efficient scheduling of part-time employees · 2015-02-06 · minimizing the costs of employee...

OMEGA Int. J. of Mgmt Sci., Vol. 20, No. 2, pp. 201-213, 1992 0305-0483/92 $5.00 + 0.00 Printed in Great Britain. All fights reserved Copyright © 1992 Pergamon Press plc

Efficient Scheduling of Part-Time Employees

AJ VAKHARIA

HS S E L I M University of Arizona, Tucson, U S A

R R H U S T E D

Hewlet t -Packard, Inc., Colorado Springs, Colorado, U S A

(Received April 1991; in revised form June 1991)

A staff scheduling model and heuristic procedure is proposed to aid in developing schedules for part-time employees. The model and heuristic method explicitly incorporates employee time preferences (i.e. times at which the employees are available to work) as well as employee wages in developing a weekly work schedule. Data collected from a local restaurant and a computer lab is used to validate and illustrate the model. The results show that the use of the model considerably simplifies the task of the restanrant/lab manager in developing weekly work schedules. Further, this application also demonstrates the trade offs between the "benefits" of maximizin~ employee satisfaction versus minimizing the costs of employee wages.

Key words--scheduling, multiple criteria programming

1. INTRODUCTION part-time help to meet variations in daily demand. For example, restaurants encounter peak

ONE OF THE MAJOR PROBLEMS encountered demands during the lunch and dinner hours. In by operating managers of service delivery sys- an attempt to minimize operating costs (of terns is that of matching capacity with demand, employee wages and fringe benefits) as well as This problem arises due to the fact that a service meet periods of peak demand, the restaurants cannot be inventoried. Further, since demand resort to using part-time help. A complication for a service is typically variable this also creates which arises when using such help is that these a problem of capacity utilization. Faced with employees typically hold multiple jobs or have variable demand and a perishable service, the other commitments/responsibilities. Conse- operations manager has three basic options [6]: quently, in developing work schedules for such (a) smooth demand (e.g. through a reser- employees, there is a need to incorporate the vations/appointments system, by demarketing work time preferences for such employees. peak times, or using price incentives in non- In this paper, we explore the problem of peak times); (b) adjust capacity (e.g. by using scheduling part-time employees so as to part time help, or by scheduling workshifts to maximize employee work-time preferences and match demand or by increasing the self service simultaneously minimize employee wages. content of a service); (c) letting the customer Although the problem of identifying appropri- wait. ate shifts/tours of full-time employees to match

A large number of service organizations (e.g. demand as well as the assignment of full-time restaurants, professional services, etc.) are using employees to shifts has received considerable

201

202 Vakharia et al.--Efficient Scheduling of Part-Time Employees

attention in the literature (see, for example, but would prefer not to work; and the not [1-3, 8, 10, 11]), the scheduling of part-time era- available hours are those times at which the ployees does not seem to have been addressed, employee has other committments and hence Thus, the purpose of this paper is to: cannot be scheduled to work. Further, the

model developed in this section focuses on (1) Develop a model and heuristic pro- hourly work periods (e.g. an employee works

cedure for generating part-time em- for a full hour starting at 9 a.m.) but can be ployee work schedules by explicitly easily modified to incorporate longer or shorter considering employee costs, time pref- units of time measurement. Finally, before erences and minimum staffing require- applying the model, the user is required to ments, specify the hourly wages per employee, the

(2) Illustrate the use of the model and number of employees required during different heuristic method by applying it to two times of each day, and the relative "weights" to data sets. be assigned to the wage minimization and pref-

erence maximization objectives. Please refer to The remainder of this paper is organized as the Appendix for the model formulation.

follows. In Section 2, we describe the math- In the model, the objective function [equation ematical model for scheduling part-time em- (4)] attempts to simultaneously minimize the ployees and in Section 3, we propose a heuristic employee wages and maximize the employees procedure for modifying the solution obtained preferences for working times. While the wages using the model. In Section 4, we discuss the can be minimized by scheduling all the era- data sets used to illustrate the model and in ployees with the lowest wage to work before Section 5, we present the results of applying the those who are paid a higher wage; the maximum model to the data. Finally, in Section 6, we value for the second part of the objective func- discuss the implications and conclusions of this tion can be achieved if all employees are sched- research, uled at their preferred times. The parameters C~,

C2 and D are input parameters and serve the 2. A MODEL FOR PART-TIME following functions. The parameter D (D > 0)

EMPLOYEE SCHEDULING can be regarded as a constant which converts the preference matrix into a contribution (i.e.

In this section, an optimization model for dollar) matrix. The parameters C~ and C2 (C~, scheduling part-time employees (referred to as (:72/> 0) can be treated as the "weights" that model PREFWAGE) is presented. In Section reflect the relative importance of the two objec- 2.1, we describe the model. This is followed by tives of wages and preferences, respectively. a short discussion in Section 2.2 of issues which In terms of the constraints, equation set (5) provide some face validity to the model as it is ensures that the number of employees scheduled currently formulated. Section 2.3 focuses on for a particular time of each day is equal to the modifications which can be made to the input estimated number required for that time on that data in case a feasible (and hence, optimal) particular day. Equation set (6) ensures that solution cannot be obtained. The multi-objec- each employee is not scheduled for more than tive nature of the model is discussed in Section the maximum part-time hours each day (i.e. 2.4 and finally, issues related to implementation TIME) since this would result in the employee of the work schedule obtained using the model becoming a full time worker (in which case are explored in Section 2.5. the employee would be eligible to a higher salary

as well as the fringe benefits). Since all em- 2,1 Model description ployees are to be scheduled to work at either

