A model for scheduling postal distribution

374 European Journal of Operational Research 58 (1992) 374-385 North-Holland Theory and Methodology A model for scheduling postal distribution employees Manoj K. Malhotra Department of Management Science, College of Business Administration, University of South Carolina, Columbia, SC 29208, USA L a r r y P. R i t z m a n , W . C . B e n t o n a n d G. K e o n g L e o n g Academic Faculty of Management Sciences, The Ohio State University, 1775 College Road, Columbus, OH 43210, USA Received Jun
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  374European Journal of Operational Research 58 (1992) 374-385North-Holland Theory and Methodology A model for scheduling postal distributionemployees Manoj K. Malhotra Department of Management Science, College of Business Administration, University of South Carolina,Columbia, SC 29208, USA Larry P. Ritzman, W.C. Benton and G. Keong Leong Academic Faculty of Management Sciences, The Ohio State University, 1775 College Road, Columbus,OH 43210, USA Received June 1990; revised November 1990 Abstract: This paper describes the design and analysis of a generalizable linear programming model forscheduling personnel in United States postal distribution stations. The scheduling model incorporatesparameters such as different work loads along routes, equitable assignment of overtime between full-timeemployees desiring overtime, training levels of part-time employees, and management preferences fordifferent levels of customer service. A near optimal schedule specifying daily assignment of full-time andpart-time employees to different task categories is developed. The model is illustrated using data from atypical week in a large postal distribution station located in a representative American city. Analysis ofthe results shows that the proposed model captures the essence of scheduling decisions undertaken bypostal supervisors and mid-level managers from the same data set. The schedules developed by themodel are considered feasible from an implementation viewpoint, and also require less overtime than themanagement schedules. The' impact of changing certain long-term policy issues (such as the mix betweenpart-time and full-time employees, assignment policies, and training levels of part-time employees) canbe evaluated by conducting a systematic sensitivity analysis of the model's parameters. Another possiblefuture benefit lies in using the model on a microcomputer for developing actual day-to-day schedules ofpostal distribution employees. Keywords: Service, post office, scheduling, distribution 1. Introduction Research in service sector scheduling can beclassified along three levels of disaggregation assuggested by Krajewski and Ritzman (1977): thestaff sizing problem, shift scheduling and crewassignment problem, and short-term individualassignment problem. The third level of disaggre-gation, the operational level, has been scantilyaddressed in the service sector literature eventhough several theoretical and application ori-ented studies in a variety of industries have beenreported at the first two levels. Past research atthe first two levels has centered on operatorscheduling in the telephone industry (Segal, 1974;Henderson and Berry, 1976; Buffa et al., 1976;and Keith, 1979), shift scheduling in the bankingindustry (Mabert and Raedels, 1977; and Krajew-ski et al., 1980), shift scheduling in the postaldepartment (Ritzman et al., 1976; and Showalter 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved  M.K Malhotra et al. / Model for scheduling postal distribution employees 375 et al., 1978), personnel staffing and scheduling inthe health care industry (Warner and Prawda,1972; Abernathy et al., 1973; and Franz et al.,1989), and crew planning in the airline industry(Marsten et al., 1979). These studies have demon-strated the usefulness of optimization models andheuristic methods for modelling actual servicesystems. The setting for our study is the maildistribution function of post offices. It deals withthe detailed short-term employee assignment andoperations scheduling problem at the third levelof disaggregation, an underaddressed area of theservice sector scheduling literature.The constraints and objective function of ourmathematical programming model represent thescheduling process undertaken daily by postalsupervisors and other mid-level management per-sonnel in the U.S. Postal Service. The structureof the model is similar to the linear programmingformulation of aggregate production planning byHannsman and Hess (1960), whereby costs areminimized subject to a inventory balance con-straint (inventory of mail in this case), availablecapacity (total hours that employees can work),and the constraint of work getting done(processing of mail). The decision variables areundertime, regular time, and overtime allocationsof hours to employees for processing the givenwork loads. However, this model is used for ob-taining detailed level daily scheduling decisions,as opposed to longer-term aggregate productionplanning decisions.Section 2 reviews the problem setting in whichthe model is developed, as well as the assump-tions made in specifying the model constraints.Section 3 presents the modeI. The model is testedand validated with data collected from a repre-sentative station in Section 4 and the model re-suits are compared with actual management deci-sions. The final section discusses the potentialusefulness of the model for decision making inpostal organizations. 