Microcomputer-based working set generation methods for personnel scheduling

Abstract

A new approach to working set generation for personnel scheduling problems is presented. In full-time (FT) and mixed-workforce (MW) experiments, the schedules in the working sets from the use of 2-phase heuristic labor scheduling solution procedures are generated. The solution procedures were implemented on a 386 microcomputer and did not require the specification of the size of the working sets in advance. In the FT experiment, the general set-covering formulations (GscF) associated with the produced working sets were solved with integer programming. The new working set procedure yielded optimal integer solutions for all 36 test problems in the FT experiment. Owing to the size and complexity of the problem data in the MW experiment, the GSCFs associated with the working sets were solved with linear programming, and heuristic rounding procedures were applied to obtain feasible integer solutions.

Introduction

Personnel scheduling problems may be classified as days-off, shift,or tour[1 ]. Days-off scheduling isconcerned with the specification of employee non-work days across aplanning horizon[2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10 ]. Shift scheduling specifies daily startand stop times, and may also include the determination of the durationand placement of rest and/or meal breaks[11 ,12 ,13 ,14 ,15 ]. The labour tour scheduling problemconsists of the specification of both shifts and non-work days foremployees across a planning horizon which is typically one week[16 ,17 ,18 ,19 ,20 ,21 ,22 ,23 ,24 ,25 ,26 ,27 ,28 ,29 ,30 ,31 ,32 ,33 ].

It is likely that the majority of service organizations are actuallyfaced with the labour tour scheduling problem. Historically, the besttour scheduling performances, relative to a goal of minimization of thecost of scheduled labour, have been achieved by methods which have usedmini- and/or mainframe computers[20 ,21 ,23 ,24 ,25 ,26 ,27 ,28 ,29 ,33 ]. Alternatively, microcomputer-based scheduling methods typicallyhave not provided efficient labour tour scheduling solutions. This isespecially unfortunate for service organizations, since they usuallyhave neither the necessary expertise nor the mini- or mainframecomputer equipment to achieve the best solutions[19 ,20 ]. As a result, many service managersspend at least one day per week in the manual development of labourschedules[34 ].

The objective of this research was the development of efficient,microcomputer-based working set procedures for full-time (FT)and mixed-workforce (MW) tour scheduling in a discontinuous (less than24 hours/day) environment. As used in the literature, a MW is one whichencompasses both FT and part-time (PT) employees. Our previousexperience in assisting service operations managers with theirscheduling problems suggested that the new working set methodologyshould not require the service operations manager to specify a workingset size.

Unfortunately, the majority of existing working set methods dorequire the specification of the working set size. This study presentsan alternative approach in which the working set is specified as the setof tours associated with a heuristic solution to the tour-schedulingproblem. Specifically, we propose the micro-computer implementation oftwo-phased heuristic solution methods for the generation of workingsets. For the ET experiment, we utilized the two-stage solutionprocedure developed by Bechtold and Showalter[22 ]. The application of branch-and-bound integer programming(for a maximum of 1,000 integer iterations) to the resulting workingsets yielded integer-optimal solutions for all 36 FT workforce testproblems.

For the MW experiment, a two-stage solution procedure developed byBechtold and Brusco[18 ] was used togenerate the working sets for 36 test problems. Linear programming-basedheuristic methods were subsequently applied to the working set for eachtest problem. The solution costs subsequently were compared with thoseassociated with other prominent working set methods evaluated by Eastonand Rossin[25 ]. The new method producedgenerally lower solution costs in less central processing unit (CPU)time than existing methods.

The next section provides an overview of previous working setresearch and its methodological implications for this research.Subsequent sections present the tour scheduling problem formulation,describe the solution heuristics and their implementation as working setmethods, and summarize the experimental results. The article concludeswith a discussion of results and summary of conclusions.

Previous working set research

Working set methods consist of a generation phase and animplementation phase[19 ]. In thegeneration phase, a subset of schedules is selected from the master setof allowable schedules (the original problem). Optimal or heuristicprocedures subsequently are applied, in the implementation phase, toobtain labour schedules based on the tours in the working set.

