variable employee productivity in workforce scheduling

European Journal of Operational Research 170 (2006) 376–390

www.elsevier.com/locate/ejor

Discrete Optimization

Variable employee productivity in workforce scheduling

Gary M. Thompson a,1, John C. Goodale b,*

a Operations Management, School of Hotel Administration, Cornell University, Ithaca, NY 14853, USAb Department of Decision Sciences, Lundquist College of Business, University of Oregon, Eugene, OR 97403-1208, USA

Received 9 May 2000; accepted 31 March 2004Available online 17 September 2004

Abstract

This paper considers the problem of developing workforce schedules using groups of employees having different pro-ductivity. We show that the existing linear representation of this problem is often inaccurate for high-contact serviceorganizations because it ignores the stochastic nature of customer arrivals. Specifically, the existing representation com-monly overestimates the number of less productive employees necessary to deliver a specified, waiting time-based cus-tomer service level. We present a new, nonlinear representation of this staffing problem that captures its nonlinearnature and demonstrate its superiority via an extensive set of labor tour scheduling problems for the two-group case.� 2004 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Service operations; Manpower planning; Employee productivity

1. Introduction

A large determinant of the effectiveness of serv-ice organizations is how well the workforce is de-ployed. Having more staff than is necessary toprovide the desired level of customer service iscostly, since the customer-contact activities of theservice staff cannot be inventoried. Conversely,having fewer staff than is necessary risks poor cus-

0377-2217/$ - see front matter � 2004 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2004.03.048

* Corresponding author. Tel.: +1 541 346 3308.E-mail addresses: [email protected] (G.M. Thompson),

[email protected] (J.C. Goodale).1 Tel.: +1 607 255 8214.

tomer service and the loss of current or future rev-enue. Ideally, just enough staff are present toprovide the desired level of service. Temporal var-iation in customer demand, even within a day,compounds the difficulty of providing the idealnumber of staff. One solution is the use of planningintervals of an hour or less, in conjunction withoverlapping shifts, and then allowing the numberof employees scheduled to change from period toperiod. The task of providing the right numberof employees at the right time is commonly knownas workforce scheduling.

Characteristics of the service environmentdefine the nature of the workforce scheduling

ed.

mailto:[email protected]

mailto:[email protected]

G.M. Thompson, J.C. Goodale / European Journal of Operational Research 170 (2006) 376–390 377

problem. Quasi-manufacturing environments, orQMEs, are those in which the customer is in con-tact with service personnel only for a small portionof the service delivery, if at all. In QMEs, becausecustomers are not what is processed, carrying aninventory of unprocessed or partially processeditems is a useful tool for buffering the system fromthe temporal vagaries of demand. Room prepara-tion by hotel housekeeping staff is a classic exam-ple of a QME. In contrast to QMEs, pure serviceenvironments, or PSEs, are those where the cus-tomer is present for a substantial portion of theservice delivery time. In a hotel, the interactionof customers with the front-desk staff is a PSE.Using customers to buffer against the vagaries ofdemand in PSEs–in effect, forcing customers towait for service–risks dissatisfying customers andthe loss of future business. Matching capacity withdemand using effective labor scheduling techniquesis thus particularly important for maximizing prof-its in PSEs.

The purpose of this paper is to contribute to thecurrent literature on workforce scheduling in threeareas that have not yet been addressed in the liter-ature. The three areas are as follows: (1) introducethe construct of employee productivity levels in thecontext of workforce scheduling and specify experi-mental values with which to study service systems;(2) demonstrate the inaccuracies inherent in thelinear representation of sufficient staff constraintsfor the workforce scheduling problem and identifythe conditions where the inaccuracies are mostprevalent in PSEs; and (3) propose a practicalmethod for determining sets of sufficient staff con-straints that specify the number of employees fromdifferent groups providing equivalent minimumservice levels for PSEs.

The structure of the remainder of the paper is asfollows. In Section 2 we review relevant businessliterature on individual output and productivitylevels and establish experimental conditions. Wereview relevant literature on workforce schedulingin Section 3. In Section 4 we present a generalmodel from the current literature that uses a linearrepresentation of productivity levels in the suffi-cient staff constraint. We specify the sufficient staffconstraints with a homogeneous group and withtwo groups of employees. We show how the exist-

ing linear representation of the problem is ofteninaccurate for PSEs in Section 5. We report onan empirical investigation to identify the condi-tions under which the existing representation isaccurate. In Section 6 we present a new, nonlinearrepresentation of the problem that overcomes thedeficiency in the existing linear formulation. Thenew nonlinear representation is a general modelfor two or more employee groups with differingproductivity levels. We report in Section 7 on anempirical investigation to compare the effective-ness of the new formulation to that of the existingformulation when using two groups of employeeswith differing productivity levels. The paper closeswith our conclusions in Section 8.

