A Break from Tradition for the San Francisco Police

A Break from Tradition for the San FranciscoPolice: Patrol Officer Scheduling Using anOptimization-Based Decision Support SystemPHILIP E. TAYLOR McLaren College of Business

University of San Franciscoignatian HeightsSan Francisco, California 94117

STEPHEN J. HUXLEY McLaren CoUege of BusinessUniversity of San Francisco

The San Francisco Police Department (SFPD) recently imple-mented an optimization-based decision support system fordeploying patrol officers. It forecasts hourly needs, schedulesofficers to maximize coverage, and allows fine tuning to meethuman needs. The fine-tuning mode helps captains evaluateschedule changes and suggests alternatives. The system alsoevaluates policy options for strategic deployment. The integersearch procedure generates solutions that make 25 percentmore patrol units available in times of need, equivalent toadding 200 officers to the force or a savings of $11 million peryear. Response times improved 20 percent, while revenuesfrom traffic citations increased by $3 million per year.

San Francisco has a population of ap- Like most police departments, SFPD

proximately 700,000 and a police operated with schedules designed byforce of about 1,900 sworn officers, of hand. With manual methods, it was im-which 850 perform regular patrol duties. possible to know if the "hit or miss"SFPD's budget in 1986 of approximately schedules were close to optimal in terms$176 million included $161 million in per- of serving residents' needs, and it wassonnel related costs. Patrol coverage of difficult to evaluate alternative policies forthe nine police districts in San Francisco scheduling and deploying officers. Whileaccounted for $79 million of the total. a few precinct captains were skilled andCopyright © 1989, The Institute of Management Sciences DECISON ANALYSIS — SYSTEMS0091-2102/89/1901/0004501.25 GOVEitNMENT SERVICES — POLICE

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SCHEDULING SAN FRANCISCO POLICE OFFICERS

reasonably effective at scheduling and de-ploying their patrol officers, others werenot.

At a higher level, the strategic alloca-tion of patrol resources and the methodsof deployment used were at issue. Man-ual methods could not easily or accu-rately measure the impact on productivityof using four-day, ten-hour (4/10) shiftsversus five-day, eight-hour (5/8) shifts orone- versus two-officer cars. Therefore,SFPD management had difficulty reachingresolution on policy changes. To bettermeet the demands of the residents of SanFrancisco, management speculated that alarger share of the police force should bedeployed in one-officer cars instead of thetraditional two-officer cars. At the sametime, the Police Officer Association (POA)wished to implement a 4/10 workweek inplace of the 5/8 workweek. Managementfeared a repeat of previous failed at-tempts to implement a 4/10 plan usingmanually developed schedules. The con-cern was that gains from one-officer carsmight be lost in switching to the 4/10plan [Chelst 1981; Green and Kolesar1984]. As a result of this uncertainty,many of the SFPD deployment policies,which had originated in the 1950s and1960s, had continued almost unchangeddespite substantial efforts since the early1970s to make improvements.Objectives in Police Scheduling

Ail police departments face three fun-damental issues: citizen safety, cost of op-erations, and officer morale. Trade-offsamong these considerations must bemade since they conflict with each other.The fundamental problem is to determinethe schedule in terms of the tour and

shift lengths (either a 5/8 or 4/10 plan),while simultaneously determining thenumber of officers who start their work-week on given days and hours. A sched-ule's productivity is measured by howwell it matches officers on duty to officersneeded during each hour of the week.Figure 1 shows the number of officersneeded for one San Francisco precinctduring each hour of each day of theweek. Having more officers on duty thanshown (surpluses) wastes resources,while having less than required (short-ages) increases response times and over-burdens officers. Effective schedulingminimizes shortages and surpluses pro-viding the highest possible correlation be-tween the number of officers needed andthe number of officers actually on dutyduring each hour. The data for Figure 1was taken from SFPD's sophisticatedcomputer aided dispatching (CAD) sys-tem, which provides a large and rich database on resident calls for service. TheCAD system is used to dispatch patrol of-ficers to calls for service and to maintainoperating statistics, such as call types,waiting times, travel times, and total timeconsumed in servicing calls.

The long-term goal in deploying andscheduling police is to minimize the num-ber of officers needed to achieve a desiredlevel of protection, as defined by policemanagement and city government, whilebalancing the work load equitably amongofficers. Balancing the work load isclosely related to minimizing shortagesand surpluses and means that all officersshould experience about the sameamount of call time. A principal advan-tage of computer scheduling is that it

January-February 1989

TAYLOR, HUXLEY

Officers

Figure 1: SFPD officers required by hour by day in one precinct. The officer requirements areforecast based on historical calls for service, consumed times by call type, management policyon percent of lime officers should spend answering calls, percent of each call type requiringtwo or more officers, and percent of cars allocated two officers. The call and consumed timedata are downloaded from the mainframe to the microcomputer. The other data are input by thecaptains and stored in a file for reuse. The resulting table of numbers graphed here forms thetarget for scheduling officers.

reduces variation in work load among of-ficers, which helps maintain or improvemorale. In the short run, the objective formost departments becomes one of balanc-ing work loads with whatever officers areavailable by setting schedules that mini-mize shortages and surpluses during the168 hours of the week.

