The scheduler must determine the shift types (and the number of each type to .... or machine-must be allowed to edit a file of empl~yee availabilities before addressing the .... "AI/human" intuitive techniques, though sometimes in large numbers, ...
Comput.& Ops. Res.Vol. 13, No.5, pp. 563-573, 1986 Printed in Great Britain.
0305-0548/86$3.00+0.00
All rights reserved
Copyright
(i:) 1986 Pergamon Journals Lid
THE GENERAL EMPLOYEE SCHEDULING PROBLEM: AN INTEGRAnON OF MS AND AI FRED GLOYER* and CLAUDE McMILLANt Center for Applied Artificial Intelligence,Graduate School of Business,University of Colorado, Boulder,CO 803~, U.S.A. Scopeand Purpose- In certain settingsthe routine schedulingof employeesprovesto bea complex problem in combinatorics. During the past 12yearsmost efforts to automateschedulinghaverelied chiefly on one or another variant of linear programming. We show that thoseapproachesare disappointing in thecaseof the generalemployeeschedulingproblem, sincemore than 4 million integervariablesare involved. On the other hand, by relying on an integration of appropriate techniquesfrom both managementscienceand artificial intelligencewe render the generalproblem thoroughly tractable, permitting the routing schedulingof over 100employeesfor one weekon an IBM PC in minutes. In this paperwe speakof the approach which makes that possible. Abstract- The generalemployee schedulingproblem extends the standard shift scheduling problem by discardingkey limitations suchas employeehomogeneityand the absenceof connectionsacrosstime period blocks. The resulting increasedgenerality yields a schedulingmodel that applies to real world problems confronted in a wide variety of areas. The price of the increasedgenerality is a marked increase in size and complexity over related models reported in the literature. The integer programming formulation for the general employee scheduling problem, arising in typical real world settings,contains from one million to over four million zero~ne variables. By contrast, studies of specialcasesreported over the past decadehave focusedon problems involving between 100 and 500variables. Wecharacterizethe relationship betweenthe generalemployeeschedulingproblemand related problems, reporting computational resultsfor a procedurethat solvesthesemore complexproblems within 98-99 % optimality and runs on a microcomputer. We view our approach as an integration of managementscience and artificial intelligencetechniques.The benefitsof suchan integration are suggestedby the fact that other zero~ne schedulingimplementationsreported in the literature, including the one awarded the Lancaster Prize in 1984,haveobtainedcomparableapproximations of optimality only for problems from two to three orders of magnitude smaller, and then only by the use of large mainframecomputers.
INTRODUCTION
Employeeschedulingproblemsarise in a variety of servicedeliverysettings,including the scheduling of nursesin hospitals, checkencodersin banks, airline and hotel reservationpersonnel,telephone operators, patrol officers and others. In their simplest form, these problems involve only the assignmentof days-off,as in someof the lesscomplexsettingsfor the schedulingof nurses.A typical problem of this form requires the schedulerto give appropriate days off to each of a number of employeeswho work standard shifts with differing start times while assuring that the required number of employeesare on duty throughout the day and week. Variations of this type were addressedin Brownell and Lawrence [1] and Keith [2]. The shift schedulingproblem,as in the schedulingof telephoneoperators,is more complex. In shift scheduling,the schedulerworks with part-time as well asfull-time employees,and shift typescontrast with each other in the following attributes: (1) durati?n (length) (2) start tImes (3) the number of breaks (reliefs) (4) the placementof the breaks.
:";'
1
I
The schedulermustdeterminethe shift types(and the numberof eachtype to employ),and in some casesdetermine which employeeshould receivewhich setof shifts. Union rules and company policy
*Fred Glover received the doctorate from Carnegie-Mellon in 1965,and served on the faculty at the University of California, Berkeley,and the University of Texasand the University of Minnesota. He is the author of over 150published papers in ManagementScienceand related fields. tClaude McMillan receivedthe doctorate from Ohio State University in 1955,and servedon the faculty at Michigan State University until 1965.He is the author or coauthor of a number of papersand four books in the field of managementscience.