In their review and comparison of working set methods, Bechtold andBrusco[19 ] classified working setgeneration procedures as either structural, demand based, or refinementin orientation. Structural methods are limited strictly to theinformation which can be extracted from the A-matrix associated with themaster set of tours. Demand-based methods utilize information derivedfrom the demand profile as well as that based on the A-matrix whichrepresents the master set of tours. Refinement procedures begin with anindependently generated working set. This set is "refined"on an iterative basis to provide a final working set which has a linearprogramming (LP) optimal solution guaranteed to be equal to thatassociated with the master set.

Structural methods

Henderson and Berry[13 ] developed thefirst structural working set generation procedure, which they applied ina shift scheduling environment. They referred to this procedure asMAXDIF since the objective was to select a set of work schedules whichwere as different from one another as possible. MAXDIF begins withan empty working set, and selects the first work schedule according tothe uniform distribution. At each subsequent iteration, the workschedule is selected which maximizes a total difference function. Tieswere broken by choosing the first of the tied tours, following thenatural order in which tours are represented in the columns of theA-matrix.

Bechtold and Brusco[19 ] subsequentlydeveloped two structural working set generation procedures SM1 and SM2,which resulted in substantially better solutions, in a tour schedulingenvironment, than those associated with MAXDIF. SM1 is a single-phaseprocess which used the A-Matrix of the tour scheduling problem forinput. SM2 uses separate phases to develop a set of days-offcombinations and a set of shifts. These two sets are then used to createa working set of tours in a third phase. Overall, the best performancein the Bechtold and Brusco[19 ] study wasachieved by SM2.

Demand-based methods

Mabert and Watts[31 ] developed thefirst demand-based methods for working set generation. These methodsutilized biased sampling and integer linear programming (ILP) in twophases for the tour scheduling of a PT workforce. Tours were specifiedfor the working set by specification of the start times and work days inthe first and second phases, respectively. Although a variety ofgeneration methods was evaluated, the most efficient results wereobtained with their SSUP and TWAP procedures.

Bechtold and Brusco[19 ] subsequentlydeveloped a dynamic demand-based working set generation method (DDM).The term dynamic refers to the continuous adjustment of the demandprofile during the generation of the working set. In a recent study, DDMoutperformed both the SSUP and TWAP procedures of Mabert and Watts[31 ].

Working set refinement

Easton and Rossin[25 ] were the firstto develop a refinement method. This was a creative piece of researchwhich used mathematical programming to select the tours to be includedin the final working set. This procedure was dependent on an initialworking set which was required to include tours sufficient to allow afeasible solution. Easton and Rossin developed this initial set throughthe sequential addition of tours, taken in their natural order. Theyreferred to their final working set as a "sufficient workingset" (SWS) since it resulted in a set of tours which wassufficient to obtain an optimal LP solution. Hereafter, SWSLP will beused to refer to this refinement procedure. In their MW experiment, theSWSLP heuristic outperformed MAXDIF and two versions of SSUP (SSUP-O andSSUP-H). We utilized Easton and Rossin's results for our performancecomparison.

Methodological decisions

Our requirement that the size of the working set be determinedprocedurally was quite restrictive since it eliminated the use ofstructural generation methods. Moreover, the size of the working setmust also be specified for all of the Mabert and Watts[31 ] generation procedures. Indeed, at thepoint in time when this research was initiated, Easton andRossin's[25 ] SWSLP refinement procedurewas the only one which procedurally determined the size of the workingset.

Given the effectiveness of SWSLP, it seemed appropriate to look foralternatives which would determine simultaneously the size andcomposition of the working set. It was at this point that we realizedthat heuristic solution methods were being used to determine the numberof employees which should be assigned to a specific set of active tours.Thus, the working sets in our new methodology are composed of the activetours in the heuristic solution.

The objective of being able to implement the new procedures on amicro-computer suggested the use of solution heuristics which solve thetour scheduling problem using both a shift and days-off phase. Thus, wedecided to use an FT heuristic developed by Bechtold and Showalter[22 ] (hereafter, the BS heuristic) and a MWheuristic developed by Bechtold and Brusco[18 ] (hereafter, the BB heuristic). These two methods are bothtwo-phased processes which produce efficient solutions.