2. Individual productivity

Work settings with unpaced tasks exhibit varia-bility in output or task completion times that maybe due to three factors (Doerr and Arreola-Risa,2000): (1) the task, (2) the workload, and (3) theworker. Doerr and Arreola-Risa examined an un-paced flow line (fish processing) and demonstratedthat ignoring the worker as a source of variabilityleads to significant errors in projecting output.Hunter et al. (1990) surveyed the academic litera-ture that included job output data (reflecting thevariation of expected output within the entiregroup of workers) and determined that the stand-ard deviation of employee output as a percentageof mean output was 19%, 32%, and 48% for low,medium, and high complexity non-sales jobs,respectively. Values were higher for sales jobs.The authors adjusted these reported standard devi-ations for measurement error and range restric-tion. Within each of the three tiers of complexity,they further segmented jobs with classificationsthat effectively separated service industry job typesfrom manufacturing-related job types. Thus, theyreported the following average standard deviationsfor service industry jobs: (a) 17.3 for routine cleri-cal jobs (low complexity), for example, grocerychecker, telephone operator, customers inspector,clerks, and others; (b) 27.8 for clerical with deci-sion making (medium complexity), that is, maildistribution, supply specialist, claims authorizer,

Table 1A comparison of productivity ratios across productivity COVs consistent with those observed in the literature

Productivity COV 0.20 0.22 0.24 0.26 0.28 0.30

Group A productivity 1.16 1.18 1.19 1.21 1.22 1.24Group B productivity 0.84 0.82 0.81 0.79 0.78 0.76

Group B�s relative productivity 0.73 0.70 0.68 0.66 0.64 0.61

Notes: COV = productivity coefficient of variation; Group A productivity = mean productivity of the most productive 50% ofemployees; Group B productivity = mean productivity of the least productive 50% of employees; and Group B�s relative productiv-ity = ratio of the productivity of the least productive 50% of employees to that of the most productive 50% of employees.

378 G.M. Thompson, J.C. Goodale / European Journal of Operational Research 170 (2006) 376–390

and claims evaluators; (c) 46.2 for professionaljudgment (high complexity), that is, cartographictechnician, attorneys (partners), physicians, anddentists; (d) 96.6 for life insurance sales; and (e)42.3 for non-insurance sales, for example, officemachine sales, industrial sales, soft drink routesales, and others. In addition, Hunter et al. reana-lyzed reported standard deviations from Schmidtand Hunter (1983), and they found the averagestandard deviation for the productivity of salesclerks was 29.1 when corrected for measurementerror and range restriction.

A few other studies that were not reported inHunter et al. (1990) have measured differences inproductivity across individuals in a variety of set-tings. Latham and Yukl (1976) witnessed measuresof output for typists that had a standard deviationthat was 30% of the mean for the entire group.Schmidt and Hunter (1983) examined the outputof sales clerks (as was previously mentioned)and reported that ‘‘. . . researchers examining theutility of personnel programs such as selectionand training can estimate the standard deviationof employee output at 20% of mean output with-out fear of overstatement (p. 412).’’ Janaro andBechtold (1985) reported productivity levels(contribution to total work) for two sessions in acontrolled experiment, where humans riding sta-tionary bicycles simulated physical work output.Actual coefficients of variation (COVs) acrosssubjects for the output values were 0.25 and 0.32.In a related study, Bechtold et al. (1984) used thisdata and reported a COV of 0.27 for the outputvalues.

Assuming that average individual output fromone population of workers was normally distri-buted and one could divide individuals into two

groups: those with higher than mean productivity(Group A) and those with lower than meanproductivity (Group B). One could derive theexpected mean productivity of the Group Aemployees as

ffiffiffiffiffiffiffiffi2=p

pstandard deviations above

the overall mean productivity, and the expectedmean productivity of Group B as

ffiffiffiffiffiffiffiffi2=p

pstandard

deviations below the overall mean productivity.We used this result in Table 1 to provide the meanproductivity levels of Group A and Group Bacross productivity COVs consistent with thoseobserved in the productivity literature and basedon a normalized overall mean productivity of1.0. Group B�s productivity, expressed as aproportion of Group A�s productivity, rangesfrom 0.73 (with a productivity COV of 0.20) to0.61 (with a productivity COV of 0.30). So, basedon the productivity differences in the literature,Group B�s productivity will be as low asabout 60% of the productivity of Group A�sproductivity.

3. Workforce scheduling

The workforce scheduling problem for PSEshas traditionally been expressed as a general setcovering problem (Dantzig, 1954). Labor or laborcosts are minimized in this problem, subject tomeeting periodic server requirements with sched-uled workers that work a fixed, consecutive setof time periods (a shift). When the objective is todevelop the best combination of shifts over aweek-long planning interval, the problem is oneof scheduling tours that have a days-off pattern,as well. Thus, the solution to the traditional tourscheduling problem is a set of schedules that deli-


ver a managerially specified level of customer serv-ice at a minimum cost for the weekly planninginterval.

A number of research articles explored the tourscheduling problem (see, for example, Baker,1976; Bechtold et al., 1991; Brusco and Jacobs,1993; Dantzig, 1954; Keith, 1979; Thompson,1995). Models have been developed that considerfull- and part-time employees (Brusco and Johns,1996; Easton and Rossin, 1991a,b; Li et al., 1991).In addition, Beaumont (1997) scheduled employeeswhile accounting for travel times and costs, Goo-dale and Tunc (1997) considered employees withdynamic productivity, Brusco and Jacobs (1998b)scheduled employees with a limited number ofstarting times, and a number of papers focused onheuristic algorithms for solving the set coveringproblem due to its NP-hardness (Brusco andJacobs, 1993, 1998a; Brusco et al., 1999; Capraraet al., 2000; Easton and Mansour, 1999).

