Operations Management
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What if we buy a faster toaster?
Place an order
Toast buns
Add dressings
Cashier Worker 1 Worker 2 Worker 3 Worker 4 Worker 5
8s 6s 8s 6s 2s 2s 450/hr 600/hr 450/hr 600/hr 1800/hr 1800/hr
Flow time of the whole process: _________ sec
Which task is now the bottleneck? (place an order (cashier ) is a bottleneck also add
dressing(worker 2) is a bottleneck
Capacity rate of the whole process: ________ orders / hr
Cycle time of the whole process : ________ s
Add meat patties
Package Deliver
Are there any operational benefits of reducing flow time at non-bottlenecks?
Place an order
Toast buns
Add dressings
Old Flow time:
8s 10s 8s 6s 2s 2s
Old Capacity rate:
450/hr 360/hr 450/hr 600/hr 1800/hr 1800/hr
4s 10s 6s 4s 1s 1s rate:
900/hr 360/hr 600/hr 900/hr 3600/hr 3600/hr
Add meat patties
Shortening non-bottleneck tasks decreases flow time
– This improves response time (which is still important), but it does not affect capacity rate of the process.
Main Insights
! To increase the capacity rate of the entire system, increase the capacity of the bottleneck process.
! The bottleneck may change when capacity is added (i.e., a new bottleneck process may now appear).
– Important when we are justifying additional capacity
! The bottleneck is fully utilized (ρ=1) while other resources are not fully utilized (ρ≤1).
! Questions
– What does increasing the capacity of the bottleneck process do to the utilization of the bottleneck station? The bottleneck station utilization potentially decreases (if the bottleneck station changes).
» Flow time? Cycle time? (Theoretical flow time can stay unchanged (For example when adding adding a new toaster, the capacity of the process increases but theoretical flow time stays unchanged. If we increase the capacity of the bottleneck station by decreasing the flow time of the bottleneck station, then theoretical flow time may decrease. If the process is capacity constrained, the cycle time decreases.
– If we double the bottleneck capacity, does the capacity of the entire system also double?
– Not necessarily. Because another station may become the bottleneck and therefore the capacity of the whole process may not be doubled.
Unbalanced multi-stage processes
Place an order
Flow time: 8s 10s Capacity rate: 450/hr 360/hr
Process is “blocked”
When the next stage is busy, the order cannot be sent to the next stage after finishing the current stage.
Toast buns
The role of inventory buffers
Place an order
Flow time: 8s Capacity rate: 450/hr
10s 360/hr
2) seasonal demand
3) economies of scale
4) separation of steps in process
5) stochastic demand
What role does the buffer play?
Place an Toast
order buns
Buffering refers to a storage area between stages where the output of a stage is placed prior to being used in a downstream stage. Buffering allows the stages to operate independently. If there is no assumption, then the assumption is that there is a direct link. Reasons for having inventories:
1) Pipeline inventory: the time a flow unit spends in the process
Toast buns
Another “unbalanced” process
Add dressings
Add meat patties
Flow time: 8s 6s Capacity rate: 450/hr 600/hr
Process is “starved”
Buffer Examples
! Consider four consecutive stages A, B, C, and D with the following capacity rates: 12 units/hr, 15 units/hr, 11 units/hr, and 14 units/hr, respectively. Assume that the demand on the system is 13 units/hr (Short run).
– Where would you add buffers to the system (minimum number of buffer)?
– We should add a buffer before A and also a buffer between B and C.
Buffer and Bottleneck
! If a buffer is provided at some upstream stage to the bottleneck, inventory may build up at the buffer
! Inventory will not build up at the (immediately) downstream stages to the bottleneck even if buffers are provided
Today’s Lecture:
Little’s Law, Inventory Build-up &
Capacity Analysis
! We return to process analysis
! with the flavor of variability – We learn about
– Implied utilization
– Inventory Build-up diagram – Little’s Law
! It now gets a little more realistic. 12
What is Variability?
! Definition:
– The randomness associated with a process.
– The extent at which measurements of the performance of a process differ from each other.
! In reality, all processes exhibit variability as nothing is completely predictable!
Types of Variability: Thanksgiving Example
! Predictable variability refers to “knowable” changes in input and/or capacity rates
– Mean demand for turkey will go up close to Thanksgiving
! Unpredictable variability refers to “unknowable” changes in input and/or capacity rates
– Supply of turkey changes each year due to crop yield
– Exact demand for turkeys each day
! Both types of variability exist simultaneously
– Turkey sales will go up during Thanksgiving, but we do not know the exact demand for turkey
Where does it come from?
Security Checking
! Variability comes from – Variable input rate
– Variable capacity rate
Predictable Variability
! Predictable variability can be controlled by making changes to the system
– We could increase or decrease the demand for turkeys by increasing or decreasing the price
! Other examples of predictable variability
– Demand for swimsuits in the winter
– Medical procedures take longer to perform in July – Students in office hours right before the midterm – Demand for uber during rush hours
Review from Class 2:
long run, Single Stage, Single Machine/Server (no variability)
Input rate
[units/hr] … … …
Output rate
[units/hr] … …
Inventory [units]
Flow Time [hr]
Capacity Rate = 1 / FlowTime [units/hr]
§ For large input rate, only for single stage.