In developing an employee work schedule, their preferred times or the available times, each part-time employee is required to specify constraint set (7) is included in the formu- the preferred, available and not available work lation. Finally, the technological constraints on hours for each day. The preferred hours are the decision variable are incorporated in defined as times at which the employee would equation (8). like to schedule his/her work; the available The possibility of overtime work is not ac- hours are those at which the employee can work counted for in the model. This is due to two

Omega, Vol 20, No. 2 203

reasons. First, we are developing work sched- determine the work schedule (i.e. an ules for part-time employees, and typically, employee earning a lower wage will be overtime work is associated with full time em- preferred over another earning a higher ployees. Second, if one wanted to include the wage). possibility of overtime work, the wage per employee for overtime hours could be estimated 2.3 Suggested modifications to escape infeasi- and included in the model. Hence, for each bUity employee, the user would need to specify two In the preceding discussion, it was assumed wage fgures: a regular hourly wage for regular that a feasible (and hence, optimal) solution to time hours and an overtime wage for those times model PREFWAGE could be determined. in a day the employee was to be considered as However, if all the employee preferences are for working overtime, similar working times, then the minimum staff

Not that solving PREFWAGE appears to be requirements per hour per day might not be computationally difficult since a 6 day, 12 h per attainable. Thus, a feasible solution of the prob- day work schedule for ten employees would lem may not exist the following several modifi- result in a problem with 720 (i.e N x J x K) cations to the data can be carried out: 0-1 decision variables and 852 [i.e. (J x K) + (N x K) + (N x J x K)] constraints. However, (1) Reducing the staff requirements for a we can regard model PREFWAGE as a special- particular time of day. On obtaining an

infeasible solution, the user can esti- ized transportation problem. In this case, we have N x K "source" nodes each with "ca- mate the particular times of days on pacity" TIME [constraint set (6)] which have to which the staff requirements cannot be supply J x K"s ink" nodes each with "demand" met. In such a case, the staff require-

ments at these times on the specific days STREQjk [constraint set (5)]. Note that constraint set (7) simply identifies the feasible could be reduced (i.e. the STREQjk SCHEDu, values (i.e. whether a source node can values for those times are decreased) be used to supply a sink node). By exploiting and a feasible solution to the problem this special structure, "large" realistic problems can be obtained. The additional "cost" can be solved in a reasonable amount of corn- of such a solution can be estimated as puting time (see Section 5 where we demonstrate the loss of a decrease in customer set- an application of this model), vice since fewer staff are on duty at

those times.

2.2 Face validity (2) Increasing the maximum amount of time Assuming that a feasible (and hence, optimal) an employee is allowed to work each day.

solution to model PREFWAGE can be ident- For days on which staff requirements ified, there are two features which provide face have not been met, the user can attempt validity to the proposed formulation of the to increase the maximum time par- model. Assuming that Cm, C2 > 0, it can easily ameter in the model (i.e. TIME). Such be shown that: a change will identify a feasible solution

(1) An employee will always be scheduled only if one or more employees are to work at a preferred work time before available to work on those days where being scheduled to work at an available infeasibilities were observed in the first work time. This also indicates that the place. The additional "cost" of a relative magnitude of values in the em- modified solution would be the amount ployee work preference matrix are not of wage differential between full-time important as long as the value for the and part time employees. employee preferred work time is greater that the value for the employee avail- (3) Removing constraint set 7 (see Appen- able work time. dix) from the formulation and resolving

the model In such a case, it is likely that (2) Given identical work time preferences certain employees may be scheduled to

for two employees, the relative wage work at times which they are not avail- differential between the employees will able. The additional "cost" of such a


solution would be the amount of in- 0 < E; < 1) such that the greater the value of this creased wages that employees may have index, the more effective is the employee in to be paid to work at times which they performing his/her task. Then the objective were previously unavailable, function of the model [see equation (4) in the

Appendix) could be modified to include the In general, any one of the three strategies could following additional term: be implemented so as to develop a feasible

N J K

solution. Further, a combination of strategies -C3D , Z ]E Z EiSCHED~j k (1) could also be used simultaneously (e.g. decreas- ,= ~j= ~k = ing the staff requirements for particular times of In this case, D l is the parameter to convert the a day and increasing the maximum amount of employee "effectiveness" index into a dollar time an employee is allowed to work on another contribution (similar to D) value while C3 is the day). relative "weight" placed on this objective.

2.4 Multi-objective optimization issues 2.5 Implementation issues

Model PREFWAGE can also be regarded as One of the only problems with the solution a multi-objective model. As noted by Rosenthal that can be derived using model PREFWAGE [14], four "commonsense" ideas have been pro- as formulated above is that there is no guarantee posed for solving multi-objective optimization that the resulting employee work schedule can models. These are: (a) the use of weights/scores; be implemented as is. This is due to the fact that (b) the use of targets, ideals, aspiration levels, model PREFWAGE does not include any con- etc.; (c) the use of priorities; (d) the use of straints that ensure that employees are to work efficiency. Using this classification, model at least two (or more) hour stretches. Thus, it is PREFWAGE is in category (a) since we are likely that a particular employee " i" will be applying relative "weights" to the different ob- required to work on day 1 starting from 9 a.m. jectives. These "weights" are in the ratio Ct: C~ to 10 a.m. and then again from 11 a.m. to where C~ is the "weight" applied to the wage 12 noon on the same day with the intervening minimization objective and C2 is the "weight" hour (i.e. from 10 a.m. to 11 a.m.) off. Although applied to the worker preference maximization such a schedule is adequate if the employee was objective. By varying the values of the par- not available for work from 10 a.m. to 11 a.m. ameters Ct and C2, we can explore the impact (i.e. if PREF~,~o.1 = 0), it is definitely not going of relative importance of these two objectives on to improve employee satisfaction if, on the the development of work schedules. Further, other hand, PREF~.to.t > 0. Hence, a user might alternative "weights" can also be used to per- be interested in identifying an employee work form a what-if analysis before finalizing a work schedule such that employees are assigned work schedule [14]. times which at least include two (or more) hour