Variants of the tour scheduling problem areplentiful. Some operations face demand 24 hoursper day, seven days a week (continuous environ-ment), thus the appropriate formulation of thetour scheduling problem may depend on whetherthe environment is continuous, discontinuous, oramenable to a robust algorithm that will performwell for both (Brusco and Jacobs, 1995). On theother hand, other research articles focused on onlyspecifying days of the week employees work (days-off scheduling, see for example, Dowling et al.,1997; Emmons and Fuh, 1997), or considered onlystarting times for a single day (shift scheduling, seefor example, Brusco and Johns, 1998; Goodaleet al., 2003).

In this paper we extend the labor scheduling lit-erature by examining the tour scheduling formula-tion and solution procedure when employees aregrouped by productivity level. Related researchshowed that significant benefits are achieved whendifferent productivity levels of cross-trainedemployees were considered and scheduled (Bruscoand Johns, 1998), and when heterogeneous, cross-trained employees were allocated without regardto schedules (Campbell, 1999).

We consider customers� waiting time in the pre-service queue as the prime influence on satisfac-tion. There are numerous articles in the literature

that employ a statement of service level similarto ‘‘Commencing service on x percent of customerswithin y seconds of their arrival’’ (Andrews andParsons, 1989; Buffa et al., 1976; Holloran andByrn, 1986; Kolesar et al., 1975). Davis (1991) ob-served a clear relationship between satisfactionand waiting times in quick service restaurants,for example. Davis and Maggard (1990) alsofound that ‘‘customer satisfaction is more affectedby the initial wait of the customer prior to enteringthe service transformation process, than it is byany subsequent waits encountered by the customerwhile in the process itself’’ (p. 332).

4. Workforce scheduling with variable employee

productivity

Several papers have addressed the issue ofsimultaneously scheduling groups of employeesthat have different, time-independent productivity(Brusco and Johns, 1998; Easton and Rossin,1991a,b; Li et al., 1991). All used formulationssimilar to the model we shall call M1:

Minimize Z ¼X

e2E

X

j2T e

cejX ej ð1Þ

subject toX

e2E

X

j2T e

aijbeX ej P ri for i 2 I ; ð2ÞX

j2T e

X ej P pe

X

k2E

X

j2T k

X kj for e 2 E; ð3Þ

X ej P 0 and integer for e 2 E and j 2 T ; ð4Þ

where I = set of planning intervals, which areindexed {i = 1,2, . . ., jIj}; E = set of employeegroups, which are indexed {e = 1,2, . . ., jEj}; Te =set of work schedules that employees in group e

may work, which are indexed {j = 1,2, . . ., jTej};T = set of all work schedules = ¨eTe; Xej = thenumber of employees from group e assigned towork schedule j; cej = the cost of an employee ingroup e working schedule j; aij = 1 if period i is awork period of schedule j, 0 otherwise; be = therelative productivity of employees in work groupe (be 6 1); ri = the number of most-productiveemployees that are required in period i to deliverthe desired level of customer service; and pe = a


minimum percentage of total employees who are inwork group e, whereX

e2Epe 6 1:0: ð5Þ

The objective (1) seeks to find the least costlyschedule. Costs vary across employee groups andacross work schedules, but M1 assumes that allemployees within a group are equally costly. Con-straint set (2) seeks to ensure that sufficientemployees are present in all periods to deliver thedesired level of customer service. The employeerequirement in each period is expressed in termsof the number of employees having the highestproductivity level. Less productive employees arerepresented by productivity coefficients less thanunity. For example, if 10 employees with the high-est productivity are required in period i to deliverthe desired level of service, then 15 employees hav-ing a relative productivity of 0.667 (i.e., 2/3) wouldbe required in the period, assuming that no otheremployees were present.

Constraint set (3) specifies that groups may haveto comprise a minimum percentage of the totalnumber of employees scheduled. For example, itmay be desirable to have full-time employees (onegroup of employees, perhaps) comprise a certainpercentage of the workforce, even though theseemployees may be more costly, on a per-hour basis,than the part-time employees (a second group ofemployees). Constraint set (3) is similar to the con-straint sets used in the current workforce schedul-ing literature to constrain the mix of part- andfull-time employees in the workforce (see Eastonand Rossin, 1991a,b), but we generalize the con-straint to different minimum requirements for mul-tiple work categories. For example, Brusco andJohns (1998) examined a paper mill maintenancedepartment with four employee groups, where any-where from one to three of these groups might berequired to comprise at least a certain proportionof the workforce. In many service operations thebulk of the workforce must consist of a core groupto ensure that the department has a critical mass ofexpertise, while the non-core groups may also havedesignated minimum sizes to ensure. Limiting thesum of the minimum proportions to one or less(5) helps ensure that the model is feasible.