Task Force DirectivesIncreasing demands for service and de-

creasing budgets prompted SFPD man-agement to examine the feasibility of ascheduling system that could substan-tially improve productivity and responsetimes in all nine precincts while maintain-ing current officer staffing levels. A task

INTERFACES 19:1


force was assigned to evaluate currentmanual scheduling procedures and de-ployment policies. Then, if necessary, itwas to define the criteria for a new sys-tem and conduct an international searchfor available scheduling or deploymentsoftware to address the problem. The taskforce was initially headed by the currentPolice Chief Frank Jordan and was com-posed of officers from the Police OfficerAssociation, patrol management, andtechnical support. Captain Walter Cullopfollowed Jordan as the lead member ofthe task force. After reviewing the man-ual system, the task force decided tosearch for a new system. The criteria itspecified for the system included the fal-lowing factors: (1) The system must usethe CAD data on calls for service andconsumed times to establish work load byday of week and hour of day; (2) It mustgenerate optimal and realistic integerschedules that meet management policyguidelines using a microcomputer; (3) Itmust allow easy adjustment of optimal

Many SFPD deploymentpolicies, which had originatedin the 1950s and 1960s, hadcontinued almost unchanged.

schedules to accommodate human consid-erations without sacrificing productivity;(4) It must create schedules in less than30 minutes and make adjustments in lessthan 60 seconds; (5) It must be able toperform both tactical scheduling and stra-tegic policy testing in one integrated sys-tem; (6) The user interface must beflexible and easy, allowing the users

(captains) to decide the sequence of func-tions to be executed instead of forcingthem to follow a restrictive sequence.

The search included examination ofRand's PCAM [Chaiken and Dormont1978] and hypercube models [Larson 1974,1975, 1985] and linear programmingbased systems. The task force found thatRand reported the PCAM model wasused by 20 to 30 departments to help inallocating cars and officers within pre-cincts but not to schedule officers. Duringits intensive search efforts in 1986, thetask force found no integer optimizationmodels in use and no departments usinglinear programming to schedule officers.The task force decided that current soft-ware did not adequately address the crite-ria and contacted the authors to deter-mine the feasibility of meeting their crite-ria. The task force cooperated with usduring approximately 10 months of un-funded research, software development,testing, schedule benchmarking, and acost/benefit analysis. At the end of thisperiod, SFPD management became confi-dent that the system was feasible andcontracted with us to develop a system.This funding permitted further coopera-tive development of a microcomputer op-timization-based decision support systemcalled the Police Patrol Scheduling System(PPSS), which supports tactical and stra-tegic decision making.Necessary Features of the SchedulingSystem

The scheduling system had to producefeasible integer solutions, provide ameans for users to fine-tune schedules toallow for employee needs and unusualevents while maintaining high quality


TAYLOR, HUXLEY

services for residents, and compute a typ-ical schedule (90 officers) in less than 30minutes. The user interface must allowusers to run the show and yet gain ad-vice, learning the why's of that advicefrom the system.

In addition, graphical displays wereneeded to show why scheduling decisionswere good or bad and why some policieswere superior to others. This would alle-viate the confusion created during theprevious years of discussion regardingthe best policies to pursue.Past Research

Analytical approaches to the generalpersonnel scheduling problem appearedas early as 1954 [Dantzig 1954]. In theclassic LP formulation, the goal is to min-imize the number of workers hired sub-ject to constraints that at least thenumber required for a satisfactory level ofservice be on duty each period. (For ex-amples of variations on this theme, seeBaily and Field [1985]; Bartholdi [1981];Bechtold and Showalter [1985]; Browneand Tibrewala [1975]; Mabert and Watts[1982]; Morris and Showalter [1983]; andWarner and Prawda [1972]. A survey ofthe manpower scheduling problem can befound in Patrikalakis and Blesseos [1988].)

Research into police scheduling beganin the late 1960s and early 1970s, both inthe US and the UK [Comrie and Kings1974; Drossman 1974; Felkenes andWhisenand 1972; Heller 1969; Heller,McEwen, and Stenzel 1972; Larson 1972;and McLaren 1967]. The New York CityRand Institute undertook a majorresearch effort on the use of managementscience in urban emergency servicesbetween 1973 and 1975. Rand analyzed

call rates, geographic locations, and time/distance relationships related toemergency services. This work, whichincluded not only police but also fire andambulance service, is summarized byWalker [1975]. The Rand models for police

In certain dangerous areas,two-officer cars are necessary.

deployment utilized queuing theory tostudy both the spatial and temporaldistribution of patrol officers. Spatially,the hypercube model [Larson 1974, 1975,1985] represented a major advance in thegeographic allocation of patrol resources.For deployment. Rand's patrol carallocation model (PCAM) [Chaiken andDormont 1978; Chaiken 1985] focusesprimarily on the allocation of patrol carsamong precincts. It also forecasts thenumber of officers needed on duty in athree- or four-shift environment, but doesnot provide a means of optimallydetermining specific hourly schedules.