The model, in its current form, does not continuous stretches. distinguish between the employees (i.e. given One approach for developing such block equal time preferences, an employee who is work schedules is simply to operationalize earning less will always be scheduled to work model PREFWAGE for two (or more) hour before an employee earning more---see Section time blocks [i.e. the index "j" will represent two 2.2). However, it can be argued that the differen- (or more) hour time periods in each day]. tial in employee wages could be related to the Hence, if an employee "i" is scheduled to start relative effectiveness of employees performing work at time "j" on day "k", the person will be their jobs. In such a case, an organization could working through two hours. Such a change to be interested in capturing the trade-off between model PREFWAGE, however, will require a three objectives: worker time preferences, modification of the staff requirements and worker wages and individual worker effective- employee preference information obtained as ness. A simple procedure for incorporating this discussed below. in the multi-objective model developed is as In terms of the staff requirements (STREQjk) , follows. In addition to the other information, these will have to be specified in two (or more) the decision maker would also be required to hourly blocks for each day. Thus, the hourly specify an employee effectiveness index (E~; variations in demand will have to be smoothed


using some other external mechanisms. For 3. A HEURISTIC PROCEDURE implementation purposes, the user can specify that the demand for a particular two (or more) In this section, we discuss the heuristic hour block will be either the average of demand method which can be used to modify the sol- for each of the one hour periods included or the ution obtained using model PREFWAGE. This maximum of demand in the same hour blocks, modification focuses on ensuring that if an The first choice (i.e. averaging of demand) employee is assigned to work on a particular would be preferred if demand smoothing can be day, each work assignment will be for at least carried out while the second choice (i.e. demand two (or more) consecutive hourly blocks. Note for a two hour time block is the maximum of that the length of these consecutive hour blocks demand) would be preferred if customer service are assumed to be specified by the user. How- is to be maximized, ever, for exposition purposes, in the description

Further, the employee work preferences will of the heuristic below, it is assumed that the also have to be specified in two (or more) hourly minimum block length is two hours. In each blocks. This sounds reasonable for individual stage of the heuristic, the focus is on the "gaps" employees since if they prefer work schedules in the work schedules for individual employees with continuous two (or more) hour stretches, (a "gap" in a work schedule is defined as any then employee preferences should also be scheduled work time which is preceded and specified in two (or more) hour stretches. In followed by a non-scheduled work time). The terms of preference requirements across all em- "gap" in a daily (e.g. on day k) work schedule ployees, however, this may lead to two prob- for an employee (e.g. employee i) is identified

as follows: lems. First, on day "k" , an employee " i " may be available and/or prefer to work starting (1) SCHEDi, i,k= 1 and SCHEDi, j+I.,=O, hour " j " through hour "j + 1" (i.e. PREFgk -- i f j --- 1; or 1 or 2) but may be unavailable for work from hour '~j + 1" through hour '~] + 2" (i.e. (2) SCHEDij_~,k = O, SCHEDi, zk = 1 and PREFu+ t.k = 0). Hence, if employee working SCHEDg, j+ ~,k = 0, if 2 < j < J - 1; or time preferences are to be specified in two hour blocks, it is difficult for an employee to specify (3) SCHEDg, j_ ~.k = 0 and SCHED~,j, k = 1, the PREF~jk values. Second, it is also likely that i f j = J. the definition of a two (or more) hour stretch for We now describe the two stages of the proposed one employee might not coincide with a similar heuristic. The first stage of the heuristic pro- stretch for another employee (e.g. employee " i " cedure focuses on attempting to remove the may prefer to work between 2 p.m. and 4 p.m. "gaps" in the working schedule for each em- while employee " i l" may prefer to work be- ployee. This is carried out as follows. First, tween 3 p.m. and 5 p.m.). Thus, there may be a employees are ranked in decreasing order of necessity to standardize the definition of the two their wages. Next, each employee in this ranked (or more) hour blocks before collecting infor- list is sequentially considered. Assume that a mation on employee time preferences. However, "gap" in the work schedule for employee i this would lead to a loss of flexibility for the exists. Then we consider any other employee part-time employees who typically have to deal "ir' (il > i) such that PREFi~,j,k = 1 or 2. After with multiple demands on their working times ensuring that this employee has not been as- each day. signed to work for the maximum hours per day

Based on this discussion, it appears that (or per week), we check if: although model PREFWAGE could be formulated so as to accommodate longer time periods (1) SCHED~Ij, k = 0 and SCHEDi,.j ÷ 1. A-= 1, than one hour, there would be several difficulties t f j = 1.

associated with specifying the input parameters (2) SCHEDiI,j_ I,k = 1, SCHED~I,~.k = 0; or in such a model (essentially, the parameters SCHED,.j ,k=Oand SCHED,,j+~.k= 1, STREQ~k and PREFug ). Thus, in the next i f 2 < j < J - l . section, we describe a heuristic method which can be used to modify the solution to model (3) SCHEDij_ l,k = 1 and SCHEDt.j,k = O, PREFWAGE. if j = J.