The QME literature has commonly ignoredthe stochastic nature of customer arrivals andassumed that employees will never experience lackof work idle time (Krajewski et al., 1980; Mabert,1979; Mabert and Watts, 1982). With this assump-tion, constraint set (2) is valid. Assuming thatemployees never experience lack of work idle timeimplies that the employees are working at close totheir individual capacity on their assigned job, inother words, at close to 100% utilization. Forexample, in a hotel the housekeepers can processan inventory of rooms; when they are scheduledcorrectly they will not have to wait for rooms tobecome available.

Planning near-100% utilization is problematicfor PSEs, however, as doing so will result in longdelays for customers and likely lower the organiza-tion�s competitive position. Indeed, as we notedearlier, a study by Davis (1991) shows a stronglinkage between customer queue time and satisfac-tion. For a PSE to have rapid response time, thestochastic nature of customer arrivals must betaken into account. It is when the stochastic natureof customer arrivals is considered that constraintset (2) begins to break down. The stochastic natureof customer arrivals necessitates building inplanned idle time. Constraint set (2) requires thesame amount of planned idle time, regardless ofthe composition of the workforce. However, lessplanned idle time is necessary when more serversare present (a result of the nature of queueing sys-tems), even if the servers have lower individualservice rates.

Table 2 shows an example of the breakdownthat can occur in constraint set (2). It is based ona single population of workers that is separatedinto two groups, A and B. Group A employeeshave a standardized productivity of 1.0 whileGroup B employees individually have a relativeproductivity of 0.6 times that of Group A employ-ees. Table 2 gives mean customer queue times andservice levels (probability of a customer experien-cing a wait) from an M/M/s queueing systemfor a complement of 80 employees from GroupA under an arrival rate of 20 customers per minuteand various service rates. Given Group B�s relativeproductivity, constraint set (2) suggests that acomplement of zero Group A employees and

Table 2Mean queue times for full complements of Group A employees and Group B employees, based on an arrival rate of 20 customers perminute

lA lB Rho Performancewith 80Group Aemployees

Performancewith 134Group Bemployeesa

Performance with reduced number of Group B employeesb

Wq Pwait Wq Pwait # Group B employees Wq Pwait Excess Group B employeesc

0.255 0.153 0.980 2.00 0.801 1.38 0.692 134 1.38 0.692 00.265 0.159 0.943 0.418 0.502 0.278 0.364 133 0.363 0.416 10.275 0.165 0.909 0.152 0.304 0.085 0.178 132 0.137 0.244 20.285 0.171 0.877 0.064 0.178 0.028 0.081 131 0.058 0.139 30.295 0.177 0.847 0.028 0.101 0.009 0.035 130 0.026 0.078 40.305 0.183 0.820 0.013 0.056 0.003 0.014 129 0.012 0.043 5

Notes: lA = the service rate of each Group A employee; lB = the service rate of each Group B employee (= 0.6 · lGroupA);Rho = employee utilization with full complement of Group A employees; Wq = the mean customer queue time; and Pwait = prob-ability that a customer waits for service.

a Number of Group B employees specified by constraint set (2) when zero Group A employees are scheduled.b Smallest number of Group B employees with a service level matching that of 80 Group A employees.c Overestimation of Group B employees by constraint set (2).


134(=80/0.6) Group B employees would deliverthe same level of service as a complement of 80Group A employees and 0 Group B employees.As Table 2 shows, however, the service level pro-vided by the 134 Group B employees will alwaysexceed the service level provided by 80 Group Aemployees. Table 2 also shows the smallest com-plement of Group B employees who can providethe same service level as 80 Group A employees.The discrepancy is as high as 5 employees, whichtranslates into constraint set (2) overestimatingthe number of Group B employees by as muchas 3.7%.

A queueing model can be very helpful for deter-mining the number of each group of employeesneeded to provide the desired service level, whenthere are no employees from any other group pre-

sent. In other words, the queueing models can tellus that we would need x Group A employees (andzero Group B) or y Group B (and zero Group A)employees. However, we are not aware of any pub-lished queueing models that can specify the num-ber of employees needed in a PSE when there isa mix (with respect to individual output/productiv-ity) of employees on hand who have different pro-ductivity levels (i.e., a blend of employees fromdifferent groups), that is, a model that specifiesthe number of Group B employees needed givenz Group A employees (with 0 < z < x). The next

section describes our use of simulation to evaluatethe accuracy of constraint set (2) with differentblends of employees.

5. An evaluation of the existing approach

In this section we report on an empirical inves-tigation to evaluate the accuracy of constraint set(2). Our goal was to evaluate the performance ofconstraint set (2) in simulated service environ-ments. We created the set of simulated serviceenvironments to be both encompassing and repre-sentative of the variability occurring in serviceoperations. In effect, we attempted to identify theconditions under which constraint set (2) failedto correctly represent the necessary staffing levels.

5.1. Design of the evaluation process

In this investigation we assume that arrivals arePoisson with a stable mean arrival rate of two cus-tomers per minute, that service times are exponen-tially distributed, and that there are two employeegroups, with the more productive group having arelative productivity of unity. We shall considerthe more productive group to be Group A and theless-productive group to be Group B. We gener-ated and stored random numbers for determining


the arrival time and service duration for 200,000customers. In initial testing, we found the accuracyof constraint set (2) not affected by the mean cus-tomer arrival rate.