The Rand work spawned an ongoinginterest in the applications of manage-ment science approaches to police opera-hons [Chelst 1978, 1981; Green andKolesar 1984a, 1984b]. Kolesar [1982] pro-vides a comprehensive review of researchon the logistics of emergency servicesover this period. Stenzel and Buren[1983], Levin and McEwen [1985] and theNational Institute of Justice [1980] provideexcellent bibliographies on policescheduling.Policy Options

Police management can set policies on avariety of factors that affect officer

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scheduling. The primary set of decisionvariables is the number of officers to startduring each hour of the week to maxi-mize coverage. However, other factorsalso had to be considered by the system:

(1) Shift length and tour length: Mostcommon is the standard 5/8 plan, but the4/10 plan is also popular. In some cities,officers work six to nine straight days

The goal programmingobjective function is minimizethe amount of shortages(uncovered needs) and thenminimize the maximumshortage for any given hourof the week.

with two to five straight days off. In othercities, officers work different hours ondifferent days. Other constrair\ts may berequired by union contracts betweenmanagement and police officer associa-tions. Clearly, management needs toknow with some precision how switchingfrom one feasible plan to another will af-fect coverage. For SFPD, a major questionwas the impact of switching from a 5/8 toa 4/10 plan.

(2) Number of start times: Determiningthe starting days and times and the num-ber of officers to assign to each time isthe heart of the problem in patrol sched-uling. Many departments use three shiftswith different numbers of officers begin-ning their week each day of the week.Many departments include overlay orpower shifts in addition to the basicthree. These power shifts overlap theother shifts to allow peak coverage.

Others allow a one-hour staggered start-ing time, where half the officers assignedto the shift start and finish one hour laterthan the other half. For greater controland accountability in the organization,most departments use squads of aboutthree to 10 officers working together eachday as an integral unit under one super-visor. The number of supervisors avail-able limits the number of groups, whichmeans that a nonlinear integer constraintlimits the number of positive variables inthe schedule {that is, it limits the numberof variables in the final solution).

(3) Rotating vs. fixed schedules: Withfixed schedules, officers work the sameshift on the same days every week per-haps with some reassignment everythree, six, or 12 months. Rotating sched-ules, on the other hand, can mean eitherrotating shifts (for example, rotating fromthe day to the evening shift), rotatingdays worked (for example, Saturday andSunday off one week followed by Sundayand Monday off the next week), or both.In San Francisco, shifts are fixed, mean-ing that transfers from one shift time toanother are permitted only twice a yearthrough a "seniority sign-up," with shiftpreferences controlled strictly by seniority.Days worked, on the other hand, rotateto give all officers the same number ofweekends off during the work cycle. Offi-cers always work consecutive days eachweek but start their workweek one daylater each successive week. Based on theresults from the Police Patrol SchedulingSystem, the SFPD now uses a 4/10 plan,which results in three consecutive four-day weekends off, followed by four con-secutive work weekends. This cycle


TAYLOR, HUXLEY

proved very popular compared to pre-vious work schedules.

From a scheduling standpoint, rotatingschemes are inherently less efficient sincethe officers usually move in groups, andgroup sizes must therefore be approxi-mately the same. This constraint, com-mon to many police agencies, oftenresults in a need for more officers to pro-vide good coverage on peak days (orworse coverage from a given number ofofficers) than with fixed schedules. Priorto PPSS it was difficult to quantify the netimpact of fixed versus rotating schedules.

(4) Officers per car: The SFPD tradition-ally deployed two officers per car but nowuses a mix of one and two officers percar. At night two-officer cars are re-quested by approximately half the offi-cers. In certain dangerous areas, two-officer cars are necessary. The POAagreed to accept a mix of one- and two-officer cars in exchange for the 4/10 plan.It is possible to evaluate productivity un-der these varying policies by allowingSFPD management to specify the percent-age and types of calls in each priority cat-egory that require two or more officersand to specify the percentage or numberof officers to be deployed in two-officercars. Requirements for officers and carscan be accurately projected and sched-uled based on the number of calls, thecall characteristics, and the mix of one-and two-officer cars.

(5) Minimum officers on duty: Theearly morning hours, during week nightsparticularly, are times of low activity formost departments. During several hoursan average of only one or two officerswithin a precinct may be needed to

answer calls for service. However, SFPDmanagement required that a certain mini-mum number of officers be on duty at allhours to ensure adequate geographic cov-erage. Hourly estimates of the minimumrequirements for officers for these lowpeak and all other periods had to be ad-justed accordingly. The model must in-sure that these hourly minimums are metat all times.

