Post on 01-Jan-2016
description
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Safety Capacity
Effect of Buffer Capacity
Process Data– Ri = 20/hour, Tp = 2.5 mins, c = 1, K = # Lines – c
Performance Measures
K 4 5 6
Ii 1.23 1.52 1.79
Ti 4.10 4.94 5.72
Pb 0.1004 0.0771 0.0603
R 17.99 18.46 18.79
0.749 0.768 0.782
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Safety Capacity
Economics of Capacity Decisions
Cost of Lost Business Cb
– $ / customer
– Increases with competition
Cost of Buffer Capacity Ck
– $/unit/unit time
Cost of Waiting Cw
– $ /customer/unit time
– Increases with competition
Cost of Processing Cs
– $ /server/unit time
– Increases with 1/ Tp
Tradeoff: Choose c, Tp, K
– Minimize Total Cost/unit time
= Cb Ri Pb + Ck K + Cw I (or Ii) + c Cs
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Safety Capacity
Optimal Buffer Capacity
Cost Data– Cost of telephone line = $5/hour, Cost of server = $20/hour, Margin lost =
$100/call, Waiting cost = $2/customer/minuteEffect of Buffer Capacity on Total Cost
K $5(K + c) $20 c $100 Ri Pb $120 Ii TC ($/hr)
4 25 20 200.8 147.6 393.4
5 30 20 154.2 182.6 386.4
6 35 20 120.6 214.8 390.4
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Safety Capacity
Optimal Processing Capacity
c K = 6 – c Pb Ii TC ($/hr) = $20c + $5(K+c) + $100Ri Pb+
$120 Ii
1 5 0.0771 1.542 $386.6
2 4 0.0043 0.158 $97.8
3 3 0.0009 0.021 $94.2
4 2 0.0004 0.003 $110.8
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Safety Capacity
Performance Variability
Effect of Variability– Average versus Actual Flow time
Time Guarantee – Promise
Service Level– P(Actual Time Time Guarantee)
Safety Time– Time Guarantee – Average Time
Probability Distribution of Actual Flow Time– P(Actual Time t) = 1 – EXP(- t / T)
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Safety Capacity
Effect of Blocking and Abandonment
Blocking: the buffer is full = new arrivals are turned away
Abandonment: the customers may leave the process before being served
Proportion blocked Pb
Proportion abandoning Pa
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Safety Capacity
Net Rate: Ri(1- Pb)(1- Pa)
Throughput Rate:R=min[Ri(1- Pb)(1- Pa),Rp]
Effect of Blocking and Abandonment
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Safety Capacity
Example 8.8 - DesiCom Call Center
Arrival Rate Ri= 20 per hour=0.33 per min
Processing time Tp =2.5 minutes (24/hr)Number of servers c=1Buffer capacity K=5
Probability of blocking Pb=0.0771
Average number of calls on hold Ii=1.52
Average waiting time in queue Ti=4.94 minAverage total time in the system T=7.44 minAverage total number of customers in the system I=2.29
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Safety Capacity
Throughput Rate
R=min[Ri(1- Pb),Rp]= min[20*(1-0.0771),24]
R=18.46 calls/hour
Server utilization:
R/ Rp=18.46/24=0.769
Example 8.8 - DesiCom Call Center
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Safety Capacity
Example 8.8 - DesiCom Call Center
Number of lines 5 6 7 8 9 10
Number of servers c 1 1 1 1 1 1
Buffer Capacity K 4 5 6 7 8 9
Average number of calls in queue
1.23 1.52 1.79 2.04 2.27 2.47
Average wait in queue Ti (min) 4.10 4.94 5.72 6.43 7.08 7.67
Blocking Probability Pb (%) 10.04 7.71 6.03 4.78 3.83 3.09
Throughput R (units/hour) 17.99 18.46 18.79 19.04 19.23 19.38
Resource utilization .749 .769 .782 .793 .801 .807
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Safety Capacity
Capacity Investment Decisions
The Economics of Buffer Capacity
Cost of servers wages =$20/hour
Cost of leasing a telephone line=$5 per line per hour
Cost of lost contribution margin =$100 per blocked call
Cost of waiting by callers on hold =$2 per minute per customer
Total Operating Cost is $386.6/hour
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Safety Capacity
Example 8.9 - Effect of Buffer Capacity on Total Cost
Number of lines n 5 6 7 8 9
Number of CSR’s c 1 1 1 1 1
Buffer capacity K=n-c 4 5 6 7 8
Cost of servers ($/hr)=20c 20 20 20 20 20
Cost of tel.lines ($/hr)=5n 25 30 35 40 45
Blocking Probability Pb (%) 10.04 7.71 6.03 4.78 3.83
Lost margin = $100RiPb200.8 154.2 120.6 95.6 76.6
Average number of calls in queue Ii1.23 1.52 1.79 2.04 2.27
Hourly cost of waiting=120Ii147.6 182.4 214.8 244.8 272.4
Total cost of service, blocking and waiting ($/hr)
393.4 386.6 390.4 400.4 414
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Safety Capacity
Example 8.10 - The Economics of Processing Capacity
The number of line is fixed: n=6
The buffer capacity K=6-c
c K Blocking Pb(%)
Lost Calls RiPb
(number/hr)
Queue length
Ii
Total Cost ($/hour)
1 5 7.71% 1.542 1.52 30+20+(1.542x100)+(1.52x120)=386.6
2 4 0.43% 0.086 0.16 30+40+(0.086x100)+(0.16x120)=97.8
3 3 0.09% 0.018 0.02 30+60+(0.018x100)+(0.02x120)=94.2
4 2 0.04% 0.008 0.00 30+80+(0.008x100)+(0.00x120)110.8
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Safety Capacity
Variability in Process Performance
Why considering the average queue length and waiting time as performance measures may not be sufficient?
