SCHEDULING CUSTOMER SERVICE REPRESENTATIVES FOR THE WORKERS* COMPENSATION BOARD OF BRITISH COLUMBIA B y R A F A E L S A N E G R E B.Sc. (Mathematics), University of-Valencia, 1996 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E (BUSINESS A D M I N I S T R A T I O N ) In T H E F A C U L T Y O F G R A D U A T E STUDIES (Department"of Commerce and Business Administration) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A July 1998 © Rafael Sanegre, 1998 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C0rtM€&C^ The University of British Columbia -Vancouver, Canada DE-6 (2/88) Abstract The Workers' Compensations Board of British Columbia ( W C B ) operates 4 call centers in the province devoted to compensation services. This thesis describes a review of operations at the call center in the head office, located in Richmond, B . C . We identified the need for a mathematical model to schedule telephone operators (called C S R s in W C B ) in order to minimize the workforce required, while assuring minimal staffing levels to provide good customer service. We developed two integer programming models and built an interface that would enable the supervisors to use the model interactively. We compared the results from the model to one of their schedules and found that the model used 5 fewer C S R s than currently i n their schedule. The potential savings derived from such a test convinced management that W C B needed to acquire and implement software to schedule staff. ii Table of Contents Abstract Table of Contents List of Figures .i Acknowledgements A I. INTRODUCTION II. THE PROCESS BEING MODELED 11.1 Reaching the Call Center, at WCB 11.2 The Scheduling Process at WCB 11.3 Our first model 11.4 Estimating the volume of calls 11.5 Computing agent requirements II.6. Computational results II.7 Modifications to the first model II. 8 The final model III. The COE Workforce Management Tool IV. CONCLUSION Bibliography APPENDIX A. The Erlang Formulas APPENDIX B. Poisson Processes APPENDIX C. Call Center Technology APPENDIX D. GAMS Code for the Assignment Problem APPENDIX E. Model for Stage 1 of the Scheduling Problem APPENDIX F TRICKS APPENDIX G ACD OUTPUT iii List of Figures Figure 1. Diagram 6 Figure 2. Arrival Process week 14 Figure 3. Arrival Process week 2 14 Figure 4. Average Demand 15 Figure 5. Blockage ~ 18 Figure 6. Service Time 20 Figure 7. Call Length 20 Figure 8. Wrap Up 21 Figure 9. Service Time 22 Figure 10 Average Delay of Answer 23 Figure 11. Staffing Levels 24 Figure 12. Staffing to a Target : 25 Figure 13. Schedule 2 27 Figure 14. "Tricks" 29 Figure 15. Sample Tricks 30 Figure 16. The good schedule 32 Figure 17. The COE Call Center skeleton 33 Figure 18. Initial screen of the COE Call Center Toolkit 34 Figure 19 On line help 35 Figure 20 Scheduling screen 36 Figure 21 Cplex 37 Figure 22 Excel 38 iv Figure 23 C S R database screen \ Acknowledgement I would like to thank a few people and organizations, without whom the completion of this thesis would have been impossible. In the first place, I would like to thank The Center for Operations Excellence and The Workers' Compensation Board of British Columbia for giving me the opportunity to work on such an interesting project. I would like to thank Professor Martin L . Puterman, and Professor Carson Woo for supervising the project and encouraging us in the work. Special thanks go to Shelby Brumelle for his kindness towards me, and for his wise suggestions. Also , I would like to thank Professor Tom McCormick , who saved me probably a couple of months of work, when giving me a key idea for the development of the model. M y thanks also to Stephen Jones, Paul H i o m and Lauren Gray, from the Center of Operations Excellence at the University of British Columbia for managing and coordinating the project. Thank you to Danny Granot and Dean Uyeno for being external referees. Thanks to my professors in Spain specially Enriqueta, Mayte, Ramon and Vicente for following up on my progress at U B C . Although not directly involved i n this project, I would like to take the opportunity to thank Professor W i l l i a m Ziemba and his wife Sandra for their constant support and interest in me. I would like to thank my close friends both here and in Spain, for giving me such a great time. Special thanks go to my girlfriend Mayte for making life so much nicer, and encouraging me to come to Canada to do a Master's degree. Finally, I can't finish this section without thanking my family, specially my parents, who made possible my adventure in Canada, and to whom I owe, and devote this Master's degree. V I I. INTRODUCTION The study documented in this thesis is part o f a broader project undertaken by The Center for Operations Excellence, at The University of British Columbia, and the Workers' Compensation Board ( W C B ) to improve the operations at the main call center for compensation services, located in Richmond. The Workers' Compensation Board is an administrative agency dedicated to the safety, protection and good health of workers in British Columbia [22]. Its 2,400 employees: • Monitor and promote occupational safety and health practices through regulation, inspection, education, and consultation; • Provide rehabilitation and compensation, as well as vocational training to workers who are injured or suffer from an occupational disease; • Provide compensation to dependants of workers who have died as the result of a work-related injury or occupational disease; • Provide compensation and assistance to victims of criminal acts (under the authority of the Criminal Injury Compensation Act). The W C B is funded by employers in industries covered by the Workers Compensation Act , which includes virtually all employers in B . C . A call center consists of a group of agents that provide service to callers. There are many types of call centers, almost one for each type of business; but, they can be classified into three major categories: Inbound, Outbound and Inbound/Outbound call centers. 1 Inbound call centers are typically service call centers. In these call centers, agents provide service to incoming calls. Outbound call centers are mainly marketing call centers, where agents place outgoing calls, usually to advertise or sell a product. Inbound/Outbound are call centers with a component of both. Although different types of call centers have different purposes, a common goal among all these call centers is to handle large volumes of calls as efficiently as possible, and at minimum cost. Efficient staffing the call center appropriately has become crucial for the financial health o f a company. Companies can no longer afford to use gut feel when it comes to determine the shifts o f the employees, but rather need a precise methodology that w i l l guarantee minimum workforce to properly accomplish managerial goals in terms of customer service. In this thesis, we develop an integer programming model to: • Automate the task of scheduling call center staff; • Minimize the number of agents needed to satisfy the required staffing levels; • Maximize preferences of C S R ' s in choice of shift and phone times within that shift. Most of today's commercial packages [3], [16], [17], [18], [19], [20], [21] for call center management include modules to: (i) forecast the volume o f calls that a call center w i l l receive on a weekly, daily and hourly base; (ii) compute the number of agents required to attend phone calls at any given time so that certain service levels are satisfied, and (iii) assign the agents' working shifts. 2 Staff scheduling problems arise in almost every organization: doctors and nurses in hospitals, operators in telephone companies, baggage handlers, check in counter attendants in airlines, officers in police departments, etc. Many such systems operate 24 hours a day, seven days a week with varying demand for services. Full-time employees in these service organizations are often assigned to a prescribed 40-hour work schedule (eight hours per day, five consecutive days) each week. The relative complexity of the scheduling task will vary from one organization to another. Mathematical programming is often used to derive efficient schedules. They are classified as either shift, days-off or tour scheduling problems. Shift scheduling problems arise when it is necessary to determine what eight hour shift the employees should be assigned to on each working day. In certain organizations (e.g. operators in telephone companies) this is the only problem to be solved since the five working days are fixed by union or management policy. For other organizations, the days-off scheduling problem must be solved to determine which two days of the week the employee should be assigned off. For these organizations the daily shift schedule may be fixed (e.g. patrol persons work one of three fixed daily shifts during a week-periodically rotating through all three daily shifts). When both the daily shift and days-off schedules can be determined simultaneously, the problem is known as the tour scheduling problem. The objective in solving each of these problems is to minimize the number of employee hours needed to satisfy hourly demand requirements over the week There is extensive literature on scheduling issues. Morris and Showalter [15] give a good introduction and overview of the three major types of scheduling problems: shifts, days-off and tour problems. Bechtold, Brusco and Showalter [4] is also a good introduction. It reviews 3 the most commonly used methods before 1990. Gaballa and Pearce [9] present a very similar problem to the one that we study in this thesis. They developed an integer programming model to schedule staff at Qantas Airways. For a good introductory paper on queuing techniques used to compute staffing levels, refer to Harris, Hoffman and Saunders [12]. The remaining chapters of this thesis are organized as follows: Chapter II describes how callers get to the call center through the public network and internal hardware. Chapter III explains the issues involved in managing a call center and focuses on the scheduling aspect of it. We also present our model in this section. Chapter I V presents an interface to the model, called The C O E Workforce Management Tool . 4 II. T H E PROCESS BEING M O D E L E D 11.1 The W C B Call Center: Incoming Calls The subject for investigation in this thesis is an inbound call center at the Workers' Compensation Board that employs customer service representatives (CSRs) to collect information related to claims, adjudicate those claims, and report the decision to the claimant. W C B currently has two locations for telephone CSRs in the province of British Columbia; Richmond with 43 CSRs , and Kelowna with 9 CSRs. A call center is equipped with telephone lines called trunks. The number of trunk lines required depends on the number of incoming calls, the length of the calls, and the number of servers available. If a person calls while all the trunk lines are in use, he or she w i l l receive a busy signal. The caller can then redial immediately, wait and try later, or call another day. Those callers who do not try again are referred to as lost demand. If a line is free, the call is received at a switching device, called an Automatic Ca l l Distributor or A C D (see Appendix B for further information on A C D s ) which distributes the calls to available CSRs . I f al l the C S R s are busy when the call reaches the switch the caller receives a recorded message asking the caller to stay on the line until a C S R becomes available. Abandoned calls result from callers who hang up after reaching the A C D but prior to speaking to a C S R . Figure 1 illustrates how a caller reaches the call center at W C B . A new call travels through the public network and reaches (through trunk lines-arrows) the P B X at W C B . The scripting of the I V R prompts the caller to choose an option. If option " l " i s chosen, the P B X w i l l route the call to the A C D at the call center, and the call w i l l remain in the queue until a C S R is available to take the call. Incoming Phone call B C T E L PUBLIC SWITCH Figure 1. Reaching the call center at W C B I L 2 The Scheduling Process at W C B . The focus of our study is the scheduling phase of the call center management. A C S R carries out two activities, answering telephone calls and adjudicating claims. These are performed in separate fixed segments of time. The scheduling process consists of: (i) assigning a working shift to every C S R , and (ii) allocating the phone times within the shift for every C S R in such a way that the office is adequately staffed for the expected volume of calls. Scheduling is J usually very tedious and time consuming if done manually. The supervisor of the call center has to allocate phone times in such a way that certain rules or constraints imposed by the union agreement or management policy are satisfied, while assuring that there will always be enough CSRs attending calls. In order to automate the process it is essential to have a detailed description of such rules. These are as follows: • The Office is open Monday to Friday, 7:00 a.m. to 7:00 p.m. • Every CSR works Monday-Friday, 8 hours a day. i. • Every CSR attends the phone for 3.5 hours per day. The rest of the day is devoted to adjudicating claims. • The allocated time for call answering (phone time) for each CSR must be scheduled in one or two contiguous blocks. If two blocks are used then there must be at least one hour of difference between the end of one block and the start of the second block, and each block is at least 1 hour in duration. • Each CSR is allowed a 45-minute block for lunch. • CSR's can only be scheduled for the following shifts: Shift 7:00 a.m. to 3:00 p.m. Lunch sometime between 11:00 a.m. and 12:30 p.m. (scheduled by the supervisor). Shift 7:30 a.m. to 3:30 p.m. Lunch sometime between 11:30 a.m. and 1:00 p.m. Shift 8:00 a.m. to 4:00 p.m. Lunch sometime between 11:30 a.m. and 1:00 p.m. Shift 8:30 a.m. to 4:30 p.m. Lunch sometime between 1:00 a.m. and 2:00 p.m. Shift 9:00 a.m. to 5:00 p.m. Lunch sometime between 1:00 a.m. and 2:00 p.m. Shift 11:00 a.m. to 7:00 p.m. Lunch sometime between 2:00 p.m. and 4:00 p.m. • Each CSR has two 15-minute-coffee breaks during work time, one during phone time and one during claim time. The supervisor of the call center cannot schedule those two breaks. • Contractual agreement states that staff work either the morning shift or the evening shift. • There are currently 43 CSR's, 6 of whom do the evening shift. A generic way of modeling the shift, days-off, and tour scheduling problems uses the following set covering formulation [15]. n 1 n Subject to 7=1 t = l,2,...,m, Xj > 0 and integer, j = 1,2, ...,n. For the shift scheduling problem: Xj = number of employees assigned to daily shift j; rt = number of employees required to work in the t' time period; n = number of daily shifts to be considered; m = number of time periods to be scheduled over a single day; and For the days-off scheduling problem: Xj - number of employees assigned to days-off pattern j; rt = number of employees required on the t,h day of the week; n = number of days-off patterns to be considered; m = number of days per week that the system operates; and rt = number of employees required to work in the t' time period; n = number of tour types to be considered; 1, if time period t is a work period in shift j; 0, otherwise 1, / / day t is a work day in the days - off pattern j; 0, otherwise For the tour scheduling problem: Xj = number of employees assigned to tour j; 8 m = number of time periods to be scheduled over the week; and f 1, if