Personnel tour scheduling

A substantial body of research has focused on the discontinuous tourscheduling environment[18 ,19 ,20 ,21 ,22 ,24 ,25 ,26 ,27 ,28 ,30 ,31 ,33 ]. Moreover,the characteristics of this environment were most representative ofactual labour scheduling conditions for the vast majority of serviceoperations. Therefore, this environment was selected to provide thecontext for our current research.

In the discontinuous tour scheduling environment, the organizationoperates less than 24 hours per day, and employees typically arescheduled to staff labour requirements, without under staffing, across aweekly planning horizon with the objective of minimizing labour costs.Further complexity is introduced when there are multiple employeecategories with varying levels of cost and productivity. In the lattercase, it is likely that organizational constraints on the mix ofemployees would be specified.

General tour scheduling model

Dantzig[12 ] was the first to suggestthat a general set-covering formulation (GSCF) could be used torepresent labour scheduling problems. Easton and Rossin[25 ] presented a GSCF which can be used tomodel the discontinuous tour scheduling environment. Using notationconsistent with their work[25 ], wepresent a GSCF for the tour scheduling problem considered in this study.The ILP formulation for this problems is: Equation (1)

subject to: Equation (2) ,Equation (3) ,Equation (4) andEquation (5)

where Equation (6)

Environment and problem set for the FT experiment

The FT scheduling environment was originally developed by Bechtoldet al. [20 ] and subsequently usedby Bechtold and Brusco[19 ] in theirevaluation of working set methods. The environment operates 16 hours perday, seven days per week. Labour requirements are satisfied by FTemployees working nine-hour shifts with a one-hour meal break betweenthe fourth and fifth hours of work. FT employees work five days perweek; however, their days-off need not be consecutive. The master setfor the FT environment contains 168 tours (eight shift-starting times x21 days-off combinations).

In their study, Bechtold and Brusco[19 ] used 36 test problems differentiated by the distribution oflabour requirements. These distributions, originally developed byBechtold and Showalter[22 ], arerepresentative of a wide range of demand patterns across the day and theweek.

Environment and problem set for the MW experiment

The MW scheduling environment was developed in Easton and Rossin'swork[25 ] and encompasses both FT and PTemployees with varying levels of cost and productivity. An additionalconstraint in this environment was the restriction of PT employees to amaximum of 75 per cent of the total workforce. FT employees worked fiveconsecutive days per week. Their shifts were nine hours in length with aone-hour meal break between the fourth and fifth hour of work. PTemployees worked four-hour shifts, without breaks, with allowed tourlengths of two, three, four or five days per week. The productivity andcost coefficients for each class of PT employee are presented inTable I . The resulting master setcontained 1,239 allowable tours. The 36 MW test problems have the samelabour requirements distributions found in Bechtold and Brusco'swork[19 ].

Working set methods

Working set method for the FT environment

Working set generation and implementation . The BS heuristicis a two-stage solution heuristic which does not require mathematicalprogramming. In the first stage, a rule-based procedure which promotesthe utilization of the earliest and latest shift starting times is usedto solve the shift-scheduling problem associated with each operatingday. In the second stage, an optimal days-off algorithm developed byBechtold[17 ] is used to solve thedays-off problem for each shift-starting time which has at least oneemployee assignment. A detailed description of the BS heuristic isprovided by Bechtold and Showalter[22 ].

The working sets for each test problem were composed of the activetours in the BS heuristic solution. Hereafter, we refer to the use ofthe BS heuristic as a working set generation procedure as BSG. The GSCFassociated with the working set for each of the 36 test problems wasformulated and integer solutions were obtained using the LINDO[35 ] branch-and-bound routine. Thebranch-and-bound routine was limited to a maximum of 1,000 integeriterations.

Computational procedures and performance benchmarks . All BSGheuristic results for the FT experiment were obtained with the 386microcomputer (with a 387 mathematics co-processor). The BS heuristicand working set input programs were written in BASIC.

