hierarchy of abstractions that is relevant to the study of DELS (and to systems in general). If we want to be able to design a system, we must be able to control it. If we want to control a system, we must be able to predict its future state. In order to be able to predict a system's future state, we must be able to describe it. In our efforts to design, control, predict, describe systems, we will use a wide variety of models. Some of the models we will use are applications of fundamental mathematical models from statistics, stochastic processes and optimization. The challenge is to bridge from these fundamentally mathematical (and therefore very abstract) models to engineering models of DELS. What will enable us to make the necessary connections is a common semantic framework, or a way of talking about and thinking about both individual instances of DELS and the DELS domain in general.

The physical realization of each DELS is unique, but what all DELS share in common is that their physical realization can be though of in terms of product, process, resource, and facility.

• Product is what flows through the system. The external flow into the system may be, e.g., a raw material, such as rolls of aluminum, or purchased materials, such as nuts and bolts, or complex assemblies, like complete automobile seats.
• Process is how value is added to the product, by changing its condition, age, or location, or by capturing useful information about it through some sort of inspection.
• Resource is the mechanism by which process is executed, and can include reusable resources like machines, space and people, but also consumable resources, like utilities, paint, etc.
• Facility is where the process takes place, and can be within the four walls of a building, the combination of space in multiple buildings, or simply a collection of geographically distributed resources. What is important about facility is that it defines constraints on the interactions between resources or on the execution of a process. For example, if the resources executing two different conversion processes are in close proximity, the transfer between them will be quick, but if they are far apart, the transfer between them may take a significant amount of time. There are other types of important interactions.

Regardless of the type of DELS - factory, warehouse, supply chain, hospital, restaurant, bank, retail store, etc. - we can identify the product, process, resource and facility elements, essentially through direct observation. This semantic convention is the basis for a variety of descriptive models, i.e., models which allow us to capture the knowledge gained through observation, but in an abstract manner that is easier to understand, communicate, and manipulate, and therefore easier to validate with system stakeholders and to use in analyses.

A very important class of descriptive models are process maps which capture the flow of product through a sequence or network of process steps. We will discuss process maps using the “Penny Fab”, a very simple model of a production process introduced by Hopp & Spearman in their seminal textbook Factory Physics. Conceptually, penny “blanks” arrive to the fab according to some process, and proceed through a series of four workstations—each consisting of an arrival queue and a process—until a finished penny is produced. Each process in the penny fab has a constant processing time, or duration, and pennies may accumulate before a process. If they do, they enter the process on a first-come-first-served (FIFO) basis as soon as the process becomes available. When one of the four processes completes working on a penny, that penny moves immediately to the next arrival queue or is disposed (leaves the penny fab). The Penny Fab is used in Hopp & Spearman as an example to illustrate some fundamental definitions and performance metrics for DELS.

A process map for Hopp & Spearman’s Penny Fab 1 is shown below. The various types of process are labelled. Note that we have used the term “process” to describe both the lowest level of transformation that occurs to the pennies, and to the entire “production process”. This is common practice, and can lead to confusion about what, exactly, is being discussed.

In this process map there are five graphical symbols, and each embodies some semantic content, e.g. we associate meaning with each symbol as indicated by the name associated with the symbol. We also can associate properties with each symbol type, and property values with instances of a symbol type. For example, in the figure above, we could have associated a property called “process time” with the conversion process symbol, and a different value of that property for each of the four instances of conversion process.

Note that the process map doesn't really tell us how each process is executed, e.g. what resources are used in each process. Clearly, it abstracts from the “real” penny fab and therefore omits a lot of information which might be important. Nevertheless, provided we have carefully identified the important properties, and correctly identified the property values, there is much that can be learned from analyzing the process map.

In the figure above, some of the transfer processes have been elided, specifically the transfers from waiting processes to conversion processes. This is a common practice, but one which can lead to a loss of information or to miscommunication. For example, if a conversion process is a batch process, pennies would move from waiting to conversion in batches, e.g. four pennies at a time for a heat treat process. In this case, the transfer has a property, “batch size” and a property value of 4. Similarly, transfers from one conversion process to a following wait process also may be done be in batches rather than one-at-a-time. If we have elided these transfers in the process map, then we may fail to capture this important information.

The kinds of questions we want to be able to answer about the penny fab are

• How long does it take an arriving blank to complete processing and depart (the “manufacturing cycle time”)
• What is the maximum rate at which pennies can depart (the “production rate”)
• How many pennies will accumulate in the penny fab and at what locations (the “work in process” or WIP profile)

In Penny Fab 1 all conversion process times are the same, two hours, the transfer process times are zero, and the waiting processes have no capacity constraints. Suppose we pre-loaded Penny Fab 1 with a penny in each conversion process, and at the same time we started all the conversion processes, and also the arrival process with an arrival at time 2 and every 2 hours afterward. How long would it take for an arriving penny to complete processing?

Penny Fab 1 allows us to illustrate some very fundamental performance attributes of DELS. In Penny Fab 1, the bottleneck throughput rate is the minimum rate at which any of the stations. The cycle time, or the time from arrival to disposition, for Penny Fab 1 is the sum of the processing times at all the stations, plus any time the penny must wait in queue. The minimum cycle time is the sum of the individual station process times. In our example where the workstations were pre-loaded, what would be the cycle time? What would it be if there was no pre-loading prior to time 0?

A very important concept is that of critical WIP, which can be illustrated in our example by the “preloading” of the workstations. What is the smallest amount of preloading that would enable Penny Fab 1 to achieve its maximum throughput of one penny every 2 hours?

The following questions are intended to ascertain the degree to which you retain understanding of the content of <a prerequisite class on queueing theory>. A poor score on this test of your understanding will indicate a need for you to do some additional work to prepare yourself for success in <this class>.

1. Write down Little's Law, and define the terms.
2. What is the relationship between “cycle time” and “waiting time”?
3. You arrive at the movie theater and see 30 people in line waiting to buy tickets. It looks like customers are arriving at a rate of 5/minute. If you join the line, how long will you have to wait to get your ticket?
4. Imagine you enter the bank and there are 3 tellers working. Each can finish a transaction in 4 minutes. Customers are arriving at the rate of 1/minute. What is the average utilization of the tellers?
5. If the average time to complete a transaction is 5 minutes and the average utilization of the server is 90%, what is the expected cycle time?
6. What is the squared coefficient of variation for the exponential distribution?
7. If the arrival rate of jobs is 12/hr and the utilization of the server is 50%, what is the average service time?

Develop a process map for the flow of customers through the Student Center food court. Assume customers only enter through <one specific entrance>, and only exit through the cashier “gates”. You may use PowerPoint, Word, Visio, or pencil and paper to prepare your process map with the following processing mapping symbols.

After constructing your process map, answer the following questions briefly but clearly.

1. How many “workstations” are there?
2. Where are the routing decisions, and how are they made?
3. Can you make some reasonable assumptions about the process flows that simplify the process map?
4. At each workstation, what resources determine the throughput rate?
5. What determines the capacity of the InQueue at each workstation?