[SOLVED] CS AM-11 AM, and by appointment on zoom

Operations Management
After class slides does not include the review slides
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explanations that are added to the after class

Copyright By Assignmentchef assignmentchef

we cover at the
beginning of the lectures and only include the new
(answers to questions and extra
slides are in pink for your
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My lectures will continue to be on zoon until February 27th.
Office hours: Fridays, 9 AM-11 AM, and by appointment on zoom
https://utoronto.zoom.us/j/88361623271 (Links to an external site.) passcode: 604020 Individual Assignments:
Assignment #2: (weight: 2%) is available on utorsubmit and will be due on 2021/02/06, 11:59 PM. Please follow the instructions specified on the assignment carefully.
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Midterm Exam: Feb 18, 7pm-9pm, (2 hrs) on Quercus
If you have academic conflicts you should have emailed me.
The students who already emailed me should receive an email from me regarding the makeup Midterm Exam by the end of this week.
The makeup exam will be available only to students with conflicts (The students who informed us will receive an email from us with more information
Case(group) work will be available on Quercus under modules, case 1 by the end of this week.
The instructions on how to submit your work will be provided on the file (and also will be announced on Quercus).
The case study is due Sunday Feb. 27th,11:59PM.
). If you have
a conflict and did not let us know, you should email us as soon as possible.

! Inventory can build up (If Input Rate > Capacity Rate) ! Inventory Build-up diagram
Shows graph of inventory versus time
Build-up Rate = Input Rate Capacity Rate
Using the inventory build up diagram we can compute average inventory
Setting: Predictable Variability, Short Run Analysis (therefore, ok to have Input Rate > Capacity Rate), Variable rates ok

Insights from Littles Law
Key Relationship:
! Throughput rate, waiting time and inventory are
intimately related:
Depending on the situation, a manager can influence any
one of these measures by controlling the other two: Once two are chosen, the third is determined.

Littles Law:
Shipping Containers
! You are managing the construction of a new container terminal at the Port of Vancouver. You expect that containers will spend about 2 days waiting to be shipped, and you have promised customers to process 2000 containers/day.
Containers to be shipped
Containers shipped

Littles Law:
Shipping Containers
! On average, your container storage yard can hold 3000 containers.
! Question:
Is your yard big enough? Justify!
Avg Inv= 2000 x 2 = 4000 containers

Unpredictable Variability
Unpredictable variability refers to unknowable changes in input and/or capacity rates
Supply of pumpkins changes each year due to crop yield Exact demand for pumpkins each day

Basic Questions
What are the effects of variability on processes In particular, how does variability affect
Average Throughput Rate
Average Inventory
Iftheeffectsarenegative,howcanwedealwithit? 9
Average Flow Time

Consider a process with no variability
(1 person/min)
Throughput Rate? 1 person/min
Assume that all customers are identical
Customers arrive exactly 1 minute apart
The service time is exactly 1 minute for all the customers
Service time
(exactly 1 min)

