O PERATİONS & L OGİSTİCS M ANAGEMENT İN A İR T RANSPORTATİON P ROFESSOR D AVİD G İLLEN (U NİVERSİTY OF B RİTİSH C OLUMBİA ) & P ROFESSOR B ENNY M ANTİN (U NİVERSİTY OF W ATERLOO ) Istanbul Technical University Air Transportation Systems and Infrastructure Air Transportation Management Strategic Planning M.Sc. Program Module 5-6 : 11 June 2014
VARIABILITY IN PROCESSES AND QUEUES 2
L EARNİNG O BJECTİVES • Variability and Process Analysis – What is variability ? – What impact does variability have on processes? – How can we quantify the impact of variability on processes? – How can we manage variability in processes? 3
W HAT İS V ARİABİLİTY ? Security check Variability comes from … Variable Variable input Capacity 4
T YPES OF V ARİABİLİTY Predictable Variability Unpredictable Variability Predictable Variability … Unpredictable Variability … … refers to “knowable” … refers to “unknowable” changes in input and/or changes in input and/or capacity rates capacity rates Demand of pumpkins will go up Supply of pumpkins will go down if during Thanksgiving the crop fails • Both types of variability exist simultaneously – Pumpkin sales will go up during Thanksgiving, but we do not know the exact sales of pumpkins 5
Predictable Variability Unpredictable Variability Can be controlled by making Is the result of the lack of changes to the system knowledge or information • We could increase or decrease the • Usually can be expressed with a demand for pumpkins by increasing probability distribution or decreasing the price • E.g., Express the probability that the • Restaurants will add staff during pumpkin crop will fail using a peak demand (lunch, dinner, etc.) probability distribution Can be reduced by gaining more knowledge or collection information • By paying close attention to weather patterns, we could increase the accuracy of our prediction that the pumpkin crop will fail 6
S HORT R EVİEW ON P ROBABİLİTY (1) Discrete Random Variable and Probability Throw a dice; the number you get is a discrete random variable: 1, w.p. 1/6 2, w.p. 1/6 Probability 3, w.p. 1/6 (Probability X = 1/6 4, w.p. 1/6 mass) x 1 2 3 4 5 6 5, w.p. 1/6 Cumulative 6, w.p. 1/6 1 5/6 probability 2/3 P{X = 2} = 1/6 1/2 P(X≤ x ) 1/3 P{X ≤ 2} = P{X=1 or X=2} = 1/3 1/6 x P{X ≤ 2.1} = P{X=1 or X=2} = 1/3 1 2 3 4 5 6 P{X ≤ x } is a function of x , called the cumulative distribution function (CDF) 7
S HORT R EVİEW ON P ROBABİLİTY (2) Continuous Random Variable and Probability The time between two customers’ arrival times is a continuous random variable Probability f ( x ) density x 0 Cumulative distribution 1 function (CDF) x P(X ≤ x ) = f (s) ds 0 0 x 8
B ASİC Q UESTİONS • What are the effects of variability on processes – In particular, how does variability affect Average Average Average Flow Throughput Inventory Time Rate • If the effects are negative, how can we deal with it? 9
T HE “M AKE -T O -O RDER ” D İCE G AME • Retail store will roll dice first to observe demand, Retail Store which will be communicated to the factory Factory • Factory will roll dice to observe capacity • Factory will satisfy retailer demand, but is constrained by realized capacity – For example, if demand is 3 and capacity is 4, then factory will give the retail store 3 units What is the average – But if demand is 5 and capacity is 4, then demand? factory will give the retail store only 4 units What is the average – No backlog capacity? • Assume 1 roll of demand and capacity = 1 day 10
C ONSİDER A PROCESS WİTH NO VARİABİLİTY Throughput Input ATM Rate ? (1 person/min) Service time (exactly 1 min) • Assume that all customers are identical • Customers arrive exactly 1 minute apart • The service time is exactly 1 minute for all the customers 11
E FFECT OF I NPUT V ARİABİLİTY ( NO BUFFER ) Random Input Throughp ATM 0, 1, 2 ut customers/min Rate ? (with equal probability) Service time (exactly 1 min) • Assume that customers who find the ATM busy do not wait time 1 2 3 4 5 6 7 12
E FFECT OF I NPUT V ARİABİLİTY ( NO BUFFER ) • When a process faces input variability, and a buffer cannot 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 • Little’s Law holds 13
