The problem with waiting time Why the only way to real optimization of any process requires discrete event simulation Bill Nordgren, MS CIM, FlexSim Software Products Over the years there have been many management techniques introduced to reduce and manage the waste in a process. These include: just-in-time, Kanban, business process re-engineering, lean manufacturing, and six-sigma. There is no doubt that these techniques and processes are useful when it comes to efficiency and decisiveness. However, the most important factor, profitability, which should be considered first, is often lost in the process because most engineers are not directly responsible for company profits. These management techniques have one goal: to reduce the waiting time (waste) in a process. Waiting time in a process can be grouped into three categories: customer waiting time, product waiting time, and machine waiting time. The problem with these waiting times is that they result in a tug of war that pits customers, products, and machines against each other. The difficult question to answer is what kind of waiting time reduction saves the most money. To get an understanding of the difficulty to solve the question at hand you have to understand the dynamics of the process itself and how the three types of waiting time impact the total cost of the process. Where does waiting time or queuing begin? Between each transformation step in any process is a period in which the item being produced must wait. This time will be defined as the product waiting time. There are cases when waiting is the objective, such as the money supply of a bank: money held overnight or for a period of days pays interest. But in most cases, waiting time and queuing is a recurring evil that can suck away the profits in a process. The fact is that waiting time can comprise 90 percent of the total throughput time of a process. If this seems high, take a walk through your production facility and measure the time product sits around versus the time it is in actual production or being worked on. You will see very quickly the seriousness of the problem. Waiting time occurs in every industry, including manufacturing, warehousing, supply chain, services, and healthcare. When was the last time you were annoyed by waiting in the line at the supermarket or delays at the doctor s office? Although difficult to quantify in terms of money, these waiting times create irritation and inconvenience for customers that can be qualitatively high. In many cases the translation from the physical world into a queuing model is made easy by analyzing the constraints (processes with high utilization) of a Page 1 of 8
system. In other cases the translation is not so evident. For example it may be easy to calculate the cost of floor space used to queue parts in production, but it is another matter to calculate the cost of customers who have to wait for service that might never return because of the frustration of waiting. The cost of waiting time Every process system contains the three types of waiting time. If you understand and are able to weight the cost of each type of waiting time, it is possible to determine the optimum cost of the total waiting time. Let s take a look at each type and weight the cost of each. Customer waiting time The customer is king. Without customers you have no business, and in every business the goal is to service the customer as quickly as possible. If a customer has to wait for products or services they will always look for another source that can service them in a timely manner. As a result, the cost of customer waiting time is the most expensive. If you have no customers the amount of waiting time for products or machines is moot. Product waiting time Product waiting time includes all the waiting time a product (raw material, work-in-process, and finished goods) spends in all its different states. This waiting time will be weighted as the second most expensive waiting time of the three. When you consider the cost of raw material storage and finished good storage (warehouses) you quickly see the cost involved. This cost or waste as seen by the lean engineer is the main target for kaizen events. possible utilization. Because the financial side of a business does not understand the effects high resource utilization (95 percent load factor) has on the system as a whole, process efficiency may be doomed even before the first product rolls off the line. Let the tug of war begin Every process or logistics problem can be viewed as a distribution problem of the cost of waiting times. Factors that impact these waiting times include but are not limited to variability, quality, equipment reliability, staff reliability, and system load factors. When each type of waiting time increases or decreases it has an impact on the other; this creates a tug of war between the costs of waiting. For example, if you wanted to satisfy the demand of every customer immediately you would have to maintain a finished goods supply of every product you make. The cost of product waiting would sky rocket. On the other hand, if you wanted to reduce your work in process you run the risk of not having product at the right place at the right time; your machines may be idle waiting for product, thus increasing the cost of waiting for machines. This tug of war is not just