Safety Capacity Planning in Services Industry 1 v

Safety Capacity Planning in Services Industry 1 v Matching Supply and Demand in Service Processes v Performance Measures v Causes of Waiting v Economics of Waiting v Management of Waiting Time v The Sof-Optics Case

Safety Capacity Make to stock vs. Make to Order 2 Made-to-stock operations (Chapters 6&7) § Product is manufactured and stocked in advance of demand § Inventory permits economies of scale and protects against stockouts due to variability of inflows and outflows v Make-to-order process (Chapter 8) § Each order is specific, cannot be stored in advance § Process Manger needs to maintain sufficient capacity § Variability in both arrival and processing time § Role of capacity rather than inventory § Safety inventory vs. Safety Capacity § Example: Service operations v

Safety Capacity Examples 3 v. Banks (tellers, ATMs, drive-ins) v. Fast food restaurants (counters, drive-ins) v. Retail (checkout counters) v. Airline (reservation, check-in, takeoff, landing, baggage claim) v. Hospitals (ER, OR, HMO) v. Service facilities (repair, job shop, ships/trucks load/unload) v. Some production systems- to some extend (Dell computer) v. Call centers (telemarketing, help desks, 911 emergency)

Safety Capacity The Desi. Talk Call Center 4 The Call Center Process Incoming Calls (Customer Arrivals) Calls on Hold (Service Inventory) Blocked Calls Abandoned Calls (Due to busy signal) (Due to long waits) Sales Reps Processing Calls (Service Process) Calls In Process (Due to long waits) Answered Calls (Customer Departures)

Safety Capacity Service Process Attributes 5 Ri : customer arrival (inflow) rate inter-arrival time = 1/Ri: Tp : processing time processing rate per recourse = R’p = 1/Tp Rp: process capacity with c recourses, Rp = c/Tp Throughput (flow rate), R = Min(Ri, Rp) Utilization: r = R/Rp Safety Capacity: Rs = Rp-Ri Ti: waiting time in the inflow buffer Ii: number of customers waiting in the inflow buffer K: buffer capacity

Safety Capacity Operational Performance Measures 6 Flow time T = Ti + Tp Inventory I = Ii + Ip Flow Rate R = Min (Ri, Rp) Stable Process = Ri < Rp, , so that R = Ri Safety Capacity Rs = Rp - Ri I = Ri T Ii = Ri Ti Ip = Ri Tp = Ip / c = Ri Tp / c = Ri / Rp < 1 Number of Busy Servers = Ip= c = Ri Tp Fraction Lost Pb = P(Blocking) = P(Queue = K)

Safety Capacity Financial Performance Measures 7 Sales – Throughput Rate – Abandonment Rate – Blocking Rate Cost – Capacity utilization – Number in queue / in system Customer service – Waiting Time in queue /in system

Safety Capacity Flow Times with Arrival Every 4 Secs 8 Customer Number Arrival Time Departure Time in Process 1 0 5 5 2 4 10 6 3 8 15 7 4 12 20 8 5 16 25 9 6 20 30 10 7 24 35 11 8 28 40 12 9 32 45 13 10 36 50 14 What is the queue size? What is the capacity utilization?

Safety Capacity Flow Times with Arrival Every 6 Secs 9 Customer Number Arrival Time Departure Time in Process 1 0 5 5 2 6 11 5 3 12 17 5 4 18 23 5 5 24 29 5 6 30 35 5 7 36 41 5 8 42 47 5 9 48 53 5 10 54 59 5 What is the queue size? What is the capacity utilization?

Safety Capacity Effect of Variability 10 Customer Number Arrival Time Processing Time in Process 1 -A 0 7 7 2 -B 10 1 1 3 -C 20 7 7 4 -D 22 2 7 5 -E 32 8 8 6 -F 33 7 14 7 -G 36 4 15 8 -H 43 8 16 9 -I 52 5 12 10 -J 54 1 11 What is the queue size? What is the capacity utilization?