2. Problem setting For a given city, mail is delivered to finaldestinations from several different stations, andeach station is split into a specific number ofroutes. For each route, employees first 'case' themail (arrange according to delivery sequence) be-fore starting the street delivery. 'Scheme knowl-edge' is required for casing the mail for any givenroute. This means that an employee must betrained before becoming fully proficient in casingthe mail for any specific route. The work forceconsists of permanent full-time (FT) employeeswho work eight hours a day, and part-time flexi-ble (PTF) employees who can be scheduled for alimited amount of time each week, and need notbe called in every day. The work shift for PTFemployees can start at any time of the day. Themix between the full-time and part-time workersdetermines the scheduling flexibility available forbuffering against demand uncertainties, and otherunforeseen circumstances such as absenteeism,sick leave, and attrition. Scheduling flexibility isconstrained by a union rule, which currently stip-ulates that at least 90% of the work force shouldbe comprised of FT employees.Generally, full-time employees 'own the bid' todeliver mail on a given route. The FT employeesusually case the mail on their owned routes, butalso can be used for delivering mail on alternateroutes. Alternately, FT employees may be usedonly for delivery purposes on their own routes ifthe work load for the route is heavy. If a FTemployee is unavailable due to vacation or sickleave, the route is temporarily assigned duringthe absence period to a senior PTF employeetrained for that route. Regular delivery for thatroute is again carried out by the FT employee onreturning for work after the break. FT employeeswork for five days only in a week, whereas thepost office delivers mail six days a week. Thework for the sixth day on each route is assignedto a FT employee called a 'T6' worker. The T6employee works on five different routes in a givenweek, with each work day corresponding to theoff day for the bid owner of that particular route.Mail volume can fluctuate significantly be-tween days of a week and across different routes.When the casing and delivery work load for aroute exceeds an eight-hour work equivalent, ei-ther overtime is incurred, or assistance must beprovided to the FT employee who owns the bidfor that route. This assistance can be in the formof providing full time 'routers' to that route for agiven day. Routers assist only in casing the mail,thereby leaving the regular owner of that routemore time for delivery on the streets by reducingthe casing work load. Each router is typicallytrained for casing mail on about ten routes. An-  376 M.K~ Malhotra et al. / Model for scheduling postal distribution employees other way to help out an overloaded FT em-ployee on a particular route is to assign a PTFemployee trained for that route for casing andauxiliary delivery assistance on the streets. Onthe other hand, FT employees whose routes areunderloaded can also be used for delivering mailon other overloaded routes after completing theirown delivery schedule. Some FT employees donot own the bid to deliver on a specific route.They are known as 'unassigned regulars'. Corre-spondingly, they also can be used for deliverypurposes on any route which is overloaded, andfor casing on those routes for which they possessscheme knowledge from past training.Now the problem arises on assigning work tothe available employees in such a way that over-time is minimized. A customer service standardmust also be satisfied: All first class mail receivedat a station in the morning must be delivered thesame day. The bulk business class mail must bedelivered within 48 hours, which means that alimited amount of maiI can be inventoried. Eachcategory of mail can be physically identified be-fore the casing operation begins.This scheduling problem is an involved one,because a station manager does not know whichPTF employee should be used in what capacity toobtain the best results. The options open are touse a PTF for casing help, auxiliary delivery assis-tance on the road, or both casing and delivery ona specific route (to cover for those FT employeeswho are unavailable on a given day due to vaca-tion or sick leave). Moreover, each PTF employeeis trained for casing mail 'on different routes.Which route must he/she be assigned to on agiven day so as to minimize overtime hours, andalso meet the service standards? Other objectivesare to minimize the idle time for FT employees,and to equitably distribute the overtime hoursamong all FT employees desiring overtime. Theimportance of these objectives is captured byassigning different relative weights to them in theobjective function, and by adding appropriateconstraints. This problem is modeled as a linearprogram, as presented next. 3. Staff scheduling model The constraint set and the objective functionof