If this is true, we set SCHEDij, k=O and fCl WAGEi-C2 D PREF~j-I k SCHEDiIj, k= 1. Note that each time the em- ,~ if SCHED~j_I,k= 1 " ' ployee working schedule is changed in this 61 = ]C~ WAGEi-C2D PREF~,j+I,k manner, the change in the value of the objective L ifSCHED~j+ I,k = 1 function can be computed as:

As with the previous step, the value of the 6 = C I ( W A G E i l = WAGE,) objective function will increase if61 > 0 and the

-C2D(PREF, mj, k-PREF,.j.k) (2) reverse is true if 6 1 > 0. A final comment regarding the heuristic pro-

If 6 < 0, solution quality has improved while if cedure proposed in this section is as follows. It 6 > 0, the reverse is true. is possible that the heuristic solution will gener-

At the second stage of the heuristic, we relax ate a solution which is "better" [in terms of the strict equality constraint for staff require- the objective function--see equation (4) in ments (constraint set 5 in model PREFWAGE). the Appendix] than that derived using model Thus, we allow more staff to be on duty at PREFWAGE. This could occur due to one of specific times of a day if this will result in two reasons. First, if C~ = 0 and C2 = 1, then employee work schedules of at least two hour every time "gap" is eliminated in stage one of blocks. Hence, we are increasing customer ser- the heuristic and PREF~I,j,k >, PREF~,j,k, then vice (i.e. by having more than the minimum an improvement in the objective function will necessary employees working) at the expense of take place. Second, with the same value of the increased wage costs and/or decreasing era- "weights" as before, an improvement in the ployee preferences. In direct contrast to the objective function will always take place if previous step, the employees are ranked in "gaps" are filled in at stage two of the heuristic increasing order of their wages and the working procedure. In sum, the heuristic procedure de- schedule for each day for each employee in scribed in this section can be used to derive this ranked list is sequentially considered. As employee schedules which contain at least 2 hr before, assume that for a "gap" in the work consecutive blocks of working times. We now schedule for employee " i " exists. In such a case, proceed to present an illustration of the model we set: and the heuristic of two data sets.

(1) SCHEDt, j+I.k = 1, if j = 1 and PREF~j + t, k = 1 or 2 (if PREFi, j + i, k = 0, 4. DATA SETS USED TO ILLUSTRATE THE then there is no need to change the MODEL AND HEURISTIC

employee schedule); or In this section, we supply details of the data sets used to illustrate the model and heuristic

(2) SCHED~j_ 1, k = 1 or SCHED~.j+ 1,~ = 1, procedure developed in the prior sections. Data if 2 ___j < J - 1 (the choice between was obtained from two sources: a restaurant whether to choose hour " j - 1 " or operating from 10a.m. to 2a.m. Mondays hour "j + 1" can be made on the through Saturdays (referred to as the Restau- basis of the employee preferences, i.e. rant Data Set); and a computer lab operated if PREF~,j_~,k>PREFi, j+I. k then we from l l a . m , through 7p.m. Monday through choose the first option and otherwise Friday (referred to as the Computer Lab Data the latter. Once again, as before, we Set). The characteristics of each data sets are are assuming that either PREF~j_ ~,k, described below. and/or PREF~,j+ ~,k are not equal to 0);

or 4.1 Restaurant data set

(3) SCHEDij_ t, k = 1 if j = J (this is car- The restaurant for which data was obtained is ried out as long as PREFij_ l,k is not located adjacent to a university and several equal to 0). commercial centers and thus, the major cus-

tomers are students and young professionals. At As before, the total increase in the objective the time this study was carried out, the restau-

function at each step in this heuristic can be rant had six full-time employees (one bartender, computed as: two chefs, one cashier, one kitchen manager and


one restaurant manager). The scheduling prob- the time varying demand per day for the restau- lem addressed in this study focused on the wait rant. Based on forecast customer demand, the person staff which consisted of ten part-time restaurant manager estimated the wait-staff re- employes. These employees were responsible for quirements shown in Panel B of Table 1. This taking orders, serving customers and cleaning information shows that Monday through Fri- up after the customers leave, day, the peak demand occurs during the lunch

Since all these employees were students at- hours (i.e. 11 a.m. through 2 p.m.) when 4 wait tending the university, they were only available persons are required. On the other hand, the at particular times of the day. By dividing up the slowest times during these days are between restaurant operating hours into one hour time 2 p.m. and 5 p.m. so that only 1 wait person is blocks, each wait person was asked to supply needed through these hours. In the evening information as to the preferred work times, hours, demand picks up again and hence, 3 wait available (but not necessarily preferred) work persons are needed from 5 p.m. through 2 a.m. times and unavailable work times. This infor- Finally, the highest demand occurs on Saturday mation, collected from one of the employees since the restaurant promotes special sports (employee number 3), is shown in an "Employee events (e.g. a basketball game between the uni- Work Preference Matrix" in Panel A of Table 1. versity team and visitors) using several large In this case, a "2" indicates the preferred work screen TV monitors on this day. Thus, on times; a "1" indicates the available (but not Saturday, 3 wait persons are needed in the lunch preferred) work times; and a "0" indicates times hours (i.e. between 11 a.m. and 2 p.m.) while 5 at which the employee will not be available, wait persons are needed from 2 p.m. through Hence, for example, on Monday, this employee 2 a.m. All the information shown in Panel B of prefers to work between 1 p.m. and 7 p.m.; is Table 1 was used to specify the STREQ~k values available (but does not prefer to work) between used to operationalize model PREFWAGE for 10 p.m. and 2 a.m.; and is unavailable for work this data set. between 10 a.m. and 1 p.m. and from 7 p.m. and In addition to the information regarding em- 10 p.m. The preference matrix for each employee time preferences and staffing require- ployee was used to estimate the PREFo~ values ments, the restaurant manager also supplied used to operationalize model PREFWAGE for information regarding employee wages and this data set. maximum work time per day for each of the