between two sides it is a 3-way war. Each can affect the other two. For example, a machine that is loaded too high will cause a large queue to build in front of the machine. It may also starve those processes downstream. Thus, the product waiting cost increases and deliveries to the customer may fall behind, increasing the cost of customer waiting time. This complex dance of waiting times can be difficult to understand and control unless you know how to analyze and experiment with the correct variables. Before we look at the tools used for such analysis and experimentation, we need to understand the dynamic behavior of what causes waiting time. Machine waiting time Machine waiting time is the time a resource (machine or human) is idle waiting to work on a product. This cost is ranked number 3 because it relates only to the cost of an idle resource. It can be costly if the resource is very expensive to purchase. The ROI of equipment is always scrutinized by the financial side of the business to insure equipment has the highest How does a queue start? Queues start the moment the supply of products or customers is greater than the processing capacity. As a result, an unstable situation develops and, theoretically, the queue will increase indefinitely as the utilization of the system approaches 100 percent. Queues build when supply exceeds capacity and decrease when Page 2 of 8
supply is less than capacity. Another factor that can influence the queue is variation in the processing time. We all know what it is like to get behind the person in the checkout line who has all kinds of special requests that cause you to wait endlessly. When viewed over a long period of time the average utilization of a machine or resource may be less than 100 percent, but variation can cause queues to build temporarily or you could have a situation where machines are starved for a period of time. The equation for the calculation of average waiting time ( W t ) is shown in 1; the relationship between utilization (ρ) and variability is expressed in terms of the variance coefficient (c v ) and waiting time (W t ). Average processing time is shown as ( P t ). ρ W t = 0.5(1 + c v )( 1 ρ )( P t ) (1) When the utilization of a machine increases to more than 70 percent and there is a high degree of variation in arrivals, a situation can develop where a queue can build. The queue can only be eliminated if arrivals are limited or there is added capacity. The relationship between utilization, variability, and average waiting of the product or customer is shown in Figure 1. of variability in arrivals or process time. By reducing both factors, the waiting time of the product or customer can be reduced. However, when you only reduce machine utilization, machine waiting time can become a factor. In a system where you have a large degree of variability, increasing capacity is the only way to reduce the utilization of a machine and the number in the queue. Many processes show a lot more variation than you might think. Breakdowns, pauses, re-sets, waste, operator error, and restocking make a seemingly constant process extremely variable. Over time the process might seem constant, but on a weekly, daily, or hourly basis production has a very stochastic nature. With the numerous uncertainties, one can assume your system is a Poisson system and will have a variance coefficient of one. Figure 2 shows the relationship of utilization and waiting time of products and machines. Figure 2: The relationship among utilization and waiting times with a Poisson process. Figure 1: The relationship between utilization and variability. Note that c v = σ Average. A constant arrival pattern has a variance coefficient of zero. An independent arrival pattern (Poisson distribution) has a variance coefficient of one. Figure 1 shows that there is only a small area where long waiting times occur. Waiting time is only a problem when there is high utilization and a large degree By adding the waiting time of the products and machine you can find the optimum utilization for the machine. The minimum cost is derived by calculating the weighted cost factors related to each waiting time. The exponential form of the waiting time graph often confines the optimum utilization of a machine between 70 and 90 percent. Waiting Time Law: The higher the utilization of a machine or resource, waiting time increases exponentially. Page 3 of 8
When you consider the dynamics of utilization, waiting time, and the cost of waiting, you can see the inherent problems designed into systems where equipment is purchased with an ROI basis of 90 percent (or higher) load factor. The process of buying equipment already has built in bottlenecks and waste that cost real money and time on the production floor. Figure 3: Wt = 0.5(1 + 1)( 0.9 1 0.9 )(5) = 45 minutes. Waiting Time 88% 12% Process Time Minimizing queues Queuing problems develop from a combination of high utilization and variability in the arrival pattern and the process time of a machine. Another factor that can cause queuing is batch processing, which increases the process time because the batch cannot move until the last piece in the batch is completed. Batch processing is used to minimize the movement steps between processes and setup times at the process. Using techniques to minimize the setup time when changing part