time period t is a work period tour j; [0, otherwise Next we present the model we used for the W C B scheduling problem. II.3 Our first model The W C B scheduling problem could be classified as a shift scheduling problem. However in our case, not only the shifts had to be determined, but also the allocation of phone times within the span of the shift and the assignment of CSRs to specific shifts. Although aware of the models described above, we initially proposed a different formulation: Parameters n = number of CSR's available to work on the given day. Py = parameter that expresses the preference of C S R i to be assigned to shift j, i = 1,2,...,n j= 1,2,...,6. r(k) - number of C S R ' s required for time slot k, k = 1,2,...,24. A s mentioned in the previous section, W C B uses 6 different shifts: Shift 1 spans from 7:00 a.m. to 3:00 p.m., shift two goes from 7:30 a.m. to 3:30 p.m., shift 3 from 8:00 a.m. to 4:00 p.m., shift 4 from 8:30 a.m. to 4:30 p.m., shift 5 from 9:00 a.m. to 5:00 p.m. and shift 6, often referred as night shift, that goes from 11:00 a.m. to 7:00 p.m. A day is divided into 24 half-hour time slots; 7:00 a.m. to 7:30 a.m., 7:30 a.m. to 8:00 a.m.,etc. Although the quantification of the parameter to express preferences would need experimentation, a first approach was to discretize the possible values of this parameters into 9 5 possibilities: -10 expressing strong unwillingness to do that shift, -5 expressing less unwillingness, 0 being neutral, +5 showing a preference for that shift, and +10 expressing a strong preference. Variables [l, if CSR i is assigned to shift j; [0, otherwise [l, if CSR i is assigned to phones at time slot t; " [0, otherwise Obj ective Function (1) Max Y.Pyy-j •j Constraints • For every time slot, the number o f CSR's answering the phones has to be greater than or equal to the number of CSR's required, r(k). (2) £X fork = \,2,-,2A i • Every C S R has to be on phones for three and a half hours. (3) ^xik = 7 for i = 1,2,...,n . Note that each time slot is 30 minutes long. k • Every C S R has to be assigned to a shift (4) i > f f =1 Mi = l2,.,n 7 = 1 • If C S R /' is assigned to shift 1 (that is 7:00-15:00) then C S R i won't be available after 15:00. i . e. 10 If yn = 1 => xn7 = xns =... = xi24 = 0. This can be modeled using the following inequalities: (5) xg <1-v , , for i = l ,2, . . . ,n; for j = 17,18 24 • If C S R / is assigned to shift 2 (7:30-15:30) then C S R i won't be available before 7:30, or after 15:30. That is: If y.a = 1 => xn = xns =... = x m = 0 . (6) Xy <\-yn V i = 1,2,...,«; for j = 1,18,19,...,2'4 • If C S R i is assigned to shift 3 (8:00-16:00) then C S R /' won't be available neither before 8:00, nor after 16:00. If y.s = 1 ^ xn =x2= xil9 =... = xi24 = 0 . (l)Xy <l-yl3 for i = 1,2,...,n; for j = 1,2,19,20...,24 • If C S R i is assigned to shift 4 (8:30-16:30) then C S R / won't be available before 8:30, or after 16:30. If yi4 = 1 => xn =... = x3 = xi20 =... = xi2A = 0. (S)xs <1 - y a for i = 1,2,...,/!; for j = 1,2,19,20,...,24 • If C S R i is assigned to shift 5 (9:00-17:00) then C S R / won't be available before 9:00, or after 17:00. If y . s = 1 => xn =... = x4 = xi2l =... = xi24 = 0 . (9) X i j < l - y i 4 for f = 1,2 n; for j = 1,2,...,4,21,22,...,24 • If C S R i is assigned to shift 5 (11:00-19:00) then C S R i won't be available before 11:00. If y . s =1 => xn = ... = xs = 0 . (10) xg <l-yi5 for i = 1,2,...,n; for j = 1,2,...,8 11 • If phone times are split, there must be at least 1 hour of difference in between. If xik = 0 => + xjk + xik+l < 1 for all i,k. As before, we use the following inequalities: (11) + xik + xjk+i < M xik +1, where M is a large enough number. M=2 is the lowest possible bound. We picked this number for computational reasons. • Lunch times. Depending on the shift, there is a period for lunch. At the time of developing this first model, we did not have the lunch times correctly. We picked from 11:00 to 14:00 as the time to allocate lunch for the shifts 1,2,3,4,5, and 12:00 to 14:00 for the night shift. These time slots were corrected in the successive models. The constraints follow: (12) xl9 + xm + xiU + xa2 + x„ 3 + xM < 4 + M(\ - y..) for i = 1,2,..., n for j = 1,2,3,4,5 ;M=2 • If xi9 = 0 => xl0 = 0 We model this with (13) x,,0 <xig +M(l-yv) for z = 1,2,...,« for j = 1,2,...,5; M=l • If *ao = 0 = > J C 9 + X I 0 + X „ ^ 1 (14) xl9 + xno + xin < 3xno +1 + M ( l - yy ) Vi = 1,2,..., n Vj = 1,2 5 ; M=2 • If xiu=Q^> xw+xn+xxl<\ (\5)xil0+xin+xii2 <3xiu +l + M(l-yiJ) V ? = l,2,...,« V y = 1,2,...,5; M=2 • If xn2 = 0=>xn +xn +xn<l (\6)xin +xn2 +x,,3 <3xin +1 + M(1-^. . ) Vi =1,2 /i V/ = 1,2,...,5;M=2 • If xin=0z^> xX2+xu+xu<\ (\l)xn2 +xiu +xiH <3x,I3 +l + M(l-y,j) V / = l,2,..,/i V y = 1,2,...,5 ; M=2 12 If x,14 = 0 => xu = 0 (18) x,,3 <x,14 +M(\-yy) for i = \,2,...,n for y+ 1,2,...,5 ;M=7 For the night shift we have: (19) xn, + x„ 2 + x,,3 + x n 4 < 2 + M ( l - yi6) /or / = 1,2,...,n;M = 2 (20) xin <xin + M(l-yi6) for i = 1,2,...,«;M = 1 (21) *.,, + x,,2 +x,,3 < 3x,,2 +l + M ( l - x - 6 ) /or z = 1,2,...,«;M = 1 (22) x,,2 +x„3 +xM <3x,,3 +l + M ( l - x - 6 ) /or 1 = l,2,. . . ,/i;M = 1 (23) x,,3 < x , , 4 + M ( l - ^ / 6 ) V/ = 1,2,...,«;M = 1 Notice that equations (14), (15), (16), (17), (21) and (22) are redundant with (11) The following two sections describe the procedures followed to estimate the parameters of the model. In order to do so, we first have to estimate the volume of incoming calls that the call center receives, and then use queuing theory to compute the agent requirements, r(k). 11:4 Estimating the volume of calls ' The first step to be taken when managing a call center is to have an accurate overview of the pattern and volume of calls that the call center usually gets. We did not use an elaborate forecasting model, but rather a simple estimate of demand based on the small amount of data collected from the A C D . However, we acknowledge the importance of employing precise forecasting algorithms in order to have an accurate estimate of the volume and pattern of incoming calls, which will be crucial in implementing this work. The following graphs show 13 two weeks of data collected from the A C D . We were unable to get more historical data due to storage capacity constraints on their system. Demand, Week 19/01/98-23/01/98 o o o o o o o o o o o o o o o o o o o O o o o o 00 p CO o CO p CO p CO p p o p p CO o CO o CO O p o CO o CO CO 6S CO o o Cvi CO CO •if •if in CO <b I"-* 00 CO cri o o o o o o CO o CO o o o o o o o o o o o o o o o o o o o o 00 00 cn 00 o CO p CO o CO o CO p CO p CO o CO p p o CO cti o o c\i CM CO CO Tf •if in in 0D <b CO 00 Monday,19/01/98 . Tuesday,20/01/98 .. . A . . . Wednesday 21/01/98 Thursday 22/01/98 Friday, 23/01/98 Figure 2. Arrival Process week / .Arrival pattern of incoming calls for the week of 19/01/98 Demand, Week 26/01/98-30/01/98 o o o o o o o o o o o o o o o o O o o o o o o o p p p p p o CO p p p p p CO p CO p p p CO p p p p p I"-* CO* CO cn cn o o c\i CO CO Tf •if tin in CD 03 I"-* 00 CO cn o o o o o o CO o p o o o o o o o o o o o o o O O o o o o o N* CO CO cn p p p p p p p p p p p p p O p p p p p cn o o c\i cvi CO CO •if •if i n in 00 CO 1-^ 1 ^ CO CO ^ Monday, 26/01/98 Tuesday, 27/01/98 .. . A . . . Wednesday, 28/01/98 * Thursday, 29/01/98 x Friday, 30/01/98 Figure 3. Arrival Process week 2. Arrival pattern of incoming calls for the week of 26/01/98 14 We found that the pattern of arriving calls is quite constant throughout the week. It has two peak periods; one in the morning, from 9:30 to 11:00, and one in the afternoon, from 14:30 to 16:00. After 17:00, and before 8:00, the office hardly receives any calls. According to the call center supervisor, Monday is usually the busiest day of the week. The graph in Figure 4 shows the difference between Mondays and the rest of the working days based on our two-week data sample. Average Demand 120 : 100 : o o o o o o o o o o o o o o o o o o o o o o o o cq P co; P co p co p co p o p c o p c o p c o p c o p c o p c o p o o o o o T T T T ' T T 7 T V ' V r r r r T r r r ' V T 7 r 7 T 7 T 7 T 7 T 7 r r o c o o c o o o o o o o o o o o o o o o o o o o o o ^ ^ ( ^ ^ ^ c o p c o o c o o c o p c o o c o p c o o c o o c o o c o c i o o ^ ^ c M c \ i c o c o ^ ^ i r i i b ^ < b i ^ r ^ c o c o Time + Mondays . . . . . . . Tuesday-Friday Figure 4. Average Demand. Mondays are slightly busier than the rest of the working days Several CSRs that we interviewed seemed to agree that other factors, such as weather conditions and month of the year also had an effect on the volume of calls received. This however, we were unable to check because of lack of data and the fact that the A C D reports don't include information regarding the weather or other external factors. Finally, we were informed that there would be some changes in the tasks to be performed by the CSRs , which 15 would probably increase the number of calls handled by the call center. When those changes take place, a new estimation of the demand will have to be done. II.5 Computing agent requirements. In this section we describe how we computed our staffing levels, also referred as agent requirements. We start by highlighting the importance of accurately computing this numbers. Incoming calls are distributed to CSRs over PBX (Personal Branch Exchange) station lines. If the CSRs are busy, calls are held in the A C D queue. This allows the caller to wait for an answer rather than being blocked from entering the PBX, and having to repeatedly re-dial the number to find a CSR. Each call that is either being serviced, or is waiting in the A C D queue takes up one of these lines. If all lines are busy, any new caller will receive a busy signal, and will have to call later. If the number of trunk lines devoted to the calf center is too high, customers will probably experience long waiting times. On the contrary, if there are not enough trunk lines, many callers will get a busy signal. Either way, customers will get frustrated with service quality. As with trunk lines, it is crucial to staff the office optimally. An understaffed office will provide poor customer service, whereas an overstaffed office will result in high expenses to the company. Approximately 60% of call center's operating costs are personnel related [21]. The difference then between operating in the red or in the black can be the ability to staff efficiently. 1 6 We used a commercial package (TCS's C3 Calculator [21]) to compute the optimal staffing levels for the call center at W C B . The determination of the right number of trunk lines devoted to the call center was out of the scope of this project. Most of the commercial workforce management packages, including the C3 Calculator, have the following formulas built in. For trunk lines computations, an Erlang-B loss system (i.e. M/G/c /c ) * is generally used, while an Erlang-C delay system (i.e. M/M/c /oo)** is usually used to model the process related to agents requirements [12]. The Erlang-B formula is used to determine the number of trunk lines required. The main problem with this procedure is that implicitly assumes that the number of trunk lines can be calculated independently of the number of CSR's answering phone calls. In addition, service times for this model are the summation of the time spent waiting in the A C D queue for an available agent plus the time spent talking to the agent. Waiting times are highly correlated, and hence destroy the hypothesis of independent service times assumed by the M/G/c/c model. The formula also implicitly embodies the assumption that callers w i l l not redial immediately i f getting busy signal, but w i l l wait for a while before trying again. If this is not the case then the Poisson assumption fails. * Mathematical notation to represent a queuing system with Poisson arrivals, general service times, c servers and queue with holding capacity of c units * * Queuing system with Poisson arrival process, exponential service times, c servers and infinite capacity for the queue. 17 The Erlang-C model has traditionally been used to determine the number of CSR's required. The servers are the CSR's , the queue length is the number of callers connected and waiting to talk to a C S R , and service does not begin until the caller is connected to a C S R . Arrivals are callers who find a trunk line available at the time of placing the call. Arrivals that find all c servers busy join a queue with an unlimited number of waiting positions. Note that this assumption is realistic only i f the blocking probability is small. We computed the estimated the blockage probability at W C B in about 10%. Also , the arrivals wait in the queue as long as necessary for service. The abandon rate at the W C B call center was minimum as the graph in figure 5 shows. Finally, the Erlang-C formula assumes [12] that the mean arrival rate, X, is strictly less than the mean system service rate cu. (for stationarity). Calls Offered Vs . Calls Answered 100 .. Figure 5. Blockage. Difference between calls offered and calls answered on 19/01/98. For the other days the difference is minimum as well 18 Recall that the M/M/c/oo assumes a Poisson arrival process and exponentially distributed service times. The arrival process to the A C D queue is the true arrival process to the WCB system filtered by the PBX blockage. In other words, suppose N(t) denotes the true demand, that is the number of phone calls that wish to speak to a CSR in the time interval (0,f]. Some of those calls will find all the PBX lines busy and therefore will be rejected. If X(t) denotes the number of calls that get through the PBX, and Y(t) denotes the calls that are blocked, then, N(t) = X(t) + Y(t) and it can be proved (see Appendix B) that if N(t) is a Poisson process with parameter X and p is the blockage probability, then X(t) is a Poisson process with parameter (1-p) X. It remains to prove that {N(t), t > 0 } is a Poisson process, but that can be justified with Grigelionis' theorem [ 1 0 ] , which states that the sum of independent infinitesimal processes converge to a Poisson process. The number of calls that arrive at WCB are the summation of calls placed by independent individual callers whose behavior can be modeled as a Poisson process. The Erlang-C formula may overestimate the number of CSR's needed. Nonetheless, these formulas have been used successfully for a long time, and are the standard procedures for computing staffing levels and planning the networking of the call center. For more complex call centers (virtual call centers, several queues, etc..) other techniques are being used, the latest of which is simulation. We now focus on the agent model and estimate parameters needed to compute the right number of agents. We start with the Service Time. In the call center environment, the service time is made of two components: Call Handle Time and Wrap-up time. Call Handle Time is the length of the call, that is, from the time a CSR answers, to the time the caller hangs up. At 19 this point, however, the C S R might not be ready for answering another call , for he/she might have some work to do related to the previous call. Wrap-up is the time the C S R is unavailable because of work related to the call. ^ service lime w *. •4 Call handle Time ^ ^ — Wrap-up —• Figure 6. Service Time. The total service time is composed of the call handle time (conversation time) plus the wrap up time (call related work). The following three graphs show our estimation of Service Times, Ca l l Handle Times and Wrap-up Times. These estimates are based on our two-week data sample provided by the A C D . Please refer to Appendix G to see some of the printouts from the A C D . Call Length 250 _ 200 150 v« - • Second: 100 . ML • - m- • ••" . m. / -• Second: 50 0 o CO o o O co o o o CO o o o CO o o o co o o o co o o o co o o o co o o o CO o o o CO o o o co o o o CO o o co CO cri cri o o .