Capacity and Bottlenecks
1Introduction
This chapter continues the analysis of process capacity begun in Chapter 4. This time we concentrate on the analysis of those activities that are the bottlenecks. This chapter discusses several ways to evaluate bottlenecks and how to design systems that minimize the effect of bottleneck areas. The chapter also presents a powerful tool on the analysis of bottlenecks in service business using operations research.
2 Bottlenecks, Near Bottlenecks, and Non-Bottlenecks
A bottleneck is defined as a resource which has a demand requirement greater than its capacity and which limits the output capacity of the complete system (planned utilization larger than 100% of capacity). A bottleneck, as the name implies, is the point of a system where the “space narrows” and the flow is constrained. If you think in term of highways, traffic problems typically occur when the number of lanes is reduced (changing the capacity) and thus becoming a bottleneck. Other resources can be close to becoming bottlenecks given they are used close to full utilization (or above), but are not the process bottleneck since there is another activity with a lower capacity. However, if the current bottleneck somehow had an increase in capacity, this resource could become the bottleneck. Finally, resources whose capacity is greater than demand by a reasonable margin are called non-bottlenecks.
Identifying Bottlenecks
The capacity analysis process discussed in Chapter 4 allows us to understand the utilization of resources/activities in a process. Based on a forecasted or historical product demand, and the times to perform activities (standard times) we can define the utilization of each area and what are possible bottlenecks. However, in systems with multiple products, the product mix can change from month to month, changing the bottleneck resource. This situation is called a shifting bottleneck problem and is addressed by demand management and the theory of constraints (TOC) techniques discussed in this chapter.
Example
A bakery makes bagels and French bread for sale to local restaurants and coffee shops. The process and standard times for both is in Figure 1. There is only one machine for each activity in the process and the bakery is open 24 hours. The standard time for the dough preparation process is based on the production of 10 units.
Figure 1
If the demand for bagels and French Bread is 2,000 per day and 4,000 per day respectively, the number of hours required in the dough preparation section will add up to 20 and 26.67 for the oven, thus the bottleneck is the oven. If the demand was the inverse; 2,000 for French bread and 4,000 bagels, the number of hours required in the dough preparation activity will add up to 25 and 23.3 for the oven, thus the bottleneck is the dough preparation activity. To minimize the effect of the shifting bottleneck problem, a product mix should be established based on the profit per product type (objective – to maximize profit) and some minimum type quotas. Product mix problems can be solved mathematically using linear programming, an operations research tool covered in other business courses.
DILBERT reprinted by permission of United Feature Syndicate, Inc
5.2Working with Bottlenecks
Recognizing what activities and resources are bottlenecks or near bottlenecks is important in order to devise ways to reduce the effect they have on the total output. Let’s look at the first scenario in the example above. If the bakery had a scheduling problem and no dough was prepared for two hours, but there was enough dough in the cooling area, nothing will be lost, as the oven will continue production (dough preparation has a slack of about 5 hours). However, if the oven was shut down for 2 hours, production will completely stop for two hours. The critical message here is: time lost in the bottleneck is lost production time and thus lost output. On the other hand, time lost on a near bottleneck or on a non-bottleneck may not affect the output.
This message is very important as we analyze process capacity and look for ways to improve the output capabilities of a process – improving the bottleneck improves the system capacity, improving the non-bottlenecks will have no effect in system capacity. The focus is on either improving the capacity of the bottleneck activity or in maintaining the bottleneck activity operating at all times. While there are several methods to increase the capacity of an activity (discussed later in the text), this is sometimes infeasible due to physical, spatial, cost, or other requirements/limitations. For example, the oven cannot cook quicker (bread and bagels will burn), and the space cannot be expanded. The only way to add capacity then is to buy another oven, but this may be financially infeasible due to cash flow or facility space. Thus the capacity is fixed. The question is then, how to maintain the oven operating to its fullest capacity.