563
564
FR'EDGLOVER and CLAUDE McMILLAN
",t,;,,;o", are handlod;na I;m;too '"h;on V',;at;o", on th;, problem ha', "," ,dd"",";" .,r, [2-8] Th,obj"t;,,;" tho ,h;ft "hoo";"g problem g,"",lIy;, to appro,;,.,re ""'"'oIy a, po,,;blo tho d,,;,oo n"mb" of,mploy", M d"ty ';th" by m;";m;';"8 th,0",a8' 0' m;";m;,;"gth,'hortag'/ o"'ag'" m;,
Th, morecomplo,day,~lIa"d ,hi/\ "h,d";"gproblom [7] com" "t,p""'" to 'h, ,'"'rnl ,mploy" ',hOO";", probl=, ,"d rom, 'a,;at;o", [3, 4, 8],o",;d" alrotho",';gnm'"t of,hifu to ,""al,mploy", botd,,1 w;th th,t ",,;g"m'"t o',h;fu to ,mploY"'Mly aft',th"h;ft typ" (,"d ""mb" of ""h) ha" boo"d,t,=;"OO T,bl, I "",;fi" ,h;/\ ~hoo"li"g problom,", theyared,~,;b,d;" tholi""till', ,"d ;"d;"", tho d;lI,re"", ;" problem,;" a"d compl";ty tho 'ppro"h employed,a"d the"t,", to wh;,hthe appro"h w", ,""ally bci", """ ;" ;"d","y
THE GENERAL
EMPLOYEE
SCHEIDULING
PROBLEM
The more complexschedulingproblem which we addr~s has wide applicability, especiallyin the supermarket,reservationoffice and fast food fields. It differs rather dramatically from the days-off and the shift schedulingproblems by including important real world features that resist practical solution by methods of formal analysis.We first describethe problem informally and indicate the featuresthat a practical solution systemmust have in ord,r to deal with the problem effectively.For comparativereference,weindicate overlapsand contrastslwith other employeeschedulingproblems previously examined.We report the result of applying o~r approach to problems from real world settingsand discussthe implications of our empirical re~ults.
FULL-TIMEjPART-TIME
EiMPLOYEES
As in the shift schedulingproblem, it is assumedin the generalemployeeschedulingproblem that some fraction of the work will be done by full-time employeesand the remainder by part-time employees.Full-time employeesare those entitled to work a staI)dard number of hours each week (commonly 40) and generallywork shifts of standard duration (commonly 8 hours each).The start
High:
Ours> 1,000,000shifts
Heuristics-blend or
MSjAI
540
Supennkt Fast food
Definitions: LOW means: no linking constraints betweenblocks of time periods, a small number of shift types, and homogeneousemployeeswith unlimited availabilities. MEDIUM availabilities,
means: no linking constraints between blocks of time, a small number of shift types, homogeneous employees with unlimited and management rules.
HIGH means: linking constraints betweenblocks of time periods,a large numb~r of shift types, non-homogeneousemployeeswith limited availabilities, and managementrules. U. formula (uniform formula) means: each shift is characterized by the same rule, such as: every employee works 5 days a week (time periods are in days).
The word "shift" in the foregoingtable translatesinto the word "variable" in an integerprogrammingformulation. (Somereferencesusethe word "trick" rather than "shift".)