— T— c\i CO CO -ir in CD CO 66 cri o o o CO 8:00-8:30-9:00- 1 o co • o o • o CO 1 o o o co • o o o co o o o o co o • o CO i o o • o CO 1 o o • o CO I o o 1 o co i o o i o co 8:00-8:30-9:00-in o o T— 1^ tSi CN co CO it ur> cb 66 66 Mondavs Tues-Fr idavs Figure 7. Call Length. The call length measure is almost constant throughout the day, averaging two and a half minutes 20 As mentioned earlier, several interviews with CSRs let us to think that Mondays were different from the rest of the working days. We decided to analyze these two groups separately. Figure 7 shows that there is no difference in the call length between Mondays and the other days, except between 7:00 a.m. and 7:30 a.m. This was a good sign as CSRs should not condition the service provided to the amount of incoming calls. The next graph however, shows that there is a difference in the wrap-up times. Agents either do the wrap-ups faster, or postpone them to the end of therr phone shift. Wrap-up Times 200 180 o o o o o o o o o o o o o o o o o o o o c o o o o c o o c o o n o c o o o o c o o o o c o o o o o o o ' T T T T T T T T T T T T T ' T T o c o o c o o o o o o o o o o o o o o o o o I ^ I ^ ^ Q d Q Q J t O O C O O C O O C O O C O O C O O C O O . C O + Mondays . . . . . . . Tuesday-Friday Figure 8. Wrap Up. Wrap Up times are shorter on Mondays than on the other days. 21 As can be seen in the graph the average wrap-up time increases considerably towards the end of the day. Asked about this, the manager of the call center told us that this is probably due to the fact that most of the other divisions in WCB are closed at that time of the day. The CSRs therefore don't have anybody to consult with, which results in more time to solve the problems. The end result is that the overall Service Time is slightly lower on Mondays than the rest of the days. The graph in Figure 9 'shows that difference. Service Times 5 0 O T~ _» Mondays .. . a_._ Tuesday-Fridays Figure 9. Service Time. The overall service time is shorter on Mondays than on the rest of the working week. However, the fact that Mondays are busier than the other days levels it off and the customer service level is the same throughout the week. In the implementation phase of this work, we would use different numbers for Mondays than for the rest of the days. 22 Average Delay o f Answer 160 n : 140 o o o o o o o o o o o o o o o o o o o o o o o o c o o c o o c o o c o o c o o c o p r o p o o c o o e o o c o o c o o o c o o c o o o o o o o o o o o o o o o o o o o o o I Mondays .. . m . . . Tuesdat-Fridays I Figure 10. Average Delay of Answer. The Average delay of answer is about the same throughout the week. When talking about customer service, we must specify how we measure customer satisfaction. Apart from the "soft" issues, such as being polite to the customer, there are several standard ways of measuring performance in call centers. Two of the most frequently used are the Telephony Service Factor (TSF), and the Average Speed of Answer ( A S A ) . The Telephony Service Factor measures the percentage of Calls answered within a given number of seconds. A standard TSF is the "TSF-80". It stands for 80 percent of the incoming calls answered in less than 20 seconds. The other measure of performance, A S A , measures the average waiting time for incoming calls. A standard A S A is the 30 seconds average delay. Both performance measures are important, and both performance measures should be taken into account when setting managerial targets. 23 The previous graph showed that the delays in the afternoon were quite high. For illustrative purposes, we chose the 30 seconds average delay as a target, and computed the number of CSRs needed at any given time to meet that target. TSF-20 had similar agent requirements. We compared the results to the actual staffing levels at WCB. Figure 3 shows the results. Time Slot ASA-30 WCB Staffing (avg.) 7:00-7:30 ' 0.6 1 7:30-8:00 1.9 2 8:00-8:30 5.3 7.4 8:30-9:00 8.3 9.2 9:00-9:30 9.6 8 9:30-10:00 11.2 11.1 10:00-10:30 11.9 10.7 10:30-11:00 13.6 16.1 11:00-11:30 14.7 17.2 11:30-12:00 13.1 11.4 12:00-12:30 11.4 10 12:30-13:00 10.5 10.5 13:00-13:30 11 11.4 13:30-14:00 12 11.2 14:00-14:30 11.8 9.9 14:30-15:00 11.8 10.7 15:00-15:30 11.2 8.7 15:30-16:00 10.1 7.9 16:00-16:30 8.4 6.7 16:30-17:00 5.4 5.7 17:00-17:30 3.8 3.9 17:30-18:00 3.4 3 18:00-18:30 2.7 3.1 18:30-19:00 3 2 Figure 11. Staffing Levels. The ASA 30 column was computed using a commercial package (TCS C3 Calculator) that incorporates Erlang C formulas. The fractional numbers indicates that an agent would only be needed part of the time. For example between 12:30 and 13:00 we would need 10 CSR's answering phones and another CSR during 15 minutes. 24 Staffing to a target o o o o o o o o o o o o o o o o o o o o o o o o c o o c o o c o o c o o c o o n o r o o c o o e o o c o o co o co o t ^ 0 3 c b c n c S o o ^ ^ c s i < N i < 0 ( 0 T r ^ o c o o c o o o o o o o o o o o o o o o o o o o o o Average Delay of Answer n Staff Required . . . A . . . WCB Staffing Figure 12. Staffing to a Target. Whenever the "WCB Staffing" line is above the "Staff Required" line, the "Average Delay of Answer" is within the threshold established. The legends Staff Required and WCB Staffing are measured on the left axis. The right axis measures time in seconds and is used as the scale for the Average Delay of Answer legend. The thick black horizontal line represents the target. We would like to keep the ASA bellow that line all the time. As we can see, whenever the actual staffing levels are above the required levels, the delay in answer is "under control", and within the limits established; but, if the office is understaffed, like in the afternoon, the answering delay surpasses the threshold. 25 In the next section we describe the computational results after setting the parameters to our estimates and solving the model. II.6. Computational results After preprocessing to eliminate redundant constraints the size of the model was 1141 variables and 3445 constraints. We coded the model using G A M S and solved it with Cplex 4.0. A t the time of the experiment, we did not have the CSRs preferences for the shifts; so we set them to zero. Figure 13 shows the results we got. We showed the schedule to the manager of the call center, and although it was a good start, we all identified the need for further work on the model. The main problem with the schedule obtained was the fact that the phone times were too spread out throughout the shift. We were informed that it would be desirable that the phone times were either straight through, or split into two blocks at most, no matter where, as long as we satisfied the general constraints, expressed earlier in the model. 26 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 CSR Shift 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 16:30 19:00 1 1 i 2 2 . . . • 3 1 J 4 1 t 5 1 i < ! "Si 6 6 mm ms mm ms 7 4 warn • 8 4 9 5 •i, ' 10 4 :-- ".. 11 1 wm u s ; 12 1 f 13 2 14 6 E«sr E Mt;S 15 3 «nv 16 6 asn me 17 3 f c 18 5 rSiV 3808c. • 19 4 20 6 SSSi 21 6 22 1 B5SS "S3 23 6 • 24 1 * K • . • 25 1 f si 26 1 •SS? ta 27 1 tm M S ?* 28 6 1 J 29 1 •^: - - , S i 30 1 83S m& 31 5 sm. M S (3*8? . • mm 32 3 ass am SIS « S mm 33 3 SSSffi 34 3 SSS5B 35 5 ass SSSISS: Si6*. 36 1 Bits Bass iSSS 37 1 C S S SI?*? jess Figure 13. Schedule 2. Our first schedule computed automatically using a mathematical model II.7 Modifications to the first model We proposed the following modification of the original model to fix the problem. M a x Z^-^- + Z j r * j r / ( * + n Subject to: (2), (3), (4) , (5), (6), (7), (8), (9), (10), (11), (12), (13), (18), (19), (20), and (23). 27 The second term in the summation expresses the number of consecutive time slots in the schedule. Such a term will encourage the model to look for compact schedules. Obviously the formulation like this makes the model nonlinear, increasing therefore the complexity of it. Nonetheless, we can rewrite that model introducing new variables and constraints; so that linearity is kept. Define the new variables njk = xlkxi{k+i), and introduce the following constraints: nik ^ * , ( * + . ) f°r a l l i ' k nik ^xik +xi(k+i} - 1 M all i,k We included these modifications in our model, encoded it and tried to solve it. We found that the solution time was prohibitively high (6+ hours). We tried all sorts of possible parameter settings for the solver, but never got a solution from it. We suspect that the linear relaxation was not giving good bounds, making it impossible to cut the tree at early stages. This, plus the fact that we had over two thousand decision variables seemed to be a convincing explanation as to why we never solved this refined model. This also helped us realize that the first model would have never worked either after plugging in the real coefficients for the preferences. With the p^'s = 0 however, any feasible solution is optimal, and therefore all Cplex needed to solve the problem was to find a feasible solution. 28 II.8 The final model At this point, we had two choices: we could either strengthen the formulation of the model by introducing cuts that exclude fractional extreme points, or we could change the entire model. We decided to proceed with the second option and presented a more standard scheduling model. Following the philosophy of the scheduling models presented in section II.2, the scheduling of phone times for the CSRs is now computed in two stages: In the first stage, we compute the tricks that ought to be used. In the second stage, we solve an assignment problem trying to maximize the CSR's preferences for "tricks". The word trick is a standard term in the call center jargon that stands for a established pattern of work time and rest time. In the context of WCB one trick could be: "Start working at 7:00 a.m. Do phones from 7:00 a.m. to 10:30 a.m., with a 15 minute break in between. Do claims from 10:30 a.m. to 11:30 a.m. Have lunch from 11:30 a.m. to 12:15 p.m. Do claims until 15:00p.m., with a 15 minute break in between." Phones • • • • • ^ • ^ • • • • • • i — ' Claims ^ ^ Rest 1 " m"^ m ^ ^ • • • • • i ^ ^ ^ ^ ^ ^ ^ ^ ™ 7:00 10:30 11:30 12:15 15:00 19:00 Figure 14. "Tricks". Example of one of the tricks used at WCB 29 We listed all the possible tricks splitting phone times in two at most, and assigned each one a variable. Please refer to Appendix F for a sample list of the "tricks" ~Tou / : 3 U 8:00 6:30 9 (XJ 9:30 10:00 10:30 11:00 11 30 12:00 12:30 TITO 13:30 14:00 14:30 15 00| 15:30j 1tf:00| 15:30| 17:00| 1 7:30| 1B:00| 18:30 variable BIOCK lyrxr Shift B:00 b 3u y U O 11:30 10 30 It 00 11 30 12 30 13 X 13:30 14:00 14.30 15:00 .5:30[ 16:00| 16:30| 17:0o| 17:30| 15:00| 1B:30| 15:00 x2 740 — x3 74D 7 xb 6 4 ' — 16 6 4 : V K/5 S42 7 ""*"" ,75 S U 7 xtl S42 7 "—** x78 4 4 3 7 x79 1 4 1 7 xHO 4 4 3 ' —-xSl 4 1 3 7 Figure 15. Sample Tricks. Phone times are indicated with stars. The dark shaded area indicates non-working time, whereas the light shaded area indicates the time to allocate lunch. There has to be two consecutive empty boxes somewhere in that shaded area Although listed, we do not include the 6&1 (see previous graph or Appendix E) tricks i n the optimization. The manager of the call center said it would be preferable not to use those. Using the 7&0, 6 & 1 , 5&2 and 4&3 block types, we found 712 possible tricks. After excluding the 6 & l s we had 558 variables. The new model for stage l i s as follows: Variables x, = Number of CSR's assigned to that trick, i=l, 2, n, where n is the number of possible tricks. Objective Function (24) M n ^ x , . 30 Constraints "Staffing level constraints" (See Appendix D) X/ integer for all i Please refer to Appendix E for a complete typing of the model. Although the number of variables had decreased from a couple of thousand to several hundred, we were concerned by the fact that these new variables were no longer binary variables, but non-negative general integer, and therefore harder to fix to its optimal value. Thankfully, we found that Cplex had no trouble at all solving this new model, as it would reach optimality very fast. Once stage l i s solved, we go to stage 2, where we maximize preferences. We do that with the following assignment model. Variables 1, if CSR i is assigned to "trick" j; for i = 1,2,...,n for j = 1,2,...,558 0, otherwise Objective Function Max^Pjjyjj, where p^ is as before a parameter for the preferences. Constraints for j = l , 2 558 Yjy>J - 1 fori = l,2,...,n 31 Although the number of variables for this second problem is very large ( 18702 variables and 749 constraints), this problem is easy to solve. The LP matrix of the problem is totally unimodular, and therefore, since our right hand side coefficients are all integer, the extreme points of the LP relaxation are integer as well. We only have to solve the LP relaxation to get the optimum point for the IP. We encoded both models and solved them to get the following schedule. 7:00 7:30 7:30 8:00 6:00 8:30 8:30 9:00 9:00 9:30 9:30 10:00 10:00 10:30 10:30 11:00 11:00 11:30 11:30 12:00 12:00 12:30 12:30 13:00 13:00 13:30 13:30 14:00 14:00 14:30 14:30 15:00 15:00 15:30 15:30 16:00 16:00 16:30 16:30 17:00 17:00 17:30 17:30 16:00 16:00 18:30 .16:30 19:00 CSR sum VERONICA 6:00 *>-• KELLY 7:30 • , < ANN H 8:00 SUEC 7:30 SUEZ 6:30 GILL 7:00 RAJ 6:30 < JANICE 7:30 VANCE 9:00 RON 9:00 KATHY 6:00 PAULINE 7:30 r MONA 6:30 ANNE F 7:30 PAT 9:00 MEL 6:00 KARI 7:30 4.1 -BRENDAN 6:30 » y SHAUNA 6:30 BETH 9:00 EARL 6:00 i " f t - ' CORNNE 9:00 ELIZABETH 7:30 LYNN 7:00 LINDA 7:30 LORRAINE 6:00 -IRMA 7:30 <• y JOHN 6:00 r " CINDY 8:30 ARLENE 7:30 SHANNON 9:00 a m MARILYN 6:30 1 - If}, MOE 11:00 ts DIANE 11:00 •> DIANA 11:00 EVELYN 11:00 HEATHER 11:00 i Figure 16. The good schedule. Finally, after several trials we achieved the desired schedule. 32 This solution was acceptable to the call center manager. After the model was finaly giving acceptable schedules, we built an interface to be used by the call center supervisor to compute weekly and daily schedules. III. The C O E Workforce Management Tool After building a model to do the scheduling, we had to build an interface to be used by the supervisor of the call center. We built the interface using Microsoft Access 97 for data entry and Excel 97 for output display. The end result is what we call the COE Workforce Management Tool. The Tool is more than just a scheduling interface. It is intended to be a complete workforce management tool, with its 3 key modules: Forecasting, Staffing levels and Scheduling. At the time of writing this thesis, only the Scheduling module was fully functional. Forecasting Module Scheduling Module Additional Modules Figure 17. The skeleton of The COE Workforce management Tool. Staffing Levels Module COE Call Center Toolkit 33 COE Call Center Toolkit - (Start: Form] E l File Edit View Insert Format Records look Window Help COE Toolkit J f f J x J ; C®6 COE Cj/Center Toeiki Product developed for The Workers' Compensation Board ol British Columbia Version Beta 1.0 March 11th. 1998 Form View MJM Figure 18. Opening screen. Initial screen of the COE interface. The minimum hardware, software and operating requirements to run the C O E Call Center Toolkit are Intel Pentium Processor or higher, A C C E S S 97, E X C E L 97, G A M S , the professional license of Cplex 4.0 or higher, and Windows 95/MSDOS. Minor changes would have to be done if the user had any other solver linkable with G A M S . If the user has other modeling language, such as A M P L or L I N G O , then the models would have to encoded again, but the Toolkit would equally work. 34 The toolkit comes with online help explaining what the purpose of each module is, and what steps are to be taken when computing a new schedule. The text of the help was carefully examined to make it as simple as possible. COE Call Cenlei Toolkit - [Start : Form] Edft View Insert Format Records Took Wjndow Hefcij COE Toolkit S3 Help Scheduing Module m Main) I Forecasting | Staffing Levels | Help Scheduling J E J The Scheduling Module is intended to help in the task of allocating CSR's phone times. To compute a new schedule for a given week or day. follow the steps: 1 . Select the CSR's available foe that week or day. 2. Enter the number of CSR's required for every half hour fr*ou should get these numbers from the Staffing Levels Module). 