To create a wide variety of environments, wevaried three factors, the employee requirements(EMRQ), the mean utilization of employees(UTIL), and the relative productivity of Group Bemployees (RLPR). We selected employee require-ments of 5, 20, and 80 of Group A employees torepresent small, medium and large workforces,respectively. We selected relative productivity lev-els for Group B employees of 0.90, 0.80, 0.70,and 0.60. Finally, we selected utilization levels of95%, 90%, 85%, 80%, 70%, and 60% to cover therange of expected staff utilization. The utilizationprovided the mean service rate of the Group Aemployees from the following relationship:

mean service rate of Group A employees

¼ mean arrival rate

utilization � employee requirements:

To ensure that we were not examining environ-ments with unrealistically high service levels, weexcluded any combination of Group A size andemployee utilization level where the probabilitythat a customer would wait for service was lowerthan 5%. We thus excluded utilization of 0.60and utilization levels of 0.80 and under for 20and 80 Group A employees, respectively.

We selected two service criteria, or measures ofcustomer service: the average wait for service(AVWT) and the probability of a customer waitingfor service (PRBW). We first simulated the systemwith a full complement of Group A employees andmeasured the actual level of service, which wedefine as the base level. We then used an iterativeprocess to identify the comparable number ofGroup B employees for each partial complementof Group A employees. We first calculated thenumber of Group B employees suggested by con-straint set (2) given the partial complement ofGroup A employees. We then simulated the systemwith this staffing mix and measured the level ofservice provided. If the service level exceeded thebase level, we reduced the number of Group Bemployees by one and again simulated the system.

We continued to reduce the number of Group Bemployees until the actual level of service waslower than the base level, at which point we exam-ined another partial complement of Group Aemployees.

For thoroughness, we also considered the effectof three common work dispatching rules (manage-rial strategies) in assigning customers to servers.Our first rule assigned a customer to the employeeidle the longest (S-IDLE). This rule focuses on thepractical issue of fairness and distributing theworkload among the workers so that individualworkers do not have long periods of idle time com-pared to other workers in the system. The secondrule randomly assigned a customer to an employ-ee. So the customer was assigned to an employeefrom a particular group, assuming one is available,with a probability based on the proportion of totalavailable employees comprised by such employeesfrom that group (S-PROB). The third rule alwaysassigned a customer to one of the Group Aemployees if one was available (S-FAST). Thiswould represent a policy of minimizing the timea job (customer) is in the system.

5.2. Results of the evaluation process

Table 3 presents specific examples from theexperiment that shows the breakdown that occursin constraint set (2) under an EMRQ of 20 using aservice criterion of AVWT. For example, considera UTIL of 70 and a RLPR of 0.80. Constraint set(2) suggests that 12, 17, 22, and 23 Group Bemployees would be required with 11, 7, 3, and 2Group A employees present, respectively. How-ever, Table 3 shows the desired level of service isactually achieved using only 11, 16, 21, and 22Group B employees.

Table 4 summarizes all of the instances whenconstraint set (2) overestimated the requirementsfor all three EMRQ levels, based on the AVWTservice criterion. Table 4 shows that inaccuraciesoccurred in 4 of 24, 12 of 20, and 10 of 12 scenar-ios with mean requirements of 5, 20, and 80employees, respectively. In general, the numberof inaccuracies is higher with lower utilization,with greater differences in the productivity of thetwo groups of employees, and with a greater

Table 3Examples of the conditions under and the amount by which constraint set (2) overestimated the number of Group B employees

UTIL RLPR Number of Group A employees present

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

70 0.80 1 1 1 170 0.60 1 1 1 1 1 1 1 2 1 1 2 1 2 2

Notes: Results are for the equivalent of 20 Group A employees with the S-PROB dispatch rule; UTIL = utilization; andRLPR = relative productivity of the less productive employees.

Table 4Number of occurrences of overestimated staff sizesa

Factor EM: S-PROB S-IDLE S-FAST

EMRQ UTIL RLPR OE: 1 2 3 4 1 2 3 4 1 2 3 4

5 0.70 0.70 1 1 15 0.70 0.60 1 1 15 0.60 0.70 1 1 15 0.60 0.60 1 1 2

20 0.90 0.70 1 1 220 0.90 0.60 1 1 220 0.85 0.70 3 3 520 0.85 0.60 4 4 720 0.80 0.90 1 1 220 0.80 0.80 1 2 320 0.80 0.70 7 7 920 0.80 0.60 8 9 1220 0.70 0.90 3 3 420 0.70 0.80 4 4 720 0.70 0.70 11 11 15 120 0.70 0.60 10 4 11 4 10 8

80 0.95 0.70 14 15 2180 0.95 0.60 16 18 3280 0.90 0.90 7 7 1480 0.90 0.80 19 19 3380 0.90 0.70 45 47 1 65 580 0.90 0.60 39 15 37 19 35 4080 0.85 0.90 18 18 2980 0.85 0.80 42 1 44 67 380 0.85 0.70 33 27 3 33 31 2 15 53 880 0.85 0.60 24 16 20 6 20 19 24 7 6 14 42 16

Notes: EMRQ = staffing level with a full complement of Group A employees; UTIL = base utilization proportion with a full com-plement of Group A employees; RLPR = the relative productivity of Group B employees; EM = estimation method; OE = over-estimation (number of Group B employees).