'he general employee schedulililg problem
565
times and the numberand location of breaksmay vary. In $.ddition,a full-time employeemayor may not be entitled to the same start time each day worked. In the general problem, as handled by our method, the useris given the ability to specify shift featuressuchas duration, start time, and the numberand placementof breaks.The useris also ableto specifythe number of days off and whether they should beiconsecutive,and to changethosefeatures as the systemis routinely used from week to week. In addition to the above,eachemployeecan specify(or t~e schedulercan specifyfor the employee): (1) Minimum and maximum hours to be worked durihg the week. (2) Days and hours of availability during the week. Item (2) above meansthat eachemployeehas an availability preference,specifying which days of the weekand the hours that employeecan work. Sinceaniemployee'savailability changes(in some settings,about 20% of the work force changestheir availabilities eachweek),the scheduler-human or machine-must be allowed to edit a file of empl~yee availabilities before addressingthe composition of a new scheduleeachweek. i Another important featureof the generalproblem, not treated in the standardschedulingcontexts, is that employeesare non-homogeneousin ways beyon~ their availability preferencesand thus cannot be treated as interchangea?leentities. EmployeesIhavediffering skill types, skill levelsand status attributes which limit the scheduler'sfreedomin as~igningshifts. Theseinclude, for example, the training required to work at various work stations. ~ince thesechange from time to time, the systemmust allow the schedulerto edit eachemployee'sIprofile data. In the generalemployeeschedulingproblem, it may al$ be necessaryto observeseniority rules, suchas those specifyingthat employeeswith more seniority must get 1l1l0re hours of work and start earlier (an early start may be considered desirable), exdept when other requirements would be I
violated. Union/managementrules,in the generalproblem, canr~uire that a specifiedminimum amountof time must elapsebetweenthe time an employeeworks on~ day and begins again the next. In some settings they may also require that certain employees,S4chas students, may not work beyond a specified hour more than one night during the week. i
IMPLICATIONS
OF THE NON-~OMOGENEOUS
EMPLOYEE
POOL
In the simpler shift schedulingproblems,it is possibleto ttesignshifts {and to determinethe desired numberof eachshift type) without regardto employeeavailabilities,employeeskills and skill levelsor employeestatus. Descriptionsof the shift schedulingproblem in the literature, on the other hand, sometimesappearto involve a more generalsolution capability-for example,indicating that shifts are generatedto conformto union rules, companypolicy, etc. What this means,in practice,is that the shifts generatedrepresentcategoriesthat are potentially acceptable,but there is no control over whetherthose selectedas a "solution" have an appropriate composition. This is entirely reasonable for settingswhere restrictions are loose enough that employeescan simply "sign-up" for whatever scheduleis posted, but in broader settings, sucha disregard of individual differencescan have dire
consequences.
In the general schedulingproblem, therefore,the desi~nof the shifts and their assignmentto specificemployeesmustbecoordinated.Designingshifts w\thout consideringwhetherempolyeescan be found to take those shifts will yield either a poor fit (a poor match of employeesassignedto employeeswanted on duty) or, still worse, an infeasiblesolution. LINKING
CONSTRAINTS
BETWEEN
l1lME PERIOD
BLOCKS
Another significant aspectof the generalemployeesch~uling problcm is the existenceof linking constraints betweentime periods. In typical applications pf the generalproblem, each day may be construedasa block consistingof96 fifteen-minuteperiod~,and the assignmentof employeesto duty periodsduring that block is not independentof assignmentsto employeesin other blocks. In addition to restrictions governing admissibleassignmentson any given day, theselinking constraints imply that there are also restrictionsgoverningthe total numberof periods throughout the entire week,as
566
FRED GLOVER and CLAUDE McMILLAN
well as governing the selectionof certain types of assignmentssuccessively,or cumulatively, across days of the week,e.g. limitations on the selectionof opening and closing assignments. In standard shift schedulingproblems,by contrast, blocks suchasdaysare construedas essentially independent.Linking conditions either do not exist or are innocuous enoughto be ignored while composinga schedulefor any given day (where,for example\the endingconditions for one day may be used to give starting conditions for the next, which again is treated independently). The days-off and shift schedulingproblems can be readily treated as specialcasesof the general problem,allowing many of the more difficult conditions to be relaxed. By assuming,during the shift designphase,a universe of homogeneousemployeeswithout linking constraints across blocks of time, the shift design problem is made relatively simple. Thus a systemfor the generalscheduling problem handlestheselessrestrictive problems as a special!case. THE REAL WORLD SETlfING