D EALİNG WİTH V ARİABİLİTY • 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) 14
E FFECT OF I NPUT V ARİABİLİTY ( WİTH BUFFER ) Random Input Throughput ATM 0, 1, 2 Buffer Rate ? customers/min (with equal probability) Waiting time Service time (exactly 1 min) • Now assume that customers wait We can build-up an inventory buffer 15
E FFECT OF I NPUT V ARİABİLİTY ( WİTH BUFFER ) • If we can build up an inventory buffer , variability leads to – An increase in the average inventory in the process – An increase in the average flow time • Little’s Law holds 16
T HE OM T RİANGLE If a firm is striving to meet the CAPACITY random demand, then it can use capacity , inventory and information (variability reduction) as substitutes You cannot have low inventory, low capacity, and low information acquisition effort INVENTORY INFORMATION at the same time. This is a (Variability Reduction) trade-off. 17
O PERATİONS AT D ELL • Inventory as “the physical embodiment of bad information” (a senior exec at Dell) • Substitute information for inventory • Less inventory =>higher inventory turns Fast Tech 18
Q UANTİFYİNG V ARİABİLİTY • So far, we focused on qualitative effect of variability – Without buffer, input may get lost and throughput may decrease – With buffer, queue may build up, flow time may increase • But … – How long is the queue on average? – How long does a customer have to wait? 19
W HY İS İT İMPORTANT TO QUANTİFY VARİABİLİTY AND İTS İMPACT ? These quantitative measures of process performance are important to any functions of a company Finance Marketing Wants to attract investors based Wants to use the short waiting on excellent operations time as a selling point performance Accounting Operations Wants to know how much Wants to shorten the queue, and wants money is tied up in the queue to quantify the trade-offs between capacity, inventory and variability What is the impact (on inventory and flow time) of increasing/decreasing capacity by 10%? 20
F İRST S TEPS İN Q UANTİFYİNG V ARİABİLİTY • Probability Statements P(X=4) P(20 < T ≤ 30) • Variances and Standard deviation – These lead to probability statements • Coefficient of variation (CV) Standard Deviation CV Mean 21
A S İNGLE S ERVER P ROCESS Long-run average input rate Server (Average) Customer inter-arrival Buffer 1/ time Process Long-run average processing rate of Boundary a single server A queue forms in a buffer Average processing time by one 1/ server A single phase service system is stable Note: We are focusing on long- whenever < run averages, ignoring the predictable variability that may Buffer capacity (for now, let K = ) K be occurring in the short run. In Number of servers in the resource c reality, we should be concerned pool (for now, let c=1) with both types of variability 22
W HAT ARE WE TRYİNG TO QUANTİFY ? I Little’s Law holds Throughput I q I s Server I q = T q rate = Buffer I s = T s Waiting time T q Service time T s I = T T System Characteristics Performance Measures T q Average waiting time (in queue) Utilization (In a stable system, I q Average queue length = / < 100%) T s Average time spent at the server Safety - Capacity I s Average number of customers being served T=T q +T s Average flow time (in process) I=I q +I s Average number of customers in the process 23
Q UİCK “Q UİZ ” Service rate: Assumption: ≤ persons/min (average capacity rate) Arrival rate: Average persons/min throughput rate Server (average input Buffer persons/min rate) • Average number of persons in the system: I = I q + I s • Question: I s =??? (Express I s in terms of and ) • Answer: I s = / 24
S İNGLE -S ERVER Q UEUİNG M ODEL Service rate: persons/min Assumption: ≤ (average capacity rate) Arrival rate: Average persons/min throughput rate Server (average input Buffer persons/min rate) On average, 1 person arrives every E{a} min. Thus, = 1 / E{ a} … Inter-arrival Time a 1 a 2 a 3 a 4 a 5 a 6 a 7 … times: Service s 1 s 2 s 3 s 4 s 5 s 6 s 7 times: On average, 1 person can be served every E{s} min. Thus, = 1 / E{s} 25
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