Effect of Input Variability (no buffer)
Random Input
0, 1, 2 customers/min (with equal probability)
Service time
(exactly 1 min)
Throughput Rate? < 1 person per minute Assume that customers who find the ATM busy do not wait 1234567Effect of Input Variability (no buffer) When a process faces input variability, and a buffercannot be built, some input may get lost Input variability can reduce the throughput Lower throughput means Lost customers; lost revenue Customer dissatisfaction Less utilization of resources Dealing with Variability When the arrival rate of customers is unpredictable, what could you do to increase throughput? Add Buffer Increase Capacity(e.g., Add another ATM;Decrease the time it takes the ATM to serve a customer) Dealing with Variability When the arrival rate of customers is unpredictable and contains variability, what could you do to increase throughput? Add a buffer before the process. Staffing, equipment, physical space, etc. Decrease the variability via information. For example, messaging demand to match supply (appointment systems) Effect of Input Variability (with buffer) Random Input0, 1, 2 customers/min (with equal probability)Throughput Rate?Waiting time Service time (exactly 1 min) Now assume that customers wait We can build-up an inventory bufferEffect of Input Variability (with buffer) Random Input0, 1, 2 customers/min (with equal probability)Throughput Rate?Waiting time Service time (exactly 1 min) 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 Littles Law still holds Quantifying Variability So far, we focused on qualitative effect of variability Without buffer, input may get lost and throughput may With buffer, queue may build up, flow time may increase How long is the queue on average? How long does a customer have to wait? We would like to quantify average inventory, average waiting time, and average throughput rate. We first learn how to represent variability. We then discuss some measures of variability that we will need. We then look at a single server Queuing model. We first introduce a few standard notations We then introduce PK formula to calculate average inventory as a function of utilization and variability measure (and by using Littles law average waiting time). Representing variability:Short Review on Probability (1)Discrete Random Variable and ProbabilityThrow a dice; the number you get is a discrete random variable:1, w.p. 1/62, w.p. 1/63, w.p. 1/64, w.p. 1/65, w.p. 1/66, w.p. 1/6P{X = 2} = 1/6Probability(Probability mass)Cumulative probability P{X 2} = P{X=1 or X=2} = 1/3 P{X 2.1} = P{X=1 or X=2} = 1/3P(Xx) 1234561/6 1/3 1/2 P{X x} is a function of x, called the cumulative distribution function (CDF) 20 Representing variability:Short Review on Probability (2)Continuous Random Variable and ProbabilityThe time between two customers arrival times is acontinuous random variableProbability densityCumulative distribution function (CDF)ox P(Xx)= 0 f(s)dsVariability Measures Source of variability:Variability from inflow of flow units.Measures of VariabilityVariability in processing time.For a given random variable X with mean where mean denotes the expected value ofa random variable., E[X] we define Variance V[X]: is a measure of variability which is equal to the expected value of the square of deviations of the random variable around its mean, i.e., Var[X] = E[(X )2]. Also sometimes referred to as X2 or 2.Standard deviation denoted by : A measure of variability, which is equal to square root of vpariance. It shows how spread out the instances of the random variable. X = Var[X].Coecient of Variation CV Standard deviation divided by the mean, i.e.,CV = . It is a relative measure of uncertainty in a random variable. Since both standard deviation and mean have the same measurement unit, CV is a unitless measure. Based on these measures of variability we introduce a simple formula to calculate the average inventory. Single Server Queuing Model As we will see, average inventory is a function of Average input rate Average capacity Utilization of the server The variation of the interarrival times More precisely the coefficient of variation of interarrival time The variation of the service time More precisely the coefficient of variation of service time A Single Server Process (a few notations)Long-run average input rate Process BoundaryA queue forms in a bufferAverage processing time by one server(Average) Customer inter-arrival timeLong-run average processing rate of a single server A single phase service system is stable whenever l < We