§ The maximum possible output rate that can be achieved.
Output rate = min{Input Rate, Capacity Rate} [units/hr]
Utilization = Throughput Rate / Capacity Rate (dimensionless) 17
So far we focused on the long run
Input rate
[units/hr]
… … …
Output rate
[units/hr]
Coffee Shop
Capacity rate is defined as the maximum possible output rate.
! But, we also discussed that in the short run: – Input Rate > Capacity Rate
» Throughput rate = Capacity rate
– Input rate < Capacity rate» Throughput rate = Capacity rate if Inventory>0 » Throughput rate = Input rate if Inventory=0
Short Run Analysis: Funnel Analogy
• In the short run, the input rate can be larger than the capacity rate for a period of
§ A properly sized buffer is
needed to store units waiting Buffer to be processed (build-up
inventory)
• Utilization only carries information about the excess demand in which case the utilization is strictly less than 100 percent. In contrast we cannot infer from utilization by how much demand exceeds the capacity of the process. This is why we need to introduce an additional measure.
Implied Utilization vs. Utilization
! Utilization = Throughput Rate / Capacity Rate
= Actual output rate / maximum output rate
! Implied Utilization= Input Rate / Capacity Rate
! Note that in the short-run, the implied utilization can be greater than 100% (i.e., utilization > 1.0).
! In the long run, Implied Utilization = Utilization. 20
Utilization Profile
! Assume “short run” input rate = 7 units/hr à throughput rate = 5 units/hr
Capacity rate
Utilization
“Short run” Implied utilization
The utilization of a particular resource is throughput divided by the capacity of that resource.
What we miss is that we can not infer how much demand is exceeding from capacity using utilization in the short run. Therefore we introduce implied utilization,
which represents the mismatch between what could flow through the resource and what the resource can provide(capacity). Any excess above 100% reflects that a resource does not have the capacity available to meet demand.
Note that if a resource has an implied utilization above 100% it does not mean it is a bottleneck.
Inventory In Process Analysis
! Flow units (ex. Customers, raw materials, vehicles) awaiting their turn to enter the process as well as units that are currently being served.
! They are waiting in a queue or a buffer.
! Flow units can be extracted from the buffer or
served in a variety of ways.
– The way in which flow units are served is called the service policy (FCFS, LCFS, Priority Service).
! How do we track the queued inventory? 22
Build-up Diagrams
Think of work as being liquid
• Predictable Variability
• Short Run Analysis
• Input Rate > Capacity Rate ok • Variable rates ok
• Build-up Rate = Input Rate – Capacity Rate
Inventory build-up: example 1 , Coffeeshop
co↵ee shop opens at 11AM and the sta↵ reported that the demand rate between 11AM till noon is 75 customers per hour and from noon to 1PM it is at the rate of 100 customers per hour. Also they reported that their serving capacity is at the constant rate of 50 customers per hour. Draw the inventory build up diagram from 11AM until 1PM.
Inventory Buildup:
Coffee Example #1
– Input Rate = 75 units/hour, 11:00am – 12:00noon = 100 units/hour, 12:00noon – 1:00pm
– Capacity Rate = 50 units/hour
– Build-up Rate = 11-12: 25 units/hour. 12-1: 50 units/hour
Inventory (or backlog)
11am 12pm 1pm
Metropolis co↵ee shop opens at 11AM, and the sta↵ reported that the demand rate between 11-12noon is 75 customers/hour, and from 12-1PM it is at the rate of 100
co↵ee shop opens at 10AM, and the sta↵ reported that the demand rate between 10-12noon is 50 customers/hour, from 12-2PM it is at the rate of 200 customers/hour, and from 2-6PM it is at the rate of 50 customers/hour. Also, they reported that their serving capacity is the constant rate of 100 customers/hour. Draw the inventory build-up diagram between 10-6PM. Can they serve all demand before closing at 6PM?
city is the constant rate
of 50 customers/hour. Draw the inventory build-up diagram between 11-1PM.
Inventory Buildup: Example #2
Input Rate
10am 12pm 2pm
Inventory 200
10am 12pm 2pm
Customers/hour
Capacity rate = 100 units/hour
10am 12pm 2pm 6pm
Inventory Buildup: Example #2
Customers/hour 200
Capacity rate = 100 units/hour
Input Rate
10am 12pm 2pm 6pm
Inventory 200
Inventory Buildup: Cranberry Example
During harvesting season, a processing factory works around the clock.
! Farmers deliver their loads of cranberries from 12am to 12pm (last
truck arrives at 11 am) at a constant rate of 2 tons/hour.
! The fruits are dumped into a big storage bin and processed at a rate of 1 ton/hour.
• Draw an Inventory buildup diagram.
(Assume the flow unit is cranberries and that the they arrive
at the station at a constant pace all day).
• What is the average inventory of cranberries in the factory?