Safety Capacity Effect of Synchronization 11 Customer Number Arrival Time Processing Time in Process 1 -E 0 8 8 2 -H 10 8 8 3 -D 20 2 2 4 -A 22 7 7 5 -B 32 1 1 6 -J 33 1 1 7 -C 36 7 7 8 -F 43 7 7 9 -G 52 4 4 10 -I 54 5 7 What is the queue size? What is the capacity utilization?

Safety Capacity Conclusion 12 If inter-arrival and processing times are constant, queues will build up if and only if the arrival rate is greater than the processing rate If there is (unsynchronized) variability in inter-arrival and/or processing times, queues will build up even if the average arrival rate is less than the average processing rate If variability in interarrival and processing times can be synchronized (correlated), queues and waiting times will be reduced

Safety Capacity Causes of Delays and Queues 13 High, unsynchronized variability in - Interarrival times - Processing times High capacity utilization ρ= Ri / Rp or low safety capacity Rs =Ri - Rp due to : - High inflow rate Ri - Low processing rate Rp=c / Tp, which may be due to smallscale c and/or slow speed 1 / Tp

Safety Capacity Drivers of Process Performance 14 Two key drivers of process performance, Stochastic variability Capacity utilization They are determined by two factors: 1. The mean and variability of interarrival times (measured by total # of arrival over a fixed period of time) 2. The mean and variability of processing times (measured for different customers) Variability in the interarrival and processing times can be measured using standard deviation. Higher standard deviation means greater variability. – Not always an accurate picture of variability Coefficient of Variation: the ratio of the standard deviation to the mean. Ci = coefficient of variation for interarrival times Cp = coefficient of variation for processing times

Safety Capacity The Queue Length Formula 15 Utilization effect x Variability effect Ri / Rp, where Rp = c / Tp Ci and Cp are the Coefficients of Variation (Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

Safety Capacity Factors affecting Queue Length 16 This part factor captures the capacity utilization effect, which shows that queue length increases rapidly as the capacity utilization p increases to 1. The second factor captures the variability effect, which shows that the queue length increases as the variability in interarrival and processing times increases. Whenever there is variability in arrival or in processing queues will build up and customers will have to wait, even if the processing capacity is not fully utilized.

Safety Capacity Throughput- Delay Curve 17 Average Flow Time T Variability Increases Tp Utilization (ρ) 100%

Safety Capacity Example 8. 4 18 A sample of 10 observations on Interarrival times and processing times 7, 1, 7 2, 8, 7, 4, 8, 5, 1 10, 2, 10, 1, 3, 7, 9, 2 Tp= 5 seconds =AVERAGE () Avg. interarrival time = 6 Rp = 1/5 processes/sec. Ri = 1/6 arrivals / sec. Std. Deviation = 2. 83 =STDEV() Std. Deviation = 3. 94 Cp = 2. 83/5 = 0. 57 Ci = 3. 94/6 = 0. 66 Ri =1/6 < RP =1/5 R = Ri = R/ RP = (1/6)/(1/5) = 0. 83 With c = 1, the average number of passengers in queue is as follows: Ii = [(0. 832)/(1 -0. 83)] ×[(0. 662+0. 572)/2] = 1. 56 On average 1. 56 passengers waiting in line, even though safety capacity is Rs= RP - Ri = 1/5 - 1/6 = 1/30 passenger per second, or 2 per minutes

Safety Capacity Example 8. 4 19 Other performance measures: Ti=Ii/R = (1. 56)(6) = 9. 4 seconds Since TP= 5 T = Ti + TP = 14. 4 seconds Total number of passengers in the process is: I = R T = (1/6) (14. 4) = 2. 4 C=2 Rp = 2/5 ρ = (1/6)/(2/5) = 0. 42 Ii = 0. 08 c ρ Rs Ii Ti T I 1 0. 83 0. 03 1. 56 9. 38 14. 38 2. 4 2 0. 42 0. 23 0. 08 0. 45 5. 45 0. 91

Safety Capacity Exponential Model 20 In the exponential model, the interarrival and processing times are assumed to be independently and exponentially distributed with means 1/Ri and Tp. Independence of interarrival and processing times means that the two types of variability are completely unsynchronized. Complete randomness in interarrival and processing times. Exponentially distribution is Memoryless: regardless of how long it takes for a person to be processed we would expect that person to spend the mean time in the process before being released.