3. Click on the 5j I icon on the toolbar. 4. Click on the S£| icon on the right toolbar. 5. Cickonthe icon on the recasting | Staffing Levels | i to the COE Call Center DI intended to help managers it of Cal Centers to achieve gets. kit is divided in three sections; ting Module . the Stalling ule and the Scheduling information on those modules lorresponding buttons. • 4 •I M |Form View NUM • Figure 19. On line help. The On line help screens of the Toolkit carry the user through the different screens, explaining the purpose of each one, and the steps to be taken to compute a new schedule. 1. Help button. Access to the Help menu. 2. Information Button. By pressing this button, the user can get detailed information about C O E . The Scheduling Screen A t the scheduling screen, the user is prompted to: 1. Enter the staff available for the given day/week. 2. Enter the date for which that schedule is being derived 35 3. Input the staffing levels required for every time slot of the working day. 0 _ A 3^ COE Call Cenlei Toolkit - [Scheduling Module] l i d @e Etft Vjew Insert Format Records Iools tfrndcw Help I COE Toolkit mm ir-|fl| x e Please select the CSR * available site CSRs • , 1 ^ CSR available Moe Ahamed Lorraine Davies Marilyn Elias Anne Fillon Pauline Fitton Linda Galatiuk John Goddard GiP Hannah Ann Harder me 4" I ^ ^ ; Please enter the number of CSR s required for every time si 3t Week' 1/19/98, 7:00 7:30 7:30 8:00 800 830 .8:30 9:00 9:00 9:30 9:30*1 10:00] 10:00 •10:301 10t30 11:00 11:00?, 11:30J 11:30' ;12Q0< 1200? 1230 1230? 1300 1 1 3, •V 12 :J< . 16 , ft _2? _ « J t U J #"» '-^ 13:00 13:30 13:30 14:00 14:00 14:30 14:30 15:00 15:00 15:30 1530 1G:00< 16:00, 16:30 16:30 .17:00 1700 17 30 1730 (iaoo: 1800 1830 1830 19 005 I ' S S 4 |FomnView INUMI Figure 20. Scheduling screen. The left hand side toolbar is the general toolbar. The right hand side toolbar is the module-specific toolbar, and changes from module to module. 1. Select the staff available using the list box. 2. Enter the date for which a new schedule is going to be computed. 3. Enter the staffing levels for every time slot. After inputting the data, all the user has to do is press the solve button, on the right hand side toolbar. The toolkit will automatically open the MSDOS window, write the GAMS file, call Cplex and solve the problem. 36 :i]fiJx| Fjle Edit View Insert Format Records Iools. Wjndow Help | COE Toolkit s DDI oat m ® e Please select the GSR's available CSR's -CSR's available Moe Ahamed Lorraine Davies Marilyn Elias Anne Fitton Pauline Fitton Linda Galatiuk John Goddard GiD Hannah Ann Harder 3 Lorraine Davies Marilyn Elias Anne Fitton Pauline Fitton Linda Galatiuk ...... • Z l gamscmex (IIP Solution : Final LP : 37.000000 37.0110000 (210 iterations, <M nodes) (0 iterations) Please enter the number of CSR's 7:00 7:30 7:30 8:00 8:00 8:30 8:30 f 9:00 I 1 1 3- 12 r 13:00 13:30 13:30 14:00 14:00 14:30 14:30 P 15:00 | I 13 14 .16 is r jFormView Best integer solution possible : 37.000000 Relative gap : 0 — Restarting execution — SCHED.GNS1291) — Reading solution for nodel SCHED — SCHED.GflSOOll — fill done — Erasing scratch files i C:\Call\UCB>oans assgnnt.gns I'.'fitIS 2.25.DQ2 Copuright (0 1988-1996 GfiMS Developnent. Rll rights reserved Licensee: miliar,"T. Zienba • G970?ll:1252fif-URT University of British Colunbia — Starting conpilation — .SCHED. INCU425) — R33GNNT.GH3127) — Starting execution — HSSGHHT.GHSI1117) — Generatina nodel flSSGHNT — RSSGrlllT.GriSllW) Figure 21. Cplex. The Toolkit opens the MSDOS window and calls Cplex to solve the scheduling model Once Cplex has finished solving the scheduling problem, the user can display the output using Excel. This can be accessed from within the Toolkit. There is no need to open Excel separately, for this is done by clicking on the "Open Excel" button, on the right hand side toolbar of the scheduling module. To view the new schedule all we have to do is press the "View new schedule" button, Excel will automatically read the output file from Cplex and display the new schedule in a predefined format. 37 r El File E * Vjew Insert Format Records Tods Window Help | COE Toolkit P J « - M * J 2<C Microsoft Excel - Schedule.xls j l ^ Hie Edit View Inser t Format Tools Data Window Help | WCB 1 0 0 % " i*ii R3C2 352 v " • 1 1 2 3 4 5 6 7 8 9 10 11 ! 12 13 14 15 11 1. 2 3 13201 7:00 7:30 7:30 8:00 8:00 6:30 8 3 0 9 0 0 9:00 9:30 9:30 10:00 10:00 10:30 10:30 11:00 11:00 11:30 11:30 1 2 0 0 12:00 12:30 1 2 3 0 13:00 13:00 13:30 13: 14: 1 352 A! 2 231 5-: 3 309 6'. 4 124 7 , S 455 8 6 104 9 " 7 442 10 8 175 1 1 - 9 504 12 10 496 13- 11 341 14, 12 175 15 13 425 16 14 207 17 15 500 18~ 16 309 19 17 175 I<I<IMHI\ Original" Tricks / Tricks / y \ S c h e d u l e / 5hee t3 / |_< R e a d y •at 2H! 9f NUM| /A • < H 14 • * m /A Figure 22. Excel. After computing a new schedule, the user can display the output using Microsoft Excel. The user can then print the new schedule and distribute it among the CSR's. Also included in the Toolkit is a customized database for the supervisor and/or the manager of the call center. In that module, the supervisor can keep a record of the CSR's preferences for the different "tricks". In addition, standard database fields have been included. 38 ^ COE Call Centet Toolki t - [CSR Database] EHHE3 ||n @e Edit View Jnsert'Fprmat Records Iools «findow J i e t y ^ C O E TocM -v -> >--- | g ' - ^ ] | bt. • Of not a •9 til e Name:- Photo Team: I Vancouver South Charge tt: ||11199 1 Preference: II Emait • of H Address: Phone ' | | _ Office T f f Home 7T7| T~ 231 8810 - ( / T h t s ts the mailing address ^ | Moble F f | ~ Notes: Form View I' r 0 . Figure 23. CSR database screen. Fields included are: Name, Team, Charge #, Preferences, Email, Photo, Address, Phone (office, home, fax, mobile, contact person), Notes 39 IV. C O N C L U S I O N This thesis was developed simultaneously with a broader project undertaken by a team from the Center for Operations Excellence at The University of British Columbia. By the end of phase one of the project, only the scheduling module of the COE workforce management tool was operational, except for fixing the scale of the parameter that measures preferences for shifts. Management from WCB.was highly satisfied by the model and interface. We tested the scheduling module against one of their schedules and found that our model needed 5 CSRs less than what they had scheduled. Impressed by the potential savings, WCB decided to contract the services of a Call Center software provider that could guarantee maintenance and training. 40 Bibliography [I] Andrews, B.H., Parsons, H.L. (1989) "L.L. Bean chooses a telephone agent scheduling system" Interfaces 19, n 6, pp. 1-9. [2] Andrews, B.H., Parsons, H.L., Quinn, P. (1991) "Allocating telecommunication resources at L.L. Bean, Inc" Interfaces 21, n 1, pp.75-91. [3] Aspect telecommunications, http://www.aspect.com [4] Bechtold, S.E., Brusco, M.J. & Showalter, MJ. (1991")" A comparative evaluation of labor tour scheduling methods" Decision Sciences 22, pp. 683-699. [5] Brusco, M.J., Jacobs, L.W., Bongiorno, R.J., Lyons, D.V. & Tang, B. (1995) "Improving personnel scheduling at airline stations" Operations Research 43, n.5., pp. 741-751. [6] Call Center magazine, http://www.callcentermagazine.com [7] Call Center 98. http://www.callvoice.com/callcenter98/ [8] Fischer, M.J., Garbin, D.A., Gharakhanian, A. (1998) "Performance modeling of distributed automatic call distribution systems" Accepted by Telecommunications Systems. [9] Gabella, A., Pearce, W. (1979) "Telephone sales manpower planning at Quantas" Interfaces 9, n 3, pp. 1-12. [10] Gnedenko, B.V., Kovalenko, I.N. (1968). "Introduction to Queueing Theory" Israel Program for Scientific Translations. [II] Gross, D., Harris, C M . (1974) "Fundamentals of Queueing Theory" John Wiley & Sons. [12] Harris, C M . , Hoffman, K.L., and Saunders, P.B. (1987) "Modeling the IRS telephone taxpayer information system" Operations Research 35, n 4, pp. 504-523. [13] Intecom, Inc. http://www.intecom.com (1997). Call Centre Infosource. [14] Mehrotra, V. (1997) "Ringing up big business" OR/MS Today August 1997, pp.18-24. [15] Morris, J.G. & Showalter, M.J. (1983) "Simple approaches to shift, days-off and tour scheduling problems" Management Science 29, n 8, pp. 942-950. [16] Northern Telecom, http://www.nortel.com/home/home.html ' [17] Pipkins, Inc. http://www.pipkins.com 41 [18] Portage Communications, Inc. http://ww.portagecorrimunications.com [19] Rockwell, http://www.ec.rockwell.com [20] Taske technology, http://www.taske.com/products.html [21] TCS Management Group, Inc. http://www.tcsmgmt.com [22] The Workers' Compensation Board of British Columbia, http://www.wcb.bc.ca 42 APPENDIX A. The Erlang Formulas The unit of measure by which the intensity of workload, or telephone traffic is expressed is called an Erlang. One Erlang is equal to one hour of one telephone conversation, ocurring over a one-hour period time. The name was chosen at a contest held in 1943 by the editors of a Swedish telephony journal to honor of Agner Krarup Erlang (1887-1929). Erlang was a mathematician who worked for the Copenhagen Telephone Company. He published several important works on the theory of telephone traffic, the most important of which was "Solution of Some Problems in Theory of Probabilities of Significance in Automatic Telephone Exchanges." The formula used to compute the Erlangs in any segment of time is the following: Suppose x denotes the intensity of telephone traffic that we wish to compute for a segment of time of length t (measured in minutes). If n denotes the number of calls that were received during that period of time, and denotes the length of call i (in seconds) during segment t, then x equals the following formula: h n n 1 n^n_ x - —— = — Erlangs t t Example: Suppose we are interested in computing the intensity of traffic over a period of time of 50 minutes in a call center (see graph below). During that period of time three phone calls were registered. 43 time t = 50 minutes Figure . Calls start before or carry. Calls c,, c2 and c3 were offered during /, two of starting before the segment of time we are interested, and the third finishing some time after. However, we only consider the. length of those calls that overlap the segment of time that interests us. Suppose those lengths are: /, =1560, l2 =1020 and /3 =1200. Then applying the above formula we wold have that the intensity of telephone traffic is: 1560 + 1020 + 1200 6 0 = 1.26 Erlangs 50 Analogously, one person speaking to another person over the phone for one hour would result in one Erlang of traffic intensity. In the context of trunk lines, the formula changes. Now the value is equal to the number of Calls Offered times the average Trunk Hold Time divided by 60 (seconds) divided by the number of minutes in the Time Segment. The name of Erlang also appears in the famous Erlang formulas, Erlang B (M/G/c/c) and Erlang C (M/M/c/oo), discussed below 44 Definition: We say that the Markov Chain {N(t), t > 0} ,with continuous parameter and discrete-state space S = {0,1,2,...}, is a birth and death process with birth rate {Xn,n = 0,1,2,...}, A. > 0 Vi and death rate {//„,« = 0,1,2,... },juQ = 0,//,. > 0 V* * 0 if the following conditions are met: (c.l) P[N(t + h) = n +11 N(t) = n] = Xnh + o{h), V« > 0 (c.2) P[N(t + h) = n-l[ N(t) = n] = nnh + o(h), V« > 1 (c.3) P[N(t + h) = n\ N{t) = n] = 1 - {Xn + //„)/, + o(h) (c.4) P[N(t + h) = j\ N(t) = n] = o(h), Vjn,n-l,n +1 For ease of notation, let's denote Pn{t) = P[N(t) = n], and Pin (t) = P[N(t) = n \ N(0) = i] then Pin (t + h) = Pt[N(t + h) = n] = P.[N(t + h) = n\ N(t) = n- l]P,[N(t) = n -1] + P;[N(t + h) = n\ N(t) = n]P,[N(t) = n] + Pt[N(t + h) = n\ N(t) = n + l]P,[N(t) = n +1] + + h) = n\ N(t) = k]P,[N(t) = k] = {X„_xh + o(hj)Pi(n_x) {t) + k*n,n-\,n+\ + {l-(Xn+Mn)h + o(h))Pin +(^h + o(h))Pi{n+l)(t) + o(h) and therefore: pin{t+h)-pin{t)=^,,^(0+^,^(0- ^ +»n)pin(o+ x ^ , ( o " k=n-\,n,n+\ " If we now take limits for h —> 0 then, we obtain the desired equations. (25) P:0(t) = -X0Pi0(t) + MlPn(t) (26) />; co=xn_{pi(n_n (o - (A. + //„)/>„ c o + ( o , v/i > i In order to compute the limiting distribution, we let n - » co and note (r) -> 0. We obtain the birth and death equations: 45 (27) 0 = - W ^ P , (28) 0 = An_,P„_, - {XN + M„)P„ + VN+IPN+I, Vn > 1 From equation (27) we get: PX = —P0, and setting n - 1 in equation (28) and operating we /t /I would get: P2 = 0 1 P0. In general we prove by induction that MvM2 ( 2 9 ) P „ = - ^ — P 0 We have already proved that the relationship is correct for n = 1. Let's suppose now that the hypothesis holds for n, and let's see that it is then true for n+1. From equation (28) we have: MN+\P„+\ = -X„_XP„^ + (/t„ + juN)PN, and therefore, by the induction hypothesis, n-2 n - l m n * ru M,.,P„, = - V , S—Pv +K P0 + Jf—P0, which after simplification Ek Ek ; = i 7=1 y=i becomes the expression we were looking for. n- l n - I 00 OO X A ' 00 X X * 1 = ^ y°„ = Po'YJ—— therefore if ] T — — < oo then there exists a unique limiting -TlM, "flMj 7=1 7 = 1 distribution, which has the following expression: 46 p 1 0 i=0 n n- l p. 0 M/M/c/oo Suppose we now have c servers in the system. Since the arrival process is assumed to be Poisson with parameter X, the birth rates are An = A. To determine the death rates, we observe that if there are more than c customers in the system, all servers are busy and each putting out at a mean rate u, and the mean system output rate is thus cu. When there are fewer than c customers in the system, say n < c, only n of the c servers are busy and the system is working at rate n\x. Hence jun may be written as Replacing these values for Xn and /un in equation (29), we get P0 (n>c) (1 < n < c) In order to find P0 we must again use the boundary condition 47 which gives c-l nn co on c" ^c\n X r X If we define r = — and p = — = — , then we have // c c^i (30) P0 c-l „n oo n = 1 but. oo .^n 1 §c-cd dSU dSU; c!(l-r/c) and we can rewrite (30) as (r/c = p TC 1) ^ 0 = c-l « 1^7 cr %&n\ c!