A powerful approach to address bottlenecks (and system improvement) is the Theory of Constraints developed by Eliyahu Goldratt (1980). The TOC basics are:
- Identify the system constraints. (No improvement is possible unless the constraint or weakest link is found.)
- Decide how to exploit the system constraint. (Make the constraints as effective as possible.)
- Subordinate everything else to that decision. (Align every other part of the system to support the constraints even if this reduces the efficiency of non-constraint resources.)
- Elevate the system constraints. (If output is still inadequate, acquire more of this resource so it no longer becomes a constraint.)
- If, in the previous steps, the constraints have been broken, go back to step 1, but do not let the inertia become the system constraint. (After this constraint problem is solved, go back to the beginning and start over.)
The TOC proposes a method to maintain the bottleneck activity operating to capacity called the Buffer, Rope, Drum system. The buffer refers to having an amount of work in process inventory always in front of the bottleneck resource. The rope refers to the signal sent upstream from the bottleneck calling for additional production (quantity Q) once the buffer has reached a predetermined level (reorder point R). The drum refers to the bottleneck that maintains the production rhythm (alignment of all other activities to the bottleneck activity). Figure 2 illustrates a generic process using a Buffer, Rope, Drum system.
Figure 2
4Modeling A Simplified Process: Queuing Theory
Queuing Theory is the body of knowledge that analyzes waiting lines (a queue). We all have been in lines, at the supermarket, Disney, or at the doctor’s office. Other type of lines include phone calls made to a reservation center, requests for an internet page to a server, a broken elevator in a building waiting for the technician. In all cases there are entities that require a service (entities) and a service provider (server).
Queuing Theory answers questions related to the size of the line, the number in line, and the probability that a certain number of entities are in the line. Why is this useful? It helps organizations recognize customer service costs, space requirements, and how service options compare. This section presents three basic models out of a large number (not within the scope of this course) of QT models.
The tradeoffs of Queuing Theory
QT allows the analysis of process capacity by determining the characteristics of the waiting line that will form before the process. The analysis of a service system is based on the cost of service (cost of providing the service) and the costs associated with customers (or entities) waiting in line: cost of waiting. While the costs of providing a service are relatively easy to compute, the cost of waiting are not easily determined. Waiting costs are related to lost customers, customers that will not return, and customers lost due to word of mouth. The longer the line (and time waiting for service) the higher the waiting costs.
Figure 3 Tradeoff Costs
The relationship between service and waiting costs is illustrated in Figure 3. As capacity increases, the cost of providing the service (given a constant set of customers) increases given additional staff, equipment, etc. On the other hand as capacity increases, the cost of waiting decreases due to a smaller time waiting by customers. The objective of capacity analysis using QT is to determine the capacity level (or capacity option) that results in the lowest total cost.
Characteristics of a Queuing System
A queuing system is a process model that has been condensed to a single activity. This single activity assumption is valid in many cases as it typically reflects the initial point of service or the task in the process that is the bottleneck.
Figure 4. Basic QT process model
As Figure 4 shows, a queuing system is made from four components: Entity arrivals, a Waiting Line, a Server(s) (activity), and a Departure area. Each of these four components will be further described next.
Entity Arrivals
This component is based on these assumptions:
1. The arrival of customers is modeled by a Poisson distribution with parameter (lambda). This parameter, called the arrival rate, is the mean (and variance) of the rate of arrivals. The arrival is always a rate, for example 10 per hour (10/hr), 56 per minute (56/min), etc. By using a Poisson distribution, we can account for actual variability in the interarrival time. For example, about 100 consumers enter a burger restaurant per hour from 2-5PM. This means that, every 36 seconds a customer walks in. However, in reality arrivals are random, one customer may enter 5 seconds after the previous, then no customer may walk in for 1 minute followed by three quick arrivals and so on. Poisson arrivals is a good representation of real life arrivals in service systems. The arrival rate could be directly linked to forecasts for customer demand.