a Values in the table represent the number of occurrences of each level of overestimation. See the text for a description of S-PROB,S-IDLE, and S-FAST.


requirement for employees. In the case of a com-plement of 80 Group A employees, there are 81possible linear combinations of the employees thatdeliver the service criterion. Since the full comple-ment of Group A employees is, by definition, accu-rate, there are only 80 combinations where

constraint set (2) could be inaccurate. At the high-est level of employee requirements (EMRQ = 80),lowest utilization for that staff size (UTIL = 0.85),and greatest difference in productivity (RLPR =0.60), most estimations of the number of GroupB employees required are inaccurate. That is, of


the 80 possible combinations where constraint set(2) could misspecify the number of Group Bemployees needed, it did so in 66, 70, and 78 ofthe cases for S-PROB, S-IDLE, and S-FAST,respectively. As seen in Table 4, the overestimationwas as great as four employees for all threedispatching rules.

Constraint set (2)�s inaccuracy varied acrosswork-dispatching rules. Of the three rules, S-PROB came closest to a linear relationship (closestin a relative, not absolute sense) while S-FASTyielded the least linear results and S-IDLEwas intermediate. The differences across work-dispatching rules suggest that managers need tobe aware of their work-dispatching rule in thisenvironment.

Although Table 4 reports the results only forthe AVWT service criterion, PRBW yielded simi-lar staffing inaccuracies (as was seen in a smallscale in Table 2). This suggests that inaccuraciesexist in constraint set (2) regardless of one�swait-related service criterion. In other words, theproblem is with constraint set (2), not with one�schoice of a service criterion. Thus, in Section 6we propose a new approach for modeling variableemployee productivity when these inaccuraciesexist.

2 Start with the number of employees suggested by con-straint set (2). Iteratively decrease the number of employees byone and simulate the system. The staffing level of constraint set(6) is the smallest number of employees that delivers the desiredservice level.

6. A new workforce scheduling approach

The true staffing function for groups of employ-ees having different productivity levels is nonlinearin nature, as the results in Table 4 clearly show.Thus, constraint set (2) must be considered a linearapproximation to a true, nonlinear relationship.To better capture the nonlinearity, we propose anew constraint set to replace (2):X

j2T jEj

aijX jEj;j P F ðdsl;ki;l1; . . . ;ljEj;W 1;i; . . . ;W jEj�1;iÞ

for i2 I ; ð6Þ

where dsl is the desired service level, ki is the ex-pected arrival rate in period i, le is the service rateof group e employees, and Wei is the number ofgroup e employees currently scheduled in periodi, where Wei =

Pj2Te

aijXej.

Constraint set (6) specifies the minimum accept-able number of employees in each period for theleast productive category of employees, jEj. Forany particular period, the minimum acceptablenumber of the least-productive employees is thesmallest number of such employees that will deli-ver an actual service that meets or exceeds the de-sired service level. This number is a function of thedesired service level (dsl), the expected customerarrival rate (ki) for the period, the service rates(le) of employees in the different groups, and thenumber of employees (We,i) from the other groupsthat are currently working in the period.

The complete form of our proposed model forworkforce scheduling using employees havingvariable productivity is

Min (1)

subject to (3); (4); and (6):

We shall refer to our proposed model as M2. Inthe next section we report on the results of experi-ment designed to evaluate the effectiveness of M2compared to M1.

In practice, we implemented constraint set (6) asa lookup table. Table 5 compares the number ofless productive employees specified by the tradi-tional, linear approach and the number specifiedusing constraint set (6). Constraint set (6)�s staffinglevel was found using the iterative process de-scribed earlier. 2 Table 5 uses the example fromTable 3, where the employee requirements are for20 of the more productive employees, where em-ployee utilization is 70% and the less-productiveemployees have a relative productivity of 0.60.Heuristic solution approaches such as simulatedannealing, which have proved very effective fortour scheduling (Brusco and Jacobs, 1993), areeasily modified to represent constraint set (6) usinga lookup function approach. This is the course wefollowed, as discussed in the next section.

Table 5A comparison of lookup function representations of constraint sets (2) and (6)a

a For a period in which 20 of the more productive employees are necessary based on a utilization of 70% and the relative productivityof the less productive employees is 0.60. Shaded rows indicate differences between the staffing levels suggested by the constraint sets.


7. A comparison of schedules

This section describes and presents the resultsfrom a set of test problems designed to evaluatethe benefit of using M2 in scheduling problems.Our goal was to answer the question, How muchbetter might M2 be in practice than M1? Thisinvestigation is necessary since the inaccuracies ofM1 identified earlier considered only single plan-ning periods. In practice, day-long or week-longscheduling horizons are broken into many plan-ning intervals, thus making it difficult to predictthe net benefit of using M2 in shift or tour schedul-ing problems. Although M2 is designed to repre-sent the nonlinear staffing relationships that existbetween multiple groups of employees, for simplic-ity we evaluate its merit in the two-group case.