(c-r) - l c-l 1 n=0 -T + — — c! i^ c/ - X (rlc = p%\) As we have seen, the condition for existence of a steady-state solution is A,/(cu)<l; that is the mean arrival rate must be less than the mean service rate of the system. Let's now compute the expected queue size Lq. ( 3 1 ) £ , =±(n-c)Pn=±^r»P0-±-±-r"P0 c~c\ v t^c"-cc\ Now oo (r\ "~C~l » fr\ r I>-c) r c n-c n=c \CJ n~c-\ c! c+l 1 • + -c2/r (1-r/c)2 1 - r / c 48 also, P0crc c! ( l - r / c ) Substituting these expressions in (31) we get rc+1 U 1 c2 Ir cl Ir (l-r/c)2 l-r/c l-r/t -c+]/c [ c ! ( l - r / c ) 2 l(c-\)\(c^-X)2 (M/M/c/K). An improvement over the Eflang C formula. This model is more realistic. It assumes finite capacity, whereas Erlang C assumes infine capacity, which might not be very realistic if the blockage probability is high. Using the following birth and death rates K = \X (0<n<K) [0 (n>K) nju (0<n<c) \cfi (c<n<K) And arguing similarly we would get (32) P„ = P0 (0<nnc) 1 c c\ f-T P0 (c<n<K) The boundary condition = 1 will give us the expression for P, n=0 7^n\\fi) tic" ^ i r^" c"-cc\ 49 hence ^ 0 = c - l i (A — + > KM) ^ c"~cci n=c L ( M / M / c / c ) The special case of AT = c is known as the Erlang's first formula, or Erlang-B formula. It is immediate from (32) to see that (A/juy/n\ P. = (0 < n < c) (=0 A very important result for this formula is that the above expression is also valid for M/G/c/c queues, that is, the result is independent of the distribution used for service times. 50 APPENDIX B. Poisson Processes Definition: We say that the Markov chain {N(t),t > 0} ,with continuous parameter and discrete-state space S - {0,1,2,...}, is a counting process if N(t) counts the number of occurrences in (§,t\. Hence a counting process must satisfy: (i) . N(t) > 0 (ii) N(t) is integer valued (iii) If s < t, then N(s) < N(t) (iv) For s <t, N(t) - N(s) equals the number of events that have occurred in the interval (s,t] A counting process is said to possess independent increments if the number of events that occur in disjoint time intervals are independent. A counting process is said to possess stationary increments if the distribution of the number of events that occur in any interval of time depends only on the length of the interval. Definition: The counting process {N(t), t > 0} is said to be a Poisson process having rate X, » 0 , if: (i) N(0)=0 (ii) The process has independent increments (iii) The number of events in any interval of length t is Poisson distributed with mean Xt. That is, for all s, t >0, -*<*>: P{N(t + s)-N(s) = n) =e , n = 0,1,... 51 Let's now prove the following result, used in section II.5 of this thesis. The result says: Suppose {N(t), t > 0} is a Poisson process with rate X, X>0. Suppose that every time there is an occurrence in this process, a new process {X(t), t>0} increments its value by one with probability p, and process {Y(t), t>0} increments its value with probability 1-p. Suppose also that N(t) = X(t) + Y(t) . Then, {X{t), t > 0} is a Poisson process with parameter Ap and {Y(t), t > 0} is a Poisson process with parameter A.(l-p). We will prove only the case for X(t) • Obviously X(0) = 0 • It is straightforward that the stationarity and independence properties hold. • Let's see that X(t) is Poisson distributed with parameter Apt P[X(t) = k] = fj P[X(t) = k | N(t) = n]P[N(t) = n] = f\n]pk(l-p) n-k tt n=k oo k\'n-k)\ Pk(l-P) n-k = (At) ke~"p k\ {\-p)"-k{At) {n-k)\ n-k (At)ke'A'pk k\ e^'jAptf k\ 52 APPENDIX C. Call Center Technology There are four populations within any call center environment: callers, agents, supervisors and management. Each of these groups has their own set of equipment requirements to ensure the call process evolves quickly and with few problems. Caller requirements Depending on the purpose of the call center, the caller may be requesting information, making a purchase or checking on the status of an order or account. Regardless of the purpose of the call, the caller's first requirement is to exchange some kind of information as quickly and friendly as possible. Agent requirements Agents are at the heart of call center activity. They are responsible for receiving incoming calls, and, at times, for placing outgoing calls to customers. Their requirements focus on the features needed to do the job as effectively as possible. Supervisor requirements Supervisors are tasked with assuring that management's goals of superior service at the lowest cost are achieved. They are concerned with the details of agent's activity, individually and as part of a group. Therefore supervisors need to be able to identify agent's current status/activity, provide assistance to agents on requests, assure that the call center is staffed appropriately during peak and low periods, and report results to management. 53 Management requirements Management must ensure that the call center provides the best possible level of service, while keeping costs at minimum. For management, customer satisfaction is critical. This means carefully balancing personnel, equipment, call volume and caller expectations. There are many different types of call centers, almost one for each type of business; but, we could group them in three major categories: Inbound, Outbound and Inbound/Outbound call centers. Inbound call centers are typically service call centers. Outbound call centers are usually marketing call centers, and Inbound/Outbound are call centers with a component of both. Although different types of call centers have different purposes, it is acceptable to say that a common goal among all these call centers is to handle large volumes of calls as efficiently as possible, and at minimum cost. Different types of call centers need different equipment, as well as different performance measures. Common equipment in a call center includes: • Automatic Call Distributor (ACD) The goal of automated call distributed technology has always been to spread incoming calls among call center agents so that each agent handles an equitable share of the load and the caller has the best chance of being served quickly. Most of the current ACD's offer the following features: 54 Anticipated Hold Time. This feature provides an announcement to the caller stating the estimated time he/she will have to wait before a CSR is available to answer their call. Frequently this anticipated hold time announcement is combined with an option to leave a message. Queue statistics display. This feature provides live statistics on how the call center is performing: number of people waiting in the queue, average speed of answer, longest waiting time, etc.... This is especially useful for the supervisor, because it helps him/her monitor the call center, and make adjustments if the office is not performing as it should. Skills based routing. Some call centers have only one queue, that is all the callers sit in one queue, and all the agents are devoted to that queue, so that whenever an agent becomes available to answer new calls, the ACD routes the first caller in the queue to that agent. Some other call centers have however, multiple queues. Whenever new callers enter the system, they are routed to a particular queue, depending on the purpose of the call, or the caller itself. Different queues are handled by different agents. The purpose of this approach is to route the call to the available agent whose skills are best matched to the callers needs. Skills based routing has to incorporate another the overflow feature. This feature recognizes when agents in a group are backlogged and reroutes calls to another group that may have fewer calls waiting. Network Automatic Connection (NAC). Some organizations have several call centers that perform the same task but are split for geographical, or strategic 55 reasons. It could happen however, that while one of the call centers is experiencing high volume of calls, another is not busy at all. NAC allows for managing all the call centers as one virtual call center, combining together all the agents in a single pool. Personal Branch Exchange (PBX) The PBX is the switching device that routes calls to the next available agent. Some ACD' s have this capability (hardware & software). Others, are just a piece of software that sit on the PBX making it a PABX (Personal Automatic Branch Exchange). Interactive Voice Response (IVR) The interactive voice response is a crucial complement to the ACD. In Skills based routing call centers, the routing of the calls to the corresponding group of agents is done through the IVR. The caller is receives recorded messages and is prompted to press buttons on his/her touch-tone telephone. Computer Telephony Integration (CTI) This is a fairly new, emerging feature very useful in marketing call centers. With the use of IVR, the ACD can retrieve information from the caller even before he/she speaks with an agent. The use of CTI permits the ACD retrieve important information about the caller, stored in a database, and pop it up on the agent screen that is answering the call. This can save important seconds per call, most importantly in rush 56 periods. It also provides a good impression to the caller, because agents have instant access to the caller records, and don't have to ask about it. Credit card companies are an example of companies that use CTI for better customer service. 57 APPENDIX D. GAMS Code for the Assignment Problem SETS i / l * 3 7 / j / l * 7 1 1 / ; VARIABLES y ( i / j ) z; PARAMETER x ( j ) $include c:\call\wcb\sched.inc PARAMETER p ( i , j ) ; *$include c:\call\wcb\asgnpref.inc BINARY VARIABLES y; EQUATIONS obj c s r (i) s h i f t s ( j ) ; obj..z =e= s u m ( i , s u m ( j , p ( i , j ) * y ( i , j ) ) ) ; c s r (i) . . s u m ( j , y ( i , j ) ) =e= 1; s h i f t s ( j ) . . s u m ( i , y ( i , j ) ) =e= x ( j ) ; y . f x ( i , j ) $ ( o r . d ( i ) l e 32 and ord( j ) ge 562)=0; y . f x ( i , j ) $ ( o r d ( i ) ge 33 and or d ( j ) l e 561)=0; model assgmnt/all/; solve assgmnt maximizing z using mip; f i l e eassgmnt/assgmnt.xls/; put eassgmnt; loop(i,loop(j, put i.tlj.tl, y.l(ij)//)); 58 APPENDIX E. Model for Stage 1 of the Scheduling Problem 1)+ x ( 3 6 ) + x ( 3 7 ) + x ( 3 8 ) + x ( 3 9 ) + x ( 4 0 ) + x ( 4 1 ) + x ( 5 7 ) + x ( 6 2 ) + 70)+ x ( 7 8 ) + x ( 7 9 ) + x ( 8 0 ) + x (81 ) + x ( 8 2 ) + x ( 9 9 ) + x ( 1 0 3 ) + 109) + x ( 1 1 6 ) > 1 1) + x ( 2 ) + x ( 3 6 ) + x ( 3 7 ) + x (38 ) + x ( 3 9 ) + x ( 4 0 ) + x ( 4 1 ) + 42) + x ( 4 3 ) + x ( 4 4 ) + x ( 4 5 ) + x (46 ) + x ( 5 7 ) + x ( 6 2 ) + x ( 6 3 ) + 70) + x ( 7 1 ) + x (78 ) + x ( 7 9 ) + x ( 8 0 ) + x ( 8 1 ) + x ( 8 2 ) + x ( 8 3 ) x ( 8 4 ) + x ( 8 5 ) + x (86 ) + x ( 9 9 ) + x ( 1 0 3 ) + x ( 1 0 4 ) + x ( 1 0 9 ) + 110) + x ( 1 1 6 ) + x ( 1 1 7 ) + x ( 1 2 3 ) + x ( 1 5 9 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + 162) + x ( 1 6 3 ) + x(l"64) + x ( 1 8 0 ) + x ( 1 8 5 ) + x ( 1 9 2 ) + x ( 2 0 0 ) + 201) + x ( 2 0 2 ) + x ( 2 0 3 ) + x ( 2 0 4 ) + x ( 2 2 1 ) + x ( 2 2 5 ) + x ( 2 3 1 ) + 238) > 3 x ( 4 0 ) + + x ( 4 8 ) x ( 7 2 ) + + x ( 8 5 ) x ( 1 0 3 ) + x ( 1 1 7 ) + x ( 1 6 2 ) + x ( 1 6 9 ) + x ( 2 0 1 ) + x ( 2 0 8 ) + x ( 2 3 9 ) + x ( 2 8 7 ) + x ( 3 2 8 ) + x 1) + x ( 2 ) + x (3 ) + x ( 3 6 ) + x ( 3 7 ) + x ( 3 8 ) + x ( 3 9 ) + X (41) + x ( 4 2 ) + x ( 4 3 ) + x ( 4 4 ) + x ( 4 5 ) + x ( 4 6 ) + x ( 4 7 ) + x ( 4 9 ) + x ( 5 0 ) + x ( 5 1 ) + x ( 6 3 ) + x ( 6 4 ) + x ( 7 1 ) + X (78) + x ( 7 9 ) + x (80 ) + x ( 8 1 ) + x (82 ) + x ( 8 3 ) + x ( 8 4 ) + x (86) + x ( 8 7 ) + x ( 8 8 ) + x ( 8 9 ) + x (90) + x ( 9 9 ) + X (104) + x ( 1 0 5 ) + x ( 1 0 9 ) + x ( 1 1 0 ) + x ( l l l ) + x ( 1 1 6 ) + X (118) + x ( 1 2 3 ) + x ( 1 2 4 ) + x ( 1 5 9 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + X (163) + x ( 1 6 4 ) + x ( 1 6 5 ) + x ( 1 6 6 ) + x (167) + x ( 1 6 8 ) + X (180) + x ( 1 8 5 ) + x ( 1 8 6 ) + x ( 1 9 2 ) + x ( 1 9 3 ) + x ( 2 0 0 ) + X (202) + x ( 2 0 3 ) + x (204) + x ( 2 0 5 ) + x (206) + x ( 2 0 7 ) + X (221) + x ( 2 2 5 ) + x ( 2 2 6 ) + x ( 2 3 1 i + x (232) + x ( 2 3 8 ) + X (244) + x (282) + x ( 2 8 3 ) + x ( 2 8 4 ) + x ( 2 8 5 ) + x ( 2 8 6 ) + X (305) + x ( 3 1 1 ) + x (318) + x ( 3 2 5 ) + x ( 3 2 6 ) + x ( 3 2 7 ) + X (329) + x ( 3 4 9 ) + x (354) + x (360) + x ( 3 6 6 ) > 9 X '1) + x ( 2 ) + x ( 3 ) + x ( 3 6 ) + x ( 3 7 ) + x ( 3 8 ) + x ( 3 9 ) + x ( 4 0 ) + X (41) + x (42 ) + x ( 4 3 ) + x ( 4 4 ) + x ( 4 5 ) + x ( 4 6 ) + x ( 4 7 ) + x ( 4 8 ) + x ( 4 9 ) + X (50) + x ( 5 1 ) + x ( 5 2 ) + x ( 5 3 ) + x ( 5 4 ) + x ( 5 5 ) + X (56) + x ( 6 4 ) + x ( 6 5 ) + x ( 7 2 ) + x ( 7 3 ) + x ( 7 8 ) + x ( 7 9 ) + x ( 8 0 ) + x ( 8 1 ) + X (82) + x ( 8 3 ) + x ( 8 4 ) + x ( 8 5 ) + x ( 8 6 ) + x ( 8 7 ) + X (88) + x ( 8 9 + x (90) + x 91) + x (92 ) + x ( 9 3 ) + x ( 9 4 ) + x ( 1 0 4 ) + x ( 1 0 5 ) + x (106 ) + x ( 1 1 0 ) + x ( l l l ) + x ( 1 1 2 ) + x ( 1 1 7 ) + x ( 1 1 8 ) + x ( 1 1 9 ) + X (123 ) + x ( 1 2 4 ) + x ( 1 2 5 ) + x ( 1 5 9 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + x (162 ) + X (163 ) + x ( 1 6 4 ) + x (165) + x ( 1 6 6 ) + x ( 1 6 7 ) + x ( 1 6 8 ) + x ( 1 6 9 ) + x (170 ) + x ( 1 7 1 ) + x (172) + x ( 1 7 3 ) + x ( 1 7 4 ) + x ( 1 8 6 ) + x ( 1 8 7 ) + x (193 ) + x ( 1 9 4 ) + x ( 2 0 0 ) + ;X (201) + x ( 2 0 2 ) + x (203) + x ( 2 0 4 ) + x (205 ) + x ( 2 0 6 ) + x ( 2 0 7 ) + x ( 2 0 8 ) + x ( 2 0 9 ) + x ( 2 1 0 ) + x ( 2 1 1 ) + x (212 ) + x ( 2 2 1 ) + x ( 2 2 5 ) + x ( 2 2 6 ) + x ( 2 2 7 ) + x (231) + x ( 2 3 2 ) + x (233 ) + x ( 2 3 8 ) + x (239) + x (240) + x (244) + x ( 2 4 5 ) + x ( 2 8 2 ) + x (283 ) + x ( 2 8 4 ) + x (285) + x ( 2 8 6 ) + x ( 2 8 7 ) + x (288) + x ( 2 8 9 ) + x (290 ) + x ( 2 9 1 ) + x (292) + x ( 2 9 3 ) + x ( 3 0 5 ) + x ( 3 0 6 ) 5 9 + x ( 3 1 1 ) + x ( 3 1 2 ) + x ( 3 1 8 ) + x ( 3 1 9 ) + x ( 3 2 5 ) + x ( 3 2 6 ) + x ( 3 2 7 ) + x ( 3 2 8 ) + x ( 3 2 9 ) + x ( 3 3 0 ) + x ( 3 3 1 ) + x ( 3 3 2 ) + x ( 3 3 3 ) + x ( 3 3 4 ) + x ( 3 4 9 ) + x ( 3 5 0 ) + x ( 3 5 4 ) + x ( 3 5 5 ) + x ( 3 6 0 ) + x ( 3 6 1 ) + x ( 3 6 6 ) + x ( 3 6 7 ) + x ( 3 7 2 ) + x ( 3 9 7 ) + x ( 3 9 8 ) + x ( 3 9 9 ) + x ( 4 0 0 ) + x ( 4 0 1 ) + x ( 4 1 5 ) + x ( 4 2 0 ) + x ( 4 2 8 ) + x ( 4 2 9 ) + x ( 4 3 0 ) + x ( 4 3 1 ) + x ( 4 4 4 ) + x ( 4 4 8 ) + x ( 4 5 5 ) > 12 X i 1) + x ( 2 ) + x ( 3 ) + x ( 3 6 ) + x ( 3 7 ) + x ( 3 8 ) + x ( 3 9 ) + x ( 4 0 ) + x ( 41) + x ( 4 2 ) + x ( 4 3 ) + x ( 4 4 ) + x ( 4 5 ) + x ( 4 6 ) + x ( 47) + x ( 4 8 ) + x ( 4 9 ) + x ( 5 0 ) + x ( 5 1 ) + x ( 5 2 ) + x ( 5 3 ) + x ( 5 4 ) + x ( 5 5 ) + x 56) + x ( 5 7 ) + x ( 5 8 ) + x ( 5 9 ) + x ( 6 0 ) + x (61) + x ( 65) + x ( 6 6 ) + x ( 7 3 ) + x ( 7 4 ) + x ( 8 3 ) + x ( 8 4 ) + x ( 8 5 ) + x ( 8 6 ) + x ( 8 7 ) + X 88) + x ( 8 9 ) + x(90Y + x ( 9 1 ) + x ( 9 2 ) + x (93) + x ( 94) + x ( 9 5 ) + x ( 9 6 ) + x ( 9 7 ) + x (98) + x ( 1 0 5 ) + x (106) + x (107) + x (111) + X 112) + x ( 1 1 3 ) + x ( 1 1 8 ) + x ( 1 1 9 ) + x ( 1 2 0 ) + x ( 1 2 3 ) + x (124) + X 125) + x ( 1 5 9 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + x (162) + x ( 1 6 3 ) + x ( 1 6 4 ) + X 165) + x ( 1 6 6 ) + x (167) + x ( 1 6 8 ) + x ( 1 6 9 ) + x ( 1 7 0 ) + x ( 1 7 1 ) + X 172) + x ( 1 7 3 ) + x ( 1 7 4 ) + x ( 1 7 5 ) + x ( 1 7 6 ) + x ( 1 7 7 ) + x ( 1 7 8 ) + X 179) + x ( 1 8 7 ) + x ( 1 8 8 ) + x ( 1 9 4 ) + x ( 1 9 5 ) + x ( 2 0 0 ) + x ( 2 0 1 ) + X 202) + x ( 2 0 3 ) + x (204) + x ( 2 0 5 ) + x ( 2 0 6 ) + x (207) + x ( 2 0 8 ) + X 209) + x ( 2 1 0 ) + x ( 2 1 1 ) + x ( 2 1 2 ) + x ( 2 1 3 ) + x (214) + x ( 2 1 5 ) + X .216) + x ( 2 2 6 ) + x ( 2 2 7 ) + x ( 2 2 8 ) + x ( 2 3 2 ) + x ( 2 3 3 ) + x ( 2 3 4 ) + X (239) + x ( 2 4 0 ) + x ( 2 4 1 ) + x ( 2 4 4 ) + x ( 2 4 5 ) + x (282) + x ( 2 8 3 ) + X (284) + x ( 2 8 5 ) + x ( 2 8 6 ) + x ( 2 8 7 ) + x(288) + x (289) + x ( 2 9 0 ) + X (291) + x ( 2 9 2 ) + x ( 2 9 3 ) + x ( 2 9 4 ) + x ( 2 9 5 ) + x (296) + x ( 2 9 7 ) + X (298) + x ( 2 9 9 ) + x ( 3 0 6 ) + x ( 3 0 7 ) + x ( 3 1 2 ) + x (313) + x ( 3 1 9 ) + X (320) + x ( 3 2 5 ) + x ( 3 2 6 ) + x ( 3 2 7 ) + x ( 3 2 8 ) + x ( 3 2 9 ) + x ( 3 3 0 ) + X (331) + x ( 3 3 2 ) + x ( 3 3 3 ) + x ( 3 3 4 ) + x ( 3 3 5 ) + x (336) + x ( 3 3 7 ) + X (338) + x ( 3 3 9 ) + x ( 3 4 9 ) + x ( 3 5 0 ) + x (351) + x ( 3 5 4 ) + x ( 3 5 5 ) + X (356) + x ( 3 6 0 ) + x ( 3 6 1 ) + x ( 3 6 2 ) + x ( 3 6 6 ) + x ( 3 6 7 ) + x ( 3 6 8 ) + X (372) + x ( 3 7 3 ) + x ( 3 9 7 ) + x ( 3 9 8 ) + x ( 3 9 9 ) + x ( 4 0 0 ) + x ( 4 0 1 ) + X (403) + x ( 4 0 4 ) + x ( 4 0 5 ) + x ( 4 0 6 ) + x ( 4 1 5 ) + x ( 4 2 0 ) + x ( 4 2 1 ) + X (428) + x ( 4 2 9 ) + x ( 4 3 0 ) + x ( 4 3 1 ) + x ( 4 3 2 ) + x (433) + x ( 4 3 4 ) + X (444) + x ( 4 4 8 ) + x ( 4 4 9 ) + x ( 4 5 5 ) + x ( 4 5 6 ) + x (462) + x (490) + X (491) + x ( 4 9 2 ) + x ( 4 9 3 ) + x ( 4 9 4 ) + x ( 5 1 0 ) + x ( 5 1 7 ) + x ( 5 2 4 ) + X (525) + x (526) + x ( 5 2 7 ) + x (544) + x ( 5 5 0 ) + x (556) 14 X (1) + x ( 2 ) + x ( 3 ) + x ( 4 2 + x ( 4 3 ) + x ( 4 4 ) + x ( 4 5 ) + x ( 4 6 ) + X (47) 4 x ( 4 8 ) + x ( 4 9 ) + x ( 5 0 ) + x ( 5 1 ) + x ( 5 2 ) + x 53) + x ( 5 4 ) + x ( 5 5 ) + x ( 5 6 ) + x ( 5 7 ) + x ( 5 8 ) + x ( 5 9 ) + x (60) + x ( 6 1 ) + X (66) + x ( 6 7 ) + x ( 7 4 ) + x ( 7 5 ) + x ( 8 7 ) + x ( 8 8 ) + x 89) + x ( 9 0 ) + x ( 9 1 ) + x ( 9 2 ) + x ( 9 3 ) + x ( 9 4 ) + x ( 9 5 ) + x ( 9 6 ) + x ( 9 7 ) + X (98) + x ( 9 9 ) + x ( 1 0 0 ) + x (101) + x ( 1 0 2 ) + x ( 1 0 6 ) + x ( 1 0 7 ) + X (108) + x ( 1 1 2 ) + x ( 1 1 3 ) + x ( 1 1 4 ) + x ( 1 1 9 ) + x (120) + x ( 1 2 1 ) + X (123) + x ( 1 2 4 ) + x ( 1 2 5 ) + x ( 1 5 9 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + x ( 1 6 2 ) + X (163) + x ( 1 6 4 ) + x ( 1 6 5 ) + x ( 1 6 6 ) + x ( 1 6 7 ) + x(168) + x ( 1 6 9 ) + X (170) + x ( 1 7 1 ) + x ( 1 7 2 ) + x ( 1 7 3 ) + x ( 1 7 4 ) + x(175) + x ( 1 7 6 ) + 60 X 177) + X 178) + X 179) + x ( 1 8 0 ) + x ( 181) + X 182) + X ,183) + X 184) + X 188) + X 189) + x ( 1 9 5 ) + X 1 196) + X 205) + X 206) + X 207) + X 208) + X 209) + x ( 2 1 0 ) x ( 211) + X 212) + X 213) + X 214) + X 215) + X 216) + x (217) + x ( 218) + X 219) + X 220) + X 227) + X 228) + X (229) + x ( 2 3 3 ) + x ( 234) + X 235) + X 240) + X 241) + X 242) + X (244) + x ( 2 4 5 ) + X 1 282) + X 283) + X 284) + X 285) + X 286) + X (287) + x ( 2 8 8 ) + x ( 289) + X 290) + X (291) + X 292) + X (293) + X (294) + x ( 2 9 5 ) + x ( 296) + X (297) + X (298) + X ,299) X (300) + X (301) + x ( 3 0 2 ) + x ( 303) + X (304) + X (307) + X (308) + X (313) + X (314) + x ( 3 2 0 ) + X i 321) + X (325) + X (326) + X (327) + X (328) + X (329) + x ( 3 3 0 ) + X 331) + X (332) + X (333) + X (334) + X (335) + X (336) + x ( 3 3 7 ) + x{338) + X (339) + X (340) + X (341) + X (342) + X (343) + x (344) + X ( 350) + X (351) + X (352) + X (355) + X (356) + X (357) + x ( 3 6 1 ) + X ( 362) + X (363) + X (367) + X (368) + X (369) + X (372) + x ( 3 7 3 ) + X 374) + X (397) + X (398) + X (399) + X (400) + X (401) + x(403) + X 404) + X (405) + X (406) + X (407) + X (408) + X (409) + x ( 4 1 0 ) + X 421) + X (422) + X (428) •+ X (429) + X (430) + X (431) + x ( 4 3 2 ) + X 433) + X (434) + X (435) + X (436) + X (437) + X (444) + x (448) + X 449) + X (450) + X (455) + X (456) + X (457) + X (462) + x ( 4 6 3 ) + X 490) + X (491) + X (492) + X (493) + X (494) + X (495) + x (496) + X 497) + X (498) + X (499) + X (510) + X (511) + X (517) + x ( 5 1 8 ) + X 524) + X (525) + X (526) + X (527) + X (528) + X (529) + x ( 5 3 0 ) + X 531) + X (544) + X (545) + X (550) + X (551) + X ,556) + x ( 5 5 7 ) > 16 1) + x (2 ) + x (3 ) + x ( 4 7 ) + x ( 4 8 ) + x ( 4 9 ) + x ( 5 0 ) + x ( 5 1 52) + x ( 5 3 ) + x ( 5 4 ) + x ( 5 5 ) + x ( 5 6 ) + x ( 5 7 ) + x ( 5 8 ) + x ( 5 9 ) x ( 6 0 ) + x ( 6 1 ) + x ( 6 7 ) + x ( 6 8 ) + x ( 7 5 ) '+ x ( 7 6 ) + x ( 7 8 x ( 9 1 ) + x ( 9 2 ) + x ( 9 3 ) + x ( 9 4 ) + x ( 9 5 ) + x (96) + x ( 9 7 ) + x ( 9 8 ) + x (99) + x (100) + x (101) + x ( 1 0 2 ) + x ( 1 0 7 ) + x ( 1 0 8 ) + X (113) + X ,114) + x ( 1 1 5 ) + X (120) + X (121) + x ( 122) + x ( 1 2 3 ) + X (124) + X (125) + x ( 1 6 5 ) + X (166) + X (167) + x ( 168) + x ( 1 6 9 ) + X (170) + X (171) + x ( 1 7 2 ) + X (173) + X (174) + x ( 175) + x ( 1 7 6 ) + X (177) + X (178) + x ( 1 7 9 ) + X (180) + X (181) + X 182) + x ( 1 8 3 ) + X (184) + X (189) + x ( 1 9 0 ) + X (196) + X (197) x ( 209) + x ( 2 1 0 ) + X (211) + X (212) + x ( 2 1 3 ) + X (214) + X (215) + x ( 216) + x ( 2 1 7 ) + X (218) + X (219) + x ( 2 2 0 ) + X (221) + X (222) + X ( 223) + x ( 2 2 4 ) + X (228) + X (229) + x ( 2 3 0 ) + X (234) + X (235) + X 236) + x ( 2 4 1 ) + X (242) + X (243) + x ( 2 4 4 ) + X (245) + X (282) + X 283) + x ( 2 8 4 ) + X (285) + X (286) + x ( 2 8 7 ) + X (288) + X (289) + X ( 290) + x ( 2 9 1 ) + X (292) + X (293) + x ( 2 9 4 ) + X (295) + X (296) + X 297) + x ( 2 9 8 ) + X (299) + X (300) + x (301) + X (302) + X (303) + X 304) + x ( 3 0 8 ) + X (309) + X (314) + x ( 3 1 5 ) + X (321) + X (322) + X 330) + x ( 3 3 1 ) + X (332) + X (333) + x ( 3 3 4 ) + X (335) + X (336) + X 337) + x ( 3 3 8 ) + X (339) + X (340) + x ( 3 4 1 ) + X (342) + X (343) + X 344) + x ( 3 4 5 ) + X (346) + X (347) + x ( 3 5 1 ) + X (352) + X (353) + X 356) + x ( 3 5 7 ) + X (358) + X (362) + x ( 3 6 3 ) + X (364) + X (368) + X 369) + x (370) + X (372) 61 + x ( 3 7 3 ) + x ( 3 7 4 ) + x ( 3 9 7 ) + x ( 3 9 8 ) + x ( 3 9 9 ) + x ( 4 0 0 ) + x (401) + x ( 4 0 3 ) + x ( 4 0 4 ) + x ( 4 0 5 ) + x ( 4 0 6 ) + x (407) + x ( 4 0 8 ) + x ( 4 0 9 ) + x ( 4 1 0 ) + x ( 4 1 1 ) x ( 4 1 2 ) + x ( 4 1 3 ) + x (414) + x ( 4 2 2 ) + x ( 4 2 3 ) + x ( 4 2 8 ) + x ( 4 2 9 ) + x ( 4 3 0 ) + x ( 4 3 1 ) + x ( 4 3 2 ) + x ( 4 3 3 ) + x ( 4 3 4 ) + x ( 4 3 5 ) + x ( 4 3 6 ) + x ( 4 3 7 ) + x ( 4 3 8 ) + x ( 4 3 9 ) + x ( 4 4 0 ) + x ( 4 4 9 ) + x ( 4 5 0 ) + x ( 4 5 1 ) + x ( 4 5 6 ) + x ( 4 5 7 ) + x ( 4 5 8 ) + x ( 4 6 2 ) + x (463) + x ( 4 9 0 ) + x ( 4 9 1 ) + x ( 4 9 2 ) + x ( 4 9 3 ) + x (494) + x ( 4 9 5 ) + x (496) + x ( 4 9 7 ) + x ( 4 9 8 ) + x ( 4 9 9 ) + x ( 5 0 0 ) + x ( 5 0 1 ) + x (502) + x ( 5 0 3 ) + x ( 5 0 4 ) + x ( 5 1 1 ) + x ( 5 1 2 ) + x ( 5 1 8 ) + x ( 5 1 9 ) + x ( 5 2 4 ) + x ( 5 2 5 ) + x ( 5 2 6 ) + x ( 5 2 7 ) + x ( 5 2 8 ) + x ( 5 2 9 ) + x ( 5 3 0 ) + x ( 5 3 1 ) + x ( 5 3 2 ) + x (533 ) + x (534) + x ( 5 3 5 ) + x (544) + x ( 5 4 5 ) + x (546) + x ( 5 5 0 ) + x ( 5 5 1 ) + x ( 5 5 2 ) + x ( 5 5 6 ) + x(557)- + -x (558) > 17 x (2 ) + x ( 3 ) + x ( 3 6 ) + x ( 5 2 ) + x ( 5 3 ) + x ( 5 4 ) + x (55) + x ( 5 6 ) + x (57 ) + x ( 5 8 ) + x ( 5 9 ) + x ( 6 0 ) + x ( 6 1 ) + x ( 6 8 ) + x ( 6 9 ) + x ( 7 6 ) + x ( 7 7 ) + x ( 7 8 ) + x ( 9 5 ) + x ( 9 6 ) + x ( 9 7 ) + x ( 9 8 ) + x ( 9 9 ) + x ( 1 0 0 ) + x ( 1 0 1 ) + x ( 1 0 2 ) + x ( 1 0 8 ) + x ( 1 1 4 ) + x ( 1 1 5 ) + x ( 1 2 1 ) + x (122) + x ( 1 2 3 ) + x ( 1 2 4 ) + x ( 1 2 5 ) + x ( 1 7 0 ) + x ( 1 7 1 ) + x ( 1 7 2 ) + x (173) + x ( 1 7 4 ) + x ( 1 7 5 ) + x ( 1 7 6 ) + x ( 1 7 7 ) + x ( 1 7 8 ) + x ( 1 7 9 ) + x ( 1 8 0 ) + x ( 1 8 1 ) + x ( 1 8 2 ) + x ( 1 8 3 ) + x ( 1 8 4 ) + x ( 1 9 0 ) + x ( 1 9 1 ) + x ( 1 9 7 ) + x ( 1 9 8 ) + x ( 2 0 0 ) + x ( 2 1 3 ) + x ( 2 1 4 ) + x ( 2 1 5 ) + x ( 2 1 6 ) + x ( 2 1 7 ) + x ( 2 1 8 ) + x ( 2 1 9 ) + x ( 2 2 0 ) + x ( 2 2 1 ) + x ( 2 2 2 ) +"'x(223) + x ( 2 2 4 ) + x ( 2 2 9 ) + x ( 2 3 0 ) + x ( 2 3 5 ) + x ( 2 3 6 ) + x ( 2 3 7 ) + x ( 2 4 2 ) + x ( 2 4 3 ) + x ( 2 4 4 ) + x ( 2 4 5 ) + x ( 2 8 8 ) + x ( 2 8 9 ) + x ( 2 9 0 ) + x ( 2 9 1 ) + X (292) + X (293) + X (294) + X (295) + X '296) + X (297) + X (298) + X (299) + X (300) + X (301) + X (302) + X f 303) + X (304) + X (309) + X (310) + X (315) + X (316) + X (322) + X (323) + X (335) + X (336) + X (337) + X (338) + X (339) + X (340) + X (341) + X (342) + X (343) + X (344) + X (345) + X (346) + X (347) + X (352) + X (353) + X (357) + X (358) + X (359) + X (363) + X (364) + X (365) + X (369) + X (370) + X (371) + X (372) + X (373) + X (374) + X (397) + X (398) + X (399) + X (400) + X (401) + X (403) + X (404) + X (405) + X (406) + X (407) + X (408) + X (409) + X (410) + X (411) + X (412) + X (413) + X (414) + X (415) + X (416) + X (417) + X (418) + X (419) + X (423) + X (424) + X (432) + X (433) + X (434) + X (435) + X (436) + X (437) + X (438) + X (439) + X (440) + X (441) + X (442) + X (443) + X (450) + X (451) + X (452) + X 457) + X I 458) + x ( 459) + x ( 4 6 2 ) + x ( 4 6 3 ) + X (490) + X (491) + X (492) + X (493) + X (494) + X (495) + X (496) + X (497) + X (498) + X (499) + X (500) + X (501) + X (502) + X (503) + X (504) + X (505) + X (506) + X (507) + X (508) + X (509) + X (512) + X (513) + X (519) + X (520) + X (524) + X (525) + X (526) + X (527) + X (528) + X (529) + X (530) + X (531) + X (532) + X (533) + X (534) + X (535) + X (536) + X (537) + X (538) + X (539) + X (545) + X (546) + X (547) + X (551) + X (552) + X (553) + X (557) + X (558) + X (559) > 22 62 X 3) + x ( 3 6 ) + x ( 5 7 ) + x ( 5 8 ) + x ( 5 9 ) + x (60 ) + x ( 6 1 ) + x ( 6 9 ) + X 77) + x ( 7 8 ) + x ( 9 9 ) + X (100) + x ( 1 0 1 ) + x ( 1 0 2 ) + x \ 115) + X 122) + X 124) + x ( 125) + x ( 159 + X 175) + X 176) + X (177) + X 178) + X 179) + x ( 180) + X 1 181 + X 182) + X 183) + X (184) + X 191) + X 198) + X 1 199) + X 1 200 + X 217) . + X 218) + X (219) + X 220) + X 221) + x ( 222) + X i 223 + X 224) + X 230) + X (236) + X 237) + X 243) + X 1 244) + X I 245 + X 294) + X 295) + X (296) X 297) + X 298) + X 1 299) + X I 300 + X (301) + X 302) + X (303) + X ,304) + X ,310 ; + x ( 316) + X 317 + X (323) + X 324) + X (340) + X (341) + X (342 + X 1 343) + x ( 344 + X (345) + X ,346) + X (347) + X (353) + X (358 + X I 359) + X 364 + X (365) + X (370) + X (371) + X (372) + X (373 + x ( 374) + X 403 + X H 0 4 ) • + X (405) + X* (406) + X (407) + X 408) + x ( 4 09) + x ( 410) + x ( 411) + x ( 4 1 2 ) + x ( 4 1 3 ) + X (414) + X (415 + x ( 416) + x ( 417 + X (418) + X (419) + X (424) + X (425) + X (435 + X 436) + X 437 + X (438) + X (439) + X (440) + X (441) + X (442 + x ( 443) + X 444 + X (445) + X (446) + X (447) + X (451) + X (452 + X 453) + X 458 + X (459) X (460) + X (462) + X (463) + X (490 + X 491) + X 492 ) + X (493) + X (494) + X (495) + X (496) + X (497 + X 498) + X 499 ) + X (500) + X (501) + X (502) + X (503) + X (504 + X 505) + X 506 ) + X (507) + X (508) + X (509) + X (513) + X (514 + X 520) + X 521 ) + X (528) + X (529) + X (530) + X (531) + X (532 + X 533) + X 534 ) + X (535) + X (536) + X (537) + X (538) + X (539 + X 540) + X 541 ) + X (542) + X (543) + X (546) + X (547) + X (548 + X 552) + X 553 ) + X (554) + X (558) + X (559) + X (560) + X (562 + X 608) + X 609 ) + X (610) + X (611) + X (612) + X (613) + X (614 ) + X 633) + X 638 ) + X (644) + X (651) + X (658) + X (659) + X (660 ) + X 661) + X 662 ) + X (663) + X (685) + X (689) + X (694) + X (700 + x ( 706) > 21 X (125) + X (159 ) + X 180) + X (181 ) + X (182) + X (183) X (184) + X (199) + X (200 ) + X 221) + X (222 ) + X (223) + X (224) + X (237) + X (245) + X (300 ) + X 301) + X (302 ) + X (303) + X (304) + X (317) + X (324) + X (345 + x ( 346) + x ( 347) + X I 359) + x ( 3 6 5 ) + x ( 3 7 1 ) + X (372) + x ( 3 7 3 ) + x ( 3 7 4 ) + x (407) + x ( 4 0 8 ) + x ( 4 0 9 ) + X (410) + X (411) + x(4-12) + x ( 4 1 3 ) + x ( 4 1 4 ) + x(415) + x ( 4 1 6 ) + X (417) + X (418) + X (419) + x ( 4 2 5 ) + x (426) + x ( 4 2 8 ) + x (438) + x ( 4 3 9 ) + X (440) + X (441) + x ( 4 4 2 ) + x ( 4 4 3 ) + x ( 4 4 4 ) + x ( 4 4 5 ) + x ( 4 4 6 ) + X (447) + X (452) + x ( 4 5 3 ) + x ( 4 5 4 ) + x ( 4 5 9 ) + x ( 4 6 0 ) + x ( 4 6 1 ) + X (462) + X (463) + x ( 4 9 5 ) + x ( 4 9 6 ) + x (497) + x (498) + x ( 4 9 9 ) + X (500) + X (501) + x ( 5 0 2 ) + x ( 5 0 3 ) + x ( 5 0 4 ) + x ( 5 0 5 ) + x (506) + X (507) + X (508) + x ( 5 0 9 ) + x ( 5 1 4 ) + x ( 5 1 5 ) + x ( 5 2 1 ) + x ( 5 2 2 ) + X (532) + X (533) + x ( 5 3 4 ) + x ( 5 3 5 ) + x ( 5 3 6 ) + x ( 5 3 7 ) + x ( 5 3 8 ) + X (539) + X (540) + x ( 5 4 1 ) + x ( 5 4 2 ) + x ( 5 4 3 ) + x (547) + x ( 5 4 8 ) + X (549) X (553) + x ( 5 5 4 ) + x (555) + x ( 5 5 9 ) + x(560) + x ( 5 6 1 ) + X (562) + X (563) + x ( 6 0 8 ) + x ( 6 0 9 ) + x ( 6 1 0 ) + x(611) + x ( 6 1 2 ) + X (613) + X (614) + x ( 6 1 5 ) + x ( 6 1 6 ) + x ( 6 1 7 ) + x ( 6 1 8 ) + x ( 6 1 9 ) + X (620) 63 + x ( 6 2 1 ) + x ( 6 3 3 ) + x ( 6 3 4 ) + x ( 6 3 8 ) + x ( 6 3 9 ) + x ( 6 4 4 ) + x ( 6 4 5 ) + x ( 6 5 1 ) + x ( 6 5 2 ) + x ( 6 5 8 ) + x ( 6 5 9 ) + x ( 6 6 0 ) + x ( 6 6 1 ) + x ( 6 6 2 ) + x ( 6 6 3 ) + x ( 6 6 4 ) + x ( 6 6 5 ) + x ( 6 6 6 ) + x ( 6 6 7 ) + x ( 6 6 8 ) + x ( 6 6 9 ) + x ( 6 8 5 ) + x ( 6 8 6 ) + x ( 6 8 9 ) + x ( 6 9 0 ) +. x ( 6 9 4 ) + x ( 6 9 5 ) + x ( 7 0 0 ) + x ( 7 0 1 ) + x ( 7 0 6 ) + x ( 7 0 7 ) > 13 x ( 3 7 ) + x ( 4 2 ) + x ( 4 7 ) + x ( 5 2 ) + x (62 ) + x ( 6 3 ) + x'(64) + x ( 6 5 ) + x ( 6 6 ) + x ( 6 7 ) + x ( 6 8 ) + x ( 6 9 ) + x ( 7 9 ) + x ( 8 3 ) + x ( 8 7 ) + x ( 9 1 ) + x ( 9 5 ) + x ( 1 0 3 ) + x ( 1 0 4 ) + x ( 1 0 5 ) + x ( 1 0 6 ) + x ( 1 0 7 ) + x ( 1 0 8 ) + x ( 3 7 3 ) + x ( 3 7 4 ) + x ( 3 9 7 ) + x ( 4 1 1 ) + x ( 4 1 2 ) + x ( 4 1 3 ) + x (414) + x ( 4 1 5 ) + x ( 4 1 6 ) + x ( 4 1 7 ) + x ( 4 1 8 ) + x ( 4 1 9 ) + x ( 4 2 6 ) + x ( 4 2 7 ) - + x ( 4 2 8 ) + x ( 4 4 1 ) + x ( 4 4 2 ) + x ( 4 4 3 ) + x ( 4 4 4 ) + x ( 4 4 5 ) + x ( 4 4 6 ) + x ( 4 4 7 ) + x ( 4 5 3 ) + x ( 4 5 4 ) + x ( 4 6 0 ) + x ( 4 6 1 ) + x ( 4 6 2 ) + x ( 4 6 3 ) + x ( 5 0 0 ) + x ( 5 0 1 ) + x ( 5 0 2 ) + x ( 5 0 3 ) + x ( 5 0 4 ) + x ( 5 0 5 ) + x ( 5 0 6 ) + x ( 5 0 7 ) + x ( 5 0 8 ) + x ( 5 0 9 ) + x ( 5 1 5 ) + x ( 5 1 6 ) + x ( 5 2 2 ) + x ( 5 2 3 ) + x ( 5 3 6 ) + x ( 5 3 7 ) + x ( 5 3 8 ) + x ( 5 3 9 ) + x ( 5 4 0 ) + x ( 5 4 1 ) + x ( 5 4 2 ) ' + x ( 5 4 3 ) + x ( 5 4 8 ) + x ( 5 4 9 ) + x ( 5 5 4 ) + x ( 5 5 5 ) + x ( 5 6 0 ) + x ( 5 6 1 ) + x ( 5 6 2 ) + x ( 5 6 3 ) + x ( 6 0 8 ) + x ( 6 0 9 ) + x ( 6 1 0 ) + x ( 6 1 1 ) + x ( 6 1 2 ) + x ( 6 1 3 ) + x (614) + x ( 6 1 5 ) + x ( 6 1 6 ) + x ( 6 1 7 ) + x ( 6 1 8 ) + x ( 6 1 9 ) + x ( 6 2 0 ) + x ( 6 2 1 ) + x ( 6 2 2 ) + x ( 6 2 3 ) + x ( 6 2 4 ) + x ( 6 2 5 ) + x ( 6 2 6 ) + x ( 6 2 7 ) + x ( 6 3 4 ) + x ( 6 3 5 ) + x ( 6 3 9 ) + x ( 6 4 0 ) + x ( 6 4 5 ) + x ( 6 4 6 ) + x ( 6 5 2 ) + x ( 6 5 3 ) + x ( 6 5 8 ) + x ( 6 5 9 ) + x ( 6 6 0 ) + x ( 6 6 1 ) x (662) + x ( 6 6 3 ) + x ( 6 6 4 ) + x ( 6 6 5 ) + x ( 6 6 6 ) + x ( 6 6 7 ) + x (668) + x ( 6 6 9 ) + x ( 6 7 0 ) + x ( 6 7 1 ) + x ( 6 7 2 ) + x ( 6 7 3 ) + x ( 6 7 4 ) + x ( 6 7 5 ) + x ( 6 8 5 ) + x ( 6 8 6 ) + x ( 6 8 7 ) + x ( 6 8 9 ) + x ( 6 9 0 ) + x ( 6 9 1 ) + x ( 6 9 4 ) + x ( 6 9 5 ) + x ( 6 9 6 ) + x ( 7 0 0 ) + x ( 7 0 1 ) + x (702) + x ( 7 0 6 ) + x ( 7 0 7 ) + x-(708) > 10 x (37) + x ( 3 8 ) + x ( 4 2 ) + x ( 4 3 ) + x ( 4 7 ) + x ( 4 8 ) + x 52) + x ( 5 3 ) + x (58) + x ( 6 2 ) + x ( 6 3 ) + x ( 6 4 ) + x (65 ) + x ( 6 6 ) + x ( 6 7 ) + X (68) + x ( 6 9 ) + x ( 7 0 ) + x ( 7 1 ) + x (72 ) + x ( 7 3 ) + x 74) + x ( 7 5 ) + x ( 7 6 ) + x ( 7 7 ) + x ( 7 9 ) + x ( 8 0 ) + x ( 8 3 ) + x ( 8 4 ) + x ( 8 7 ) + X (88) + x ( 91) + x ( 9 2 ) + x ( 9 5 ) + x ( 9 6 ) + x ( 1 0 0 ) + x ( 1 0 3 ) + X (104) + X (105) + x ( 1 0 6 ) + x ( 1 0 7 ) + X (108) + x ( 1 0 9 ) + x ( 1 1 0 ) + X (111) + X (112) + x ( 1 1 3 ) + x ( 1 1 4 ) + X (115) + x ( 1 6 0 ) + x ( 1 6 5 ) + X (170) + X (175) + x ( 1 8 5 ) + x ( 1 8 6 ) + X (187) + x (188) + x ( 1 8 9 ) + X (190) + X (191) + x ( 2 0 1 ) + x ( 2 0 5 ) + X (209) + x ( 2 1 3 ) + x (217) + X (225) + X (226) + x (227) + x ( 2 2 8 ) + X (229) + x ( 2 3 0 ) + x ( 2 4 6 ) + X (282) + X (288) + x ( 2 9 4 ) + x(305^ + X (306) + x ( 3 0 7 ) + x ( 3 0 8 ) + X (309) + X (310) + x ( 3 2 5 ) + x ( 3 3 0 ) + X (335) + x ( 3 4 0 ) + x ( 3 4 9 ) + X (350) + X (351) + x ( 3 5 2 ) + x ( 3 5 3 ) + X (374) + x ( 3 9 7 ) + x ( 4 1 5 ) + X (416) + X (417) + x ( 4 1 8 ) + x ( 4 1 9 ) + X (427) + x ( 4 2 8 ) + x ( 4 4 4 ) + X (445) + X (446) + x ( 4 4 7 ) + x ( 4 5 4 ) + X (461) + x ( 4 6 3 ) + x ( 5 0 5 ) + X (506) + X (507) + x ( 5 0 8 ) + x ( 5 0 9 ) + X (516) + x ( 5 2 3 ) + x ( 5 4 0 ) + X (541) + X (542) + x ( 5 4 3 ) + x ( 5 4 9 ) + X (555) + x(561) + x ( 5 6 2 ) + X (563) + X (608) + x ( 6 0 9 ) + x ( 6 1 0 ) + X (611) + x ( 6 1 2 ) + x ( 6 1 3 ) + 64 x ( 6 1 4 ) + x ( 6 1 5 ) + x ( 6 1 6 ) + x ( 6 1 7 ) + x ( 6 1 8 ) + x ( 6 1 9 ) + x ( 6 2 0 ) + x ( 6 2 1 ) + x (622) + x ( 6 2 3 ) + x ( 6 2 4 ) + x ( 6 2 5 ) + x ( 6 2 6 ) + x ( 6 2 7 ) + x ( 6 2 8 ) + x ( 6 2 9 ) + x ( 6 3 0 ) x ( 6 3 1 ) + x ( 6 3 2 ) + x ( 6 3 5 ) + x ( 6 3 