(6) Non-patrol duty: Although most pa-trol officers are assigned to cars or walkbeats and deal with calls for service,some time is spent on other tasks: train-ing programs, station duty, or dealingwith calls from schools (schools cars).These hourly requirements are added to

The system producedapproximately a 50 percentreduction in shortages andsurpluses.

the patrol requirements to arrive at an es-timate of total personnel needs for eachhour at each station.The Model, Its Data Base, and Its HumanInterface

PPSS consists of three integratedphases: forecasting hourly requirements(Figure 1), mathematically determiningstart times and the number of officers,and fine-tuning the results for special cir-cumstances (Figure 2).Phase 1 — Forecasting

From the CAD data on number of calls,the percentage or kinds of calls in eachpriority type requiring two officers, thepercentage mix of one- and two-officercars deployed, and the time spent per

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call, a "consumed time" and officer needis calculated for each hour of the week,and this becomes the basic building blockfor the forecast (Figure 1). These 168 val-ues are then adjusted to allow for non-patrol duty requirements and a rate ofgrowth (or decline) in calls for service.Also, management supplies a policy tar-get as to the percentage of time officersshould spend answering calls. This leadsto a table and graph that shows how

many officers are needed each hour ofthe week for a precinct.Phase 2 — Scheduling

A specially developed integer searchprocedure is used to find the schedulethat best fits the hourly requirements forthe number of officers available. Themathematical mode! of the problem isprovided in the appendix.

The model may be described as follows:the goal programming objective function

CAD System maintainshourly data onhistorical patrol

activity

Compute LPto gain lower

bound

Producefinal schedule

of officers

Forecast officersrequired by hour

by day

Integheuristicmodel produces initial

solution

Expert Systemprovides fine tuning

feedback on managementmodifications

Management suppliespolicy decisions

Managementreviews and

modifies

Produce batteryof detail, statistical,

and summary reportswith graphical support

Figure 2; The flow of information and decisions through the Police Patrol Scheduling System.The CAD data are downloaded from the mainframe to the forecasting program, and manage-ment supplies the policy decisions. Then an optional LP model can be run to determine a lowerbound on the objective function based on the forecasted needs. The IH model then develops theschedule, which management can review and fine-tune using the captains' expert system.Finally, reports and schedules are generated on screen and selected for printing.

January-February 1989 11

TAYLOR, HUXLEY

is to minimize the amount of shortages{uncovered needs) and then minimize themaximum shortage for any given hour ofthe week. For example, a schedule thathas one officer short during 15 well scat-tered hours (that is, 15 total hours short)is superior to one that has 15 officersshort during a single hour (also 15 totalhours short). A schedule that simply min-imizes shortages may cause serious lapsesin coverage. To accommodate both theminimization of shortages and the minim-ization of the maximum shortage, themodel is structured with a nonlinear goalprogramming objective function.

The model has several constraint sets:(1) a limitation on the number of officersavailable; (2) minimum staffing require-ments for all hours of the day; (3) softconstraints on the number of officersneeded for each hour of the day, whichrequire that officers scheduled plus officershortages must equal or exceed officersrequired; and (4) a limited number ofstart-time groups, which is dictated bythe number of supervisors available, anonlinear constraint.

The decision variables are the shift starttimes and the integer number of officersstarting at each time. If N start times canbe utilized, then the combination of 168times (24 hours and seven days) taken Nat a time represents the number of possi-ble start time patterns. The other aspectthat must be considered simultaneously ishow many officers should be assigned toeach of the N start times. This problem isNP-complete (Bartholdi [1981], Mabertand Watts [19821, and Morris andShowalter [1983]), and algorithms thatguarantee optimality are unlikely. We

coined the term "integheuristic" (IH) todescribe the approach of the stepwise in-teger-heuristic search. The IH-generatedsolutions can be compared to the optimalLP to determine the effectiveness of theprocedure.

The IH model uses a primal-dual inte-ger step-search philosophy. The IHmethod tries at each step to take the bestoverall integer unit step towards a feasi-ble and optimal solution by examining allconstraints. For a simple illustration, as-sume the only goal is minimizing short-ages. Then a step towards feasibility orreducing shortages is also a step towardsoptimality since the objective function isto minimize shortages. Hence, the primalobjective (optimality) and the dual objec-tive (feasibility) are compatible. IH thenworks on multiple constraints by examin-ing the possible increase of each variable

We ran the IH model for 36different schedulingscenarios.

by one unit and selecting as the variableto elevate the one that reduces the maxi-mum number of shortages. The basicpremise of the approach is to work onboth optimality and feasibility at the sametime. Intuitively, the IH procedure aimsat confining the iterations to those con-straints that will be most binding in theoptimal solution. This is done by also ex-amining the impact of increasing eachvariable on the magnitude of the hourlyshortages when ties occur in the reduc-tion of total shortages. The consequenceis to try to obtain feasibility and optimal-

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SCHEDULING SAN ERANCISCO POLICE OFFICERS

ity. Of course, this does not always occurand then further search is necessary inthe area of the first solution generated, asdiscussed below. The general flow of theIH search is provided in the appendix.

Taylor [1975] applied the IH concept toan air traffic flow control problem. Taylor,Luman, and Sciullo [1978] applied a simi-lar approach to nurse scheduling. Gloverand McMillan [1986] have applied theconcept to a class of personnel schedulingproblems. Glover [1986 and 1988] devel-oped "tabu search," which utilizes theprinciples for a more general class of NP-complete problems.