types has reduced the variation in batching processes. Optimization of scheduling batches that reduce setup based on similar part makeup is vital to prevent unnecessary delays in processing. There are three ways to add capacity to a process that will help reduce the utilization of the process and the product waiting time. The first is to reduce the stochastic nature of the arrival rate of products, the second would be to add a machine, and the third is to reduce the process time on the machine. It is not easy to fix the stochastic nature of arrivals; adding expensive machines or resources may not be cost effective. There is a persistent notion that a minor reduction in process time has no significant influence on waiting time and throughput. The reasoning is that process time only accounts for 5 to 10 percent of the total throughput time. However, this reasoning ignores the relationship between the utilization of a machine and the wait time in the queue associated with the machine. Suppose the processing time of a production step is five minutes and the utilization of the machine is 90 percent. Using the Pollaczek-Kyntchin equation for M/G/1 queuing systems, one can calculate that the average waiting time is 45 minutes (Figure 3). The total throughput time is 50 minutes, resulting in 90 percent waiting time and 10 percent processing time. If you could optimize the working process and re- duce the time by 10 percent (30 seconds), would the average waiting time be reduced significantly? At first look, it doesn t appear that the waiting time would be reduced by much. However, as a result of the decrease in processing time, the utilization would drop to 81 percent. Since waiting time increases exponentially with utilization, the waiting time could be reduced significantly (Figure 4). With this value, the waiting time is 19 minutes. The total throughput decreases from 50 minutes to 23.5 minutes, a reduction of approximately 50 percent. This seemingly small change in processing time results in a significant reduction of waiting time. Figure 4: Wt = 0.5(1 + 1)( 0.81 1 0.81 )(4.5) = 19 minutes. Process Time Waiting Time 48% 52% Using math to calculate waiting time is easy for simple M/G/1 systems. What about more complex systems that are commonplace in today s manufacturing and logistic environment? Mathematical analysis and spreadsheet calculations fall far short of measuring and understanding the behavior of complex stochastic processing and logistic systems. Understanding and managing the competing costs of waiting is critical to the profits of every company, yet very few understand the relationship and effect that waiting has on costs and customer satisfaction, let alone have any idea how to measure the effects so they can manage the process. Page 4 of 8
Discrete Event Simulation The experiment The only tool ever developed to analyze the stochastic nature and tug of war impact, relationship, and cost of customer, machine, and product waiting time is discrete event simulation. With the use of simulation you don t have to guess how your system is going to behave before or after change. Simulation allows you to scientifically experiment with your system before you make real changes so you know with certainty the changes you make will save real money. How would you like to know with 95 percent confidence that you will save 22 percent of the cost of work-inprocess storage costs? Or, that you have 95 percent confidence that you can reduce the delivery time of a product to your customer by 65 percent? You can with the use of simulation. Proof waiting time exponentially increases with machine utilization Simulation can be used to prove waiting time exponentially increases as machine utilization increases. We will experiment with an M/G/3. A source will generate the arrival of a box every 10 seconds exponentially distributed (to provide some stochastic behavior to the model). Boxes then move to a queue before moving to one of 3 machines for processing. The processing time at each machine will be set to 30 seconds to balance the system. Boxes are sent to a sink after processing and leave the model. A picture of the model is in Figure 5. Figure 5: M/G/3 queuing system. To run the experiment we will look at two scenarios (see Figure 6). (a) Performance measures (b) Scenario configuarion (c) Experiment configuration Figure 6: Simulation experiment configuration for two scenarios. Scenario 1: Baseline The baseline will establish the utilization percentage of the machine and the average wait time of boxes in the queue with an arrival rate of 1 box every 10 seconds exponentially distributed (Poisson process, c v = 1) and a cycletime of 30 seconds at each machine. We will run the model for 8 hours of production and we will run the model 20 times and average the results. Results will be based on a 95% confidence interval. Performance measures will be the average utilization for all machines, the average wait time of boxes in the queue, and the maximum content in the queue. Results of the baseline experiment are shown in Figure 7. Results show an average of 98.72% utilization at each machine, an average wait time of 228.75 seconds for each box in the queue, and the average maximum content of the queue as 59.65. All averages are based on a 95% percent confidence interval (the min and max range and standard deviation is shown for each average). Page 5 of 8