Average waiting time includes both customers with very long wait and customers with short or no wait.
We would like to look at the entire probability distribution of the waiting time across all customers.
Thus we need to focus on the upper tail of the probability distribution of the waiting time, not just its average value.
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Safety Capacity
Example 8.11 - WalCo Drugs
One pharmacist, DaveAverage of 20 customers per hourDave takes Average of 2.5 min to fill prescriptionProcess rate 24 per hourAssume exponentially distributed interarrival and
processing time; we have single phase, single server exponential model
Average total process is;T = 1/(Rp – Ri) = 1/(24 -20) = 0.25 or 15 min
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Safety Capacity
Example 8.11 - Probability distribution of the actual time customer spends in process
(obtained by simulation)
0
2000
4000
6000
8000
10000
12000
14000
Total Time in Process
Fre
qu
ency
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Safety Capacity
Example 8.11 - Probability Distribution Analysis
65% of customers will spend 15 min or less in process
95% of customers are served within 40 min
5% of customers are the ones who will bitterly complain. Imagine if they new that the average customer spends 15 min in the system.
35% may experience delays longer than Average T,15min
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Safety Capacity
Service Promise:Tduedate , Service Level & Safety Time
SL; The probability of fulfilling the stated promise. The Firm will set the SL and calculate the Tduedate from the probability distribution of the total time in process (T).
Safety time is the time margin that we should allow over and above the expected time to deliver service in order to ensure that we will be able to meet the required date with high probability
Tduedate = T + Tsafety
Prob(Total time in process <= Tduedate) = SL
Larger SL results in grater probability of fulfilling the promise.
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Safety Capacity
Due Date Quotation
Due Date Quotation is the practice of promising a time frame within which the product will be delivered.
We know that in single-phase single server service process; the Actual total time a customer spends in the process is exponentially
distributed with mean T.
SL = Prob(Total time in process <= Tduedate) = 1 – EXP( - Tduedate /T)
Which is the fraction of customers who will no longer be delayed more than promised.
Tduedate = -T ln(1 – SL)
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Safety Capacity
Example 8.12 - WalCo Drug
WalCo has set SL = 0.95Assuming total time for customers is exponential
Tduedate = -T ln(1 – SL)
Tduedate = -T ln(0.05) = 3TFlow time for 95 percentile of exponential distribution is three times
the average T
Tduedate = 3 * 15 = 4595% of customers will get served within 45 min
Tduedate = T + Tsafety
Tsafety = 45 – 15 = 30 min30 min is the extra margin that WalCo should allow as protection
against variability
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Safety Capacity
Relating Utilization and Safety Time: Safety Time Vs. Capacity Utilization
Capacity utilization ρ 60 % 70% 80% 90%
Waiting time Ti 1.5Tp 2.33Tp 4Tp 9Tp
Total flow time T= Ti + Tp 2.5Tp 3.33Tp 5Tp 10Tp
Promised time Tduedate 7.7Tp 10Tp 15Tp 30Tp
Safety time Tsafety = Tduedate – T 5Tp 6.67Tp 10Tp 20Tp
Higher the utilization; Longer the promised time and Safety time
Safety Capacity decreases when capacity utilization increases
Larger safety capacity, the smaller safety time and therefore we can promise a shorter wait
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Safety Capacity
Managing Customer Perceptions and Expectations
Uncertainty about the length of wait (Blind waits) makes customers more impatient.
Solution is Behavioral Strategies
Making the waiting customers comfortable
Creating distractions
Offering entertainment
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Safety Capacity
Thank you
Questions?