In actual operations, schedules shouldnot vary drastically from scheduling pe-riod to scheduling period (for example,quarter to quarter). The sociology of a po-lice department and of most organizations

Even the best schedule forany two-officer car systemrequires more officers thanthe worst schedule for a mixof one- and two-officer cars.

can tolerate some changes (for example,15 to 25 percent) in the schedules fromperiod to period, but major changes (forexample, over 40 percent) are not accepta-ble. We refer to this as "dynamic socialfeasibility" and address this problem byusing the existing schedule from the cur-rent ending scheduling period as thestarting solution for the next schedulingperiod. Then we pursue the logic of mod-ifying the existing schedule step by step,while temporarily disallowing certainmoves from recurring (that is, making

them "tabu" [Glover 1988]), and at eachstep either improving the objective func-tion and also the feasibility or reducingthe number of officers scheduled withminimum loss of shortages covered. Bymodifying the existing schedule in an in-telligent manner, we generate a newschedule that is close in starting times tothe existing schedule in its social impactbut substantially reduces shortages andbetter matches the requirements for thenew scheduling period.

Prior research into the formulation as-pects of the personnel-scheduling prob-lem using LP or other mathematical-programming techniques has not resultedin widespread use of these techniques bypolice departments. This is perhaps be-cause the optimal LP solution regularlycalls for large numbers of weekly startingtimes (for example, 30 to 40) with toomany patrol "squads" of only one or twoofficers (plus or minus a fraction). An-other characteristic of LP is that smallchanges in the hourly officer require-ments from scheduling period to periodcan often cause dramatic reshuffling ofstart times and officers assigned to eachstart time; such solutions are socially in-feasible. Although general integer pro-gramming techniques are available, theycan often require large amounts of timeon high speed mainframes and some-times do not produce a solution in thetime allotted. The use of general integerprogramming was determined to be im-practical due to computation time andconvergence reliability on amicrocomputer.

A battery of comparisons were madefor small (that is, 40 to 50 officers).


TAYLOR, HUXLEY

medium {60 to 90 officers), and large pre-cincts (100 to 120 officers) between the IHsolution procedure and a linear program(LP), which could not generate feasibleinteger solutions but did provide a lowerbound on the optimal integer solutionvalue. In all cases the IH solution valueswere within one percent of the LP nonin-teger solution values in terms of totalshortages. The computation times for theLP of as much as seven to nine hours onLINDO-based software were at least 10-fold greater than the IH computationtimes on the same microcomputer. PPSScan produce a 90-officer schedule in 15 to20 minutes on an 80286 12-megahertzmicrocomputer.

In the flowchart of PPSS (Figure 2), anoptional LP model run establishes a lowerbound on the solution value. Since theoptional LP solution usually requires toomany start times with unacceptably smallgroup sizes, does not consider minimiz-ing the maximum shortages, regularlysuggests fractional officers, and requireshours to compute the solution, it has noother practica! value.

To understand the effect of using PPSSin a hypothetical situation, look at thenumber of officers required to be on dutyduring each hour of the week based onforecasted calls for service (Figure 1).Figure 3 shows the difference betweenthe officers scheduled and those re-quired. From such data, shortages, sur-pluses, correlations, and any othermeasure of schedule quality can be calcu-lated for the department. Also a simplequeuing/simulation model can be used tosimulate travel times, response times, to-tal consumed times, and other statistics

based on the scheduled staffing versusthe officers required.Phase 3 — Fine Tuning

After generating initial schedules, theuser can use a fine-tuning subsystem inan interactive mode. Police personnel us-ing this adjustment feature can get rec-ommendations on alternate schedules orgain almost immediate feedback on their

These results stopped thespeculation that hadfrustrated the departmentand the Police OfficerAssociation for 20 years.

own adjustments. This subsystem meas-ures the impact on schedule quality (thatis, shortages and maximum single short-age) if a change were made to (1) add anofficer to any schedule, (2) delete an offi-cer from any schedule, (3) modify thenumber of officers on station duty or anyother nonpatrol duty, (4) open a new shiftstarting time or close an old one, or (5)increase or decrease the percentage ofone-officer cars.

This combination of generating solu-tions and integrating human factors givesthe system the flexibility to consider thehuman aspects of work schedules. By em-ploying the add and delete capabilities,the user can restructure a computer-gen-erated schedule to meet one-time needswithout substantially sacrificing schedulequality. Various major or minor changesin the solution can be tested quickly {themicrocomputer response time is 20 sec-onds). This feature can be used to evalu-ate special circumstances, for example.

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SurplusesShortages

Figure 3: The shortages and surpluses minimized by the computer generated schedule. Short-ages occur wben fewer officers than needed are scheduled for a particular hour. Surpluses resultwhen too many officers are scheduled for a particular bour. Surpluses contribute to idle time,while shortages often result in poor response times, and in some cases, officer overload.

how should last month's schedules bemodified to meet this month's needs, orhow should an injured or vacationing offi-cer's time/day slot be handled to mini-mize the impact on shortages in anexisting schedule.