Integer optimal solutions associated with the master set of tourswere obtained, for all 36 test problems, by Bechtold and Brusco[19 ] using the GOMORY option of theMulti-Purpose Optimization System (MPOS)[36 ]. This cutting-plane package was implemented on a CDC CYBER 830mainframe computer.

Working set method for the MW environment

Working set generation and implementation . The BB heuristicis a two-stage solution heuristic based on principles originallydeveloped by Bechtold and Showalter[22 ].In the first stage, shift scheduling solutions are developed, for eachday of the week, using a pre-emptive goal programming model (PGPM). Theprimary goal of the PGPM is the minimization of shift scheduling cost.The secondary goal is the utilization of the earliest and latestallowable shift starting times. If the solution to PGPM contains anyfractional decision variables, heuristic rounding[30 ] and improvement[13 ]procedures are applied to obtain feasible integer solutions.

In the second stage, optimal days-off algorithms[5 ,17 ] are applied across eachshift-starting time which has at least one employee assignment. Theintegration of shifts with days-off was conducted in precisely the samemanner as described by Bechtold and Showalter[22 ]. Specifically, tours are specified as a direct result ofthe days-off solution since all shifts within a tour are required tohave the same length and start time. For a complete description of theBB heuristic, refer to Bechtold and Brusco's work[18 ].

The working sets for each test problem were composed of the activetours in the BB heuristic solution. Hereafter, we refer to the use ofthe BB heuristic as a working set generation procedure as BBG. The GSCFassociated with the working set for each of the 36 test problems wasformulated. Unfortunately, we were unable to solve these problems usingthe LINDO[25 ] branch-and-bound code.Therefore, we solved the LP-relaxation of the GSCF associated with eachworking set. The round-down-build-up (RDBU) procedure[30 ] was subsequently applied to obtainfeasible tour scheduling solutions. This was followed by a dropprocedure which examined tours, in their natural order, to determine ifan employee could be dropped from a tour without creating anyunderstaffing.

Computational procedures and performance benchmarks for BBG .All BBG heuristic results for the MW experiment were obtained with the386 microcomputer (with a 387 mathematics co-processor) as follows:*

All LP problems were solved using LINDO[35 ].

*

The RDBU and drop heuristic, as well as all input generationprograms were written and executed in BASIC.

Benchmark solutions for the MW test problems were obtained by Eastonand Rossin[25 ] using a VAX mini-computer.Integerization and improvement were accomplished via the application ofRDBU and the linear-programming-vector-exchange (LPVE) procedure[13 ] to the LP solution of the GSCF associatedwith the master set.

Experimental results

Results for the FT experiment

A comparison of the performances of BSG with the procedures evaluatedby Bechtold and Brusco[19 ] is containedin Table II . The BSG heuristicgenerated working sets which resulted in global ILP optimal solutionsfor all 36 FT workforce test problems. It should be noted that thebranch-and-bound routine confirmed only 16 of the 36 solutionsas optimal within the integer iteration limit. For the remaining 20problems, an integer optimal solution was identified within 1,000integer iterations. This identification was based on the list of integeroptimal solutions found by MPOS. The only other method of Bechtold andBrusco[19 ] which yielded 36 optimalsolutions was SM2. However, unlike BSG, this method requires the workingset size to be predetermined.

The BSG procedure compares very favourably with the methods ofBechtold and Brusco[19 ] with respect tothe mean CPU time required to generate the working sets. The total CPUtime for BSG is inflated by the large CPU times associated with thebranch-and-bound algorithm in the implementation phase. The BSGheuristic generated an average of 14.8 fewer tours than the 50 toursallowed for MAXDIF, SM1, SM2, SSUP, and TWAP procedures, and one lessthan that of the DDM procedure. Across the 36 test problems, BSGgenerated a minimum of 21 and maximum of 53 tours. It is important torecognize the difference between the number of tours in the working setand the number of active (x_j > 0) tours in the final ILPsolution. The mean working set size for BSG was 35.2 tours; however, theILP solutions had a mean of only 25.2 active tours.

