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

2. 2 Safety Capacity Make to stock vs. Make to Order  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  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

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

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

5. 5 Safety Capacity Service Process Attributes R i : customer arrival (inflow) rate inter-arrival time = 1/R i: T p : processing time processing rate per recourse = R’ p = 1/T p R p : process capacity with c recourses, R p = c/T p Throughput (flow rate), R = Min(R i, R p ) Utilization:  = R/R p Safety Capacity: R s = R p -R i T i : waiting time in the inflow buffer I i : number of customers waiting in the inflow buffer K: buffer capacity

6. 6 Safety Capacity Operational Performance Measures Flow time T = T i + T p Inventory I = I i + I p Flow Rate R = Min (R i, R p  Stable Process = R i < R p,, so that R = R i Safety Capacity R s = R p - R i I = R i  T  I i = R i  T i  I p = R i  T p  = I p / c   = R i  T p / c   = R i / R p < 1 Number of Busy Servers = I p = c  = R i  T p Fraction Lost P b = P(Blocking) = P(Queue = K)

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

8. 8 Safety Capacity Flow Times with Arrival Every 4 Secs Customer NumberArrival Time Departure Time Time in Process 1055 24106 38157 412208 516259 6203010 7243511 8284012 9324513 10365014 What is the queue size? What is the capacity utilization?

9. 9 Safety Capacity Customer NumberArrival Time Departure Time Time in Process 1055 26115 312175 418235 524295 630355 736415 842475 948535 1054595 Flow Times with Arrival Every 6 Secs What is the queue size? What is the capacity utilization?

10. 10 Safety Capacity Customer NumberArrival TimeProcessing Time Time in Process 1-A077 2-B1011 3-C2077 4-D2227 5-E3288 6-F33714 7-G36415 8-H43816 9-I52512 10-J54111 Effect of Variability What is the queue size? What is the capacity utilization?

11. 11 Safety Capacity Customer NumberArrival TimeProcessing Time Time in Process 1-E088 2-H1088 3-D2022 4-A2277 5-B3211 6-J3311 7-C3677 8-F4377 9-G5244 10-I5457 Effect of Synchronization What is the queue size? What is the capacity utilization?

12. 12 Safety Capacity Conclusion 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

13. 13 Safety Capacity Causes of Delays and Queues High, unsynchronized variability in - Interarrival times - Processing times High capacity utilization ρ= R i / R p or low safety capacity R s =R i - R p due to : - High inflow rate R i - Low processing rate R p=c / T p, which may be due to small- scale c and/or slow speed 1 / T p

14. 14 Safety Capacity Drivers of Process Performance 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. C i = coefficient of variation for interarrival times C p = coefficient of variation for processing times

15. 15 Safety Capacity The Queue Length Formula Utilization effect Variability effect x  R i / R p, where R p = c / T p C i and C p are the Coefficients of Variation (Standard Deviation/Mean) of the inter-arrival and processing times (assumed independent)

16. 16 Safety Capacity Factors affecting Queue Length 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.

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

18. 18 Safety Capacity Example 8.4 10,10,2,10,1,3,7,9, 2 =AVERAGE ()  Avg. interarrival time = 6 R i = 1/6 arrivals / sec. =STDEV()  Std. Deviation = 3.94 C i = 3.94/6 = 0.66 7,1,7 2,8,7,4,8,5, 1 T p = 5 seconds R p = 1/5 processes/sec. Std. Deviation = 2.83 C p = 2.83/5 = 0.57 A sample of 10 observations on Interarrival times and processing times R i =1/6 < R P =1/5  R = R i  = R/ R P = (1/6)/(1/5) = 0.83 With c = 1, the average number of passengers in queue is as follows: I i = [(0.83 2 )/(1-0.83)] ×[(0.66 2 +0.57 2 )/2] = 1.56 On average 1.56 passengers waiting in line, even though safety capacity is Rs= R P - Ri = 1/5 - 1/6 = 1/30 passenger per second, or 2 per minutes

19. 19 Safety Capacity Example 8.4 Other performance measures: T i =I i /R = (1.56)(6) = 9.4 seconds Since T P = 5  T = T i + T P = 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  I i = 0.08 cρRsRs IiIi TiTi TI 10.830.031.569.3814.382.4 20.420.230.080.455.450.91

20. 20 Safety Capacity Exponential Model 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.