6 ) + x ( 6 4 0 ) + x ( 6 4 1 ) + x ( 6 4 6 ) + x ( 6 4 7 ) + x ( 6 5 3 ) + x ( 6 5 4 ) + x ( 6 5 8 ) + x ( 6 5 9 ) + x ( 6 6 0 ) + x ( 6 6 1 ) + x ( 6 6 2 ) + x ( 6 6 3 ) + x ( 6 6 4 ) + x ( 6 6 5 ) + x ( 6 6 6 ) + x(667) + x ( 6 6 8 ) + x ( 6 6 9 ) + x ( 6 7 0 ) + x ( 6 7 1 ) + x ( 6 7 2 ) + x ( 6 7 3 ) + x(674) + x ( 6 7 5 ) + x ( 6 7 6 ) + x ( 6 7 7 ) + x ( 6 7 8 ) + x ( 6 7 9 ) + x ( 6 8 0 ) + x ( 6 8 6 ) + x ( 6 8 7 ) + x ( 6 8 8 ) + x ( 6 9 0 ) + x ( 6 9 1 ) + x ( 6 9 2 ) + x ( 6 9 5 ) + x ( 6 9 6 ) + x ( 6 9 7 ) + x ( 7 0 1 ) + x ( 7 0 2 ) + x ( 7 0 3 ) + x ( 7 0 7 ) + x ( 7 0 8 ) + x(709) > 11 x (38) + x ( 3 9 ) + x ( 4 3 ) + x (4 4 ) + x ( 4 8 ) - + x (4 9) •' + x ( 53) + x ( 5 4 ) + x ( 5 8 ) + x{59) + x l(62) + x ( 6 3 ) + x (64) + x ( 6 5 ) + x ( 6 6 ) + x ( 6 7 ) + x (68 ) + x ( 6 9 ) + x ( 7 0 ) + x ( 7 1 ) + x ( 7 2 ) + x ( 73) + x ( 7 4 ) + x ( 7 5 ) + x ( 7 6 ) + x ( 7 7 ) + x ( 7 9 ) + x (80) + x ( 8 1 ) + x ( 8 3 ) + x ( 8 4 ) + x ( 8 5 ) + x ( 8 7 ) + x ( 8 8 ) + x (89) + x ( 9 1 ) + x ( 92) + x ( 9 3 ) + x ( 9 5 ) + x ( 9 6 ) + x ( 9 7 ) + x ( 1 0 0 ) + x ( 1 0 1 ) + x ( 1 0 3 ) + x ( 1 0 4 ) x ( 1 0 5 ) + x (106) + x ( 1 0 7 ) + x ( 1 0 8 ) + x ( 1 0 9 ) + x (110) + x ( l l l ) + x ( 1 1 2 ) + x ( 1 1 3 ) + x ( 1 1 4 ) + x ( 1 1 5 ) + x ( 1 1 6 ) + x ( 1 1 7 ) + x ( 1 1 8 ) + x ( 1 1 9 ) + x ( 1 2 0 ) + x ( 1 2 1 ) + x ( 1 2 2 ) + x ( 1 6 0 ) + x ( 1 6 1 ) + x ( 1 6 5 ) + x ( 1 6 6 ) + x ( 1 7 0 ) + x ( 1 7 1 ) + x ( 1 7 5 ) + x ( 1 7 6 ) + x ( 1 8 1 ) + x ( 1 8 5 ) + x (186) + x (187) + x ( 1 8 8 ) + x ( 1 8 9 ) + x ( 1 9 0 ) + x ( 1 9 1 ) + x ( 1 9 2 ) + x ( 1 9 3 ) + x ( 1 9 4 ) + x ( 1 9 5 ) + x ( 1 9 6 ) + x ( 1 9 7 ) + x ( 1 9 8 ) + x ( 1 9 9 ) + x ( 2 0 1 ) + x ( 2 0 2 ) + x ( 2 0 5 ) + x ( 2 0 6 ) + x ( 2 0 9 ) + x ( 2 1 0 ) + x ( 2 1 3 ) + x ( 2 1 4 ) + x ( 2 1 7 ) + x ( 2 1 8 ) + x (222) + x ( 2 2 5 ) + x ( 2 2 6 ) + x ( 2 2 7 ) + x (228) + x ( 2 2 9 ) + x ( 2 3 0 ) + x ( 2 3 1 ) + x ( 2 3 2 ) + x (233) + x ( 2 3 4 ) + x (235) + x (236) + x ( 2 3 7 ) + x ( 2 4 6 ) + x ( 2 8 2 ) + x (283) + x ( 2 8 8 ) + x (289) + x ( 2 9 4 ) + x ( 2 9 5 ) + x ( 3 0 0 ) + x ( 3 0 5 ) + x (306) + x ( 3 0 7 ) + x (308) + x ( 3 0 9 ) + x ( 3 1 0 ) + x (311) + x ( 3 1 2 ) x ( 3 1 3 ) + x ( 3 1 4 ) + x (315) + x ( 3 1 6 ) + x ( 3 1 7 ) + x ( 3 2 5 ) + x ( 3 2 6 ) + x ( 3 3 0 ) + x ( 3 3 1 ) + X (335) + X (336) + X (340) + x ( 3 4 1 ) + x ( 3 4 5 ) + x (349) + x (350) + X (351) + X (352) + X (353) + x ( 3 5 4 ) + x ( 3 5 5 ) + x ( 3 5 6 ) + X (357) + X (358) + X (359) + X (562) + x ( 5 6 3 ) + x ( 6 0 8 ) + x ( 6 0 9 ) + X (610) + X (611) + X (612) + X (613) + x ( 6 1 4 ) + x ( 6 1 5 ) + x ( 6 1 6 ) + X (617) + X (618) + X (619) + X (620) + x ( 6 2 1 ) + x ( 6 2 2 ) + x ( 6 2 3 ) + X (624) + X (625) + X (626) + X (627) + x ( 6 2 8 ) + x (629) + x ( 6 3 0 ) + X (631) + X (632) + X (636) + X (637) + x ( 6 4 1 ) + x ( 6 4 2 ) + x ( 6 4 7 ) + X (648) + X (654) + X (655) + x ( 6 6 4 ) + x ( 6 6 5 ) + x ( 6 6 6 ) + X (667) + X (668) + X (669) + X (670) + x ( 6 7 1 ) + x ( 6 7 2 ) + x ( 6 7 3 ) + X (674) + X (675) + X (676) + X (677) + x ( 6 7 8 ) + x ( 6 7 9 ) + x ( 6 8 0 ) + X (681) + X (682) + X (683) + X (684) + x ( 6 8 7 ) + x ( 6 8 8 ) + x ( 6 9 1 ) + X (692) + X (693) + X (696) + X (697) + x ( 6 9 8 ) + x ( 7 0 2 ) + x ( 7 0 3 ) + X (704) + X (708) + X (709) + X (710) > 13 x ( 3 9 ) + x ( 4 0 ) + x ( 4 4 ) + x ( 4 5 ) + x ( 4 9 ) + x ( 5 0 ) + x ( 5 4 ) + x ( 5 5 ) + x ( 5 9 ) + x ( 6 0 ) + x ( 6 2 ) + x ( 6 3 ) + x ( 6 4 ) + x ( 6 5 ) + x ( 6 6 ) + x (67 ) + x ( 6 8 ) + x ( 6 9 ) + x ( 7 0 ) + x ( 7 1 ) + x ( 7 2 ) + x ( 7 3 ) + x ( 7 4 ) 65 + x ( 7 5 ) + x ( 7 6 ) + x ( 7 7 ) f x (80) + x(81) + x (82) + x ( 8 4 ) + x ( 8 5 ) 4 x ( 8 6 ) + x ( 8 8 ) + x ( 8 9 ) + x ( 9 0 ) + x ( 9 2 ) + x ( 93) + x (94) + x ( 9 6 ) + x(97) + x ( 9 8 ) + x ( 1 0 0 ) + x ( 1 0 1 ) + x ( 1 0 2 ) + x ( 103) + X 104) + X (105) + X (106) + X (107) + X (108) + x ( 1 0 9 ) + x ( 110) + X 111) + X (112) + X (113) + X (114) + X (115) + x ( 1 1 6 ) + x ( 117) + X 118) + X (119) + X (120) + X (121) + X (122) + x (161) + x ( 162) + X 166) + X (167) + X (171) + X (172) + X (176) + x (177) + x ( 181) + X 182) + X (185) + X (186) + X (187) + X (188) + x ( 1 8 9 ) + x ( 190) + X 191) + X (192) + X (193) + X (194) + X (195) + x ( 1 9 6 ) + X 1 197) + X 198) + X (199) + X (201) + X (202) + X (203) + x (205) + x ( 206) + X ,207) + X (209) + X (210) + X (211) + X (213) + x (214) + x ( 215) + X 217) + X (218) + X (219) + X (222) + X (-223) x ( 2 2 5 ) + x ( 226) ' + X '227) + X (228) + X (229) + X (230) + X (231) + x ( 2 3 2 ) + x ( 233) + X ,234) + X (235) + X (236) + X (237) X (238) + x (239) + x ( 240) + X (241) + X (242) + X (243) + X (246) + X (283) + x (284) + x ( 289) + X (290) + X (295) + X (296) + X (300) + X (301) + x ( 3 0 5 ) + x ( '306) + X (307) + X (308) + X (309) + X (310) + X (311) + x ( 3 1 2 ) + x ( 313) + X (314) + X (315) + X (316) + X (317) + X (318) + x ( 3 1 9 ) + X 320) + X (321) + X (322) + X (323) • + X (324) + X (325) + x ( 3 2 6 ) + x ( 327) + X (330) + X (331) + X (332) + X (335) + X (336) + x ( 3 3 7 ) + X 340) + X (341) + X (342) + X (345) + X (346) + X (349) + x ( 3 5 0 ) + X 351) + X (352) + X (353) + X (354) + X (355) + X (356) + x ( 3 5 7 ) + X 358) + X (359) + X (360) + X (361) + X (362) + X (363) + ^x (364) + X 365) + X (562) + X (563) + X (615) + X (616) + X (617) + x ( 6 1 8 ) + X 619) + X (620) + X (621) + X (622) + X (623) + x ( 6 2 4 ) + x ( 6 2 5 ) + x (626) + x ( 6 2 7 ) + x ( 6 2 8 ) + x ( 6 2 9 ) + x ( 6 3 0 ) + x ( 6 3 1 ) + x ( 6 3 2 ) + x ( 6 3 7 ) + x ( 6 4 2 ) + x ( 6 4 3 ) + x ( 6 4 8 ) + x ( 6 4 9 ) + x ( 6 5 5 ) + x ( 6 5 6 ) + x ( 6 7 0 ) + x ( 6 7 1 ) + x ( 6 7 2 ) + x ( 6 7 3 ) + x ( 6 7 4 ) + x ( 6 7 5 ) + x ( 6 7 6 ) + x ( 6 7 7 ) + x ( 6 7 8 ) + x ( 6 7 9 ) + x ( 6 8 0 ) + x ( 6 8 1 ) + x (682) + x ( 6 8 3 ) + x ( 6 8 4 ) + x(688) + x ( 6 9 2 ) + x ( 6 9 3 ) + x ( 6 9 7 ) + x ( 6 9 8 ) + x ( 6 9 9 ) + x ( 7 0 3 ) + x ( 7 0 4 ) + x ( 7 0 5 ) + x ( 7 0 9 ) + x ( 7 1 0 ) + x ( 7 1 1 ) > 14 x ( 4 0 ) + x ( 4 1 ) + x ( 4 5 ) + x ( 4 6 ) + x ( 5 0 ) + x ( 5 1 ) + x ( 5 5 ) + x ( 5 6 ) + x ( 6 0 ) + x ( 6 1 ) + x ( 6 2 ) + x ( 6 3 ) + x ( 6 4 ) + x ( 6 5 ) + x ( 6 6 ) + x ( 6 7 ) + x ( 6 8 ) + x ( 6 9 ) + x ( 7 0 ) + x ( 7 1 ) + x ( 7 2 ) + x ( 7 3 ) + x ( 7 4 ) + x ( 7 5 ) + x ( 7 6 ) + x ( 7 7 ) + x ( 8 1 ) + x ( 8 2 ) + x ( 8 5 ) + x ( 8 6 ) + x (89 ) + x ( 9 0 ) + x ( 9 3 ) + x ( 9 4 ) + x ( 9 7 ) + x ( 9 8 ) + x ( 1 0 1 ) + x ( 1 0 2 ) + x ( 1 0 9 ) + x ( 1 1 0 ) + x ( l l l ) + x ( 1 1 2 ) + x ( 1 1 3 ) + x ( 1 1 4 ) + x ( 1 1 5 ) + x ( 1 1 6 ) + x ( 1 1 7 ) + x ( 1 1 8 ) + x ( 1 1 9 ) + x ( 1 2 0 ) + x ( 1 2 1 ) + x ( 1 2 2 ) + x ( 1 6 2 ) + x ( 1 6 3 ) + x ( 1 6 7 ) + x ( 1 6 8 ) + x ( 1 7 2 ) + x ( 1 7 3 ) + x ( 1 7 7 ) + x ( 1 7 8 ) + x ( 1 8 2 ) + x ( 1 8 3 ) + x ( 1 8 5 ) + x ( 1 8 6 ) + x ( 1 8 7 ) + x ( 1 8 8 ) + x ( 1 8 9 ) + x ( 1 9 0 ) + x ( 1 9 1 ) + x ( 1 9 2 ) + x ( 1 9 3 ) + x ( 1 9 4 ) + x ( 1 9 5 ) + x ( 1 9 6 ) + x ( 1 9 7 ) + x ( 1 9 8 ) + x ( 1 9 9 ) + x ( 2 0 2 ) + x ( 2 0 3 ) + x ( 2 0 4 ) + x ( 2 0 6 ) + x ( 2 0 7 ) + x ( 2 0 8 ) + x ( 2 1 0 ) + x ( 2 1 1 ) + x ( 2 1 2 ) + x (214) + x ( 2 1 5 ) + x ( 2 1 6 ) + x ( 2 1 8 ) + x ( 2 1 9 ) + x ( 2 2 0 ) + x ( 2 2 2 ) + x ( 2 2 3 ) + x ( 2 2 4 ) + x ( 2 2 5 ) + x ( 2 2 6 ) + x ( 2 2 7 ) + x ( 2 2 8 ) + x ( 2 2 9 ) + x (230) + x ( 2 3 1 ) + x ( 2 3 2 ) + x ( 2 3 3 ) + x ( 2 3 4 ) + x ( 2 3 5 ) + x ( 2 3 6 ) 66 + X 237) + x ( 2 3 8 ) + x ( 239) + X 240) + X 241) + x ( 242) + X 243) + x 246) + x ( 2 8 4 ) + x ( 285) + x ( 290) + X 291) + X 296) + X 297) + X 301) + x ( 3 0 2 ) + x ( 305) + X 306) + X 307) + X 308) + X 309) + X 310) + x ( 3 1 1 ) + x ( 312) + X 313) + X 314) + X 315) + X 316) + X 317) + x ( 3 1 8 ) + x ( 319) + X 320) + X 321) + X 322) + X 323) + X 324) + x ( 3 2 6 ) + X 1 327) + X 328) + X 331) + X 332) + X 333) + X 336) + x ( 3 3 7 ) + x ( 338) + X 341) + X ,342) + X 343) + X (345) + X (346) + x ( 3 4 7 ) + x ( 349) + X (350) + X (351) + X 352) + X (353) + X (354) + x ( 3 5 5 ) + X 1 356) + X (357) + X (358) + X ,359) + X (360) + X (361) + x ( 3 6 2 ) + x ( 363) + X (364) + X (365) + X (366) + X (367) + X (368) + x ( 3 6 9 ) + X I 370) + X (371) + X (398) + X (403) + X (407) + X (411) + x ( 4 1 6 ) + X 420) + X (421) X (422) + ' x' (423) + X (424) + X (425) + x ( 4 2 6 ) + xM 427) + X (429) + X (432) + X (435) + X (438) + X (441) + x ( 4 4 5 ) + X 448) + X (449) + X (450) + X (451) + X (452) + X (453) + x ( 4 5 4 ) + X 490) + X (495) + X (500) + X (505) + X (510) + X (511) + x ( 5 1 2 ) + X 513) + X (514) + X (515) + X (516) + X (524) + X (528) + x ( 5 3 2 ) + X 536) + X (540) + X (544) + X (545) + X (546) + X (547) + x ( 5 4 8 ) + X 549) + X (562) + X (563) X (622) + X (623) + X (624) + x ( 6 2 5 ) + X 626) + X (627) + X (628) + X (629) + X (630) + X (631) + x ( 6 3 2 ) + X (643) + X (649) + X (650) + X (656) + X (657) + X (676) + x ( 6 7 7 ) + X (678) + X (679) + X (680) + X (681) + X (682) + X (683) + x ( 6 8 4 ) + X (693) + X (698) + X (699) + X (704) + X (705) + X (710) + x ( 7 1 1 ) > 16 x ( 4 1 ) + x ( 4 6 ) + x ( 5 1 ) + x ( 5 6 ) + x ( 6 1 ) + x ( 7 0 ) + x ( 7 1 ) + x ( 7 2 ) + x ( 7 3 ) + x ( 7 4 ) + x ( 7 5 ) + x ( 7 6 ) + x ( 7 7 ) + x ( 8 2 ) + x ( 8 6 ) + x ( 9 0 ) + x ( 9 4 ) + x ( 9 8 ) + x ( 1 0 2 ) + x ( 1 1 6 ) + x (117) + x ( 1 1 8 ) + x ( 1 1 9 ) + x ( 1 2 0 ) + x ( 1 2 1 ) + x ( 1 2 2 ) + x ( 1 6 3 ) + x ( 1 6 4 ) + x ( 1 6 8 ) + x ( 1 6 9 ) + x ( 1 7 3 ) + x ( 1 7 4 ) + x ( 1 7 8 ) + x ( 1 7 9 ) + x ( 1 8 3 ) + x ( 1 8 4 ) + x ( 1 8 5 ) + x ( 1 8 6 ) + x ( 1 8 7 ) + x ( 1 8 8 ) + x ( 1 8 9 ) + x ( 1 9 0 ) + x ( 1 9 1 ) + x ( 1 9 2 ) + x ( 1 9 3 ) + x ( 1 9 4 ) + x ( 1 9 5 ) + x ( 1 9 6 ) + x ( 1 9 7 ) + x ( 1 9 8 ) + x ( 1 9 9 ) + x ( 2 0 3 ) + x ( 2 0 4 ) + X 207) + x ( 2 0 8 ) + x ( 2 1 1 ) + x ( 2 1 2 ) + x ( 2 1 5 ) + x ( 2 1 6 ) + x ( 2 1 9 ) + X 220) + x ( 2 2 3 ) + x ( 2 2 4 ) + x ( 2 3 1 ) + x ( 2 3 2 ) + x ( 2 3 3 ) + x (234) + X (235) + x ( 2 3 6 ) + x ( 2 3 7 ) + x ( 2 3 8 ) + x ( 2 3 9 ) + x ( 2 4 0 ) + x ( 2 4 1 ) + X (242) + x ( 2 4 3 ) + x ( 2 4 6 ) + x ( 2 8 5 ) + x ( 2 8 6 ) + x ( 2 9 1 ) + x ( 2 9 2 ) + X (297) + x ( 2 9 8 ) + x ( 3 0 2 ) + x ( 3 0 3 ) + x ( 3 0 5 ) + x ( 3 0 6 ) + x ( 3 0 7 ) + X (308) + x ( 3 0 9 ) + x ( 3 1 0 ) + x (311) + x ( 3 1 2 ) + x ( 3 1 3 ) + x ( 3 1 4 ) + X (315) + x ( 3 1 6 ) + x ( 3 1 7 ) + x ( 3 1 8 ) + x ( 3 1 9 ) + x ( 3 2 0 ) + x ( 3 2 1 ) + X (322) + x ( 3 2 3 ) + x ( 3 2 4 ) + x ( 3 2 7 ) + x ( 3 2 8 ) + x ( 3 2 9 ) + x ( 3 3 2 ) + X (333) + x ( 3 3 4 ) + x ( 3 3 7 ) + x ( 3 3 8 ) + x ( 3 3 9 ) + x ( 3 4 2 ) + x ( 3 4 3 ) + X (344) + x ( 3 4 6 ) + x ( 3 4 7 ) + x ( 3 5 4 ) + x ( 3 5 5 ) + x ( 3 5 6 ) + x ( 3 5 7 ) + X (358) + x (359) + x ( 3 6 0 ) + x ( 3 6 1 ) + x ( 3 6 2 ) + x ( 3 6 3 ) + x ( 3 6 4 ) + X (365) + x ( 3 6 6 ) + x ( 3 6 7 ) + x ( 3 6 8 ) + x ( 3 6 9 ) + x ( 3 7 0 ) + x ( 3 7 1 ) + X (398) + x ( 3 9 9 ) + x ( 4 0 3 ) + x (404) + x ( 4 0 7 ) + x ( 4 0 8 ) + x ( 4 1 1 ) + X (412) + x (416) + x ( 4 1 7 ) + x ( 4 2 0 ) + x ( 4 2 1 ) + x ( 4 2 2 ) + x ( 4 2 3 ) + X (424) + x ( 4 2 5 ) + x ( 4 2 6 ) + x ( 4 2 7 ) + x ( 4 2 9 ) + x ( 4 3 0 ) + x ( 4 3 2 ) + x ( 433) + x ( 4 3 5 ) + x ( 4 3 6 ) + x (438) 67 + x ( 4 3 9 ) + x ( 4 4 1 ) + x ( 4 4 2 ) + x ( 4 4 5 ) + x (446) + x ( 4 4 8 ) + x (449) + x(450) + x ( 4 5 1 ) + x ( 4 5 2 ) + x ( 4 5 3 ) + x ( 4 5 4 ) + x ( 4 5 5 ) + x (456) + x(457) + x ( 4 5 8 ) + x ( 4 5 9 ) + x ( 4 6 0 ) + x ( 4 6 1 ) + x ( 4 9 0 ) + x ( 4 9 1 ) + x(495) + x ( 4 9 6 ) + x ( 5 0 0 ) + x ( 5 0 1 ) + x ( 5 0 5 ) + x ( 5 0 6 ) + x (510) + x ( 5 1 1 ) + x ( 5 1 2 ) + x ( 5 1 3 ) + x ( 5 1 4 ) + x ( 5 1 5 ) + x ( 5 1 6 ) + x ( 5 1 7 ) + x(518) + x ( 5 1 9 ) + x ( 5 2 0 ) + x ( 5 2 1 ) + x ( 5 2 2 ) + x ( 5 2 3 ) + x ( 5 2 4 ) + x(525) + x ( 5 2 8 ) + x ( 5 2 9 ) + x ( 5 3 2 ) + x ( 5 3 3 ) x ( 5 3 6 ) + x ( 5 3 7 ) + x(540) + x ( 5 4 1 ) + x ( 5 4 4 ) + x ( 5 4 5 ) + x ( 5 4 6 ) + x ( 5 4 7 ) + x ( 5 4 8 ) + x(549) + x ( 5 5 0 ) + x ( 5 5 1 ) + x ( 5 5 2 ) + x ( 5 5 3 ) + x ( 5 5 4 ) + x ( 5 5 5 ) + x(563) + x ( 6 2 8 ) + x ( 6 2 9 ) + x ( 6 3 0 ) + x ( 6 3 1 ) + x ( 6 3 2 ) + x ( 6 5 0 ) + x ( 6 5 7 ) + x ( 6 8 1 ) + x ( 6 8 2 ) + x ( 6 8 3 ) + x ( 6 8 4 ) + x ( 6 9 9 ) x ( 7 0 5 ) + x(711) > 18 x (164) + x ( 1 6 9 ) + x ( 1 7 4 ) + x ( 1 7 9 ) + x ( 1 8 4 ) + x ( 1 9 2 ) + x ( 1 9 3 ) + x ( 1 9 4 ) + x ( 1 9 5 ) + x ( 1 9 6 ) + x ( 1 9 7 ) + x ( 1 9 8 ) + x ( 1 9 9 ) + x ( 2 0 4 ) + x(208) + x ( 2 1 2 ) + x ( 2 1 6 ) + x ( 2 2 0 ) + x ( 2 2 4 ) + x ( 2 3 8 ) + x (239) + x(240) + x ( 2 4 1 ) + x ( 2 4 2 ) + x ( 2 4 3 ) + x (246) + x ( 2 8 6 ) + x ( 2 8 7 ) + x(292) + x ( 2 9 3 ) + x ( 2 9 8 ) + x ( 2 9 9 ) + x ( 3 0 3 ) + x ( 3 0 4 ) + x ( 3 1 1 ) + x(312) + x ( 3 1 3 ) + x (314) + x ( 3 1 5 ) + x ( 3 1 6 ) + x ( 3 1 7 ) + x ( 3 1 8 ) + x ( 3 1 9 ) + x ( 3 2 0 ) + x ( 3 2 1 ) x ( 3 2 2 ) + x ( 3 2 3 ) + x (324) + x ( 3 2 8 ) + x ( 3 2 9 ) + x ( 3 3 3 ) + x ( 3 3 4 ) + x ( 3 3 8 ) + x ( 3 3 9 ) + x ( 3 4 3 ) + x ( 3 4 4 ) + x ( 3 4 7 ) + x ( 3 6 0 ) + x ( 3 6 1 ) + x ( 3 6 2 ) + x ( 3 6 3 ) + x ( 3 6 4 ) + x ( 3 6 5 ) + x(366) + x ( 3 6 7 ) + x ( 3 6 8 ) + x ( 3 6 9 ) + x ( 3 7 0 ) + x ( 3 7 1 ) + x ( 3 9 9 ) + x(400) + x (404) + x ( 4 0 5 ) + x (408) + x ( 4 0 9 ) + x ( 4 1 2 ) + x ( 4 1 3 ) + x(417) + x ( 4 1 8 ) + x ( 4 2 0 ) + x (421) + x ( 4 2 2 ) + x ( 4 2 3 ) + x ( 4 2 4 ) + x(425) + x ( 4 2 6 ) + x (427) + x ( 4 2 9 ) + x(430) + x ( 4 3 1 ) + x ( 4 3 2 ) + x ( 4 3 3 ) + x (434) + x ( 4 3 5 ) x ( 4 3 6 ) + x ( 4 3 7 ) + x ( 4 3 8 ) + x ( 4 3 9 ) + x(440) + x (441) + x ( 4 4 2 ) + x ( 4 4 3 ) + x ( 4 4 5 ) + x ( 4 4 6 ) + x (447) + x(448) + x (449) + x ( 4 5 0 ) + x ( 4 5 1 ) + x (452) + x ( 4 5 3 ) • + x ( 4 5 4 ) + x(455) + x ( 4 5 6 ) + x ( 4 5 7 ) + x ( 4 5 8 ) + x ( 4 5 9 ) + x ( 4 6 0 ) + x ( 4 6 1 ) + x(491) x ( 4 9 2 ) + x ( 4 9 6 ) + x ( 4 9 7 ) + x ( 5 0 1 ) + x ( 5 0 2 ) + x ( 5 0 6 ) + x ( 5 0 7 ) + x ( 5 1 0 ) + x ( 5 1 1 ) + x ( 5 1 2 ) + x ( 5 1 3 ) + x ( 5 1 4 ) + x ( 5 1 5 ) + x ( 5 1 6 ) + x ( 5 1 7 ) + x ( 5 1 8 ) + x ( 5 1 9 ) + x ( 5 2 0 ) + x ( 5 2 1 ) + x ( 5 2 2 ) + x ( 5 2 3 ) + x ( 5 2 4 ) + x ( 5 2 5 ) + x ( 5 2 6 ) + x ( 5 2 8 ) + x ( 5 2 9 ) + x ( 5 3 0 ) + x ( 5 3 2 ) + x ( 5 3 3 ) + x ( 5 3 4 ) + x ( 5 3 6 ) + x ( 5 3 7 ) + x ( 5 3 8 ) + x ( 5 4 0 ) + x ( 5 4 1 ) + x ( 5 4 2 ) + x ( 5 4 4 ) + x ( 5 4 5 ) + x ( 5 4 6 ) + x ( 5 4 7 ) + x ( 5 4 8 ) + x(549) + x ( 5 5 0 ) + x ( 5 5 1 ) + x ( 5 5 2 ) + x ( 5 5 3 ) + x ( 5 5 4 ) + x ( 5 5 5 ) + x(556) + x ( 5 5 7 ) + x ( 5 5 8 ) + x ( 5 5 9 ) + x ( 5 6 0 ) + x ( 5 6 1 ) + x ( 5 6 4 ) + x(608) + x ( 6 1 5 ) + x ( 6 3 3 ) + x ( 6 3 4 ) + x ( 6 3 5 ) + x ( 6 3 6 ) + x ( 6 3 7 ) + x(658) + x ( 6 6 4 ) + x ( 6 7 0 ) + x ( 6 8 5 ) + x ( 6 8 6 ) + x ( 6 8 7 ) + x ( 6 8 8 ) > 11 x (246) + x (320) + x (339) + x (371) + x (413) + x ( 2 8 7 ) + x ( 3 2 1 ) + x ( 3 4 4 ) + x ( 4 0 0 ) + x ( 4 1 4 ) + x ( 2 9 3 ) + x ( 3 2 2 ) + x ( 3 6 6 ) + x ( 4 0 1 ) + x ( 4 1 8 ) + x ( 2 9 9 ) + x ( 3 2 3 ) + x ( 3 6 7 ) + x ( 4 0 5 ) + x ( 4 1 9 ) + x ( 3 0 4 ) + x ( 3 2 4 ) + x ( 3 6 8 ) + x ( 4 0 6 ) + x ( 4 2 0 ) + x ( 3 1 8 ) + x ( 3 2 9 ) + x ( 3 6 9 ) + x ( 4 0 9 ) + x ( 4 2 1 ) + x ( 3 1 9 ) + x ( 3 3 4 ) + x ( 3 7 0 ) + x ( 4 1 0 ) + x ( 4 2 2 ) + 68 x ( 4 2 3 ) + x ( 4 2 4 ) + x ( 4 2 5 ) + x ( 4 2 6 ) + x (427) + x ( 4 3 0 ) 4- X ( 4 3 1 ) 4-x ( 4 3 3 ) + x ( 4 3 4 ) + x ( 4 3 6 ) 4- x ( 4 3 7 ) + x ( 4 3 9 ) + x ( 4 4 0 ) 4- x ( 4 4 2 ) 4-x ( 4 4 3 ) + x ( 4 4 6 ) + x ( 4 4 7 ) + x ( 4 4 8 ) + x ( 4 4 9 ) + x (450) 4- x ( 4 5 1 ) 4-x ( 4 5 2 ) + x ( 4 5 3 ) + x ( 4 5 4 ) 4- x ( 4 5 5 ) +• x ( 4 5 6 ) + x ( 4 5 7 ) + x ( 4 5 8 ) + x ( 4 5 9 + x ( 4 6 0 ) + x ( 4 6 1 + x ( 4 9 2 + x ( 4 9 3 + x ( 4 9 7 4- x (498) + x ( 5 0 2 4- x ( 5 0 3 ) + x ( 5 0 7 + x ( 