In real world settings,the manual scheduler(generallya supervisor)works in a highly dynamic mode. Each week,and sometimesmore frequently, a new schedulemust be created to reflect the altered employeeavailabilities and changesin the forecastedvolume of business.To control costs, managementfrequently requires that the person-hoursof work assignedmust not call for a dollar expenditure out of proportion to the dollar salesforecasted. The manual production of a schedulethat respectslimited and varying employeeavailabilities,and yet matches the requirements,is very difficult. The consequenceis that the manually produced schedulegenerallycalls for overages(too many on duty) at certain times during the day and week, and- to keeplabor costswithin acceptableboundaries- producescorrespondingshortages(too few on duty) in others. This results in poor service:a condition which managersmust seekactively to avoid in highly competitive serviceindustries. Producinga schedulemanuallyalso requiresa good dealo£time. For example,in the supermarket and fastfood industries,our investigationsindicate that it takes from 8 to 14hours for a managerto schedulefrom 70 to 100employeesfor one week,dependingon the seriousnessdevotedto the task. The resulting scheduleis likely to be substantiallylessthan optimal. Evenwhen all specialconditions may be met (which is often not the casefor schedulesproduced manually), shortagesand overages often combine to yield less than desirableserviceor an inflated payroll. IMPLEMENTING
OUR APP~OACH
To apply our solution method to the generalemployeeschedulingproblem. we designshifts with deferenceto featuresspecifiedby the user,and with deferenceto employeeavailabilities. Whena shift is selected,to augmentthe growing set of shifts which is generated with the goal of meeting the requirements,the identity of the employeeto take that shift is specified.This assuresthat employees are only assignedshifts theyareavailablefor. In casethe problemlacksa schedulethat is feasiblein all respects,our approachgeneratesa schedulethat neverthelesscomesascloseas possibleto achieving feasibility, then helpsthe useridentify alternative ways to deal with the limitations that createdthe infeasiblesituations. As we subsequentlydocument, our proceduresucceedsin solving problems in the range of 1000 times larger than those related scheduling problems previously studied in the literature. Fundamentally,we view our procedureas an integration of managementscience(MS) and artificial intelligence(AI). Among the levels of proceduralgeneralityto which suchan integration is relevant, from the micro levelof computer codingto the macro levelof global strategies,it is the higherlevels that havethe greatestimpact on solution quality and efficienc)j,and accountfor what we believemay be unique to our approach. Evidently, orders of magnitude of difference in the size of combinatorial problems that are successfully treated cannot be explained by clever coJDPuter coding. Intermediate level considerationsof specificchoicerules are more relevantto achieving suchsuccesses. The "structure" on which the primary choice rules are superimposedconsists of a procedure for building and amendingemployeeshifts which hasits roots in the alternating assignmentideasof Glover [9], and which is characterized more broadly in the context of tabu search in Glover [10]. For this application, we conceptually view each stage of generating!a partial (or complete) set of duty
The generalemployeeschedulingiproblem
567
assignments as creating a trial solution, which is modified or elaborated by transition rules: a standard framework. (Those interested in further details of the system at an intermediate level of implementation are invited to contact the authors.) The k~y ingredient of our approach lies in the macro level strategies, which combine the perspectives of management science and artificial intelligence in ways not commonly done. At this level, our approach consists of three main components. First, following an MS based perspective, we develop nu~erical criteria for evaluating the moves that define possible transitions from one trial solution to apother. We do not, however, settle ona single criterion for evaluating a particular type of move. In*tead, interlinking criteria and are based on separate evaluation functions are generated which reflect both