The integer and feasibility require-ments, the necessary user interaction,and the computation time constraint com-bine to produce a very challenging com-

puter modeling and integer searchtechnique development problem. By in-corporating management science model-ing, decision support system modeling,fine-tuning capabilities, and a seamlessuser environment, PPSS allows the userto control the final schedule. The user in-terface was designed to include (1) datadisplays that are easy to view and under-stand; (2) graphical views of the data;


TAYLOR, HUXLEY

(3) decision variables that are clear, intui-tive, and can be easily modified; (4) dataand decision variables that can be viewedin many ways; (5) fast response (seconds)to frequently used tasks and reasonableresponse time (minutes) for complextasks; (6) flexible user selection and se-quencing of tasks; (7) advice as to thenext suggested function; (8) flexible filesaving and retrieval that allows alternativeschedules to be saved and recalled; (9)screens showing final reports for previewbefore printing; and (10) a means for thesystem to learn from the user what is ac-ceptable and what is unacceptable (for ex-ample, start times).

Prior to implementation, SFPD testedthe system against manual schedules us-ing the existing two-officer car, 5/8 sched-ule strategy. The system producedapproximately a 50 percent reduction inshortages and surpluses. Studies con-ducted by SFPD management showedthat PPSS added 176,000 productive hoursto the patrol staff per year. At $30 perhour, that is a savings of $5.2 million peryear improved scheduling. Since then,the Sacramento Police Department hasused PPSS to modify its 4/10 plan. PPSSproduced schedules under the samescheduling policies that reduce shortagesand surpluses approximately 45 percentwhen compared to manual schedules.How PPSS Supports Practical StrategicPolicy Decisions

PPSS helped to answer a number ofstrategic questions facing SFPD manage-ment: (1) How much extra manpowerwould be needed if the department useda 4/10 work schedule instead of the 5/8system? (2) How much better coverage

could be attained if management assignedmore one-officer cars? (3) How much ex-tra manpower would be needed if the de-partment continued to use rotatingschedules instead of fixed schedules? (4)How much would varying the numbers ofshift starting times affect personnel needsunder various scheduling policies?

To get answers, we ran the IH modelfor 36 different scheduling scenarios.These included three possible officer-per-car configurations (all one-officer cars, alltwo-officer cars, and a mix consisting ofall one-officer cars during daylight hoursand 75 percent two-officer cars duringdarkness), four possible shift plans (5/8plan with fixed days off, 5/8 plan withrotating days off, 4/10 plan with fixeddays off, and 4/10 plan with rotating daysoff), and three station sizes (small, me-dium, and large). Based on manage-ment's existing policies for dispatchingone- and two-officer cars for various typesof calls for service, we developed an offi-cer requirements table for each scenarioand produced schedules. The least effi-cient schedule (or the worst case mathe-matically) turned out to be all two-officercars with a rotating 4/10 plan, which re-quired 112 officers. Using this as a bench-mark, we determined the percentagesavings associated with the other sched-ules for a medium-sized station (Figure4). Management had been concerned thatthe switch to a 4/10 plan would offset thegains from using one-officer cars. How-ever, switching from any two-officer-carschedule to a mix of one- and two-officercars would result in so much improve-ment that it more than offsets any declineresulting from switching from a 5/8 to a

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4/10 plan. Even the best schedule for anytwo-officer car system requires more offi-cers than the worst schedule for a mix ofone- and two-officer cars. For all stationscenarios, the policy change improve-ments averaged 25 percent.

Of significant interest is the fact that100 percent one-officer cars do not in-crease efficiency a great deal compared toa mix. This was important, since it meant

that the common-sense policy of using amix with two-officer cars during somehours of the night and in certain geo-graphic areas was not unduly damagingto productivity compared to the probableimprovement in officer safety. Of course,the viability of each of these scenarios de-pended on quality schedules producedunder the scenario.

Also relevant is the impact of

% Gain inOfficers

50 Available

40

30

20

10

FIXED 5/8

FIXED 4/10

ROTATE 5/8

ROTATE 4/10

Figure 4: PPSS computed percent gain in officers available under various deployment strategies.This enabled SFPD management to examine the trade-offs between officer preferences and pro-ductivity. Management chose to switch from a 5/8 rotating two-officer car policy to a 4/10 rotat-ing one- and two-officer car mix.


TAYLOR, HUXLEY

scheduling additional shift starting timesunder four different scheduling strategies{Figure 5). From three to 24 start timesare possible, and SFPD historically usedthree. They discovered that the 4/10 rotat-ing plan with five start times was supe-rior to the 5/8 rotating plan with three orfour start times.

These results ended the speculationthat had frustrated the department and

the POA for 20 years. They convincedmanagement to implement the 4/10 planand schedules with a mix of one- andtwo-officer cars. SFPD management con-ducted a formal analysis that showed thatthese changes contributed $5.8 million toimproved availability of officers whenneeded. Combined with the savings fromimproved scheduling the total savingswere $11 million per year.