The problem of scheduling the restaurant employees. The employee wages were based wait person staff was further compounded by on seniority and were as follows: two of the

Table I. Restaurant data set

Panel A. Employee work preference matrix Panel B. Minimum employee requirements

Time Day of week Time Day of the week of of day Mon. Tues. Wed. Thur. Fri. Sat. day Mon. Tues. Wed. Tbur. Fri. Sat.

10 a.m. 0 0 0 0 0 10 a.m. 2 2 2 2 2 2 11 a.m. 0 0 0 0 0 I1 a.m. 4 4 4 4 4 3 12 noon 0 0 0 0 0 12 noon 4 4 4 4 4 3

1 p.m. 2 2 2 2 2 1 p.m. 4 4 4 4 4 3 2p.m. 2 2 2 2 2 2p.m. I 1 1 1 I 5 3p.m. 2 2 2 2 2 3p.m. I 1 1 1 I 5 4p.m. 2 2 2 2 2 4p.m. I l 1 1 I 5 5 p.m. 2 2 2 2 2 5 p.m. 3 3 3 3 3 5 6p.m. 2 2 2 2 2 6p.m. 3 3 3 3 3 5 7p.m. 0 0 0 0 0 7p.m. 3 3 3 3 3 5 8 p.m. 0 0 0 0 0 8 p.m. 3 3 3 3 3 5 9p.m. 0 0 0 0 0 9p.m. 3 3 3 3 3 5

10 p.m. 1 1 1 1 1 10 p.m. 3 3 3 3 3 5 11 p.m. 1 1 1 I 1 11 p.m. 3 3 3 3 3 5 12 mid. 1 1 1 1 1 12 mid. 3 3 3 3 3 5

1 a.m. 1 1 1 1 1 1 a.m. 3 3 3 3 3 5

Notes (a) In both panels, the times of day indicate the starting time for that particular hour. (b) In Panel A, a "2" indicates the preferred working times; a "1" indicates the available (but not preferred) working

times; a "0" indicates times at which the employee is not available. (c) In Panel B, the "2" at I0 a.m. on Monday indicates that 2 employees are required to be at work from 10 a.m.

to 11 a.m. on that day.


employees were paid a wage of $5.00 per hr; work times. This information, collected from three of the employees were paid a wage of $4.00 one of the assistants (assistant number 2), is per hr; and the remaining five employees were shown in N "Employee Work Preference paid the minimum wage of $3.35 per hr. Based Matrix" in Panel A of Table 2. The preference on this information, WAGEi was set to $5.00 for matrix for each assistant was used to estimate i = 1, 2; WAGEi was set to $4.00 for i = 3, 4, 5; the PREFo~ values used to opcrationalize model and WAGE~ was set to $3.35 for i = 6 . . . . . 10 P R E F W A G E for this data set. for this data set. In terms of the maximum work Although the demand for the computer lab time per day, the restaurant manager would like was highly variable throughout the day, it ex- to ensure that each employee was not scheduled ceeded the available capacity (i.e. 14 PCs and 6 to work more than 8 hr per day. This was based terminals which could access the mainframe on the fact that if an employee worked more computer) at most times each day. Hence, these than 8 hr, he/she would be considered full-time capacity restrictions dictated the number of lab and hence, would be eligible to fringe benefits, assistants required. Based on this, the lab man- Thus, the parameter T I M E was set equal to 8 in ager estimated the lab assistant requirements operationalizing model P R E F W A G E for this shown in Panel B of Table 2. This panel shows data set. that 2 lab assistants are needed to be on duty at

all times Monday through Thursday and also 4.2 Computer lab data set between 11 a.m. and 1 p.m. on Friday. It is only

The computer lab for which data was between 1 p.m. and 7p.m. on Friday that de- collected is operated by the MIS department at mand is considerably lower than capacity and a large state university. It is staffed by graduate hence, the lab manager felt that only 1 lab students and reversed for undergraduate assistant was required during those times. All students in the College of Business. In addition the information shown in Panel B of Table 2 to the computer lab manager (a full-time was used to specify the STREQjk parameter employee), there were five lab assistants who values. had been trained to assist the undergraduate Finally, the lab manager provided estimates students using the lab. Since these five lab of the hourly wages for each lab assistant as well assistants were all graduate students simul- as the maximum available time per day each taneously attending the university and working assistant can bc scheduled to work. Two of the in the computer lab, there were certain times of lab assistants were senior graduate students, and the day when they were unavailable for work. hence their wages were higher than those for the As with the restaurant data set, the operating other three assistants. Hence, the wages for the hours were divided into one hour time blocks senior graduate students were approximately and each lab assistant was asked to specify the $5.75 per hour while the wages for the other preferred work times, available (but not necess- three students were estimated to be $4.75 per arily preferred) work times and unavailable hour (i.e. WAGEi=$5.75 for i = 1,2 and

Table 2. Computer lab data set Panel A. Employee work preference matrix Panel B. Minimum employee requirements

Time Day of week Time Day of week of of day Mon. Tues. Wed. Thur. Fri. day Mon. Tues. Wed. Thur. Fri.