feasibility and optimality goals. Our use of multiple evaluative criteria presents a new difficulty that MS based heuristics traditionally have not had to face: the fact that a local optimum relative to one criterion may not be a local opt~mum relativ~ to ano~her. Rather than a st.umbling blockl, we found this difficulty to be a source of fertile opportUnIty, leading to ways out of blInd alleys encountered by other procedures. To exploit this opportunity, we formulated our approach so that e~ch criterion was allowed to "vote" on alternative moves, initially assigning equal weight to the different votes. When a "deadlock" (local optimum) was reached, we increased the weight of those votes that would find a different solution preferable, thereby allowing the procedure to find new trial solutions. The procedure was further endowed with a memory to prevent reversing the direction in which weights were changed, but allowing the memory to decay so that choices would not be unduly influenced by decisions that should be regarded ancient history. This method for managing memory was implemented by the use of tabu lists as described in Glover [1(;1]and Glover et at. [11]. Decay factors that "forgot" decisions beyond five to twelve moves earlier all succeededin avoiding cycling and producing good choices. The combined effect of these features is implicitly to create a tolerance for "bad moves", but only to the extent required to avoid becoming mired down at local optima, resulting in a highly effective strategy. Our second main component employed on AI based perspective to identify patterns in the ways trial solutions are configured. However, in a departure from standard AI perspective, our goal was not merely to recognize patterns but to create them. In developing this approach, we were guided by the supposition that certain configurations-as manifested in the magnitude and distribution of residual requirements, and available means of meeting them-would ultimately lead to better solutions by our tools of analysis than others. Thus, we adopted the goal of identifying moves that would lead to configurations we judged intuitively promising. This led to creating additional criteria, based on means-end analysis to arrive at exploitable patterns, independent of whether moves to attain these patterns might contribute to the objectives embodied in other criteria. These new criteria were then incorporated into our first component strategy. The third major component of our approach was to iden~ify significant segments of the problem, and then to subject each segment to its own sequence o1 solution phases. In this approach, we implicitly perform what might be called a conceptual d omposition. While the problem is an indivisible whole, we nevertheless artifically break it apa~t. After each round of evaluations and modifications, we put it all back together-a "Humpty Dumpty" process that characteristically yields gains throughout several repetitions. For example, a~tera global evaluation of an employee's availability versus needs yet to be filled over all periods (rel*tive to the current state of the solution), bias factors are generated for and against scheduling th~ employee in particular period blocks. Thereupon, the blocks of time periods are treated as t~ough independent for the purpose of generating the next move. The succeeding global evaluatio~ then fulfills the function of restoring the links between different blocks. A similar procedure is applie~ to meeting union and seniority rules as the solution process evolves, temporarily decoupling theseiconsiderations by bias factors and then restoring them by global review. ; Viewing our approach in terms of these macro level strategies, it is clear there is nothing to the AI nor MS oriented approaches that would not, in theory, provide support for our undertaking. In practice, however, AI and MS approaches are usually imp~emented in a narrower fashion. It is our impression that the approaches most often described in the literature either employ rather shallow "AI/human" intuitive techniques, though sometimes in large numbers, or employ slightly deeper MS heuristic techniques, but in very small numbers. In neither instance do we find great interlinking and overlaying of alternative criteria. Certainly, we have not seenwidespread implementation of multiple
568
FRED GLOVER and CLA~DE MCMILLAN
evaluation functions,integrated and controlled to ov~rcomelocal optimality and cycling, nor have we seenthe active useof pattern creation (in addition to recognition) using notions of exploitability instead of objectivegain. Formal mathematicaldecomposition,although too rigid and inapplicable to discreteproblems of the type we examine,has sompresemblanceto the third componentof our approach, if one views the decompositionas susceptipleto being carried out in different ways. THE QUALITY