Safety Capacity The Exponential Model 21 Poisson Arrivals – Infinite pool of potential arrivals, who arrive completely randomly, and independently of one another, at an average rate Ri constant over time Exponential Processing Time – Completely random, unpredictable, i. e. , during processing, the time remaining does not depend on the time elapsed, and has mean Tp Computations – Ci = Cp = 1 – K = ∞ , use Ii Formula – K < ∞ , use Performance. xls

Safety Capacity Example 22 Interarrival time = 6 secs Ri = 10/min Tp = 5 secs Rp = 12/min for 1 server and 24 /min for 2 servers Rs = 12 -10 = 2 c ρ Rs Ii Formula Ti= Ri / Ii T= Ti+ 5/60 I= Ii + c ρ 1 0. 83 2 4. 16 0. 42 0. 5 5 2 0. 42 14 0. 18 0. 02 0. 1 1

Safety Capacity t ≤ t in Exponential Distribution 23 Mean inter-arrival time = 1/Ri Probability that the time between two arrivals t is less than or equal to a specific vaule of t P(t≤ t) = 1 - e-Rit, where e = 2. 718282, the base of the natural logarithm Example 8. 5: If the processing time is exponentially distributed with a mean of 5 seconds, the probability that it will take no more than 3 seconds is 1 - e-3/5 = 0. 451188 If the time between consecutive passenger arrival is exponentially distributed with a mean of 6 seconds ( Ri =1/6 passenger per second) The probability that the time between two consecutive arrivals will exceed 10 seconds is e-10/6 = 0. 1888

Safety Capacity Performance Improvement Levers 24 – Decrease variability in customer inter-arrival and processing times. – Decrease capacity utilization. – Synchronize available capacity with demand.

Safety Capacity Variability Reduction Levers 25 Customers arrival are hard to control – Scheduling, reservations, appointments, etc…. Variability in processing time – Increased training and standardization processes – Lower employee turnover rate = more experienced work force – Limit product variety

Safety Capacity Utilization Levers 26 If the capacity utilization can be decreased, there will also be a decrease in delays and queues. Since ρ=Ri/RP, to decrease capacity utilization there are two options: – Manage Arrivals: Decrease inflow rate Ri – Manage Capacity: Increase processing rate RP Managing Arrivals – Better scheduling, price differentials, alternative services Managing Capacity – Increase scale of the process (the number of servers) – Increase speed of the process (lower processing time)

Safety Capacity Synchronizing Capacity with Demand 27 Capacity Adjustment Strategies – Personnel shifts, cross training, flexible resources – Workforce planning & season variability – Synchronizing of inputs and outputs

Safety Capacity Effect of Pooling 28 Ri/2 Server 1 Queue 1 Ri Ri/2 Server 2 Queue 2 Server 1 Ri Queue Server 2

Safety Capacity Effect of Pooling 29 Under Design A, – We have Ri = 10/2 = 5 per minute, and TP= 5 seconds, c =1 and K =50, we arrive at a total flow time of 8. 58 seconds Under Design B, – We have Ri =10 per minute, TP= 5 seconds, c=2 and K=50, we arrive at a total flow time of 6. 02 seconds So why is Design B better than A? – Design A the waiting time of customer is dependent on the processing time of those ahead in the queue – Design B, the waiting time of customer is only partially dependent on each preceding customer’s processing time – Combining queues reduces variability and leads to reduce waiting times

Safety Capacity Effect of Buffer Capacity 30 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

Safety Capacity Economics of Capacity Decisions 31 Cost of Lost Business Cb – $ / customer – Increases with competition Cost of Buffer Capacity Ck – $/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

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/minute Effect 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 32

Safety Capacity Optimal Processing Capacity 33 c K=6–c Pb Ii TC ($/hr) = $20 c + $5(K+c) + $100 Ri 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

Safety Capacity Performance Variability 34 Effect of Variability – Average versus Actual Flow time Time Guarantee – Promise Service Level – P(Actual Time Guarantee) Safety Time – Time Guarantee – Average Time Probability Distribution of Actual Flow Time – P(Actual Time t) = 1 – EXP(- t / T)