7.1. Design of the experiment

We developed a set of 3240 tour-schedulingproblems, differing on seven problem characteris-

tics, as outlined in Table 6. For all problems,each Group A employee worked five consecutive9-hour shifts (including an unpaid break in thefifth hour), starting the same period each workday. Each Group B employee worked 4-hourshifts on some number of consecutive days (asdiscussed below), starting work at the same timeeach day. There was no constraints on the start-ing times for Group A and Group B employ-ees—all periods were available to be scheduledstarting times.

Our goal in selecting the variable problem char-acteristics, and their levels, was to reflect therange of operating conditions occurring in serviceorganizations. The first characteristic was thelength of operating day (ODL). We used operatingdays (ODLs) of 12, 16, and 20 hours. We used fiveweekly demand (requirements) patterns, varyingon the type of within-day employee requirementspattern (ERP): unimodal, bimodal, trimodal, andtwo ‘‘random’’ patterns. All test problems had amean requirement of 50 employees. There were

Table 6Problem characteristics

Problem characteristics Number of levels Levels

Operating day length (ODL) 3 12, 16, and 20 hoursEmployee requirements pattern (ERP) 5 Unimodal, bimodal, trimodal,

random A, random BAmplitude of the employee requirements (AMP) 3 10, 20, and 30 employeesMinimum percentage of total work hours comprisedby Group A employees (MIX)

3 0%, 33.33%, and 66.67%

Number of days worked each weekby the Group B employees (BDW)

4 5, 4, 3, and 2

Relative cost of the Group B employees,per work hour (BRC)

3 0.8, 1.0, and 1.2

Group B�s relative productivity (BRP) 2 0.8300 and 0.6667


three levels of amplitude in the requirements(AMP): plus or minus 10, 20, and 30 employees.

We specified three mixes of staff (MIX): thosewhere the Group A employees had to compriseat least 0%, at least 33.33%, and at least 66.67%of the total work hours. Levels one through fourof Group B days worked (BDW) respectivelyrequire that the Group B employees work 5, 4, 3,and 2 days per week. There were three levels ofthe relative cost of the Group B employees, orBRC. These were hourly labor costs of 0.8, 1.0,and 1.2 times the hourly labor cost of the GroupA employees. For example, with the three-daywork week for Group B employees and a BRCof 0.8, while the tour cost for Group A employeesis normalized to 1.00, the cost of each Group Btour is ((3 · 4)/40) · 0.8 = 0.24.

We used two levels for the productivity of theGroup B employees, or BRP. In BRP level oneGroup B employees had a relative productivityof 0.830, while in BRP level 2 the Group Bemployees had a relative productivity level of0.6667. The lower level is well within the rangeseen in the literature on individual productivity,as Table 1 shows. The results reported in Table 3suggest that constraint set (2) will be more accu-rate for BRP level one, and thus we expect the ben-efit of M2 to be greater with BRP level two.

The combinations of the levels of each experi-mental factor resulted in a total of 3240 test envi-ronments. For each of the 3240 environments, wedeveloped implementations of M1 and M2. Thenonlinear nature of M2 requires a heuristic solu-tion procedure, as does the sheer volume of prob-

lems for which we desire solutions to both M1 andM2. We each developed a heuristic based on thesimulated-annealing heuristic developed by Bruscoand Jacobs (1993). Simulated annealing has beenused with great success in workforce scheduling.Brusco and Jacobs�s simulated annealing heuristicfound the best solution in 94% of their test prob-lems. Thompson (1996a)�s simulated annealingheuristic yielded solutions that were 0.29% morecostly than optimal, on average. Thompson(1997)�s simulated annealing heuristic yielded solu-tions approximately 0.11% more costly than thebest known solutions to his problems.

The pseudo-code for the SA heuristics is pro-vided in Fig. 1. We ran the heuristics on a Pen-tium-based personal computer. We define themetaheuristic, SA, to be the union of two heuris-tics (SA1 and SA2), which were the same heuristiccoded by different co-authors. Thus, the best solu-tion to M1 (M2) using SA would be the better ofthe solutions to M1 (M2) yielded by SA1 andSA2. Both SA1 and SA2 accounted for the mul-ti-period impact of service according to Thomp-son�s (1993) method. As seen in Fig. 1, the SAheuristics for M1 and M2 are similar, but differin the approach used to satisfy the sufficient staffconstraint, that is, Eq. (2) for M1 and Eq. (6) forM2.

7.2. Results of the experiment

Table 7 presents the results of using SA withM1 and M2 on the set of 3240 test problems.The results in Table 7 show that the average labor

Fig. 1. Pseudo-code for workforce scheduling simulated annealing heuristics used in the experiment.


savings yielded by M2 was greater with longeroperating days, lower amplitude of the employeerequirements, lower relative productivity of theGroup B employees, and when the Group Aemployees had to comprise less of the total workhours. Although SA solutions to M2 were superior

to the SA solutions to M1 across all levels of em-ployee requirement patterns, days worked perweek by Group B employees, and relative cost ofthe Group B employees, no consistent patternemerged across the levels of these factors. The dif-ference in labor cost savings provided by M2 was

Table 7Schedule cost saving yielded by M2 for the complete problem seta

Factor Factor level (average percentage labor savings)

Group B�s relative productivity High (0.635) Low (2.325)Operating day length 12 hours (1.328) 16 hours (1.531) 20 hours (1.581)Amplitude of the employee requirements Low (1.616) Medium (1.532) High (1.292)Minimum % of total hours workedby Group A employees