(a) State Pie average replication plot (b) State Pie average data summary (c) Average wait time in queue data summary (d) Average wait time in queue replication plot (e) Maximum content of queue replication plot (f) Maximum content of queue data summary Figure 7: Results of the baseline run. Scenario 2: Proposed change In Scenario 2, the utilization percent of the machines will be lowered by a 10% reduction in cycletime. This will result in a 27 second process time at each machine. The model will be run again with the same settings as the baseline with the only change being the cycletime. The results are displayed and compared to the baseline run in Figure 8. Results show a 10% reduction of utilization of each machine as a result of the 10% reduction in cycletime. Baseline model showed a utilization percent of 98.72% and the proposed model shows an 89.8% average machine utilization. The average wait time of each box fell sharply from 228.75 in the baseline model to 32.05 in the proposed model. The maximum content of the queue also fell sharply from 59.65 in the baseline model to 19.4 in the proposed model. This data confirms the hypothesis that waiting time rises exponentially as machine utilization increases. The baseline model shows a 90% waiting time versus a 10% processing time. The proposed system reduced the waiting time percent to about a 50-50 percentage. The real story is to look at the average waiting time for the boxes in the queue for each scenario. For the baseline model, where machine utilization was approaching 99%, the average waiting time was almost ten times the processing time for each box. The proposed model, with the cycletime reduced by 10%, reduced the machine utilization by 10% and also reduced the average waiting time for each box to only 32.05 seconds. In Figure 9 you can see that the relationship between machine utilization and waiting time in the queue is exponential. Page 6 of 8
(a) State Pie average replication plot (b) State Pie average data summary (c) Average wait time in queue data summary (d) Average wait time in queue replication plot (e) Maximum content of queue replication plot (f) Maximum content of queue data summary Figure 8: Results of the proposed run compared to the baseline run. Conclusion As you can see, the use of simulation made it easy to understand the relationship of machine utilization and product waiting. Simulation provides an array of data and charting options to easily identify and experiment with bottlenecks (areas of high utilization) to understand the best way to solve problems. What we have looked at so far is very simplistic but would have taken hours to calculate by hand or using spreadsheets. The real benefit of simulation is the ability to build models of real life complexities and relationships that are impossible to analyze any other way. For example, how would the system change if people were introduced to move the boxes and process them at the machines? Add more uncertainty and the waiting time will increase. Simulation is the only tool that allows you to build a model, experiment, and make informed decisions before you implement. How do you know your lean initiatives will work or save as much money as presented? Build a model of the current and proposed system and find out. Most organizations do not understand or measure their process to even know that they have a problem; nothing is done to improve the process other than the most obvious. Sitting by and doing nothing may cost millions each year. It s possible that a fourth type of waiting time that costs your company money is waiting to use simulation. No matter what your process, simulation is the only tool that can help you Page 7 of 8
Figure 9: The exponential relationship between machine utilization and waiting time. classes in the use of simulation software. He received a Bachelor of Science in Manufacturing Engineering Technology, and a Master of Science in CIM (Computer Integrated Manufacturing) from Brigham Young University. make the right decision every time. The tug of war created by the cost of waiting customers, products, and machines is complex and sometimes very hard to understand. Methodologies, techniques, and heuristics all play a role in process and system optimization. Simulation is the only way to know if those methodologies, techniques, and heuristics are of real value to the bottom line of the organization. Simulation helps you understand the behavior of current systems and processes and predict with confidence what will happen when proposed changes are implemented, without the risks of actual physical change. Said another way, speculation dies when simulation is employed. About the Author Bill Nordgren is the President and CEO of FlexSim Software Products, Inc. In 1998, he founded ProModel Corporation and was Vice President until he left in 1992. In 1993, Bill founded F&H Simulation, Inc. (now FlexSim Software Products, Inc.) and introduced Taylor II, Taylor ED, and FlexSim into the market. Bill has authored several papers dealing with simulation project management and queuing theory, co-wrote the textbook Applied Simulation: Modeling and Analysis using FlexSim, and has taught hundreds of This document Copyright c 2014 FlexSim Simulation Software, Inc. All rights reserved. Page 8 of 8