5 0 8 + x ( 5 1 0 + x ( 5 1 1 4- x ( 5 1 2 ) + x ( 5 1 3 + x ( 5 1 4 ) + x (515 + x (516 + x ( 5 1 7 + x ( 5 1 8 4- x ( 5 1 9 ) 4- x ( 5 2 0 4- x ( 5 2 1 ) 4- x ( 5 2 2 + x ( 5 2 3 + x ( 5 2 5 4- x ( 5 2 6 4- x ( 5 2 7 ) + x ( 5 2 9 + x ( 5 3 0 ) + x ( 5 3 1 + x ( 5 3 3 + x ( 5 3 4 + x ( 5 3 5 4- x ( 5 3 7 ) + x (538 + x ( 5 3 9 ) + x (541 + x ( 5 4 2 + x ( 5 4 3 + x ( 5 4 4 4- x ( 5 4 5 ) + x ( 5 4 6 + x ( 5 4 7 ) + x ( 5 4 8 + x (549 + x ( 5 5 0 + x ( 5 5 1 4- x ( 5 5 2 ) + x ( 5 5 3 + x ( 5 5 4 ) + x ( 5 5 5 + x ( 5 5 6 + ~x (557 + x(558 4- x ( 5 5 9 ) + x (560 + x ( 5 6 1 ) + x'(564 + x (565 + x ( 6 0 8 + x (609 4- x ( 6 1 5 ) + x ( 6 1 6 + x ( 6 2 2 ) 4- x ( 6 3 3 + x (634 + x ( 6 3 5 + x (636 4- x ( 6 3 7 ) + x ( 6 3 8 + x ( 6 3 9 ) + x ( 6 4 0 + x ( 6 4 1 + x ( 6 4 2 4- x ( 6 4 3 4- x ( 6 5 8 ) + x ( 6 5 9 + x ( 6 6 4 ) + x ( 6 6 5 + x ( 6 7 0 + x ( 6 7 1 4- x ( 6 7 6 4- x ( 6 8 5 ) + x ( 6 8 6 + x ( 6 8 7 ) + x ( 6 8 8 + x ( 6 8 9 + x ( 6 9 0 4- x ( 6 9 1 4- x ( 6 9 2 ) + x ( 6 9 3 > 11 X (401) + x (406) + x (410) + X (414) + x ( 4 1 9 ) + X (420) 4- X (421) + X (422) + X (423) + X (424) + X (425) + x ( 4 2 6 ) + X (427) 4- X (431) + X (434) + X (437) + X (440) + X (443) + x (447) + X (455) 4- X (456) 4-X (457) + X (458) + X (459) + X (460) + x ( 4 6 1 ) + X (493) 4- X (494) + X (498) + X (499) + X (503) + X (504) + x ( 5 0 8 ) + X (509) 4- X (510) 4-X (511) + X (512) + X (513) + X (514) + x ( 5 1 5 ) + X (516) 4- X (517) 4-X (518) + X (519) + X (520) + X (521) + x ( 5 2 2 ) + X (523) 4- X (526) + X (527) + X (530) + X (531) + X (534) + x ( 5 3 5 ) + X (538) 4- X (539) 4-X (542) + X (543) + X (550) + X (551) + x ( 5 5 2 ) + X (553) 4- X (554) + X (555) + X (556) + X (557) + X (558) + x ( 5 5 9 ) + X (560) 4- X (561) 4-X (564) + X (565) + X (609) + X (610) + x ( 6 1 6 ) + X (617) 4- X (622) + X (623) + X (628) + X (633) + X (634) + x ( 6 3 5 ) + X (636) 4- X (637) + X (638) + X (639) + X (640) + X (641) + x ( 6 4 2 ) + X (643) 4- X (644) 4-X (645) + X (646) + X (647) + X (648) + x ( 6 4 9 ) + X (650) 4- X (658) + X (659) + X (660) + X (664) + X (665) + x ( 6 6 6 ) + X (670) 4- X (671) 4-X (672) + X (676) + X (677) + X (681) + x ( 6 8 5 ) + X (686) 4- X (687) + X (688) + X (689) + X (690) + X (691) + x (692) + X (693) 4- X (694) + X (695) + X (696) + X 697) + x ( 698) + x ( 6 9 9 ) > 9 X (494) + X (499) + X (504) + X (509) + x (517) + X (518) 4- X (519) 4-X (520) + X (521) + X (522) + X (523) + x (527) + X (531) 4- X (535) + X (539) + X (543) + X (556) + X (557) + x ( 5 5 8 ) + X (559) 4- X (560) + X (561) + X (564) + X (565) + X (610) + x ( 6 1 1 ) + X (617) 4- X (618) + X (623) + X (624) + X (628) + X (629) + x ( 6 3 3 ) + X (634) 4- X (635) + X (636) + X (637) + X (638) + X (639) + x ( 6 4 0 ) + X (641) 4- X (642) + X (643) + X (644) + X (645) + X (646) + x ( 6 4 7 ) + X (648) 4- X (649) + X (650) + X (651) + X (652) + X (653) + x ( 6 5 4 ) + X (655) 4- X (656) + X (657) + X (659) + X (660) + X (661) + x ( 6 6 5 ) + X (666) 4- X (667) 4-69 X 1 671) + X 1 672) + x ( 673) + X 1 676) + X x ( 682) + x ( 685) + x ( 686) + X 1 687) + X X 1 691) + x ( 692) + x ( 693) + x ( 694) + X x ( 698) + x ( 699) + X 1 700) + x ( 701) + X x ( 705) > 8 x ( 564) + x ( 565) + x ( 611) + x ( 612) + X x ( 625) + x ( 629) + x ( 630) + x ( 633) + X X 637) x ( 638) + x ( 639) + X 640) + X X 644) + X 645) + x ( 646) + X 647) + X X 651) + X 652) + X 653) + X 654) + X X 660) + X 661) + x ( 662) + X 666) + X X 673) + X 674) + X 677) + X 678) + X X 683) + X 689) + X 690) + X 691) + X X 695) + X 696) + X 697) X 698) + X X 702) + X 703) + X 704) + X 705) + X X 709) + X 710) + x ( 711) > 4 X [564) + X (565) + X 612) + X (613) + X X (626) + X (630) + X 631) + X (638) + X X (642) + X (643) + X L.644) + X (645) + X X (649) + X (650) + X (651) + X (652) + X X (656) + X (657) + X (661) + X (662) + X X (669) + X (673) + X (674) + X (675) + X X (682) + X (683) + X (684) + X (694) + X X (698) + X (699) + X (700) + X (701) + X X (705) + X (706) + x ( 707) + x ( 708) + X > 3 X (564) + X (565) + X (613) + X (614) + X X (627) + X (631) + X (632) + X (644) + X X (648) + X (649) + X (650) + X (651) + X X (655) + X (656) + X (657) + X (662) + X X (674) + X (675) + X (679) + X (680) + X X (701) + X (702) + X (703) + X (704) + X X (708) + X (709) + X ,710) + X 711.) > 2 X (565) + X (614) + X (621) + X (627) + X X (653) + X (654) + X (655) + X (656) + X X (675) + X (680) + X (684) + X (706) + X X (710) + X (711) > 1 X (562) + X (563) + X (564) + X (565) + X X (611) + X (612) + X (613) + X (614) + X X (618) + X (619) + X (620) + X (621) + X X (625) + X (626) + X (627) + X (628) + X X (632) + X (633) + X (634) + X (635) + X 677) + x ( 6 7 8 ) 688) + x ( 6 8 9 ) 695) + x ( 6 9 6 ) 702) + x ( 7 0 3 ) + x ( 6 8 1 ) + + x ( 6 9 0 ) + + x ( 6 9 7 ) + + x ( 7 0 4 ) + 618) 634) 641) 648) 655) C667) 679) 692) 699) 706) 619) 639) 646) 653) 663) 678) 695) 702) 709) x ( 6 1 9 ) x ( 6 3 5 ) x ( 6 4 2 ) x ( 6 4 9 ) x ( 6 5 6 ) x ( 6 6 8 ) x ( 6 8 1 ) + x ( 6 9 3 ) + x ( 7 0 0 ) + x ( 7 0 7 ) x ( 6 2 0 ) x ( 6 4 0 ) x ( 6 4 7 ) x ( 6 5 4 ) x (667) x ( 6 7 9 ) x (696) x ( 7 0 3 ) x ( 7 1 0 ) + x ( 6 2 4 ) + x ( 6 3 6 ) + x ( 6 4 3 ) + x ( 6 5 0 ) + x ( 6 5 7 ) + x ( 6 7 2 ) + x ( 6 8 2 ) + x ( 6 9 4 ) + x ( 7 0 1 ) + x ( 7 0 8 ) + x ( 6 2 5 ) + x ( 6 4 1 ) + x ( 6 4 8 ) + x ( 6 5 5 ) + x ( 6 6 8 ) + x ( 6 8 0 ) + x ( 6 9 7 ) + x ( 7 0 4 ) + x ( 7 1 1 ) 620) + x ( 6 2 1 ) 645) + x ( 6 4 6 ) 652) + x ( 6 5 3 ) 663) + x ( 6 6 8 ) 683) + x ( 6 8 4 ) 632) + x ( 6 5 1 ) 657) + x ( 6 6 3 ) 707) + x ( 7 0 8 ) 608) + x ( 6 0 9 ) 615) + x ( 6 1 6 ) 622) + x ( 6 2 3 ) 629) + x ( 6 3 0 ) 636) + x ( 6 3 7 ) + + + + + + + + + + + + + + + + + + + x ( 6 2 6 ) + + x ( 6 4 7 ) + + x ( 6 5 4 ) + + x ( 6 6 9 ) + + x ( 7 0 0 ) + 705) + x ( 7 0 6 ) + x ( 7 0 7 ) + + x ( 6 5 2 ) + + x ( 6 6 9 ) + + x ( 7 0 9 ) + + x ( 6 1 0 ) + + x ( 6 1 7 ) + + x ( 6 2 4 ) + + x ( 6 3 1 ) + + x ( 6 3 8 ) + 70 x ( 6 3 9 ) + x ( 6 4 0 ) + x ( 6 4 1 ) + x ( 6 4 2 x ( 6 4 6 ) + • x ( 6 4 7 ) + x ( 6 4 8 ) + x (649 x ( 6 5 3 ) + x ( 6 5 4 ) + x ( 6 5 5 ) + x ( 6 5 6 x ( 6 6 0 ) + x ( 6 6 1 ) + x(662) ' + x ( 6 6 3 x ( 6 6 7 ) + x ( 6 6 8 ) + x ( 6 6 9 ) + x (670 x ( 6 7 4 ) + x ( 6 7 5 ) + x ( 6 7 6 ) + x ( 6 7 7 x ( 6 8 1 ) + x ( 6 8 2 ) + x ( 6 8 3 ) + x (684 x ( 6 8 8 ) + x ( 6 8 9 ) + x ( 6 9 0 ) + x (691 x ( 6 9 5 ) + x ( 6 9 6 ) + x ( 6 9 7 ) + x ( 6 9 8 x ( 7 0 2 ) + x ( 7 0 3 ) + x ( 7 0 4 ) + x ( 7 0 5 x ( 7 0 9 ) + x (710) + x ( 7 1 1 ) > 5 + x ( 6 4 3 ) + x ( 6 4 4 ) + x ( 6 4 5 ) + + x ( 6 5 0 ) + x ( 6 5 1 ) + x ( 6 5 2 ) + + x ( 6 5 7 ) + x (658) + x ( 6 5 9 ) + + x ( 6 6 4 ) + x ( 6 6 5 ) + x ( 6 6 6 ) + + x ( 6 7 1 ) + x ( 6 7 2 ) + x ( 6 7 3 ) + + x ( 6 7 8 ) + x ( 6 7 9 ) + x ( 6 8 0 ) + + x ( 6 8 5 ) + x ( 6 8 6 ) + x ( 6 8 7 ) + + x ( 6 9 2 ) + x ( 6 9 3 ) + x ( 6 9 4 ) + + x ( 6 9 9 ) + x ( 7 0 0 ) + x ( 7 0 1 ) + + x (706) + x (707) + x (708) + 71 A P P E N D I X F . T R I C K S 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 Variable Block type Shift 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 x1 7&0 7 ******* ******* ******* ******* ******* ******* ******* x2 7&0 7 ******* ******* ******* ******* ******* ******* ******* x3 7&0 7 ******* ******* ******* ******* ******* ******* i * . . . * * . BB x4 6&1 7 ******* ******* ******* ******* ******* ******* !*******. L . 1 x5 6&1 7 ******* ******* ******* ******* ******* ******* ••>:******* x6 6&1 7 ******* ******* ******* ******* ******* **'***** x7 6&1 7 ******* ******** ******* ******* ******* ******* x8 6&1 7 ******* ******* ******* ******* ******* ******* x9 6&1 7 ******* ******* ******* ******* ******* ******* x10 6&1 7 ******* ******* ******* ******* ******* ******* X11 6&1 7 ******* ******* ******* ******* ******* ******* ' ******* ' x12 6&1 7 ******* ******* ******* ******* ******* ******* x13 6&1 7 ******* ******* ******* ******* ******* ******* x14 6&1 7 ******* ******* ******* ******* ******* ******* x15 6&1 7 ******* ******* ******* ******* ******* ******* x16 6&1 7 ******* ******* ******* ******* ******* ******* x17 6&1 7 ******* ******* ******* ....... ******* ******* -*******. x18 6&1 7 ******* ******* ******* ******* ******* ******* . . . . . . \«~ : . -x19 6&1 7 ******* ******* ******* ******* ******* ******* x20 6&1 7 ******* ******* ******* ******* ******* ******* x21 6&1 7 ******* ******* ******* ******* ******* ******* • x22 6&1 7 ******* ******* ******* ....... ******* ******* x23 6&1 7 ******* ******* ******* ******* ******* * ******* x24 6&1 7 ******* ******* ******* ******* ******* • ******* -x25 6&1 7 ******* ******* ******* ******* ******* ******* x26 6&1 7 ******* ******* ******* ******* ******* j*i * * * * * * * x27 6&1 7 ******* ******* ******* ******* ******* '&•*******• x28 6&1 7 ******* x29 6&1 7 ....... • ******* x30 6&1 7 ******* ******* x31 6&1 7 ******* „. ******* x32 6&1 7 ******* ******* x33 6&1 7 ******* :******* x34 6&1 7 ******* x35 6&1 7 ******* ******* x36 5&2 7 ******* ******* ******* ******* ******* ******* •;. .«****. x37 5&2 7 ******* ....... ******* ******* ******* ,*******: x38 5&2 7 ******* ....... ******* ******* ******* x39 5&2 7 ******* ******* ******* ******* ******* x40 5&2 7 ******* ******* ******* ******* ******* x41 5&2 7 ******* ******* ******* ******* ******* • x42 5&2 7 ******* ******* ******* ******* ******* ******* x43 5&2 7 ******* ******* ******* ******* ******* 72 12:30 13:001 13:00 13:30 13:30 14:00 14:00 14:30 14:30 15:00 15:00 15:30 15:30 16:00 16:00 16:30 16:30 17:00 MM 17:00 17:30 17:30 18:00 18:00 18:30 18:30 19:00 mm mm mm.. m « 2 mm mm Wife mm M I mm ' . " i i , - i f T i f — n r v n H P - j r T — o- 1|- y; —-j; 73 x44 5&2 7 ******* ....... ******* ....... ******* x45 5&2 7 ******* ....... ******* ******* ....... 7:00 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 Variable Block type Shift 7:30 8:00 8:30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 x46 5&2 7 ******* ******* ******* ******* ******* x47 5&2 7 ******* ******* ******* ******* ******* ******* x48 5&2 7 ******* ******* ******* ******* ******* x49 5&2 7 ******* ******* ******* ******* ******* x50 5&2 7 ....... ******* ******* ******* ******* x51 5&2 7 ******* ******* ******* ******* ******* x52 5&2 7 ******* ******* ******* ******* ******* ******* x53 5&2 7 ******* ******* ******* ******* ******* x54 5&2 7 ******* ******* ******* ******* ******* x55 5&2 7 ******* ******* ******* ******* ******* x56 5&2 7 ******* ******* ******* ******* ******* x57 5&2 7 ******* ******* ******* ******* ******* ******* ******* x58 5&2 7 ******* ******* ******* ******* ******* x59 5&2 7 ******* ******* ******* ******* ******* x60 5&2 7 ******* ******* ******* ******* ******* x61 5&2 7 ******* ******* ******* ....... ******* x62 5&2 7 ******* ******* ******* x63 5&2 7 ....... ******* ******* x64 5&2 7 ....... ....... ******* x65 5&2 7 ******* * * * * * * * x66 5S.2 7 ******* ******* ******* x67 5&2 7 ******* ******* ******* x68 5&2 7 ******* ******* ******* x69 5&2 7 ******* ******* ******* x70 5&2 7 ******* ******* x71 5&2 7 ******* x72 5&2 7 ....... ....... x73 5&2 7 ******* ******* x74 5&2 7 ******* ******* x75 5&2 7 ******* ******* x76 5&2 7 ******* ******* x77 5&2 7 ******* ******* x78 4&3 7 ******* ******* ******* ******* ******* ******* ....... x79 4&3 7 x80 4&3 7 ******* ....... * * * * * * * ******* x81 4&3 7 ******* ******* ******* ******* x82 4&3 7 ******* ******* ******* ******* x83 4&3 7 ******* ******* ******* ******* - ******* x84 4&3 7 ******* ******* ******* ******* x85 4&3 7 ******* ******* ******* ******* x86 4&3 7 ******* ******* ******* ....... x87 4&3 7 ******* ******* ******* ******* *******. x88 4&3 7 ******* ******* ******* ******* x89 4&3 7 ******* ******* ******* ******* x90 4&3 7 ******* ******* ******* ******* 74 12:30 13:00 13:30 14:00 14:30 13:00 13:30 14:00 14:30 15:00 ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* | ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* : ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* ******* mm 75 APPENDIX G. ACD OUTPUT CALL CENTRE ACD QUEUE STATS -INT/DAY Interval Report WCB RICHMOND Date: 19/01/98 . Time: 22:00:30 Intervals: 07:00.-19:00 Day: 19/01/98 ACD-DN INTVL AVG CALLS TOT TOT AVG SRV ACD AVG AVG AVG Number AGT ACCPT CALLS CALLS ANS LVL CALLS ACD NOT HOLD ANSWD ABAND DEL. % XFERD TALK RDY TIME TIME TIME SEC SEC SEC SEC 7:00 2 2 2 0 11 100 0 152 49 19 7:30 5 3 3 0 3 100 1 167 409 1 8:00 8 19 18 1 5 100 6 95 123 12 8:30 10 46 46 0 9 96 15 162 71 8 9:00 12 67 66 1 9 96 24 154 68 20 9:30 14 84 84 0 6 100 22 138 67 13 10:00 13 73 73 0 6 97 28 142 54 19 10:30 17 73 73 0 7 96 21 136 91 18 11:00 17 82 79 3 6 95 31 151 107 18 11:30 14 81 79 2 10 90 21 141 47 17 12:00 11 65 61 4 48 68 9 219 53 19 12:30 13 54 54 0 6 100 18 159 114 15 13:00 14 59 57 2 8 95 16 143 124 25 13:30 13 68 65 3 9 93 13 177 112 24 14:00 13 68 68 0 37 66 23 188 91 34 14:30 13 55 51 4 20 73 19 161 158 25 15:00 15 58 58 0 9 90 14 184 141 22 15:30 14 46 46 0 8 96 17 233 71 28 16:00 6 54 51 3 44 54 6 132 50 6 16:30 5 14 14 0 55 71 4 168 96 18 17:00 3 8 8 0 7 100 0 70 117 0 17:30 4 10 10 0 7 100 0 95 124 0 18:00 3 5 5 0 18 80 0 108 350 0 18:30 2 1 1 0 6 100 0 102 1420 0 19:00 1 0 0 0 0 0 0 0 20 0 10 1095 1072 23 15 88 308 159 92 19 76
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Scheduling customer service representatives for the...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Scheduling customer service representatives for the Workers’ Compensation Board of British Columbia Sanegre, Rafael 1998
pdf
Warning
You are currently on our download blacklist and unable to view media. You will be unbanned within an hour.
To un-ban yourself please visit the following link and solve the reCAPTCHA, we will then redirect you back here.
Page Metadata
Item Metadata
Title | Scheduling customer service representatives for the Workers’ Compensation Board of British Columbia |
Creator |
Sanegre, Rafael |
Date Issued | 1998 |
Description | The Workers' Compensations Board of British Columbia (WCB) operates 4 call centers in the province devoted to compensation services. This thesis describes a review of operations at the call center in the head office, located in Richmond, B.C. We identified the need for a mathematical model to schedule telephone operators (called CSRs in WCB) in order to minimize the workforce required, while assuring minimal staffing levels to provide good customer service. We developed two integer programming models and built an interface that would enable the supervisors to use the model interactively. We compared the results from the model to one of their schedules and found that the model used 5 fewer CSRs than currently in their schedule. The potential savings derived from such a test convinced management that WCB needed to acquire and implement software to schedule staff. |
Extent | 6690662 bytes |
Subject |
Workers’ Compensation Board of British Columbia -- Customer services Scheduling -- Computer programs |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-05-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0088624 |
URI | http://hdl.handle.net/2429/8261 |
Degree |
Master of Science in Business - MScB |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
GraduationDate | 1998-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-ubc_1998-0607.pdf [ 6.38MB ]
- Metadata
- JSON: 831-1.0088624.json
- JSON-LD: 831-1.0088624-ld.json
- RDF/XML (Pretty): 831-1.0088624-rdf.xml
- RDF/JSON: 831-1.0088624-rdf.json
- Turtle: 831-1.0088624-turtle.txt
- N-Triples: 831-1.0088624-rdf-ntriples.txt
- Original Record: 831-1.0088624-source.json
- Full Text
- 831-1.0088624-fulltext.txt
- Citation
- 831-1.0088624.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0088624/manifest