are focusing on long-run averages, Ignoring the predictable variability thatmay be occurring in the short run. Single-Server ProcessAssumption: l < Arrival rate:l persons/min (average input rate)On average, 1 person arrives every E{a} unit of time.Service rate: persons/min (average capacity rate)Average throughput rate l persons/minThus, l = 1 / E{a} …On average, 1 person can be served every E{s} unit of time. Thus, = 1 / E{s}Inter-arrival times:Service times:a1a2 a3 a4a5 a6 a7…s1 s2 s3 s4 s5 s6 s7 ATM Example (adjusting to notations)Suppose on average 40 people arrive at the ATM per hour.The ATM can serve on average 45 people per hour. Throughput rate. Utilization Average interarrival time . Average service time. l = 40 people/hour r = l/=40/45 E[a]= 1/l = 1/40 *60=3/2 minutes E[s]= 1/ = 1/45 *60= 4/3 minutes What are we trying to quantify? (a few notations) Avg inventory: IAvg queue length IqAvg number of peoplebeing served: Is Littles Law holds Iq = lTq Throughput rate = l Waiting time Tq Service time Ts TSystem CharacteristicsPerformance MeasuresAverage queue lengthAverage number of customers being served Average number of customers in the processUtilization(In a stable system, r = l/ < 100%) Average waiting time (in queue)Is I=Iq+Is Average time spent at the server Average flow time (in process) Average InventoryAssumption: l < Arrival rate:l persons/min (average input rate)Service rate: persons/min (average capacity rate)Average throughput rate l persons/min Average number of persons in the system: I = Iq + Is Question: Is=??? (Express Is in terms of l and )Answer: Is=l/ Pollaczek-Khinchin (PK) Formula I @ 2 C 2a + C 2s What is the relationship among variability,qinventory (queue length) and utilization? 1- 2 = for special cases in general Average queue length (excl. the one in service) (Long run) Average utilization= Average Throughput / Average Capacity = l / Ca = s{a}/E{a} Coefficient of variation (CV) of inter-arrival timesCs = s{s}/E{s} Coefficient of variation (CV) of service times Performance Measure: Average queue length: Pollaczek-Khinchin (PK) Formula Ca= coefficient of variation of inter-arrival time = s{a}/E{a} average queue length (excl. the one in service) C 2a + C 2s 2 coefficient of variation of service times = s{s}/E{s} average utilization=average throughput / average capacity = l / ATM ExampleSuppose on average 40 people arrive at the ATM per hour.The ATM can serve on average 45 people per hour. Suppose the variance of interarrival times is 1 minute squared, and the standard deviation of service time is 1/3 minute. What are the coefficients of variation of interarrival time and service time?Calculate Utilization. Calculate the average queue length. Calculate the average waiting time.E[a]= 1/l=1/40*60=1.5 minutes, Ca=1/1.5=2/3 E[s]= 1/ =1/45*60=4/3 minutes, Cs=(1/3)/(4/3)=1/4 = l/ =40/45 Iq @ 2C2a+C2s 1- 2 Iq= (8/9)2/(1-8/9)x( (2/3)^2 +(1/4)^2)/2 Tq=Iq/ l=Iq/40 hours Another Example Customers arrive at rate 4/hour, and mean service time is 10 minutes Assume that standard deviation of inter-arrival times equals 5 minutes, and the standard deviation of service time equals 3 minutes What is the average size of the queue? What is the average time that a flow unit spends in the queue? l = 4 E[a] =1/ 4 hour =6 E[s]=1/6hourr=l=46=23 s[a]=1/12hourCa =s[a]=1/12=1 E[a] 1/4 3s[s]=1/20hourI @ 2C2+C2=(23)2 (13)2+(310)2Cs =s[s]=1/20= 3 E[s] 1/6 101- 2 13 2T=I l=I 4 qqq PK Formula and OM Triangle2 C2 +C2 l l C2 +C21- 2 -l 2 I@ a s= a s= Capacity Rate l = Input RateINFORMATIONVariability Impact of Utilization ( = l/) Impact on Queue Length (Inventory) I @ 2C2+C2 as Impact on Waiting Time (Flow Time) T = I l Littles Law qq Queue LengthWaiting TimeUtilization 0% 100%Impact of VariabilityIq @ 2 1- Queue length or waiting timeIncreasing variabilityUtilization rC 2a + C 2s 2 Impact of VariabilityInventory orLead Time bHigh InventoryI q @ 2 C 2a + C 2s 1- 2Extra CapacityLow Variability50% 100% Utilization r ! Firm with large amount of capacity (e.g., utilization = 0.5) is at point (a)! If you reduce capacity, then you increase the systems utilization (e.g., utilization= 0.9) and move up the curve to point (b).! Thus, inventory (or units in queue) and lead-time are dramatically increased. The OM Triangle Inventory or Lead Time Information c Firms choice of optimal position on OM triangle is determined by: Its strategic (business) objective consideration of the relative costs of capacity, information, and inventory (NPV, cost-benefit analysis, project management, etc.) 