MGTSC 352 Lecture 23 Inventory Management Big Blue
MGTSC 352 Lecture 23: Inventory Management Big Blue Congestion Management Introduction: Asgard Bank example Simulating a queue Types of congested systems, queueing template
Pg. 161 Big. Blue. Pills, Inc. • • Expensive drug treatments Perishable – last only 3 months Order once every 3 months Regular cost: $400 per treatment If demand > order size, place rush order Rush cost: $1, 000 per treatment Price to patient: $650 How much should they order?
Single period models • Perishable product • Past demand data • Must decide how much to order before knowing actual demand for the period • Must live with the consequences • We’ve seen this before: it’s called the “newsvendor problem”
Five Years of Demand Data
Solution 1 • Average demand = 18, so … – … let’s order 18 each quarter – Profit = 18 (650 – 400) = $4, 500 – Right? • Q < D lose ? / unit • Q > D lose ? / unit • Do these cancel out on average?
Tradeoffs • Q>D wasted product – Lose $400 – On product bought but not sold • Q<D loss on Sale – Lose $1000 - $650 = $350 – On Rush order (no lost sales) • If order = average demand each outcome will occur half the time. • Is that what you want?
Solution 2: simulation Solution 1 predicted this profit with Q = 18
The Flaw of Averages When input is uncertain. . . output given average input may not equal the average output
Huh? • Average Big. Blue. Pill demand = 18 • Profit for Q = 18, given D = 18: $4, 500 • Average profit with Q = 18: $1, 740 – Less than half • Optimal Q = 20, with avg. profit: $1, 786 • Using avg. demand (ignoring variability) • Seriously overestimates profit • Results in a suboptimal decision
How bad can it get? • What if rush cost is $1, 800 (instead of $1, 000) • The “averaging analyst” will still recommend Q = 18 and estimate P = $4, 500. • The actual profit with Q = 18 will be -$565. • Using Q = 20 generates P = $1, 217 • Using average inputs is a bad idea. “How bad” will depend on data.
In general Profit(AVERAGE(Demand 1, Demand 2, …, Demandn)) AVERAGE(Profit(Demand 1), Profit(Demand 2), …, Profit(Demandn))
Simple example of the flaw of averages: • A drunk on a highway • Random walk
Consider the drunk’s condition • The AVERAGE location of the drunk – Middle of the road • The outcome at the middle of the road ALIVE • What do you think the average outcome for the drunk is? DEAD Average inputs do not result in average outputs.
Congestion Asgard Bank ATM Pg. 168
pg. 168 Asgard Bank: Times Between Arrivals (pg. 173)
Asgard Bank: Arrival Rate • Given: avg. time between arrivals = 1. 00 minute • average arrival rate per hour = =?
Asgard Bank: Service Times
Asgard Bank: Service Rate • Given: avg. service time = 0. 95 minutes • average service rate per hour (if working continuously) = = • Note: – the service rate is not the rate at which customers are served – it’s the rate at which customers could be served, if there were enough customers – service rate = capacity of a server
Asgard Bank – Collecting Data Real system: • Record arrival time, service start, service end • Compute inter-arrival times, service times, waiting time, time in system Simulating the system: • Simulate inter-arrival times, service times • Compute arrival time, service start, service end, time in system, waiting time
Why Do Customers Wait? Given: Average inter-arrival time = 1. 00 min. Average service time = 0. 95 min. 0. 95 Customer leaves Next customer arrives and begins service Customer arrives and begins service What’s missing from this picture? time
What’s missing from this Picture? VARIABILITY!
Including Randomness: Simulation • Service times: Normal distribution, mean = 57/3600 of an hour stdev = 10/3600 of an hour MAX(NORMINV(RAND(), 57/3600, 10/3600), 0) • Inter-arrival times: Exponential distribution, mean = 1/ 60 of an hour. (1/60)*LN(RAND()) To Excel …
Simulated Lunch Hour 1: 71 arrivals
Simulated Lunch Hour 2: 50 arrivals
Simulated Lunch Hour 3: Unused capacity
Causes of Congestion • Higher than average number of arrivals • Lower than average service capacity • Lost capacity due to timing Lesson: For a service where customers arrive randomly, it is not a good idea to operate the system close to its average capacity
Anatomy of a Congested System (pg. 172) waiting room = queue potential customers parallel servers
Notation: M/M/s/K/N 1. Inter-arrival time distribution 2. Service time distribution • • M = exponential distribution G = general distribution 3. s = number of servers 4. K = max. # of customers allowed in system • being served + waiting 5. N = population size • K and N are left out when they are infinite
Types of Congested Systems We Will Analyze (pg. 173) Type Service Inter-arrival # of # that time dist’n servers can wait # of potential customers M/M/s Exponential exponential s M/M/s/s+C exponential s C Finite Population exponential s M M/G/1 general exponential 1
Analyzing a Congested System (pg. 174) Inputs System Description Model of the System Outputs Measures of Quality of Service Measures important to Servers
System Description · · (mu) = service rate (per server per time unit) (lambda) = arrival rate (per time unit) s = number of servers C = maximum number that can wait in line = queue capacity · M = number of potential customers · s (sigma sub S) = standard deviation of service times S
Measures of Quality of Service · Wq = average time in queue · W = average time in system · Lq = average number of customers in queue · L = average number of customers in system · SL = service level = fraction of customers that wait less than some given amount of time · Pr. Balk = fraction of customers that balk (do not enter system) · Pr. Wait = fraction of customers that wait
Measures Important to Servers · r (rho) = utilization = fraction of time each server is busy
And now the formulas for the simplest case: M / 1 Or would you prefer an Excel template?
Template. xls • Does calculations for – – • M/M/s/s+C M/M/s/ /M M/G/1 Want to know more? Go to http: //www. bus. ualberta. ca/aingolfsson/qtp/ • Asgard Bank Data – – Model: M/G/1 Arrival rate: 1 per minute Average service time: 57/60 min. St. dev of service time: 10/60 min.
Asgard Conclusions • The ATM is busy 95% of the time. • Average queue length = 9. 3 people • Average no. in the system = 10. 25 (waiting, or using the ATM) • Average wait = 9. 3 minutes • What if the arrival rate changes to … – 1. 05 / min. ? – 1. 06 / min. ?
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