2. There is an infinite population of customers. The system will not run out of customers (an average of will always arrive into the system.
3. Each arrival is by a single entity.
Waiting Line
This component is based on these assumptions:
1. Customers are organized by FCFS (First Come First Served).
2. Once a customer enters the line, it will not leave the system at any time (regardless of line size or waiting time).
3. The line has infinite space.
4. There is a single line (like in a bank) even when there are multiple servers.
Server
This component is based on these assumptions:
1. One entity is serviced by server at a particular instance.
2. One or more servers can work in parallel (like tellers in a bank). Figure 4 shows a process with two servers.
3. Service time is based on a rate (mu) similar to the arrival rate. This rate represents the number of entities that can be processed by the server in a specific amount of time (its capacity). For example 12 per hour (10/hr) = service time of 5 minutes, 60 per minute (56/min) = service time of 1 minute, etc.
4. The service time is either constant or variable. The constant service time case represents processes with little or no variability in the time required, for example automated machine processes or the process of selling tickets on a both. Variable service times are based on an Exponential distribution with parameter (mu). This represents cases where the service time may take 5 minutes in some cases, 4.3 in another, 7.3 the next and so on.
Departure
This component is based on this assumption:
1. Once the service is completed entities leave the system. No additional processes are required.
Figure 5 A QT process with two servers.
As mentioned, the objective of QT modeling is to understand the behavior of the line in order to determine the capacity option that results in the lowest total cost. To determine this we need two cost measures: the cost of each capacity option (cost of service) and the cost of waiting.
Total Cost = Cost of Service + Cost of Waiting
Cost of waiting (per day) = Number of visitors that visit the facility per day * average time each visitor waits * cost of waiting per time unit
The cost of service is often based on the cost of servers or equipment, while the cost of waiting is based on lost sales per time unit of customer wait. In cases where the cost of waiting cannot be established, for example what is the cost of waiting for an ambulance, a tradeoff between time and cost is the basis for the analysis. The example section will demonstrate such a case.
The performance of a waiting line system is based on four measures:
Wq = Average time an entity waits in the Line (Queue)
Ws = Average time an entity spends in the complete system (Wq + Expected Time in the Server)
Lq = Average length of the line (number of entities on the queue)
Ls = Average total number in the system (Lq + Expected number in the Server)
Determining these measures of performance depends on the characteristics of the process being analyzed and driven by the arrival rate () and the service rate ().
Model 1: Constant Service Processes and Single Server
Wq / Ws / Lq / Ls / Wq + / 1 / Wq / Lq + /
2( - ) / /
Model 2: Exponential Service Processes and Single Server
Wq / Ws / Lq / Ls / Wq + / 1 / Wq / Lq + /
( - ) / /
Model 3: Exponential Service Processes and Multiple Servers
Wq / Ws / Lq / LsLq / Wq + / 1 / CC* / Lq + /
/ /
CC* = Complex Calculations. Will use a spreadsheet to solve.
Example
The visitor center at the Kennedy Space center is facing increased attendance and lines. One of the predicted problem areas is the bus system that takes visitors to the launch area and to the Saturn V complex. The bus system can handle about 520 customers per hour with exponential service times. A visitor arrives to the bus area about every 7 seconds also exponentially distributed. The initial questions are: what is the average time in line to take the bus and what is the average size of the line?
First, we must determine what are and .
While is given, 520 visitors /hr, must be calculated by
= 1 visitor / 7 seconds (3600 sec/ 1 hour) = 514.3 visitors/hr which is a rate.
Second, is the selection of the model to use. Given both the arrival rate and the service rate are exponentially distributed, model 2 is the correct choice. Now we can calculate the average waiting time and visitors in the line.
Wq = 514.3 / (520 (520 - 514.3)) = 0.173 hours
Lq = 514.3 * 0.173 hours = 89 visitors
While the line is quite long on the average, the average time in the line is small (the line moves fast as the average time waiting is 10.4 minutes).