II a.m. l 0 I 0 I II a.m. 2 2 2 2 12 noon 1 0 l 0 1 12 noon 2 2 2 2

l p.m. 1 2 I 2 1 I p.m. 2 2 2 2 2 p.m. 1 2 2 2 1 2 p.m. 2 2 2 2 3p.m. 1 2 2 2 1 3p.m. 2 2 2 2 4 p.m. 0 2 0 2 2 4 p.m. 2 2 2 2 5 p.m. 0 I 0 1 2 5 p.m. 2 2 2 2 6 p.m. 1 I 2 1 2 6 p.m. 2 2 2 2 7 p.m. 1 1 2 1 2 7 p.m. 2 2 2 2

Notes: (a) In both panels, the times of day indicate the starting time for that particular hour. (b) In Panel A, a "2" indicates the preferred working times; a " l " indicates the available (but not

preferred) working times; a "0" indicates times at which the employee is not available. (c) In Panel B, the "2" at 11 a.m. on M o n d a y indicates that 2 employees are required to be at work

from 10a.m. to l l a.m. on that day.


WAGE i = $4.75 for i = 3, 4, 5 in operationaliz- puted in relation to the actual value for ing model PREFWAGE for this data set). In C1:C2 = 0: l . Note that the closer the terms of the maximum work time per day for value of RES to 100%, the better is the each lab assistant, the lab manager did not have solution quality. daily restrictions. However, each lab assistant could only be scheduled to work 20 hr per week (2) Relative Employee Wages (REW). This since this was the terms of the "half-time" is computed as follows. First, the value contract for Graduate Assistants. Hence, of total wages at C1:C2 = 1:0 is stan- constraint set 6 (in the Appendix), was replaced dardized at 100 performance measure. by the constraint set 3 shown below when Next, the REW for alternative settings operationalizing model PREFWAGE for this of the Ct:C2 parameter values is corn- data set: puted in relation to the actual value for

K C1 : C2 = 1 : 0. As with the RES measure, ~ SCHED~k <_ 20 Vi (3) the closer the value of REW to 100%,

j~ k~ ~ the better is the solution quality. Three further points need to clarified before

we discuss the performance measures used to (3) Total Costs (TC). These are computed evaluate the solution quality. First, in both for each solution as the value of the cases, the tasks carried out by the employees objective function presented in the were found to be fairly simplistic and hence, no Appendix [see equation (4)].

information of relative employee effectiveness The first two measures (i.e. RES and REW) (i.e. Ei values--see Section 2.4) was included in are used to investigate the relative impact of the our illustration. Second, for these applications, weights C1 and C~ while the change in the TC the preference matrix was converted to a dollar measure is used to evaluate the quality of the contribution matrix by setting the value of D at solutions generated by the heuristic. We now 1.00. Hence, the results discussed in the next turn to a discussion of the results of applying the section should be interpreted in light of this model and heuristic procedure. parameter value. Finally, several alternative values of the relative weights (C~ and C2) were investigated. The value of one parameter was set 5. RESULTS

at 1.00 and the values for the other parameter In this section, the results of applying the were set at 0, 0.50 and 1.00. Thus, we investi- model to the restaurant and computer lab gated five combinations of theparametervalues data sets are discussed. These results were (i.e. C~:C2= l:0, 1:0.50, l : l , 0.50:1 and 0:l). obtained using the GAMS/ZOOM software These particular combinations are useful in package [16] and the ZOOM mathematical pro- investigating the impact of minimizing em- gramming library [15]. All the solutions for ployee wages versus maximizing work prefer- model PREFWAGE for either data set were ences. The results obtained from the model and obtained in a maximum of 16 sec of CPU time heuristic for each data set are evaluated in terms on a VAX 8600 computer with no floating of the following performance measures: points accelerator. The modified solutions using

(I) Relative Employee "Satisfaction" the heuristic method were obtained in a maxi- (RES). This is computed as follows, mum of 3 seconds on the same machine. Fur- First, the ratio of the total number of ther, the computation time did not vary times all the employees are scheduled to significantly with alternative values of the rela- work at their preferred time divided by tive weights of C~ and C2. In presenting the the model number of times all the em- results, we discuss two major aspects. ployees are scheduled to work at their preferred times plus their available 5.1 Impact of the relative weights Ct :(?2 times is computed. Next, the value of In investigating the impact of the relative this ratio for CI : C2 = 0:1 is standard- weights on the quality of the solutions, we only ized at 100 obtainable by this ratio, focus on the solutions derived using model Finally, the RES for alternative settings PREFWAGE and the RES and REW perform- of the C~: C2 parameter values is com- ance measures. The impact of the alternative


P a n e l 1 : Restaurant data set • Rest data set + Comp lab data set

110 -

• e J ° 3

®'° . o I Q- 80

0

70 I I I I -I I I :0 1:0.50 1:1 0.50:1 0:1 1:0 1:0 0:1

Ratio of C 1 • C 2 Ratio of C 1 : C 2

Fig. 2. A comparison of heuristic and optimal solutions.