0.00% (1.566) 33.33% (1.505) 66.67% (1.369)

Relative cost of theGroup B employees

Low (1.542) Medium (1.403) High (1.495)

Days worked per week by Group Bemployees

5 days (1.487) 4 days (1.457) 3 days (1.475) 2 days (1.502)

Employee requirements pattern Unimodal (1.561) Bimodal (1.696) Trimodal (1.593) RandA (1.280) RandB(1.270)

a Measured as a percentage of the labor cost for SA solutions to M1.


much greater across the levels of relative produc-tivity of the Group B employees than across thelevels of any other factor. At the low and highlevels of relative productivity of the Group Bemployees, M2 yielded labor savings of 2.33%and 0.64%, respectively.

8. Conclusions

We observed the existing linear representationof staffing using employee groups which differ inproductivity (i.e., constraint set (2)) to be fre-quently inaccurate in pure service environments(PSEs). This inaccuracy takes the form of overesti-mating the number of less productive employeesrequired to deliver a wait-based service criterion.It occurred both for planning periods consideredindependently and in the environment of labortour scheduling, where planning periods are notindependent. We presented M2, a new, nonlinearrepresentation of the problem, designed to betterreflect the true staffing function, which we imple-mented using a lookup table approach. In anextensive experiment of PSEs with two employeegroups, M2 delivered the desired, waiting time-based service level at a measurably lower laborcost than the existing model of the problem, M1(proposed by Easton and Rossin, 1991a,b; Li etal., 1991). We observed larger differences in thecosts of solutions to M1 and M2 with larger differ-ences in productivity across employee groups.That is, M1 was less accurate with more diverse

employee groups. In fact, M2 yielded average sav-ings of 2.33% compared to M1 when employeesin the less productive group were individuallytwo-thirds as productive as employees in the moreproductive group. Since this level of relative pro-ductivity is well within the range observed in theliterature we expect our results to be typical ofthe savings offered by our model in practice.

8.1. Implications

Managers of PSEs typically use commercialworkforce scheduling systems to facilitate thescheduling of their staff. Because correctly mode-ling variable employee productivity can increasetheir operation�s profitability, managers shouldquestion the developers of their systems to deter-mine whether their systems are based on modelsconsistent with M2 or models such as M1. Also,the method of assigning customers to servers hasa clear effect on the acceptable staffing blends, sothe dispatching rule that is used needs to be spec-ified accurately.

Our findings also have important implicationsfor researchers in workforce scheduling. Our find-ings suggest that solution procedures developedfor M1 may not be completely accurate for PSEs.It makes little sense to expend a great deal of effortin the search for optimal solutions to M1 whenthese optimal solutions might be inferior to heuris-tically developed solutions to M2. Obviously,developers of solution procedures for the problemenvironment we considered should ensure their


procedures are consistent with M2�s nonlinearstaffing constraint, Eq. (6).

8.2. Extensions

A great deal of computational effort may haveto be expended when using simulation to identifythe inputs to M2, specifically the employee blendsthat deliver the desired service level. In fact, it maytake much more time to identify the acceptablestaffing level combinations (the inputs to M2) thanit does to develop the labor schedule proper (solv-ing M2). The effort involved in determining M2�sinputs will grow combinatorially as more thantwo employee groups are considered. There are anumber of avenues for future research that canfocus on improving the search for solutions. Forexample, can one make the simulation as efficientas possible by predetermining the minimum num-ber of customers to be simulated while yieldingaccurate estimates of the acceptable staffing levelcombinations. Is there a way of linking the devel-opment of allowable staffing combinations withthe development of the schedule proper, so thatone does not evaluate staffing level combinationsthat are not under consideration? Perhaps thismay be accomplished by making the simulation acallable routine that is only used as needed. Lastly,can one develop a fast closed-form solution orapproximation so that one does not have to resortto a relatively slow simulation to identify theacceptable staffing level combinations. The varia-bility of results across work dispatching rules sug-gests this will be challenging.

The issue of computational effort of determin-ing M2�s inputs may pose more of a problem in re-search settings than in practical settings, however.In research settings, one may be solving hundredsor thousands of diverse problems. In practical set-tings, however, similar problems are resolved peri-odically. Thus, in a practical setting one may beable to develop a priori the necessary informationand store it in a database to be used as needed.

In addition, the use of part-time employees inpractice and research is widespread, howeverincreasing the number of part-time workers in-creases the managerial burden of implementingthe schedule. Examining this trade-off is outside

the scope of this paper. However, another possibleextension of this project is to use Thompson�s(1996b) methods to determine whether controllingaction times will mitigate the cost savings we foundin this study. Furthermore, extending M2 to useeach employee�s unique productivity poses excitingchallenges. Although M2 as presented holds formore than two groups, the process of specifyingconstraint (6) increases geometrically. In this envi-ronment, it is quite likely that one would have tointegrate the identification of all staffing level com-binations with the development of the schedule. Itis quite possible that the best approach lies some-where between using two employee groups andusing each employee�s unique productivity.

Acknowledgements

The authors are grateful to the referees andeditor for their many helpful comments.

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