Inventory Buildup: Cranberry Example
Average Inventory in the System (Area Under the Curve) / (Time)
(12*24/2)/24=6 average inventory
12:00pm 12:00am
Inventory Buildup: Cranberry Example Continued
Suppose the storage bin has room to hold only 6 tons of cranberries. Once this space is filled, the farmers’ trucks must wait to dump their contents
! Notice there are now two buffers: » Cranberry buffer
» Truck buffer
Questions:
What would happen if there was no truck buffer? We would lose a part of the input.
What is the average inventory of cranberries
• In the bin? (12+24)*6/2/24=4.5
• In the trucks? (12*6)/2/24=1.5
• In both? 6
Do we lose any cranberries? no
Inventory Buildup: Cranberry Example
Average Inventory in the System (Area Under the Curve) / (Time)
Storage bin
Inventory Buildup: Cranberry Example Continued
Now let us change the flow unit to a “truck”
! Assumptions:
– Each truck carries 1 ton of cranberries, i.e., two trucks arrive
every hour between 12:00am to 12:00pm.
– The storage bin has a capacity of 6 tons.
– At the start of every hour, the processor takes 1 ton of cranberries from the storage bin.
• Draw an Inventory buildup diagram of trucks.
• At what time will the trucks likely start to wait to unload? 6 AM
• What is the “average” inventory of trucks waiting? • (1+2+3+4+5+6+5+4+3+2+1)*1/24=36/24=1.5
Inventory Buildup: Cranberry Example Continued
7 6 5 4 3 2 1 0
Inventory of trucks
0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00
10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Number of trucks waiting
Little’s Law: Introduction & Capacity Analysis
! Establishes a relationship between:
! Average Inventory
! Average Throughput Rate
! Average Flow Time
… … …
Average Inventory (I)
Average Throughput Rate
(R) [units/hr]
… … …
Average Flow Time (T) [hr]
Little’s Law:
Coffee Example
… … … … …
Average Waiting Time (T) [hr]
! Time to make one coffee: 30 seconds
What is the throughput rate in minutes?
! 60 customers in the system (Inventory) Waiting time:
Waiting Time = 60 customers * 0.5 (min/customer) Waiting Time = 60 customers / 2 (customer/min) Waiting Time = Inventory / Throughput Rate Inventory = Throughput Rate x Waiting Time
Coffee Shop
Little’s Law:
Big Reveal & Key Equation
Inventory = Throughput Rate x Waiting Time I= RxT
! We are talking about an inventory of flow units: – Could be customers in a restaurant
– Claims in an insurance company
– Materials in a manufacturing industry
! This equation can be re-arranged to solve for other quantities depending on the question!
Little’s Law:
Remember the Units
Inventory = Throughput Rate x Waiting Time I= RxT
All units should match:
– Inventory (flow units)
– Throughput rate (flow units / unit time) – Waiting time (time)
Summary: Variability (with buffer)
! If we can build-up inventory (i.e., insert a buffer into the process) variability leads to
– An increase in the average inventory in the process
– An increase in the average flow time
– We are not immediately losing customers due to abandonment (although they may still by unhappy)
– Fewer customers may be unhappy
– More utilization of resources
! Little’s Law still holds
Unpredictable Variability
! Unpredictable variability is the result of a lack of knowledge or information
– Usually can be expressed with a probability distribution, e.g., we can express the probability that a certain percentage of the pumpkin crop will fail using a probability distribution.
! Unpredictable variability can be reduced by gaining more knowledge or collecting information
– By paying close attention to weather patterns, we could increase the accuracy of our prediction that the pumpkin crop will fail.
Readings & More…
! For a better grasp of the key concepts, the following readings are only recommended:
– Chapter 2 of the textbook.
! For next class, please:
– Don’t forget to submit Assignment 1 (Jan 30th).
– Form your groups (Jan 30th).
– Let me know if you have midterm conflict.
– Assignment 2 will be available and due on Feb 6th. – Keep track of the Quercus site for the course.
! Next Class: Queueing
Appendix: Another Example for PA
! What if both Worker 1(toast buns) and worker 3(add meat patties) need to share the toaster (one toaster is available and toasting buns takes 10 seconds on toaster and toasting patties takes 6 seconds)?
! We compute the capacity of resources. Toaster is a resource. The flow time on this stage is 10+6 second=16 seconds. Therefore, capacity of this stage is 1/(10+6) order/second=3600/16 order/hour =150 order/hour, therefore the bottleneck resource is the toaster, and the capacity of the whole process is 150 order/hour. The theoretical flow time assuming that each stage takes the same time as before is still 36.
Appendix: Buffer Examples
! Consider four consecutive stages A, B, C, and D with the following capacity rates: 12 units/hr, 15 units/hr, 11 units/hr, and 14 units/hr, respectively. Assume that the demand on the system is 13 units/hr (Short run).
– Where would you add buffers to the system (minimum number of buffer)?
(before A and between B and C)
– What if stage C had a variable capacity rate of 13+-1 units/hr, instead of the original given 11 units/hr. Would you add or remove any buffer from the system compared to the previous part of the question.
– We still need a buffer before A but we no longer need a buffer between B and C.
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