OF THIE SCHEDUljE
We describenow the resultsof 10 computer solutiob tests, producing one schedulefor one week, for eachof 10 restaurantproblems,involving 100emplfyees,from a real world application in the fast food industry. Thesetestswereconducted on problemsselectedfor benchmarkruns by McDonald's Corporation Headquarters, Oak Brook, Illinois. These problems correspond to integer programming problems involving roughly from 1,000,000to 4,000,000variables,and from 3400to 9000 constraints, as noted in the integer program~ing formulation described subsequently. A summary of the test results is provided in Table 2. ! The percentof optimality figuresindicated in Table :1werearrived at asfollows. With eachrun, no shortageswere produced. That is, during no 15-minu~eperiod during the weekwas the number of employeesassignedto be on duty lessthan the desirednumber. In one run, during three I5-minute periods throughout the week there was an excessof Iione employee,over and above the number desired.This doesnot meanthat feweremployeescould be used,sinceeliminating anemployeewould then create shortages in all other I5-minute periods!the employee worked (assumingthe same employeewas on duty in thesethree I5-minute periodsduring the week).In two other runs, there was an excessof one employee,during eight I5-minute periods during the week, over and above the number desired. For the other seventhe averagewas ,betweenthree and eight. In a perfectschedule,the shortagesand the overage~would all be zero for all periods during the week. In the ten problems tested,periods designatedifor schedulingequalled 540 (out of 672 15minute periods for the week).Thus, in the worst casequr method achievedzero shortagesand zero overagesfor 98 % (532 out of 540)of the total periodslunderconsideration. We did not attempt to obtain the very bestsolutionspossibleby our approac~(if better solutions did in fact exist), but used an automatic cut-off rule to terminate search, which !accountsfor the similarity of run times on problems of different sizes. : In the paperscited that report on an application inl a practical setting, comparisonsof solution quality are difficult due to different types of objective functions and a frequent lack of explicit performancedata. Partial evidenceof the quality of performanceis offered in Segal[8] by reporting executiontime on an IBM 360/67,which ranged from about 40secondsto something over a minute (the maximum was not specified) for problems with 300 to 400 variables. The other papers, addressinga problem that comes closestto our general problem structure [2, 3], likewise report reliance on a mainframe computer, but do not offer computation times. In typical real world settingsfor fastfood stores (supermarkets,banks, reservationofficesand the
Table 2. Test results in a t~n casestudy
I
2 3 4
5 6 7 8 9 10
24 min 24 min 23 min 24 min 23 min 23 min 23 min 22 min 24 min 22 min
"All runs were executed on an IBM PC, 128 K memory. +Determined by program counters, explained in formulation
2,64p,250 I ,35! 1,15 ,772 ,117
3,25
,222
2,.14
.605
4",9~,580 I ,j6~.1 4,005,202 2,613,800 1.215,641
section.
~Solutions verified to be at least the indicated percent of optimality.
00
3400 3400 3400 4732 4732 4732 4732 4732 9004 9004
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CURRENTUSAPE i
Our systemhas beenused in severalsettings for overi2 years, producing schedulessubstantially superior to those an experiencedschedulercanprOducef' any amount of time. Applying our system to problems involving approx. 100 employees,the useris able in one to two hours to update the forecast for the coming week, to modify the employ availabilities, and to make "manual" assignments(usingthe computer)to selectedpersonnelf r whom specificwork schedulesare wanted (e.g. managers).The computer then completesthe pr4cess by designing and assigning shifts to approximate the optimal match of employeeson duty io employeeswanted, while respectingthe constraints describedabove. Figure 1 showsone of the printed outputs which the usermay select, indicating by meansof a bar graph the shifts assignedto employeeson Sunday. No human intervention is required during the scheduling.Disregarding the improvement in the schedulesproduced,the processtrims 6 to 12hoursoffth9 time required eachweekfor a supervisorto preparea schedulemanually,and this labor savingsby itsFlf can pay for the computerin a reasonable period of time. THE INTEGER PROGRAMMINO FORMULATION I
A moredetaileddescriptionof the conditions whichwererespectedin the solutionscited above (the schedulesproduced using our approach)is as follows: : !
(1) The numberof employeeson duty (and not takingja lunch or quarter-hour break) in each15-
570
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