Panel 2: Computer lab data set

110 - compare the TC of the optimal solution with 100 "J t - - - ,I, ,t

® 9o , ~ that of the heuristic solution. The comparison of = TC values is carried out as follows• For each 0~ a0 "® C~:C2 ratio, we computed the Relative Change p 70 ® . RES in TC ( R C T C ) = [(TChcur- TCopt)/(TCopt)] x

60 + REW 100. These RCTC values for both data set sets 50 40 I I I I are shown in Fig. 2. In the restaurant data set,

:0 1:0.50 1:1 0.50"1 0:1 the heuristic solution deviated by at approxi- Ratio of C 1 : C 2 merely 4% from the optimal solution while for

Fig. 1. RES and REW values for alternative ratios of C~: C 2. the computer lab data set, this figure was approximately 2%. As noted in Section 3, it is possible that the heuristic solution could provide

values of the ration C~ : C2 on these two perform- lower value for the TC measure than the opti- ance measures is shown in Panels 1 and 2 of real solution. This occurred only for the com- Fig. 1. Based on this figure, the following purer lab data set when the ratio of CI : C2 = l '0. general results hold: Based on this discussion, it appears that the

• As would be expected, the greater the heuristic appears to provide "good" quality weight assigned to (72 as compared to solutions. Finally, both the panels in this figure CI, the greater is the value of RES. also show that there is no specific interaction Further, RES is maximized for all val- between the values of the relative weights C~ and ues of C2 which are at least two times C2 on the performance of the heuristic. the value assigned to C~. To further evaluate the heuristic, we also

focused on the individual employee work sched- • On the other hand, the greater the ules that were developed. To illustrate this

weight assigned to Ct as compared to aspect, we present the optimal and heuristic C2, the smaller is the value of RTW. In work schedule for employee number 3 for the addition, RTW is minimized for all restaurant data set on a Monday and for em- values of C~ which are at least two times ployee number 2 for the computer lab data set the value assigned to C2. on a Friday in Panels A and B of Table 3.

• Although these results in are predic- Additionally, these schedules are presented for table, they do provide some face three alternative values of C~ : C2 = 1 : 0, 1 : 1 and validity to the model and they also 0:1. In this table, the values indicated with an illustrate the conflict between the objec- * indicates those times at which the employee rives of maximizing worker preferences has been scheduled to work. On examining this versus minimizing employee wages, table, note the following:

• For both data sets, as the ratio of Cl: C 2 5.2 Performance o f the heuristic changes from 1:0 to 0:1 the individual

To evaluate how the quality of the solution employees are scheduled to work more deteriorates through the use of the heuristic, we at their preferred work times. Thus, for


Table 3. Employee work schedule derived using the model and heuristic

Panel A. Restaurant Panel B. Computer lab data set

Day of week--Monday Day of week--Tuesday

Time C2:C2= 1:0 CI:C2= 1:1 CI:C~=O:I Time C2:C2 = 1:0 Cl:C2ffi 1:1 Cj:C2ffi0:I of of day Model Heuris. Model Heuris. Model Heuris. day Model Heuris. Model Heuris. Model Hem'is.

10 a.m. 0 0 0 0 0 0 II a.m. 0 0 0 0 0 O II a.m. 0 0 0 0 0 0 12 noon 0 0 0 0 0 0 12 noon 0 0 0 0 0 0 I p.m. 2 2 2* 2* 2* 2*

1 p.m. 2* 2 2* 2* 2* 2* 2 p.m. 2 2 2* 2* 2* 2* 2 p.m. 2 2 2 2* 2* 2* 3 p.m. 2 2 2* 2* 2* 2* 3 p.m. 2 2* 2* 2* 2* 2* 4 p.m. 2* 2* 2 2 2* 2* 4p .m. 2* 2* 2 2* 2 2 5p.m. I I* I 1 I* 1" 5p.m. 2 2 2* 2* 2* 2* 6p.m. I* 1" 1 1 I 1" 6p.m. 2 2 2 2 2 2* 7p.m. 1 1" 1 1 1" I* 7 p.m. 0 0 0 0 0 0 8 p.m. 0 0 0 0 0 0 9 p.m. 0 0 0 0 0 0

10 p.m. 1" 1" 1 1 1 1" 11 p.m. 1" 1" 1 I 1 1" 12 mid. I* I* 1 1 1 1

1 a.m. 1 1" 1 1 1 1

Notes: (a) In both panels, the times of day indicate the starting time for that particular hour. (b) In each panel, a "2" indicates the preferred working times; a "1'" indicates the available (but not preferred) working times; a

"0" indicates times at which the employee is not available. (c) In each panel, an asterisk next to the working time preference indicates that the employee has been scheduled to work starting

at that time of the day.

example, in Panel A, the solution from between (i.e. 5 p.m. and 6 p.m.), the the model indicated that when the modified heuristic solution scheduled ratio of C~ : C: was 1:0, employee 3 this same employee from 4 p.m. was scheduled to work for a total of through 8 p.m. Similar examples 2 preferred hr and 3 available hr. can be observed in Panel A of the However, as the ratio changed to same table for the Computer Lab Data C~: C2 = 0:1, the same employee was Set.

scheduled to work 4 preferred hr and 0 One final result that was observed but is available hr. Note that similar results not presented due to space constraints is as can be observed for the solutions de- follows. As would be expected, it was found rived using the heuristic in Panel A as that when the minimization of employee well as for the computer lab data in Panel B. Hence, worker satisfaction wages was prioritized over maximizing worker with the work schedule can be maxi- preferences with the work schedule, most

of the highly paid workers were scheduled mized as we emphasize C2 in relation to work the minimum number of hours. This to C~. could be one potential explanation of why