21. 21 Safety Capacity The Exponential Model Poisson Arrivals –Infinite pool of potential arrivals, who arrive completely randomly, and independently of one another, at an average rate R i  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 T p Computations –C i = C p = 1 – K = ∞, use I i Formula –K < ∞, use Performance.xls

22. 22 Safety Capacity Example Interarrival time = 6 secs  R i = 10/min T p = 5 secs  R p = 12/min for 1 server and 24 /min for 2 servers Rs = 12-10 = 2 cρRsRs I i  Formula T i = R i / I i T= Ti+ 5/60I= Ii + c ρ 10.8324.160.420.55 20.42140.180.020.11

23. 23 Safety Capacity t ≤ t in Exponential Distribution 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 - R i t, 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

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

25. 25 Safety Capacity Variability Reduction Levers 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

26. 26 Safety Capacity Capacity Utilization Levers If the capacity utilization can be decreased, there will also be a decrease in delays and queues. Since ρ=R i /R P, to decrease capacity utilization there are two options: –Manage Arrivals: Decrease inflow rate R i –Manage Capacity: Increase processing rate R P 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)

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

28. 28 Safety Capacity Server 1 Queue 1 Server 2 Queue 2 Server 1 Queue Server 2 Effect of Pooling RiRi RiRi R i /2

29. 29 Safety Capacity Effect of Pooling Under Design A, –We have R i = 10/2 = 5 per minute, and T P = 5 seconds, c =1 and K =50, we arrive at a total flow time of 8.58 seconds Under Design B, –We have R i =10 per minute, T P = 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

30. 30 Safety Capacity Effect of Buffer Capacity Process Data –R i = 20/hour, T p = 2.5 mins, c = 1, K = # Lines – c Performance Measures K456 IiIi 1.231.521.79 TiTi 4.104.945.72 PbPb 0.10040.07710.0603 R17.9918.4618.79  0.7490.7680.782

31. 31 Safety Capacity Economics of Capacity Decisions Cost of Lost Business C b –$ / customer –Increases with competition Cost of Buffer Capacity C k –$/unit/unit time Cost of Waiting C w –$ /customer/unit time –Increases with competition Cost of Processing C s –$ /server/unit time –Increases with 1/ T p Tradeoff: Choose c, T p, K –Minimize Total Cost/unit time = C b R i P b + C k K + C w I (or I i ) + c C s

32. 32 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 R i P b $120 I i TC ($/hr) 42520200.8147.6393.4 53020154.2182.6386.4 63520120.6214.8390.4

33. 33 Safety Capacity Optimal Processing Capacity cK = 6 – cPbPb IiIi TC ($/hr) = $20c + $5(K+c) + $100R i P b + $120 I i 150.07711.542$386.6 240.00430.158$97.8 330.00090.021$94.2 420.00040.003$110.8

34. 34 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)

35. 35 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 P b Proportion abandoning P a

36. 36 Safety Capacity Net Rate: Ri(1- P b )(1- P a ) Throughput Rate: R=min[Ri(1- P b )(1- P a ),R p ] Effect of Blocking and Abandonment

37. 37 Safety Capacity Example 8.8 - DesiCom Call Center Arrival Rate R i = 20 per hour=0.33 per min Processing time T p =2.5 minutes (24/hr) Number of servers c=1 Buffer capacity K=5 Probability of blocking P b =0.0771 Average number of calls on hold I i =1.52 Average waiting time in queue T i =4.94 min Average total time in the system T=7.44 min Average total number of customers in the system I=2.29