this linear programming model are representa-tive of the actual operating environment of apostal distribution system. It can be solved toyield daily schedules, which identify what routeseach employee must work on, in what capacity(overtime or regular), and how many hours mustbe spent on each work option. For the PTFemployees, the schedule specifies the routes onwhich a PTF employee works for casing mail orrendering auxiliary delivery assistance on thestreets. It also shows which PTF employees shouldbe used from those available on a given day. Themodel outputs the amount of uncased or cur-tailed mail on each route for that day. The sta-tion manager can then either hold back the bulk-business class mail on that route for the next day,or authorize more overtime. The curtailed mail isgiven the same priority as first class mail arrivingon the following morning. This procedure permitsthe model to meet the service standards of deliv-ering the bulk business class mail also within 48hours, and in fact is identical to the way thedecisions are actually made by the postal supervi-sors.Currently the scheduler assigns workers for agiven day to their respective routes on the basisof employee availability. No forecasting of mail isdone, since the actual mail volume figures areavailable at the time that scheduling decisionsmust be made. The PTF workers are called in onthe basis of actual mail volumes experienced. Thebid owners of over-loaded routes request addi-tional assistance on a daily basis. The supervisorcan then assign them some PTF worker from thepool available, assign a FT employee from anunder-loaded route, or authorize overtime. Acombination of these alternatives may also beconsidered. 3.1. Staff scheduling model notation The staff scheduling model parameters anddecision variables are defined below.Parameters:D; ---Demand of mail in feet to be deliveredon route i.pc i = Normal productivity of casing, measuredas feet of mail cased per hour by a fullytrained employee on route i.j1 = The ordered set of FT employees at thestation. This includes unassigned regu-  M.K. Malhotra et al. /Model for scheduling postal distribution employees 377 j2 j3jR lu Qi CiUq~. tj q2 tj lars, as well as the employees who ownbids for their routes.The ordered set of PTF employees.The ordered set of FT employees willingto work overtime. This is a subset of j1The ordered set of employees who onlydo router work on some specified routes.The set of underloaded routes (havingless than an eight-hour equivalent ofwork) on a given day.'Office hours standardization' for route i,defined as the number of hours spent forcasing the reference volume of mail in anormal eight-hour day.'Street hours standardization' for route i,defined as the number of hours spent fordelivering the reference volume of mail ina normal eight-hour day.Maximum amount of mail in hours equiv-alent which can be curtailed on route ion any single day.An upper bound on the total mail cur-tailed across all the routes. This parame-ter can be set by the management, andserves as a surrogate measure of cus-tomer service.The set of all the routes to be serviced ata station.1 if a FT employee j owns the bidto deliver mail on route i;0 otherwise.1 if a FT employee j, who serves as arouter, is trained for casing mail onroute i;0 otherwise.1 if a PTF employee j is trained forcasing mail on route i;0 otherwise.1 if a FT employee j is willing tobj = work overtime;0 otherwise.10 if the FT employee j is availabler) -- for work on thegivenday;otherwise.i if the FT employee j, who serves asthe router, is available for work onthe given day;otherwise.0if the PTF employee j is availablefor work on the given day;otherwise.f = Overtime limit, in hours per day, for eachemployee. It is normally two hours ofovertime, which represents the industrystandard of a 25% overtime limit peremployee.p; = Relative casing efficiency of an untrainedPTF employee working as a substitute bidowner on an untrained route. This vari-able is a real number greater than 1.0 foran untrained PTF employee, and equals1.0 for a fully trained employee. { 1/pi if the route i is carried tempo-rarily by a substitute PTF em- Ci ployee on a given day;1 otherwise.Decision variables:D~j = Labor hours of FT employee j at regularwage rate assigned for delivery on route i.DO~j = Labor hours of FT employee j at over-time wage rate assigned for delivery onroute i.D 2. = Labor hours of PTF employee j at regu-lar wage rate assigned for delivery onroute i.DO 2 = Labor hours of PTF employee j at over-time wage rate assigned for delivery onroute i.Ci~ = Labor hours of FT employee j at regularwage rate assigned for casing mail onroute i.CO/lj = Labor hours of FT employee j at over-time wage rate assigned for casing mailon route i.C~j = Labor hours of PTF employee j at regu-lar wage rate assigned for casing mail onroute i.CO 2 -- Labor hours of PTF employee j at over-time wage rate assigned for casing mailon route i. R~s = Labor hours of FT employee j at regularwage rate assigned as router time forcasing mail on route i.Uj I = Number of idle labor hours for FT em-ployee j.Uj R = Number of idle labor hours for router FTemployee j.
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