% Gain inOfficers

Available

HOUR

HOUR

Figure 5: PPSS also enabled SFPD management to evaluate alternative numbers of shift startingtimes. Management chose to increase the number of start times based on this analysis. Eachprecinct was evaluated separately by PPSS to determine the best number of start times and theexact times each shift should start.

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Implementation ChallengesThe environment for change was excel-

lent, but due to Police Officer Association(POA) pressure for the 4/10 and one-offi-cer cars, the relationship between the Po-lice Officers Association and managementwas intense. The POA had been involvedin the project prior to funding and hadbeen supportive to the point of consider-ing POA funding of the project. The in-tensity of the management/laborrelationship was due in part to frustrationwith the long-running battles and failedattempts at implementing a 4/10 plan.Management meetings and management/POA meetings on the subject of patrol de-ployment had usually been burdenedwith wide and heated speculation on theimpacts of various policies since accurateevidence was unavailable. Therefore,gaining support for changing schedulingpolicies that had been in effect for so longwas a major challenge. Working withmanagement we decided to phase in thechanges one station at a time. Park Sta-tion was selected as the initial test site forthe new PPSS schedules. It had sufficientparking for the additional vehicles neededfor the one-officer cars that were "do-nated" by other stations.

Working with management we drew upa timetable for starting the new sched-ules. It required the cooperation of sev-eral other city departments: payroll had toadjust its recordkeeping procedures tocover 10-hour days, and communicationshad to assign new radio identificationcodes so that dispatchers would knowwhich officers were just starting andwhich were ending their shifts, whichcars had two officers, and so forth. The

new plan went on line Sunday, November2, 1986.

Management anticipated a problemwith convincing the city to purchaseenough extra cars to support the one-offi-cer car policy. With its perennial budgetproblems, the city would have to havestrong empirical evidence to persuade itto fund the additional vehicles. SFPDneeded to track Park's experience care-fully to convince the city to invest in newequipment. The results at Park Stationwere persuasive, and the city approvedan initial sum of $470,000 for additionalcars, added communications equipment,and more parking space at severalstations.

Acceptance of the computerized sched-uling system was facilitated by the use ofanimation and 3-D graphics for trainingpolice officers on how to exploit its capa-bilities. Graphics were also useful for ex-plaining why one set of policies wassuperior to another. Training time for thepolice officers averaged only two daysdue to the extremely intuitive interfaceand a customized animated computergraphics tutorial that took the userthrough the system's features. Also, thegraphical displays in the fine tuner ex-plaining why a decision is good or badassisted in the training. Most important,the system provided an environment inwhich trainees could leam by challengingthe system's schedules to see if theycould do better. The system's use of callsfor service as the primary definition ofwork loaci has encouraged the users tobecome more oriented towards meetingthe safety needs of San Franciscoresidents.


TAYLOR, HUXLEY

Tangible and Intangible BenefitsIntangible benefits include potential

crime reduction due to improved re-sponse times. The California CareerCriminal Apprehension Program qualifiedthe system for program funds because itaddresses three major areas: "analysissupport of patrol decision-ma king, inte-grating patrol assignments, and managingthe patrol work load," all concerned with"the day, time, location, staff allocations,and patterns for implementing tactical pa-trol activities."

SFPD used the system to evaluate andimplement new policies, converting to the4/10 plan and to a better mix of one- andtwo-officer cars. Management conductedseveral studies over a period of a yearand found that response times improveddramatically, dropping an average of 20percent for A-Priority calls {life threaten-ing situations), 18 percent for B-Prioritycalls (crime committed but no currentthreat), and 20 percent for C-Priority calls(minor citizen complaints). Additionalmeasures of improvement addressed inthe evaluation included a 21 percent in-crease in self-initiated officer activities, a32 percent increase in traffic citations,and a 36 percent reduction in days lost tosick leave. The increased traffic citationsrepresent approximately $3 million in po-tential revenues to the city treasury.

A cost benefit analysis of the schedul-ing aspects of PPSS shows a one-timecost of $50,000 and a benefit of $5.2 mil-lion per year or a payback period of 3.5days. If we include the policy changesthat required $470,000 in cars and equip-ment, the total cost of the system is$530,000. The savings from the

scheduling and policy changes is $11 mil-lion plus $3 million in increased citationrevenues for a total $14 million per year.Therefore, the payback on the entirepackage is 14 days.

In addition to tracking these statistics,SFPD management also conducted aquestionnaire survey of the officers them-selves to check their attitudes about the

A cost benefit analysis of thescheduling aspects of PPSSshows a one-time $50,000 anda benefit of $5.2 million peryear.

new schedules and their equitability. Theresponses indicated overwhelming ap-proval (96 percent). The department con-siders this improved morale an extremelyvaluable intangible benefit that affects thequality of service that police officers pro-vide to the city's residents.