Results for the MW experiment

A comparison of the performances of BBG with the procedures evaluatedin Easton and Rossin's work[25 ] iscontained in Table III . The BBGheuristic generated working sets which resulted in near-optimalrelaxed-LP solutions for the MW test problems. Encouraging results wereprovided by the integer (RDBU/LPVE) solution summaries inTable III . The mean RDBU/LPVE solutioncost associated with the BBG procedure was superior to those produced byall other working set procedures of Easton and Rossin[25 ]. This cost was also 0.69 per cent less(better) than the RDBU/LPVE mean solution cost associated with themaster set.

Owing to differences in computer hardware and software, a directcomparison of CPU times was not possible. However, the CPU timesassociated with BBG, which averaged less than 1.5 minutes on amicrocomputer, appear to compare very favourably with the VAX 8800mini-computer working set generation times reported by Easton andRossin[25 ]. The BBG heuristic generatedan average of 41 fewer tours than the 100 tours allowed for MAXDIF,SSUP-O, and SSUP-H procedures, and 11 more tours than the mean of 48generated by the SWSLP procedure. Across the 36 test problems, BBGgenerated a minimum of 41 and a maximum of 76 tours. The mean workingset size for BBG was 59 tours, however, the RDBU/LPVE solutions had amean of only 48 active tours.

Discussion and conclusions

Relative performance in the FT experiment

The BSG working set method yielded a mean solution cost equivalent tothat of the best performing method of Bechtold and Brusco[19 ], SM2. It is clear that both these methodsare very efficient and amenable to microcomputer implementation. Theprimary advantage of BSG is that it does not require the working setsize to be specified in advance.

Relative performance in the MW experiment

The BBG working set method resulted in a lower mean labour cost thanall other methods. Specifically, the mean labour scheduling cost of BBGwas 7.8, 4.4, 3.9, and 0.7 per cent less than those associated withMAXDIF, SSUP-O, SSUP-H, and SWSLP, respectively. Moreover, the BBGprocedure was substantially more efficient than the other methodsincluded in this study. Specifically, the mean CPU time for the methodsof Easton and Rossin[25 ] ranged from69.05 (SSUP-H) to 10,007.66 (SSUP-O) on a VAX 8800 mini-computer, whileBBG required a mean CPU time of 88.12 seconds on a 386 microcomputer(with a 387 mathematics co-processor).

Numbers of tours within the working sets

The number of tours in the working sets is important for severalreasons. A larger number of tours in the working set increases thelikelihood of solutions which offer work schedules that are preferableto the employees. While this is likely to reduce the difficulty ofassigning employees to tours, it may increase the manager's workloadwith respect to monitoring the arrival and departure of employees. Anin-depth discussion of these and other issues is provided by Bechtoldet al. [20 ].

The number of tours in the working sets produced by the SWSLP and BBGmethods appeared to be within reasonable bounds. In general, the workingsets associated with BBG were somewhat larger than those produced bySWSLP. It has been noted that the number of tours in a working set maybe increased or decreased through the application of structural workingset methods[19 ].

Use of new working set methods to schedule an existing,heterogeneous workforce

This process need not involve any modification of the new method; itcan be initiated by exchanging the master set of tours preferred bymanagement with the set of tours desired/required by the currentworkforce. If this latter set of tours does not allow a feasiblesolution to be obtained, it can be augmented with additional tours whichare preferred by management. These results should be compared with thoseof the general employee scheduling methods of Love and Hoey[34 ], and Glover and McMillan[37 ].

Conclusions

The new working set generation procedure, as implemented on a 386microcomputer, achieved very impressive results. In the FT workforceexperiment, the BSG procedure yielded optimal solutions to all 36 testproblems while using generally smaller working sets than the methods ofBechtold and Brusco[19 ]. In the MWexperiment, the BBG procedure produced lower mean scheduled labour costand required considerably less CPU time than all the other methodscompared by Easton and Rossin[25 ].

Table I. Mixed workforce environment

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Source : [25 ]

Table II. Performance summary for the FT experiment

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Table III. Performance summary for the MW experiment

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Equation (1)

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Equation (2)

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Equation (3)

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Equation (4)

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Equation (5)

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Equation (6)

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References

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