• Table 3 also illustrates the changes in there is such a high turnover in part-time the work schedule that are made when

employees for service businesses. Hence, once the heuristic is applied to the schedule an employee is paid a higher wage (could be due derived from using using the model, to seniority), there is a possibility that such an Note that in both panels, the heuristic employee will be given extremely poor work solution always ensures that the em- schedules and this may lead to his/her leaving ployee is schedule to work for at least the organization. two hour time blocks. For example, in Panel B, when C~:C2 = 1:0, the model

6. CONCLUSIONS solution shows that employee 2 was scheduled to work from 4p.m. to In this paper, a model and heuristic for 5 p.m. and subsequently from 6 p.m. to developing work schedules for part-time era- 7 p.m. Since this schedule would force ployees was presented. This model incorporates the employee to be idle for the hour in worker time preferences as well as employee

OME 20/2--F


wages in specifying an appropriate work sched- D = parameter which converts the ule. Using data collected from a local restau- worker preference matrix into a rant, the model was validated and illustrated. "dollar" contribution matrix.

Based on the illustration of the model and Decision variable: heuristic, we can draw the following general conclusions. First, worker time preferences ( 1 if employee i is scheduled to can be maximized at the expense of higher SCHED°~ = 1 work starting hour j on day k, wages paid by the employers. Second, the 0 otherwise;

choice of an appropriate work schedule seems Using this notation, model P R E F W A G E is to involve a trade-off between the "benefits formulated as: of maximizing the satisfaction of part-time employees with their work times versus the Minimize

"costs" of employee wages. Finally, the z = [ ~ ~ ~ CI(WAGE~SCHEDok ) results also indicate that the high turnover of L , = , j ~ , k = ,

part-time employees in service establishments -C2D(PREFukSCHED~) ] (4) could be partly due to the fact that these organizations focus more on wage minimiz- Subject to: ation. However, if employee turnover is to be minimized, the implementation of a work sched- N ule needs to be done by considering the "costs" 5", SCHED~ = STREQ~ Vj, k (5)

i=1

of employee dissatisfaction and employee z wages, simultaneously, y. SCHED~ < TIME Vi, k (6)

i = 1

SCHED~ < PREFuk Vi, j, k (7)

A P P E N D I X A SCnED,~=[O, 1] Vi, j , k (8)

Mode l Formulation

Define: R E F E R E N C E S

i = index for employees (i = 1 , . . . , N); 1. Baker KR (1974) Scheduling a full time workforce to j = index for hour of the day ( j = 1, J); meet cycling staffing requirements. Mgmt Sci. 20(12),

• " " ' 1561-1568. k = index for day of the week 2. Baker KR (1976) Workforce allocation in cyclical

(k = 1 . . . . . K). scheduling problems: A survey. Opl Res. Q. 27(1), 155-167.

Input parameters: 3. Baker KR and Magazine MJ (1977) Workforce scheduling with cyclic demands and days-offconstraints. Mgmt

S T R E Q j k = n u m b e r of employees required Sci. 2A(2), 161-167. starting at hour j on day k; 4. Beehtold SE and Showalter MJ (1977) A methodology

for labor scheduling in a service operating system. i if employee i cannot work Decis. Sci. 18(1), 89-107.

starting hour j on day k, 5. Buffa ES, Cosgrove MJ and Luce BJ (1976) An inte- grated work shift scheduling system. Decis. Sci. 7(4),

PREFok = if employee i is available for 620--630. work starting hour j on day 6. Fitzsimmons JA and Sullivan RS (1982) Service Oper-

ations Management. McGraw-Hill, New York. k, if employee i prefers to 7. Glover F, McMillan C and Glover R (1984) A heuristic work starting h o u r j o n d a y k; programming approach to the employee scheduling

problem. J. Opns Mgmt 4(2), 113-128. W A G E i = hourly wage paid to employee i; 8. Henderson WB and Berry WL (1976) Henristic methods T I M E = maximum number of hours a for telephone operator shift scheduling: An experimen-

tal analysis. Mgmt Sci. 22(12), 1372-1380. part-time employee can be sched- 9. Keith EG (1979) Operator scheduling. AIIE Trans. uled to work per day without 11(1), 37-41.. changing the status of the em- 10. Mabert VA (1979) Chemical Bank's encoder daily shift

scheduling system. J. Bank Res. 10(3), 173-180. ployee to a full-time level; 1 I. Mabert VA and Watts CA (1982) A simulation analysis

Cj, (72 = relative "weights" assigned to the of tour-shift construction procedures. Mgrat Sci. 2g(5), wage minimization and prefer- 521-532.

12. McGinnis LF, Culver WD and Dearie RH (1978) One- ence maximization objectives, re- and two-phase heuristics for workforce scheduling. spectively; Comput. Ind. Engng 2(1), 7-15.


13. Morris JG and Showalter MJ (1983) Simple approaches 16. World Bank (1987) GAMS/ZOOM: Version 2.0. The to shift, days-off and tour scheduling problems. Mgmt World Bank, Washington, DC. Sci. 29(8), 942-950.

14. Rosenthal RE (1985) Principles of multiobjective optimization. Decis. Sci. 16(2), 133-152. ADDRESS FOR CORRESPONDENCE: Professor AJ Vakharia,

15. Singhal J, Marsten RE and Morin TE (1989) Fixed Decision Science Group, Department of Management order branch-and-bound methods for mixed integer Information Systems, Karl Eller Graduate School of programming: The ZOOM system. ORSA J. Comput. Management, University of Arizona, Tucson, AZ 85721, 1(1), 44-51. USA.

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