Safety Capacity Effect of Blocking and Abandonment 35 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

Safety Capacity Effect of Blocking and Abandonment 36 Net Rate: Ri(1 - Pb)(1 - Pa) Throughput Rate: R=min[Ri(1 - Pb)(1 - Pa), Rp]

Safety Capacity Example 8. 8 - Desi. Com Call Center 37 Arrival Rate Ri= 20 per hour=0. 33 per min Processing time Tp =2. 5 minutes (24/hr) Number of servers c=1 Buffer 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 min Average total time in the system T=7. 44 min Average total number of customers in the system I=2. 29

Safety Capacity Example 8. 8 - Desi. Com Call Center 38 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

Safety Capacity Example 8. 8 - Desi. Com Call Center 39 Number of lines 5 6 7 8 9 10 Number of servers c 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

Safety Capacity Investment Decisions 40 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

Safety Capacity Example 8. 9 - Effect of Buffer Capacity on Total Cost 41 Number of lines n 5 6 7 8 9 Number of CSR’s c 1 1 1 Buffer capacity K=n-c 4 5 6 7 8 Cost of servers ($/hr)=20 c 20 20 20 Cost of tel. lines ($/hr)=5 n 25 30 35 40 45 Blocking Probability Pb (%) 10. 04 7. 71 6. 03 4. 78 3. 83 Lost margin = $100 Ri. Pb 200. 8 154. 2 120. 6 95. 6 76. 6 Average number of calls in queue Ii 1. 23 1. 52 1. 79 2. 04 2. 27 Hourly cost of waiting=120 Ii 147. 6 182. 4 214. 8 244. 8 272. 4 Total cost of service, blocking and waiting ($/hr) 393. 4 386. 6 390. 4 400. 4 414

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 Ri. Pb (number/hr) Queue length Ii Total Cost ($/hour) 1 5 7. 71% 1. 542 1. 52 30+20+(1. 542 x 100)+(1. 52 x 120)=386. 6 2 4 0. 43% 0. 086 0. 16 30+40+(0. 086 x 100)+(0. 16 x 120)=97. 8 3 3 0. 09% 0. 018 0. 02 30+60+(0. 018 x 100)+(0. 02 x 120)=94. 2 4 2 0. 04% 0. 008 0. 00 30+80+(0. 008 x 100)+(0. 00 x 120)110. 8 42

Safety Capacity Variability in Process Performance 43 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.

Safety Capacity Example 8. 11 - Wal. Co Drugs 44 One pharmacist, Dave Average of 20 customers per hour Dave takes Average of 2. 5 min to fill prescription Process rate 24 per hour Assume 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

Safety Capacity Example 8. 11 - Probability distribution of the actual time customer spends in process (obtained by simulation) 45

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, 15 min 46

Safety Capacity Tduedate Service Promise: , Service Level & Safety Time 47 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.

Safety Capacity Due Date Quotation 48 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)

Safety Capacity Example 8. 12 - Wal. Co Drug 49 Wal. Co has set SL = 0. 95 Assuming total time for customers is exponential Tduedate = -T ln(1 – SL) Tduedate = -T ln(0. 05) = 3 T Flow time for 95 percentile of exponential distribution is three times the average T Tduedate = 3 * 15 = 45 95% of customers will get served within 45 min Tduedate = T + Tsafety = 45 – 15 = 30 min is the extra margin that Wal. Co should allow as protection against variability

Safety Capacity Relating Utilization and Safety Time: Safety Time Vs. Capacity Utilization 50 Capacity utilization ρ Waiting time Ti Total flow time T= Ti + Tp Promised time Tduedate Safety time Tsafety = Tduedate – T 60 % 1. 5 Tp 2. 5 Tp 7. 7 Tp 5 Tp 70% 2. 33 Tp 3. 33 Tp 10 Tp 6. 67 Tp 80% 4 Tp 5 Tp 10 Tp 90% 9 Tp 10 Tp 30 Tp 20 Tp 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

Safety Capacity Managing Customer Perceptions and Expectations 51 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

Safety Capacity Thank you 52 Questions?