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 Littles Law still holds Summary: Intro to Queueing In systems with unpredictable variability, averages dont tell the whole story Variability can cause loss of throughput rate Inventory buffers or increased capacity may be needed to deal with variability In variable systems, inventory and flow time increase non-linearly with utilization (see the PK formula) The impact of variability (on inventory and flow time) can be quantified using the PK formula, Littles Law, and assumptions about the probability distributions of variability. Summary: Intro to Queueing What are all the components to a queueing model? Input process Service process Performance measures There is a cost/performance trade-off between inventory, capacity and information. All three can improve the operational performance of your business. Depending on the industry, it may be more cost-effective to spend money on one (or a combination) versus another. Greek Letters used in Queuing Models – lambda: input rate- mu: service rate (capacity)- rho: server utilization (single-server model) – sigma: standard deviation Readings & More… For a better grasp of the key concepts, the following readings are only recommended: Chapter 4 of Textbook Keep track of the Quercus site for the course. Next Class: The rest of Queueing Model/ Midterm preparation. Dont forget: Assignment #2: (weight: 2%) is available on utorsubmit and will be due on 2021/02/06, Assignment #3 (weight 2%) will be available on utorsubmit by Friday and will be due on 2021/02/13, 11:59 PM.! Midterm Exam: Feb 18, 7pm-9pm, (2 hrs) If you have academic conflicts you should have emailed me. Makeup Midterm Exam: The exam will be available only to students with conflicts (The students who informed us will receive an email from us with more information). If you have a conflict and did not let us know, you should email us as soon as possible. Sample midterms are available on Quercus Case(group) work will b available on Quercus by the end of this week under modules, case 1. The instructions on how to submit your work are provided on the file (and also will be announced on Quercus). The case study is due Sunday Feb. 27th,11:59PM. Our lectures will continue to be on zoom until February 27th. Appendix: Another Example Customers arrive at rate 4/hour, and mean service time is 10 minutes Assume that standard deviation of inter-arrival times equals 5 minutes, and the standard deviation of service time equals 3 minutes What is the average size of the queue? What is the average time that a flow unit spends in the queue? l = 4 E[a] =1/ 4 hour =6 E[s]=1/6hourr=l=46=23 s[a]=1/12hourCa =s[a]=1/12=1 E[a] 1/4 3s[s]=1/20hourI @ 2C2+C2=(23)2 (13)2+(310)2Cs =s[s]=1/20= 3 E[s] 1/6 101- 2 13 2T=I l=I 4 qqq Appendix: Practice Problem (play the animation)Professor Longhair holds office hours everyday to answer students questions.Students arrive at an average rate of 50 per hour.Professor Longhair can process students at an average rate of 60 per hour.What is the average number of students waiting outside Professor Longhairs office, and how long do they wait on average?Assume the inter-arrival time and the service time are both exponentially distributed(We can also say that the arrival rate follows a Poisson distribution) l = 50 = 60r = l = 50 60 = 5 6 Iq=l2 =502 =25(-l) 60(60-50) 6T = Iq = 25 6 = 1Appendix: The OM Triangle: Example Webers Hamburger on Highway 11 Inventory: Customers line up for burgers, but this inventory is costly, because customers may get annoyed that they waited so long and decide not to come next time. Capacity: If you add another worker (i.e., more capacity), you can quickly cook up a larger number of burgers when the place gets busy. Information: Have people pre-order their burgers from the roard. Create a phone application so that customers can fill in exactly what they want and their distance from Webers. The orders are completed right before the customer arrives! You have a choice of spending money on either inventory, capacity, or information (or a combination) Appendix: Managing the Psychology of Queuing Unoccupied time feels longer than occupied time Process waits feel longer than in process waits Anxiety makes waits seem longer Uncertain waits seem longer than known, finite waits Unexplained waits are longer than explained Unfair waits are longer than equitable waits The more valuable the service, the longer the customer will Solo waits feel longer than group waits CS: assignmentchef QQ: 1823890830 Email: [email protected]