The center estimates a per hour cost (operations and investment payback) for this entry system at $340. The center also estimates that for each minute a visitor waits in line, the center losses $0.30 in future sales and visits. The next question is then, what are the total costs of this system.
TC = cost of service + cost of waiting = $340/hr + ?
Cost of waiting per customer is = 0.173 hours * ($0.30/min * 60 hrs/min) = $3.11
Per hour the center has: 514.3 visitors and each costs $3.11 thus the cost of waiting per hour is: $3.11 * 514.3 = $1602.2/hr.
TC = $340 + $1603.2 = $1943.2/hr
The center is considering opening an additional bus entry point on the bus system station. This additional entrance will have a service rate of 290 customers per hour for each entry and an operational cost (total for both) of $1000/hr. and costs. Using the spreadsheet (for model 3), we get a Wq and Lq of 0.01268 and 6.52 respectively. The total cost for this option is:
Cost of waiting per customer is = 0.01268 hours * ($0.30/min * 60 hrs/min) = $0.29 and the total cost per hour = 514.3 * $0.29 = $117.4
TC = $1000 + $117.4 = $1117.4/hr.
Given the total cost is lower for option 2, the center should pursue the new design for the system.
Activities
- Describe process bottlenecks that you have encountered in banks, airports, or even (hope is not often) the university. Why do you think they form at this point? What is the constraint?
- Recommend ways to improve the situations described in activity 1.
- Select a service operation during a peak time- a cafeteria or bank during lunch, the gym at 6PM, or the movies at 8PM.
a)Track the inter-arrival time of people at this location. Make at least 50 measurements.
b)Plot the times and calculate the average.
c)Track the service time of the process. Make at least 50 measurements.
d)Plot the times and calculate the average.
e)Describe the behavior of the line during your observations.
Exercises
- Determine all the waiting line measures for the following systems. Calculate all time measures in hours.
a)Arrival rate = 5/min, Service Rate = 6/min, Constant service, 1 server
b)Arrival rate = 300/hr, Service Rate = 180/hr, Exponential service, 2 servers
c)Time between arrivals = 12 seconds, Service time = 10 seconds, Exponential service, 1 server
- The managers of an amusement park want to determine what system will result in the smallest average line for a new food selling both (experience tells them that small lines make people want to join the line). A human operated kiosk can serve 90 customers per hour exponentially distributed and about 80 customers per hour want to buy something from those items sold in the booth. An advanced automated selling machine can serve 80 customers and about 75 per hour want something sold in the machine. What is the best choice?
- An order-taking center receives about 285 orders per hour (Poisson) and can serve 300 orders per hour exponentially distributed (as a single server). The order-taking center is considering splitting the center into two locations. Each will then handle 150 orders per hour. The cost of the current system is $100 per hour and $125 if split. For each hour an order waits it costs the company $5 in future sales. Should they split the operation?
- A new people-moving system is being designed for the Tampa Expo 2004. Three designs for the stations' entry system are being considered.
.
Option 1, there will be two separate entrances (100' feet between each), thus each will have with its own line. About 500 visitors will arrive per hour for each line, and the service rate for each entrance will be 650 per hour (exponential). The service cost of this facility is $500K per year year. / Option 2 will have two entrances next to each other, feeding from a single line. Each will have a service rate of 600 visitors per hour (exponential). Arrivals at this single entrance will be about 1,000 per hour. The service cost of this facility is $700K per year. / Option 3 is to have a single entrance using an automated system that can process 1,100 per hour at a constant service rate. Arrivals at this single entrance will be about 1,000 per hour. The service cost of this facility is $800K per year.The Tampa-Expo will be open for 240 days and 20 hours per day. Find the option that minimizes total cost per year. The cost of lost goodwill is $25/hr of waiting time.