38. 38 Safety Capacity Throughput Rate R=min[Ri(1- P b ),R p ]= min[20*(1-0.0771),24] R=18.46 calls/hour Server utilization: R/ R p =18.46/24=0.769 Example 8.8 - DesiCom Call Center

39. 39 Safety Capacity Example 8.8 - DesiCom Call Center Number of lines 5678910 Number of servers c 111111 Buffer Capacity K 456789 Average number of calls in queue 1.231.521.792.042.272.47 Average wait in queue Ti (min) 4.104.945.726.437.087.67 Blocking Probability Pb (%) 10.047.716.034.783.833.09 Throughput R (units/hour) 17.9918.4618.7919.0419.2319.38 Resource utilization.749.769.782.793.801.807

40. 40 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

41. 41 Safety Capacity Example 8.9 - Effect of Buffer Capacity on Total Cost Number of lines n 56789 Number of CSR’s c 11111 Buffer capacity K=n-c 45678 Cost of servers ($/hr)=20c 20 Cost of tel.lines ($/hr)=5n 2530354045 Blocking Probability P b (%) 10.047.716.034.783.83 Lost margin = $100R i P b 200.8154.2120.695.676.6 Average number of calls in queue I i 1.231.521.792.042.27 Hourly cost of waiting=120I i 147.6182.4214.8244.8272.4 Total cost of service, blocking and waiting ($/hr) 393.4386.6390.4400.4414

42. 42 Safety Capacity Example 8.10 - The Economics of Processing Capacity The number of line is fixed: n=6 The buffer capacity K=6-c cKBlocking P b (%) Lost Calls R i P b (number/hr) Queue length I i Total Cost ($/hour) 157.71%1.5421.5230+20+(1.542x100)+(1.52x120)=386.6 240.43%0.0860.1630+40+(0.086x100)+(0.16x120)=97.8 330.09%0.0180.0230+60+(0.018x100)+(0.02x120)=94.2 420.04%0.0080.0030+80+(0.008x100)+(0.00x120)110.8

43. 43 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.

44. 44 Safety Capacity Example 8.11 - WalCo Drugs 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/(R p – R i ) = 1/(24 -20) = 0.25 or 15 min

45. 45 Safety Capacity Example 8.11 - Probability distribution of the actual time customer spends in process (obtained by simulation)

46. 46 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

47. 47 Safety Capacity Service Promise: T duedate, Service Level & Safety Time SL; The probability of fulfilling the stated promise. The Firm will set the SL and calculate the T duedate 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 T duedate = T + T safety Prob(Total time in process <= T duedate ) = SL Larger SL results in grater probability of fulfilling the promise.

48. 48 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 <= T duedate ) = 1 – EXP( - T duedate /T) Which is the fraction of customers who will no longer be delayed more than promised. T duedate = -T ln(1 – SL)

49. 49 Safety Capacity Example 8.12 - WalCo Drug WalCo has set SL = 0.95 Assuming total time for customers is exponential T duedate = -T ln(1 – SL) T duedate = -T ln(0.05) = 3T Flow time for 95 percentile of exponential distribution is three times the average T T duedate = 3 * 15 = 45 95% of customers will get served within 45 min T duedate = T + T safety T safety = 45 – 15 = 30 min 30 min is the extra margin that WalCo should allow as protection against variability

50. 50 Safety Capacity Relating Utilization and Safety Time: Safety Time Vs. Capacity Utilization Capacity utilization ρ 60 % 70% 80% 90% Waiting time T i 1.5T p 2.33T p 4T p 9T p Total flow time T= T i + T p 2.5T p 3.33T p 5T p 10T p Promised time T duedate 7.7T p 10T p 15T p 30T p Safety time T safety = T duedate – T 5T p 6.67T p 10T p 20T p 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

51. 51 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

52. 52 Safety Capacity Thank you Questions?