Other intangible benefits include im-proved consistency of schedule quality inall precincts. PPSS also allows manage-ment to adjust schedules easily andquickly to meet the seasonal changes inthe pattern of crime. A major intangiblebenefit is that PPSS helped to resolvesome of the policy questions that for-merly consumed so much energy andtime.Further Efforts

Since the installation of PPSS, the com-putation time for the IH portion of themodel has decreased from approximately45 minutes for a 100-officer problem tousually less than 15 minutes. We have in-stalled functions that make the system

INTERFACES 19:1 20


easier to use and more powerful.PPSS spawned the development of an-

other system. Since the scheduling sys-tem addresses the "when" of patroldeployment and not the "where," in 1987we developed another system, the PatrolBeat Design System (PBDS), which rede-signs the geographic areas (beats or pa-trol car areas) that an officer patrols.Manually redesigning these areas is atime-consuming "hit or miss" chore oftentaking many people-months. Because ofthis, beats are rarely changed, with somedepartments revising them once everyfive to 10 years. Seasonal crime patternscannot be incorporated into these manualarea designs. PPSS now uses the beat de-signs from PBDS to define precinct workload. The user interface of PBDS is verysimilar to the PPSS interface, which short-ens the learning time for users of bothsystems.

Both systems are portable. The policedepartments in Sacramento, VirginiaBeach, and Richmond, California haveimplemented the scheduling and the beatdesign systems, using them on a regularbasis to redesign their substation (that is,precinct) boundaries and their patrolschedules.Summary

This project appears to be the first sig-nificant advance in police deploymenttechnology since the pioneering efforts ofthe Rand Institute in the early to mid-1970s, and based on SFPD research, it isthe only optimization-based model cur-rently used to create integer police officerwork schedules. Speed and the ability toconsider a nonlinear objective functionwith discontinuous constraints was a

major consideration in developing the IHmodel. The seamless integration of the IHmodel and the fine tuner into a fast andeasy-to-use decision support system en-courages management to evaluate deploy-ment and schedule changes. This hasraised management's perspective, takingit from speculation about the impacts ofschedule changes to a sound explorationof the best schedules and policies for de-ploying patrol officers. The implementa-tion of the Police Patrol SchedulingSystem at the San Francisco Police De-partment improved productivity by $11million per year. Response limes have de-clined by approximately 20 percent. Pro-ductivity measures are significantlyimproved, city revenues are increased by$3 million per year, and officer morale ispositive. The payback period on the proj-ect, including hardware, software, andequipment, was less than two weeks. TheSan Francisco Police Department consid-ers it "an outstanding service to thecause of law enforcement."APPENDIX

Algebraic Statement of the San FranciscoPolice Department Problem

The following indices and index sets aredefined:i = hour,; = day,h = hours in shift,d - days in shift,TA = set of start time indices i,j that are

acceptable, andS,i = set of start times that contribute per-

sonnel to the coverage of /,;' givenh,d.

The variables are defined as follows:NO,, = number of officers starting the


TAYLOR, HUXLEY

work week at /,/ and

T^the set of selected start times.

The input parameters are defined as

follows:

K = minimum number of officers on duty

any i,j,

M = smallest group size,

N/4 = number of officers available,

NP-maximum number of patrol groups,

R,, = number of officers required each

hour and day,

P = a value 3= 1 where higher values re-

flect a high degree of management

concern for large hourly shortages,

a = a value that balances the relative im-

portance of total shortage minimiza-

tion to minimization of maximum

hourly shortage.

The symbols used in defining the

model are defined as follows:

Number(s) is the number of elements in

the set s,

P,̂ = officers on duty at i,j,

U,, = shortage of officers at /,;, and

O,, = surplus of officers at /,;.

The problem is defined as follows:

minimize

subject to

,̂ + a .

Number

TeTA,

ijtT

V NO,,>O and V//.

O and integer Vij,

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travel times," Management Science, Vol. 27,No. 3 (March), pp. 213-230.

Comrie, M. and Kings, E. 1974, "Urban work-loads," Police Research Bulletin, No. 23(Spring), pp. 32-38.

Comrie, M. and Kings, E. 1974, "The flexibleuse of manpower," Police Research Bulletin,No. 25 (Autumn), pp. 27-33.

Dantzig, George 1954, "A comment on Edie's'traffic delays at toll booths, '" Operations Re-search, Vol. 2, No. 2, pp. 339-341.

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John J. Jordan, Deputy Chief of the SanFrancisco Police Department, attests that:"As both a strategic and tactical tool,PPSS helped us. We used the system as atactical tool. The precinct captains used itin scheduling their patrol officers and of-fice staffs. The schedules they producedusing the system were well received bythe officers, with sick leave decreasing byapproximately 50 percent. Response timesto calls for service decreased in the rangeof 20 percent. Citation revenues increaseddramatically, enabling the city to realizean immediate financial gain.

The system also offered us a quick and

easy way to test a variety of deploymentstrategies {4/10 vs. 5/8 plans, one vs. twoofficer cars, etc.); it helped us select thosestrategies which increased our productiv-ity in meeting calls for services by ap-proximately 25 percent. This increase inpolice coverage would have requiredmany additional officers using previousdeployment methods.

In short, we saw improvements in offi-cer morale, decreased response time, im-proved revenues, decreased officer sickleave, and the system paid for itself al-most immediately."

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Documents

A Break from Tradition for the San Francisco Police